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GENERALIZATION OF BACKPROPAGATION
TO
RECURRENT AND HIGHER ORDER NEURAL NETWORKS
Fernando J. Pineda
Applied Physics Laboratory, Johns Hopkins University
Johns Hopkins Rd., Laurel MD 20707
Abstract
A general method for deriving backpropagation algorithms for networks
with recurrent and higher order networks is introduced. The propagation of activation
in these networks is determined by dissipative differential equations. The error signal
is backpropagated by integrating an associated differential equation. The method is
introduced by applying it to the recurrent generalization of the feedforward
backpropagation network. The method is extended to the case of higher order
networks and to a constrained dynamical system for training a content addressable
memory. The essential feature of the adaptive algorithms is that adaptive equation has
a simple outer product form.
Preliminary experiments suggest that learning can occur very rapidly in
networks with recurrent connections. The continuous formalism makes the new
approach more suitable for implementation in VLSI.
Introduction
One interesting class of neural networks, typified by the Hopfield neural
networks (1,2) or the networks studied by Amari(3,4) are dynamical systems with three
salient properties. First, they posses very many degrees of freedom, second their
dynamics are nonlinear and third, their dynamics are dissipative. Systems with these
properties can have complicated attractor structures and can exhibit computational
abilities.
The identification of attractors with computational objects, e.g. memories at d
rules, is one of the foundations of the neural network paradigm. In this paradigl n,
programming becomes an excercise in manipulating attractors. A learning algorithm is
a rule or dynamical equation which changes the locations of fixed points to encode
information. One way of doing this is to minimize, by gradient descent, some
function of the system parameters. This general approach is reviewed by Amari(4)
and forms the basis of many learning algorithms. The formalism described here is a
specific case of this general approach.
The purpose of this paper is to introduce a fonnalism for obtaining adaptive
dynamical systems which are based on backpropagation(5,6,7). These dynamical
systems are expressed as systems of coupled first order differential equations. The
formalism will be illustrated by deriving adaptive equations for a recurrent network
with first order neurons, a recurrent network with higher order neurons and finally a
recurrent first order associative memory.
Example 1: Recurrent backpropagation with first order units
Consider a dynamical system whose state vector x evolves according to the
following set of coupled differential equations
? American Institute ofPhvsics 1988
603
dx?/dt
= -x'1 + g'(LW"X')
I
1. IJ J + I?I
(1)
J
where i=l, ... ,N. The functions g' are assumed to be differentiable and may have
different forms for various populations of neurons. In this paper we shall make no
other requirements on gi' In the neural network literature it is common to take these
functions to be Sigmoid shaped functions. A commonly used form is the logistic
function,
(2)
This form is biologically motivated since it attempts to account for the refractory phase
of real neurons. However, it is important to stress that there is nothing in the
mathematical content of this paper which requires this form -- any differentiable
function will suffice in the formalism presented in this paper. For example, a choice
which may be of use in signal processing is sin(~).
A necessary condition for the learning algorithms discussed here to exist is that the
system posesses stable isolated attractors, i.e. fixed points. The attractor structure of
(1) is the same as the more commonly used equation
du/dt = -ui +~Wijg(Uj) + Ki'
(3)
J
Because (1) and (3) are related by a simple linear transformation. Therefore results
concerning the stability of (3) are applicable to (1). Amari(3) studied the dynamics of
equation (3) in networks with random conections. He found that collective variables
corresponding to the mean activation and its second moment must exhibit either stable
or bistable behaviour. More recently, Hopfield(2) has shown how to construct content
addressable memories from symmetrically connected networks with this same
dynamical equation. The symmetric connections in the network gaurantee global
stability. The solution of equation (1) is also globally asymptotically stable if w can be
transformed into a lower triangular matrix by row and column exchange operations.
This is because in such a case the network is a simply a feedforward network and the
output can be expressed as an explicit function of the input. No Liapunov function
exists for arbitrary weights as can be demonstrated by constructing a set of weights
which leads to oscillation. In practice, it is found that oscillations are not a problem
and that the system converges to fixed points unless special weights are chosen.
Therefore it shall be assumed, for the purposes of deriving the backpropagation
equations, that the system ultimately settles down to a fixed point.
Consider a system of N neurons, or units, whose dynamics is determined by
equation (1). Of all the units in the network we will arbitrarily define some subset of
them (A) as input units and some other subset of them (0) as output units. Units
which are neither members of A nor 0 are denoted hidden units. A unit may be
simultaneously an input unit and an output unit. The external environment influences
the system through the source term, I. If a unit is an input unit, the corresponding
component of I is nonzero. To make this more precise it is useful to introduce a
notational convention. Suppose that <I> represent some subset of units in the network
then the function 8i<I> is defined by
8'm= {
1'V
1
if i-th unit is a member of <I>
0
o therwise
(4)
In terms of this function, the components of the I vector are given by
(5)
604
where ~i is detennined by the external environment.
Our goal will be to fmd a local algorithm which adjusts the weight matrix w so that
a given initial state XO = x(to)' and a given input I result in a fixed point, xoo= x(too ),
whose components have a desired set of values Ti along the output units. This will be
accomplished by minimizing a function E which tneasures the distance between the
desired fixed point and the actual fixed point i.e.,
1 N
E = - :E Ji2
(6)
2 i=l
where
J.I -- (T.I - xoo.I ) e'n
I.u.
(7)
E depends on the weight matrix w through the fixed point Xoo(w). A learning
algorithm drives the fixed points towards the manifolds which satisfy xi00 = Ti on the
output units. One way of accomplishing this with dynamics is to let the system evolve
in the weight space along trajectories which are antiparallel to the gradient of E. In
. other words,
dE
(8)
dWi/dt = - T\ -
dw ..
IJ
where T\ is a numerical constant which defines the (slow) time scale on which w
changes. T\ must be small so that x is always essentially at steady state, i.e.
x(t) == xoo. It is important to stress that the choice of gradient descent for the learning
dynamics is by no means unique, nor is it necessarily the best choice. Other learning
dynamics which employ second order time derivatives (e.g. the momentum
method(5? or which employ second order space derivatives (e.g. second order
backpropagation(8? may be more useful in particular applications. However, equation
(8) does have the virtue of being the simplest dynamics which minimizes E.
On performing the differentiations in equation (8), one immediately obtains
dwrs/dt = T\ 1: Jk
k
dx ook
awrs
(9)
The derivative of xook with respect to wrs is obtained by first noting that the fixed
points of equation (1) satisfy the nonlinear algebraic equation
oo .) + J.
Xoo.I = g?(:Ewoox
I . IJ
J
I'
(10)
J
differentiating both sides of this equation with respect to Wrs and finally solving for
dxooId dW rs ' The result is
dXook
= (L- 1)kr gr'(Ur)xoo s
(11)
dWrs
where gr' is the derivative of gr and where the matrix L is given by
(12)
Bii is the Kroneker Bfunction ( BU= 1 if i=j, otherwise Bij = 0). On substituting (11)
into (9) one obtains the remarkablY simple form
605
where
dWrsldt
=11 YrXoo s
(13)
(14)
Yr = gr'(u r ) LJk(L -1)kr
k=
Equations (13) and (14) specify a fonnallearning rule. Unfortunately, equation
(14) requires a matrix inversion to calculate the error signals Yk' Direct matrix
inversions are necessarily nonlocal calculations and therefore this learning algorithm is
not suitable for implementation as a neural network. Fortunately, a local method for
calculating Yr can be obtained by the introduction of an associated dynamical system.
To obtain this dynamical system fIrst rewrite equation (14) as
LLrk (Yr / gr'(ur )} = Jk .
(15)
r
Then multiply both sides by fk'(uk)' substitute the explicit form for L and finally sum
over r. The result is
o= - Yk + gk'(uk){ LWrkYr + Jk} .
(16)
r
One now makes the observation that the solutions of this linear equation are the fIxed
points of the dynamical system given by
dYk/dt = - Yk +gk'(uk){LWrkYr + Jk} .
(17)
r
This last step is not unique, equation (16) could be transformed in various ways
leading to related differential equations, cf. Pineda(9). It is not difficult to show that
the frrst order fInite difference approximation (with a time step ~t = 1) of equations
(1), (13) and (17) has the same form as the conventional backpropagation algorithm.
Equations (1), (13) and (17) completely specify the dynamics for an adaptive
neural network, provided that (1) and (17) converge to stable fixed points and
provided that both quantities on the right hand side of equation (13) are the steady
state solutions of (1) and (17).
It was pointed out by Almeida(10) that the local stability of (1) is a sufficient
condition for the local stability of (17). To prove this it suffices to linearize equation
(1) about a stable fixed point. The resulting linearized equation depends on the same
matrix L whose transpose appears in the derivation of equation (17), cf. equation
(15). But Land LT have the same eigenValues, hence it follows that the fIXed points
of (17) must also be locally stable if the fIxed points of (1) are locally stable.
Learning multiple associations
It is important to stress that up to this point the entire discussionhas assumed that I
and T are constant in time, thus no mechanism has been obtained for learning multiple
input/output associations. Two methods for training the network to learn multiple
associations are now discussed. These methods lead to qualitatively different learning
behaviour.
Suppose that each input/output pair is labeled by a pattern label n, i.e. {In ,Tn}.
Then the energy function which is minimized in the above discussion must also
depend on this label since it is an implicit function of the In ,Tn pairs. In order to
learn multiple input/output associations it is necessary to minimize all the E[n]
simultaniously. In otherwords the function to minimize is
(18)
606
where the sum is over all input/output associations. From (18) it follows that the
gradient for Etotal is simply the sum of the gradients for each association, hence the
corresponding gradient descent equation has the form,
= 11 L yOOi[a] xOOia] .
(19)
a
In numerical simulations, each time step of (19) requires relaxing (1) and (17) for each
pattern and accumulating the gradient over all the patterns. This fonn of the algorithm
is deterministic and is guaranteed to converge because, by construction, Etotal is a
Liapunov function for equation (19). However, the system may get stuck m a local
minimum. This method is similar to the master/slave approach of Lapedes and
Farber(1l). Their adaptive equation, which plays the same role as equation (19), also
has a gradient form, although it is not strictly descent along the gradient. For a
randomly or fully connected network it can be shown that tbe number of oper~tions
required per weight update in the master/slave fonnalis~ is proportional to N where
N is the number of units. This is because there are O(N ) update equations and each
equation requires O(N) operations (assuming some precomputation). On the other
hand, in the backpropagation formalism each update equation re~uires only 0(1)
operations because of their trivial outer product form. Also O(N ) operations are
required t~ precompute XOO and yoo. The result is that each weight update requires
only O(N ) operations. It is not possible to conclude from this argument that one or
the other approach will be more efficient in a particular application because there are
other factors to consider such as the number of patterns and the number of time steps
required for x and y to converge. A detailed comparison of the two methods is in
preparation.
A second approach to learning multiple patterns is to use (13) and to change the
patterns randomly on each time step. The system therefore receives a sequence of
random impulses each of which attempts to minimize E[ex] for a single pattern. One
can then defme L(w) to be the mean E[a] (averaged over the distribution of patterns).
dWijldt
L(w) = <E [w, la,Ta ]>
(20)
Amari(4) has pointed out that if the sequence of random patterns is stationary and if
L(w) has a unique minimum then the theory of stochastic approximation guarantees
that the solution of (13) wet) will converge to the minimum point '!min of L(w) to
within a small fluctuating tenn which vanishes as 11 tends to zero. hVlaently 11 is
analogous to the temperature parameter in simulated annealing. This second approach
generally converges more slowly than the first, but it will ultimately converge (in a
statistical sense) to the global minimum.
In principle the fixed points, to which the solutions of (1) and (17) eventually
converge, depend on the initial states. Indeed, Amari's(3) results imply that equation
(1) is bistable for certain choices of weights. Therefore the presentation of multiple
patterns might seem problematical since in both approaches the final state of the
previous pattern becomes the initial state of the new pattern. The safest approach is to
reinitialize the network to the same initial state each time a new pattern is presented.
e.g. xi(t~ = 0.5 for all i. In practice the system learns robustly even if the initial
conditIons are chosen randomly.
Example 2: Recurrent higher order networks
It is straightforward to apply the technique of the previous section to a dynamical
system with higher order units. Higher order systems have been studied by
Sejnowski (12) and Lee et al.(13). Higher order networks may have definite advantages
607
over networks with first order units alone A detailed discussion of the
backpropagation fonnalism applied to higher order networks is beyond the scope of
this paper. Instead, the adaptive equations for a network with purely n-th order units
will be presented as an example of the fonnalism. To this end consider a dynamical
system of the fonn
dx?/dt
-- -x'1 + g'(lI!)
I
1-:1 + I?1
(21)
where
(22)
and where there are n+ 1 indices and the summations are over all indices except i. The
superscript on the weight tensor indicates the order of the correlation. Note that an
additional nonlinear function f has been added to illustrate a further generalization.
Both f and g must be differentiable and may be chosen to be sigmoids. It is not
difficult, although somewhat tedious, to repeat the steps of the previous example to
derive the adaptive equations for this system. The objective function in this case is the
same as was used in the fIrst example, i.e. equation (6). The n-th order gradient
descent equation has the fonn
(23)
Equation (23) illustrates the major feature of backpropagation which distinguishes it
from other gradient descent algorithms or similar algorithms which make use of a
gradient. Namely, that the gradient of the objective function has a very trivial outer
product fonn. y (n)oo is the steady state solution of
dy(n)k/dt = - y(n)k + gk'(uk) {fk'(xk)Ly(n)rkY (n)r + Jk }.
(24)
r
The matrix v(n) plays the role of w in the previous example, however v(n) now
depends on the state of the network according to
y(n)ij
= L'"
L s<n)ijk"'l ( f(xk) ... f(xI)}
(25)
k I
where is s(n) a tensor which is symmetric with respect to the exchange of the second
index and all the indices to the right, i.e.
S(n)..IJk"1 --
w(n)
ijk"'l
+ w(n)
ikj"'l
+ ... + w(n)
ijl"'k .
(26)
Finally, it should be noted that: 1) If the polynomial ui is not homogenous, the
adaptive equations are more complicated and involve cross tenns between the various
orders and that: 2) The local stability of the n-th order backpropagation equations now
depends on the eigenvalues of the matrix
L .. = 0" - g.'(u?) f.'(x?) y(n) ..
IJ
IJ
1
1
1
1
IJ'
(27)
As before, if the forward propagation converges so will the backward propagation.
Example 3: Adaptive content addressable memory
In this section the adaptive equations for a content addressable memory
(CAM) are derived as a fmal illustration of the generality of the formalism. Perhaps
608
the best known (and best studied) examples of dYnamical systems which exhibit CAM
behaviour are the systems discussed by Hopfield(l). Hopfield used a nonadaptive
method for programming the symmetric weight matrix. More recently Lapedes and
Farber<ll) have demonstrated how to contruct a master dynamical system which can be
used to train the weights of a slave system which has the Hopfield fonn. This slave
system then performs the CAM operation. The resulting weights are not symmetric.
The learning proceedure presented in this section is most closely related to the
method of Lapedes and Farber in that a master network is used to adjust the weights of
a slave network. In constrast to the afforementioned formalism, which requires a
very large associated weight matrix for the master network, both the master and slave
networks of the following approach make use of the same weight matrix. The CAM
under consideration is based on equation (1). However, the interpretation of the
dynamics will be somewhat different from the first section. The main difference is that
the dynamics in the learning phase is constrained. The constrained dynamical system
is denoted the master network. The unconstrained system is denoted the slave
network. The units in the network are divided into only two sets: the set of visible
units (V) and the set of internal or hidden units (H). There will be no distinction made
between input and output units. Thus, I will generally be zero unless an input bias is
needed in some application.
The dynamical system will be used as an autoassociative memory, thus the
memory recall is performed by starting the network at a particular initial state which
represents partial information about a stored memory. More precisely, suppose that
there exists a subset K of the visible units whose states are known to have values Ti'
Then the initial state of the network is
(28)
where the bi are arbitrary. The CAM relaxes to the previously stored memory whose
basin of attraction contains this partial state.
Memories are stored by a master network whose topology is exactly the same
as the slave network, but whose dynamics is somewhat modified. The state vector z
of the master network evolves according to the equation
N
d~/dt = -~
+ gi(LwikZk) + Ii
(29)
k=l
where Z is defmed by
Z,1 = T?1 E)?V
+ z?1 E)'H
1
1
?
(30)
The components of Z along the visible units are just the target value specified by T.
This equation is useful as a master equation because if the weights can be chosen so
that the zi of the visible units relax to the target values Ti,: then a fixed point of (29) is
also a fixed point of (1). It can be concluded therefore, that by training the weights of
the master network one is also training the weights of the slave network. Note that the
form of Z implies that equation (29) can be rewritten as
(31)
where
9 i = - LWikTk .
(32)
keY
From equations (31) and (32) it is clear that the dynamics of the master system is
driven by the thresholds which depend on the targets.
609
To derive the adaptive equations consider the objective function
1 N
2
E master =2" 1: Ii
(33)
i=l
where
(34)
It is straightforward to apply the steps discussed in previous sections to EJ1Iaster' This
results in adaptive equations for the weights. The mathematical details Will be omitted
since they are essentially the same as before, the gradient descent equation is
dWi/dt = 11yooiZOOj
(35)
where yOO is the steady state solution of
where
dyk"dt = - Yk +g'k(vkHeiHLwrkYr + Ik}
r
vi i ~ikZook .
(36)
(37)
Equations (31), and (35)-37) define the dynamics of the master network. To train the
slave network to be an autoassociative memory it is necessary to use the stored
memories as the initial states of the master network, i.e.
z?(t
1 0 ) = T?1 e?1V + b?1 eiH
(39)
where bi is an arbitrary value as before. The previous discussions concerning the
stability of the three equations (1), (13) and (17) apply to equations (31) (35) and (36)
as well. It is also possible to derive the adaptive equations for a higher order
associative network, but this will not be done here.
Only preliminary computer simulations have been performed with this
algorithm to verify their validity, but more extensive experiments are in progress. The
fIrst simulation was with a fully connected network with 10 visible units and 5 hidden
units. The training set consisted of four random binary vectors with the magnitudes of
the vectors adjusted so that 0.1 ~ Ti S; 0.9. The equations were approximated by first
order fmite difference equations with ~t = 1 and 11 = 1. The training was performed
with the detenninistic method for learning multiple associations. Figure 1. shows
Etotal as a function of the number of updates for both the master and slave networks.
Etota! for the slave exhibits discontinous behaviour because the trajectory through the
weight space causes x(to ) to cut across the basins of attraction for the fixed points of
equation (1).
The number of updates required for the network to learn the patterns is
relatively modest and can be reduced further by increasing 11. This suggests that
learning can occur very rapidly in this type of network.
Discussion
The algorithms presented here by no means exhaust the class of possible
adaptive algorithms which can be obtained with this formalism. Nor is the choice of
gradient descent a crucial feature in this formalism. The key idea is that it is possible
to express the gradient of an objective function as the outer product of vectors which
can be calculated by dynamical systems. This outer produc2,form is also responsible
for the fact that the gradient can be calculated with only O(N ) operations in a fully
connected or randomly connected network. In fact the number of operations per
610
weight update is proportional to the number of connections in the network. The
methods used here will generalize to calculate higher order derivatives of the objective
function as well.
The fact that the algorithms are expressed as differential equations suggests
that they may be implemented in analog electronic or optical hardware.
2.00 . . . . . - - - - - - - - - - - - - - ,
~
Master
-.- Slave
1.00 --"",.. ~
20
40
60
Updates
80
100
figure 1. Etota! as a function of the the number of updates.
References
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
I. J. HopfieJd. Neural Networks as Physical Systems with Emergent Collective
Computational Abilities. Proc. Nat. Acad. Sci. USA. Bio.79. 2554-2558.
(1982)
1. I. Hopfield. Neurons with graded response have collective computational
properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA. Bio. .8l,
3088-3092. (1984)
Shun-Ichi Amari. IEEE Trans. on Systems Man and Cybernetics. 2.643-657.
(1972)
Shun-Ichi Amari. in Systems Neuroscience. ed. Jacqueline Metzler.
Academic press. (1977)
D. E. Rumelhart. G. E. Hinton and R.I. Williams. in Parallel Distributed
Processing. edited by D. E. Rumelhart and 1. L. McClelland.
M.LT. press. (1986)
David B. Parker. Learning-Logic, Invention Report. S81-64. File 1.
Office of Technology Licensing. Stanford University. October. 1982
Y. LeChun. Proceedings of Cognitiva. 85. p. 599. (1985)
David B. Parker. Second Order Backpropagation: Implementing an Optimal
O(n) Approximation to Newton's Method as an Artificial Neural Network.
submitted to Computer. (1987)
Fernando J. Pineda. Generalization ofbackpropagation to recurrent neural
networks, Phys. Rev. Lett.? l.8. 2229-2232. (1987)
(10) Luis B. Almeida. in the Proceedings of the IEEE First Annual International
Conference on Neural Networks. San Diego. California. June 1987. edited by
(9)
611
M. Caudil and C. Butler (to be published This is a discrete version of the
algorithm presented as the fIrst example
(11) Alan Lapedes and Robert Farber, A self-optimizing, nonsymmetrical neural net
for content addressable memory and pattern recognition, Physica, D22,
247-259, (1986), see also, Programming a Massively Parallel, Computation
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Snowbird, UT 1986, AIP Conference Proceedings, 151, (1986),
edited by John S. Denker
(12) Terrence J. Sejnowski, Higher-order Boltzmann Machines, Draft preprint
obtained from author
(13) Y.C. Lee, Gary Doolen, H.H. Chen, G.Z. Sun, Tom Maxwell, H.Y. Lee and
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Physica D22, 276-306, (1986)
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6,301 | 670 | A Model of Feedback to the Lateral
Geniculate Nucleus
Carlos D. Brody
Computation and Neural Systems Program
California Institute of Technology
Pasadena, CA 91125
Abstract
Simplified models of the lateral geniculate nucles (LGN) and striate cortex illustrate the possibility that feedback to the LG N may
be used for robust, low-level pattern analysis. The information
fed back to the LG N is rebroadcast to cortex using the LG N 's full
fan-out, so the cortex-LGN-cortex pathway mediates extensive
cortico-cortical communication while keeping the number of necessary connections small.
1
INTRODUCTION
The lateral geniculate nucleus (LGN) in the thalamus is often considered as just a
relay station on the way from the retina to visual cortex, since receptive field properties of neurons in the LGN are very similar to retinal ganglion cell receptive field
properties. However, there is a massive projection from cortex back to the LGN:
it is estimated that 3-4 times more synapses in the LG N are due to corticogeniculate connections than those due to retinogeniculate connections [12]. This suggests
some important processing role for the LGN, but the nature of the computation
performed has remained far from clear.
I will first briefly summarize some anatomical facts and physiological results concerning the corticogeniculate loop, and then present a simplified model in which its
function is to (usefully) mediate communication between cortical cells.
409
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Brody
1.1
SOME ANATOMY AND PHYSIOLOGY
The LG N contains both principal cells, which project to cortex, and inhibitory
interneurons. The projection to cortex sends collaterals into a sheet of inhibitory
cells called the perigeniculate nucleus (PGN). PGN cells, in turn, project back to
the LGN. The geniculocortical projection then proceeds into cortex, terminating
principally in layers 4 and 6 in the cat [11, 12]. Areas 17, 18, and to a lesser extent,
19 are all innervated. Layer 6 cells in area 17 of the cat have particularly long,
non-end-stopped receptive fields [2]. It is from layer 6 that the corticogeniculate
projection back originates. 1 It, too, passes through the PGN, sending collaterals
into it, and then cont.acts both principal cells and interneurons in the LGN, mostly
in the more distal parts of their dendrites [10, 13]. Both the forward and the
backward projection are retinotopically ordered.
Thus there is the possibility of both excitatory and inhibitory effects in the corticogeniculate projection, which is principally what shall be used in the model.
The first attempts to study the physiology of the corticogeniculate projection involved inactivating cortex in some way (often cooling cortex) while observing geniculate responses to simple visual stimuli. The results were somewhat inconclusive:
some investigators reported that the projection was excitatory, some that it was inhibitory, and still others t.hat it had no observable effect at all. [1, 5,9] Later studies
have emphasized the need for using stimuli which optimally excite the cortical cells
which project to the LGN; inactivating cortex should then make a significant difference in the inputs to geniculate cells. This has helped to reveal some effects:
for example, LGN cells with corticogeniculate feedback are end-stopped (that is,
respond much less to long bars than to short bars), while the end-stopping is quite
dearly reduced when the cortical input is removed [8].
One study [13] has used cross-correlat.ion analysis between cortical and geniculate
cells to suggest that there is spatial structure in the corticogeniculate projection: an
excitatory corticogeniculate interaction was found if cells had receptive field centers
that were close to each other, while an inhibitory interaction was found if the centers
were farther apart. However, the precise spatial structure of the projection remains
unknown.
2
A FEEDBACK MODEL
I will now describe a simplified model of the LGN and the corticogeniculate loop.
The very simple connection scheme shown in fig 1 originated in a suggestion by
Christof Koch [3] that the long receptive fields in layer 6 might be used to facilitate
contour completion at t.he LGN level. In the model, then, striate cortex simple
cells feed back positively to the LGN, enhancing the conditions which gave rise
to their firing. This reinforces, or completes, the oriented bar or edge patterns to
which they are tuned. Assuming that the visual features of interest are for the most
part oriented, while much of the noise in images may be isotropic and unoriented,
enhancing the oriented features improves the signal-to-noise ratio.
1
In all areas innervated by the LGN.
A Model of Feedback to the Lateral Geniculate Nucleus
OO[i]
///~
~~
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~ 00
RETINA
LGN
CELLS
[f]
~~
VI CELLS
Figure 1: Basic model connectivity: A schematic diagram showing the connections
between different. pools of units in the single spatial frequency channel model. LGN cells
first filter the image linearly through a center-surround filter (V 2 G), the result of which
is then passed through a sigmoid nonlinearity (tanh). (In the simulations presented here
G was a Gaussian with standard deviation 1.4 pixels.) VI cells then provide oriented
filtering, which is also passed through a nonlinearity (logistic; but see details in text) and
fed back positively to the LGN to reinforce detected oriented edges. VI cells excite LGN
cells which have excitat.ory connections to them, and inhibit those t.hat have inhibitory
connections to them. Inhibition is implicitly assumed to be mediated by interneurons.
(Note that there are no intracortical or intrageniculate connections: communication takes
place entirely through the feedback loop.) See text for further details.
For simplicity, only striate cortex simple "edge-detecting" cells were modeled. Two
models are presented. In the first one, all cortical cells have the same spatial
frequency characteristics. In the second one, two channels, a high frequency channel
and a low frequency channel, interact simultaneously.
2.1
SINGLE SPATIAL FREQUENCY CHANNEL MODEL
A srhematic diagram of the model is shown in figure 1. The retina is used simply
as an input layer. To each input position (pixel) in the retina there corresponds
one LGN unit. Linear weights from the retina to the LGN implement a '\l2C
filter, where G(x,y) is a two-dimensional Gaussian. The LGN units then project to
eight. different pools of "orientation-t.uned" cells in VI. Each of these pools has as
many units as t.here are pixels in the input "retina". The weights in the projection
forward to VI represent eight rotations of the template shown in figure 2a, covering
360 degrees. This simulates basic orientation tuning in VI. Cortical cells then feed
411
412
Brody
back positively to the geniculus, using rotations of the template shown in fig 2(b).
The precise dynamics of the model are as follows: Ri are real-valued retinal inputs,
Li are geniculate unit outputs, and V; are cortical cell outputs. Gji represent weights
from retina - LGN, Fji forward weights from LGN - VI, and Bji backward
weights from VI - LGN. o:,/3,,,,(,TCl and TC2 are all constants. For geniculate
units:
dl_J
dt
=-"111.+
I J
L G .. Q? + L B ' LVIe
JI~~
L j = tanh(/j
J~
i
)
Ie
While for cortical cell units:
dVj
dt = -o:v'J
+ L.....J
~ Y?L? - /3(~
L.....J IY?IL ?)2
JI
I
i
Here
JI
i
I
V;={
g(Vj - Tcd
o
if vi > TC2
otherwise
gO is the logistic function.
"receptIYe field
le.atII"
???????
???????
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0000000
0000000
0000000
? ? ? ? ? ? ?
? ? ? ? ? ? ?
???????
0000000
o
o
0
0
0
000
0
0
0
000
(b)
Figure 2: Weights between the LGN and VI. Figure 2(a): Forward weights, from the
LGN to VI. Each circle represents the weight from a cell in the LGN; dark circles represent
positive weights, light circles negative weights (assumed mediated by interneurons). The
radius of each circle represent.s the strength of the corresponding weight. These weights
create "edge-detecting" neurons in VI. Figure 2(b): Backwards weights, from VI back to
the LGN. Only cells close to the contrast edge receive strong feedback.
In the scheme described above many cortical cells have overlapping receptive fields,
both in the forward projection from the geniculus and in the backwards projection
from cortex. A cell which is reinforcing an edge within its receptive field will also
partially reinforce the edge for retinotopically nearby cortical cells. For nearby cells
with similar orientation tuning, the reinforcement will enhance their own firing;
they will then enhance the firing of ot.her, similar, cells farther along; and so on.
That is, the overlapping feedback fields allow the edge detection process to follow
contours (note that the process is tempered at the geniculate level by actual input
from the retina). This is illustrated in figure 3.
A Model of Feedback to the Lateral Geniculate Nucleus
Figure 3: Following contours: This figure shows the effect on the LGN of the feedback
enhancement. The image on the left is the retinal input: a very weak, noisy horizontal edge.
The center image is the LGN after two iterations of the simulation. Note that initially
only certain sectors of the edge are detected (and hence enhanced) . The rightmost image
is the LGN after 8 iterations: the enhanced region has spread to cover the entire edge
through the effect of horizontally oriented, overlapping receptive fields. This is the final
stable point of the dynamics.
2.2
MULTIPLE SPATIAL FREQUENCY CHANNELS MODEL
In the model described above the LGN is integrating and summarizing the information provided by each of the orientation-tuned pools of cortical cells. 2 It does so in
a way which would easily ext.end to cover other types of cortical cells (bar or grating "detectors" , or varying spatial frequency channels). To experiment simply with
this possibility, an extra set of eight pools of orientation-tuned "edge-detecting"
cortical cells was added. The new set's weights were similar to the original weights
described above, except t.hey had a "receptive field length" (see figure 2) of 3 pixels:
the original set had a "receptive field length" of 9 pixels.
Thus one set was tuned for detecting short edges, while the other was tuned for
detecting long edges. The effect of using both of these sets is illustrated in figure 4.
Both sets interact nonlinearly to produce edge detection rather more robust than
either set used alone: the image produced using both simultaneously is not a linear
addit.ion of those produced using each set separately. Note how little noise is accepted as an edge. The same model, running with the same parameters but more
pixels, was also tested on a real image. This is shown in figure 5.
3
DISCUSSION ON CONNECTIVITY
A major function fulfilled by the LG N in this model is that of providing a communicat.ions pathway between cortical cells, both between cells of similar orientation but
different location or spatial frequency tuning, and between cells of different orienta2 A function not unlike that suggested by Mumford [7], except that here the "experts"
are extremely low-level orient.ation-tuned channels.
413
414
Brody
~::. ' ''1
:r"
Figure 4: Combined spatial frequency channels: The leftmost image is the retinal
input, a weak noisy edge. (The other three images are "summary outputs", obtained as
follows: the model produces activations in many pools of cortical cell units; the activations
from all VI units corresponding to a particular retinotopic position are added together to
form a real-valued number corresponding to that position; and this is then displayed as
a grey-scale pixel. Since only "edge-detecting" units were used, this provides a rough
estimate of the certainty of there being an edge at that point.) Second from left we see the
summary output of the model after 20 iterations (by which time it has stabilized), using
only the low spatial frequency channel. Only a single segment of the edge is detected.
Third from left is the output after 20 iterations using only the high frequency channel.
Only isolat.ed, short, segment.s of the edge are detected. The rightmost image is the output
using both channels simultaneously. Now the segments detected by the high frequency
channel can combine with the original image to provide edges long enough for the low
frequency channel t.o detect and complete into a single, long continuous edge.
tion tuning: for example, these last compete to reinforce their particular orientation
preference on the geniculus. The model qualitatively shows that such a pathway,
while mediated by a low-level representation like that of the LGN, can nevertheless be used effectively, producing contour-following and robust edge-detection. We
must now ask whether such a function could not be performed without feedback.
Clearly, it could be done without feedback to the LGN, purely through intracortical
connections, since any feedback net.work can in principle be "unfolded in time" into
a feedforward network which performs the same computation- provided we have
enough units and connections available.
In other words, any sugg{'st.ed functional role for corticogeniculate feedback must not
only include an account of the proposed computation performed, but also an account
of why it is preferable to perform that computation through a feedback process, in
terms of some efficiency measure (like the number of cells or synapses necessary,
for example). There can be no other rationale, apart from fortuitous coincidence,
for constructing an elaborate feedback mechanism to perform a computation that
could just as well be done without it.
With this view in mind, it. is worth re-stating that in this model any two cortical cells
whose receptive fields overlap are connected (disynaptically) through the LG N. How
many connections would we require in order to achieve similar communication if we
only used direct connections between cortical orientation-tuned cells instead? In
monkey, each cell's receptive field overlaps with approximately 10 6 others [4]- thus,
A Model of Feedback to the Lateral Geniculate Nucleus
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Figure 5: A real image: The top image is the retinal input. Stippling is due to printing
only. The center image is that obtained through detecting the zero-crossings of v 2 c.
To reduce spurious edges. a minimum slope threshold was placed on the point of the
zero-crossing below which edges were not accepted. The image shown here was the best
that could be obtained through varying both the width of the Gaussian G and the slope
threshold value. The last image shows the summary output from the model, using two
simultaneous spatial frequency cha.nnels. Note how noise is reduced compared to the center
image, straight lines are smoother, and resolution is not impaired, but is better in places
(group of people at lower left. or "smoke stacks" atop launcher).
415
416
Brody
any cortical cell would need to synapse onto at least 106 cells. If the information
can be sent via the LGN, geniculate cell fan-out can reduce the number of necessary
synapses by a significant factor. It is estimated that geniculate cells (in the cat)
synapse onto at least 200 cortical cells (probably more) [6], reducing the number of
necessary connections considerably.
4
BIOLOGY AND CONCLUSIONS
In section 1.1 I noted one important study [8J which established that corticogeniculate input reduces firing of geniculate cells for long bars; this is in direct contradiction to the prediction which would be made by this model, where the feedback
enhances firing for long features (here, edges). Thus, the model does not agree with
known physiology.
However, the model's value lies simply in clearly illustrating the possibility
that feedback in a hierarchical processing scheme like the corticogeniculate loop
can be utilized for robust, low-level pattern analysis, through the use of the
cortex-+LGN-+cortex communications pathway. The possibility that a great deal
of different types of information could be flowing through this pathway for this
purpose should not be left unconsidered.
Acknowledgements
The author is supported by fellowships from the Parsons Foundation and from
CONACYT (Mexico). Thanks are due to Michael Lyons for careful reading of the
manuscript.
References
[IJ Baker, F. H. and Malpeli, J. G. 1977 Exp. Brain Res. 29 pp. 433-444
[2J Gilbert, C.D. 1977, J. Physiol., 268, pp. 391-421
[3J Koch, C. 1992, personal communication.
[4J Hubel, D.H. and Wiesel, T. N. 1977, Proc. R. Soc. Lond. (B) 198 pp. 1-59
[5] Kalil, R. E. and Chase, R. 1970, J. Neurophysiol. 33 pp. 459-474
[6] Martin, K.A.C. 1988, Q. J. Exp. Phy. 73 pp. 637-702
[7] Mumford, D. 1991 Bioi. Cybern. 65 pp. 135-145
[8] Murphy, P.C. and Sillito, A.M. 1987, Nature 329 pp. 727-729
[9] Richard. D. et. al. 1975, Exp. Brain Res. 22 pp. 235-242
[10] Robson, J. A. 1983. J. Compo Neurol. 216 pp. 89-103
[11] Sherman, S. M. 1985. Prog. in Psychobiol. and Phys. Psych. 11 pp. 233-314
[12J Sherman, S.M. and Koch, C. 1986, Exp. Brain Res. 63 pp. 1-20
[13J Tsumoto, T. et. al. 1978, Exp. Brain Res. 32 pp. 345-364
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6,302 | 6,700 | SchNet: A continuous-filter convolutional neural
network for modeling quantum interactions
?
K. T. Sch?tt1?, P.-J. Kindermans1 , H. E. Sauceda2 , S. Chmiela1
A. Tkatchenko3 , K.-R. M?ller1,4,5?
1
Machine Learning Group, Technische Universit?t Berlin, Germany
2
Theory Department, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany
3
Physics and Materials Science Research Unit, University of Luxembourg, Luxembourg
4
Max-Planck-Institut f?r Informatik, Saarbr?cken, Germany
5
Dept. of Brain and Cognitive Engineering, Korea University, Seoul, South Korea
[email protected] ? [email protected]
Abstract
Deep learning has the potential to revolutionize quantum chemistry as it is ideally
suited to learn representations for structured data and speed up the exploration
of chemical space. While convolutional neural networks have proven to be the
first choice for images, audio and video data, the atoms in molecules are not
restricted to a grid. Instead, their precise locations contain essential physical
information, that would get lost if discretized. Thus, we propose to use continuousfilter convolutional layers to be able to model local correlations without requiring
the data to lie on a grid. We apply those layers in SchNet: a novel deep learning
architecture modeling quantum interactions in molecules. We obtain a joint model
for the total energy and interatomic forces that follows fundamental quantumchemical principles. Our architecture achieves state-of-the-art performance for
benchmarks of equilibrium molecules and molecular dynamics trajectories. Finally,
we introduce a more challenging benchmark with chemical and structural variations
that suggests the path for further work.
1
Introduction
The discovery of novel molecules and materials with desired properties is crucial for applications
such as batteries, catalysis and drug design. However, the vastness of chemical compound space
and the computational cost of accurate quantum-chemical calculations prevent an exhaustive exploration. In recent years, there have been increased efforts to use machine learning for the accelerated
discovery of molecules and materials with desired properties [1?7]. However, these methods are
only applied to stable systems in so-called equilibrium, i.e., local minima of the potential energy
surface E(r1 , . . . , rn ) where ri is the position of atom i. Data sets such as the established QM9
benchmark [8] contain only equilibrium molecules. Predicting stable atom arrangements is in itself
an important challenge in quantum chemistry and material science.
In general, it is not clear how to obtain equilibrium conformations without optimizing the atom
positions. Therefore, we need to compute both the total energy E(r1 , . . . , rn ) and the forces acting
on the atoms
?E
Fi (r1 , . . . , rn ) = ?
(r1 , . . . , rn ).
(1)
?ri
One possibility is to use a less computationally costly, however, also less accurate quantum-chemical
approximation. Instead, we choose to extend the domain of our machine learning model to both
compositional (chemical) and configurational (structural) degrees of freedom.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this work, we aim to learn a representation for molecules using equilibrium and non-equilibrium
conformations. Such a general representation for atomistic systems should follow fundamental
quantum-mechanical principles. Most importantly, the predicted force field has to be curl-free.
Otherwise, it would be possible to follow a circular trajectory of atom positions such that the energy
keeps increasing, i.e., breaking the law of energy conservation. Furthermore, the potential energy
surface as well as its partial derivatives have to be smooth, e.g., in order to be able to perform geometry
optimization. Beyond that, it is beneficial that the model incorporates the invariance of the molecular
energy with respect to rotation, translation and atom indexing. Being able to model both chemical
and conformational variations constitutes an important step towards ML-driven quantum-chemical
exploration.
This work provides the following key contributions:
? We propose continuous-filter convolutional (cfconv) layers as a means to move beyond
grid-bound data such as images or audio towards modeling objects with arbitrary positions
such as astronomical observations or atoms in molecules and materials.
? We propose SchNet: a neural network specifically designed to respect essential quantumchemical constraints. In particular, we use the proposed cfconv layers in R3 to model
interactions of atoms at arbitrary positions in the molecule. SchNet delivers both rotationally
invariant energy prediction and rotationally equivariant force predictions. We obtain a
smooth potential energy surface and the resulting force-field is guaranteed to be energyconserving.
? We present a new, challenging benchmark ? ISO17 ? including both chemical and conformational changes3 . We show that training with forces improves generalization in this setting
as well.
2
Related work
Previous work has used neural networks and Gaussian processes applied to hand-crafted features to
fit potential energy surfaces [9?14]. Graph convolutional networks for circular fingerprint [15] and
molecular graph convolutions [16] learn representations for molecules of arbitrary size. They encode
the molecular structure using neighborhood relationships as well as bond features, e.g., one-hot
encodings of single, double and triple bonds. In the following, we briefly review the related work that
will be used in our empirical evaluation: gradient domain machine learning (GDML), deep tensor
neural networks (DTNN) and enn-s2s.
Gradient-domain machine learning (GDML) Chmiela et al. [17] proposed GDML as a method
to construct force fields that explicitly obey the law of energy conservation. GDML captures the
relationship between energy and interatomic forces (see Eq. 1) by training the gradient of the energy
estimator. The functional relationship between atomic coordinates and interatomic forces is thus
learned directly and energy predictions are obtained by re-integration. However, GDML does not
scale well due to its kernel matrix growing quadratically with the number of atoms as well as the
number of examples. Beyond that, it is not designed to represent different compositions of atom
types unlike SchNet, DTNN and enn-s2s.
Deep tensor neural networks (DTNN) Sch?tt et al. [18] proposed the DTNN for molecules that
are inspired by the many-body Hamiltonian applied to the interactions of atoms. They have been
shown to reach chemical accuracy on a small set of molecular dynamics trajectories as well as QM9.
Even though the DTNN shares the invariances with our proposed architecture, its interaction layers
lack the continuous-filter convolution interpretation. It falls behind in accuracy compared to SchNet
and enn-s2s.
enn-s2s Gilmer et al. [19] proposed the enn-s2s as a variant of message-passing neural networks that
uses bond type features in addition to interatomic distances. It achieves state-of-the-art performance
on all properties of the QM9 benchmark [19]. Unfortunately, it cannot be used for molecular dynamics
predictions (MD-17). This is caused by discontinuities in their potential energy surface due to the
3
ISO17 is publicly available at www.quantum-machine.org.
2
Figure 1: The discrete filter (left) is not able to capture the subtle positional changes of the atoms
? (bottom left). The continuous filter captures these
resulting in discontinuous energy predictions E
changes and yields smooth energy predictions (bottom right).
discreteness of the one-hot encodings in their input. In contrast, SchNet does not use such features
and yields a continuous potential energy surface by using continuous-filter convolutional layers.
3
Continuous-filter convolutions
In deep learning, convolutional layers operate on discretized signals such as image pixels [20, 21],
video frames [22] or digital audio data [23]. While it is sufficient to define the filter on the same
grid in these cases, this is not possible for unevenly spaced inputs such as the atom positions of a
molecule (see Fig. 1). Other examples include astronomical observations [24], climate data [25]
and the financial market [26]. Commonly, this can be solved by a re-sampling approach defining
a representation on a grid [7, 27, 28]. However, choosing an appropriate interpolation scheme is
a challenge on its own and, possibly, requires a large number of grid points. Therefore, various
extensions of convolutional layers even beyond the Euclidean space exist, e.g., for graphs [29, 30]
and 3d shapes[31]. Analogously, we propose to use continuous filters that are able to handle unevenly
spaced data, in particular, atoms at arbitrary positions.
Given the feature representations of n objects X l = (xl1 , . . . , xln ) with xli ? RF at locations
R = (r1 , . . . , rn ) with ri ? RD , the continuous-filter convolutional layer l requires a filter-generating
function
W l : RD ? RF ,
that maps from a position to the corresponding filter values. This constitutes a generalization of a
filter tensor in discrete convolutional layers. As in dynamic filter networks [32], this filter-generating
function is modeled with a neural network. While dynamic filter networks generate weights restricted
to a grid structure, our approach generalizes this to arbitrary position and number of objects. The
output xl+1
for the convolutional layer at position ri is then given by
i
X
xl+1
= (X l ? W l )i =
xlj ? W l (ri ? rj ),
(2)
i
j
where "?" represents the element-wise multiplication. We apply these convolutions feature-wise
for computational efficiency [33]. The interactions between feature maps are handled by separate
object-wise or, specifically, atom-wise layers in SchNet.
4
SchNet
SchNet is designed to learn a representation for the prediction of molecular energies and atomic
forces. It reflects fundamental physical laws including invariance to atom indexing and translation, a
smooth energy prediction w.r.t. atom positions as well as energy-conservation of the predicted force
fields. The energy and force predictions are rotationally invariant and equivariant, respectively.
3
Figure 2: Illustration of SchNet with an architectural overview (left), the interaction block (middle)
and the continuous-filter convolution with filter-generating network (right). The shifted softplus is
defined as ssp(x) = ln(0.5ex + 0.5).
4.1
Architecture
Fig. 2 shows an overview of the SchNet architecture. At each layer, the molecule is represented atomwise analogous to pixels in an image. Interactions between atoms are modeled by the three interaction
blocks. The final prediction is obtained after atom-wise updates of the feature representation and
pooling of the resulting atom-wise energy. In the following, we discuss the different components of
the network.
Molecular representation A molecule in a certain conformation can be described uniquely by a set
of n atoms with nuclear charges Z = (Z1 , . . . , Zn ) and atomic positions R = (r1 , . . . rn ). Through
the layers of the neural network, we represent the atoms using a tuple of features X l = (xl1 , . . . xln ),
with xli ? RF with the number of feature maps F , the number of atoms n and the current layer l. The
representation of atom i is initialized using an embedding dependent on the atom type Zi :
x0i = aZi .
(3)
The atom type embeddings aZ are initialized randomly and optimized during training.
Atom-wise layers A recurring building block in our architecture are atom-wise layers. These are
dense layers that are applied separately to the representation xli of atom i:
xl+1
= W l xli + bl
i
These layers is responsible for the recombination of feature maps. Since weights are shared across
atoms, our architecture remains scalable with respect to the size of the molecule.
Interaction The interaction blocks, as shown in Fig. 2 (middle), are responsible for updating the
atomic representations based on the molecular geometry R = (r1 , . . . rn ). We keep the number of
feature maps constant at F = 64 throughout the interaction part of the network. In contrast to MPNN
and DTNN, we do not use weight sharing across multiple interaction blocks.
The blocks use a residual connection inspired by ResNet [34]:
xl+1
= xli + vil .
i
As shown in the interaction block in Fig. 2, the residual vil is computed through an atom-wise layer,
an interatomic continuous-filter convolution (cfconv) followed by two more atom-wise layers with a
softplus non-linearity in between. This allows for a flexible residual that incorporates interactions
between atoms and feature maps.
4
(a) 1st interaction block
(b) 2nd interaction block
(c) 3rd interaction block
Figure 3: 10x10 ? cuts through all 64 radial, three-dimensional filters in each interaction block of
SchNet trained on molecular dynamics of ethanol. Negative values are blue, positive values are red.
Filter-generating networks The cfconv layer including its filter-generating network are depicted
at the right panel of Fig. 2. In order to satisfy the requirements for modeling molecular energies,
we restrict our filters for the cfconv layers to be rotationally invariant. The rotational invariance is
obtained by using interatomic distances
dij = kri ? rj k
as input for the filter network. Without further processing, the filters would be highly correlated since
a neural network after initialization is close to linear. This leads to a plateau at the beginning of
training that is hard to overcome. We avoid this by expanding the distance with radial basis functions
ek (ri ? rj ) = exp(??kdij ? ?k k2 )
located at centers 0? ? ?k ? 30? every 0.1? with ? = 10?. This is chosen such that all distances
occurring in the data sets are covered by the filters. Due to this additional non-linearity, the initial
filters are less correlated leading to a faster training procedure. Choosing fewer centers corresponds
to reducing the resolution of the filter, while restricting the range of the centers corresponds to the
filter size in a usual convolutional layer. An extensive evaluation of the impact of these variables is
left for future work. We feed the expanded distances into two dense layers with softplus activations
to compute the filter weight W (ri ? rj ) as shown in Fig. 2 (right).
Fig 3 shows 2d-cuts through generated filters for all three interaction blocks of SchNet trained on
an ethanol molecular dynamics trajectory. We observe how each filter emphasizes certain ranges of
interatomic distances. This enables its interaction block to update the representations according to the
radial environment of each atom. The sequential updates from three interaction blocks allow SchNet
to construct highly complex many-body representations in the spirit of DTNNs [18] while keeping
rotational invariance due to the radial filters.
4.2
Training with energies and forces
As described above, the interatomic forces are related to the molecular energy, so that we can obtain
an energy-conserving force model by differentiating the energy model w.r.t. the atom positions
?
? i (Z1 , . . . , Zn , r1 , . . . , rn ) = ? ? E (Z1 , . . . , Zn , r1 , . . . , rn ).
F
?ri
(4)
Chmiela et al. [17] pointed out that this leads to an energy-conserving force-field by construction.
As SchNet yields rotationally invariant energy predictions, the force predictions are rotationally
equivariant by construction. The model has to be at least twice differentiable to allow for gradient
descent of the force loss. We chose a shifted softplus ssp(x) = ln(0.5ex + 0.5) as non-linearity
throughout the network in order to obtain a smooth potential energy surface. The shifting ensures that
ssp(0) = 0 and improves the convergence of the network. This activation function shows similarity
to ELUs [35], while having infinite order of continuity.
5
Table 1: Mean absolute errors for energy predictions in kcal/mol on the QM9 data set with given
training set size N . Best model in bold.
N
SchNet
DTNN [18]
enn-s2s [19]
enn-s2s-ens5 [19]
50,000
100,000
110,462
0.59
0.34
0.31
0.94
0.84
?
?
?
0.45
?
?
0.33
We include the total energy E as well as forces Fi in the training loss to train a neural network that
performs well on both properties:
!
2
n
X
?
?
?
E
2
? (E, F1 , . . . , Fn )) = kE ? Ek
? +
(5)
`(E,
.
Fi ? ?
n
?Ri
i=0
This kind of loss has been used before for fitting a restricted potential energy surfaces with MLPs [36].
In our experiments, we use ? = 0 in Eq. 5 for pure energy based training and ? = 100 for combined
energy and force training. The value of ? was optimized empirically to account for different scales of
energy and forces.
Due to the relation of energies and forces reflected in the model, we expect to see improved generalization, however, at a computational cost. As we need to perform a full forward and backward
pass on the energy model to obtain the forces, the resulting force model is twice as deep and, hence,
requires about twice the amount of computation time.
Even though the GDML model captures this relationship between energies and forces, it is explicitly
optimized to predict the force field while the energy prediction is a by-product. Models such as
circular fingerprints [15], molecular graph convolutions or message-passing neural networks[19] for
property prediction across chemical compound space are only concerned with equilibrium molecules,
i.e., the special case where the forces are vanishing. They can not be trained with forces in a similar
manner, as they include discontinuities in their predicted potential energy surface caused by discrete
binning or the use of one-hot encoded bond type information.
5
Experiments and results
In this section, we apply the SchNet to three different quantum chemistry datasets: QM9, MD17 and
ISO17. We designed the experiments such that each adds another aspect towards modeling chemical
space. While QM9 only contains equilibrium molecules, for MD17 we predict conformational
changes of molecular dynamics of single molecules. Finally, we present ISO17 combining both
chemical and structural changes.
For all datasets, we report mean absolute errors in kcal/mol for the energies and in kcal/mol/? for
the forces. The architecture of the network was fixed after an evaluation on the MD17 data sets for
benzene and ethanol (see supplement). In each experiment, we split the data into a training set of
given size N and use a validation set of 1,000 examples for early stopping. The remaining data is
used as test set. All models are trained with SGD using the ADAM optimizer [37] with 32 molecules
per mini-batch. We use an initial learning rate of 10?3 and an exponential learning rate decay with
ratio 0.96 every 100,000 steps. The model used for testing is obtained using an exponential moving
average over weights with decay rate 0.99.
5.1
QM9 ? chemical degrees of freedom
QM9 is a widely used benchmark for the prediction of various molecular properties in equilibrium [8,
38, 39]. Therefore, the forces are zero by definition and do not need to be predicted. In this setting,
we train a single model that generalizes across different compositions and sizes.
QM9 consists of ?130k organic molecules with up to 9 heavy atoms of the types {C, O, N, F}. As the
size of the training set varies across previous work, we trained our models each of these experimental
settings. Table 1 shows the performance of various competing methods for predicting the total energy
(property U0 in QM9). We provide comparisons to the DTNN [18] and the best performing MPNN
6
Table 2: Mean absolute errors for energy and force predictions in kcal/mol and kcal/mol/?, respectively. GDML and SchNet test errors for training with 1,000 and 50,000 examples of molecular
dynamics simulations of small, organic molecules are shown. SchNets were trained only on energies
as well as energies and forces combined. Best results in bold.
N = 1,000
GDML [17]
forces
N = 50,000
SchNet
energy both
DTNN [18]
energy
SchNet
energy both
Benzene
energy
forces
0.07
0.23
1.19
14.12
0.08
0.31
0.04
?
0.08
1.23
0.07
0.17
Toluene
energy
forces
0.12
0.24
2.95
22.31
0.12
0.57
0.18
?
0.16
1.79
0.09
0.09
Malonaldehyde
energy
forces
0.16
0.80
2.03
20.41
0.13
0.66
0.19
?
0.13
1.51
0.08
0.08
Salicylic acid
energy
forces
0.12
0.28
3.27
23.21
0.20
0.85
0.41
?
0.25
3.72
0.10
0.19
Aspirin
energy
forces
0.27
0.99
4.20
23.54
0.37
1.35
?
?
0.25
7.36
0.12
0.33
Ethanol
energy
forces
0.15
0.79
0.93
6.56
0.08
0.39
?
?
0.07
0.76
0.05
0.05
Uracil
energy
forces
0.11
0.24
2.26
20.08
0.14
0.56
?
?
0.13
3.28
0.10
0.11
Naphtalene
energy
forces
0.12
0.23
3.58
25.36
0.16
0.58
?
?
0.20
2.58
0.11
0.11
configuration denoted enn-s2s and an ensemble of MPNNs (enn-s2s-ens5) [19]. SchNet consistently
obtains state-of-the-art performance with an MAE of 0.31 kcal/mol at 110k training examples.
5.2
MD17 ? conformational degrees of freedom
MD17 is a collection of eight molecular dynamics simulations for small organic molecules. These
data sets were introduced by Chmiela et al. [17] for prediction of energy-conserving force fields
using GDML. Each of these consists of a trajectory of a single molecule covering a large variety
of conformations. Here, the task is to predict energies and forces using a separate model for each
trajectory. This molecule-wise training is motivated by the need for highly-accurate force predictions
when doing molecular dynamics.
Table 2 shows the performance of SchNet using 1,000 and 50,000 training examples in comparison
with GDML and DTNN. Using the smaller data set, GDML achieves remarkably accurate energy and
force predictions despite being only trained on forces. The energies are only used to fit the integration
constant. As mentioned before, GDML does not scale well with the number of atoms and training
examples. Therefore, it cannot be trained on 50,000 examples. The DTNN was evaluated only on
four of these MD trajectories using the larger training set [18]. Note that the enn-s2s cannot be used
on this dataset due to discontinuities in its inferred potential energy surface.
We trained SchNet using just energies and using both energies and forces. While the energy-only
model shows high errors for the small training set, the model including forces achieves energy
predictions comparable to GDML. In particular, we observe that SchNet outperforms GDML on the
more flexible molecules malonaldehyde and ethanol, while GDML reaches much lower force errors
on the remaining MD trajectories that all include aromatic rings.
The real strength of SchNet is its scalability, as it outperforms the DTNN in three of four data sets
using 50,000 training examples using only energies in training. Including force information, SchNet
consistently obtains accurate energies and forces with errors below 0.12 kcal/mol and 0.33 kcal/mol/?,
respectively. Remarkably, when training on energies and forces using 1,000 training examples, SchNet
performs better than training the same model on energies alone for 50,000 examples.
7
Table 3: Mean absolute errors on C7 O2 H10 isomers in kcal/mol.
mean predictor
5.3
energy
SchNet
energy+forces
known molecules /
unknown conformation
energy
forces
14.89
19.56
0.52
4.13
0.36
1.00
unknown molecules /
unknown conformation
energy
forces
15.54
19.15
3.11
5.71
2.40
2.18
ISO17 ? chemical and conformational degrees of freedom
As the next step towards quantum-chemical exploration, we demonstrate the capability of SchNet
to represent a complex potential energy surface including conformational and chemical changes.
We present a new dataset ? ISO17 ? where we consider short MD trajectories of 129 isomers, i.e.,
chemically different molecules with the same number and types of atoms. In contrast to MD17, we
train a joint model across different molecules. We calculate energies and interatomic forces from short
MD trajectories of 129 molecules drawn randomly from the largest set of isomers in QM9. While
the composition of all included molecules is C7 O2 H10 , the chemical structures are fundamentally
different. With each trajectory consisting of 5,000 conformations, the data set consists of 645,000
labeled examples.
We consider two scenarios with this dataset: In the first variant, the molecular graph structures
present in training are also present in the test data. This demonstrates how well our model is able to
represent a complex potential energy surface with chemical and conformational changes. In the more
challenging scenario, the test data contains a different subset of molecules. Here we evaluate the
generalization of our model to previously unseen chemical structures. We predict forces and energies
in both cases and compare to the mean predictor as a baseline. We draw a subset of 4,000 steps from
80% of the MD trajectories for training and validation. This leaves us with a separate test set for each
scenario: (1) the unseen 1,000 conformations of molecule trajectories included in the training set and
(2) all 5,000 conformations of the remaining 20% of molecules not included in training.
Table 3 shows the performance of the SchNet on both test sets. Our proposed model reaches chemical
accuracy for the prediction of energies and forces for the test set of known molecules. Including
forces in the training improves the performance here as well as on the set of unseen molecules. This
shows that using force information does not only help to accurately predict nearby conformations of
a single molecule, but indeed helps to generalize across chemical compound space.
6
Conclusions
We have proposed continuous-filter convolutional layers as a novel building block for deep neural
networks. In contrast to the usual convolutional layers, these can model unevenly spaced data as
occurring in astronomy, climate reasearch and, in particular, quantum chemistry. We have developed
SchNet to demonstrate the capabilities of continuous-filter convolutional layers in the context of
modeling quantum interactions in molecules. Our architecture respects quantum-chemical constraints
such as rotationally invariant energy predictions as well as rotationally equivariant, energy-conserving
force predictions.
We have evaluated our model in three increasingly challenging experimental settings. Each brings us
one step closer to practical chemical exploration driven by machine learning. SchNet improves the
state-of-the-art in predicting energies for molecules in equilibrium of the QM9 benchmark. Beyond
that, it achieves accurate predictions for energies and forces for all molecular dynamics trajectories in
MD17. Finally, we have introduced ISO17 consisting of 645,000 conformations of various C7 O2 H10
isomers. While we achieve promising results on this new benchmark, modeling chemical and
conformational variations remains difficult and needs further improvement. For this reason, we expect
that ISO17 will become a new standard benchmark for modeling quantum interactions with machine
learning.
8
Acknowledgments
This work was supported by the Federal Ministry of Education and Research (BMBF) for the Berlin
Big Data Center BBDC (01IS14013A). Additional support was provided by the DFG (MU 987/20-1)
and from the European Union?s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement NO 657679. K.R.M. gratefully acknowledges the BK21 program
funded by Korean National Research Foundation grant (No. 2012-005741) and the Institute for
Information & Communications Technology Promotion (IITP) grant funded by the Korea government
(no. 2017-0-00451).
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11
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6,303 | 6,701 | Active Bias: Training More Accurate Neural
Networks by Emphasizing High Variance Samples
Haw-Shiuan Chang, Erik Learned-Miller, Andrew McCallum
University of Massachusetts, Amherst
140 Governors Dr., Amherst, MA 01003
{hschang,elm,mccallum}@cs.umass.edu
Abstract
Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on
lightweight estimates of sample uncertainty in stochastic gradient descent (SGD):
the variance in predicted probability of the correct class across iterations of minibatch SGD, and the proximity of the correct class probability to the decision
threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains
on top of other popular training techniques, such as residual learning, momentum,
ADAM, batch normalization, dropout, and distillation.
1
Introduction
Learning easier material before harder material is often beneficial to human learning. Inspired by
this observation, curriculum learning [5] has shown that learning from easier instances first can also
improve neural network training. When it is not known a priori which samples are easy, examples
with lower loss on the current model can be inferred to be easier and can be used in early training.
This strategy has been referred to as self-paced learning [25]. By decreasing the weight of difficult
examples in the loss function, the model may become more robust to outliers [33], and this method
has proven useful in several applications, especially with noisy labels [36].
Nevertheless, selecting easier examples for training often slows down the training process because
easier samples usually contribute smaller gradients, and the current model has already learned how to
make correct predictions on these samples. On the other hand, and somewhat ironically, the opposite
strategy (i.e., sampling harder instances more often) has been shown to accelerate (mini-batch)
stochastic gradient descent (SGD) in some cases, where the difficulty of an example can be defined by
its loss [18, 29, 44] or be proportional to the magnitude of its gradient [51, 1, 12, 13]. This strategy is
sometimes referred to as hard example mining [44].
In the literature, we can see that these two opposing strategies work well in different situations.
Preferring easier examples may be effective when either machines or humans try to solve a challenging
task containing more label noise or outliers. On the other hand, focusing on harder samples may
accelerate and stabilize SGD in cleaner data by minimizing the variance of gradients [1, 12]. However,
we often do not know how noisy our training dataset is. Motivated by this practical need, this paper
explores new methods of re-weighting training examples that are effective in both scenarios.
Intuitively, if a model has already predicted some examples correctly with high confidence, those
samples may be too easy to contain useful information for improving that model further. Similarly,
if some examples are always predicted incorrectly over many iterations of training, these examples
may just be too difficult/noisy and may degrade the model. This suggests that we should somehow
prefer uncertain samples that are predicted incorrectly sometimes during training and correctly at
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Cat
Images
P(class | image)
Cat?
0.9
Cat?
0.8
Cat?
0.4
Dog?
0.2
Frog?
0.2
Emphasize?
Frog?
0.3
Methods
Easy
Self Paced
Learning
Uncertain
Active
Learning
Hard
Hard Example
Mining
Training Mini-batch Iterations
Figure 1: The proposed methods emphasize uncertain samples based on previous prediction history.
other times, as illustrated in Figure 1. This preference is consistent with common variance reduction
strategies in active learning [43].
Previous studies suggest that finding informative unlabeled samples to label is related to selecting
already-labeled samples to optimize the model parameters [14]. As reported in the previous studies [42, 6], models can sometimes achieve lower generalization error after being trained with only
a subset of actively selected training data. In other words, focusing on informative samples can be
beneficial even when all labels are available.
We propose two lightweight methods that actively emphasize uncertain samples to improve minibatch SGD for classification. One method measures the variance of prediction probabilities, while the
other one estimates the closeness between the prediction probabilities and the decision threshold. For
logistic regression, both methods can be proven to reduce the uncertainty in the model parameters
under reasonable approximations.
We present extensive experiments on CIFAR 10, CIFAR 100, MNIST (image classification), Question
Type (sentence classification), CoNLL 2003, and OntoNote 5.0 (Named entity Recognition), as well
as on different architectures, including multiple class logistic regression, fully-connected networks,
convolutional neural networks (CNNs) [26], and residual networks [16]. The results show that
active bias makes neural networks more robust without prior knowledge of noise, and reduces the
generalization error by 1% ?18% even on training sets having few (if any) annotation errors.
2
Related work
As (deep) neural networks become more widespread, many methods have recently been proposed to
improve SGD training. When using (mini-batch) SGD, the randomness of the gradient sometimes
slows down the optimization, so one common approach is to use the gradient computed in previous
iterations to stabilize the process. Examples include momentum [38], stochastic variance reduced
gradient (SVRG) [21], and proximal stochastic variance reduced gradient (Prox-SVRG) [49]. Other
work proposes variants of semi-stochastic algorithms to approximate the exact gradient direction and
reduce the gradient variance [47, 34]. More recently, supervised optimization methods like learning
by learning [3] also show great potential in this problem.
In addition to the high variance of the gradient, another issue with SGD is the difficulty of tuning the
learning rate. Like Quasi-Newton methods, several methods adaptively adjust learning rates based on
local curvature [2, 40], while ADAGRAD [11] applies different learning rates to different dimensions.
ADAM [23] combines several of these techniques and is widely used in practice.
More recently, some studies accelerate SGD by weighting each class differently [13] or weighting
each sample differently as we do [18, 51, 29, 12, 1, 44], and their experiments suggest that the methods
are often compatible with other techniques such as Prox-SVRG, ADAGRAD, or ADAM [29, 13].
Notice that Gao et al. [12] discuss the idea of selecting uncertain examples for SGD based on active
learning, but their proposed methods choose each sample according to the magnitude of its gradient
as in ISSGD [1], which actually prefers more difficult examples.
The aforementioned methods focus on accelerating the optimization of a fixed loss function given a
fixed model. Many of these methods adopt importance sampling. That is, if the method prefers to
2
select harder examples, the learning rate corresponding to those examples will be lower. This makes
gradient estimation unbiased [18, 51, 1, 12, 13], which guarantees convergence [51, 13].
On the other hand, to make models more robust to outliers, some approaches inject bias into the
loss function in order to emphasize easier examples [37, 48, 27, 35]. Some variants of the strategy
gradually increase the loss of hard examples [32], as in self-paced learning [25]. To alleviate the local
minimum problem during training, other techniques that smooth the loss function have been proposed
recently [8, 15]. Nevertheless, to our knowledge, it remains an unsolved challenge to balance the
easy and difficult training examples to facilitate training while remaining robust to outliers.
3
Methods
In this section, we first discuss the baseline methods against which we shall compare and introduce
some notations which we are going to use later on. We then present our two active bias methods
based on prediction variance and closeness to the decision threshold.
3.1
Baselines
Due to its simplicity and generally good performance, the most widely used version of SGD samples
each training instance uniformly. This basic strategy has two variants. The first samples with
replacement. Let D = (xi , yi )i indicate the training dataset. The probability of selecting each sample
1
is equal (i.e., Ps (i|D) = |D|
), so we call it SGD Uniform (SGD-Uni). The second samples without
replacement. Let Se be the set of samples we have already used in the current epoch. Then, the
1
sampling probability Ps (i|Se , D) would become ( |D|?|S
)1i?S
/ e , where 1 is an indicator function.
e|
This version scans through all of the examples in each epoch, so we call it SGD-Scan.
We propose a simple baseline which selects harder examples with higher probability, as done by
Loshchilov and Hutter [29]. Specifically, we let Ps (i|H, Se , D) ? 1 ? p?H t?1 (yi |xi ) + D , where
i
Hit?1 is the history of prediction probability which stores all p(yi |xi ) when xi is selected to train
S
the network before the current iteration t, H = i Hit?1 , p?H t?1 (yi |xi ) is the average probability
i
of classifying sample i into its correct class yi over all the stored p(yi |xi ) in Hit?1 , and D is a
smoothness constant. Notice that by only considering p(yi |xi ) in Hit?1 , we won?t need to perform
extra forward passes. We refer to this simple baseline as SGD Sampled by Difficulty (SGD-SD).
In practice, SGD-Scan often works better than SGD-Uni because it ensures that the model sees all of
the training examples in each epoch. To emphasize difficult examples while applying SGD-Scan,
we weight
P each sample differently in the loss function. That is, the loss function is modified as
L =
i vi ? lossi (W ) + ?R(W ), where W are the parameters in the model, lossi (W ) is the
prediction loss, and ?R(W ) is the regularization term of the model. The weight of the ith sample vi
can be set as N1D (1 ? p?H t?1 (yi |xi ) + D ), where ND is a normalization constant making the average
i
of vi equal to 1. We want to keep the average of the vi fixed so that we do not change the global
learning rate. We denote this method SGD Weighted by Difficulty (SGD-WD).
Models usually cannot fit outliers well, so SGD-SD and SGD-WD would not be robust to noise. To
make a model unbiased, importance sampling can be used. That is, we can let Ps (i|H, Se , D) ?
1 ? p?H t?1 (yi |xi ) + D and vi = ND (1 ? p?H t?1 (yi |xi ) + D )?1 , which is similar to an approach
i
i
used by Hinton [18]. We refer to this as SGD Importance-Sampled by Difficulty (SGD-ISD).
In addition, we propose two simple baselines that emphasize easy examples, as in self-paced
learning. Based on the same naming convention, SGD Sampled by Easiness (SGD-SE) denotes
that Ps (i|H, Se , D) ? p?H t?1 (yi |xi ) + E , while SGD Weighted by Easiness (SGD-WE) sets
i
vi = N1E (?
pH t?1 (yi |xi ) + E ), where NE normalizes the vi ?s to have unit mean.
i
3.2
Prediction Variance
In the active learning setting, the prediction variance can be used to measure the uncertainty of each
sample for either a regression or classification problem [41]. In order to gain more information at
each SGD iteration, we choose samples with high prediction variances.
3
Since the prediction variances are estimated on the fly, we would like to balance exploration and
exploitation. Adopting the optimism in face of uncertainty heuristics of bandit problems [7], we draw
the next sample based on the estimated prediction variance plus its confidence interval. Specifically,
for SGD Sampled by Prediction Variance (SGD-SPV), we let
s
conf
ci
Ps (i|H, Se , D) ? std
(H) + V , where
conf
ci
std
(H) =
vd
ar(pH t?1 (yi |xi )) +
vd
ar(pH t?1 (yi |xi ))2
i
i
|Hit?1 | ? 1
,
(1)
t?1
t?1
vd
ar pH t?1 (yi |xi ) is the prediction variance estimated by history Hi , and |Hi | is the number
i
of stored prediction probabilities. Assuming pH t?1 (yi |xi ) is normally distributed under the unceri
tainty of model parameters w, the variance of prediction variance estimation can be estimated by
2
2 ? vd
ar pH t?1 (yi |xi ) (|Hit?1 | ? 1)?1 . As we did in the baselines, adding the smoothness constant
i
V prevents the low variance instances from never being selected again. Similarly, another variant of
conf
c (H) + V ), where NV normalizes vi like other weighted methods;
the method sets vi = N1V (std
i
we call this SGD Weighted by Prediction Variance (SGD-WPV).
As in SGD-WD, SGD-WE or self-paced learning [4], we train an unbiased model for several burnin epochs at the beginning so as to judge the sampling uncertainty reasonably and stably. Other
implementation details will be described in the first section of the supplementary material.
Using a low learning rate, model parameters w would be close to a good local minimum after
sufficient burn-in epochs, and thus the posterior distribution of w can be locally approximated
by a Gaussian distribution. Furthermore, the prediction distribution p(yi |xi , w) is often locally
smooth with respect to the model parameters w (i.e., small changes of model parameters only induce
small changes in the prediction distribution), so a Gaussian tends to approximate the distribution of
pH t?1 (yi |xi ) well in practice.
i
Example: logistic regression
Given a Gaussian prior P r(W = w) = N (w|0, s0 I) on the parameters, consider the probabilistic
interpretation of logistic regression:
? log(P r(Y, W = w|X)) = ?
X
log(p(yi |xi , w)) ?
i
where p(yi |xi , w) =
1
,
1+exp(?yi wT xi )
c
||w||2 ,
s0
(2)
and yi ? {1, ?1}.
Since the posterior distribution of W is log-concave [39], we can use P r(W = w|Y, X) ?
N (w|wN , SN ), where wN is maximum a posteriori (MAP) estimation, and
?1
SN
= 5w 5w ? log(P r(Y, W |X)) =
X
p(yi |xi ) (1 ? p(yi |xi )) xi xi T +
i
2c
I.
s0
(3)
Then, we further approximate p(yi |xi , W ) using the first order Taylor expansion p(yi |xi , W ) ?
p(yi |xi , w) + gi (w)T (W ? w), where gi (w) = p(yi |xi , w) (1 ? p(yi |xi , w)) xi . We can compute
the prediction variance [41] with respect to the uncertainty of W
V ar(p(yi |xi , W )) ? gi (w)T SN gi (w).
(4)
These approximations tell us several things. First, V ar(p(yi |xi , W )) is proportional to
p(yi |xi , w)2 (1 ? p(yi |xi , w))2 , so the prediction variance is larger when the sample i is closer
to the boundary. Second, when we have more sample points close to the boundary, the variance of
the parameters SN is lower. That is, when we emphasize samples with high prediction variances,
the uncertainty of parameters tends to be reduced, akin to the variance reduction strategy in active
learning [30]. Third, with a Gaussian assumption on the posterior distribution P r(W = w|Y, X) and
the Taylor expansion, the distribution of p(yi |xi , W ) in logistic regression becomes Gaussian, which
justifies our previous assumption of pH t?1 (yi |xi ) for the confidence estimation of the prediction
i
variance. Notice that there are other methods that can measure the prediction uncertainty, such as the
mutual information between labels and parameters [19], but we found that the prediction variance
works better in our experiments.
4
7
7
6
6
5
5
4
4
3
3
2
5
0
5
10
15
w[1]
1.27
0.05
1.26
0.00
1.25
1.24
0.05
1.23
0.10
1.22
0.15
2.35
2.30
2.25
b
2.20
2.15
2.10
(e) SGD-WD
parameters space
0.98
0.05
0.96
5
7
0.97
6
0.95
0.00
0.94
5
0.93
4
0.92
0.91
0.15
2.35
2.30
2.25
10
b
2.20
2.15
3
2.10
0
(c) SGD-Scan
parameters space
(b) Training Samples
5
10
(d) SGD-Scan boundaries
0.15
0.840
7
0.10
6
0.05
5
0.00
0.795
5
4
0.05
0.780
4
3
0.10
1.21
w[1]
1.28
0.99
0.10
0.10
1.29
0.10
0.15
0.05
0
(a) Sampling distribution
0.15
w[1]
8
0
5
(f) SGD-WD sample
weights and boundaries
6
0.810
0.765
3
0.750
0.15
2.35
10
7
0.825
2.30
2.25
b
2.20
2.15
2.10
0
(g) SGD-WPV
parameters space
5
10
(h) SGD-WPV sample
weights and boundaries
Figure 2: A toy example which compares different methods in a two-class logistic regression model.
To visualize the optimization path for the classifier parameters (the red paths in (c), (e), and (g)) in
two dimensions, we fix the weight corresponding to the x-axis to 0.5 and only show the weight for
y-axis w[1] and bias term b. The ith sample size in (f) and (h) is proportional to vi . The toy example
shows that SGD-WPV can train a more accurate model in a noisy dataset.
Figure 2 illustrates a toy example. Given the same learning rate, we can see that the normal SGD in
Figure 2c and 2d will have higher uncertainty when there are many outliers, and emphasizing difficult
examples in Figure 2e and 2f makes it worse. On the other hand, the samples near the boundaries
would have higher prediction variances (i.e., larger circles or crosses in Figure 2h) and thus higher
impact on the loss function in SGD-WPV.
After burn-in epochs, w becomes close to a local minimum using SGD. Then, the parameters
estimated in each iteration can be viewed, approximately, as samples drawn from the posterior
distribution
of the parameters
P r(W = w|Y, X) [31]. Therefore, after running SGD long enough,
vd
ar pH t?1 (yi |xi ) can be used to approximate V ar (p(yi |xi , W )). Notice that if we directly apply
i
bias at the beginning without running burn-in epochs, incorrect examples might be emphasized,
which is also known as the local minimum problem in active learning [14]. For instance, in Figure 2,
if burn-in epochs are not applied and the initial w is a vertical line on the left, the outliers close to the
initial boundary would be emphasized, which slows down the convergence speed.
In this simple example, we can also see that the gradient magnitude is proportional to the difficulty
because ?5w log(p(yi |xi , w)) = (1 ? p(yi |xi , w)) xi . This is why we believe the SGD acceleration
methods based on gradient magnitude [1, 13] can be categorized as variants of preferring difficult
examples, and thus more vulnerable to outliers (like the samples on the left or right in Figure 2).
3.3
Threshold Closeness
Motivated by the previous analysis, we propose a simpler and more direct approach to select samples
whose correct class probability is close to the decision threshold.
SGD Sampled by Threshold
Closeness (SGD-STC) makes Ps (i|H, Se , D) ? p?H t?1 (yi |xi ) 1 ? p?H t?1 (yi |xi ) + T , where
i
i
p?H t?1 (yi |xi ) is the average probability of classifying sample i into its correct class yi over all the
i
stored p(yi |xi ) in Hit?1 . When there are multiple classes, this measures the closeness of the threshold
for distinguishing the correct class out of the union of the rest of the classes (i.e., one-versus-rest).
The method is similar to an approximation of the optimal allocation in stratified sampling proposed
by Druck and McCallum [10].
Similarly, SGD Weighted by Threshold Closeness (SGD-WTC)
P chooses the weight of ith sample vi =
1
1
t?1 (yi |xi )(1? p
t?1 (yi |xi ))+T , where NT =
p
?
?
?H t?1 (yj |xj )(1? p?H t?1 (yj |xj ))+T .
jp
Hi
NT Hi
|D|
j
j
The weighting can be viewed as combining the SGD-WD and SGD-WE by multiplying their weights
5
Table 1: Model architectures. Dropouts and L2 reg (regularization) are only applied to the fullyconnected (FC) layer(s).
Dataset
MNIST
CIFAR 10
CIFAR 100
Question Type
CoNLL 2003
OntoNote 5.0
MNIST
# Conv
layers
2
0
26 or
62
1
Filter
size
5x5
N/A
Filter
number
32, 64
N/A
# Pooling
layers
2
0
# FC
layers
2
1
Dropout
keep probs
0.5
1
L2
reg
0.0005
0.01
1
1
0
1
# BN
layers
0
0
13 or
31
0
3X3
16, 32, 64
0
(2,3,4)x1
64
1
0.5
0.01
3
3x1
100
0
0
1
0.5, 0.75
0.001
0
N/A
N/A
0
0
2
1
0
Table 2: Optimization hyper-parameters and experiment settings
Momentum
SGD
Batch
size
64
100
Learning
rate
0.01
1e-6
CIFAR 100
Momentum
128
0.1
Question Type
CoNLL 2003
OntoNote 5.0
MNIST
ADAM
64
0.001
Learning
rate decay
0.95
0.5 (per 5 epochs)
0.1 (at 80, 100,
120 epochs)
1
ADAM
128
0.0005
SGD
128
0.1
Dataset
Optimizer
MNIST
CIFAR 10
150
# Burn-in
epochs
2
10
90 or
50
20
1
200
30
10
1
60
20
10
# Epochs
80
30
150
# Trials
20
30
20
100
together. Although other uncertainty estimates such as entropy are widely used in active learning and
can also be viewed as a measure of boundary closeness, we found the proposed formula works better
in our experiments.
When using logistic regression, after injecting the bias vi into the loss function, approximating the
prediction probability based on previous history, removing the regularization and smoothness constant
(i.e., p(yi |xi , w) ? p?H t?1 (yi |xi ), 1/s0 = 0, and T = 0), we can show that
i
X
i
V ar(p(yi |xi , W )) ?
X
gi (w)T SN gi (w) ? NT ? dim(w),
(5)
i
where dim(w) is the dimension of parameters w. This will ensure that the average prediction variance
drops linearly as the number of training instance increases. The derivation could be seen in the
supplementary materials.
4
Experiments
We test our methods on six different datasets. The results show that the active bias techniques
constantly outperform the standard uniform sampling (i.e., SGD-Uni and SGD-Scan) in the deep
models as well as the shallow models. For each dataset, we use an existing, publicly available
implementation for the problem and emphasize samples using different methods. The architectures
and hyper-parameters are summarized in Table 1. All neural networks use softmax and cross-entropy
loss at the last layer. The optimization and experiment setups are listed in Table 2. As shown in the
second column of the table, SGD in CNNs and residual networks actually refers to momentum or
ADAM instead of vanilla SGD. All experiments use mini-batch.
Like most of the widely used neural network training techniques, the proposed techniques are not
applicable to every scenario. For all the datasets we tried, we found that the proposed methods are not
sensitive to the hyper-parameter setup except when applying a very complicated model to a relatively
smaller dataset. If a complicated model achieves 100% training accuracy within a few epochs, the
most uncertain examples would often be outliers, biasing the model towards overfitting.
To avoid this scenario, we modify the default hyper-parameters setup in the implementation of the text
classifiers in Section 4.3 and Section 4.4 to achieve similar performance using simplified models. For
all other models and datasets, we use the default hyper-parameters of the existing implementations,
which should favor the SGD-Uni or SGD-Scan methods, since the default hyper-parameters are
6
Table 3: The average of the best testing error rates for different sampling methods and datasets (%).
The confidence intervals are standard errors. LR means logistic regression.
Datasets
MNIST
Noisy MNIST
CIFAR 10
QT
Model
CNN
CNN
LR
CNN
SGD-Uni
0.55?0.01
0.83?0.01
62.49?0.06
2.19?0.02
SGD-SD
0.52?0.01
1.00?0.01
63.14?0.06
2.03?0.02
SGD-ISD
0.57?0.01
0.84?0.01
62.48?0.07
2.20?0.02
SGD-SE
0.54?0.01
0.69 ?0.01
60.87?0.06
2.28?0.02
SGD-SPV
0.51 ?0.01
0.64?0.01
60.66?0.06
2.08 ?0.02
SGD-STC
0.51?0.01
0.63?0.01
61.00?0.06
2.08?0.02
optimized for these cases. To show the reliability of the proposed methods, we do not optimize the
hyper-parameters for the proposed methods or baselines.
Due to the randomness in all the SGD variants, we repeat experiments and list the number of trials
in Table 2. At the beginning of each trial, network weights are trained with uniform sampling
SGD until validation performance starts to saturate. After these burn-in epochs, we apply different
sampling/weighting methods and compare performance. The number of burn-in epochs is determined
by cross-validation, and the number of epochs in each trial is set large enough to let the testing error
of most methods converge. In Tables 3 and 4, we evaluate the testing performance of each method
after each epoch and report the best testing performance among epochs within each trial.
As previously discussed, there are various versions preferring easy or difficult examples. Some
of them require extra time to collect necessary statistics such as the gradient magnitude of each
sample [12, 1], change the network architecture [15, 44], or involve an annealing schedule like selfpaced learning [25, 32]. We tried self-paced learning on CIFAR 10 but found that performance usually
remains the same and is sometimes sensitive to the hyper-parameters of the annealing schedule. This
finding is consistent with the results from [4]. To simplify the comparison, we focus on testing the
effects of steady bias based on sample difficulty (e.g., compare with SGD-SE and SGD-SD) and do
not gradually change the preference during the training like self-paced learning.
It is not always easy to change the sampling procedure because of the model or implementation
constraints. For example, in sequence labeling tasks (CoNLL 2003 and OntoNote 5.0), the words in
the same sentence need to be trained together. Thus, we only compare methods which modify the
loss function (SGD-W*) with SGD-Scan for some models. For the other experiments, re-weighting
examples (SGD-W*) generally gives us better performance than changing the sampling distribution
(SGD-S*). It might be because we can better estimate the statistics of each sample.
4.1
MNIST
We apply our method to a CNN [26] for MNIST1 using one of the Tensorflow tutorials.2 The dataset
has high testing accuracy, so most of the examples are too easy for the model after a few epochs.
Selecting more difficult instances can accelerate learning or improve testing accuracy [18, 29, 13].
The results from SGD-SD and SGD-WD confirm this finding while selecting uncertain examples can
give us a similar or larger boost. Furthermore, we test the robustness of our methods by randomly
reassigning the labels of 10% of the images, and the results indicate that the SGD-WPV improves the
performance of SGD-Scan even more while SGD-SD overfits the data seriously.
4.2
CIFAR 10 and CIFAR 100
We test a simple multi-class logistic regression3 on CIFAR 10 [24].4 Images are down-sampled
significantly to 32 ? 32 ? 3, so many examples are difficult, even for humans. SGD-SPV and
SGD-SE perform significantly better than SGD-Uni here, consistent with the idea that avoiding
difficult examples increases robustness to outliers.
For CIFAR 100 [24], we demonstrate that the proposed approaches can also work in very deep
residual networks [16].5 To show the method is not sensitive to the network depth and the number of
burn-in epochs, we present results from the network with 27 layers and 90 burn-in epochs as well
1
http://yann.lecun.com/exdb/mnist/
https://github.com/tensorflow/models/blob/master/tutorials/image/mnist
3
https://cs231n.github.io/assignments2016/assignment2/
4
https://www.cs.toronto.edu/~kriz/cifar.html
5
https://github.com/tensorflow/models/tree/master/resnet
2
7
Table 4: The average of the best testing error rates and their standard errors for different weighting
methods (%). For CoNLL 2003 and OntoNote 5.0, the values are 1-(F1 score). CNN, LR, RN 27,
RN 63 and FC mean convolutional neural network, logistic regression, residual networks with 27
layers, residual network with 63 layers, and fully-connected network, respectively.
Datasets
MNIST
Noisy MNIST
CIFAR 10
CIFAR 100
CIFAR 100
QT
CoNLL 2003
OntoNote 5.0
MNIST
MNIST (distill)
Model
CNN
CNN
LR
RN 27
RN 63
CNN
CNN
CNN
FC
FC
SGD-Scan
0.54?0.01
0.81?0.01
62.48?0.06
34.04?0.06
30.70?0.06
2.24?0.02
11.62?0.04
17.80?0.05
2.85?0.03
2.27?0.01
SGD-WD
0.48?0.01
0.92?0.01
63.10?0.06
34.55?0.06
31.57?0.09
1.93?0.02
11.50?0.05
17.65?0.06
2.17?0.01
2.13?0.02
SGD-WE
0.56?0.01
0.72?0.01
60.88?0.06
33.65?0.07
29.92?0.09
2.30?0.02
11.73?0.04
18.40?0.05
3.08?0.03
2.35?0.01
SGD-WPV
0.48?0.01
0.61?0.02
60.61?0.06
33.69?0.07
30.02?0.08
1.99?0.02
11.24?0.06
17.82?0.03
2.68?0.02
2.18?0.02
SGD-WTC
0.48?0.01
0.63?0.01
61.02?0.06
33.64?0.07
30.16?0.09
2.02?0.02
11.18?0.03
17.51?0.05
2.34?0.03
2.07?0.02
as the network with 63 layers and 50 burn-in epochs. Without changing architectures, emphasizing
uncertain or easy examples gains around 0.5% in both settings, which is significant considering the
fact that the much deeper network shows only 3% improvement here.
When training a neural network, gradually reducing the learning rate (i.e., the magnitude of gradients)
usually improves performance. When difficult examples are sampled less, the magnitude of gradients
would be reduced. Thus, some of the improvement of SGD-SPV and SGD-SE might come from
using a lower effective learning rate. Nevertheless, since we apply the aggressive learning rate decay
in the experiments of CIFAR 10 and CIFAR 100, we know that the improvements from SGD-SPV
and SGD-SE cannot be entirely explained by its lower effective learning rate.
4.3
Question Type
To investigate whether our methods are effective for smaller text datasets, we apply them to a sentence
classification task, which we refer to as the Question Type (QT) dataset [28].6 We use the CNN
architecture proposed by Kim [22].7 Like many other NLP tasks, the dataset is relatively small and
this CNN classifier does not inject noise to inputs like the implementation of residual networks in
CIFAR 100, so this complicated model reaches 100% training accuracy within a few epochs.
To address this, we reduced the model complexity by (i) decreasing the number of filters from 128 to
64, (ii) decreasing convolutional filter widths from 3,4,5 to 2,3,4, (iii) adding L2 regularization with
scale 0.01, (iv) performing PCA to reduce the dimension of pre-trained word embedding from 300 to
50 and fixing the word embedding during training. The smaller model achieves better performance
compared with the results from the original paper [22]. As with MNIST, most examples are too easy
for the model, so preferring hard examples is effective while the proposed active bias methods can
achieve comparable performance, and are better than SGD-Uni and SGD-Scan.
4.4
Sequence Tagging Tasks
We also test our methods on Named Entity Recognition (NER) in CoNLL 2003 [46] and OntoNote
5.0 [20] datasets using the CNN from Strubell et al. [45].8 Similar to Question Type, the model is too
complex for our approaches. So we (i) only use 3 layers instead of 4 layers, (ii) reduce the number of
filters from 300 to 100, (iii) add 0.001 L2 regularization, (iv) make the 50 dimension word embedding
from Collobert et al. [9] non-trainable. The micro F1 of this smaller model only drops around 1%-2%
from the original big model. Table 4 shows that our methods achieve the lowest error rate (1-F1) in
both benchmarks.
6
https://cogcomp.cs.illinois.edu/Data/QA/QC/
https://github.com/dennybritz/cnn-text-classification-tf
8
https://github.com/iesl/dilated-cnn-ner
7
8
4.5
Distillation
Although state-of-the-art neural networks in many applications memorize examples easily [50], much
simpler models can usually achieve similar performance like those in the previous two experiments.
In practice, such models are often preferable due to their low computation and memory requirements.
We have shown that the proposed method can improve these smaller models as distillation did [17], so
it is natural to check whether our methods can work well with distillation. We use an implementation9
that distills a shallow CNN with 3 convolution layers to a 2 layer fully-connected network in MNIST.
The teacher network can achieve 0.8% testing error, and the temperature of softmax is set as 1.
Our approaches and baselines simply apply the sample dependent weights vi to the final loss function
(i.e., cross-entropy of the true labels plus cross-entropy of the prediction probability from the teacher
network). In MNIST, SGD-WTC and SGD-WD can achieve similar or better improvements compared
with adding distillation into SGD-Scan. Furthermore, the best performance comes from the distillation
plus SGD-WTC, which shows that active bias is compatible with distillation in this dataset.
5
Conclusion
Deep learning researchers often gain accuracy by employing training techniques such as momentum,
dropout, batch normalization, and distillation. This paper presents a new compatible sibling to these
methods, which we recommend for wide use. Our relatively simple and computationally lightweight
techniques emphasize the uncertain examples (i.e., SGD-*PV and SGD-*TC).
The experiments confirm that the proper bias can be beneficial to generalization performance. When
the task is relatively easy (both training and testing accuracy are high), preferring more difficult
examples works well. On the contrary, when the dataset is challenging or noisy (both training and
testing accuracy are low), emphasizing easier samples often lead to a better performance. In both
cases, the active bias techniques consistently lead to more accurate and robust neural networks as
long as the classifier does not memorize all the training samples easily (i.e., training accuracy is high
but testing accuracy is low).
Acknowledgements
This material is based on research sponsored by National Science Foundation under Grant No.
1514053 and by DARPA under agreement number FA8750-1 3-2-0020 and HRO011-15-2-0036.
The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes
notwithstanding any copyright notation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing the official policies or
endorsements, either expressed or implied, of DARPA or the U.S. Government.
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6,304 | 6,702 | Differentiable Learning of Submodular Models
Andreas Krause
Department of Computer Science
ETH Zurich
[email protected]
Josip Djolonga
Department of Computer Science
ETH Zurich
[email protected]
Abstract
Can we incorporate discrete optimization algorithms within modern machine learning models? For example, is it possible to incorporate in deep architectures a layer
whose output is the minimal cut of a parametrized graph? Given that these models
are trained end-to-end by leveraging gradient information, the introduction of such
layers seems very challenging due to their non-continuous output. In this paper we
focus on the problem of submodular minimization, for which we show that such
layers are indeed possible. The key idea is that we can continuously relax the output
without sacrificing guarantees. We provide an easily computable approximation
to the Jacobian complemented with a complete theoretical analysis. Finally, these
contributions let us experimentally learn probabilistic log-supermodular models
via a bi-level variational inference formulation.
1
Introduction
Discrete optimization problems are ubiquitous in machine learning. While the majority of them are
provably hard, a commonly exploitable trait that renders some of them tractable is that of submodularity [1, 2]. In addition to capturing many useful phenomena, submodular functions can be minimized
in polynomial time and also enjoy a powerful connection to convex optimization [3]. Both of these
properties have been used to great effect in both computer vision and machine learning, to e.g. compute the MAP configuration in undirected graphical models with long-reaching interactions [4]
and higher-order factors [5], clustering [6], to perform variational inference in log-supermodular
models [7, 8], or to design norms useful for structured sparsity problems [9, 10].
Despite all the benefits of submodular functions, the question of how to learn them in a practical
manner remains open. Moreover, if we want to open the toolbox of submodular optimization to
modern practitioners, an intriguing question is how to to use them in conjunction with deep learning
networks. For instance, we need to develop mechanisms that would enable them to be trained
together in a fully end-to-end fashion. As a concrete example from the computer vision domain,
consider the problem of image segmentation. Namely, we are given as input an RGB representation
x ? Rn?3 of an image captured by say a dashboard camera, and the goal is to identify the set of pixels
A ? {1, 2, . . . , n} that are occupied by pedestrians. While we could train a network ? : x ? v ? Rn
to output per-pixel scores, it would be helpful, especially in domains with limited data, to bias the
predictions by encoding some prior beliefs about the expected output. For example, we might prefer
segmentations that are spatially consistent. One common approach to encourage such configurations
is to first define a graph over the image G = (V, E) by connecting neighbouring pixels, specify
weights w over the edges, and then solve the following graph-cut problem
X
X
A? (w, v) = arg min F (A) = arg min
wi,j JA ? {i, j} = 1K +
vi . (1)
|{z}
|
{z
}
A?V
A?V
{i,j}?E
1 iff the predictions disagree
i?A pixel score
While this can be easily seen as a module computing the best configuration as a function of the edge
weights and per-pixel scores, incorporating it as a layer in a deep network seems at a first glance to
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
be a futile task. Even though the output is easily computable, it will be discontinuous and have no
Jacobian, which is necessary for backpropagation. However, as the above problem is an instance of
submodular minimization, we can leverage its relationship to convexity and relax it to
X
1
1
y? (w, v) = arg min f (y) + kyk2 = arg min
wi,j |yi ? yj | + vT y + kyk2 . (2)
2
2
y?Rn
y?Rn
{i,j}?E
In addition to having a continuous output, this relaxation has a very strong connection with the discrete
problem as the discrete optimizer can be obtained by thresholding y? as A? = {i ? V | yi? > 0}.
Moreover, as explained in ?2, for every submodular function F there exists an easily computable
convex function f so that this relationship holds. For general submodular functions, the negation of
the solution to the relaxed problem (2) is known as the min-norm point [11]. In this paper we consider
the problem of designing such modules that solve discrete optimization problems by leveraging this
continuous formulation. To this end, our key technical contribution is to analyze the sensitivity of the
min-norm point as a function of the parametrization of the function f . For the specific case above we
will show how to compute ?y? /?w and ?y? /?v.
Continuing with the segmentation example, we might want to train a conditional model P (A | x) that
can model the uncertainty in the predictions to be used in downstream decision making. A rich class
of models are log-supermodular models, i.e., those of the form P (A | x) = exp(?F?(x) (A))/Z(?)
for some parametric submodular function F? . While they can capture very useful interactions, they
are very hard to train in the maximum likelihood setting due to the presence of the intractable
normalizer Z(?). However, Djolonga and Krause [8] have shown that for any such distribution we
can find the closest fully factorized distribution Q(? | x) minimizing a specific information theoretic
divergence D? . In other words, we can exactly compute Q(? | x) = arg minQ?Q D? (P (? | x) k Q),
where Q is the family of fully factorized distributions. Most importantly, the optimal Q can also be
computed from the min-norm point. Thus, a reasonable objective would be to learn a model ?(x) so
that the best approximate distribution Q(? | x) gives high likelihood to the training data point. This is
a complicated bi-level optimization problem (as Q implicitly depends on ?) with an inner variational
inference procedure, which we can again train end-to-end using our results. In other words, we can
optimize the following algorithm end-to-end with respect to ?.
?(x)
xi ???? ?i ?? P = exp(?F?i (A))/Z(?i ) ?? Q = arg min D? (P k Q) ?? Q(Ai | xi ). (3)
Q?Q
Related work. Sensitivity analysis of the set of optimal solutions has a long history in optimization
theory [12]. The problem of argmin-differentiation of the specific case resulting from graph cuts (i.e.
eq. (2)) has been considered in the computer vision literature, either by smoothing the objective [13],
or by unrolling iterative methods [14]. The idea to train probabilistic models by evaluating them using
the marginals produced by an approximate inference algorithm has been studied by Domke [15] for
tree-reweighted belief propagation and mean field, and for continuous models by Tappen [16]. These
methods either use the implicit function theorem, or unroll iterative optimization algorithms. The
benefits of using an inconsistent estimator, which is what we do by optimizing eq. (3), at the benefit
of using computationally tractable inference methods has been discussed by Wainwright [17]. Amos
and Kolter [18] discuss how to efficiently argmin-differentiate quadratic programs by perturbing
the KKT conditions, an idea that goes back to Boot [19]. We make an explicit connection to their
work in Theorem 4. In Section 4 we harness the connection between the min-norm problem and
isotonic regression, which has been exploited to obtain better duality certificates [2], and by Kumar
and Bach [20] to design an active-set algorithm for the min-norm problem. Chakravarti [21] analyzes
the sensitivity of the optimal isotonic regression point with respect to perturbations of the input, but
does not discuss the directional derivatives of the problem. Recently, Dolhansky and Bilmes [22]
have used deep networks to parametrize submodular functions. Discrete optimization is also used
in structured prediction [23, 24] for the computation of the loss function, which is closely related
to our work if we use discrete optimization only at the last layer. However, in this case we have the
advantage in that we allow for arbitrary loss functions to be applied to the solution of the relaxation.
Contributions. We develop a very efficient approximate method (?4) for the computation of the
Jacobian of the min-norm problem inspired by our analysis of isotonic regression in ?3, where
we derive results that might be of independent interest. Even more importantly, from a practical
perspective, this Jacobian has a very nice structure and we can multiply with it in linear time. This
2
means that we can efficiently perform back-propagation if we use these layers in a modern deep
architectures. In ?5 we show how to compute directional derivatives exactly in polynomial time, and
give conditions under which our approximation is correct. This is also an interesting theoretical result
as it quantifies the stability of the min-norm point with respect to the model parameters. Lastly, we
use our results to learn log-supermodular models in ?6.
2
Background on Submodular Minimization
Let us introduce the necessary background on submodular functions. They are defined over subsets of
some ground set, which in the remaining of this paper we will w.l.o.g. assume to be V = {1, 2, . . . , n}.
Then, a function F : 2V ? R is said to be submodular iff for all A, B ? V it holds that
F (A ? B) + F (A ? B) ? F (A) + F (B).
(4)
We will furthermore w.l.o.g. assume that F is normalized so that F (?) = 0. A very simple family of
submodular functions are modular functions. These, seen P
as discrete analogues of linear functions,
satisfy the above with equality and are given as F (A) = i?A mi for some real numbers mi . As
common practice in combinatorial optimization,
we will treat any vector m ? Rn as a modular
P
V
function 2 ? R defined as m(A) = i?A mi . In addition to the graph cuts from the introduction
(eq. (1)), another widely used class of functions are concave-of-cardinality functions, i.e. those of the
form F (|A|) = h(|A|) for some concave h : R ? R [5]. From eq. (4) we see that if we want to define
a submodular function over a collection D ( 2V it has to be closed under union and intersection.
Such collections are known as lattices, and two examples that we will use are the simple lattice 2V
and the trivial lattice {?, V }. In the theory of submodular minimization, a critical object defined by a
pair consisting of a submodular function F and a lattice D ? {?, V } is the base polytope
B(F | D) = {x ? Rn | x(A) ? F (A) for all A ? D} ? {x ? Rn | x(V ) = F (V )}.
(5)
V
We will also use the shorthand B(F ) = B(F | 2 ). Using the result of Edmonds [25], we know
how to maximize a linear function over B(F ) in O(n log n) time with n function evaluations of F .
Specifically, to compute maxy?B(F ) zT y, we first choose a permutation ? : V ? V that sorts z, i.e.
so that z?(1) ? z?(2) ? ? ? ? ? z?(n) . Then, a maximizer f (?) ? B(F ) can be computed as
[f (?)]?(i) = F ({?(i)} | {?(1), . . . , ?(i ? 1)}),
(6)
where the marginal gain of A given B is defined as F (A | B) = F (A ? B) ? F (B). Hence, we
know how to compute the support function f (z) = supy?B(F ) zT y, which is known as the Lov?sz
extension [3]. First, this function is indeed an extension as f (1A ) = F (A) for all A ? V , where
1A ? {0, 1}n is the indicator vector for the set A. Second, it is convex as it is a supremum of linear
functions. Finally, and most importantly, it lets us minimize submodular functions in polynomial
time with convex optimization because minA?2V F (A) = minz?[0,1] f (z) and we can efficiently
round the optimal continuous point to a discrete optimizer. Another problem, with a smooth objective,
which is also explicitly tied to the problem of minimizing F is that of computing the min-norm point,
which can be defined in two different ways as
1
1
y? = arg min kyk2 , or equivalently as ? y? = arg min f (y) + kyk2 ,
(7)
2
2
y
y?B(F )
where the equivalence comes from strong Fenchel duality [2]. The connection with submodular
minimization comes from the following lemma, which we have already hinted at in the introduction.
Lemma 1 ([1, Lem. 7.4]). Define A? = {i | yi? < 0} and A0 = {i | yi? ? 0}. Then A? (A0 ) is the
unique smallest (largest) minimizer of F .
Moreover, if instead of hard-thresholding we send the min-norm point through a sigmoid, the result
has the following variational inference interpretation, which lets us optimize the pipeline in eq. (3).
Lemma 2 ([8, Thm. 3]). Define the infinite
R?nyi divergence
between any distributions P and Q over
2V as D? (P || Q) = supA?V log P (A)/Q(A) . For P (A) ? exp(?F (A)), the distribution Q?
minimizing D? over all fully factorized distributions Q is given as
Y
Y
Q(A) =
?(?yi? )
?(yi? ),
i?A
i?A
/
where ?(u) = 1/(1 + exp(?u)) is the sigmoid function.
3
3
Argmin-Differentiation of Isotonic Regression
We will first analyze a simpler problem, i.e., that of isotonic regression, defined as
1
y(x) = arg min ky ? xk2 ,
2
y?O
(8)
where O = {y ? Rn | yi ? yi+1 for i = 1, 2, . . . , n ? 1}. The connection to our problem will be
made clear in Section 4, and it essentially follows from the fact that the Lov?sz extension is linear
on O. In this section, we will be interested in computing the Jacobian ?y/?x, i.e., in understanding
how the solution y changes with respect to the input x. The function is well-defined because of
the strict convexity of the objective and the non-empty convex feasible set. Moreover, it can be
easily computed in O(n) time using the pool adjacent violators algorithm (PAVA) [26]. This is
a well-studied problem in statistics, see e.g. [27]. To understand the behaviour of y(x), we will
start by stating the optimality conditions
of problem (8). To simplify the notation, for any A ? V
P
1
x
we will define Meanx (A) = |A|
i?A i . The optimality conditions can be stated via ordered
partitions ? = (B1 , B2 , . . . , Bm ) of V , meaning that the sets Bi are disjoint, ?kj=1 Bj = V , and
? is ordered so that 1 + maxi?Bj i = mini?Bj+1 i. Specifically, for any such partition we define
y? = (y1 , y2 , . . . , ym ), where yj = Meanx (Bj )1|Bj | and 1k = {1}k is the vector of all ones. In
other words, y? is defined by taking block-wise averages of x with respect to ?. By analyzing the
KKT conditions of problem (8), we obtain the following well-known condition.
Lemma 3 ([26]). An ordered partition ? = (B1 , B2 , . . . , Bm ) is optimal iff the following hold
1. (Primal feasibility) For any two blocks Bj and Bj+1 we have
Meanx (Bj ) ? Meanx (Bj+1 ).
(9)
2. (Dual feasibility) For every block B ? ? and each i ? B define PreB (i) = {j ? B | j ? i}.
Then, the condition reads
Meanx (PreB (i)) ? Meanx (B) ? 0.
(10)
Points where eq. (9) is satisfied with equality are of special interest, because of the following result.
Lemma 4. If for some Bj and Bj+1 the first condition is satisfied with equality, we can merge the
two sets so that the new coarser partition ?0 will also be optimal.
Thus, in the remaining of this section we will assume that the sets Bj are chosen maximally. We will
now introduce a notion that will be crucial in the subsequent analysis.
Definition 1. For any block B, we say that i ? B is a breakpoint if Meanx (PreB (i)) = Meanx (B)
and it is not the right end-point of B (i.e., i < maxi0 ?B i0 ).
From an optimization perspective, any breakpoint is equivalent to non-strict complementariness of the
corresponding Lagrange multiplier. From a combinatorial perspective, they correspond to positions
where we can refine ? into a finer partition ?0 that gives rise to the same point, i.e., y? = y?0 (if we
merge blocks using Lemma 4, the point where we merge them will become a breakpoint). We can now
discuss the differentiability of y(x). Because projecting onto convex sets is a proximal operator and
thus non-expansive, we have the following as an immediate consequence of Rademacher?s theorem.
Lemma 5. The function y(x) is 1-Lipschitz continuous and differentiable almost everywhere.
We will denote by ?x?k and ?x+k the left and right partial derivatives with respect to xk . For any index
k we will denote by u(k) (l(k)) the breakpoint with the smallest (largest) coordinate larger (smaller)
than k. Define it as +? (??) if no such point exists. Moreover, denote by ?(z) the collection of
indices where z takes on distinct values, i.e., ?(z) = ?ni=1 {{i0 ? V | zi = zi0 }}.
Theorem 1. Let k be any coordinate and let B ? ?(y(x)) be the block containing coordinate i.
Also define B+ = {i ? B | i ? u(k)} and B? = {i ? B | i ? l(k)}. Hence, for any i ? B
?x+k (yi ) = Ji ? B \ B? K/|B \ B? |,
and
4
?x?k (yi ) = Ji ? B \ B+ K/|B \ B+ |.
Note that all of these derivatives will agree iff there are no breakpoints, which means that the existence
of breakpoints is an isolated phenomenon due to Lemma 5. In this case the Jacobian exists and has a
very simple block-diagonal form. Namely, it is equal to
?y
= ?(y(x)) ? blkdiag(C|B1 | , C|B2 | , . . . , C|Bm | ),
(11)
?x
where Ck = 1k?k /k is the averaging matrix with elements 1/k. We will use ?(z) for the matrix
taking block-wise averages with respect to the blocks ?(z). As promised in the introduction, Jacobian
multiplication ?(y(x))u is linear as we only have to perform block-wise averages.
4
Min-Norm Differentiation
In this section we will assume that we have a function F? parametrized by some ? ? Rd that we seek
to learn. For example, we could have a mixture model
F? (A) =
d
X
?j Gj (A),
(12)
j=1
for some fixed submodular functions Gj : 2V ? R. In this case, to ensure that the resulting function
is submodular we also want to enforce ?j ? 0 unless Gj is modular. We would like to note that the
discussion in this section goes beyond such models. Remember that the min-norm point is defined as
1
y? = ? arg min f? (y) + kyk2 ,
(13)
2
y
where f? is the Lov?sz extension of F? . Hence, we want to compute ?y/??. To make the connection
with isotonic regression, remember how we evaluate the Lov?sz extension at y. First, we pick a
permutation ? that sorts y, and then evaluate it as f? (y) = f? (?)T y, where f? (?) is defined in
eq. (6). Hence, the Lov?sz extension is linear on the set of all vectors that are sorted by ?. Formally,
for any permutation ? the Lov?sz extension is equal to f? (?)T y on the order cone
O(?) = {y | y?(n) ? y?(n?1) ? . . . ? y?(1) }.
Given a permutation ?, if we constrain eq. (13) to O(?) we can replace f? (y) by the linear function
f? (?)T , so that the problem reduces to
1
y? (?) = ? arg min ky + f? (?)k2 ,
(14)
y?O(?) 2
which is an instance of isotonic regression if we relabel the elements of V using ?. Roughly, the idea
is to instead differentiate eq. (14) with f? (?) computed at the optimal point y? . However, because
we can choose an arbitrary order among the elements with equal values, there may be multiple
permutations that sort y? , and this extra choice we have seems very problematic. Nevertheless, let us
continue with this strategy and analyze the resulting approximations to the Jacobian. We propose the
following approximation to the Jacobian
?y?
?f? (?)
? Jb? ? ?(y? ) ?
= ?(y? ) ? [??1 f? (?) | ??2 f? (?) |
| {z }
??
??
???
| ??d f? (?)] ,
?y (?)
? ?f ?(?)
?
where ?(y? ) is used as an approximation of a Jacobian which might not exist. Fortunately, due to the
special structure of the linearizations, we have the following result that the gradient obtained using
the above strategy does not depend on the specific permutation ? that was chosen.
Theorem 2. If ??k F (A) exists for all A ? V the approximate Jacobians Jb? are equal and do not
depend on the choice of ?. Specifically, the j-th block of any element i ? B ? ?(y? ) is equal to
1
?? F? (B | {i0 | [y? ]i0 < [y? ]i }).
(15)
|B| j
Proof sketch, details in supplement. Remember that ?(y? ) averages f? (?) within each B ? ?(y? ).
Moreover, as ? sorts y? , the elements in B must be placed consecutively. The coordinates of f? (?)
are marginal gains (6) and they will telescope inside the mean, which yields the claimed quantity.
5
Graph cuts. As a special, but important case, let us analyze how the approximate Jacobian looks
like for a cut function (eq. (1)), in which case eq. (13) reduces to eq. (2). Let ?(y(w, v)) =
(B1 , B2 , . . . , Bm ). For any element i ? V we will denote by ?(i) ? {1, 2, . . . , m} the index of the
block where it belongs to. Then, the approximate Jacobian Jb at ? = (w, v) has entries
?bvj (yi ) = J?(i) = ?(j)K/|B?(i) |,
?
1
?
?sign(yi ? yj ) |B?(k) |
1
?bwi,j (yk ) = sign(yj ? yi ) |B?(k)
|
?
?
0
and
if ?(k) = ?(i), or
if ?(k) = ?(j), and
otherwise,
where the sign function is defined to be zero if the argument is zero. Intuitively, increasing the
modular term vi by ? will increase all the coordinates B in y that are in the same segment by ?/|B|.
On the other hand, increasing the weight of an edge wi,j will have no effect if i and j are already on y
in the same segment, and otherwise it will pull the segments containing i and j together by increasing
the smaller one and decreasing the larger one. In the supplementary we provide a pytorch module
that executes the back propagation pass in O(|V | + |E|) time in about 10 lines of code, and we also
derive the approximate Jacobians for concave-of-cardinality and facility location functions.
5
Analysis
We will now theoretically analyze the conditions under which our approximation is correct, and then
give a characterization of the exact directional derivative together with a polynomial algorithm that
computes it. The first notion that will have implications for our analysis is that of (in)separability.
Definition 2. The function F : 2V ? R is said to be separable if there exists some B ? V such that
B?
/ {?, V } and F (V ) = F (B) + F (V \ B).
The term separable is indeed appropriate as it implies that F (A) = F (A ? B) + F ((V \ B) ? A) for
all A ? V [2, Prop. 4.3], i.e., the function splits as a sum of two functions on disjoint domains. Hence,
we can split the problem into two (on B and V \ B) and analyze them independently. We would
like to point out that separability is checkable in cubic time using the algorithm of Queyranne [28].
To simplify the notation, we will assume that we want to compute the derivative at point ? 0 ? Rd
which results in the min-norm point y0 = y? ? Rn . We will furthermore assume that y0 takes on
unique values ?1 < ?2 < ? ? ? < ?k on sets B1 , B2 , . . . , Bk respectively, and we will define the chain
? = A0 ? A1 ? A2 ? ? ? ? ? Ak = V by Aj = ?jj 0 =1 Bj 0 . A central role in the analysis will be
played by the set of constraints in B(F? ) (see (5)) that are active at y? , which makes sense given that
we expect small perturbations in ? 0 to result in small changes in y?0 as well.
Definition 3. For any submodular function F : 2V ? R and any point z ? B(F ) we shall denote by
DF (z) the lattice of tight sets of z on B(F ), i.e.
DF (z) = {A ? V | z(A) = F (A)}.
The fact that the above set is indeed a lattice is well-known [1]. Moreover, note that DF (z) ? {?, V }.
We will also define D0 = DF?0 (y0 ), i.e., the lattice of tight sets at the min-norm point.
5.1
When will the approximate approach work?
We will analyze sufficient conditions so that irrespective of the choice of ?, the isotonic regression
problem eq. (14) has no breakpoints, and the left and right derivatives agree. To this end, for any
j ? {1, 2, . . . , k} we define the submodular function Fj : 2Bj as Fj (H) = F?0 (H | Aj?1 ), where
we have dropped the dependence on ? 0 as it will remain fixed throughout this section.
Theorem 3. The approximate problem (14) is argmin-continuously differentiable irrespective of the
chosen permutation ? sorting y? if and only if any of the following equivalent conditions hold.
(a) arg minH?Bj Fj (H) ? Fj (Bj )|H|/|Bj | = {?, Bj }.
0
0
(b) yB
? relint(B(Fj )), i.e. DFj (yB
) = {?, Bj }, which is only possible if Fj is inseparable.
j
j
6
In other words, we can equivalently say that the optimum has to lie on the interior of the face.
Moreover, if ? ? y? is continuous1 , this result implies that the min-norm point is locally defined as
averaging within the same blocks using (15), so that the approximate Jacobian is exact.
We would like to point out that one can obtain the same derivatives as the ones suggested in ?4, if we
perturb the KKT conditions, as done by Amos and Kolter [18]. If we use that approach, in addition to
the computational challenges, there is the problem of non-uniqueness of the Lagrange multiplier, and
moreover, some valid multipliers might be zero for some of the active constraints. This would render
the resulting linear system rank deficient, and it is not clear how to proceed. Remember that when we
analyzed the isotonic regression problem in ?3 we had non-differentiability due to the exactly same
reason ? zero multipliers for active constraints, which in that case correspond to the breakpoints.
Theorem 4. For a specific Lagrange multiplier there exists a solution to the perturbed KKT conditions
derived by [18] that gives rise to the approximate Jacobians from Section 4. Moreover, this multiplier
is unique if the conditions of Theorem 3 are satisfied.
5.2
Exact computation
Unfortunately, computing the gradients exactly seems very complicated for arbitrary parametrizations
F? , and we will focus our attention to mixture models of the form given in eq. (12). The directions v
where we will compute the directional derivatives will have in general non-negative components vj ,
unless Fj is modular. By leveraging the theory of Shapiro [29], and exploiting the structure of both
the min-norm point and the polyhedron B(Fv | D0 ) we obtain at the following result.
Theorem 5. Assume that F?0 is inseparable and let v be any direction so that Fv is submodular. The
directional derivative ?y/??j at ? 0 in direction v is given by the solution of the following problem.
1
kdk2 ,
2
d
subject to d ? B(Fv | D0 ), and
d(Bj ) = Fv (Aj ) for j ? {1, 2, . . . , k}.
minimize
(16)
First, note that this is again a min-norm problem, but now defined over a reduced lattice D0 with k
additional equality constraints. Fortunately, due to these additional equalities, we can split the above
problem into k separate min-norm problems. Namely, for each j ? {1, 2, . . . , k} collect the lattice
of tight sets that intersect Bj as Dj0 = {H ? Bj | H ? D0 }, and define the function Gj : 2Bj ? R
as Gj (A) = Fv (A | Aj?1 ), where note that the parameter vector ? is taken as the direction v in
which we want to compute the derivative. Then, the block of the optimal solution of problem (16)
corresponding to Bj is equal to
d?Bj =
1
kyj k2 ,
yj ?B(Gj |Dj0 ) 2
arg min
(17)
which is again a min-norm point problem where the base polytope is defined using the lattice Dj0 . We
can then immediately draw a connection with the results from the previous subsection.
Corollary 1. If all latices are trivial, the solution of (17) agrees with the approximate Jacobian (15).
How to solve problem (16)? Fortunately, the divide-and-conquer algorithm of Groenevelt [30]
can be used to find the min-norm point over arbitrary lattices. To do this, we have to compute for
each i ? Bj the unique smallest set Hi? in arg minHj 3i Fj (Hj ) ? y 0 (Hj ), which can be done using
submodular minimization after applying the reduction of Schrijver [31].
To highlight the difference with the approximation from section 4, let us consider a very simple case.
Lemma 6. Assume that Gj is equal to Gj (A) = Ji ? AK for some i ? Bj . Then, the directional
derivative is equal to 1|D| /|D| where D = {i0 | i ? Hi?0 }.
Hence, while the approximate directional derivative would average over all elements in Bj , the
true one averages only over a subset D ? Bj and is possibly sparser. Lemma 6 gives us the exact
directional derivatives for the graph cuts, as each component Gj will be either a cut function on
1
For example if the correspondence ? B(F? ) is hemicontinuous due to Berge?s theorem.
7
two vertices, or a function of the form in Lemma 6. In the first case the directional derivative is
zero because 0 ? B(Gj ) ? B(Gj | Dj0 ). In the second case, we can can either solve exactly
using Lemma 6 or use a more sophisticated approximation, generalizing the result from [32] ?
given that Fj is separable over 2Bj iff the graph is disconnected, we can first separate the graph into
connected components, and then take averages within each connected component instead of over Bj .
5.3
Structured attention and constraints
Recently, there has been an interest in the design of structured attention mechanisms, which, as we
will now show, can be derived and furthermore generalized using the results in this paper. The first
mechanism is the sparsemax of Martins and Astudillo [33]. It takes as input a vector and projects
it to the probability simplex, which is the base polytope corresponding to G(A) = min{|A|, 1}.
Concurrently with this work, Niculae and Blondel [32] have analyzed the following problem
1
y? = min f (y) + ky ? zk2 ,
(18)
2
y?B(G)
for the special case when B(G) is the simplex and f is the Lov?sz extension of one of two specific
submodular functions. We will consider the general case when G can be any concave-of-cardinality
function and F is an arbitrary submodular function. Note that, if either f (y) or the constraint were
not present in problem (18), we could have simply leveraged the theory we have developed so far
to differentiate it. Fortunately, as done by Niculae and Blondel [32], we can utilize the result of Yu
[34] to significantly simplify (18). Namely, because projection onto B(G) preserves the order of the
coordinates [35, Lemma 1], we can write the optimal solution y? to (18) as
1
1
y? = min ky ? y0 k, where y0 = arg min f (y) + ky ? zk2 .
2
x?B(G) 2
y
We can hence split problem (18) into two subtasks ? first, compute y0 and then project it onto
B(G). As each operation can reduces to a minimum-norm problem, we can differentiate each of
them separately, and thus solve (18) by stacking two submodular layers one after the other.
6
Experiments
We consider the image segmentation tasks
CNN
CNN+GC
from the introduction, where we are given
n?3
Mean
Std.
Dev.
Mean
Std. Dev.
an RGB image x ? R
and are supposed
n
to predict those pixels y ? {0, 1} con- Accuracy
0.8103 0.1391
0.9121 0.1034
taining the foreground object. We used the
0.3919 0.1911
0.2681 0.2696
NLL
Weizmann horse segmentation dataset [36], # Fg. Objs. 96.9
65.8
25.3
30.6
which we split into training, validation and
test splits of sizes 180, 50 and 98 respec- Figure 1: Test set results. We see that incorporating a
tively. The implementation was done in graph cut solver improves both the accuracy and negpytorch2 , and to compute the min-norm ative log-likelihood (NLL), while having consistent
point we used the algorithm from [37]. To segmentations with fewer foreground objects.
make the problem more challenging, at training time we randomly selected and revealed only 0.1%
of the training set labels. We first trained a convolutional neural network with two hidden layers that
directly predicts the per-pixel labels, which we refer to as CNN. Then, we added a second model,
which we call CNN+GC, that has the same architecture as the first one, but with an additional graph
cut layer, whose weights are parametrized by a convolutional neural network with one hidden layer.
Details about the architectures can be found in the supplementary. We train the models by maximizing
the log-likelihood of the revealed pixels, which corresponds to the variational bi-level strategy (eq. (3))
due to Lemma 2. We trained using SGD, Adagrad [38] and Adam [39], and chose the model with
the best validation score. As evident from the results presented in Section 6, adding the discrete
layer improves not only the accuracy (after thresholding the marginals at 0.5) and log-likelihood,
but it gives more coherent results as it makes predictions with fewer connected components (i.e.,
foreground objects). Moreover, if we have a look at the predictions themselves in Figure 2, we
can observe that the optimization layer not only removes spurious predictions, but there is is also a
qualitative difference in the marginals as they are spatially more consistent.
2
The code will be made available at https://www.github.com/josipd/nips-17-experiments.
8
Figure 2: Comparison of results from both models on four instances from the test set (up: CNN,
down: CNN+GC). We can see that adding the graph-cut layers helps not only quantitatively, but also
qualitatively, as the predictions are more spatially regular and vary smoothly inside the segments.
7
Conclusion
We have analyzed the sensitivity of the min-norm point for parametric submodular functions and
provided both a very easy-to-implement practical approximate algorithm for general objectives, and
strong theoretical result characterizing the true directional derivatives for mixtures. These results
allow the use of submodular minimization inside modern deep architectures, and they are also
immediately applicable to bi-level variational learning of log-supermodular models of arbitrarily
high order. Moreover, we believe that the theoretical results open the new problem of developing
algorithms that can compute not only the min-norm point, but also solve for the associated derivatives.
Acknowledgements. The research was partially supported by ERC StG 307036 and a Google
European PhD Fellowship.
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6,305 | 6,703 | Inductive Representation Learning on Large Graphs
William L. Hamilton?
[email protected]
Rex Ying?
[email protected]
Jure Leskovec
[email protected]
Department of Computer Science
Stanford University
Stanford, CA, 94305
Abstract
Low-dimensional embeddings of nodes in large graphs have proved extremely
useful in a variety of prediction tasks, from content recommendation to identifying
protein functions. However, most existing approaches require that all nodes in the
graph are present during training of the embeddings; these previous approaches are
inherently transductive and do not naturally generalize to unseen nodes. Here we
present GraphSAGE, a general inductive framework that leverages node feature
information (e.g., text attributes) to efficiently generate node embeddings for
previously unseen data. Instead of training individual embeddings for each node,
we learn a function that generates embeddings by sampling and aggregating features
from a node?s local neighborhood. Our algorithm outperforms strong baselines
on three inductive node-classification benchmarks: we classify the category of
unseen nodes in evolving information graphs based on citation and Reddit post
data, and we show that our algorithm generalizes to completely unseen graphs
using a multi-graph dataset of protein-protein interactions.
1
Introduction
Low-dimensional vector embeddings of nodes in large graphs1 have proved extremely useful as
feature inputs for a wide variety of prediction and graph analysis tasks [5, 11, 28, 35, 36]. The basic
idea behind node embedding approaches is to use dimensionality reduction techniques to distill the
high-dimensional information about a node?s neighborhood into a dense vector embedding. These
node embeddings can then be fed to downstream machine learning systems and aid in tasks such as
node classification, clustering, and link prediction [11, 28, 35].
However, previous works have focused on embedding nodes from a single fixed graph, and many
real-world applications require embeddings to be quickly generated for unseen nodes, or entirely new
(sub)graphs. This inductive capability is essential for high-throughput, production machine learning
systems, which operate on evolving graphs and constantly encounter unseen nodes (e.g., posts on
Reddit, users and videos on Youtube). An inductive approach to generating node embeddings also
facilitates generalization across graphs with the same form of features: for example, one could train
an embedding generator on protein-protein interaction graphs derived from a model organism, and
then easily produce node embeddings for data collected on new organisms using the trained model.
The inductive node embedding problem is especially difficult, compared to the transductive setting,
because generalizing to unseen nodes requires ?aligning? newly observed subgraphs to the node
embeddings that the algorithm has already optimized on. An inductive framework must learn to
?
The two first authors made equal contributions.
While it is common to refer to these data structures as social or biological networks, we use the term graph
to avoid ambiguity with neural network terminology.
1
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Visual illustration of the GraphSAGE sample and aggregate approach.
recognize structural properties of a node?s neighborhood that reveal both the node?s local role in the
graph, as well as its global position.
Most existing approaches to generating node embeddings are inherently transductive. The majority
of these approaches directly optimize the embeddings for each node using matrix-factorization-based
objectives, and do not naturally generalize to unseen data, since they make predictions on nodes in a
single, fixed graph [5, 11, 23, 28, 35, 36, 37, 39]. These approaches can be modified to operate in an
inductive setting (e.g., [28]), but these modifications tend to be computationally expensive, requiring
additional rounds of gradient descent before new predictions can be made. There are also recent
approaches to learning over graph structures using convolution operators that offer promise as an
embedding methodology [17]. So far, graph convolutional networks (GCNs) have only been applied
in the transductive setting with fixed graphs [17, 18]. In this work we both extend GCNs to the task
of inductive unsupervised learning and propose a framework that generalizes the GCN approach to
use trainable aggregation functions (beyond simple convolutions).
Present work. We propose a general framework, called GraphSAGE (SAmple and aggreGatE), for
inductive node embedding. Unlike embedding approaches that are based on matrix factorization,
we leverage node features (e.g., text attributes, node profile information, node degrees) in order to
learn an embedding function that generalizes to unseen nodes. By incorporating node features in the
learning algorithm, we simultaneously learn the topological structure of each node?s neighborhood
as well as the distribution of node features in the neighborhood. While we focus on feature-rich
graphs (e.g., citation data with text attributes, biological data with functional/molecular markers), our
approach can also make use of structural features that are present in all graphs (e.g., node degrees).
Thus, our algorithm can also be applied to graphs without node features.
Instead of training a distinct embedding vector for each node, we train a set of aggregator functions
that learn to aggregate feature information from a node?s local neighborhood (Figure 1). Each
aggregator function aggregates information from a different number of hops, or search depth, away
from a given node. At test, or inference time, we use our trained system to generate embeddings for
entirely unseen nodes by applying the learned aggregation functions. Following previous work on
generating node embeddings, we design an unsupervised loss function that allows GraphSAGE to be
trained without task-specific supervision. We also show that GraphSAGE can be trained in a fully
supervised manner.
We evaluate our algorithm on three node-classification benchmarks, which test GraphSAGE?s ability
to generate useful embeddings on unseen data. We use two evolving document graphs based on
citation data and Reddit post data (predicting paper and post categories, respectively), and a multigraph generalization experiment based on a dataset of protein-protein interactions (predicting protein
functions). Using these benchmarks, we show that our approach is able to effectively generate
representations for unseen nodes and outperform relevant baselines by a significant margin: across
domains, our supervised approach improves classification F1-scores by an average of 51% compared
to using node features alone and GraphSAGE consistently outperforms a strong, transductive baseline
[28], despite this baseline taking ?100? longer to run on unseen nodes. We also show that the new
aggregator architectures we propose provide significant gains (7.4% on average) compared to an
aggregator inspired by graph convolutional networks [17]. Lastly, we probe the expressive capability
of our approach and show, through theoretical analysis, that GraphSAGE is capable of learning
structural information about a node?s role in a graph, despite the fact that it is inherently based on
features (Section 5).
2
2
Related work
Our algorithm is conceptually related to previous node embedding approaches, general supervised
approaches to learning over graphs, and recent advancements in applying convolutional neural
networks to graph-structured data.2
Factorization-based embedding approaches. There are a number of recent node embedding
approaches that learn low-dimensional embeddings using random walk statistics and matrix
factorization-based learning objectives [5, 11, 28, 35, 36]. These methods also bear close relationships to more classic approaches to spectral clustering [23], multi-dimensional scaling [19],
as well as the PageRank algorithm [25]. Since these embedding algorithms directly train node
embeddings for individual nodes, they are inherently transductive and, at the very least, require
expensive additional training (e.g., via stochastic gradient descent) to make predictions on new nodes.
In addition, for many of these approaches (e.g., [11, 28, 35, 36]) the objective function is invariant
to orthogonal transformations of the embeddings, which means that the embedding space does not
naturally generalize between graphs and can drift during re-training. One notable exception to this
trend is the Planetoid-I algorithm introduced by Yang et al. [40], which is an inductive, embeddingbased approach to semi-supervised learning. However, Planetoid-I does not use any graph structural
information during inference; instead, it uses the graph structure as a form of regularization during
training. Unlike these previous approaches, we leverage feature information in order to train a model
to produce embeddings for unseen nodes.
Supervised learning over graphs. Beyond node embedding approaches, there is a rich literature
on supervised learning over graph-structured data. This includes a wide variety of kernel-based
approaches, where feature vectors for graphs are derived from various graph kernels (see [32] and
references therein). There are also a number of recent neural network approaches to supervised
learning over graph structures [7, 10, 21, 31]. Our approach is conceptually inspired by a number of
these algorithms. However, whereas these previous approaches attempt to classify entire graphs (or
subgraphs), the focus of this work is generating useful representations for individual nodes.
Graph convolutional networks. In recent years, several convolutional neural network architectures
for learning over graphs have been proposed (e.g., [4, 9, 8, 17, 24]). The majority of these methods
do not scale to large graphs or are designed for whole-graph classification (or both) [4, 9, 8, 24].
However, our approach is closely related to the graph convolutional network (GCN), introduced by
Kipf et al. [17, 18]. The original GCN algorithm [17] is designed for semi-supervised learning in a
transductive setting, and the exact algorithm requires that the full graph Laplacian is known during
training. A simple variant of our algorithm can be viewed as an extension of the GCN framework to
the inductive setting, a point which we revisit in Section 3.3.
3
Proposed method: GraphSAGE
The key idea behind our approach is that we learn how to aggregate feature information from a
node?s local neighborhood (e.g., the degrees or text attributes of nearby nodes). We first describe
the GraphSAGE embedding generation (i.e., forward propagation) algorithm, which generates
embeddings for nodes assuming that the GraphSAGE model parameters are already learned (Section
3.1). We then describe how the GraphSAGE model parameters can be learned using standard
stochastic gradient descent and backpropagation techniques (Section 3.2).
3.1
Embedding generation (i.e., forward propagation) algorithm
In this section, we describe the embedding generation, or forward propagation algorithm (Algorithm
1), which assumes that the model has already been trained and that the parameters are fixed. In
particular, we assume that we have learned the parameters of K aggregator functions (denoted
AGGREGATE k , ?k ? {1, ..., K}), which aggregate information from node neighbors, as well as a set
of weight matrices Wk , ?k ? {1, ..., K}, which are used to propagate information between different
layers of the model or ?search depths?. Section 3.2 describes how we train these parameters.
2
In the time between this papers original submission to NIPS 2017 and the submission of the final, accepted
(i.e., ?camera-ready?) version, there have been a number of closely related (e.g., follow-up) works published on
pre-print servers. For temporal clarity, we do not review or compare against these papers in detail.
3
Algorithm 1: GraphSAGE embedding generation (i.e., forward propagation) algorithm
Input : Graph G(V, E); input features {xv , ?v ? V}; depth K; weight matrices
Wk , ?k ? {1, ..., K}; non-linearity ?; differentiable aggregator functions
AGGREGATE k , ?k ? {1, ..., K}; neighborhood function N : v ? 2V
Output : Vector representations zv for all v ? V
1
2
3
4
5
6
7
8
9
h0v ? xv , ?v ? V ;
for k = 1...K do
for v ? V do
hkN (v) ? AGGREGATEk ({hk?1
u , ?u ? N (v)});
k
k
k?1
hv ? ? W ? CONCAT(hv , hkN (v) )
end
hkv ? hkv /khkv k2 , ?v ? V
end
zv ? hK
v , ?v ? V
The intuition behind Algorithm 1 is that at each iteration, or search depth, nodes aggregate information
from their local neighbors, and as this process iterates, nodes incrementally gain more and more
information from further reaches of the graph.
Algorithm 1 describes the embedding generation process in the case where the entire graph, G =
(V, E), and features for all nodes xv , ?v ? V, are provided as input. We describe how to generalize
this to the minibatch setting below. Each step in the outer loop of Algorithm 1 proceeds as follows,
where k denotes the current step in the outer loop (or the depth of the search) and hk denotes a node?s
representation at this step: First, each node v ? V aggregates the representations of the nodes in its
k?1
immediate neighborhood, {hk?1
u , ?u ? N (v)}, into a single vector hN (v) . Note that this aggregation
step depends on the representations generated at the previous iteration of the outer loop (i.e., k ? 1),
and the k = 0 (?base case?) representations are defined as the input node features. After aggregating
the neighboring feature vectors, GraphSAGE then concatenates the node?s current representation,
hk?1
, with the aggregated neighborhood vector, hk?1
v
N (v) , and this concatenated vector is fed through a
fully connected layer with nonlinear activation function ?, which transforms the representations to
be used at the next step of the algorithm (i.e., hkv , ?v ? V). For notational convenience, we denote
the final representations output at depth K as zv ? hK
v , ?v ? V. The aggregation of the neighbor
representations can be done by a variety of aggregator architectures (denoted by the AGGREGATE
placeholder in Algorithm 1), and we discuss different architecture choices in Section 3.3 below.
To extend Algorithm 1 to the minibatch setting, given a set of input nodes, we first forward sample
the required neighborhood sets (up to depth K) and then we run the inner loop (line 3 in Algorithm
1), but instead of iterating over all nodes, we compute only the representations that are necessary to
satisfy the recursion at each depth (Appendix A contains complete minibatch pseudocode).
Relation to the Weisfeiler-Lehman Isomorphism Test. The GraphSAGE algorithm is conceptually
inspired by a classic algorithm for testing graph isomorphism. If, in Algorithm 1, we (i) set K = |V|,
(ii) set the weight matrices as the identity, and (iii) use an appropriate hash function as an aggregator
(with no non-linearity), then Algorithm 1 is an instance of the Weisfeiler-Lehman (WL) isomorphism
test, also known as ?naive vertex refinement? [32]. If the set of representations {zv , ?v ? V} output
by Algorithm 1 for two subgraphs are identical then the WL test declares the two subgraphs to be
isomorphic. This test is known to fail in some cases, but is valid for a broad class of graphs [32].
GraphSAGE is a continuous approximation to the WL test, where we replace the hash function
with trainable neural network aggregators. Of course, we use GraphSAGE to generate useful node
representations?not to test graph isomorphism. Nevertheless, the connection between GraphSAGE
and the classic WL test provides theoretical context for our algorithm design to learn the topological
structure of node neighborhoods.
Neighborhood definition. In this work, we uniformly sample a fixed-size set of neighbors, instead of
using full neighborhood sets in Algorithm 1, in order to keep the computational footprint of each batch
4
fixed.3 That is, using overloaded notation, we define N (v) as a fixed-size, uniform draw from the set
{u ? V : (u, v) ? E}, and we draw different uniform samples at each iteration, k, in Algorithm 1.
Without this sampling the memory and expected runtime of a single batch is unpredictable and in
the worst case O(|V|). In contrast, the per-batch space and time complexity for GraphSAGE is fixed
QK
at O( i=1 Si ), where Si , i ? {1, ..., K} and K are user-specified constants. Practically speaking
we found that our approach could achieve high performance with K = 2 and S1 ? S2 ? 500 (see
Section 4.4 for details).
3.2
Learning the parameters of GraphSAGE
In order to learn useful, predictive representations in a fully unsupervised setting, we apply a
graph-based loss function to the output representations, zu , ?u ? V, and tune the weight matrices,
Wk , ?k ? {1, ..., K}, and parameters of the aggregator functions via stochastic gradient descent. The
graph-based loss function encourages nearby nodes to have similar representations, while enforcing
that the representations of disparate nodes are highly distinct:
>
JG (zu ) = ? log ?(z>
(1)
u zv ) ? Q ? Evn ?Pn (v) log ?(?zu zvn ) ,
where v is a node that co-occurs near u on fixed-length random walk, ? is the sigmoid function,
Pn is a negative sampling distribution, and Q defines the number of negative samples. Importantly,
unlike previous embedding approaches, the representations zu that we feed into this loss function
are generated from the features contained within a node?s local neighborhood, rather than training a
unique embedding for each node (via an embedding look-up).
This unsupervised setting emulates situations where node features are provided to downstream
machine learning applications, as a service or in a static repository. In cases where representations
are to be used only on a specific downstream task, the unsupervised loss (Equation 1) can simply be
replaced, or augmented, by a task-specific objective (e.g., cross-entropy loss).
3.3
Aggregator Architectures
Unlike machine learning over N-D lattices (e.g., sentences, images, or 3-D volumes), a node?s
neighbors have no natural ordering; thus, the aggregator functions in Algorithm 1 must operate over
an unordered set of vectors. Ideally, an aggregator function would be symmetric (i.e., invariant to
permutations of its inputs) while still being trainable and maintaining high representational capacity.
The symmetry property of the aggregation function ensures that our neural network model can
be trained and applied to arbitrarily ordered node neighborhood feature sets. We examined three
candidate aggregator functions:
Mean aggregator. Our first candidate aggregator function is the mean operator, where we simply
take the elementwise mean of the vectors in {hk?1
u , ?u ? N (v)}. The mean aggregator is nearly
equivalent to the convolutional propagation rule used in the transductive GCN framework [17]. In
particular, we can derive an inductive variant of the GCN approach by replacing lines 4 and 5 in
Algorithm 1 with the following:4
hkv ? ?(W ? MEAN({hk?1
} ? {hk?1
v
u , ?u ? N (v)}).
(2)
We call this modified mean-based aggregator convolutional since it is a rough, linear approximation of
a localized spectral convolution [17]. An important distinction between this convolutional aggregator
and our other proposed aggregators is that it does not perform the concatenation operation in line
5 of Algorithm 1?i.e., the convolutional aggregator does concatenate the node?s previous layer
representation hk?1
with the aggregated neighborhood vector hkN (v) . This concatenation can be
v
viewed as a simple form of a ?skip connection? [13] between the different ?search depths?, or ?layers?
of the GraphSAGE algorithm, and it leads to significant gains in performance (Section 4).
LSTM aggregator. We also examined a more complex aggregator based on an LSTM architecture
[14]. Compared to the mean aggregator, LSTMs have the advantage of larger expressive capability.
However, it is important to note that LSTMs are not inherently symmetric (i.e., they are not permutation invariant), since they process their inputs in a sequential manner. We adapt LSTMs to operate on
an unordered set by simply applying the LSTMs to a random permutation of the node?s neighbors.
3
4
Exploring non-uniform samplers is an important direction for future work.
Note that this differs from Kipf et al?s exact equation by a minor normalization constant [17].
5
Pooling aggregator. The final aggregator we examine is both symmetric and trainable. In this
pooling approach, each neighbor?s vector is independently fed through a fully-connected neural
network; following this transformation, an elementwise max-pooling operation is applied to aggregate
information across the neighbor set:
pool
k
AGGREGATE k = max({? Wpool hui + b , ?ui ? N (v)}),
(3)
where max denotes the element-wise max operator and ? is a nonlinear activation function. In
principle, the function applied before the max pooling can be an arbitrarily deep multi-layer perceptron, but we focus on simple single-layer architectures in this work. This approach is inspired by
recent advancements in applying neural network architectures to learn over general point sets [29].
Intuitively, the multi-layer perceptron can be thought of as a set of functions that compute features for
each of the node representations in the neighbor set. By applying the max-pooling operator to each of
the computed features, the model effectively captures different aspects of the neighborhood set. Note
also that, in principle, any symmetric vector function could be used in place of the max operator
(e.g., an element-wise mean). We found no significant difference between max- and mean-pooling in
developments test and thus focused on max-pooling for the rest of our experiments.
4
Experiments
We test the performance of GraphSAGE on three benchmark tasks: (i) classifying academic papers
into different subjects using the Web of Science citation dataset, (ii) classifying Reddit posts as
belonging to different communities, and (iii) classifying protein functions across various biological
protein-protein interaction (PPI) graphs. Sections 4.1 and 4.2 summarize the datasets, and the
supplementary material contains additional information. In all these experiments, we perform
predictions on nodes that are not seen during training, and, in the case of the PPI dataset, we test on
entirely unseen graphs.
Experimental set-up. To contextualize the empirical results on our inductive benchmarks, we
compare against four baselines: a random classifer, a logistic regression feature-based classifier
(that ignores graph structure), the DeepWalk algorithm [28] as a representative factorization-based
approach, and a concatenation of the raw features and DeepWalk embeddings. We also compare four
variants of GraphSAGE that use the different aggregator functions (Section 3.3). Since, the ?convolutional? variant of GraphSAGE is an extended, inductive version of Kipf et al?s semi-supervised
GCN [17], we term this variant GraphSAGE-GCN. We test unsupervised variants of GraphSAGE
trained according to the loss in Equation (1), as well as supervised variants that are trained directly
on classification cross-entropy loss. For all the GraphSAGE variants we used rectified linear units as
the non-linearity and set K = 2 with neighborhood sample sizes S1 = 25 and S2 = 10 (see Section
4.4 for sensitivity analyses).
For the Reddit and citation datasets, we use ?online? training for DeepWalk as described in Perozzi et
al. [28], where we run a new round of SGD optimization to embed the new test nodes before making
predictions (see the Appendix for details). In the multi-graph setting, we cannot apply DeepWalk,
since the embedding spaces generated by running the DeepWalk algorithm on different disjoint
graphs can be arbitrarily rotated with respect to each other (Appendix D).
All models were implemented in TensorFlow [1] with the Adam optimizer [16] (except DeepWalk,
which performed better with the vanilla gradient descent optimizer). We designed our experiments
with the goals of (i) verifying the improvement of GraphSAGE over the baseline approaches (i.e.,
raw features and DeepWalk) and (ii) providing a rigorous comparison of the different GraphSAGE
aggregator architectures. In order to provide a fair comparison, all models share an identical implementation of their minibatch iterators, loss function and neighborhood sampler (when applicable).
Moreover, in order to guard against unintentional ?hyperparameter hacking? in the comparisons between GraphSAGE aggregators, we sweep over the same set of hyperparameters for all GraphSAGE
variants (choosing the best setting for each variant according to performance on a validation set). The
set of possible hyperparameter values was determined on early validation tests using subsets of the
citation and Reddit data that we then discarded from our analyses. The appendix contains further
implementation details.5
5
Code and links to the datasets: http://snap.stanford.edu/graphsage/
6
Table 1: Prediction results for the three datasets (micro-averaged F1 scores). Results for unsupervised
and fully supervised GraphSAGE are shown. Analogous trends hold for macro-averaged scores.
Citation
Name
Reddit
PPI
Unsup. F1
Sup. F1
Unsup. F1
Sup. F1
Unsup. F1
Sup. F1
Random
Raw features
DeepWalk
DeepWalk + features
GraphSAGE-GCN
GraphSAGE-mean
GraphSAGE-LSTM
GraphSAGE-pool
0.206
0.575
0.565
0.701
0.742
0.778
0.788
0.798
0.206
0.575
0.565
0.701
0.772
0.820
0.832
0.839
0.043
0.585
0.324
0.691
0.908
0.897
0.907
0.892
0.042
0.585
0.324
0.691
0.930
0.950
0.954
0.948
0.396
0.422
?
?
0.465
0.486
0.482
0.502
0.396
0.422
?
?
0.500
0.598
0.612
0.600
% gain over feat.
39%
46%
55%
63%
19%
45%
Figure 2: A: Timing experiments on Reddit data, with training batches of size 512 and inference
on the full test set (79,534 nodes). B: Model performance with respect to the size of the sampled
neighborhood, where the ?neighborhood sample size? refers to the number of neighbors sampled at
each depth for K = 2 with S1 = S2 (on the citation data using GraphSAGE-mean).
4.1
Inductive learning on evolving graphs: Citation and Reddit data
Our first two experiments are on classifying nodes in evolving information graphs, a task that is
especially relevant to high-throughput production systems, which constantly encounter unseen data.
Citation data. Our first task is predicting paper subject categories on a large citation dataset. We
use an undirected citation graph dataset derived from the Thomson Reuters Web of Science Core
Collection, corresponding to all papers in six biology-related fields for the years 2000-2005. The
node labels for this dataset correspond to the six different field labels. In total, this is dataset contains
302,424 nodes with an average degree of 9.15. We train all the algorithms on the 2000-2004 data
and use the 2005 data for testing (with 30% used for validation). For features, we used node degrees
and processed the paper abstracts according Arora et al.?s [2] sentence embedding approach, with
300-dimensional word vectors trained using the GenSim word2vec implementation [30].
Reddit data. In our second task, we predict which community different Reddit posts belong to.
Reddit is a large online discussion forum where users post and comment on content in different topical
communities. We constructed a graph dataset from Reddit posts made in the month of September,
2014. The node label in this case is the community, or ?subreddit?, that a post belongs to. We sampled
50 large communities and built a post-to-post graph, connecting posts if the same user comments
on both. In total this dataset contains 232,965 posts with an average degree of 492. We use the first
20 days for training and the remaining days for testing (with 30% used for validation). For features,
we use off-the-shelf 300-dimensional GloVe CommonCrawl word vectors [27]; for each post, we
concatenated (i) the average embedding of the post title, (ii) the average embedding of all the post?s
comments (iii) the post?s score, and (iv) the number of comments made on the post.
The first four columns of Table 1 summarize the performance of GraphSAGE as well as the baseline
approaches on these two datasets. We find that GraphSAGE outperforms all the baselines by a
significant margin, and the trainable, neural network aggregators provide significant gains compared
7
to the GCN approach. For example, the unsupervised variant GraphSAGE-pool outperforms the
concatenation of the DeepWalk embeddings and the raw features by 13.8% on the citation data
and 29.1% on the Reddit data, while the supervised version provides a gain of 19.7% and 37.2%,
respectively. Interestingly, the LSTM based aggregator shows strong performance, despite the fact
that it is designed for sequential data and not unordered sets. Lastly, we see that the performance of
unsupervised GraphSAGE is reasonably competitive with the fully supervised version, indicating
that our framework can achieve strong performance without task-specific fine-tuning.
4.2
Generalizing across graphs: Protein-protein interactions
We now consider the task of generalizing across graphs, which requires learning about node roles
rather than community structure. We classify protein roles?in terms of their cellular functions from
gene ontology?in various protein-protein interaction (PPI) graphs, with each graph corresponding
to a different human tissue [41]. We use positional gene sets, motif gene sets and immunological
signatures as features and gene ontology sets as labels (121 in total), collected from the Molecular
Signatures Database [34]. The average graph contains 2373 nodes, with an average degree of 28.8.
We train all algorithms on 20 graphs and then average prediction F1 scores on two test graphs (with
two other graphs used for validation).
The final two columns of Table 1 summarize the accuracies of the various approaches on this
data. Again we see that GraphSAGE significantly outperforms the baseline approaches, with the
LSTM- and pooling-based aggregators providing substantial gains over the mean- and GCN-based
aggregators.6
4.3
Runtime and parameter sensitivity
Figure 2.A summarizes the training and test runtimes for the different approaches. The training time
for the methods are comparable (with GraphSAGE-LSTM being the slowest). However, the need to
sample new random walks and run new rounds of SGD to embed unseen nodes makes DeepWalk
100-500? slower at test time.
For the GraphSAGE variants, we found that setting K = 2 provided a consistent boost in accuracy of
around 10-15%, on average, compared to K = 1; however, increasing K beyond 2 gave marginal
returns in performance (0-5%) while increasing the runtime by a prohibitively large factor of 10-100?,
depending on the neighborhood sample size. We also found diminishing returns for sampling
large neighborhoods (Figure 2.B). Thus, despite the higher variance induced by sub-sampling
neighborhoods, GraphSAGE is still able to maintain strong predictive accuracy, while significantly
improving the runtime.
4.4
Summary comparison between the different aggregator architectures
Overall, we found that the LSTM- and pool-based aggregators performed the best, in terms of both
average performance and number of experimental settings where they were the top-performing
method (Table 1). To give more quantitative insight into these trends, we consider each of the
six different experimental settings (i.e., (3 datasets) ? (unsupervised vs. supervised)) as trials and
consider what performance trends are likely to generalize. In particular, we use the non-parametric
Wilcoxon Signed-Rank Test [33] to quantify the differences between the different aggregators across
trials, reporting the T -statistic and p-value where applicable. Note that this method is rank-based and
essentially tests whether we would expect one particular approach to outperform another in a new
experimental setting. Given our small sample size of only 6 different settings, this significance test is
somewhat underpowered; nonetheless, the T -statistic and associated p-values are useful quantitative
measures to assess the aggregators? relative performances.
We see that LSTM-, pool- and mean-based aggregators all provide statistically significant gains over
the GCN-based approach (T = 1.0, p = 0.02 for all three). However, the gains of the LSTM and
pool approaches over the mean-based aggregator are more marginal (T = 1.5, p = 0.03, comparing
6
Note that in very recent follow-up work Chen and Zhu [6] achieve superior performance by optimizing
the GraphSAGE hyperparameters specifically for the PPI task and implementing new training techniques (e.g.,
dropout, layer normalization, and a new sampling scheme). We refer the reader to their work for the current
state-of-the-art numbers on the PPI dataset that are possible using a variant of the GraphSAGE approach.
8
LSTM to mean; T = 4.5, p = 0.10, comparing pool to mean). There is no significant difference
between the LSTM and pool approaches (T = 10.0, p = 0.46). However, GraphSAGE-LSTM is
significantly slower than GraphSAGE-pool (by a factor of ?2?), perhaps giving the pooling-based
aggregator a slight edge overall.
5
Theoretical analysis
In this section, we probe the expressive capabilities of GraphSAGE in order to provide insight into
how GraphSAGE can learn about graph structure, even though it is inherently based on features.
As a case-study, we consider whether GraphSAGE can learn to predict the clustering coefficient of
a node, i.e., the proportion of triangles that are closed within the node?s 1-hop neighborhood [38].
The clustering coefficient is a popular measure of how clustered a node?s local neighborhood is, and
it serves as a building block for many more complicated structural motifs [3]. We can show that
Algorithm 1 is capable of approximating clustering coefficients to an arbitrary degree of precision:
Theorem 1. Let xv ? U, ?v ? V denote the feature inputs for Algorithm 1 on graph G = (V, E),
where U is any compact subset of Rd . Suppose that there exists a fixed positive constant C ? R+
such that kxv ? xv0 k2 > C for all pairs of nodes. Then we have that ? > 0 there exists a parameter
setting ?? for Algorithm 1 such that after K = 4 iterations
|zv ? cv | < , ?v ? V,
where zv ? R are final output values generated by Algorithm 1 and cv are node clustering coefficients.
Theorem 1 states that for any graph there exists a parameter setting for Algorithm 1 such that it can
approximate clustering coefficients in that graph to an arbitrary precision, if the features for every
node are distinct (and if the model is sufficiently high-dimensional). The full proof of Theorem 1 is
in the Appendix. Note that as a corollary of Theorem 1, GraphSAGE can learn about local graph
structure, even when the node feature inputs are sampled from an absolutely continuous random
distribution (see the Appendix for details). The basic idea behind the proof is that if each node has a
unique feature representation, then we can learn to map nodes to indicator vectors and identify node
neighborhoods. The proof of Theorem 1 relies on some properties of the pooling aggregator, which
also provides insight into why GraphSAGE-pool outperforms the GCN and mean-based aggregators.
6
Conclusion
We introduced a novel approach that allows embeddings to be efficiently generated for unseen nodes.
GraphSAGE consistently outperforms state-of-the-art baselines, effectively trades off performance
and runtime by sampling node neighborhoods, and our theoretical analysis provides insight into
how our approach can learn about local graph structures. A number of extensions and potential
improvements are possible, such as extending GraphSAGE to incorporate directed or multi-modal
graphs. A particularly interesting direction for future work is exploring non-uniform neighborhood
sampling functions, and perhaps even learning these functions as part of the GraphSAGE optimization.
Acknowledgments
The authors thank Austin Benson, Aditya Grover, Bryan He, Dan Jurafsky, Alex Ratner, Marinka
Zitnik, and Daniel Selsam for their helpful discussions and comments on early drafts. The authors
would also like to thank Ben Johnson for his many useful questions and comments on our code. This
research has been supported in part by NSF IIS-1149837, DARPA SIMPLEX, Stanford Data Science
Initiative, Huawei, and Chan Zuckerberg Biohub. W.L.H. was also supported by the SAP Stanford
Graduate Fellowship and an NSERC PGS-D grant. The views and conclusions expressed in this
material are those of the authors and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the above funding agencies, corporations, or
the U.S. and Canadian governments.
9
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6,306 | 6,704 | Subset Selection and Summarization
in Sequential Data
Ehsan Elhamifar
Computer and Information Science College
Northeastern University
Boston, MA 02115
[email protected]
M. Clara De Paolis Kaluza
Computer and Information Science College
Northeastern University
Boston, MA 02115
[email protected]
Abstract
Subset selection, which is the task of finding a small subset of representative items
from a large ground set, finds numerous applications in different areas. Sequential
data, including time-series and ordered data, contain important structural relationships among items, imposed by underlying dynamic models of data, that should
play a vital role in the selection of representatives. However, nearly all existing
subset selection techniques ignore underlying dynamics of data and treat items
independently, leading to incompatible sets of representatives. In this paper, we
develop a new framework for sequential subset selection that finds a set of representatives compatible with the dynamic models of data. To do so, we equip items with
transition dynamic models and pose the problem as an integer binary optimization
over assignments of sequential items to representatives, that leads to high encoding,
diversity and transition potentials. Our formulation generalizes the well-known
facility location objective to deal with sequential data, incorporating transition
dynamics among facilities. As the proposed formulation is non-convex, we derive a
max-sum message passing algorithm to solve the problem efficiently. Experiments
on synthetic and real data, including instructional video summarization, show that
our sequential subset selection framework not only achieves better encoding and
diversity than the state of the art, but also successfully incorporates dynamics of
data, leading to compatible representatives.
1
Introduction
Subset selection is the task of finding a small subset of most informative items from a ground set.
Besides helping to reduce the computational time and memory of algorithms, due to working on a
much smaller representative set [1], it has found numerous applications, including, image and video
summarization [2, 3, 4], speech and document summarization [5, 6, 7], clustering [8, 9, 10, 11, 12],
feature and model selection [13, 14, 15, 16], sensor placement [17, 18], social network marketing [19]
and product recommendation [20]. Compared to dictionary learning methods such as Kmeans [21],
KSVD [22] and HMMs [23], that learn centers/atoms in the input-space, subset selection methods
choose centers/atoms from the given set of items.
Sequential data, including time-series such as video, speech, audio and sensor measurements as well
as ordered data such as text, form an important large part of modern datasets, requiring effective subset
selection techniques. Such datasets contain important structural relationships among items, often
imposed by underlying dynamic models, that should play a vital role in the selection of representatives.
For example, there exists a logical way in which segments of a video or sentences of a document are
connected together and treating segments/sentences as a bag of randomly permutable items results
in losing the semantic content of the video/document. However, existing subset selection methods
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: We propose a framework for the summarization of sequential data. Given a source set of items
{x1 , . . . , xM }, with a dynamic transition model, and a target set of sequential items (y 1 , . . . , y T ), we find a
sequence of representatives from the source set with high transition probability that well encode the target set.
ignore these relationships and treat items independent from each other. Thus, there is a need for
sequential subset selection methods that, instead of treating items independently, use the underlying
dynamic models of data to select high-quality, diverse and compatible representatives.
Prior Work: A subset selection framework consists of three main components: i) the inputs to the
algorithm; ii) the objective function to optimize, characterizing the informativeness and diversity of
selected items; iii) the algorithm to optimize the objective function. The inputs to subset selection
algorithms are in the form of either feature vector representations or pairwise similarities between
items. Several subset selection criteria have been studied in the literature, including maximum cut
objective [24, 25], maximum marginal relevance [26], capacitated and uncapacitated facility location
objectives [27, 28], multi-linear coding [29, 30] and maximum volume subset [6, 31], which all try to
characterize the informativeness/value of a subset of items in terms of ability to represent the entire
distribution and/or having minimum information overlap among selected items. On the other hand,
optimizing almost all subset selection criteria is, in general, NP-hard and non-convex [25, 32, 33, 34],
which has motivated the development and study of approximate methods for optimizing these criteria.
This includes greedy approximate algorithms [28] for maximizing submodular functions, such as
graph-cuts and facility location, which have worst-case approximation guarantees, as well as sampling
methods from Determinantal Point Process (DPP) [6, 31], a probability measure on the set of all
subsets of a ground set, for approximately finding the maximum volume subset. Motivated by the
maturity of convex optimization and advances in sparse and low-rank recovery, recent methods have
focused on convex relaxation-based methods for subset selection [8, 9, 2, 35, 36].
When it comes to sequential data, however, the majority of subset selection methods ignore the
underlying dynamics and relationships among items and treat items independent from each other.
Recent results in [37, 3] have developed interesting extensions to DPP-based subset selection, by
capturing representatives in a sequential order such that newly selected representatives are diverse
with respect to the previously selected ones. However, sequential diversity by itself is generally
insufficient, especially, when the sequence of diverse selected items are unlikely to follow each other
according to underlying dynamic models. For example, in a video/document on a specific topic
with intermediate irrelevant scenes/sentences to the topic, promoting sequential diversity results
in selecting irrelevant scenes/sentences. [38] extends submodular functions to capture ordered
preferences among items, where ordered preferences are represented by a directed acyclic graph over
items, and presents a greedy algorithm to pick edges instead of items. The method, however, cannot
deal with arbitrary graphs, such as Markov chains with cycles. On the other hand, while Hidden
Markov Models (HMMs) [23, 39] and dynamical systems [40, 41] have been extensively studied for
modeling sequential data, they have not been properly exploited in the context of subset selection.
Paper Contributions: In this paper we develop a new framework for sequential subset selection that
incorporates the dynamic model of sequential data into subset selection. We develop a new class
of objective functions that promotes to select not only high-quality and diverse items, but also a
sequence of representatives that are compatible with the dynamic model of data. To do so, we propose
a dynamic subset selection framework, where we equip items with transition probabilities and design
objective functions to select representatives that well capture the data distribution with a high overall
transition probability in the sequence of representatives, see Figure 1. Our formulation generalizes
the facility location objective [27, 28] to sequential data, by incorporating transition dynamics among
facilities. Since our proposed integer binary optimization is non-convex, we develop a max-sum
message passing framework to solve the problem efficiently. By experiments on synthetic and real
data, including instructional video summarization, we show that our method outperforms the state of
the art in terms of selecting representatives with better encoding, diversity and dynamic compatibility.
2
2
Subset Selection for Sequential Data
Sequential data, including time-series and ordered data contain important structural relationships
among items, often imposed by underlying dynamic models of data, that should play a vital role in
the selection of representatives. In this section, we develop a new framework for sequential subset
selection that incorporates underlying dynamic models and relationships among items into subset
selection. More specifically, we propose a dynamic subset selection framework, where we equip
items with transition probabilities and design objectives to select representatives that capture the data
distribution with a high transition probability in the sequence of representatives. In the next section,
we develop an efficient algorithm to solve the proposed optimization problem.
2.1
Sequential Subset Selection Formulation
Assume we have a source set of items X = {x1 , . . . , xM }, equipped with a transition model,
p(xi0 |xi1 , . . . , xin ), between items, and a target set of sequential items Y = (y 1 , . . . , y T ). Our
goal is to find a small representative subset of X that well encode Y, while the set of representatives
are compatible according to the dynamic model of X. Let xrt be the representative of y t for
t 2 {1, . . . , T }. We propose a potential function (r1 , . . . , rT ) whose maximization over all
possible assignments (r1 , . . . , rT ) ? {1, . . . , M }T , i.e.,
max
(r1 ,...,rT )?{1,...,M }T
(1)
(r1 , . . . , rT )
achieves the three goals of i) minimizing the encoding cost of Y via the representative set; ii) selecting
a small set of representatives from X; iii) selecting an ordered set of representatives (xr1 , . . . , xrT )
that are compatible with the dynamics on X.
To tackle the problem, we consider a decomposition of the potential function
three potentials, corresponding to the three aforementioned objectives, as
(r1 , . . . , rT ) ,
enc (r1 , . . . , rT )
?
card (r1 , . . . , rT )
?
into the product of
dyn (r1 , . . . , rT ),
(2)
where enc (r1 , . . . , rT ) denotes the encoding potential that favors selecting a representative set from
X that well encodes Y, card (r1 , . . . , rT ) denotes the cardinality potential that favors selecting a
small number of distinct representatives. Finally, dyn (r1 , . . . , rT ) denotes the dynamic potential
that favors selecting an ordered set of representatives that are likely to be generated by the underlying
dynamic model on X. Next, we study each of the three potentials.
Encoding Potential: Since the encoding of each item of Y depends on its own representative, we
assume that the encoding potential function factorizes as
enc (r1 , . . . , rT )
=
T
Y
enc,t (rt ),
(3)
t=1
where enc,t (i) characterizes how well xi encodes y t and becomes larger when xi better represents
y t . In this paper, we assume that enc,t (i) = exp( di,t ), where di,t indicates the dissimilarity of xi
to y t . A lower dissimilarity di,t means that xi better encodes/represents y t .
Cardinality Potential: Notice that maximizing the encoding potential alone results in selecting
many representatives. Hence, we consider a cardinality potential to restrict the total number of
representatives. Denoting the number of representatives by |{r1 , . . . , rT }|, we consider
card (r1 , . . . , rT )
= exp(
|{r1 , . . . , rT }|),
(4)
which promotes to select a small number of representatives. The parameter > 0 controls the effect
of the cardinality on the global potential , where a close to zero ignores the effect of cardinality
potential, resulting in many representatives, and a larger results in a smaller representative set.
Dynamic Potential: While encoding and cardinality potentials together promote selecting a few
representatives from X that well encode Y, there is no guarantee that the sequence of representatives
(xr1 , . . . , xrT ) is compatible with the underlying dynamic of X. Thus, we introduce a dynamic
potential that measures the compatibility of the sequence of representatives. To do so, we consider
3
an n-th order Markov Model to represent the dynamic relationships among the items in X, where
the selection of the representative xrt depends on the m previously selected representatives, i.e.,
xrt 1 , . . . , xrt n . More precisely, we consider
dyn (r1 , . . . , rT )
n
Y
=
t=1
T
Y
pt (xrt ) ?
t=n+1
pt (xrt |xrt 1 , . . . , xrt
n
)
!
(5)
,
where pt (xi ) indicates the probability of selecting xi as the representative of y t and
pt (xi0 |xi1 , . . . , xin ) denotes the probability of selecting xi0 as the representative of y t given that
xi1 , . . . , xin has been selected as the representative of y t 1 , . . . , y t n , respectively. The regularization parameter > 0 determines the effect of the dynamic potential on the overall potential , where
a close to zero results in discounting the effect of the dynamic of X. As a result, maximizing the
dynamic potential promotes to select a sequence of representatives that are highly likely to follow the
dynamic model on the source set.
2.2
Optimization Framework for Sequential Subset Selection
In the rest of the paper, we consider a first order Markov model, which performs well in the application
studied in the paper (our proposed optimization can be generalized to n-th order Markov models as
well). Putting all three potentials together, we consider maximization of the global potential function
=
T
Y
enc,t (rt )
t=1
?
card (r1 , . . . , rT )
?
T
Y
p1 (xr1 ) ?
t=2
pt (xrt |xrt 1 )
!
(6)
.
over all possible assignments (r1 , . . . , rT ) ? {1, . . . , M }T . To do so, we cast the problem as an
integer binary optimization. We define binary assignment variables {zi,t }t=1,...,T
i=1,...,M , where zi,t 2
{0, 1} indicates if xi is a representative of y t . Since each item y t is associated with only a single
PM
representative, we have i=1 zi,t = 1. Also, we define variables { i }i=1,...,M and {uti0 ,i }t=1,...,T
i,i0 =1,...,M ,
t
where i 2 {0, 1} indicates if xi is a representative of y 1 and ui0 ,i 2 {0, 1} indicates if xi0 is a
representative of y t given that xi is a representative of y t 1 . As we will show, { i } and {uti0 ,i } are
related to {zi,t }, hence, the final optimization only depends on {zi,t }.
Using the variables defined above, we can rewrite the global potential function in (6) as
=
T Y
M
Y
enc,t (i)
zi,t
t=1 i=1
?
card (r1 , . . . , rT )
?
We can equivalently maximize the logarithm of
T X
M
X
zi,t di,t + log
card (r1 , . . . , rT ) +
t=1 i=1
M
Y
p1 (xi )
i=1
i
?
T Y
M Y
M
Y
t=2 i0 =1 i=1
pt (xi0 |xi )
uti0 ,i
. (7)
, which is to maximize
M
X
i log p1 (xi ) +
T X
M
X
t=2 i,i0 =1
i=1
ui0 ,i log pt (xi0 |xi ), (8)
where we used log enc,t (i) = di,t . Notice that { i } and {uti0 ,i } can be written as functions of the
assignment variables {zi,t }. Denoting the indicator function by 1(?) , which is one when its argument
is true and is zero otherwise, we can write i = 1(r1 =i) and uti0 ,i = 1(rt =i0 ,rt 1 =i) . Hence, we have
i
= zi,1 , uti0 ,i = zi,t
(9)
1 zi0 ,t .
As a result, we can rewrite the maximization in (8) as the equivalent optimization
max
{zi,t }
T X
M
X
zi,t di,t + log
card (r1 , . . . , rT ) + (
t=1 i=1
+
T X
M
X
t=2 i,i0 =1
M
X
zi,1 log p1 (xi )
i=1
zi,t
1 zi0 ,t
log pt (xi0 |xi )) s. t. zi,t 2 {0, 1},
4
M
X
i=1
(10)
zi,t = 1, 8 i, t.
?1C
?11
?2C
?12
?3C
z11
D
?11;12
z12
D
?12;13
z13
?21
..
?22
..
?23
z21
?31
z31
D
?21;12
z22
..
?32
D
?31;12
z32
..
D
?22;13
z23
..
D
?32,13
z33
?it
?tC
???
?it
?2R
D
?1(t
???
0
it,1(t+1)
..
???
..
..
?iD0(t
1);it
D
?M
(t
1);it
0
0 ( t+
?it
it,1(t+1)
1)
it,i 0
(t+
it,i
..
it
?iR
zit
1);it
?3R
?33
..
..
?1R
?13
0 ,M
it
(t
+
it,
1)
M
(t
+
1)
D
?it;1(t+1)
1)
..
..
D
?it;i
0 (t+1)
D
?it;M
(t+1)
Figure 2: Left: Factor graph representing (12). Right: Messages from each factor to a variable node zi,t .
It is important to note that if xi becomes a representative of some items in Y, then k [zi,1 ? ? ? zi,T ] k1
PM
would be 1. Hence, the number of representatives is given by i=1 k [zi,1 ? ? ? zi,T ] k1 . As a result,
we can rewrite the cardinality potential in (4) as
card (r1 , . . . , rT )
M
X
= exp(
i=1
k [zi,1 ? ? ? zi,T ] k1 ).
(11)
Finally, considering a homogeneous Markov Model on the dynamics of the source set, where
pt (?|?) = p(?|?), i.e., transitioning from xi as the representative of y t 1 to xi0 as the representative of
y t does not depend on t, we propose to solve the optimization
max
{zi,t }
T X
M
X
zi,t di,t
t=1 i=1
+
T X
M
X
t=2 i,i0 =1
M
X
i=1
zi,t
1 zi0 ,t
k [zi,1 ? ? ? zi,T ] k1 + (
M
X
zi,1 log p1 (xi )
i=1
log p(xi0 |xi )) s. t. zi,t 2 {0, 1},
M
X
i=1
(12)
zi,t = 1, 8 i, t.
In our proposed formulation above, we assume that the dissimilarities {di,t } and the dynamic models,
i.e., the probabilities p1 (?) and p(?|?), are known. These models can be given by prior knowledge
or by learning from training data, as we show in the experiments. It is important to notice that the
optimization in (12) is non-convex, due to binary optimization variables and quadratic terms in the
objective function, which is not necessarily positive semi-definite (this can be easily seen when
p(xi0 |xi ) 6= p(xi |xi0 ) for some i, i0 ). In the next section, we treat (12) as a MAP inference on binary
random variables and develop a message passing algorithm to find the hidden values {zi,t }.
Once we solve the optimization in (12), we can obtain the representatives as the items of X for which
zi,t is non-zero for some t. Moreover, we can obtain the segmentation of the sequential items in Y
according to their assignments to the representatives. In fact, the sequence of representatives obtained
by our proposed optimization in (12) not only corresponds to diverse items that well encode the
sequential target data, but also is compatible with the underlying dynamic of the source data.
Remark 1 Without the dynamic potential, ie, with = 0, our proposed optimization in (12) reduces
to the uncapacitated facility location objective. Hence, our framework generalizes the facility location
to sequential data by considering transition dynamics among facilities (source set items).
3
Message Passing for Sequential Subset Selection
In this section, we develop an efficient message passing algorithm to solve the proposed optimization
in (12). To do so, we treat the sequential subset selection as a MAP inference, where {zi,t } correspond
to binary random variables whose joint log-likelihood is given by the objective function in (12). We
represent the log-likelihood, i.e., the objective function in (12), with a factor graph [42], which is
shown in Figure 2. Recall that a factor graph is a bipartite graph that consists of variable nodes and
factor nodes, where every factor evaluates a potential function over variables it is connected to. The
log-likelihood is then proportional to the sum of all factor potentials.
5
To form the factors corresponding to the objective function in (12), we define mi,i0 , log p(xi0 |xi )
and d?i,t , di,t log p1 (xi ) if t = 1 and d?i,t , di,t for all other values of t. Denoting z i,: ,
>
>
[zi,1 ? ? ? zi,T ] and z :,t , [z1,t ? ? ? zM,t ] , we define factor potentials corresponding to our
framework, shown in Figure 2. More specifically, we define the encoding and dynamic potentials,
D
respectively, as ?i,t (zi,t ) , d?i,t zi,t and ?i,t
1;i0 ,t (zi,t 1 , zi0 ,t ) , mi,i0 zi,t 1 zi0 ,t . Moreover we
define the cardinality and constraint potentials, respectively, as
(
?
PM
, kz i,: k1 > 0
0,
R
C
i=1 zi,t = 1 .
?i (z i,: ) ,
, ?t (z :,t ) ,
0,
otherwise
1, otherwise
The MAP formulation of our sequential subset selection is then given by
max
{zi,t }
T X
M
X
?i,t (zi,t ) +
t=1 i=1
M
X
?iR (z i,: ) +
T
X
T
M X
M
X1 X
?lC (z :,t ) +
1;i0 ,t (zi,t 1 , zi0 ,t ).
(13)
t=1 i0 =1 i=1
t=1
i=1
D
?i,t
To perform MAP inference, we use the max-sum message passing algorithm, which iteratively
updates messages between variable and factor nodes in the graph. In our framework, the incoming
messages to each variable node zi,t are illustrated in Figure 2. Messages are computed as follows
(please see the supplementary materials for the derivations).
d?i,t
(14)
i,t
max{0, mi,j + ?}
i,t;j,t+1
0
i,t 1;j,t
d?i0 ,1 +
k6=i ,i
?i,t
(15)
0
(16)
max{0, mi,j + ? }
max
{?i0 ,1
0
?i,t
max{0, ?}
0
min{0,
+
M
X
max{0, ? }
i0 ,t;j,t+1
j=1
X
+
M
X
j=1
max{0, d?i,k + ?i,k +
where, for brevity of notation, we have defined ? and ?0 as
4
M
X
(
d?j,t+1 + ?j,t+1 + ?j,t+1 +
M
X
j,t+1;k,t+2
k=1
4
?0 =
d?i,t
1
+ ?i,t
1
i,k;j,k+1
(17)
+
0
j,k 1;i,k )}}
(18)
j=1
k6=t
?=
0
j,t 1;i0 ,t }
+ ?i,t
1
+
X
k6=i
+
X
0
k,t;j,t+1 ,
(19)
0
k,t 2;i,t 1 .
(20)
k6=i
i,t 1;k,t
+
M
X
k=1
The update of messages continues until convergence, when each variable zi,t is assigned to the value
that maximizes the sum of its incoming messages. It is important to note that the max-sum algorithm
always converges to the optimal MAP assignment on trees, and has shown good performance on
graphs with cycles in many applications, including our work. We also use a dampening factor 2
[0, 1) on message updates as so that a message ? is computed as ?(new)
?(old) + (1
)?(update) .
4
Experiments
In this section, we evaluate the performance of our proposed method as well as the state of the art
for subset selection on synthetic and real sequential data. For real applications, we consider the task
of summarizing instructional videos to learn the key steps of the task described in the videos. In
addition to our proposed message passing (MP) algorithm, we have implemented the optimization
in (12) using an ADMM framework [43], where we have relaxed the integer binary constraints to
zi,t 2 [0, 1]. In practice both MP and ADMM algorithms achieve similar results.
We compare our proposed method, Sequential Facility Location (SeqFL), with several subset selection
algorithms. Since we study the performance of methods as a function of the size of the representative
set, we use the fixed-size variant of DPP, called kDPP [44]. In addition to kDPP, we evaluate the
performance of Markov kDPP (M-kDPP) [37], in which successive representatives are diverse among
themselves and with respect to the previously selected representatives, as well as Sequential kDPP
6
20
10
0
0
5
10
15
20
kDPP
M-kDPP
Seq-kDPP
Greedy
DS3
SeqFL
250
200
150
100
0
5
Number of representatives
10
15
50
kDPP
M-kDPP
Seq-kDPP
Greedy
DS3
SeqFL
40
30
20
10
0
kDPP
M-kDPP
Seq-kDPP
Greedy
DS3
SeqFL
0.8
0.6
0.4
0.2
0
0
20
1
Diversity score
30
300
Encoding + *Dynamic cost
kDPP
M-kDPP
Seq-kDPP
Greedy
DS3
SeqFL
40
Dynamic cost
Encoding cost
50
5
10
15
20
0
5
10
15
20
Number of representatives
Number of representatives
Number of representatives
Figure 3: Encoding cost, dynamic cost, total cost and diversity score of different algorithms as a function of the
15
10
5
0
0
2
4
6
8
10
12
300
15
=0
=0.01
=0.02
=0.06
=0.08
10
5
0
0
2
4
6
8
10
Dynamic cost
=0
=0.01
=0.02
=0.06
=0.08
20
200
150
100
0
12
=0
=0.01
=0.02
=0.06
=0.08
250
2
4
6
8
10
12
1
Diversity score
20
25
Encoding cost
Number representatives
number of selected representatives. The size of the source set is M = 50.
=0
=0.01
=0.02
=0.06
=0.08
0.5
0
0
2
4
6
8
10
12
Figure 4: Number of representatives, encoding cost, dynamic cost and diversity score of our proposed method
(SeqFL) as a function of the parameters ( , ).
(Seq-kDPP) [3], which divides a time-series into multiple windows and successively selects diverse
samples from each window conditioned on the previous window.1 We also compare our method
against DS3 [8] and the standard greedy method [28], which optimize the conventional facility
location, which has no dynamic cost, via convex relaxation and greedy selection, respectively.
To compare the performance of different methods, we evaluate several costs and scores that
demonstrate the effectiveness of each method in terms of encoding, diversity and dynamic compatibility of the set of selected representatives. More specifically, given dissimilarities {di,t },
the dynamic model p1 (?) and p(?|?), representative set ?, and the assignment of points to repPT PM
?
?
resentatives {zi,t
}, we compute the encoding cost as t=1 i=1 di,t zi,t
, the dynamic cost as
PM
P
P
T
M
?
?
?
0
log
p
(x
)z
log
p(x
|x
)z
z
and
the
diversity
score as det(K ? ).
0
1
i i,1
i
i i,t 1 i ,t
i=1
t=2
i,i0 =1
Here, K corresponds to the kernel matrix, used in DPP and its variants, and K ? denotes the submatrix of K indexed by ?. In this paper, we use Euclidean distances as dissimilarities and compute the
corresponding inner-product kernel to run DPPs. Notice that the diversity score, which is the volume
of the parallelotope spanned by the representatives, is what DPP methods aim to (approximately)
maximize. As DPP methods only find representatives and not assignment of points, we compute
?
zi,t
?s by assigning each point to the closest representative in ?, according to the kernel.
4.1
Synthetic Data
To demonstrate the effectiveness of our proposed method for sequential subset selection, we generate
synthetic data where, for a source set X with M items corresponding to means of M Gaussians, we
generate a transition probability matrix among items and an initial probability vector. We draw a
sequence of length T from the corresponding Markov model to form the target set Y and run different
algorithms to generate k representatives. We then compute the average encoding and transition
costs as well as the diversity scores for sequences drawn from the Markov model, as a function of
k 2 {1, 2, . . . , M }. In the experiments we set M = 50, T = 100. For a fixed , we run SeqFL for
different values of to select different number of representatives.
Figure 3 illustrates encoding and transition costs and the diversity scores of different methods, where
for SeqFL we have set = 0.02. Notice that our proposed method consistently obtains lower
encoding, dynamic and total costs for all numbers of representatives, demonstrating its effectiveness
for obtaining a sequence of high-quality and compatible representatives according to the dynamics. It
is important to notice that although our method does not maximize the diversity score, used in kDPP
and its variants, it achieves slightly better diversity scores (higher is better) than kDPP and M-kDPP.
Figure 4 demonstrates the effect of the parameters ( , ) on the solution of our proposed method.
Notice that for a fixed , as increases, we select a smaller number of representatives, hence, the
encoding cost increases. Also, for a fixed , as increases, we put more more emphasis on dynamic
compatibility of representatives, hence, the dynamic cost decreases. On the other hand, the diversity
score decreases for smaller , as we select more representatives which become more redundant. The
results in Figure 4 also demonstrate the robustness of our method to the change of parameters.
1
To have a fair comparison and to select a fixed number of representatives, we modify the SeqDPP method
[3] and implement Seq-kDPP where k representatives are chosen in each window.
7
Task
Change
tire
Make
coffee
CPR
Jump
car
Repot
plant
All tasks
(P, R)
F-score
(P, R)
F-score
(P, R)
F-score
(P, R)
F-score
(P, R)
F-score
(P, R)
F-score
kDPP
(0.56, 0.50)
0.53
(0.38, 0.33)
0.35
(0.71, 0.71)
0.71
(0.50, 0.50)
0.50
(0.57, 0.67)
0.62
(0.54, 0.54)
0.54
M-kDPP
(0.55, 0.60)
0.57
(0.50, 0.44)
0.47
(0.71, 0.71)
0.71
(0.56, 0.50)
0.53
(0.60, 0.50)
0.55
(0.58, 0.55)
0.57
Seq-kDPP
(0.44, 0.40)
0.42
(0.63, 0.56)
0.59
(0.71, 0.71)
0.71
(0.56, 0.50)
0.53
(0.57, 0.67)
0.62
(0.58, 0.57)
0.57
DS3
(0.56, 0.50)
0.53
(0.50, 0.56)
0.53
(0.71, 0.71)
0.71
(0.50, 0.50)
0.50
(0.57, 0.67)
0.62
(0.57, 0.59)
0.58
SeqFL
(0.60, 0.60)
0.60
(0.50, 0.56)
0.53
(0.83, 0.71)
0.77
(0.60, 0.60)
0.60
(0.80, 0.67)
0.73
(0.67, 0.63)
0.65
Table 1: Precision (P), Recall (R) and F-score for summarization of instructional videos for five tasks.
Ground
Truth
Open Airway
Check Breathing
Give Breath
Give Compression
Give Breath
Give Compression
Give Compression
Check Breathing
Give Breath
Give Compression
Give Breath
Give Compression
Check Response
SeqFL
1
Figure 5: Ground-truth and automatic summarization
results of our method for the CPR task.
4.2
Instructional Video Summarization
We apply SeqFL to the task of summarization of intructional videos to automatically learn the
sequence of key actions to perform a task. We use videos from the instructional video dataset [45],
which consists of 30 instructional videos for each of five activities. The dataset also provides labels
for frames which contain the main steps required to perform that task. We preprocess the videos by
segmenting each video into superframes [46] and obtain features using a deep neural network that
we have constructed for feature extraction for summarization tasks. We use 60% of the videos from
each task as the training set to build an HMM model whose states form the source set, X. Using
the learned dynamic model, we apply our method to summarize each of the remaining videos. The
summaries for each video are sets of elements of X, states in the HMM model. For evaluation, we
map the representative states into actions in the ground truth by using the labels of the five nearest
neighbors in the training set to each state. The summary for each video is an assignment of each
superframe in the video to one of the action labels in the training set. Since each video may have
shown each action performed for a different length of time, we remove consecutive repeated labels
to form a list of actions performed, removing the length of time each action was performed. To
construct the final summary for a task, we align the lists of actions obtained by each method for all
the test videos, following the alignment method of [45] for several number of slots. For each method,
we choose the number of HMM states and the alignment of videos that give the best performance
(see the supplementary materials for more details). Table 1 shows the precision, recall and the
F-score of various methods. Notice that existing methods perform similar to each other for most
tasks, suggesting that sequential diversity promoted by Seq-kDPP and M-kDPP is not sufficient for
capturing the important steps of tasks. On the other hand, for most tasks and over the entire dataset,
our method (SeqFL) significantly outperforms other algorithms, better producing the sequence of
important steps to perform a task, thanks to the ability of our framework to incorporate the underlying
dynamics of the data. Figure 5 shows the ground-truth and the automatic summary produced by our
method for the CPR task, demonstrating that we can sufficiently well capture the main steps and the
sequence of steps to perform the task.
5
Conclusions
We developed a new framework for sequential subset selection that takes advantage of the underlying
dynamic models of data, promoting to select a set of representatives that are compatible according to
the dynamic models of data. By experiments on synthetic and real data, we showed the effectiveness
of our method for summarization of sequential data. Our ongoing research include development of
fast greedy algorithms for our sequential subset selection objective that can also deal with n-th order
dynamic models as well as investigation of the theoretical guarantees of our proposed formulations.
8
Acknowledgements
This work is supported by NSF IIS-1657197 award and startup funds from the Northeastern University,
College of Computer and Information Science.
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10
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6,307 | 6,705 | Question Asking as Program Generation
Anselm Rothe1
[email protected]
1
Brenden M. Lake1,2
[email protected]
Todd M. Gureckis1
[email protected]
2
Department of Psychology
Center for Data Science
New York University
Abstract
A hallmark of human intelligence is the ability to ask rich, creative, and revealing
questions. Here we introduce a cognitive model capable of constructing humanlike questions. Our approach treats questions as formal programs that, when executed on the state of the world, output an answer. The model specifies a probability
distribution over a complex, compositional space of programs, favoring concise
programs that help the agent learn in the current context. We evaluate our approach by modeling the types of open-ended questions generated by humans who
were attempting to learn about an ambiguous situation in a game. We find that our
model predicts what questions people will ask, and can creatively produce novel
questions that were not present in the training set. In addition, we compare a number of model variants, finding that both question informativeness and complexity
are important for producing human-like questions.
1
Introduction
In active machine learning, a learner is able to query an oracle in order to obtain information that is
expected to improve performance. Theoretical and empirical results show that active learning can
speed acquisition for a variety of learning tasks [see 21, for a review]. Although impressive, most
work on active machine learning has focused on relatively simple types of information requests
(most often a request for a supervised label). In contrast, humans often learn by asking far richer
questions which more directly target the critical parameters in a learning task. A human child might
ask ?Do all dogs have long tails?? or ?What is the difference between cats and dogs?? [2]. A long
term goal of artificial intelligence (AI) is to develop algorithms with a similar capacity to learn
by asking rich questions. Our premise is that we can make progress toward this goal by better
understanding human question asking abilities in computational terms [cf. 8].
To that end, in this paper, we propose a new computational framework that explains how people construct rich and interesting queries within in a particular domain. A key insight is to model questions
as programs that, when executed on the state of a possible world, output an answer. For example,
a program corresponding to ?Does John prefer coffee to tea?? would return True for all possible
world states where this is the correct answer and False for all others. Other questions may return
different types of answers. For example ?How many sugars does John take in his coffee?? would
return a number 0, 1, 2, etc. depending on the world state. Thinking of questions as syntactically
well-formed programs recasts the problem of question asking as one of program synthesis. We show
that this powerful formalism offers a new approach to modeling question asking in humans and may
eventually enable more human-like question asking in machines.
We evaluate our model using a data set containing natural language questions asked by human
participants in an information-search game [19]. Given an ambiguous situation or context, our
model can predict what questions human learners will ask by capturing constraints in how humans
construct semantically meaningful questions. The method successfully predicts the frequencies of
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
human questions given a game context, and can also synthesize novel human-like questions that
were not present in the training set.
2
Related work
Contemporary active learning algorithms can query for labels or causal interventions [21], but they
lack the representational capacity to consider a richer range of queries, including those expressed
in natural language. AI dialog systems are designed to ask questions, yet these systems are still
far from achieving human-like question asking. Goal-directed dialog systems [25, 1], applied to
tasks such as booking a table at a restaurant, typically choose between a relatively small set of
canned questions (e.g., ?How can I help you??, ?What type of food are you looking for??), with
little genuine flexibility or creativity. Deep learning systems have also been developed for visual
?20 questions? style tasks [22]; although these models can produce new questions, the questions
typically take a stereotyped form (?Is it a person??, ?Is it a glove?? etc.). More open-ended question
asking can be achieved by non-goal-driven systems trained on large amounts of natural language
dialog, such as the recent progress demonstrated in [20]. However, these approaches cannot capture
intentional, goal-directed forms of human question asking.
Recent work has probed other aspects of question asking. The Visual Question Generation (VQG)
data set [16] contains images paired with interesting, human-generated questions. For instance,
an image of a car wreck might be paired with the question, ?What caused the accident?? Deep
neural networks, similar to those used for image captioning, are capable of producing these types of
questions after extensive training [16, 23, 11]. However, they require large datasets of images paired
with questions, whereas people can ask intelligent questions in a novel scenario with no (or very
limited) practice, as shown in our task below. Moreover, human question asking is robust to changes
in task and goals, while state-of-the-art neural networks do not generalize flexibly in these ways.
3
The question data set
Our goal was to develop a model of context-sensitive, goal-directed question asking in humans,
which falls outside the capabilities of the systems described above. We focused our analysis on a
data set we collected in [19], which consists of 605 natural language questions asked by 40 human
players to resolve an ambiguous game situation (similar to ?Battleship?).1 Players were individually presented with a game board consisting of a 6?6 grid of tiles. The tiles were initially turned
over but each could be flipped to reveal an underlying color. The player?s goal was to identify as
quickly as possible the size, orientation, and position of ?ships? (i.e., objects composed of multiple
adjacent tiles of the same color) [7]. Every board had exactly three ships which were placed in nonoverlapping but otherwise random locations. The ships were identified by their color S = {Blue,
Red, Purple}. All ships had a width of 1, a length of N = {2, 3, 4} and orientation O = {Horizontal,
Vertical}. Any tile that did not overlap with a ship displayed a null ?water? color (light gray) when
flipped.
After extensive instructions about the rules and purpose of the game and a number of practice rounds
[see 19], on each of 18 target contexts players were presented with a partly revealed game board
(similar to Figure 1B and 1C) that provided ambiguous information about the actual shape and
location of the ships. They were then given the chance to ask a natural-language question about
the configuration. The player?s goal was to use this question asking opportunity to gain as much
information as possible about the hidden game board configuration. The only rules given to players
about questions was that they must be answerable using one word (e.g., true/false, a number, a color,
a coordinate like A1 or a row or column number) and no combination of questions was allowed. The
questions were recorded via an HTML text box in which people typed what they wanted to ask. A
good question for the context in Figure 1B is ?Do the purple and the red ship touch??, while ?What
is the color of tile A1?? is not helpful because it can be inferred from the revealed game board
and the rules of the game (ship sizes, etc.) that the answer is ?Water? (see Figure 3 for additional
example questions).
Each player completed 18 contexts where each presented a different underlying game board and
partially revealed pattern. Since the usefulness of asking a question depends on the context, the data
1
https://github.com/anselmrothe/question_dataset
2
!"#$%&!
a)
1
2
3
4
5
6
ABCDEF
Hidden gameboard
b)
1
2
3
4
5
6
ABCDEF
c)
1
2
3
4
5
6
ABCDEF
Partially revealed gameboard
Figure 1: The Battleship game used to obtain the question data set by Rothe et al. [19]. (A) The hidden
positions of three ships S = {Blue, Red, Purple} on a game board that players sought to identify. (B) After
observing the partly revealed board, players were allowed to ask a natural language question. (C) The partly
revealed board in context 4.
set consists of 605 question-context pairs hq, ci, with 26 to 39 questions per context.2 The basic
challenge for our active learning method is to predict which question q a human will ask from the
given context c and the overall rules of the game. This is a particularly challenging data set to model
because of the the subtle differences between contexts that determine if a question is potentially
useful along with the open-ended nature of human question asking.
4
A probabilistic model of question generation
Here we describe the components of our probabilistic model of question generation. Section 4.1
describes two key elements of our approach, compositionality and computability, as reflected in the
choice to model questions as programs. Section 4.2 describes a grammar that defines the space of
allowable questions/programs. Section 4.3 specifies a probabilistic generative model for sampling
context-sensitive, relevant programs from this space. The remaining sections cover optimization,
the program features, and alternative models (Sections 4.4-4.6).
4.1
Compositionality and computability
The analysis of the data set [19] revealed that many of the questions in the data set share similar
concepts organized in different ways. For example, the concept of ship size appeared in various
ways across questions:
?
?
?
?
?
?
?How long is the blue ship??
?Does the blue ship have 3 tiles??
?Are there any ships with 4 tiles??
?Is the blue ship less then 4 blocks??
?Are all 3 ships the same size??
?Does the red ship have more blocks than the blue ship??
As a result, the first key element of modeling question generation was to recognize the compositionality of these questions. In other words, there are conceptual building blocks (predicates
like size(x) and plus(x,y)) that can be put together to create the meaning of other questions (plus(size(Red), size(Purple))). Combining meaningful parts to give meaning
to larger expressions is a prominent approach in linguistics [10], and compositionality more generally has been an influential idea in cognitive science [4, 15, 14].
The second key element is the computability of questions. We propose that human questions are
like programs that when executed on the state of a world output an answer. For example, a program
that when executed looks up the number of blue tiles on a hypothesized or imagined Battleship
game board and returns said number corresponds to the question ?How long is the blue ship??. In
this way, programs can be used to evaluate the potential for useful information from a question
by executing the program over a set of possible or likely worlds and preferring questions that are
informative for identifying the true world state. This approach to modeling questions is closely
2
Although each of the 40 players asked a question for each context, a small number of questions were
excluded from the data set for being ambiguous or extremely difficult to address computationally [see 19].
3
related to formalizing question meaning as a partition over possible worlds [6], a notion used in
previous studies in linguistics [18] and psychology [9]. Machine systems for question answering
have also fruitfully modeled questions as programs [24, 12], and computational work in cognitive
science has modeled various kinds of concepts as programs [17, 5, 13]. An important contribution
of our work here is that it tackles question asking and provides a method for generating meaningful
questions/programs from scratch.
4.2
A grammar for producing questions
To capture both compositionality and computability, we represent questions in a simple programming language, based on lambda calculus and LISP. Every unit of computation in that language is
surrounded by parentheses, with the first element being a function and all following elements being
arguments to that function (i.e., using prefix notation). For instance, the question ?How long is the
blue ship?? would be represented by the small program (size Blue). More examples will be
discussed below. With this step we abstracted the question representation from the exact choice of
words while maintaining its meaning. As such the questions can be thought of as being represented
in a ?language of thought? [3].
Programs in this language can be combined as in the example (> (size Red) (size
Blue)), asking whether the red ship is larger than the blue ship. To compute an answer, first
the inner parentheses are evaluated, each returning a number corresponding to the number of red
or blue tiles on the game board, respectively. Then these numbers are used as arguments to the >
function, which returns either True or False.
A final property of interest is the generativity of questions, that is, the ability to construct novel
expressions that are useful in a given context. To have a system that can generate expressions in this
language we designed a grammar that is context-free with a few exceptions, inspired by [17]. The
grammar consists of a set of rewrite rules, which are recursively applied to grow expressions. An
expression that cannot be further grown (because no rewrite rules are applicable) is guaranteed to be
an interpretable program in our language.
To create a question, our grammar begins with an expression that contains the start symbol A and
then rewrites the symbols in the expression by applying appropriate grammatical rules until no
symbol can be rewritten. For example, by applying the rules A ? N, N ? (size S), and S ? Red,
we arrive at the expression (size Red). Table SI-1 (supplementary materials) shows the core
rewrite rules of the grammar. This set of rules is sufficient to represent all 605 questions in the
human data set.
To enrich the expressiveness and conciseness of our language we added lambda expressions, mapping, and set operators (Table SI-2, supplementary material). Their use can be seen in the question
?Are all ships the same size??, which can be conveniently represented by (= (map (? x (size
x)) (set Blue Red Purple))). During evaluation, map sequentially assigns each element
from the set to x in the ?-part and ultimately returns a vector of the three ship sizes. The three ship
sizes are then compared by the = function. Of course, the same question could also be represented
as (= (= (size Blue) (size Red)) (size Purple)).
4.3
Probabilistic generative model
An artificial agent using our grammar is able to express a wide range of questions. To decide which
question to ask, the agent needs a measure of question usefulness. This is because not all syntactically well-formed programs are informative or useful. For instance, the program (> (size
Blue) (size Blue)) representing the question ?Is the blue ship larger than itself?? is syntactically coherent. However, it is not a useful question to ask (and is unlikely to be asked by a human)
because the answer will always be False (?no?), no matter the true size of the blue ship.
We propose a probabilistic generative model that aims to predict which questions people will ask and
which not. Parameters of the model can be fit to predict the frequency that humans ask particular
questions in particular context in the data set by [19]. Formally, fitting the generative model is a
problem of density estimation in the space of question-like programs, where the space is defined by
the grammar. We define the probability of question x (i.e., the probability that question x is asked)
4
with a log-linear model. First, the energy of question x is the weighted sum of question features
E(x) = ?1 f1 (x) + ?2 f2 (x) + ... + ?K fK (x),
(1)
where ?k is the weight of feature fk of question x. We will describe all features below. Model
variants will differ in the features they use. Second, the energy is related to the probability by
p(x; ?) = P
exp(?E(x))
exp(?E(x))
=
,
exp(?E(x))
Z
x?X
(2)
where ? is the vector of feature weights, highlighting the fact that the probability is dependent on
a parameterization of these weights, Z is the normalizing constant, and X is the set of all possible
questions that can be generated by the grammar in Tables SI-1 and SI-2 (up to a limit on question
length).3 The normalizing constant needs to be approximated since X is too large to enumerate.
4.4
Optimization
The objective is to find feature weights that maximize the likelihood of asking the human-produced
questions. Thus, we want to optimize
arg max
?
N
X
log p(d(i) ; ?),
(3)
i=1
where D = {d(1) , ..., d(N ) } are the questions (translated into programs) in the human data set. To
optimize via gradient ascent, we need the gradient of the log-likelihood with respect to each ?k ,
which is given by
?log p(D; ?)
= N Ex?D [fk (x)] ? N Ex?P? [fk (x)].
(4)
??k
PN
The term Ex?D [fk (x)] = N1 i=1 fk (d(i) ) is the expected
P (average) feature values given the empirical set of human questions. The term Ex?P? [fk (x)] = x?X fk (x)p(x; ?) is the expected feature
values given the model. Thus, when the gradient is zero, the model has perfectly matched the data
in terms of the average values of the features.
Computing the exact expected feature values from the model is intractable, since there is a very large
number of possible questions (as with the normalizing constant in Equation 2). We use importance
sampling to approximate this expectation. To create a proposal distribution, denoted as q(x), we
use the question grammar as a probabilistic context free grammar with uniform distributions for
choosing the re-write rules.
The details of optimization are as follows. First, a large set of 150,000 questions is sampled in order
to approximate the gradient at each step via importance sampling.4 Second, to run the procedure for
a given model and training set, we ran 100,000 iterations of gradient ascent at a learning rate of 0.1.
Last, for the purpose of evaluating the model (computing log-likelihood), the importance sampler is
also used to approximate the normalizing constant in Eq. 2 via the estimator Z ? Ex?q [ p(x;?)
q(x) ].
4.5
Question features
We now turn to describe the question features we considered (cf. Equation 1), namely two features
for informativeness, one for length, and four for the answer type.
Informativeness. Perhaps the most important feature is a question?s informativeness, which we
model through a combination of Bayesian belief updating and Expected Information Gain (EIG).
To compute informativeness, our agent needs to represent several components: A belief about the
current world state, a way to update its belief once it receives an answer, and a sense of all possible
3
We define X to be the set of questions with 100 or fewer functions.
We had to remove the rule L ? (draw C) from the grammar and the corresponding 14 questions from
the data set that asked for a demonstration of a colored tile. Although it is straightforward to represent those
questions with this rule, the probabilistic nature of draw led to exponentially complex computations of the set
of possible-world answers.
4
5
Empirical question frequency
context: 3
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context: 4
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context: 5
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context: 7
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10
context: 8
? = 0.62
0
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context: 11
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context: 10
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context: 9
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Negative energy
Figure 2: Out-of-sample model predictions regarding the frequency of asking a particular question. The y-axis
shows the empirical question frequency, and x-axis shows the model?s energy for the question (Eq. 1, based on
the full model). The rank correlation ? is shown for each context.
answers to the question.5 In the Battleship game, an agent must identify a single hypothesis h (i.e.,
a hidden game board configuration) in the space of possible configurations H (i.e., possible board
games). The agent can ask a question x and receive the answer d, updating its hypothesis space
by applying Bayes? rule, p(h|d; x) ? p(d|h; x)p(h). The prior p(h) is specified first by a uniform
choice over the ship sizes, and second by a uniform choice over all possible configurations given
those sizes. The likelihood p(d|h; x) ? 1 if d is a valid output of the question program x when
executed on h, and zero otherwise.
The Expected Information Gain (EIG) value of a question x is the expected reduction in uncertainty
about the true hypothesis h, averaged across all possible answers Ax of the question
h
i
X
EIG(x) =
p(d; x) I[p(h)] ? I[p(h|d; x)] ,
(5)
d?Ax
where I[?] is the Shannon entropy. Complete details about the Bayesian ideal observer follow the
approach we used in [19]. Figure 3 shows the EIG scores for the top two human questions for
selected contexts.
In addition to feature fEIG (x) = EIG(x), we added a second feature fEIG=0 (x), which is 1 if EIG
is zero and 0 otherwise, to provide an offset to the linear EIG feature. Note that the EIG value
of a question always depends on the game context. The remaining features described below are
independent of the context.
Complexity. Purely maximizing EIG often favors long and complicated programs (e.g., polynomial
questions such as size(Red)+10*size(Blue)+100*size(Purple)+...). Although a
machine would not have a problem with answering such questions, it poses a problem for a human
answerer. Generally speaking, people prefer concise questions and the rather short questions in the
data set reflect this. The probabilistic context free grammar provides a measure of complexity that
favors shorter programs, and we use the log probability under the grammar fcomp (x) = ? log q(x)
as the complexity feature.
Answer type. We added four features for the answer types Boolean, Number, Color, and Location.
Each question program belongs to exactly one of these answer types (see Table SI-1). The type
Orientation was subsumed in Boolean, with Horizontal as True and Vertical as False.
This allows the model to capture differences in the base rates of question types (e.g., if people prefer
true/false questions over other types).
Relevance. Finally, we added one auxiliary feature to deal with the fact that the grammar can
produce syntactically coherent programs that have no reference to the game board at all (thus are
not really questions about the game; e.g., (+ 1 1)). The ?filter? feature f? (x) marks questions
5
We assume here that the agent?s goal is to accurately identify the current world state. In a more general
setting, the agent would require a cost function that defines the helpfulness of an answer as a reduced distance
to the goal.
6
that refer to the Battleship game board with a value of 1 (see the
otherwise.6
4.6
b
marker in Table SI-1) and 0
Alternative models
To evaluate which features are important for human-like question generation, we tested the full
model that uses all features, as well as variants in which we respectively lesioned one key property.
The information-agnostic model did not use fEIG (x) and fEIG=0 (x) and thus ignored the informativeness of questions. The complexity-agnostic model ignored the complexity feature. The type-agnostic
model ignored the answer type features.
5
Results and Discussion
The probabilistic model of question generation was eval- Table 1: Log likelihoods of model variants
uated in two main ways. First, it was tasked with predict- averaged across held out contexts.
ing the distribution of questions people asked in novel
scenarios, which we evaluate quantitatively. Second, it
Model
LL
was tasked with generating genuinely novel questions
that were not present in the data set, which we evaluate
Full
-1400.06
qualitatively. To make predictions, the different candiInformation-agnostic
-1464.65
date models were fit to 15 contexts and asked to predict
Complexity-agnostic -22993.38
the remaining one (i.e., leave one out cross-validation).7
Type-agnostic
-1419.26
This results in 64 different model fits (i.e., 4 models ?
16 fits).
First, we verify that compositionality is an essential ingredient in an account of human question asking. For any given context, about 15% of the human questions did not appear in any of the other contexts. Any model that attempts to simply
reuse/reweight past questions will be unable to account for this productivity (effectively achieving
a log-likelihood of ??), at least not without a much larger training set of questions. The grammar
over programs provides one account of the productivity of the human behavior.
Second, we compared different models on their ability to quantitatively predict the distribution of
human questions. Table 1 summarizes the model predictions based on the log-likelihood of the
questions asked in the held-out contexts. The full model ? with learned features for informativeness, complexity, answer type, and relevance ? provides the best account of the data. In each case,
lesioning its key components resulted in lower quality predictions. The complexity-agnostic model
performed far worse than the others, highlighting the important role of complexity (as opposed
to pure informativeness) in understanding which questions people choose to ask. The full model
also outperformed the information-agnostic and type-agnostic models, suggesting that people also
optimize for information gain and prefer certain question types (e.g., true/false questions are very
common). Because the log-likelihood values are approximate, we bootstrapped the estimate of the
normalizing constant Z and compared the full model and each alternative. The full model?s loglikelihood advantage over the complexity-agnostic model held in 100% of the bootstrap samples,
over the information-agnostic model in 81% of samples, and over type-agnostic model in 88%.
Third, we considered the overall match between the best-fit model and the human question frequencies. Figure 2 shows the correlations between the energy values according to the held-out predictions
of the full model (Eq. 1) and the frequencies of human questions (e.g., how often participants asked
?What is the size of the red ship?? in a particular context). The results show very strong agreement
for some contexts along with more modest alignment for others, with an average Spearman?s rank
correlation coefficient of 0.64. In comparison, the information-agnostic model achieved 0.65, the
complexity-agnostic model achieved -0.36, and the type-agnostic model achieved 0.55. One limitation is that the human data is sparse (many questions were only asked once), and thus correlations
6
The features f? (x) and fEIG=0 (x) are not identical. Questions like (size Blue) do refer to the board
but will have zero EIG if the size of the blue ship is already known.
7
For computational reasons we had to drop contexts 1 and 2, which had especially large hypothesis spaces.
However, we made sure that the grammar was designed based on the full set of contexts (i.e., it could express
all questions in the human question data set).
7
are limited as a measure of fit. However, there is, surprisingly, no correlation at all between question
generation frequency and EIG alone [19], again suggesting a key role of question complexity and
the other features.
Last, the model was tasked with generating novel, ?human-like? questions that were not part of the
human data set. Figure 3 shows five novel questions that were sampled from the model, across four
different game contexts. Questions were produced by taking five weighted samples from the set
of programs produced in Section 4.4 for approximate inference, with weights determined by their
energy (Eq. 2). To ensure novelty, samples were rejected if they were equivalent to any human
question in the training data set or to an already sampled question. Equivalence between any two
questions was determined by the mutual information of their answer distributions (i.e., their partitions over possible hypotheses), and or if the programs differed only through their arguments (e.g.
(size Blue) is equivalent to (size Red)). The generated questions in Figure 3 demonstrate
that the model is capable of asking novel (and clever) human-like questions that are useful in their
respective contexts. Interesting new questions that were not observed in the human data include
?Are all the ships horizontal?? (Context 7), ?What is the top left of all the ship tiles?? (Context 9),
?Are blue and purple ships touching and red and purple not touching (or vice versa)?? (Context 9),
and ?What is the column of the top left of the tiles that have the color of the bottom right corner
of the board?? (Context 15). The four contexts were selected to illustrate the creative range of the
model, and the complete set of contexts is shown in the supplementary materials.
6
Conclusions
People use question asking as a cognitive tool to gain information about the world. Although people
ask rich and interesting questions, most active learning algorithms make only focused requests for
supervised labels. Here were formalize computational aspects of the rich and productive way that
people inquire about the world. Our central hypothesis is that active machine learning concepts
can be generalized to operate over a complex, compositional space of programs that are evaluated
over possible worlds. To that end, this project represents a step toward more capable active learning
machines.
There are also a number of limitations of our current approach. First, our system operates on semantic representations rather than on natural language text directly, although it is possible that such
a system can interface with recent tools in computational linguistics to bridge this gap [e.g., 24].
Second, some aspects of our grammar are specific to the Battleship domain. It is often said that
some knowledge is needed to ask a good question, but critics of our approach will point out that the
model begins with substantial domain knowledge and special purpose structures. On the other hand,
many aspects of our grammar are domain general rather than domain specific, including very general
functions and programming constructs such as logical connectives, set operations, arithmetic, and
mapping. To extend this approach to new domains, it is unclear exactly how much new knowledge
engineering will be needed, and how much can be preserved from the current architecture. Future
work will bring additional clarity as we extend our approach to different domains.
From the perspective of computational cognitive science, our results show how people balance informativeness and complexity when producing semantically coherent questions. By formulating
question asking as program generation, we provide the first predictive model to date of open-ended
human question asking.
Acknowledgments
We thank Chris Barker, Sam Bowman, Noah Goodman, and Doug Markant for feedback and advice.
This research was supported by NSF grant BCS-1255538, the John Templeton Foundation Varieties
of Understanding project, a John S. McDonnell Foundation Scholar Award to TMG, and the MooreSloan Data Science Environment at NYU.
8
Figure 3: Novel questions generated by the probabilistic model. Across four contexts, five model questions are
displayed, next to the two most informative human questions for comparison. Model questions were sampled
such that they are not equivalent to any in the training set. The natural language translations of the question
programs are provided for interpretation. Questions with lower energy are more likely according to the model.
9
References
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10
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6,308 | 6,706 | Revisiting Perceptron:
Ef?cient and Label-Optimal Learning of Halfspaces
Songbai Yan
UC San Diego
La Jolla, CA
[email protected]
Chicheng Zhang?
Microsoft Research
New York, NY
[email protected]
Abstract
It has been a long-standing problem to ef?ciently learn a halfspace using as few
labels as possible in the presence of noise. In this work, we propose an ef?cient
Perceptron-based algorithm for actively learning homogeneous halfspaces under the
uniform distribution over the unit sphere. Under the bounded noise condition [49],
where each label is ?ipped with probability
at most?? < 12 , our algorithm
achieves
a
?
?
?
d
1 2
d2
?
?
near-optimal label complexity of O
in time O
. Under
2 ln
3
(1?2?)
?
?(1?2?)
?
the adversarial noise condition [6, 45, 42], where at most a ?(?)
fraction of
? labels?
? d ln 1
O
can be ?ipped,
our
algorithm
achieves
a
near-optimal
label
complexity
of
?
? ?
? d2 . Furthermore, we show that our active learning algorithm can be
in time O
?
converted to an ef?cient passive learning algorithm that has near-optimal sample
complexities with respect to ? and d.
1
Introduction
We study the problem of designing ef?cient noise-tolerant algorithms for actively learning homogeneous halfspaces in the streaming setting. We are given access to a data distribution from which we
can draw unlabeled examples, and a noisy labeling oracle O that we can query for labels. The goal is
to ?nd a computationally ef?cient algorithm to learn a halfspace that best classi?es the data while
making as few queries to the labeling oracle as possible.
Active learning arises naturally in many machine learning applications where unlabeled examples are
abundant and cheap, but labeling requires human effort and is expensive. For those applications, one
natural question is whether we can learn an accurate classi?er using as few labels as possible. Active
learning addresses this question by allowing the learning algorithm to sequentially select examples
to query for labels, and avoid requesting labels which are less informative, or can be inferred from
previously-observed examples.
There has been a large body of work on the theory of active learning, showing sharp distributiondependent label complexity bounds [21, 11, 34, 27, 35, 46, 60, 41]. However, most of these general
active learning algorithms rely on solving empirical risk minimization problems, which are computationally hard in the presence of noise [5].
On the other hand, existing computationally ef?cient algorithms for learning halfspaces [17, 29, 42,
45, 6, 23, 7, 8] are not optimal in terms of label requirements. These algorithms have different degrees
of noise tolerance (e.g. adversarial noise [6], malicious noise [43], random classi?cation noise [3],
?
Work done while at UC San Diego.
?
? (?)) := O(f (?) ln f (?)), and ?(f
? (?)) := ?(f (?)/ ln f (?)). We say f (?) = ?(g(?))
We use O(f
if f (?) =
?
?
?
?
O(g(?)) and f (?) = ? g(?)
2
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
bounded noise [49], etc), and run in time polynomial in 1? and d. Some of them naturally exploit the
utility of active learning [6, 7, 8], but they do not achieve the sharpest label complexity bounds in
contrast to those computationally-inef?cient active learning algorithms [10, 9, 60].
Therefore, a natural question is: is there any active learning halfspace algorithm that is computationally
ef?cient, and has a minimum label requirement? This has been posed as an open problem in [50].
In the realizable setting, [26, 10, 9, 56] give ef?cient algorithms that have optimal label complexity
? ln 1 ) under some distributional assumptions. However, the challenge still remains open in
of O(d
?
the nonrealizable setting. It has been shown that learning halfspaces with agnostic noise even under
Gaussian unlabeled distribution is hard [44]. Nonetheless, we give an af?rmative answer to this
question under two moderate noise settings: bounded noise and adversarial noise.
1.1
Our Results
We propose a Perceptron-based algorithm, ACTIVE -P ERCEPTRON, for actively learning homogeneous halfspaces under the uniform distribution over the unit sphere. It works under two noise
settings: bounded noise and adversarial noise. Our work answers an open question by [26] on
whether Perceptron-based active learning algorithms can be modi?ed to tolerate label noise.
In the ?-bounded noise setting (also known as the Massart noise model [49]), the label of an example
x ? Rd is generated by sign(u ? x) for some underlying
halfspace
u, and ?ipped
?
? with probabil?
?
d2
d
1
?
?
O
ity ?(x) ? ? < 1 . Our algorithm runs in time O
,
and
requires
3
2 ? ln
2
(1?2?) ?
(1?2?)
?
labels. We show that this label complexity
is nearly ?optimal by providing an almost matching
?
d
information-theoretic lower bound of ? (1?2?)2 ? ln 1? . Our time and label complexities substan?
tially improve over the state of the art result of [8], which runs in time O(d
1
O(
)
1
4
?
(1?2?)
O(d
ln ) labels.
1
O( (1?2?)
4)1
?)
and requires
?
Our main theorem on learning under bounded noise is as follows:
Theorem 2 (Informal). Suppose the labeling oracle O satis?es the ?-bounded noise condition with
respect to u, then for ACTIVE -P ERCEPTRON, with probability at least 1??: (1) The output halfspace
v is such? that P[sign(v?? X) ?= sign(u ? X)] ? ?; (2) The number of label queries to? oracle O?is at
d
1
d
?
?
most O
(1?2?)2 ? ln ? ; (3) The number of unlabeled examples drawn is at most O (1?2?)3 ? ; (4)
?
?
d2
?
The algorithm runs in time O
(1?2?)3 ? .
In addition, we show that our algorithm also works in a more challenging setting, the ?-adversarial
noise setting [6, 42, 45].3 In this setting, the examples still come iid from a distribution, but the
assumption on the labels is just that P[sign(u ? X) ?= Y ] ? ? for some halfspace u. Under this
assumption, the Bayes classi?er may not be a halfspace.
We?show that our algorithm achieves an
?
? ?
?
? d2 , and requires
error of ? while tolerating a noise level of ? = ? ln d +ln ln 1 . It runs in time O
?
?
?
?
?
1
? d ? ln
only O
? labels which is near-optimal. ACTIVE -P ERCEPTRON has a label complexity bound
that matches the state of the art result of [39]4 , while having a lower running time.
Our main theorem on learning under adversarial noise is as follows:
Theorem 3 (Informal). Suppose the labeling oracle O satis?es the ?-adversarial noise condition
?
with respect to u, where ? < ?( ln d +ln
). Then for ACTIVE -P ERCEPTRON, with probability at
ln 1
?
?
least 1 ? ?: (1) The output halfspace v is such
? that?P[sign(v ? X) ?= sign(u ? X)] ? ?; (2) The number
? d ? ln 1 ; (3) The number of unlabeled examples drawn is
O
of label queries
to
oracle
O
is
at
most
? ? ?
? ?
? d ; (4) The algorithm runs in time O
? d2 .
at most O
?
?
3
Note that the adversarial noise model is not the same as that in online learning [18], where each example
can be chosen adversarially.
4
The label complexity bound is implicit in [39] by a re?ned analysis of the algorithm of [6] (See their Lemma
8 for details).
2
Table 1: A comparison of algorithms for active learning of halfspaces under the uniform distribution,
in the ?-bounded noise model.
Algorithm
[10, 9, 60]
Label Complexity
d
1
?
O(
)
2 ln
[8]
Our Work
?
O(d
?
O(
Time Complexity
(1?2?)
?
1
O( (1?2?)
4)
d
(1?2?)2
? ln 1? )
ln 1? )
superpoly(d, 1? ) 5
1
? O( (1?2?)4 ) ? 1 )
O(d
? ?
?
d2
1
?
O
3
(1?2?) ?
Table 2: A comparison of algorithms for active learning of halfspaces under the uniform distribution,
in the ?-adversarial noise model.
Algorithm
Noise Tolerance
[60]
[39]
Our Work
? = ?(?)
? = ?(?)
?
)
? = ?( ln d+ln
ln 1
Label Complexity
? ln 1 )
O(d
?
? ln 1 )
O(d
?
? ln 1 )
O(d
?
?
Time Complexity
superpoly(d, 1? )
1
poly(d,
? 2 1? ?)
?
O d ??
Throughout the paper, ACTIVE -P ERCEPTRON is shown to work if the unlabeled examples are drawn
uniformly from the unit sphere. The algorithm and analysis can be easily generalized to any spherical
symmetrical distributions, for example, isotropic Gaussian distributions. They can also be generalized
to distributions whose densities with respect to uniform distribution are bounded away from 0.
In addition, we show in Section 6 that ACTIVE -P ERCEPTRON can be converted to a passive learning
algorithm, PASSIVE -P ERCEPTRON, that has near optimal sample complexities with respect to ? and
d under the two noise settings. We defer the discussion to the end of the paper.
2
Related Work
Active Learning. The recent decades have seen much success in both theory and practice of active
learning; see the excellent surveys by [54, 37, 25]. On the theory side, many label-ef?cient active
learning algorithms have been proposed and analyzed. An incomplete list includes [21, 11, 34, 27,
35, 46, 60, 41]. Most algorithms relies on solving empirical risk minimization problems, which are
computationally hard in the presence of noise [5].
Computational Hardness of Learning Halfspaces. Ef?cient learning of halfspaces is one of
the central problems in machine learning [22]. In the realizable case, it is well known that linear
programming will ?nd a consistent hypothesis over data ef?ciently. In the nonrealizable setting,
however, the problem is much more challenging.
A series of papers have shown the hardness of learning halfspaces with agnostic noise [5, 30, 33, 44,
23]. The state of the art result [23] shows that under standard complexity-theoretic assumptions, there
exists a data distribution, such that the best linear classi?er has error o(1), but no polynomial time
algorithms can achieve an error at most 12 ? d1c for every c > 0, even with improper learning. [44]
shows that under standard assumptions, even if the unlabeled distribution is Gaussian, any agnostic
halfspace learning algorithm must run in time ( 1? )?(ln d) to achieve an excess error of ?. These results
indicate that, to have nontrivial guarantees on learning halfspaces with noise in polynomial time, one
has to make additional assumptions on the data distribution over instances and labels.
Ef?cient Active Learning of Halfspaces. Despite considerable efforts, there are only a few halfspace learning algorithms that are both computationally-ef?cient and label-ef?cient even under the
uniform distribution. In the realizable setting, [26, 10, 9] propose computationally ef?cient active
? ln 1 ).
learning algorithms which have an optimal label complexity of O(d
?
Since it is believed to be hard for learning halfspaces in the general agnostic setting, it is natural
to consider algorithms that work under more moderate noise conditions. Under the bounded noise
5
The algorithm needs to minimize 0-1 loss, the best known method for which requires superpolynomial time.
3
setting [49], the only known algorithms that are both label-ef?cient and computationally-ef?cient are
[7, 8]. [7] uses a margin-based framework which queries the labels of examples near the decision
boundary. To achieve computational ef?ciency, it adaptively chooses a sequence of hinge loss
minimization problems to optimize as opposed to directly optimizing the 0-1 loss. It works only
when the label ?ipping probability upper bound ? is small (? ? 1.8 ? 10?6 ). [8] improves over [7]
by adapting a polynomial regression procedure into the margin-based framework. It works for any
1
O(
)
? < 1/2, but its label complexity is O(d (1?2?)4 ln 1? ), which is far worse than the informationd
1
theoretic lower bound ?( (1?2?)
2 ln ? ). Recently [20] gives an ef?cient algorithm with a near-optimal
label complexity under the membership query model where the learner can query on synthesized
points. In contrast, in our stream-based model, the learner can only query on points drawn from the
data distribution. We note that learning in the stream-based model is harder than in the membership
query model, and it is unclear how to transform the DC algorithm in [20] into a computationally
ef?cient stream-based active learning algorithm.
Under the more challenging ?-adversarial noise setting, [6] proposes a margin-based algorithm that
reduces the problem to a sequence of hinge loss minimization problems. Their algorithm achieves an
? 2 ln 1 ) labels. Later, [39] performs a
error of ? in polynomial time when ? = ?(?), but requires O(d
?
? ln 1 ), but the time complexity of
re?ned analysis to achieve a near-optimal label complexity of O(d
?
the algorithm is still an unspeci?ed high order polynomial.
Tables 1 and 2 present comparisons between our results and results most closely related to ours
in the literature. Due to space limitations, discussions of additional related work are deferred to
Appendix A.
3
De?nitions and Settings
We consider learning homogeneous halfspaces under uniform
distribution. ?
The instance space X
?
is the unit sphere in Rd , which we denote by Sd?1 := x ? Rd : ?x? = 1 . We assume d ? 3
throughout this paper. The label space Y = {+1, ?1}. We assume all data points (x, y) are drawn
i.i.d. from an underlying distribution D over X ? Y. We denote by DX the marginal of D over X
(which is uniform over Sd?1 ), and DY |X the conditional distribution of Y given X. Our algorithm is
allowed to draw unlabeled examples x ? X from DX , and to make queries to a labeling oracle O for
labels. Upon query x, O returns a label y ?
drawn from DY |X=x . The hypothesis
? class of interest is
d?1
the set of homogeneous halfspaces H := hw (x) = sign(w ? x) | w ? S
. For any hypothesis
h ? H, we de?ne its error rate err(h) := PD [h(X)
=
?
Y
].
We
will
drop
the
subscript
D in PD when
?
?
it is clear from the context. Given a dataset S = ?(X1 , Y1 ), . .?. , (Xm , Ym ) , we de?ne the empirical
?m
1
error rate of h over S as errS (h) := m
i=1 1 h(xi ) ?= yi .
De?nition 1 (Bounded Noise [49]). We say that the labeling oracle O satis?es the ?-bounded noise
condition for some ? ? [0, 1/2) with respect to u, if for any x, P[Y ?= sign(u ? x) | X = x] ? ?.
It can be seen that under ?-bounded noise condition, hu is the Bayes classi?er.
De?nition 2 (Adversarial Noise [6]). We say that the labeling oracle O satis?es the ?-adversarial
noise condition for some ? ? [0, 1] with respect to u, if P[Y ?= sign(u ? X)] ? ?.
For two unit vectors v1 , v2 , denote by ?(v1 , v2 ) = arccos(v1 ? v2 ) the angle between them. The
following lemma gives relationships between errors and angles (see also Lemma 1 in [8]).
?
?
?
?
Lemma 1. For any v1 , v2 ? Sd?1 , ?err(hv1 ) ? err(hv2 )? ? P hv1 (X) ?= hv2 (X) = ?(v1?,v2 ) .
Additionally, if? the labeling oracle? satis?es the ?-bounded
noise condition
with respect to u, then for
?
?
any vector v, ?err(hv ) ? err(hu )? ? (1 ? 2?)P hv (X) ?= hu (X) = 1?2?
? ?(v, u).
Given access to unlabeled examples drawn from DX and a labeling oracle O, our goal is to ?nd a
polynomial time algorithm A such that with probability at least 1 ? ?, A outputs a halfspace hv ? H
with P[sign(v ? X) ?= sign(u ? X)] ? ? for some target accuracy ? and con?dence ?. (By Lemma 1,
this guarantees that the excess error of hv is at most ?, namely, err(hv ) ? err(hu ) ? ?.) The desired
algorithm should make as few queries to the labeling oracle O as possible.
4
We say an algorithm A achieves a label complexity of ?(?, ?), if for any target halfspace hu ? H,
with probability at least 1 ? ?, A outputs a halfspace hv ? H such that err(hv ) ? err(hu ) + ?, and
requests at most ?(?, ?) labels from oracle O.
4
Main Algorithm
Our main algorithm, ACTIVE -P ERCEPTRON (Algorithm 1), works in epochs. It works under the
bounded and the adversarial noise models, if its sample schedule {mk } and band width {bk } are set
appropriately with respect to each noise model. At the beginning of each epoch k, it assumes an upper
bound of 2?k on ?(vk?1 , u), the angle between current iterate vk?1 and the underlying halfspace u.
As we will see, this can be shown to hold with high probability inductively. Then, it calls procedure
M ODIFIED -P ERCEPTRON (Algorithm 2) to ?nd an new iterate vk , which can be shown to have an
?
angle with u at most 2k+1
with high probability. The algorithm ends when a total of k0 = ?log2 1? ?
epochs have passed.
For simplicity, we assume for the rest of the paper that the angle between the initial halfspace v0 and
the underlying halfspace u is acute, that is, ?(v0 , u) ? ?2 ; Appendix F shows that this assumption
can be removed with a constant overhead in terms of label and time complexities.
Algorithm 1 ACTIVE -P ERCEPTRON
Input: Labeling oracle O, initial halfspace v0 , target error ?, con?dence ?, sample schedule {mk },
band width {bk }.
Output: learned halfspace v.
1: Let k0 = ?log2 1? ?.
2: for k = 1, 2, . . . , k0 do
?
3:
vk ? M ODIFIED -P ERCEPTRON(O, vk?1 , 2?k , k(k+1)
, mk , bk ).
4: end for
5: return vk0 .
Procedure M ODIFIED -P ERCEPTRON (Algorithm 2) is the core component of ACTIVE -P ERCEPTRON.
It sequentially performs a modi?ed Perceptron update rule on the selected new examples (xt , yt ) [51,
17, 26]:
(1)
wt+1 ? wt ? 21 {yt wt ? xt < 0} (wt ? xt ) ? xt
De?ne ?t := ?(wt , u). Update rule (1) implies the following relationship between ?t+1 and ?t (See
Lemma 8 in Appendix E for its proof):
cos ?t+1 ? cos ?t = ?21 {yt wt ? xt < 0} (wt ? xt ) ? (u ? xt )
(2)
This motivates us to take cos ?t as our measure of progress; we would like to drive cos ?t up to 1(so
that ?t goes down to 0) as fast as possible.
To this end, M ODIFIED -P ERCEPTRON samples
new points xt under time-varying
distributions DX |Rt
?
?
d?1 : b
and query for their labels, where Rt = x ? S
2 ? wt ? x ? b is a band inside the unit sphere.
The rationale behind the choice of Rt is twofold:
?
1. We set Rt to have a probability mass of ?(?),
so that the time complexity of rejection
1
?
sampling is at most O( ? ) per example. Moreover, in the adversarial noise setting, we set Rt
?
large enough to dominate the noise of magnitude ? = ?(?).
2. Unlike the active Perceptron algorithm in [26] or other margin-based approaches (for
example [55, 10]) where examples with small margin are queried, we query the label of the
examples with a range of margin [ 2b , b]. From a technical perspective, this ensures that ?t
decreases by a decent amount in expectation (see Lemmas 9 and 10 for details).
Following the insight of [32], we remark that the modi?ed Perceptron update (1) on distribution
DX |Rt can be alternatively viewed as performing stochastic gradient descent on a special non-convex
loss function ?(w, (x, y)) = min(1, max(0, ?1? 2b yw?x)). It is an interesting open question whether
optimizing this new loss function can lead to improved empirical results for learning halfspaces.
5
Algorithm 2 M ODIFIED -P ERCEPTRON
Input: Labeling oracle O, initial halfspace w0 , angle upper bound ?, con?dence ?, number of
iterations m, band width b.
Output: Improved halfspace wm .
1: for t = 0, 1, 2, . . . , m ?
? 1 do
?
2:
De?ne region Rt = x ? Sd?1 : 2b ? wt ? x ? b .
Rejection sample xt ? DX |Rt . In other words, draw xt from DX until xt is in Rt . Query O
for its label yt .
4:
wt+1 ? wt ? 21 {yt wt ? xt < 0} ? (wt ? xt ) ? xt .
5: end for
6: return wm .
3:
5
Performance Guarantees
We show that ACTIVE -P ERCEPTRON works in the bounded and the adversarial noise models, achieving computational ef?ciency and near-optimal label complexities. To this end, we ?rst give a lower
bound on the label complexity under bounded noise, and then give computational and label complexity
upper bounds under the two noise conditions respectively. We defer all proofs to the Appendix.
5.1
A Lower Bound under Bounded Noise
We ?rst present an information-theoretic lower bound on the label complexity in the bounded noise
setting under uniform distribution. This extends the distribution-free lower bounds of [53, 37], and
generalizes the realizable-case lower bound of [47] to the bounded noise setting. Our lower bound
can also be viewed as an extension of [59]?s Theorem 3; speci?cally it addresses the hardness under
the ?-Tsybakov noise condition where ? = 0 (while [59]?s Theorem 3 provides lower boundes when
? ? (0, 1)).
1
Theorem 1. For any d > 4, 0 ? ? < 12 , 0 < ? ? 4?
, 0 < ? ? 14 , for any active learning algorithm
d?1
A, there is a u ? S , and a labeling oracle O that satis?es ?-bounded noise condition with respect
to u, such that if with probability at least 1 ? ?, A makes at most n queries
of labels to O?and outputs
?
d log 1
? log 1
v ? Sd?1 such that P[sign(v ? X) ?= sign(u ? X)] ? ?, then n ? ? (1?2?)? 2 + (1?2?)?2 .
5.2
Bounded Noise
We establish Theorem 2 in the bounded noise setting. The theorem implies that, with appropriate
settings of input parameters, ACTIVE -P ERCEPTRON ef?ciently learns a halfspace of excess error
at most ? with probability at least 1 ? ?, under the assumption that DX is uniform over the unit
d
1
?
sphere and O has bounded noise. In addition, it queries at most O(
(1?2?)2 ln ? ) labels. This matches
the lower bound of Theorem 1, and improves over the state of the art result of [8], where a label
1
? O( (1?2?)4 ) ln 1 ) is shown using a different algorithm.
complexity of O(d
?
The proof and the precise setting of parameters (mk and bk ) are given in Appendix C.
Theorem 2 (ACTIVE -P ERCEPTRON under Bounded Noise). Suppose Algorithm 1 has inputs labeling oracle O that satis?es ?-bounded noise condition with respect to halfspace u, initial halfspace v0 such that ?(v0 , u) ? [0, ?2 ], target error ?, con?dence ?, sample
? schedule {m
? k } where
?
?
?k
2
(1?2?)
d
d
k
?
mk = ? (1?2?)
. Then with
2 (ln (1?2?)2 + ln ? ) , band width {bk } where bk = ?
d ln(km /?)
k
probability at least 1 ? ?:
1. The output halfspace v is such that P[sign(v ? X) ?= sign(u ? X)] ? ?.
?
?
??
d
1
d
1
1
?
ln
?
ln
+
ln
+
ln
ln
2. The number of label queries is O (1?2?)
.
2
?
(1?2?)2
?
?
6
3. The
is
? number of?unlabeled examples drawn ?
2
d
d
1
1
O (1?2?)3 ? ln (1?2?)2 + ln ? + ln ln ? ?
4. The algorithm runs in time O
?
d2
(1?2?)3
1
?
?
ln 1? .
?
?2
d
1
1
? ln (1?2?)
?
2 + ln ? + ln ln ?
1
?
?
ln 1? .
The theorem follows from Lemma 2 below. The key ingredient of the lemma is a delicate analysis
m
of the dynamics of the angles {?t }t=0 , where ?t = ?(wt , u) is the angle between the iterate wt
and the halfspace u. Since xt is randomly sampled and yt is noisy, we are only able to show that
?t decreases by a decent amount in expectation. To remedy the stochastic ?uctuations, we apply
m
martingale concentration inequalities to carefully control the upper envelope of sequence {?t }t=0 .
Lemma 2 (M ODIFIED -P ERCEPTRON under Bounded Noise). Suppose Algorithm 2 has inputs
labeling oracle O that satis?es ?-bounded noise condition with respect to halfspace u, initial
halfspace w0 and angle upper bound ? ? (0, ?2 ] such that ?(w0 , u) ???, con?dence
? ?, number
?(1?2?)
?
d ln(m/?)
d
d
1
of iterations m = ?( (1?2?)
2 (ln (1?2?)2 + ln ? )), band width b = ?
. Then with
probability at least 1 ? ?:
1. The output halfspace wm is such that ?(wm , u) ? ?2 .
?
?
??
d
d
1
2. The number of label queries is O (1?2?)2 ln (1?2?)2 + ln ? .
3. The number of unlabeled examples drawn is O
4. The algorithm runs in time O
5.3
?
d2
(1?2?)3
?
? ln
?
d
(1?2?)3
d
(1?2?)2
?
? ln
+ ln
1
?
?2
d
(1?2?)2
?
1
?
?
+ ln
1
?
?2
?
1
?
?
.
.
Adversarial Noise
We establish Theorem 3 in the adversarial noise setting. The theorem implies that, with appropriate
settings of input parameters, ACTIVE -P ERCEPTRON ef?ciently learns a halfspace of excess error at
most ? with probability at least 1 ? ?, under the assumption that DX is uniform over the unit sphere
?
and O has an adversarial noise of magnitude ? = ?( ln d+ln
). In addition, it queries at most
ln 1?
1
?
O(d ln ? ) labels. Our label complexity bound is information-theoretically optimal [47], and matches
the state of the art result of [39]. The bene?t of our approach is computational: it has a running time
? d2 ), while [39] needs to solve a convex optimization problem whose running time is some
of O(
?
polynomial over d and 1? with an unspeci?ed degree.
The proof and the precise setting of parameters (mk and bk ) are given in Appendix C.
Theorem 3 (ACTIVE -P ERCEPTRON under Adversarial Noise). Suppose Algorithm 1 has inputs
labeling oracle O that satis?es ?-adversarial noise condition with respect to halfspace u, initial
halfspace v0 such that ?(v0 , u) ? ?2 , target error ?, con?dence
{mk } where
?
? ?, sample schedule
mk = ?(d(ln d + ln k? )), band width {bk } where bk = ?
?
?( ln d +ln
ln
?
1
?
?
2?k
d ln(kmk /?)
). Then with probability at least 1 ? ?:
1. The output halfspace v is such that P[sign(v ? X) ?= sign(u ? X)] ? ?.
?
??
?
2. The number of label queries is O d ? ln 1? ? ln d + ln 1? + ln ln 1? .
. Additionally ? ?
? ?
?2
3. The number of unlabeled examples drawn is O d ? ln d + ln 1? + ln ln 1? ?
?
?
?2
4. The algorithm runs in time O d2 ? ln d + ln 1? + ln ln 1? ?
7
1
?
?
ln 1? .
1
?
?
ln 1? .
The theorem follows from Lemma 3 below, whose proof is similar to Lemma 2.
Lemma 3 (M ODIFIED -P ERCEPTRON under Adversarial Noise). Suppose Algorithm 2 has inputs
labeling oracle O that satis?es ?-adversarial noise condition with respect to halfspace u, initial
halfspace w0 and angle upper bound ? ? (0, ?2 ] such?that ?(w0 , ?
u) ? ?, con?dence ?, number of
iterations m = ?(d(ln d + ln 1? )), band width b = ?
?
?
d ln(m/?)
Then with probability at least 1 ? ?:
?
. Additionally ? ? ?( ln(m/?))
).
1. The output halfspace wm is such that ?(wm , u) ? ?2 .
? ?
??
2. The number of label queries is O d ? ln d + ln 1? .
? ?
?2 ?
3. The number of unlabeled examples drawn is O d ? ln d + ln 1? ? ?1
6
?
?
?2 ?
4. The algorithm runs in time O d2 ? ln d + ln 1? ? ?1 .
Implications to Passive Learning
ACTIVE -P ERCEPTRON can be converted to a passive learning algorithm, PASSIVE -P ERCEPTRON,
for learning homogeneous halfspaces under the uniform distribution over the unit sphere.
PASSIVE -P ERCEPTRON has PAC sample complexities close to the lower bounds under the two
noise models. We give a formal description of PASSIVE -P ERCEPTRON in Appendix B. We give its
formal guarantees in the corollaries below, which are immediate consequences of Theorems 2 and 3.
In the ?-bounded noise model, the sample complexity of PASSIVE -P ERCEPTRON improves over the
O(
1
)
? d (1?2?)4 ) is obtained. The bound
state of the art result of [8], where a sample complexity of O(
?
d
?
has the same dependency on ? and d as the minimax upper bound of ?(
?(1?2?) ) by [49], which is
achieved by a computationally inef?cient ERM algorithm.
Corollary 1 (PASSIVE -P ERCEPTRON under Bounded Noise). Suppose PASSIVE -P ERCEPTRON has
inputs distribution D that satis?es ?-bounded noise condition with respect
to u, initial halfspace v?0 ,
?
d
d
k
target error ?, con?dence ?, sample schedule {mk } where mk = ? (1?2?)
2 (ln (1?2?)2 + ln ? ) ,
?
?
?k
(1?2?)
band width {bk } where bk = ? ?2d ln(km
. Then with probability at least 1 ? ?: (1) The
/?)
k
output
? halfspace
? v is such that err(hv ) ? err(hu )?+ ?; 2(2) The
? number of labeled examples drawn is
d
d
?
?
O
. (3) The algorithm runs in time O
.
3
3
(1?2?) ?
(1?2?) ?
In the ?-adversarial noise model, the sample complexity of PASSIVE -P ERCEPTRON matches the
? d ) obtained in [39]. Same as in active
minimax optimal sample complexity upper bound of ?(
?
learning, our algorithm has a faster running time than [39].
Corollary 2 (PASSIVE -P ERCEPTRON under Adversarial Noise). Suppose PASSIVE -P ERCEPTRON
has inputs distribution D that satis?es ?-adversarial noise condition with respect
to u, initial
?
?
halfspace v0 , target error ?, con?dence ?, sample schedule {mk } where mk = ? d(ln d + ln k? ) ,
?
?
2?k
?
band width {bk } where bk = ?
. Furthermore ? = ?( ln ln 1?+ln d ). Then with
d ln(km /?)
k
?
?
probability at least 1 ? ?: (1) The output
v is such that err(hv ) ? err(h
u ) + ?; (2) The
?
? 2?
? halfspace
d
d
?
?
. (3) The algorithm runs in time O
.
number of labeled examples drawn is O
?
?
Tables 3 and 4 present comparisons between our results and results most closely related to ours.
Acknowledgments. The authors thank Kamalika Chaudhuri for help and support, Hongyang Zhang
for thought-provoking initial conversations, Jiapeng Zhang for helpful discussions, and the anonymous
reviewers for their insightful feedback. Much of this work is supported by NSF IIS-1167157 and
1162581.
8
Table 3: A comparison of algorithms for PAC learning halfspaces under the uniform distribution, in
the ?-bounded noise model.
Algorithm
[8]
ERM [49]
Our Work
Sample Complexity
?
O(
?
O(
1
O(
)
d (1?2?)4
?
Time Complexity
O(
)
d
(1?2?)? )
d
?
O(
(1?2?)3 ? )
1
)
? d (1?2?)4 )
O(
?
superpoly(d, 1? )
d2
1
?
O(
)
3 ?
(1?2?)
?
Table 4: A comparison of algorithms for PAC learning halfspaces under the uniform distribution, in
the ?-adversarial noise model where ? = ?( ln ln 1?+ln d ).
?
Algorithm
[39]
ERM [57]
Our Work
Sample Complexity
? d)
O(
?
? d)
O(
?
? d)
O(
?
Time Complexity
poly(d, 1? )
superpoly(d, 1? )
? d2 )
O(
?
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11
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6,309 | 6,707 | Gradient Descent Can Take Exponential Time to
Escape Saddle Points
Chi Jin
University of California, Berkeley
[email protected]
Simon S. Du
Carnegie Mellon University
[email protected]
Jason D. Lee
University of Southern California
[email protected]
Michael I. Jordan
University of California, Berkeley
[email protected]
Aarti Singh
Carnegie Mellon University
[email protected]
Barnab?s P?czos
Carnegie Mellon University
[email protected]
Abstract
Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random
initialization schemes and non-pathological functions, GD can be signi?cantly
slowed down by saddle points, taking exponential time to escape. On the other
hand, gradient descent with perturbations [Ge et al., 2015, Jin et al., 2017] is not
slowed down by saddle points?it can ?nd an approximate local minimizer in
polynomial time. This result implies that GD is inherently slower than perturbed
GD, and justi?es the importance of adding perturbations for ef?cient non-convex
optimization. While our focus is theoretical, we also present experiments that
illustrate our theoretical ?ndings.
1
Introduction
Gradient Descent (GD) and its myriad variants provide the core optimization methodology in machine
learning problems. Given a function f (x), the basic GD method can be written as:
?
?
x(t+1) ? x(t) ? ??f x(t) ,
(1)
where ? is a step size, assumed ?xed in the current paper. While precise characterizations of the
rate of convergence GD are available for convex problems, there is far less understanding of GD
for non-convex problems. Indeed, for general non-convex problems, GD is only known to ?nd a
stationary point (i.e., a point where the gradient equals zero) in polynomial time [Nesterov, 2013].
A stationary point can be a local minimizer, saddle point, or local maximizer. In recent years, there
has been an increasing focus on conditions under which it is possible to escape saddle points (more
speci?cally, strict saddle points as in De?nition 2.4) and converge to a local minimizer. Moreover,
stronger statements can be made when the following two key properties hold: 1) all local minima
are global minima, and 2) all saddle points are strict. These properties hold for a variety of machine
learning problems, including tensor decomposition [Ge et al., 2015], dictionary learning [Sun et al.,
2017], phase retrieval [Sun et al., 2016], matrix sensing [Bhojanapalli et al., 2016, Park et al., 2017],
matrix completion [Ge et al., 2016, 2017], and matrix factorization [Li et al., 2016]. For these
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
problems, any algorithm that is capable of escaping strict saddle points will converge to a global
minimizer from an arbitrary initialization point.
Recent work has analyzed variations of GD that include stochastic perturbations. It has been shown
that when perturbations are incorporated into GD at each step the resulting algorithm can escape strict
saddle points in polynomial time [Ge et al., 2015]. It has also been shown that episodic perturbations
suf?ce; in particular, Jin et al. [2017] analyzed an algorithm that occasionally adds a perturbation
to GD (see Algorithm 1), and proved that not only does the algorithm escape saddle points in
polynomial time, but additionally the number of iterations to escape saddle points is nearly dimensionindependent1 . These papers in essence provide suf?cient conditions under which a variant of GD
has favorable convergence properties for non-convex functions. This leaves open the question as to
whether such perturbations are in fact necessary. If not, we might prefer to avoid the perturbations if
possible, as they involve additional hyper-parameters. The current understanding of gradient descent
is silent on this issue. The major existing result is provided by Lee et al. [2016], who show that
gradient descent, with any reasonable random initialization, will always escape strict saddle points
eventually?but without any guarantee on the number of steps required. This motivates the following
question:
Does randomly initialized gradient descent generally escape saddle points in polynomial time?
In this paper, perhaps surprisingly, we give a strong negative answer to this question. We show
that even under a fairly natural initialization scheme (e.g., uniform initialization over a unit cube,
or Gaussian initialization) and for non-pathological functions satisfying smoothness properties
considered in previous work, GD can take exponentially long time to escape saddle points and reach
local minima, while perturbed GD (Algorithm 1) only needs polynomial time. This result shows that
GD is fundamentally slower in escaping saddle points than its perturbed variant, and justi?es the
necessity of adding perturbations for ef?cient non-convex optimization.
The counter-example that supports this conclusion is a smooth function de?ned on Rd , where GD
with random initialization will visit the vicinity of d saddle points before reaching a local minimum.
While perturbed GD takes a constant amount of time to escape each saddle point, GD will get closer
and closer to the saddle points it encounters later, and thus take an increasing amount of time to
escape. Eventually, GD requires time that is exponential in the number of saddle points it needs to
escape, thus e?(d) steps.
1.1
Related Work
Over the past few years, there have been many problem-dependent convergence analyses of nonconvex optimization problems. One line of work shows that with smart initialization that is assumed
to yield a coarse estimate lying inside a neighborhood of a local minimum, local search algorithms
such as gradient descent or alternating minimization enjoy fast local convergence; see, e.g., [Netrapalli
et al., 2013, Du et al., 2017, Hardt, 2014, Candes et al., 2015, Sun and Luo, 2016, Bhojanapalli
et al., 2016, Yi et al., 2016, Zhang et al., 2017]. On the other hand, Jain et al. [2017] show that
gradient descent can stay away from saddle points, and provide global convergence rates for matrix
square-root problems, even without smart initialization. Although these results give relatively strong
guarantees in terms of rate, their analyses are heavily tailored to speci?c problems and it is unclear
how to generalize them to a wider class of non-convex functions.
For general non-convex problems, the study of optimization algorithms converging to minimizers
dates back to the study of Morse theory and continuous dynamical systems ([Palis and De Melo, 2012,
Yin and Kushner, 2003]); a classical result states that gradient ?ow with random initialization always
converges to a minimizer. For stochastic gradient, this was shown by Pemantle [1990], although
without explicit running time guarantees. Lee et al. [2016] established that randomly initialized
gradient descent with a ?xed stepsize also converges to minimizers almost surely. However, these
results are all asymptotic in nature and it is unclear how they might be extended to deliver explicit
convergence rates. Moreover, it is unclear whether polynomial convergence rates can be obtained for
these methods.
Next, we review algorithms that can provably ?nd approximate local minimizers in polynomial time.
The classical cubic-regularization [Nesterov and Polyak, 2006] and trust region [Curtis et al., 2014]
1
Assuming that the smoothness parameters (see De?nition 2.1- 2.3) are all independent of dimension.
2
algorithms require access to the full Hessian matrix. A recent line of work [Carmon et al., 2016,
Agarwal et al., 2017, Carmon and Duchi, 2016] shows that the requirement of full Hessian access
can be relaxed to Hessian-vector products, which can be computed ef?ciently in many machine
learning applications. For pure gradient-based algorithms without access to Hessian information,
Ge et al. [2015] show that adding perturbation in each iteration suf?ces to escape saddle points in
polynomial time. When smoothness parameters are all dimension independent, Levy [2016] analyzed
a normalized form of gradient descent with perturbation, and improved the dimension dependence to
O(d3 ). This dependence has been further improved in recent work [Jin et al., 2017] to polylog(d) via
perturbed gradient descent (Algorithm 1).
1.2
Organization
This paper is organized as follows. In Section 2, we introduce the formal problem setting and
background. In Section 3, we discuss some pathological examples and ?un-natural" initialization
schemes under which the gradient descent fails to escape strict saddle points in polynomial time. In
Section 4, we show that even under a fairly natural initialization scheme, gradient descent still needs
exponential time to escape all saddle points whereas perturbed gradient descent is able to do so in
polynomial time. We provide empirical illustrations in Section 5 and conclude in Section 6. We place
most of our detailed proofs in the Appendix.
2
Preliminaries
Let ???2 denote the Euclidean norm of a ?nite-dimensional vector in Rd . For a symmetric matrix A,
let ?A?op denote its operator norm and ?min (A) its smallest eigenvalue. For a function f : Rd ? R,
let ?f (?) and ?2 f (?) denote its gradient vector and Hessian matrix. Let Bx (r) denote the dd
dimensional ?2 ball centered at x with radius r, [?1, 1] denote the d-dimensional cube centered at 0
d
with side-length 2, and B? (x, R) = x + [?R, R] denote the d-dimensional cube centered at x with
side-length 2R. We also use O(?), and ?(?) as standard Big-O and Big-Omega notation, only hiding
absolute constants.
Throughout the paper we consider functions that satisfy the following smoothness assumptions.
De?nition 2.1. A function f (?) is B-bounded if for any x ? Rd :
|f (x) | ? B.
De?nition 2.2. A differentiable function f (?) is ?-gradient Lipschitz if for any x, y ? Rd :
??f (x) ? ?f (y)?2 ? ? ?x ? y?2 .
De?nition 2.3. A twice-differentiable function f (?) is ?-Hessian Lipschitz if for any x, y ? Rd :
? 2
?
?? f (x) ? ?2 f (y)? ? ? ?x ? y? .
2
op
Intuitively, de?nition 2.1 says function value is both upper and lower bounded; de?nition 2.2 and
2.3 state the gradients and Hessians of function can not change dramatically if two points are close
by. De?nition 2.2 is a standard asssumption in the optimization literature, and de?nition 2.3 is also
commonly assumed when studying saddle points and local minima.
Our goal is to escape saddle points. The saddle points discussed in this paper are assumed to be
?strict? [Ge et al., 2015]:
De?nition 2.4. A saddle point x?? is called? an ?-strict saddle point if there exists some ? > 0 such
that ??f (x? )?2 = 0 and ?min ?2 f (x? ) ? ??.
That is, a strict saddle point must have an escaping direction so that the eigenvalue of the Hessian
along that direction is strictly negative. It turns out that for many non-convex problems studied in
machine learning, all saddle points are strict (see Section 1 for more details).
To escape strict saddle points and converge to local minima, we can equivalently study the approximation of second-order stationary points. For ?-Hessian Lipschitz functions, such points are de?ned
as follows by Nesterov and Polyak [2006]:
3
Algorithm 1 Perturbed Gradient Descent [Jin et al., 2017]
1:
2:
3:
4:
5:
6:
7:
8:
Input: x(0) , step size ?, perturbation radius r, time interval tthres , gradient threshold gthres .
tnoise ? ?tthres ? 1.
for t = 1, 2, ? ? ? do
if ??f (xt )?2 ? gthres and t ? tnoise > tthres then
x(t) ? x(t) + ? t , ? t ? unif (B0 (r)), tnoise ? t,
end if
?
?
x(t+1) ? x(t) ? ??f x(t) .
end for
De?nition
2.5.? A point x is a called a second-order stationary point if ??f (x)?2 = 0 and
?
?min ?2 f (x) ? 0. We also de?ne its ?-version, that is,
stationary point
?
? an ?-second-order
?
for some ? > 0, if point x satis?es ??f (x)?2 ? ? and ?min ?2 f (x) ? ? ??.
Second-order stationary points must have a positive semi-de?nite Hessian in additional to a vanishing
gradient. Note if all saddle points x? are strict, then second-order stationary points are exactly
equivalent to local minima.
In this paper, we compare gradient descent and one of its variants?the perturbed gradient descent
algorithm (Algorithm 1) proposed by Jin et al. [2017]. We focus on the case where the step
size satis?es ? < 1/?, which is commonly required for ?nding a minimum even in the convex
setting [Nesterov, 2013].
The following theorem shows that if GD with random initialization converges, then it will converge
to a second-order stationary point almost surely.
Theorem 2.6 ([Lee et al., 2016] ). Suppose that f is ?-gradient Lipschitz, has continuous Hessian,
and step size ? < 1? . Furthermore, assume that gradient descent converges, meaning limt?? x(t)
exists, and the initialization distribution ? is absolutely continuous with respect to Lebesgue measure.
Then limt?? x(t) = x? with probability one, where x? is a second-order stationary point.
The assumption that gradient descent converges holds for many non-convex functions (including
all
?
?
the examples considered in this paper). This assumption is used to avoid the case when ?x(t) ?2 goes
to in?nity, so limt?? x(t) is unde?ned.
Note the Theorem 2.6 only provides limiting behavior without specifying the convergence rate. On
the other hand, if we are willing to add perturbations, the following theorem not only establishes
convergence but also provides a sharp convergence rate:
Theorem 2.7 ([Jin et al., 2017]). Suppose f is B-bounded, ?-gradient Lipschitz, ?-Hessian Lipschitz.
2
For any ? > 0, ? ? ?? , there exists a proper choice of ?, r, tthres , gthres (depending on B, ?, ?, ?, ?)
such that Algorithm 1 will ?nd an ?-second-order stationary point, with at least probability 1 ? ?, in
the following number of iterations:
??
?
?
d?B
?B
4
.
log
O
?2
?2 ?
This theorem states that with proper choice of hyperparameters, perturbed gradient descent can
consistently escape strict saddle points and converge to second-order stationary point in a polynomial
number of iterations.
3
Warmup: Examples with ?Un-natural" Initialization
The convergence result of Theorem 2.6 raises the following question: can gradient descent ?nd a
second-order stationary point in a polynomial number of iterations? In this section, we discuss two
very simple and intuitive counter-examples for which gradient descent with random initialization
requires an exponential number of steps to escape strict saddle points. We will also explain that,
however, these examples are unnatural and pathological in certain ways, thus unlikely to arise in
practice. A more sophisticated counter-example with natural initialization and non-pathological
behavior will be given in Section 4.
4
(a) Negative Gradient Field of f (x) =
x21 ? x22 .
(b) Negative Gradient Field for function de?ned in Equation (2).
Figure 1: If the initialization point is in red rectangle then it takes GD a long time to escape the
neighborhood of saddle point (0, 0).
Initialize uniformly within an extremely thin band. Consider a two-dimensional function f with
a strict saddle point at (0, 0). Suppose that inside the neighborhood U = [?1, 1]2 of the saddle point,
function is locally quadratic f (x1 , x2 ) = x21 ? x22 , For GD with ? = 14 , the update equation can be
written as
(t)
(t)
x
3x
(t+1)
(t+1)
and x2
= 1
= 2 .
x1
2
2
3 ? exp( 1? ) 3 ? exp( 1? )
If we initialize uniformly within [?1, 1] ? [?( 2 )
, (2)
] then GD requires at least
exp( 1? ) steps to get out of neighborhood U , and thereby escape the saddle point. See Figure 1a
for illustration. Note that in this case the initialization region is exponentially thin (only of width
1
2 ? ( 32 )? exp( ? ) ). We would seldom use such an initialization scheme in practice.
Initialize far away. Consider again a two-dimensional function with a strict saddle point at (0, 0).
This time, instead of initializing in a extremely thin band, we construct a very long slope so that a
relatively large initialization region necessarily converges to this extremely thin band. Speci?cally,
consider a function in the domain [??, 1] ? [?1, 1] that is de?ned as follows:
? 2
2
?x1 ? x2 if ? 1 < x1 < 1
(2)
f (x1 , x2 ) = ?4x1 + x22 if x1 < ?2
?
h(x1 , x2 ) otherwise,
where h(x1 , x2 ) is a smooth function connecting region [??, ?2] ? [?1, 1] and [?1, 1] ? [?1, 1]
while making f have continuous second derivatives and ensuring x2 does not suddenly increase when
x1 ? [?2, ?1].2 For GD with ? = 14 , when ?1 < x1 < 1, the dynamics are
(t+1)
x1
(t)
=
x1
2
and
(t+1)
x2
(t)
3x2
,
2
=
and when x1 < ?2 the dynamics are
(t)
x2
.
2
Suppose we initialize uniformly within [?R ? 1, ?R + 1] ? [?1, 1] , for R large. See Figure 1b for an
(t)
illustration. Letting t denote the ?rst time that x1 ? ?1, then approximately we have t ? R and so
1
(t)
(0)
x2 ? x2 ?( 12 )R . From the previous example, we know that if ( 12 )R ? ( 32 )? exp ? , that is R ? exp 1? ,
then GD will need exponential time to escape from the neighborhood U = [?1, 1] ? [?1, 1] of the
saddle point (0, 0). In this case, we require an initialization region leading to a saddle point at distance
R which is exponentially large. In practice, it is unlikely that we would initialize exponentially far
away from the saddle points or optima.
(t+1)
x1
2
(t)
= x1 + 1
and
(t+1)
x2
=
We can construct such a function using splines. See Appendix B.
5
4
Main Result
In the previous section we have shown that gradient descent takes exponential time to escape saddle
points under ?un-natural" initialization schemes. Is it possible for the same statement to hold even
under ?natural? initialization schemes and non-pathological functions? The following theorem
con?rms this:
Theorem 4.1 (Uniform initialization over a unit cube). Suppose the initialization point is uniformly
sampled from [?1, 1]d . There exists a function f de?ned on Rd that is B-bounded, ?-gradient
Lipschitz and ?-Hessian Lipschitz with parameters B, ?, ? at most poly(d) such that:
1. with probability one, gradient descent with step size ? ? 1/? will be ?(1) distance away
from any local minima for any T ? e?(d) .
2. for any ? > 0, with probability 1 ? e?d , perturbed gradient descent (Algorithm 1) will ?nd
a point x such that ?x ? x? ?2 ? ? for some local minimum x? in poly(d, 1? ) iterations.
Remark: As will be apparent in the next section, in the example we constructed, there are 2d
symmetric local minima at locations (?c . . . , ?c), where c is some constant. The saddle points are
of the form (?c, . . . , ?c, 0, . . . , 0). Both algorithms will travel across d neighborhoods of saddle
points before reaching a local minimum. For GD, the number of iterations to escape the i-th saddle
point increases as ?i (? is a multiplicative factor larger than 1), and thus GD requires exponential
time to escape d saddle points. On the other hand, PGD takes about the same number of iterations
to escape each saddle point, and so escapes the d saddle points in polynomial time. Notice that
B, ?, ? = O(poly(d)), so this does not contradict Theorem 2.7.
We also note that in our construction, the local minimizers are outside the initialization region. We
note this is common especially for unconstrained optimization problems, where the initialization
is usually uniform on a rectangle or isotropic Gaussian. Due to isoperimetry, the initialization
concentrates in a thin shell, but frequently the ?nal point obtained by the optimization algorithm is
not in this shell.
It turns out in our construction, the only second-order stationary points in the path are the ?nal local
minima. Therefore, we can also strengthen Theorem 4.1 to provide a negative result for approximating
?-second-order stationary points as well.
Corollary 4.2. Under the same initialization as in Theorem 4.1, there exists a function f satisfying
the requirements of Theorem 4.1 such that for some ? = 1/poly(d), with probability one, gradient
descent with step size ? ? 1/? will not visit any ?-second-order stationary point in T ? e?(d) .
The corresponding positive result that PGD to ?nd ?-second-order stationary point in polynomial
time immediately follows from Theorem 2.7.
The next result shows that gradient descent does not fail due to the special choice of initializing
uniformly in [?1, 1]d . For a large class of initialization distributions ?, we can generalize Theorem
4.1 to show that gradient descent with random initialization ? requires exponential time, and perturbed
gradient only requires polynomial time.
Corollary 4.3. Let B? (z, R) = {z} + [?R, R]d be the ?? ball of radius R centered at z. Then for
any initialization distribution ? that satis?es ?(B? (z, R)) ? 1 ? ? for any ? > 0, the conclusion of
Theorem 4.1 holds with probability at least 1 ? ?.
That is, as long as most of the mass of the initialization distribution ? lies in some ?? ball, a
similar conclusion to that of Theorem 4.1 holds with high probability. This result applies to random
Gaussian initialization, ? = N (0, ? 2 I), with mean 0 and covariance ? 2 I, where ?(B? (0, ? log d)) ?
1 ? 1/poly(d).
4.1
Proof Sketch
In this section we present a sketch of the proof of Theorem 4.1. The full proof is presented in the
Appendix. Since the polynomial-time guarantee for PGD is straightforward to derive from Jin et al.
[2017], we focus on showing that GD needs an exponential number of steps. We rely on the following
key observation.
6
Key observation: escaping two saddle points sequentially.
?
2
2
if
???x1 + Lx2
2
f (x1 , x2 ) = L(x1 ? 2) ? ?x22
if
?
L(x1 ? 2)2 + L(x2 ? 2)2
if
Consider, for L > ? > 0,
x1 ? [0, 1] , x2 ? [0, 1]
x1 ? [1, 3] , x2 ? [0, 1] .
x1 ? [1, 3] , x2 ? [1, 3]
(3)
Note that this function is not continuous. In the next paragraph we will modify it to make it smooth
and satisfy the assumptions of the Theorem but useful intuition is obtained using this discontinuous
function. The function has an optimum at (2, 2) and saddle points at (0, 0) and (2, 0). We call
[0, 1] ? [0, 1] the ?neighborhood
of (0, 0) and [1, 3] ? [0, 1] the neighborhood of (2, 0). Suppose
?
(0) (0)
lies in [0, 1] ? [0, 1]. De?ne t1 = minx(t) ?1 t to be the time of ?rst
the initialization x , y
1
departure from the neighborhood of (0, 0) (thereby escaping the ?rst saddle point). By the dynamics
of gradient descent, we have
(t )
(0)
(t )
(0)
x1 1 = (1 + 2??)t1 x1 , x2 1 = (1 ? 2?L)t1 x2 .
Next we calculate the number of iterations such that x2 ? 1 and the algorithm thus leaves the
neighborhood of the saddle point (2, 0) (thus escaping the second saddle point). Letting t2 =
minx(t) ?1 t, we have:
2
(t )
(0)
x2 1 (1 + 2??)t2 ?t1 = (1 + 2??)t2 ?t1 (1 ? 2?L)t1 x2 ? 1.
We can lower bound t2 by
t2 ?
2?(L + ?)t1 + log( x10 )
2
2??
?
L+?
t1 .
?
The key observation is that the number of steps to escape the second saddle point is
number of steps to escape the ?rst one.
L+?
?
times the
Spline: connecting quadratic regions. To make our function smooth, we create buffer regions
and use splines to interpolate the discontinuous parts of Equation (3). Formally, we consider the
following function, for some ?xed constant ? > 1:
?
?
??x21 + Lx22
if x1 ? [0, ? ] , x2 ? [0, ? ]
?
?
?
?
if x1 ? [?, 2? ] , x2 ? [0, ? ]
?g(x1 , x2 )
(4)
f (x1 , x2 ) = L(x1 ? 4? )2 ? ?x22 ? ?
if x1 ? [2?, 6? ] , x2 ? [0, ? ]
?
?
2
?
L(x
?
4?
)
+
g
(x
)
?
?
if
x
?
[2?,
6?
]
,
x
?
[?,
2?
]
1
1 2
1
2
?
?
?L(x ? 4? )2 + L(x ? 4? )2 ? 2?
if x1 ? [2?, 6? ] , x2 ? [2?, 6? ] ,
1
2
where g, g1 are spline polynomials and ? > 0 is a constant de?ned in Lemma B.2. In this case, there
are saddle points at (0, 0), and (4?, 0) and the optimum is at (4?, 4? ). Intuitively, [?, 2? ] ? [0, ? ] and
[2?, 6? ] ? [?, 2? ] are buffer regions where we use splines (g and g1 ) to transition between regimes
and make f a smooth function. Also in this region there is no stationary point and the smoothness
assumptions are still satis?ed in the theorem. Figure. 2a shows the surface and stationary points of
this function. We call the union of the regions de?ned in Equation (4) a tube.
From two saddle points to d saddle points. We can readily adapt our construction of the tube
to d dimensions, such that the function is smooth, the location of saddle points are (0, . . . , 0),
(4?, 0, . . . , 0), . . ., (4?, . . . , 4?, 0), and optimum is at (4?, . . . , 4? ). Let ti be the number of step to
escape the neighborhood of the i-th saddle point. We generalize our key observation to this case and
L+? d
obtain ti+1 ? L+?
? ? ti for all i. This gives td ? ( ? ) which is exponential time. Figure 2b shows
the tube and trajectory of GD.
Mirroring trick: from tube to octopus. In the construction thus far, the saddle points are all on
the boundary of tube. To avoid the dif?culties of constrained non-convex optimization, we would like
to make all saddle points be interior points of the domain. We use a simple mirroring trick; i.e., for
every coordinate xi we re?ect f along its axis. See Figure 2c for an illustration in the case d = 2.
7
-90
15
-10
-10
0
-9
0
-110
0
-1
20
x2
10
-9
0
-80
-70
5
-110
-100
-6
0
10
-1
0
-10
-90
-80
0
-7
-5
0
-6
-5
0
-50
0
5
0
0
5
10
15
x1
(a) Contour plot of the objective
function and tube de?ned in 2D.
(b) Trajectory of gradient descent
in the tube for d = 3.
(c) Octopus de?ned in 2D.
Figure 2: Graphical illustrations of our counter-example with ? = e. The blue points are saddle
points and the red point is the minimum. The pink line is the trajectory of gradient descent.
-300
0
-200
-400
-600
0
500
1000
0
500
0
-200
-400
-600
1000
GD
PGD
0
500
Epochs
Epochs
(a) L = 1, ? = 1
Objective Function
-200
200
GD
PGD
Objective Function
-100
Objective Function
Objective Function
GD
PGD
0
-400
200
200
100
-200
-400
-600
-800
1000
GD
PGD
0
0
500
Epochs
(b) L = 1.5, ? = 1
1000
Epochs
(c) L = 2, ? = 1
(d) L = 3, ? = 1
Figure 3: Performance of GD and PGD on our counter-example with d = 5.
-400
-600
-800
0
1000
2000
Epochs
(a) L = 1, ? = 1
Objective Function
-200
500
GD
PGD
0
-500
-1000
0
1000
-500
-1000
-1500
2000
Epochs
500
GD
PGD
0
0
1000
Epochs
(b) L = 1.5, ? = 1
(c) L = 2, ? = 1
2000
Objective Function
500
GD
PGD
0
Objective Function
Objective Function
200
GD
PGD
0
-500
-1000
-1500
0
1000
2000
Epochs
(d) L = 3, ? = 1
Figure 4: Performance of GD and PGD on our counter-example with d = 10
Extension: from octopus to Rd . Up to now we have constructed a function de?ned on a closed
subset of Rd . The last step is to extend this function to the entire Euclidean space. Here we apply the
classical Whitney Extension Theorem (Theorem B.3) to ?nish our construction. We remark that the
Whitney extension may lead to more stationary points. However, we will demonstrate in the proof
that GD and PGD stay within the interior of ?octopus? de?ned above, and hence cannot converge to
any other stationary point.
5
Experiments
In this section we use simulations to verify our theoretical ?ndings. The objective function is de?ned
in (14) and (15) in the Appendix. In Figures 3 and Figure 4, GD stands for gradient descent and
1
PGD stands for Algorithm 1. For both GD and PGD we let the stepsize ? = 4L
. For PGD, we
?e
e
choose tthres = 1, gthres = 100
and r = 100
. In Figure 3 we ?x dimension d = 5 and vary L as
considered in Section 4.1; similarly in Figure 4 we choose d = 10 and vary L. First notice that in
all experiments, PGD converges faster than GD as suggested by our theorems. Second, observe the
?horizontal" segment in each plot represents the number of iterations to escape a saddle point. For
GD the length of the segment grows at a ?xed rate, which coincides with the result mentioned at the
beginning for Section 4.1 (that the number of iterations to escape a saddle point increase at each time
with a multiplicative factor L+?
? ). This phenomenon is also veri?ed in the ?gures by the fact that as
the ratio L+?
? becomes larger, the rate of growth of the number of iterations to escape increases. On
1
the other hand, the number of iterations for PGD to escape is approximately constant (? ??
).
8
6
Conclusion
In this paper we established the failure of gradient descent to ef?ciently escape saddle points for
general non-convex smooth functions. We showed that even under a very natural initialization scheme,
gradient descent can require exponential time to converge to a local minimum whereas perturbed
gradient descent converges in polynomial time. Our results demonstrate the necessity of adding
perturbations for ef?cient non-convex optimization.
We expect that our results and constructions will naturally extend to a stochastic setting. In particular,
we expect that with random initialization, general stochastic gradient descent will need exponential
time to escape saddle points in the worst case. However, if we add perturbations per iteration or
the inherent randomness is non-degenerate in every direction (so the covariance of noise is lower
bounded), then polynomial time is known to suf?ce [Ge et al., 2015].
One open problem is whether GD is inherently slow if the local optimum is inside the initialization
region in contrast to the assumptions of initialization we used in Theorem 4.1 and Corollary 4.3. We
believe that a similar construction in which GD goes through the neighborhoods of d saddle points
will likely still apply, but more work is needed. Another interesting direction is to use our counterexample as a building block to prove a computational lower bound under an oracle model [Nesterov,
2013, Woodworth and Srebro, 2016].
This paper does not rule out the possibility for gradient descent to perform well for some non-convex
functions with special structures. Indeed, for the matrix square-root problem, Jain et al. [2017] show
that with reasonable random initialization, gradient updates will stay away from all saddle points,
and thus converge to a local minimum ef?ciently. It is an interesting future direction to identify other
classes of non-convex functions that gradient descent can optimize ef?ciently and not suffer from the
negative results described in this paper.
7
Acknowledgements
S.S.D. and B.P. were supported by NSF grant IIS1563887 and ARPA-E Terra program. C.J. and
M.I.J. were supported by the Mathematical Data Science program of the Of?ce of Naval Research
under grant number N00014-15-1-2670. J.D.L. was supported by ARO W911NF-17-1-0304. A.S.
was supported by DARPA grant D17AP00001, AFRL grant FA8750-17-2-0212 and a CMU ProSEED/BrainHub Seed Grant. The authors thank Rong Ge, Qing Qu, John Wright, Elad Hazan, Sham
Kakade, Benjamin Recht, Nathan Srebro, and Lin Xiao for useful discussions. The authors thank
Stephen Wright and Michael O?Neill for pointing out calculation errors in the older version.
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6,310 | 6,708 | Union of Intersections (UoI) for Interpretable Data
Driven Discovery and Prediction
Kristofer E. Bouchard?
Alejandro F. Bujan?
Shashanka Ubaru?
Prabhat?
Edward F. Chang??
Michael W. Mahoney?
Farbod Roosta-Khorasani?
Antoine M. Snijdersk
Jian-Hua Maok
Sharmodeep Bhattacharyya??
Abstract
The increasing size and complexity of scientific data could dramatically enhance
discovery and prediction for basic scientific applications. Realizing this potential,
however, requires novel statistical analysis methods that are both interpretable
and predictive. We introduce Union of Intersections (UoI), a flexible, modular,
and scalable framework for enhanced model selection and estimation. Methods
based on UoI perform model selection and model estimation through intersection
and union operations, respectively. We show that UoI-based methods achieve
low-variance and nearly unbiased estimation of a small number of interpretable
features, while maintaining high-quality prediction accuracy. We perform extensive
numerical investigation to evaluate a UoI algorithm (U oILasso ) on synthetic and
real data. In doing so, we demonstrate the extraction of interpretable functional
networks from human electrophysiology recordings as well as accurate prediction
of phenotypes from genotype-phenotype data with reduced features. We also show
(with the U oIL1Logistic and U oICU R variants of the basic framework) improved
prediction parsimony for classification and matrix factorization on several benchmark biomedical data sets. These results suggest that methods based on the UoI
framework could improve interpretation and prediction in data-driven discovery
across scientific fields.
1
Introduction
A central goal of data-driven science is to identify a small number of features (i.e., predictor variables;
X in Fig. 1(a)) that generate a response variable of interest (y in Fig. 1(a)) and then to estimate
the relative contributions of these features as the parameters in the generative process relating the
predictor variables to the response variable (Fig. 1(a)). A common characteristic of many modern
massive data sets is that they have a large number of features (i.e., high-dimensional data), while
?
Biological Systems and Engineering Division, LBNL. [email protected]
Redwood Center, UC Berkeley. [email protected]
?
ICSI and Department of Statistics, UC Berkeley. {farbod,mmahoney}@icsi.berkeley.edu
?
Department of Computer Science and Engineering, University of Minnesota. [email protected]
?
NERSC, LBNL. [email protected]
k
Biological Systems and Engineering Division, LBNL. {AMSnijders,jhmao}@lbl.gov
??
Department of Neurological Surgery, UC San Francisco. [email protected]
??
Department of Statistics, Oregon State University. [email protected]
?
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: The basic UoI framework. (a) Schematic of regularization and ensemble methods for
regression. (b) Schematic of the Union of Intersections (UoI) framework. (c) A data-distributed
version of the U oILasso algorithm. (d) Dependence of false positive, false negatives, and estimation
variability on number of bootstraps in selection (B1 ) and estimation (B2 ) modules.
also exhibiting a high degree of sparsity and/or redundancy [2, 19, 11]. That is, while formally
high-dimensional, most of the useful information in the data features for tasks such as reconstruction,
regression, and classification can be restricted or compressed into a much smaller number of important
features. In regression and classification, it is common to employ sparsity-inducing regularization
to attempt to achieve simultaneously two related but quite different goals: to identify the features
important for prediction (i.e., model selection) and to estimate the associated model parameters
(i.e., model estimation) [2, 19]. For example, the Lasso algorithm in linear regression uses L1 regularization to penalize the total magnitude of model parameters, and this often results in feature
compression by setting some parameters exactly to zero [18] (See Fig. 1(a), pure white elements
in right-hand vectors, emphasized by ?). It is well known that this type of regularization implies a
prior assumption about the distribution of the parameter (e.g., L1 -regularization implicitly assumes
a Laplacian prior distribution) [12]. However, strong sparsity-inducing regularization, which is
common when there are many more potential features than data samples (i.e., the so-called small
n/p regime) can severely hinder the interpretation of model parameters (Fig. 1(a), indicated by less
saturated colors between top and bottom vectors on right hand side). For example, while sparsity may
be achieved, incorrect features may be chosen and parameters estimates may be biased. In addition,
it can impede model selection and estimation when the true model distribution deviates from the
assumed distribution [2, 10]. This may not matter for prediction quality, but it clearly has negative
consequences for interpretability, an admittedly not completely-well-defined property of algorithms
that is crucial in many scientific applications [9]. In this context, interpretability reflects the degree to
which an algorithm returns a small number of physically meaningful features with unbiased and low
variance estimates of their contributions.
On the other hand, another common characteristic of many state of the art methods is to combine
several related models for a given task. In statistical data analysis, this is often formalized by so-called
2
ensemble methods, which improve prediction accuracy by combining parameter estimates [12]. In
particular, by combining several different models, ensemble methods often include more features
to predict the response variables, and thus the number of data features is expanded relative to the
individuals in the ensemble. For example, estimating an ensemble of model parameters by randomly
resampling the data many times (e.g., bootstrapping) and then averaging the parameter estimates
(e.g., bagging) can yield improved prediction accuracy by reducing estimation variability [8, 12] (See
Fig. 1(a), bottom). However, by averaging estimates from a large ensemble, this process often results
in many non-zero parameters, which can hinder interpretability and the identification of the true
model support (compare top and bottom vectors on right hand side of Fig. 1(a)). Taken together, these
observations suggest that explicit and more precise control of feature compression and expansion
may result in an algorithm with improved interpretative and predictive properties.
In this paper, we introduce Union of Intersections (UoI), a flexible, modular, and scalable framework
to enhance both the identification of features (model selection) as well as the estimation of the
contributions of these features (model estimation). We have found that the UoI framework permits us
to explore the interpretability-predictivity trade-off space, without imposing an explicit prior on the
model distribution, and without formulating a non-convex problem, thereby often leading to improved
interpretability and prediction. Ideally, data analysis methods in many scientific applications should
be selective (only features that influence the response variable are selected), accurate (estimated
parameters in the model are as close to the true value as possible), predictive (allowing prediction of
the response variable), stable (e.g., the variability of the estimated parameters is small), and scalable
(able to return an answer in a reasonable amount of time on very large data sets) [17, 2, 15, 10]. We
show empirically that UoI-based methods can simultaneously achieve these goals, results supported
by preliminary theory. We primarily demonstrate the power of UoI-based methods in the context of
sparse linear regression (U oILasso ), as it is the canonical statistical/machine learning problem, it is
theoretically tractable, and it is widely used in virtually every field of scientific inquiry. However, our
framework is very general, and we demonstrate this by extending UoI to classification (U oIL1Logistic )
and matrix factorization (U oICU R ) problems. While our main focus is on neuroscience (broadly
speaking) applications, our results also highlight the power of UoI across a broad range of synthetic
and real scientific data sets.1
2
Union of Intersections (UoI)
For concreteness, we consider an application of UoI in the context of the linear regression. Specifically,
we consider the problem of estimating the parameters ? ? Rp that map a p-dimensional vector of
predictor variables x ? Rp to the observation variable y ? R, when there are n paired samples of x
and y corrupted by i.i.d Gausian noise:
y = ? T x + ?,
(1)
where ? ? N (0, ? ) for each sample. When the true ? is thought to be sparse (i.e., in the L0 -norm
? can be found by solving a constrained optimization problem
sense), then an estimate of ? (call it ?)
of the form:
n
X
?? ? argmin
(yi ? ?xi )2 + ?R(?).
(2)
p
iid
2
??R
i=1
Here, R(?) is a regularization term that typically penalizes the overall magnitude of the parameter
vector ? (e.g., R(?) = k?k1 is the target of the Lasso algorithm).
The Basic UoI Framework. The key mathematical idea underlying UoI is to perform model selection
through intersection (compressive) operations and model estimation through union (expansive)
operations, in that order. This is schematized in Fig. 1(b), which plots a hypothetical range of selected
1
More details, including both empirical and theoretical results, are in the associated technical report [4].
3
features (x1 : xp , abscissa) for different values of the regularization parameter (?, ordinate). See [4]
for a more detailed description. In particular, UoI first performs feature compression (Fig. 1(b), Step
1) through intersection operations (intersection of supports across bootstrap samples) to construct a
family (S) of candidate model supports (Fig. 1(b), e.g., Sj?1 , opaque red region is intersection of
abutting pink regions). UoI then performs feature expansion (Fig. 1(b), Step 2) through a union of
(potentially) different model supports: for each bootstrap sample, the best model estimates (across
different supports) is chosen, and then a new model is generated by averaging the estimates (i.e.,
taking the union) across bootstrap samples (Fig. 1(b), dashed vertical black line indicates the union
of features from Sj and Sj+1 ). Both feature compression and expansion are performed across all
regularization strengths. In UoI, feature compression via intersections and feature expansion via
unions are balanced to maximize prediction accuracy of the sparsely estimated model parameters for
the response variable y.
Innovations in Union of Intersections. UoI has three central innovations: (1) calculate model
supports (Sj ) using an intersection operation for a range of regularization parameters (increases
in ? shrink all values ?? towards 0), efficiently constructing a family of potential model supports
{S : Sj ? Sj?k , for k sufficiently large}; (2) use a novel form of model averaging in the union
step to directly optimize prediction accuracy (this can be thought of as a hybrid of bagging [8]
and boosting [16]); and (3) combine pure model selection using an intersection operation with
model selection/estimation using a union operation in that order (which controls both false negatives
and false positives in model selection). Together, these innovations often lead to better selection,
estimation and prediction accuracy. Importantly, this is done without explicitly imposing a prior on
the distribution of parameter values, and without formulating a non-convex optimization problem.
The U oILasso Algorithm. Since the basic UoI framework, as described in Fig. 1(c), has two main
computational modules?one for model selection, and one for model estimation?UoI is a framework
into which many existing algorithms can be inserted. Here, for simplicity, we primarily demonstrate
UoI in the context of linear regression in the U oILasso algorithm, although we also apply it to
classification with the U oIL1Logistic algorithm as well as matrix factorization with the U oICU R
algorithm. U oILasso expands on the BoLasso method for the model selection module [1], and it
performs a novel model averaging in the estimation module based on averaging ordinary least squares
(OLS) estimates with potentially different model supports. U oILasso (and UoI in general) has a
high degree of natural algorithmic parallelism that we have exploited in a distributed Python-MPI
implementation. (Fig. 1(c) schematizes a simplified distributed implementation of the algorithm;
see [4] for more details.) This parallelized U oILasso algorithm uses distribution of bootstrap data
samples and regularization parameters (in Map) for independent computations involving convex
optimizations (Lasso and OLS, in Solve), and it then combines results (in Reduce) with intersection
operations (model selection module) and union operations (model estimation module). By solving
independent convex optimization problems (e.g., Lasso, OLS) with distributed data resampling, our
U oILasso algorithm efficiently constructs a family of model supports, and it then averages nearly
unbiased model estimates, potentially with different supports, to maximize prediction accuracy while
minimizing the number of features to aid interpretability.
3
3.1
Results
Methods
All numerical results used 100 random sub-samplings with replacement of 80-10-10 cross-validation
to estimate model parameters (80%), choose optimal meta-parameters (e.g., ?, 10%), and determine
prediction quality (10%). Below, ? denotes the values of the true model parameters, ?? denotes the
estimated values of the model parameters from some algorithm (e.g., U oILasso ), S? is the support of
4
the true model (i.e., the set of non-zero parameter indices), and S?? is the support of the estimated
model. We calculated several metrics of model selection, model estimation, and prediction accuracy.
|S ? ?S? |
(1) Selection accuracy (set overlap): 1? |S ? |?0 +|S? |0 , where, ? is the symmetric set difference operator.
?
This metric ranges in [0, 1], taking a value of 0 if S? and S?? have no elements in common, and
q P
2
(?i ? ??i ) .
taking a value of 1 if and only if they are identical. (2) Estimation error (r.m.s): p1
? 2 . (4) Prediction accuracy (R2 ):
(3)
Estimation variability (parameter variance): E[??2 ] ? (E[?])
P
Pn
(yi ?y?i )2
1
? 0 log(n). For the
P
(yi ? y?i )2 ) + k?k
2 . (5) Prediction parsimony (BIC): n log(
n?1
(yi ?E[y])
i=1
?
0
experimental data, as the true model size is unknown, the selection ratio ( k?k
p ) is a measure of the
overall size of the estimated model relative to the total number of parameters. For the classification
task using U oIL1Logistic , BIC was calculated as: ?2 log ` + S?? log N , where ` is the log-likelihood
on the validation set. For the matrix factorization task using U oICU R , reconstruction accuracy was
the Frobenius norm of the difference between the data matrix A and the low-rank approximation
matrix A0 constructed from A(:, c), the reduced column matrix of A: kA ? A0 kF , where c is the set
of k selected columns.
3.2
Model Selection and Stability: Explicit Control of False Positives, False Negatives, and
Estimate Stability
Due to the form of the basic UoI framework, we can control both false negative and false positive
discoveries, as well as the stability of the estimates. For any regularized regression method like
in (2), a decrease in the penalization parameter (?) tends to increase the number of false positives,
and an increase in ? tends to increase false negatives. Preliminary analysis of the UoI framework
shows that, for false positives, a large number of bootstrap resamples in the intersection step (B1 )
produces an increase in the probability of getting no false positive discoveries, while an increase in
the number of bootstraps in the union step (B2 ) leads to a decrease in the probability of getting no
false positives. Conversely, for false negatives, a large number of bootstrap resamples in the union
step (B2 ) produces an increase in the probability of no false negative discoveries, while an increase
in the number of bootstraps in the intersection step (B1 ) leads to a decrease in the probability of no
false negatives. Also, a large number of bootstrap samples in union step (B2 ) gives a more stable
estimate. These properties were confirmed numerically for U oILasso and are displayed in Fig. 1(d),
which plots the average normalized false negatives, false positives, and standard deviation of model
estimates from running U oILasso , with ranges of B1 and B2 on four different models. These results
are supported by preliminary theoretical analysis of a variant of U oILasso (see [4]). Thus, the relative
values of B1 and B2 express the fundamental balance between the two basic operations of intersection
(which compresses the feature space) and union (which expands the feature space). Model selection
through intersection often excludes true parameters (i.e., false negatives), and, conversely, model
estimation using unions often includes erroneous parameters (i.e., false positives). By using stochastic
resampling, combined with model selection through intersections, followed by model estimation
through unions, UoI permits us to mitigate the feature inclusion/exclusion inherent in either operation.
Essentially, the limitations of selection by intersection are counteracted by the union of estimates,
and vice versa.
3.3
U oILasso has Superior Performance on Simulated Data Sets
To explore the performance of the U oILasso algorithm, we have performed extensive numerical
investigations on simulated data sets, where we can control key properties of the data. There are a
large number of algorithms available for linear regression, and we picked some of the most popular
algorithms (e.g., Lasso), as well as more uncommon, but more powerful algorithms (e.g., SCAD, a
non-convex method). Specifically, we compared U oILasso to five other model selection/estimation
5
Figure 2: Range of observed results, in comparison with existing algorithms. (a) True ? distribution (grey histograms) and estimated values (colored lines). (b) Scatter plot of true and estimated
values of observation variable on held-out samples. (c) Metrics of algorithm performance.
methods: Ridge, Lasso, SCAD, BoATS, and debiased Lasso [12, 18, 10, 5, 3, 13]. Note that BoATS
and debiased Lasso are both two-stage methods. We examined performance of these algorithms
across a variety of underlying distributions of model parameters, degrees of sparsity, and noise
levels. Across all algorithms examined, we found that U oILasso (Fig. 2, black) generally resulted
in very high selection accuracy (Fig. 2(c), right) with parameter estimates with low error (Fig. 2(c),
center-right), leading to the best prediction accuracy (Fig. 2(c), center-left) and prediction parsimony
(Fig. 2(c), left). In addition, it was very robust to differences in underlying parameter distribution,
degree of sparsity, and magnitude of noise. (See [4] for more details.)
U oILasso in Neuroscience: Sparse Functional Networks from Human Neural
Recordings and Parsimonious Prediction from Genetic and Phenotypic Data
We sought to determine if the enhanced selection and estimation properties of U oILasso also improved
its utility as a tool for data-driven discovery in complex, diverse neuroscience data sets. Neurobiology
seeks to understand the brain across multiple spatio-temporal scales, from molecules-to-minds.
We first tackled the problem of graph formation from multi-electrode (p = 86 electrodes) neural
recordings taken directly from the surface of the human brain during speech production (n = 45 trials
each). See [7] for details. That is, the goal was to construct sparse neuroscientifically-meaningful
graphs for further downstream analysis. To estimate functional connectivity, we calculated partial
correlation graphs. The model was estimated independently for each electrode, and we compared
the results of graphs estimated by U oILasso to the graphs estimated by SCAD. In Fig. 3(a)-(b), we
display the networks derived from recordings during the production of /b/ while speaking /ba/. We
found that the U oILasso network (Fig. 3(a)) was much sparser than the SCAD network (Fig. 3(b)).
Furthermore, the network extracted by U oILasso contained electrodes in the lip (dorsal vSMC), jaw
(central vSMC), and larynx (ventral vSMC) regions, accurately reflecting the articulators engaged
in the production of /b/ (Fig. 3(c)) [7]. The SCAD network (Fig. 3(d)) did not have any of these
properties. This highlights the improved power of U oILasso to extract sparse graphs with functionally
meaningful features relative to even some non-convex methods.
3.4
We calculated connectivity graphs during the production of 9 consonant-vowel syllables. Fig. 3(e)
displays a summary of prediction accuracy for U oILasso networks (red) and SCAD networks (black)
6
Figure 3: Application of UoI to neuroscience and genetics data. (a)-(f): Functional connectivity
networks from ECoG recordings during speech production. (g)-(h): Parsimonious prediction of
complex phenotypes form genotype and phenotype data.
as a function of time. The average relative prediction accuracy (compared to baseline times) for the
U oILasso network was generally greater during the time of peak phoneme encoding [T = -100:200]
compared to the SCAD network. Fig. 3(f) plots the time course of the parameter selection ratio for
the U oILasso network (red) and SCAD network (black). The U oILasso network was consistently
? 5? sparser than the SCAD network. These results demonstrate that U oILasso extracts sparser
graphs from noisy neural signals with a modest increase in prediction accuracy compared to SCAD.
We next investigated whether U oILasso would improve the identification of a small number of highly
predictive features from genotype-phenotype data. To do so, we analyzed data from n = 365 mice
(173 female, 192 male) that are part of the genetically diverse Collaborative Cross cohort. We
analyzed single-nucleotide polymorphisms (SNPs) from across the entire genome of each mouse
(p = 11, 563 SNPs). For each animal, we measured two continuous, quantitative phenotypes: weight
and behavioral performance on the rotorod task (see [14] for details). We focused on predicting
these phenotypes from a small number of geneotype-phenotype features. We found that U oILasso
identified and estimated a small number of features that were sufficient to explain large amounts
of variability in these complex behavioral and physiological phenotypes. Fig. 3(g) displays the
non-zero values estimated for the different features (e.g., location of loci on the genome) contributing
to the weight (black) and speed (red) phenotype. Here, non-opaque points correspond to the mean
? s.d. across cross-validation samples, while the opaque points are the medians. Importantly, for
both speed and weight phenotypes, we confirmed that several identified predictor features had been
reported in the literature, though by different studies, e.g., genes coding for Kif1b, Rrm2b/Ubr5,
and Dloc2. (See [4] for more details.) Accurate prediction of phenotypic variability with a small
number of factors was a unique property of models found by U oILasso . For both weight and rotorod
performance, models fit by U oILasso had marginally increased prediction accuracy compared to
other methods (+1%), but they did so with far fewer parameters (lower selection ratios). This results
in prediction parsimony (BIC) that was several orders of magnitude better (Fig. 3(h)). Together, these
results demonstrate that U oILasso can identify a small number of genetic/physiological factors that
are highly predictive of complex physiological and behavioral phenotypes.
7
Figure 4: Extension of UoI to classification and matrix decomposition. (a) UoI for classification
(U oIL1Logistic ). (b) UoI for matrix decomposition (U oICU R ); solid and dashed lines are for PAH
and dashed SORCH data sets, respectively.
3.5
U oIL1Logistic and U oICU R : Application of UoI to Classification and Matrix
Decomposition
As noted, UoI is is a framework into which other methods can be inserted. While we have primarily
demonstrated UoI in the context of linear regression, it is much more general than that. To illustrate
this, we implemented a classification algorithm (U oIL1Logistic ) and matrix decomposition algorithm
(U oICU R ), and we compared them to the base methods on several data sets (see [4] for details).
In classification, UoI resulted in either equal or improved prediction accuracy with 2x-10x fewer
parameters for a variety of biomedical classification tasks (Fig. 4(a)). For matrix decomposition
(in this case, column subset selection), for a given dimensionality, UoI resulted in reconstruction
errors that were consistently lower than the base method (BasicCUR), and quickly approached an
unscalable greedy algorithm (GreedyCUR) for two genetics data sets (Fig. 4(b)). In both cases, UoI
improved the prediction parsimony relative to the base (classification or decomposition) method.
4
Discussion
UoI-based methods leverage stochastic data resampling and a range of sparsity-inducing regularization
parameters/dimensions to build families of potential features, and they then average nearly unbiased
parameter estimates of selected features to maximize predictive accuracy. Thus, UoI separates model
selection with intersection operations from model estimation with union operations: the limitations of
selection by intersection are counteracted by the union of estimates, and vice versa. Stochastic data
resampling can be a viewed as a perturbation of the data, and UoI efficiently identifies and robustly
estimates features that are stable to these perturbations. A unique property of UoI-based methods is
the ability to control both false positives and false negatives. Initial theoretical work (see [4]) shows
that increasing the number of bootstraps in the selection module (B1 ) increases the amount of feature
compression (primary controller of false positives), while increasing the number of bootstraps in
the estimation module (B2 ) increases feature expansion (primary controller of false negatives), and
we observe this empirically. Thus, neither should be too large, and their relative values express the
balance between feature compression and expansion. This tension is seen in many places in machine
learning and data analysis: local nearest neighbor methods vs. global latent factor models; local
spectral methods that tend to expand due to their diffusion-based properties vs. flow-based methods
that tend to contract; and sparse L1 vs. dense L2 penalties/priors more generally. Interestingly, an
analogous balance of compressive and expansive forces contributes to neural leaning algorithms
based on Hebbian synaptic plasticity [6]. Our results highlight how revisiting popular methods in
light of new data science demands can lead to still further-improved methods, and they suggest
several directions for theoretical and empirical work.
8
References
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[13] A. Javanmard and A. Montanari. Confidence intervals and hypothesis testing for high-dimensional
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[16] R. E. Schapire and Y. Freund. Boosting: Foundations and Algorithms. MIT Press, Cambridge, MA, 2012.
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9
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6,311 | 6,709 | One-Shot Imitation Learning
Yan Duan?? , Marcin Andrychowicz? , Bradly Stadie?? , Jonathan Ho?? ,
Jonas Schneider? , Ilya Sutskever? , Pieter Abbeel?? , Wojciech Zaremba?
?
Berkeley AI Research Lab, ? OpenAI
?
Work done while at OpenAI
{rockyduan, jonathanho, pabbeel}@eecs.berkeley.edu
{marcin, bstadie, jonas, ilyasu, woj}@openai.com
Abstract
Imitation learning has been commonly applied to solve different tasks in isolation.
This usually requires either careful feature engineering, or a significant number of
samples. This is far from what we desire: ideally, robots should be able to learn
from very few demonstrations of any given task, and instantly generalize to new
situations of the same task, without requiring task-specific engineering. In this
paper, we propose a meta-learning framework for achieving such capability, which
we call one-shot imitation learning.
Specifically, we consider the setting where there is a very large (maybe infinite)
set of tasks, and each task has many instantiations. For example, a task could be
to stack all blocks on a table into a single tower, another task could be to place
all blocks on a table into two-block towers, etc. In each case, different instances
of the task would consist of different sets of blocks with different initial states.
At training time, our algorithm is presented with pairs of demonstrations for a
subset of all tasks. A neural net is trained such that when it takes as input the first
demonstration demonstration and a state sampled from the second demonstration,
it should predict the action corresponding to the sampled state. At test time, a full
demonstration of a single instance of a new task is presented, and the neural net
is expected to perform well on new instances of this new task. Our experiments
show that the use of soft attention allows the model to generalize to conditions and
tasks unseen in the training data. We anticipate that by training this model on a
much greater variety of tasks and settings, we will obtain a general system that can
turn any demonstrations into robust policies that can accomplish an overwhelming
variety of tasks.
1
Introduction
We are interested in robotic systems that are able to perform a variety of complex useful tasks, e.g.
tidying up a home or preparing a meal. The robot should be able to learn new tasks without long
system interaction time. To accomplish this, we must solve two broad problems. The first problem is
that of dexterity: robots should learn how to approach, grasp and pick up complex objects, and how
to place or arrange them into a desired configuration. The second problem is that of communication:
how to communicate the intent of the task at hand, so that the robot can replicate it in a broader set of
initial conditions.
Demonstrations are an extremely convenient form of information we can use to teach robots to overcome these two challenges. Using demonstrations, we can unambiguously communicate essentially
any manipulation task, and simultaneously provide clues about the specific motor skills required to
perform the task. We can compare this with an alternative form of communication, namely natural
language. Although language is highly versatile, effective, and efficient, natural language processing
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Policy for task F
Many
demonstrations
Imitation
Learning
Algorithm
Policy for
task A
Task A
e.g. stack blocks into towers of height 3
One-Shot Imitator
(Neural Network)
Meta Learning
Algorithm
obs
action
Single demonstration
for task F
Many demonstrations
for task A
Many demonstrations
for task B
obs
action
?
more demonstrations for
more tasks
Environment
Environment
(b) One-Shot Imitation Learning
Demo1
Many
demonstrations
Imitation
Learning
Algorithm
sample
Policy for
task B
action
Task B
e.g. stack blocks into towers of height 2
Many demonstrations
for task A
One-Shot Imitator
(Neural Network)
obs
Many demonstrations
for task B
predicted
action
observation from
Demo2
Environment
corresponding
action in Demo2
Supervised loss
(a) Traditional Imitation Learning
(c) Training the One-Shot Imitator
Figure 1: (a) Traditionally, policies are task-specific. For example, a policy might have been
trained through an imitation learning algorithm to stack blocks into towers of height 3, and then
another policy would be trained to stack blocks into towers of height 2, etc. (b) In this paper, we
are interested in training networks that are not specific to one task, but rather can be told (through a
single demonstration) what the current new task is, and be successful at this new task. For example,
when it is conditioned on a single demonstration for task F, it should behave like a good policy for
task F. (c) We can phrase this as a supervised learning problem, where we train this network on a set
of training tasks, and with enough examples it should generalize to unseen, but related tasks. To train
this network, in each iteration we sample a demonstration from one of the training tasks, and feed it
to the network. Then, we sample another pair of observation and action from a second demonstration
of the same task. When conditioned on both the first demonstration and this observation, the network
is trained to output the corresponding action.
systems are not yet at a level where we could easily use language to precisely describe a complex
task to a robot. Compared to language, using demonstrations has two fundamental advantages: first,
it does not require the knowledge of language, as it is possible to communicate complex tasks to
humans that don?t speak one?s language. And second, there are many tasks that are extremely difficult
to explain in words, even if we assume perfect linguistic abilities: for example, explaining how to
swim without demonstration and experience seems to be, at the very least, an extremely challenging
task.
Indeed, learning from demonstrations have had many successful applications . However, so far
these applications have either required careful feature engineering, or a significant amount of system
interaction time. This is far from what what we desire: ideally, we hope to demonstrate a certain task
only once or a few times to the robot, and have it instantly generalize to new situations of the same
task, without long system interaction time or domain knowledge about individual tasks.
In this paper we explore the one-shot imitation learning setting illustrated in Fig. 1, where the
objective is to maximize the expected performance of the learned policy when faced with a new,
previously unseen, task, and having received as input only one demonstration of that task. For the
tasks we consider, the policy is expected to achieve good performance without any additional system
interaction, once it has received the demonstration.
We train a policy on a broad distribution over tasks, where the number of tasks is potentially infinite.
For each training task we assume the availability of a set of successful demonstrations. Our learned
policy takes as input: (i) the current observation, and (ii) one demonstration that successfully solves
a different instance of the same task (this demonstration is fixed for the duration of the episode).
The policy outputs the current controls. We note that any pair of demonstrations for the same task
provides a supervised training example for the neural net policy, where one demonstration is treated
as the input, while the other as the output.
2
To make this model work, we made essential use of soft attention [6] for processing both the (potentially long) sequence of states and action that correspond to the demonstration, and for processing the
components of the vector specifying the locations of the various blocks in our environment. The use
of soft attention over both types of inputs made strong generalization possible. In particular, on a
family of block stacking tasks, our neural network policy was able to perform well on novel block
configurations which were not present in any training data. Videos of our experiments are available
at http://bit.ly/nips2017-oneshot.
2
Related Work
Imitation learning considers the problem of acquiring skills from observing demonstrations. Survey
articles include [48, 11, 3].
Two main lines of work within imitation learning are behavioral cloning, which performs supervised
learning from observations to actions (e.g., [41, 44]); and inverse reinforcement learning [37], where
a reward function [1, 66, 29, 18, 22] is estimated that explains the demonstrations as (near) optimal
behavior. While this past work has led to a wide range of impressive robotics results, it considers
each skill separately, and having learned to imitate one skill does not accelerate learning to imitate
the next skill.
One-shot and few-shot learning has been studied for image recognition [61, 26, 47, 42], generative
modeling [17, 43], and learning ?fast? reinforcement learning agents with recurrent policies [16, 62].
Fast adaptation has also been achieved through fast-weights [5]. Like our algorithm, many of the
aforementioned approaches are a form of meta-learning [58, 49, 36], where the algorithm itself is
being learned. Meta-learning has also been studied to discover neural network weight optimization
algorithms [8, 9, 23, 50, 2, 31]. This prior work on one-shot learning and meta-learning, however,
is tailored to respective domains (image recognition, generative models, reinforcement learning,
optimization) and not directly applicable in the imitation learning setting. Recently, [19] propose a
generic framework for meta learning across several aforementioned domains. However they do not
consider the imitation learning setting.
Reinforcement learning [56, 10] provides an alternative route to skill acquisition, by learning through
trial and error. Reinforcement learning has had many successes, including Backgammon [57],
helicopter control [39], Atari [35], Go [52], continuous control in simulation [51, 21, 32] and on
real robots [40, 30]. However, reinforcement learning tends to require a large number of trials and
requires specifying a reward function to define the task at hand. The former can be time-consuming
and the latter can often be significantly more difficult than providing a demonstration [37].
Multi-task and transfer learning considers the problem of learning policies with applicability and
re-use beyond a single task. Success stories include domain adaptation in computer vision [64, 34,
28, 4, 15, 24, 33, 59, 14] and control [60, 45, 46, 20, 54]. However, while acquiring a multitude of
skills faster than what it would take to acquire each of the skills independently, these approaches do
not provide the ability to readily pick up a new skill from a single demonstration.
Our approach heavily relies on an attention model over the demonstration and an attention model
over the current observation. We use the soft attention model proposed in [6] for machine translations,
and which has also been successful in image captioning [63]. The interaction networks proposed
in [7, 12] also leverage locality of physical interaction in learning. Our model is also related to
the sequence to sequence model [55, 13], as in both cases we consume a very long demonstration
sequence and, effectively, emit a long sequence of actions.
3
3.1
One Shot Imitation Learning
Problem Formalization
We denote a distribution of tasks by T, an individual task by t ? T, and a distribution of demonstrations for the task t by D(t). A policy is symbolized by ?? (a|o, d), where a is an action, o is
an observation, d is a demonstration, and ? are the parameters of the policy. A demonstration
d ? D(t) is a sequence of observations and actions : d = [(o1 , a1 ), (o2 , a2 ), . . . , (oT , aT )]. We
assume that the distribution of tasks T is given, and that we can obtain successful demonstrations for
each task. We assume that there is some scalar-valued evaluation function Rt (d) (e.g. a binary value
3
indicating success) for each task, although this is not required during training. The objective is to
maximize the expected performance of the policy, where the expectation is taken over tasks t ? T,
and demonstrations d ? D(t).
3.2
Block Stacking Tasks
To clarify the problem setting, we describe a concrete example of a distribution of block stacking
tasks, which we will also later study in the experiments. The compositional structure shared among
these tasks allows us to investigate nontrivial generalization to unseen tasks. For each task, the goal is
to control a 7-DOF Fetch robotic arm to stack various numbers of cube-shaped blocks into a specific
configuration specified by the user. Each configuration consists of a list of blocks arranged into
towers of different heights, and can be identified by a string. For example, ab cd ef gh means
that we want to stack 4 towers, each with two blocks, and we want block A to be on top of block B,
block C on top of block D, block E on top of block F, and block G on top of block H. Each of these
configurations correspond to a different task. Furthermore, in each episode the starting positions
of the blocks may vary, which requires the learned policy to generalize even within the training
tasks. In a typical task, an observation is a list of (x, y, z) object positions relative to the gripper,
and information if gripper is opened or closed. The number of objects may vary across different
task instances. We define a stage as a single operation of stacking one block on top of another. For
example, the task ab cd ef gh has 4 stages.
3.3
Algorithm
In order to train the neural network policy, we make use of imitation learning algorithms such
as behavioral cloning and DAGGER [44], which only require demonstrations rather than reward
functions to be specified. This has the potential to be more scalable, since it is often easier to
demonstrate a task than specifying a well-shaped reward function [38].
We start by collecting a set of demonstrations for each task, where we add noise to the actions in order
to have wider coverage in the trajectory space. In each training iteration, we sample a list of tasks
(with replacement). For each sampled task, we sample a demonstration as well as a small batch of
observation-action pairs. The policy is trained to regress against the desired actions when conditioned
on the current observation and the demonstration, by minimizing an `2 or cross-entropy loss based on
whether actions are continuous or discrete. A high-level illustration of the training procedure is given
in Fig. 1(c). Across all experiments, we use Adamax [25] to perform the optimization with a learning
rate of 0.001.
4
Architecture
While, in principle, a generic neural network could learn the mapping from demonstration and current
observation to appropriate action, we found it important to use an appropriate architecture. Our
architecture for learning block stacking is one of the main contributions of this paper, and we believe
it is representative of what architectures for one-shot imitation learning could look like in the future
when considering more complex tasks.
Our proposed architecture consists of three modules: the demonstration network, the context network,
and the manipulation network. An illustration of the architecture is shown in Fig. 2. We will describe
the main operations performed in each module below, and a full specification is available in the
Appendix.
4.1
Demonstration Network
The demonstration network receives a demonstration trajectory as input, and produces an embedding
of the demonstration to be used by the policy. The size of this embedding grows linearly as a function
of the length of the demonstration as well as the number of blocks in the environment.
Temporal Dropout: For block stacking, the demonstrations can span hundreds to thousands of time
steps, and training with such long sequences can be demanding in both time and memory usage.
Hence, we randomly discard a subset of time steps during training, an operation we call temporal
dropout, analogous to [53, 27]. We denote p as the proportion of time steps that are thrown away.
4
Attention
over
Current
State
Context Network
Block# A B C D E
Attention over
Demonstration
Neighborhood Attention
+
Temporal Convolution
F G H
I
J
Context Embedding
Hidden layers
Hidden layers
Temporal Dropout
Action
Demonstration Network
Manipulation Network
Demonstration
Current State
Figure 2: Illustration of the network architecture.
In our experiments, we use p = 0.95, which reduces the length of demonstrations by a factor of 20.
During test time, we can sample multiple downsampled trajectories, use each of them to compute
downstream results, and average these results to produce an ensemble estimate. In our experience,
this consistently improves the performance of the policy.
Neighborhood Attention: After downsampling the demonstration, we apply a sequence of operations, composed of dilated temporal convolution [65] and neighborhood attention. We now describe
this second operation in more detail.
Since our neural network needs to handle demonstrations with variable numbers of blocks, it must
have modules that can process variable-dimensional inputs. Soft attention is a natural operation which
maps variable-dimensional inputs to fixed-dimensional outputs. However, by doing so, it may lose
information compared to its input. This is undesirable, since the amount of information contained
in a demonstration grows as the number of blocks increases. Therefore, we need an operation that
can map variable-dimensional inputs to outputs with comparable dimensions. Intuitively, rather than
having a single output as a result of attending to all inputs, we have as many outputs as inputs, and
have each output attending to all other inputs in relation to its own corresponding input.
We start by describing the soft attention module as specified in [6]. The input to the attention includes
a query q, a list of context vectors {cj }, and a list of memory vectors {mj }. The ith attention weight
is given by wi ? v T tanh(q + ci ), where v is a learned weight vector. The output of attention is a
weighted combination of the memory content, where the weights are given by a softmax operation
P
i)
over the attention weights. Formally, we have output ? i mi Pexp(w
. Note that the output has
j exp(wj )
the same dimension as a memory vector. The attention operation can be generalized to multiple query
heads, in which case there will be as many output vectors as there are queries.
Now we turn to neighborhood attention. We assume there are B blocks in the environment. We
denote the robot?s state as srobot , and the coordinates of each block as (x1 , y1 , z1 ), . . . , (xB , yB , zB ).
in
The input to neighborhood attention is a list of embeddings hin
1 , . . . , hB of the same dimension,
which can be the result of a projection operation over a list of block positions, or the output of a
previous neighborhood attention operation. Given this list of embeddings, we use two separate linear
layers to compute a query vector and a context embedding for each block: qi ? Linear(hin
i ), and
ci ? Linear(hin
i ). The memory content to be extracted consists of the coordinates of each block,
concatenated with the input embedding. The ith query result is given by the following soft attention
in
B
operation: resulti ? SoftAttn(query: qi , context: {cj }B
j=1 , memory: {((xj , yj , zj ), hj ))}j=1 ).
Intuitively, this operation allows each block to query other blocks in relation to itself (e.g. find the
closest block), and extract the queried information. The gathered results are then combined with
each block?s own information, to produce the output embedding per block. Concretely, we have
5
outputi ? Linear(concat(hin
i , resulti , (xi , yi , zi ), srobot )). In practice, we use multiple query
heads per block, so that the size of each resulti will be proportional to the number of query heads.
4.2
Context network
The context network is the crux of our model. It processes both the current state and the embedding
produced by the demonstration network, and outputs a context embedding, whose dimension does
not depend on the length of the demonstration, or the number of blocks in the environment. Hence, it
is forced to capture only the relevant information, which will be used by the manipulation network.
Attention over demonstration: The context network starts by computing a query vector as a function
of the current state, which is then used to attend over the different time steps in the demonstration
embedding. The attention weights over different blocks within the same time step are summed
together, to produce a single weight per time step. The result of this temporal attention is a vector
whose size is proportional to the number of blocks in the environment. We then apply neighborhood
attention to propagate the information across the embeddings of each block. This process is repeated
multiple times, where the state is advanced using an LSTM cell with untied weights.
Attention over current state: The previous operations produce an embedding whose size is independent of the length of the demonstration, but still dependent on the number of blocks. We then
apply standard soft attention over the current state to produce fixed-dimensional vectors, where the
memory content only consists of positions of each block, which, together with the robot?s state, forms
the context embedding, which is then passed to the manipulation network.
Intuitively, although the number of objects in the environment may vary, at each stage of the
manipulation operation, the number of relevant objects is small and usually fixed. For the block
stacking environment specifically, the robot should only need to pay attention to the position of the
block it is trying to pick up (the source block), as well as the position of the block it is trying to place
on top of (the target block). Therefore, a properly trained network can learn to match the current
state with the corresponding stage in the demonstration, and infer the identities of the source and
target blocks expressed as soft attention weights over different blocks, which are then used to extract
the corresponding positions to be passed to the manipulation network. Although we do not enforce
this interpretation in training, our experiment analysis supports this interpretation of how the learned
policy works internally.
4.3
Manipulation network
The manipulation network is the simplest component. After extracting the information of the source
and target blocks, it computes the action needed to complete the current stage of stacking one block
on top of another one, using a simple MLP network.1 This division of labor opens up the possibility
of modular training: the manipulation network may be trained to complete this simple procedure,
without knowing about demonstrations or more than two blocks present in the environment. We leave
this possibility for future work.
5
Experiments
We conduct experiments with the block stacking tasks described in Section 3.2.2 These experiments
are designed to answer the following questions:
? How does training with behavioral cloning compare with DAGGER?
? How does conditioning on the entire demonstration compare to conditioning on the final
state, even when it already has enough information to fully specify the task?
? How does conditioning on the entire demonstration compare to conditioning on a ?snapshot?
of the trajectory, which is a small subset of frames that are most informative?
1
In principle, one can replace this module with an RNN module. But we did not find this necessary for the
tasks we consider.
2
Additional experiment results are available in the Appendix, including a simple illustrative example of
particle reaching tasks and further analysis of block stacking
6
? Can our framework generalize to tasks that it has never seen during training?
To answer these questions, we compare the performance of the following architectures:
? BC: We use the same architecture as previous, but and the policy using behavioral cloning.
? DAGGER: We use the architecture described in the previous section, and train the policy
using DAGGER.
? Final state: This architecture conditions on the final state rather than on the entire demonstration trajectory. For the block stacking task family, the final state uniquely identifies the
task, and there is no need for additional information. However, a full trajectory, one which
contains information about intermediate stages of the task?s solution, can make it easier to
train the optimal policy, because it could learn to rely on the demonstration directly, without
needing to memorize the intermediate steps into its parameters. This is related to the way in
which reward shaping can significantly affect performance in reinforcement learning [38].
A comparison between the two conditioning strategies will tell us whether this hypothesis is
valid. We train this policy using DAGGER.
? Snapshot: This architecture conditions on a ?snapshot? of the trajectory, which includes the
last frame of each stage along the demonstration trajectory. This assumes that a segmentation
of the demonstration into multiple stages is available at test time, which gives it an unfair
advantage compared to the other conditioning strategies. Hence, it may perform better than
conditioning on the full trajectory, and serves as a reference, to inform us whether the policy
conditioned on the entire trajectory can perform as well as if the demonstration is clearly
segmented. Again, we train this policy using DAGGER.
We evaluate the policy on tasks seen during training, as well as tasks unseen during training. Note
that generalization is evaluated at multiple levels: the learned policy not only needs to generalize to
new configurations and new demonstrations of tasks seen already, but also needs to generalize to new
tasks.
Concretely, we collect 140 training tasks, and 43 test tasks, each with a different desired layout of the
blocks. The number of blocks in each task can vary between 2 and 10. We collect 1000 trajectories
per task for training, and maintain a separate set of trajectories and initial configurations to be used
for evaluation. The trajectories are collected using a hard-coded policy.
5.1
Performance Evaluation
100%
Policy Type
Demo
BC
DAGGER
Snapshot
Final state
Demo
BC
DAGGER
Snapshot
Final state
80%
Average Success Rate
80%
Average Success Rate
100%
Policy Type
60%
40%
60%
40%
20%
20%
0%
0%
1
2
3
4
5
6
7
2
Number of Stages
4
5
6
7
8
Number of Stages
(a) Performance on training tasks.
(b) Performance on test tasks.
Figure 3: Comparison of different conditioning strategies. The darkest bar shows the performance of the
hard-coded policy, which unsurprisingly performs the best most of the time. For architectures that use temporal
dropout, we use an ensemble of 10 different downsampled demonstrations and average the action distributions.
Then for all architectures we use the greedy action for evaluation.
Fig. 3 shows the performance of various architectures. Results for training and test tasks are presented
separately, where we group tasks by the number of stages required to complete them. This is because
tasks that require more stages to complete are typically more challenging. In fact, even our scripted
policy frequently fails on the hardest tasks. We measure success rate per task by executing the greedy
policy (taking the most confident action at every time step) in 100 different configurations, each
conditioned on a different demonstration unseen during training. We report the average success rate
over all tasks within the same group.
7
From the figure, we can observe that for the easier tasks with fewer stages, all of the different
conditioning strategies perform equally well and almost perfectly. As the difficulty (number of stages)
increases, however, conditioning on the entire demonstration starts to outperform conditioning on the
final state. One possible explanation is that when conditioned only on the final state, the policy may
struggle about which block it should stack first, a piece of information that is readily accessible from
demonstration, which not only communicates the task, but also provides valuable information to help
accomplish it.
More surprisingly, conditioning on the entire demonstration also seems to outperform conditioning
on the snapshot, which we originally expected to perform the best. We suspect that this is due
to the regularization effect introduced by temporal dropout, which effectively augments the set of
demonstrations seen by the policy during training.
Another interesting finding was that training with behavioral cloning has the same level of performance
as training with DAGGER, which suggests that the entire training procedure could work without
requiring interactive supervision. In our preliminary experiments, we found that injecting noise into
the trajectory collection process was important for behavioral cloning to work well, hence in all
experiments reported here we use noise injection. In practice, such noise can come from natural
human-induced noise through tele-operation, or by artificially injecting additional noise before
applying it on the physical robot.
5.2
Visualization
We visualize the attention mechanisms underlying the main policy architecture to have a better
understanding about how it operates. There are two kinds of attention we are mainly interested in,
one where the policy attends to different time steps in the demonstration, and the other where the
policy attends to different blocks in the current state. Fig. 4 shows some of the attention heatmaps.
(a) Attention over blocks in the current state.
(b) Attention over downsampled demonstration.
Figure 4: Visualizing attentions performed by the policy during an entire execution. The task
being performed is ab cde fg hij. Note that the policy has multiple query heads for each type
of attention, and only one query head per type is visualized. (a) We can observe that the policy
almost always focuses on a small subset of the block positions in the current state, which allows the
manipulation network to generalize to operations over different blocks. (b) We can observe a sparse
pattern of time steps that have high attention weights. This suggests that the policy has essentially
learned to segment the demonstrations, and only attend to important key frames. Note that there are
roughly 6 regions of high attention weights, which nicely corresponds to the 6 stages required to
complete the task.
6
Conclusions
In this work, we presented a simple model that maps a single successful demonstration of a task to
an effective policy that solves said task in a new situation. We demonstrated effectiveness of this
approach on a family of block stacking tasks. There are a lot of exciting directions for future work.
We plan to extend the framework to demonstrations in the form of image data, which will allow
more end-to-end learning without requiring a separate perception module. We are also interested in
enabling the policy to condition on multiple demonstrations, in case where one demonstration does
not fully resolve ambiguity in the objective. Furthermore and most importantly, we hope to scale up
8
our method on a much larger and broader distribution of tasks, and explore its potential towards a
general robotics imitation learning system that would be able to achieve an overwhelming variety of
tasks.
7
Acknowledgement
We would like to thank our colleagues at UC Berkeley and OpenAI for insightful discussions. This
research was funded in part by ONR through a PECASE award. Yan Duan was also supported by a
Huawei Fellowship. Jonathan Ho was also supported by an NSF Fellowship.
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6,312 | 671 | An Information-Theoretic Approach to
Deciphering the Hippocampal Code
William E. Skaggs
Bruce L. McNaughton
Katalin M. Gothard
Etan J. Markus
Center for Neural Systems, Memory, and Aging
344 Life Sciences North
University of Arizona
Tucson AZ 85724
[email protected]
Abstract
Information theory is used to derive a simple formula for the
amount of information conveyed by the firing rate of a neuron about
any experimentally measured variable or combination of variables
(e.g. running speed, head direction, location of the animal, etc.).
The derivation treats the cell as a communication channel whose
input is the measured variable and whose output is the cell's spike
train. Applying the formula, we find systematic differences in the
information content of hippocampal "place cells" in different experimental conditions.
1
INTRODUCTION
Almost any neuron will respond to some manipulation or other by changing its firing
rate, and this change in firing can convey information to downstream neurons. The
aim of this article is to introduce a very simple formula for the average rate at which
a cell conveys information in this way, and to show how the formula is helpful in
the study of the firing properties of cells in the rat hippocampus. This is by no
means the first application of information theory to the study of neural coding; see
especially Richmond and Optican (1990). The thing that particularly distinguishes
1030
An Information-Theoretic Approach to Deciphering the Hippocampal Code
our approach is its simplemindedness.
To get the basic idea, imagine we are recording the activity of a neuron in the brain
of a rat, while the rat is wandering around randomly on a circular platform. Suppose
we observe that the cell fires only when the rat is on the left half of the platform,
and that it fires at a constant rate everywhere on the left half; and suppose that on
the whole the rat spends half of its time on the left half of the platform. In this case,
if we are prevented from seeing where the rat is, but are informed that the neuron
has just this very moment fired a spike, we obtain thereby one bit of information
about the current location of the rat. Suppose we have a second cell, which fires
only in the southwest quarter of the platform; in this case a spike would give us two
bits of information. If there were in addition a small amount of background firing,
the information would be slightly less than two bits. And so on.
Going back to the cell that fires everywhere on the left half of the platform, suppose
that when it is active, it fires at a mean rate of 10 spikes per second. Since it is active
half the time, it fires at an overall mean rate of 5 spikes per second. Since a spike
conveys one bit of information about the rat's location, the cell's spike train conveys
information at an average rate of 5 bits per second. This does not mean that if the
cell is observed for one second, on average 5 bits will be obtained-rather it means
that if the cell is observed for a sufficiently short time interval dt, on average 5dt
bits will be obtained. In 20 milliseconds, for example, the expected information
conveyed by the cell about the location of the rat will be very nearly 0.1 bits. The
longer the time interval over which the cell is observed, the more redundancy in the
spike train, and hence the farther below 5dt the total information falls.
The formula that leads to these numbers is
1= l.\(X) log2 .\~) p(x)dx,
(1)
where I is the information rate of the cell in bits per second, x is spatial location,
p( x) is the probability density for the rat being at location x, .\( x) is the mean firing
rate when the rat is at location x, and .\ = Jz .\(x)p(x)dx is the overall mean firing
rate of the cell. The derivation of this formula appears in the final section. (To our
knowledge the formula, though very simple, has not previously been published.)
Note that, as far as the formula is concerned, there is nothing special about spatial
location: the formula can equally well be used to define the rate at which a cell
conveys information about any aspect of the rat's state, or any combination of
aspects. The only mathematical requirement 1 is that the rat's state x and the spike
train of the cell both be stationary random variables, so that the probability density
p( x) and the expected firing rate .\( x) are well-defined.
The information rate given by formula (1) is measured in bits per second. If it is
divided by the overall mean firing rate.\ of the cell (expressed in spikes per second),
then a different kind of information rate is obtained, in units of bits per spike-let us
call it the information per spike. This is a measure of the specificity of the cell: the
more "grandmotherish" the cell, the more information per spike. For a population
lOther than obvious requirements of integrability that are sure to be fulfilled in natural
situations.
1031
1032
Skaggs, McNaughton, Gothard, and Markus
of cells, then, a highly distributed representation equates to little information per
spike.
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Figure 1: "Spot plot" of the activity of a single pyramidal cell in the hippocampus
of a rat, recorded while the rat foraged for food pellets inside a small cylinder. The
dots show locations visited by the rat, and the circles show points where the cell
fired-large circles mean that several spikes occurred within a short time. The lines
indicate which direction the rat was facing when the cell fired. The plot represents
29 minutes of data, during which the cell fired at an overall mean rate of 1.319 Hz.
Consider, as an example, a typical "place cell" (actually an especially nice place
cell) from the CAl layer of the hippocampus of a rat-Figure I shows a "spot plot"
of the activity of the cell as the rat moves around inside a 76 cm diameter cylinder
with high, opaque walls, foraging for randomly scattered food pellets. This cell, like
most pyramidal cells in CAl, fires at a relatively high rate (above 10 Hz) when the
rat is in a specific small portion of the environment-the "place field" of the cellbut at a much lower rate elsewhere. Different cells have place fields in different
locations; there are no systematic rules for their arrangement, except that there
may be a tendency for neighboring cells to have nearby place fields. The activity of
place cells is known to be related to more than just place: in some circumstances
it is sensitive to the direction the rat is facing, and it can also be modulated by
running speed, alertness, or other aspects of behavioral state. The dependence on
An Information-Theoretic Approach to Deciphering the Hippocampal Code
head direction has given rise to a certain amount of controversy, because in some
types of environment it is very strong, while in others it is virtually absent.
Table 1 gives statistics for the amount of information conveyed by this cell about
spatial location, head direction, running speed, and combinations of these variables.
Note that the information conveyed about spatial location and head direction is
hardly more than the information conveyed about spatial location alone-the difference is well within the error bounds of the calculation. Thus this cell has no
detectable directionality. This seems to be typical of cells recorded in unstructured
environments.
Table 1: Information conveyed by the cell whose activity is plotted in Figure 1.
VARIABLES
Location
Head Direction
Running Speed
Location and Head Direction
Location and Running Speed
INFO
2.40 bits/sec
0.48 bits/sec
0.03 bits/sec
2.53 bits/sec
2.36 bits/sec
INFO PER SPIKE
1.82 bits
0.37 bits
0.02 bits
1.92 bits
1.79 bits
The information-rate measure may be helpful in understanding the computations
performed by neural populations. Consider an example. Cells in the CA3 and
CAl regions of the rat hippocampal formation have long been known to convey
information about a rat's spatial location (this is discussed in more detail below).
Data from our lab suggest that, in a given environment, an average CA3 cell conveys
something in the neighborhood of 0.1 bits per second about the rat's position-some
cells convey a good deal more information than this, but many are virtually silent.
Cells in CAl receive most of their input from cells in CA3; each gets on the order of
10,000 such inputs. Question: How long must the integration time of a CAl cell be
in order for it to form a good estimate of the rat's location? Answer: With 10,000
inputs, each conveying on average 0.1 bits per second, the cell receives information
at a rate of 1000 bits per second, or 1 bit per millisecond, so in 5-10 msec the cell
receives enough information to form a moderately precise estimate of location.
2
APPLICATIONS
We now very briefly describe two experimental studies that have found differences
in the spatial information content of rat hippocampal activity under different conditions. The methods used for recording the cells are described in detail in McNaughton et al (1989)-to summarize, the cells were recorded with stereotrodes,
which are twisted pairs of electrodes, separated by about 15 microns at the tips,
that pick up the extracellular electric fields generated when cells fire. A single
stereotrode can detect the activity of as many as six or seven distinct hippocampal
cells; spikes from different cells can be separated on the basis of their amplitudes on
the two electrodes, as well as other differences in wave shape. The location of the
rat was tracked using arrays of LEDs attached to their heads and a video camera on
the ceiling. Spatial firing rate maps for each cell were constructed using an adaptive
binning technique designed to minimize error (Skaggs and McNaughton, submitted),
1033
1034
Skaggs, McNaughton, Gothard, and Markus
and information rates were calculated using these firing rate maps. As a control,
the spike train was randomly time-shifted relative to the sequence of locations; this
was done 100 times, and the cell was deemed to have significant spatial dependence
if its information rate was more than 2.29 standard deviations above the mean of
the 100 control information rates.
2.1
EXPERIMENT: PROXIMAL VERSUS DISTAL VISUAL CUES
In this study (a preliminary account of which appears in Gothard et al (1992?, the
activity of place cells was recorded successively in two environments, the first a 76 em
diameter cylinder with four patterned cue-cards on the high, opaque gray wall, the
second a cylinder of the same shape, but with a low, transparent plexiglass wall
and four patterned cue-cards on the distant black walls of the recording room. The
two environments thus had the same shape, and from any given point were visually
quite similar; the difference is that in one all of the visual cues were proximal to
the rat, while in the other many of them were distal.
DISTAL CUES
PROXIMAL CUES
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Figure 2: Firing rate maps of four simultaneously recorded cells, in the distal cue
environment (top) and proximal cue environment (bottom). The scale is identical
for all plots; black ~ 5 Hz.
Fifty cells were recorded with robust place-dependent firing in one or the other
cylinder. There was no discernable relationship between place fields in the two
environments-a cell having a place field in the proximal cue environment might
be nearly silent in the distal cue environment, and even if it did fire, its place field
would be in a different location. (Figure 2 shows firing rate maps for four of the
cells.) A substantially higher fraction of the cells had place fields in the proximal cue
environment, and overall the average information per second was almost 50% higher
An Information-Theoretic Approach to Deciphering the Hippocampal Code
in the proximal cue environment. For the cells possessing fields, the information
per spike was significantly higher in the proximal cue environment, meaning that
place fields were more compact.
These results indicate that in the proximal cue environment, spatial location is
represented by the hippocampal population more precisely, and by a larger pool of
cells, than in the distal cue environment. The most likely explanation is that, at
least in the absence of local cues, the configuration of visual landmarks controls the
activity of the place cell population.
2.2
EXPERIMENT: LIGHT VERSUS DARK
Visual cues have a great deal of influence on place fields, but they are not the
only important factor; in fact, some hippocampal cells maintain place fields even in
complete darkness (McNaughton et a/., 1989b; Quirk et a/., 1990). This experiment
(Markus et a/., 1992) was designed to examine how lack of visual cues changes the
properties of place fields. Rats traversed an eight-arm radial maze for chocolate
milk reward, with the room lights being turned on and oft' on alternate trials. (A
trial consisted of one visit to each of the eight arms of the maze.) Figure 3 shows
firing rate maps for four simultaneously recorded cells.
LIGHT
:/::;...:....:
.tt:?:
'::::.::;.:
DARK
Figure 3: Firing rate maps of four simultaneously recorded cells, with room lights
turned on (top) and off (bottom). The scale is identical for all plots; black ~ 5 Hz.
(The loops at the ends of the arms are caused by the rat turning around there.)
The most salient effect was that a much larger fraction of cells showed significant
spatially selective firing in the light than in the dark: 35% as opposed to 20%.
However, the average information per second decreased only by 15% in the dark
as compared to the light, from 0.326 bits per second in the light to 0.278 bits per
1035
1036
Skaggs, McNaughton, Gothard, and Markus
second in the dark. (These are overestimates of the population averages, because
cells silent in both light and dark were not included in the sample.)
Interestingly, the drop in information content from light to dark seemed to be much
smaller than the drop from proximal cues to distal cues in the previous experiment.
A major difference between the two experime:nts is that, in the eight-arm maze,
tactile cues potentially give a great deal of information about spatial location, but
in a cylinder they serve only to distinguish the center from the wall. While it
is dangerous to compare the two experiments, which differed methodologically in
several ways, the results suggest that tactile cues can have a very strong influence
on hippocampal firing, at least when visual cues are absent.
3
THEORY
The information-rate formula (1) is derived by considering a neuron as a "channel"
(in the information-theoretic sense) whose input is the spatial location of the rat,
and whose output is the spike train. During a sufficiently short time interval the
spike train is effectively a binary random variable (Le. the only possibilities are to
spike once or not at all), and the probability of spiking is determined by the spatial
location. The event of spiking may be indicated by a random variable S whose value
is 1 if the cell spikes and 0 otherwise. If the environment is partitioned into a set of
nonoverlapping bins, then spatial location may be represented by an integer-valued
random variable X giving the index of the currently occupied bin.
In information theory, the information conveyed by a discrete random variable X
about another discrete random variable Y, which is identical to the mutual information of X and Y, is given by
where Z i and Yi are the possible values of X and Y, and pO is probability.
If Aj is the mean firing rate when the rat is in bin j, then the probability of a spike
during a brief time interval tl.t is
P(S= 11X =j) = Ajtl.t.
Also, the overall probability of a spike is
P(S=1)
= Atl.t,
where
with Pi = P(X =j).
After these expressions are plugged in to the equation for I(Y IX) above, it is a
matter of straightforward algebra, using power series expansions of logarithms and
keeping only lower order terms, to derive a discrete approximation of equation (1).
An Information-Theoretic Approach to Deciphering the Hippocampal Code
4
DISCUSSION
In many situations, neurons must decide whether to fire on the basis of relatively
brief samples of input, often 100 milliseconds or less. A cell cannot get much
information from a single input in such a short time, so to achieve precision it needs
to integrate many inputs. Formula (1) provides a measure of how much information
a single input conveys about a given variable in such a brief time interval.
The formula can be applied to any type of cell that uses firing rate to convey
information. The only requirement is to have enough data to get good, stable
estimates of firing rates. In practice, for a hippocampal cell having a mean firing
rate of around 0.5 Hz in an environment, twenty minutes of data is adequate for
measuring position-dependence; and for a "theta cell" (an interneuron, firing at a
considerably higher rate), very clean measurements are possible.
We have used the measure in the study of hippocampal place cells, but it might
actually work better for some other types. The problem with place cells is that
they fire at low overall rates, so it is time-consuming to get an adequate sample.
Cortical pyramidal cells often have mean rates at least ten times faster, so it ought
to be easier to get accurate numbers for them. The information measure might
naturally be applied to study, for example, the changes in information content of
visual cortical cells as a visual stimulus is blurred or dimmed.
Supported by NIMH grant MH46823
References
Gothard, K. M., Skaggs, W. E., McNaughton, B. L., Barnes, C. A., and Youngs,
S. P. (1992). Place field specificity depends on proximity of visual cues. Soc
Neurosci Abstr, 18:1216. 508.10 .
Markus, E. J., Barnes, C. A., McNaughton, B. L., Gladden, V., Abel, T. W., and
Skaggs, W. E. (1992). Decrease in the information content of hippocampal
cal cell spatial firing patterns in the dark. Soc Neuroscience Abstr, 18:1216.
508.12.
McNaughton, B. L., Leonard, B., and Chen, L. (1989b). Cortical-hippocampal
interactions and cognitive mapping: A hypothesis based on reintegration of
the parietal and inferotemporal pathways for visual processing. Psychobiology,
17:230-235.
McNaughton, B. L., Barnes, C. A., Meltzer, J., and Sutherland, R. J. (1989a).
Hippocampal granule cells are necessary for normal spatial learning but not for
spatially selective pyramidal cell discharge. Exp Brain Res, 76:485-496.
Quirk, G. J., Muller, R. U., and Kubie, J. L. (1990). The firing of hippocampal
place cells in the dark depends on the rat's previous experience. J N eurosci,
10:2008-2017.
Richmond, B. J. and Optican, L. M. (1990). Temporal encoding of two-dimensional
patterns by single units in primate primary visual cortex: Ii information transmission. J Neurophysiol, 64:370-380.
1037
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6,313 | 6,710 | Learning the Morphology of Brain Signals Using
Alpha-Stable Convolutional Sparse Coding
Mainak Jas1 , Tom Dupr? La Tour1 , Umut Sim?
? sekli1 , Alexandre Gramfort1,2
1: LTCI, Telecom ParisTech, Universit? Paris-Saclay, Paris, France
2: INRIA, Universit? Paris-Saclay, Saclay, France
Abstract
Neural time-series data contain a wide variety of prototypical signal waveforms
(atoms) that are of significant importance in clinical and cognitive research. One of
the goals for analyzing such data is hence to extract such ?shift-invariant? atoms.
Even though some success has been reported with existing algorithms, they are
limited in applicability due to their heuristic nature. Moreover, they are often
vulnerable to artifacts and impulsive noise, which are typically present in raw
neural recordings. In this study, we address these issues and propose a novel
probabilistic convolutional sparse coding (CSC) model for learning shift-invariant
atoms from raw neural signals containing potentially severe artifacts. In the core of
our model, which we call ?CSC, lies a family of heavy-tailed distributions called
?-stable distributions. We develop a novel, computationally efficient Monte Carlo
expectation-maximization algorithm for inference. The maximization step boils
down to a weighted CSC problem, for which we develop a computationally efficient
optimization algorithm. Our results show that the proposed algorithm achieves
state-of-the-art convergence speeds. Besides, ?CSC is significantly more robust to
artifacts when compared to three competing algorithms: it can extract spike bursts,
oscillations, and even reveal more subtle phenomena such as cross-frequency
coupling when applied to noisy neural time series.
1
Introduction
Neural time series data, either non-invasive such as electroencephalograhy (EEG) or invasive such as
electrocorticography (ECoG) and local field potentials (LFP), are fundamental to modern experimental
neuroscience. Such recordings contain a wide variety of ?prototypical signals? that range from beta
rhythms (12?30 Hz) in motor imagery tasks and alpha oscillations (8?12 Hz) involved in attention
mechanisms, to spindles in sleep studies, and the classical P300 event related potential, a biomarker for
surprise. These prototypical waveforms are considered critical in clinical and cognitive research [1],
thereby motivating the development of computational tools for learning such signals from data.
Despite the underlying complexity in the morphology of neural signals, the majority of the computational tools in the community are based on representing the signals with rather simple, predefined
bases, such as the Fourier or wavelet bases [2]. While such bases lead to computationally efficient
algorithms, they often fall short at capturing the precise morphology of signal waveforms, as demonstrated by a number of recent studies [3, 4]. An example of such a failure is the disambiguation of
the alpha rhythm from the mu rhythm [5], both of which have a component around 10 Hz but with
different morphologies that cannot be captured by Fourier- or wavelet-based representations.
Recently, there have been several attempts for extracting more realistic and precise morphologies
directly from unfiltered electrophysiology signals, via dictionary learning approaches [6?9]. These
methods all aim to extract certain shift-invariant prototypical waveforms (called ?atoms? in this
context) to better capture the temporal structure of the signals. As opposed to using generic bases
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
that have predefined shapes, such as the Fourier or the wavelet bases, these atoms provide a more
meaningful representation of the data and are not restricted to narrow frequency bands.
In this line of research, Jost et al. [6] proposed the MoTIF algorithm, which uses an iterative strategy
based on generalized eigenvalue decompositions, where the atoms are assumed to be orthogonal to
each other and learnt one by one in a greedy way. More recently, the ?sliding window matching?
(SWM) algorithm [9] was proposed for learning time-varying atoms by using a correlation-based
approach that aims to identify the recurring patterns. Even though some success has been reported
with these algorithms, they have several limitations: SWM uses a slow stochastic search inspired by
simulated annealing and MoTIF poorly handles correlated atoms, simultaneously activated, or having
varying amplitudes; some cases which often occur in practical applications.
A natural way to cast the problem of learning a dictionary of shift-invariant atoms into an optimization
problem is a convolutional sparse coding (CSC) approach [10]. This approach has gained popularity
in computer vision [11?15], biomedical imaging [16] and audio signal processing [10, 17], due to its
ability to obtain compact representations of the signals and to incorporate the temporal structure of
the signals via convolution. In the neuroscience context, Barth?lemy et al. [18] used an extension
of the K-SVD algorithm using convolutions on EEG data. In a similar spirit, Brockmeier and
Pr?ncipe [7] used the matching pursuit algorithm combined with a rather heuristic dictionary update,
which is similar to the MoTIF algorithm. In a very recent study, Hitziger et al. [8] proposed the
AWL algorithm, which presents a mathematically more principled CSC approach for modeling
neural signals. Yet, as opposed to classical CSC approaches, the AWL algorithm imposes additional
combinatorial constraints, which limit its scope to certain data that contain spike-like atoms. Also,
since these constraints increase the complexity of the optimization problem, the authors had to resort
to dataset-specific initializations and many heuristics in their inference procedure.
While the current state-of-the-art CSC methods have a strong potential for modeling neural signals,
they might also be limited as they consider an `2 reconstruction error, which corresponds to assuming
an additive Gaussian noise distribution. While this assumption could be reasonable for several signal
processing tasks, it turns out to be very restrictive for neural signals, which often contain heavy noise
bursts and have low signal-to-noise ratio.
In this study, we aim to address the aforementioned concerns and propose a novel probabilistic
CSC model called ?CSC, which is better-suited for neural signals. ?CSC is based on a family
of heavy-tailed distributions called ?-stable distributions [19] whose rich structure covers a broad
range of noise distributions. The heavy-tailed nature of the ?-stable distributions renders our model
robust to impulsive observations. We develop a Monte Carlo expectation maximization (MCEM)
algorithm for inference, with a weighted CSC model for the maximization step. We propose efficient
optimization strategies that are specifically designed for neural time series. We illustrate the benefits
of the proposed approach on both synthetic and real datasets.
2
Preliminaries
P
1/p
Notation: For a vector v ? Rn we denote the `p norm by kvkp = ( i |vi |p ) . The convolution of
N
M
N +M ?1
two vectors v1 ? R and v2 ? R is denoted by v1 ? v2 ? R
. We denote by x the observed
signals, d the temporal atoms, and z the sparse vector of activations. The symbols U, E, N , S denote
the univariate uniform, exponential, Gaussian, and ?-stable distributions, respectively.
Convolutional sparse coding: The CSC problem formulation adopted in this work follows the Shift
Invariant Sparse Coding (SISC) model from [10]. It is defined as follows:
min
d,z
N
X
1
n=1
2
kxn ?
K
X
k=1
dk ? znk k22 + ?
K
X
kznk k1 ,
s.t. kdk k22 ? 1 and znk ? 0, ?n, k ,
(1)
k=1
where xn ? RT denotes one of the N observed segments of signals, also referred to as a trials in
this paper. We denote by T as the length of a trial, and K the number of atoms. The aim in this
model is to approximate the signals xn by the convolution of certain atoms and their respective
activations, which are sparse. Here, dk ? RL denotes the kth atom of the dictionary d ? {dk }k , and
znk ? RT+?L+1 denotes the activation of the kth atom in the nth trial. We denote by z ? {znk }n,k .
The objective function (1) has two terms, an `2 data fitting term that corresponds to assuming an
additive Gaussian noise model, and a regularization term that promotes sparsity with an `1 norm. The
2
p(x)
10-2
10-3
10
-4
10-5
-30
Trial 2 (xn,t )
?=2.0, ?=0
?=1.9, ?=0
?=1.8, ?=0
?=0.9, ?=1
Trial 1 (xn,t )
5
100
10-1
10
0
-10
0
-5
-20
-10
0
10
20
x
30
500
1000
1500
2000
2500
500
1000
1500
2000
2500
Time (t)
Time (t)
(a)
(b)
Figure 1: (a) PDFs of ?-stable distributions. (b) Illustration of two trials from the striatal LFP data,
which contain severe artifacts. The artifacts are illustrated with dashed rectangles.
regularization parameter is called ? > 0. Two constraints are also imposed. First, we ensure that dk
lies within the unit sphere, which prevents the scale ambiguity between d and z. Second, a positivity
constraint on z is imposed to be able to obtain physically meaningful activations and to avoid sign
ambiguities between d and z. This positivity constraint is not present in the original SISC model [10].
?-Stable distributions: The ?-stable distributions have become increasingly popular in modeling
signals that might incur large variations [20?24] and have a particular importance in statistics since
they appear as the limiting distributions in the generalized central limit theorem [19]. They are
characterized by four parameters: ?, ?, ?, and ?: (i) ? ? (0, 2] is the characteristic exponent and
determines the tail thickness of the distribution: the distribution will be heavier-tailed as ? gets
smaller. (ii) ? ? [?1, 1] is the skewness parameter. If ? = 0, the distribution is symmetric. (iii)
? ? (0, ?) is the scale parameter and measures the spread of the random variable around its mode
(similar to the standard deviation of a Gaussian distribution). Finally, (iv) ? ? (??, ?) is the
location parameter (for ? > 1, it is simply the mean).
The probability density function of an ?-stable distribution cannot be written in closed-form except
for certain special cases; however, the characteristic function can be written as follows:
x ? S(?, ?, ?, ?) ?? E[exp(i?x)] = exp(?|??|? [1 + i sign(?)??? (?)] + i??) ,
?
where ?? (?) = log |?| for ? = 1, ?? (?) = tan(??/2) for ? 6= 1, and i = ?1. As an important
special case of the ?-stable distributions, we obtain the Gaussian distribution when ? = 2 and ? = 0,
i.e. S(2, 0, ?, ?) = N (?, 2? 2 ). In Fig. 1(a), we illustrate the (approximately computed) probability
density functions (PDF) of the ?-stable distribution for different values of ? and ?. The distribution
becomes heavier-tailed as we decrease ?, whereas the tails vanish quickly when ? = 2.
The moments of the ?-stable distributions can only be defined up to the order ?, i.e. E[|x|p ] < ?
if and only if p < ?, which implies the distribution has infinite variance when ? < 2. Furthermore, despite the fact that the PDFs of ?-stable distributions do not admit an analytical form, it is
straightforward to draw random samples from them [25].
3
Alpha-Stable Convolutional Sparse Coding
3.1
The Model
From a probabilistic perspective, the CSC problem can be also formulated as a maximum a-posteriori
(MAP) estimation problem on the following probabilistic generative model:
k
zn,t
? E(?),
xn,t |z, d ? N (?
xn,t , 1),
where,
x
?n ,
K
X
dk ? znk .
(2)
k=1
k
Here, zn,t
denotes the tth element of znk . We use the same notations for xn,t and x
?n,t . It is easy to
verify that the MAP estimate for this probabilistic model, i.e. maxd,z log p(d, z|x), is identical to the
original optimization problem defined in (1)1 .
It has been long known that, due to their light-tailed nature, Gaussian models often fail at handling
noisy high amplitude observations or outliers [26]. As a result, the ?vanilla? CSC model turns out
to be highly sensitive to outliers and impulsive noise that frequently occur in electrophysiological
1
Note that the positivity constraint on the activations is equivalent to an exponential prior for the regularization
term rather than the more common Laplacian prior.
3
recordings, as illustrated in Fig. 1(b). Possible origins of such artifacts are movement, muscle
contractions, ocular blinks or electrode contact losses.
In this study, we aim at developing a probabilistic CSC model that would be capable of modeling
challenging electrophysiological signals. We propose an extension of the original CSC model defined
in (2) by replacing the light-tailed Gaussian likelihood (corresponding to the `2 reconstruction loss
in (1)) with heavy-tailed ?-stable distributions. We define the proposed probabilistic model (?CSC)
as follows:
?
k
zn,t
? E(?), xn,t |z, d ? S(?, 0, 1/ 2, x
?n,t ) ,
(3)
where S denotes the ?-stable distribution. While still being able to capture the temporal structure
of the observed signals via convolution, the proposed model has a richer structure and would allow
large variations and outliers, thanks to the heavy-tailed ?-stable distributions. Note that the vanilla
CSC defined in (2) appears as a special case of ?CSC, as the ?-stable distribution coincides with the
Gaussian distribution when ? = 2.
3.2
Maximum A-Posteriori Inference
Given the observed signals x, we are interested in the MAP estimates, defined as follows:
X
X
k
(d? , z ? ) = arg max
log p(xn,t |d, z) +
log p(zn,t
) .
d,z
n,t
(4)
k
As opposed to the Gaussian case, unfortunately, this optimization problem is not amenable to classical
optimization tools, since the PDF of the ?-stable distributions does not admit an analytical expression.
As a remedy, we use the product property of the symmetric ?-stable densities [19, 27] and re-express
the ?CSC model as conditionally Gaussian. It leads to:
?
?? 2/?
1
k
zn,t
? E(?), ?n,t ? S
, 1, 2(cos
) , 0 , xn,t |z, d, ? ? N x
?n,t , ?n,t , (5)
2
4
2
where ? is called the impulse variable that is drawn from a positive ?-stable distribution (i.e. ? = 1),
whose PDF is illustrated in Fig. 1(a). It can be shown that both formulations of the ?CSC model are
identical by marginalizing the joint distribution p(x, d, z, ?) over ? [19, Proposition 1.3.1].
The impulsive structure of the ?CSC model becomes more prominent in this formulation: the
variances of the Gaussian observations are modulated by stable random variables with infinite
variance, where the impulsiveness depends on the value of ?. It is also worth noting that when ? = 2,
?n,t becomes deterministic and we can again verify that ?CSC coincides with the vanilla CSC.
The conditionally Gaussian structure of the augmented model has a crucial practical implication: if
the impulse variable ? were to be known, then the MAP estimation problem over d and z in this
model would turn into a ?weighted? CSC problem, which is a much easier task compared to the
original problem. In order to be able to exploit this property, we propose an expectation-maximization
(EM) algorithm, which iteratively maximizes a lower bound of the log-posterior log p(d, z|x), and
algorithmically boils down to computing the following steps in an iterative manner:
E-Step:
B (i) (d, z) = E [log p(x, ?, z|d)]p(?|x,z(i) ,d(i) ) ,
(6)
M-Step:
(d(i+1) , z (i+1) ) = arg maxd,z B (i) (d, z).
(7)
where E[f (x)]q(x) denotes the expectation of a function f under the distribution q, i denotes the
iterations, and B (i) is a lower bound to log p(d, z|x) and it is tight at the current iterates z (i) , d(i) .
The E-Step: In the first step of our algorithm, we need to compute the EM lower bound B that has
the following form:
N q
K
K
X
X
X
(i)
(i)
+
B (d, z) = ?
k wn (xn ?
dk ? znk )k22 + ?
kznk k1 ,
(8)
n=1
k=1
k=1
where =+ denotes equality up to additive constants, denotes the Hadamard (element-wise) product,
(i)
and the square-root operator is also defined element-wise. Here, wn ? RT+ are the weights that are
(i)
defined as follows: wn,t , E [1/?n,t ]p(?|x,z(i) ,d(i) ) . As the variables ?n,t are expected to be large
when x
?n,t cannot explain the observation xn,t ? typically due to a corruption or a high noise ? the
weights will accordingly suppress the importance of the particular point xn,t . Therefore, the overall
approach will be more robust to corrupted data than the Gaussian models where all weights would be
deterministic and equal to 0.5.
4
Unfortunately, the weights w(i) cannot be
Algorithm 1 ?-stable Convolutional Sparse Coding
computed analytically, therefore we need
to resort to approximate methods. In this Require: Regularization: ? ? R+ , Num. atoms:
K, Atom length: L, Num. iterations: I , J, M
study, we develop a Markov chain Monte
1:
for
i = 1 to I do
Carlo (MCMC) method to approximately
compute the weights, where we approxi- 2: /* E-step: */
mate the intractable expectations with a finite 3: for j = 1 to J do
(i,j)
(i)
Draw ?n,t via MCMC (9)
sample average, given as follows: wn,t ? 4:
PJ
5:
end for
(i,j)
(i,j)
PJ
(i)
(i,j)
(1/J) j=1 1/?n,t , where ?n,t are some
6:
wn,t ? (1/J) j=1 1/?n,t
samples that are ideally drawn from the pos- 7: /* M-step: */
terior distribution p(?|x, z (i) , d(i) ). Unfor- 8: for m = 1 to M do
tunately, directly drawing samples from the 9:
z (i) = L-BFGS-B on (10)
posterior distribution of ? is not tractable ei- 10:
d(i) = L-BFGS-B on the dual of (11)
ther, and therefore, we develop a Metropolis- 11: end for
Hastings algorithm [28], that asymptotically 12: end for
generates samples from the target distribution 13: return w(I) , d(I) , z (I)
p(?|?) in two steps. In the j-th iteration of this
algorithm, we first draw a random sample for each n and t from the prior distribution (cf. (5)), i.e.,
?0n,t ? p(?n,t ). We then compute an acceptance probability for each ?0n,t that is defined as follows:
n
o
(i,j)
(i,j)
acc(?n,t ? ?0n,t ) , min 1, p(xn,t |d(i) , z (i) , ?0n,t )/p(xn,t |d(i) , z (i) , ?n,t )
(9)
where j denotes the iteration number of the MCMC algorithm. Finally, we draw a uniform random
(i)
number un,t ? U([0, 1]) for each n and t. If un,t < acc(?n,t ? ?0n,t ), we accept the sample and set
(i+1)
(i+1)
(i)
?n,t = ?0n,t ; otherwise we reject the sample and set ?n,t = ?n,t . This procedure forms a Markov
chain that leaves the target distribution p(?|?) invariant, where under mild ergodicity conditions, it
can be shown that the finite-sample averages converge to their true values when J goes to infinity
[29]. More detailed explanation of this procedure is given in the supplementary document.
The M-Step: Given the weights wn that are estimated during the E-step, the objective of the Mstep (7) is to solve a weighted CSC problem, which is much easier when compared to our original
problem. This objective function is not jointly convex in d and z, yet it is convex if one fix either d
or z. Here, similarly to the vanilla CSC approaches [9, 10], we develop a block coordinate descent
strategy, where we solve the problem in (7) for either d or z, by keeping respectively z and d fixed.
We first focus on solving the problem for z while keeping d fixed, given as follows:
min
z
N
K
X
X
X
?
k wn (xn ?
Dk z?nk )k22 + ?
kznk k1
n=1
k=1
s.t. znk ? 0, ?n, k .
(10)
k
Here, we expressed the convolution of dk and znk as the inner product of the zero-padded activations
z?nk , [(znk )> , 0 ? ? ? 0]> ? RT+ , with a Toeplitz matrix Dk ? RT ?T , that is constructed from dk . The
matrices Dk are never constructed in practice, and all operations are carried out using convolutions.
This problem can be solved by various constrained optimization algorithms. Here, we choose the
k
quasi-Newton L-BFGS-B algorithm [30] with a box constraint: 0 ? zn,t
? ?. This approach
only requires the simple computation of the gradient of the objective function with respect to z (cf.
supplementary material). Note that, since each trial is independent from each other, we can solve this
problem for each zn in parallel.
We then solve the problem for the atoms d while keeping z fixed. This optimization problem turns out
to be a constrained weighted least-squares problem. In the non-weighted case, this problem can be
solved either in the time domain or in the Fourier domain [10?12]. The Fourier transform simplifies
the convolutions that appear in least-squares problem, but it also induces several difficulties, such
as that the atom dk have to be in a finite support L, an important issue ignored in the seminal work
of [10] and addressed with an ADMM solver in[11, 12]. In the weighted case, it is not clear how to
solve this problem in the Fourier domain. We thus perform all the computations in the time domain.
Following the traditional filter identification approach [31], we need to embed the one-dimensional
k
signals znk into a matrix of delayed signals Znk ? RT ?L , where (Znk )i,j = zn,i+j?L+1
if L ? 1 ?
5
Heide et al (2015)
Wohlberg (2016)
M-step
M-step - 4 parallel
100
10
1
10
2
10
3
6000
Heide et al (2015)
Wohlberg (2016)
M-step
M-step - 4 parallel
5000
Time (s)
(objective - best) / best
101
4000
3000
2000
1000
0
0
2000
4000
Time (s)
(a) K = 10, L = 32.
K = 2, L = 32
K = 2, L = 128
K = 10, L = 32
(b) Time to reach a relative precision of 0.01.
Figure 2: Comparison of state-of-the-art methods with our approach. (a) Convergence plot with the
objective function relative to the obtained minimum, as a function of computational time. (b) Time
taken to reach a relative precision of 10?2 , for different settings of K and L.
i + j < T and 0 elsewhere. Equation (1) then becomes:
min
d
N
K
X
X
?
k wn (xn ?
Znk dk )k22 ,
n=1
s.t. kdk k22 ? 1 .
(11)
k=1
Due to the constraint, we must resort to an iterative approach. The options are to use (accelerated)
projected gradient methods such as FISTA [32] applied to (11), or to solve a dual problem as done
in [10]. The dual is also a smooth constraint problem yet with a simpler positivity box constraint
(cf. supplementary material). The dual can therefore be optimized with L-BFGS-B. Using such a
quasi-Newton solver turned out to be more efficient than any accelerated first order method in either
the primal or the dual (cf. benchmarks in supplementary material).
Our entire EM approach can be summarized in the Algorithm 1. Note that during the alternating
minimization, thanks to convexity we can warm start the d update and the z update using the solution
from the previous update. This significantly speeds up the convergence of the L-BFGS-B algorithm,
particularly in the later iterations of the overall algorithm.
4
Experiments
In order to evaluate our approach, we conduct several experiments on both synthetic and real
data. First, we show that our proposed optimization scheme for the M-step provides significant
improvements in terms of convergence speed over the state-of-the-art CSC methods. Then, we provide
empirical evidence that our algorithm is more robust to artifacts and outliers than three competing
CSC methods [6, 7, 12]. Finally, we consider LFP data, where we illustrate that our algorithm can
reveal interesting properties in electrophysiological signals without supervision, even in the presence
of severe artifacts. The source code is publicly available at https://alphacsc.github.io/.
Synthetic simulation setup: In our synthetic data experiments, we simulate N trials of length T by
first generating K zero mean and unit norm atoms of length L. The activation instants are integers
drawn from a uniform distribution in J0, T ? LK. The amplitude of the activations are drawn from a
uniform distribution in [0, 1]. Atoms are activated only once per trial and are allowed to overlap. The
activations are then convolved with the generated atoms and summed up as in (1).
M-step performance: In our first set of synthetic experiments, we illustrate the benefits of our
M-step optimization approach over state-of-the-art CSC solvers. We set N = 100, T = 2000 and
? = 1, and use different values for K and L. To be comparable, we set ? = 2 and add Gaussian
noise to the synthesized signals, where the standard deviation is set to 0.01. In this setting, we
have wn,t = 1/2 for all n, t, which reduces the problem to a standard CSC setup. We monitor the
convergence of ADMM-based methods by Heide et al. [11] and Wohlberg [12] against our M-step
algorithm, using both a single-threaded and a parallel version for the z-update. As the problem is
non-convex, even if two algorithms start from the same point, they are not guaranteed to reach the
same local minimum2 . Hence, for a fair comparison, we use a multiple restart strategy with averaging
across 24 random seeds.
2
Note that the M-step can be viewed as a biconvex problem, for which global convergence guarantees can be
shown under certain assumptions [33, 34]. However, we have observed that it is required to use multiple restarts
even for vanilla CSC, implying that these assumptions are not satisfied in this particular problem.
6
Brockmeier et al.
MoTIF
Brockmeier et al.
MoTIF
Brockmeier et al.
MoTIF
?CSC
CSC
?CSC
CSC
?CSC
Atom 1
Atom 2
G. Truth
CSC
(a) No corruption.
(b) 10% corruption.
(c) 20% corruption
Figure 3: Simulation to compare state-of-the-art methods against ?CSC.
During our experiments we have observed that the ADMM-based methods do not guarantee the
feasibility of the iterates. In other words, the norms of the estimated atoms might be greater than 1
during the iterations. To keep the algorithms comparable, when computing the objective value, we
project the atoms to the unit ball and scale the activations accordingly. To be strictly comparable,
we also imposed a positivity constraint on these algorithms. This is easily done by modifying the
soft-thresholding operator to be a rectified linear function. In the benchmarks, all algorithms use a
single thread, except ?M-step - 4 parallel? which uses 4 threads during the z update.
In Fig. 2, we illustrate the convergence behaviors of the different methods. Note that the y-axis is the
precision relative to the objective value obtained upon convergence. In other words, each curve is
relative to its own local minimum (see supplementary document for details). In the right subplot, we
show how long it takes for the algorithms to reach a relative precision of 0.01 for different settings
(cf. supplementary material for more benchmarks). Our method consistently performs better and the
difference is even more striking for more challenging setups. This speed improvement on the M-step
is crucial for us as this step will be repeatedly executed.
Robustness to corrupted data: In our second synthetic data experiment, we illustrate the robustness
of ?CSC in the presence of corrupted observations. In order to simulate the likely presence of high
amplitude artifacts, one way would be to directly simulate the generative model in (3). However,
this would give us an unfair advantage, since ?CSC is specifically designed for such data. Here,
we take an alternative approach, where we corrupt a randomly chosen fraction of the trials (10% or
20%) with strong Gaussian noise of standard deviation 0.1, i.e. one order of magnitude higher than
in a regular trial. We used a regularization parameter of ? = 0.1. In these experiments, by CSC we
refer to ?CSC with ? = 2, that resembles using only the M-step of our algorithm with deterministic
weights wn,t = 1/2 for all n, t. We used a simpler setup where we set N = 100, T = 512, and
L = 64. We used K = 2 atoms, as shown in dashed lines in Fig. 3.
For ?CSC, we set the number of outer iterations I = 5, the number of iterations of the M-step to
M = 50, and the number of iterations of the MCMC algorithm to J = 10. We discard the first 5
samples of the MCMC algorithm as burn-in. To enable a fair comparison, we run the standard CSC
algorithm for I ? M iterations, i.e. the total number of M-step iterations in ?CSC. We also compared
?CSC against competing state-of-art methods previously applied to neural time series: Brockmeier
and Pr?ncipe [7] and MoTIF [6]. Starting from multiple random initializations, the estimated atoms
with the smallest `2 distance with the true atoms are shown in Fig. 3.
?V
In the artifact-free scenario, all algorithms perform equally well, except for MoTIF that suffers
from the presence of activations with varying amplitudes. This is because it aligns the data using
correlations before performing the eigenvalue decomposition, without taking into account the strength
of activations in each trial. The performance of Brockmeier and Pr?ncipe [7] and CSC degrades as
400
200
0
200
400
600
8009.0
0.0
9.5
10.0
10.5
Time (s)
11.0
11.5
12.0
0.1
0.0
(a) LFP spike data from [8]
0.1
0.2
Time (s)
(b) Estimated atoms
7
0.3
Figure 4:
Atoms
learnt by ?CSC on
LFP data containing
epileptiform
spikes
with ? = 2.
0.15
High frequency
0.10
0.05
0.00
0.05
0.10
0.0
0.2
0.4
Time (s)
0.6
0.0
0.2
0.4
Time (s)
0.6
0.0
0.2
0.4
Time (s)
0.6
(a) Atoms learnt by: CSC (clean data), CSC (full data), ?CSC (full data)
175
150
125
100
75
50
25
0.005
0.004
0.003
0.002
0.001
2.5
5.0 7.5 10.0
Low frequency
0.000
(b) Comodulogram.
Figure 5: (a) Three atoms learnt from a rodent striatal LFP channel, using CSC on cleaned data, and
both CSC and ?CSC on the full data. The atoms capture the cross-frequency coupling of the data
(dashed rectangle). (b) Comodulogram presents the cross-frequency coupling intensity computed
between pairs of frequency bands on the entire cleaned signal, following [37].
the level of corruption increases. On the other hand, ?CSC is clearly more robust to the increasing
level of corruption and recovers reasonable atoms even when 20% of the trials are corrupted.
Results on LFP data In our last set of experiments, we consider real neural data from two different
datasets. We first applied ?CSC on an LFP dataset previously used in [8] and containing epileptiform
spikes as shown in Fig. 4(a). The data was recorded in the rat cortex, and is free of artifact. Therefore,
we used the standard CSC with our optimization scheme, (i.e. ?CSC with ? = 2). As a standard
preprocessing procedure, we applied a high-pass filter at 1 Hz in order to remove drifts in the signal,
and then applied a tapered cosine window to down-weight the samples near the edges. We set ? = 6,
N = 300, T = 2500, L = 350, and K = 3. The recovered atoms by our algorithm are shown in
Fig. 4(b). We can observe that the estimated atoms resemble the spikes in Fig. 4(a). These results
show that, without using any heuristics, our approach can recover similar atoms to the ones reported
in [8], even though it does not make any assumptions on the shapes of the waveforms, or initializes
the atoms with template spikes in order to ease the optimization.
The second dataset is an LFP channel in a rodent striatum from [35]. We segmented the data into 70
trials of length 2500 samples, windowed each trial with a tapered cosine function, and detrended the
data with a high-pass filter at 1 Hz. We set ? = 10, initialized the weights wn to the inverse of the
variance of the trial xn . Atoms are in all experiments initialized with Gaussian white noise.
As opposed to the first LFP dataset, this dataset contains strong artifacts, as shown in Fig. 1(b). In
order to be able to illustrate the potential of CSC on this data, we first manually identified and removed
the trials that were corrupted by artifacts. In Fig. 5(a), we illustrate the estimated atoms with CSC on
the manually-cleaned data. We observe that the estimated atoms correspond to canonical waveforms
found in the signal. In particular, the high frequency oscillations around 80 Hz are modulated in
amplitude by the low-frequency oscillation around 3 Hz, a phenomenon known as cross-frequency
coupling (CFC) [36]. We can observe this by computing a comodulogram [37] on the entire signal
(Fig. 5(b)). This measures the correlation between the amplitude of the high frequency band and the
phase of the low frequency band.
Even though CSC is able to provide these excellent results on the cleaned data set, its performance
heavily relies on the manual removal of the artifacts. Finally, we repeated the previous experiment on
the full data, without removing the artifacts and compared CSC with ?CSC, where we set ? = 1.2.
The results are shown in the middle and the right sub-figures of Fig. 5(a). It can be observed that in
the presence of strong artifacts, CSC is not able to recover the atoms anymore. On the contrary, we
observe that ?CSC can still recover atoms as observed in the artifact-free regime. In particular, the
cross-frequency coupling phenomenon is still visible.
5
Conclusion
We address the present need in the neuroscience community to better capture the complex morphology
of brain waves. Our approach is based on a probabilistic formulation of a CSC model. We propose
an inference strategy based on MCEM to deal efficiently with heavy tailed noise and take into
account the polarity of neural activations with a positivity constraint. Our problem formulation
allows the use of fast quasi-Newton methods that outperform previously proposed state-of-the-art
ADMM-based algorithms, even when not making use of our parallel implementation. Results on LFP
data demonstrate that such algorithms can be robust to the presence of transient artifacts in data and
reveal insights on neural time-series without supervision.
8
6
Acknowledgement
The work was supported by the French National Research Agency grants ANR-14-NEUC-0002-01,
ANR-13-CORD-0008-02, and ANR-16-CE23-0014 (FBIMATRIX), as well as the ERC Starting
Grant SLAB ERC-YStG-676943.
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6,314 | 6,711 | Integration Methods and Optimization Algorithms
Vincent Roulet
INRIA, ENS,
PSL Research University,
Paris France
[email protected]
Damien Scieur
INRIA, ENS,
PSL Research University,
Paris France
[email protected]
Francis Bach
INRIA, ENS,
PSL Research University,
Paris France
[email protected]
Alexandre d?Aspremont
CNRS, ENS
PSL Research University,
Paris France
[email protected]
Abstract
We show that accelerated optimization methods can be seen as particular instances
of multi-step integration schemes from numerical analysis, applied to the gradient
flow equation. Compared with recent advances in this vein, the differential equation
considered here is the basic gradient flow, and we derive a class of multi-step
schemes which includes accelerated algorithms, using classical conditions from
numerical analysis. Multi-step schemes integrate the differential equation using
larger step sizes, which intuitively explains the acceleration phenomenon.
Introduction
Applying the gradient descent algorithm to minimize a function f has a simple numerical interpretation as the integration of the gradient flow equation, written
x(t)
?
= ??f (x(t)),
x(0) = x0
(Gradient Flow)
using Euler?s method. This appears to be a somewhat unique connection between optimization and
numerical methods, since these two fields have inherently different goals. On one hand, numerical
methods aim to get a precise discrete approximation of the solution x(t) on a finite time interval. On
the other hand, optimization algorithms seek to find the minimizer of a function, which corresponds
to the infinite time horizon of the gradient flow equation. More sophisticated methods than Euler?s
were developed to get better consistency with the continuous time solution but still focus on a
finite time horizon [see e.g. S?li and Mayers, 2003]. Similarly, structural assumptions on f lead to
more sophisticated optimization algorithms than the gradient method, such as the mirror gradient
method [see e.g. Ben-Tal and Nemirovski, 2001; Beck and Teboulle, 2003], proximal gradient
method [Nesterov, 2007] or a combination thereof [Duchi et al., 2010; Nesterov, 2015]. Among
them Nesterov?s accelerated gradient algorithm [Nesterov, 1983] is proven to be optimal on the
class of smooth convex or strongly convex functions. This latter method was designed with optimal
complexity in mind, but the proof relies on purely algebraic arguments and the key mechanism behind
acceleration remains elusive, with various interpretations discussed in e.g. [Bubeck et al., 2015;
Allen Zhu and Orecchia, 2017; Lessard et al., 2016].
Another recent stream of papers used differential equations to model the acceleration behavior and
offer another interpretation of Nesterov?s algorithm [Su et al., 2014; Krichene et al., 2015; Wibisono
et al., 2016; Wilson et al., 2016]. However, the differential equation is often quite complex, being
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
reverse-engineered from Nesterov?s method itself, thus losing the intuition. Moreover, integration
methods for these differential equations are often ignored or are not derived from standard numerical
integration schemes.
Here, we take another approach. Rather than using a complicated differential equation, we use
advanced multistep discretization methods on the basic gradient flow equation in (Gradient Flow).
Ensuring that the methods effectively integrate this equation for infinitesimal step sizes is essential
for the continuous time interpretation and leads to a family of integration methods which contains
various well-known optimization algorithms. A full analysis is carried out for linear gradient flows
(quadratic optimization) and provides compelling explanations for the acceleration phenomenon.
In particular, Nesterov?s method can be seen as a stable and consistent gradient flow discretization
scheme that allows bigger step sizes in integration, leading to faster convergence.
1
Gradient flow
We seek to minimize a L-smooth ?-strongly convex function defined on Rd . We discretize the
gradient flow equation (Gradient Flow), given by the following ordinary differential equation
x(t)
?
= g(x(t)),
x(0) = x0 ,
(ODE)
where g comes from a potential ?f , meaning g = ??f . Smoothness of f means Lipschitz continuity
of g, i.e.
kg(x) ? g(y)k ? Lkx ? yk, for every x, y ? Rd ,
where k.k is the Euclidean norm. This property ensures existence and uniqueness of the solution
of (ODE) (see [S?li and Mayers, 2003, Theorem 12.1]). Strong convexity of f also means strong
monotonicity of ?g, i.e.,
?kx ? yk2 ? ?hx ? y, g(x) ? g(y)i,
for every x, y ? Rd ,
and ensures that (ODE) has a unique point x? such that g(x? ) = 0, called the equilibrium. This is the
minimizer of f and the limit point of the solution, i.e. x(?) = x? . Finally this assumption allows us
to control the convergence rate of the potential f and the solution x(t) as follows.
Proposition 1.1 Let f be a L-smooth and ?-strongly convex function and x0 ? dom(f ). Writing
x? the minimizer of f , the solution x(t) of (Gradient Flow) satisfies
f (x(t)) ? f (x? ) ? (f (x0 ) ? f (x? ))e?2?t ,
kx(t) ? x? k ? kx0 ? x? ke??t .
(1)
A proof of this last result is recalled in the Supplementary Material. We now focus on numerical
methods to integrate (ODE).
2
2.1
Numerical integration of differential equations
Discretization schemes
In general, we do not have access to an explicit solution x(t) of (ODE). We thus use integration
algorithms to approximate the curve (t, x(t)) by a grid (tk , xk ) ? (tk , x(tk )) on a finite interval
[0, tmax ]. For simplicity here, we assume the step size hk = tk ? tk?1 is constant, i.e., hk = h and
tk = kh. The goal is then to minimize the approximation error kxk ? x(tk )k for k ? [0, tmax /h].
We first introduce Euler?s method to illustrate this on a basic example.
Euler?s explicit method. Euler?s (explicit) method is one of the oldest and simplest schemes for
integrating the curve x(t). The idea stems from a Taylor expansion of x(t) which reads
x(t + h) = x(t) + hx(t)
? + O(h2 ).
When t = kh, Euler?s method approximates x(t + h) by xk+1 , neglecting the second order term,
xk+1 = xk + hg(xk ).
In optimization terms, we recognize the gradient descent algorithm used to minimize f . Approximation errors in an integration method accumulate with iterations, and as Euler?s method uses only the
last point to compute the next one, it has only limited control over the accumulated error.
2
Linear multistep methods. Multi-step methods use a combination of past iterates to improve
convergence. Throughout the paper, we focus on linear s-step methods whose recurrence can be
written
s?1
s
X
X
xk+s = ?
?i xk+i + h
?i g(xk+i ), for k ? 0,
i=0
i=0
where ?i , ?i ? R are the parameters of the multistep method and h is again the step size. Each new
point xk+s is a function of the information given by the s previous points. If ?s = 0, each new point
is given explicitly by the s previous points and the method is called explicit. Otherwise each new
point requires solving an implicit equation and the method is called implicit.
To simplify notations we use the shift operator E, which maps Exk ? xk+1 . Moreover, if we write
gk = g(xk ), then the shift operator also maps Egk ? gk+1 . Recall that a univariate polynomial is
called monic if its leading coefficient is equal to 1. We now give the following concise definition of
s-step linear methods.
Definition 2.1 Given an (ODE) defined by g, x0 , a step size h and x1 , . . . , xs?1 initial points, a
linear s-step method generates a sequence (tk , xk ) which satisfies
?(E)xk = h?(E)gk ,
for every k ? 0,
(2)
where ? is a monic polynomial of degree s with coefficients ?i , and ? a polynomial of degree s with
coefficients ?i .
A linear s?step method is uniquely defined by the polynomials (?, ?). The sequence generated by
the method then depends on the initial points and the step size. We now recall a few results describing
the performance of multistep methods.
2.2
Stability
Stability is a key concept for integration methods. First of all, consider two curves x(t) and y(t), both
solutions of (ODE), but starting from different points x(0) and y(0). If the function g is Lipchitzcontinuous, it is possible to show that the distance between x(t) and y(t) is bounded on a finite
interval, i.e.
kx(t) ? y(t)k ? Ckx(0) ? y(0)k
?t ? [0, tmax ],
where C may depend exponentially on tmax . We would like to have a similar behavior for our
sequences xk and yk , approximating x(tk ) and y(tk ), i.e.
kxk ? yk k ? kx(tk ) ? y(tk )k ? Ckx(0) ? y(0)k
?k ? [0, tmax /h],
(3)
when h ? 0, so k ? ?. Two issues quickly arise. First, for a linear s-step method, we need s
starting values x0 , ..., xs?1 . Condition (3) will therefore depend on all these starting values and
not only x0 . Secondly, any discretization scheme introduces at each step an approximation error,
called local error, which accumulates over time. We write this error loc (xk+s ) and define it as
loc (xk+s ) , xk+s ? x(tk+s ), where xk+s is computed using the real solution x(tk ), ..., x(tk+s?1 ).
In other words, the difference between xk and yk can be described as follows
kxk ? yk k ? Error in the initial condition + Accumulation of local errors.
We now write a complete definition of stability, inspired by Definition 6.3.1 from Gautschi [2011].
Definition 2.2 (Stability) A linear multistep method is stable iff, for two sequences xk , yk generated
by (?, ?) using a sufficiently small step size h > 0, from the starting values x0 , ..., xs?1 , and
y0 , ..., ys?1 , we have
kxk ? yk k ? C
tmax /h
max
i?{0,...,s?1}
kxi ? yi k +
X
kloc (xi+s )k + kloc (yi+s )k ,
(4)
i=1
for any k ? [0, tmax /h]. Here, the constant C may depend on tmax but is independent of h.
When h tends to zero, we may recover equation (3) only if the accumulated local error also tends to
zero. We thus need
1 loc
lim
k (xi+s )k = 0 ?i ? [0, tmax /h].
h?0 h
3
This condition is called consistency. The following proposition shows there exist simple conditions to
check consistency, which rely on comparing a Taylor expansion of the solution with the coefficients
of the method. Its proof and further details are given in the Supplementary Material.
Proposition 2.3 (Consistency) A linear multistep method defined by polynomials (?, ?) is consistent
if and only if
?(1) = 0
and
?0 (1) = ?(1).
(5)
Assuming consistency, we still need to control sensitivity to initial conditions, written
kxk ? yk k ? C
max
i?{0,...,s?1}
kxi ? yi k.
(6)
Interestingly, analyzing the special case where g = 0 is completely equivalent to the general case
and this condition is therefore called zero-stability. This reduces to standard linear algebra results as
we only need to look at the solution of the homogeneous difference equation ?(E)xk = 0. This is
captured in the following theorem whose technical proof can be found in [Gautschi, 2011, Theorem
6.3.4].
Theorem 2.4 (Root condition) Consider a linear multistep method (?, ?). The method is zero-stable
if and only if all roots of ?(z) are in the unit disk, and the roots on the unit circle are simple.
2.3
Convergence of the global error
Numerical analysis focuses on integrating an ODE on a finite interval of time [0, tmax ]. It studies the
behavior of the global error defined by x(tk ) ? xk , as a function of the step size h. If the global error
converges to 0 with the step size, the method is guaranteed to approximate correctly the ODE on the
time interval, for h small enough.
We now state Dahlquist?s equivalence theorem, which shows that the global error converges to zero
when h does if the method is stable, i.e. when the method is consistent and zero-stable. This naturally
needs the additional assumption that the starting values x0 , . . . , xs?1 are computed such that they
converge to the solution (x(0), . . . , x(ts?1 )). The proof of the theorem can be found in Gautschi
[2011].
Theorem 2.5 (Dahlquist?s equivalence theorem) Given an (ODE) defined by g and x0 and a consistent linear multistep method (?, ?), whose starting values are computed such that limh?0 xi =
x(ti ) for any i ? {0, . . . , s ? 1}, zero-stability is necessary and sufficient for convergence, i.e. to
ensure x(tk ) ? xk ? 0 for any k when the step size h goes to zero.
2.4
Region of absolute stability
The results above ensure stability and global error bounds on finite time intervals. Solving optimization problems however requires looking at infinite time horizons. We start by finding conditions
ensuring that the numerical solution does not diverge when the time interval increases, i.e. that the
numerical solution is stable with a constant C which does not depend on tmax . Formally, for a fixed
step-size h, we want to ensure
kxk k ? C
max
i?{0,...,s?1}
kxi k
for all k ? [0, tmax /h] and tmax > 0.
(7)
This is not possible without further assumptions on the function g as in the general case the solution
x(t) itself may diverge. We begin with the simple scalar linear case which, given ? > 0, reads
x(t)
?
= ??x(t),
x(0) = x0 .
(Scalar Linear ODE)
The recurrence of a linear multistep methods with parameters (?, ?) applied to (Scalar Linear ODE)
then reads
?(E)xk = ??h?(E)xk ? [? + ?h?](E)xk = 0,
where we recognize a homogeneous recurrence equation. Condition (7) is then controlled by the
step size h and the constant ?, ensuring that this homogeneous recurrent equation produces bounded
solutions. This leads us to the definition of the region of absolute stability, also called A-stability.
4
Definition 2.6 (Absolute stability) The region of absolute stability of a linear multistep method
defined by (?, ?) is the set of values ?h such that the characteristic polynomial
??h (z) , ?(z) + ?h ?(z)
(8)
of the homogeneous recurrence equation ??h (E)xk = 0 produces bounded solutions.
Standard linear algebra links this condition to the roots of the characteristic polynomial as recalled in
the next proposition (see e.g. Lemma 12.1 of S?li and Mayers [2003]).
Proposition 2.7 Let ? be a polynomial and write xk a solution of the homogeneous recurrence
equation ?(E)xk = 0 with arbitrary initial values. If all roots of ? are inside the unit disk and the
ones on the unit circle have a multiplicity exactly equal to one, then kxk k ? ?.
Absolute stability of a linear multistep method determines its ability to integrate a linear ODE defined
by
x(t)
?
= ?Ax(t),
x(0) = x0 ,
(Linear ODE)
where A is a positive symmetric matrix whose eigenvalues belong to [?, L] for 0 < ? ? L. In this
case the step size h must indeed be chosen such that for any ? ? [?, L], ?h belongs to the region of
absolute stability of the method. This (Linear ODE) is a special instance of (Gradient Flow) where f
is a quadratic function. Therefore absolute stability gives a necessary (but not sufficient) condition to
integrate (Gradient Flow) on L-smooth, ?-strongly convex functions.
2.5
Convergence analysis in the linear case
By construction, absolute stability also gives hints on the convergence of xk to the equilibrium in the
linear case. More precisiely, it allows us to control the rate of convergence of xk , approximating the
solution x(t) of (Linear ODE) as shown in the following proposition whose proof can be found in
Supplementary Material.
Proposition 2.8 Given a (Linear ODE) defined by x0 and a positive symmetric matrix A whose
eigenvalues belong to [?, L] with 0 < ? ? L, using a linear multistep method defined by (?, ?) and
applying a fixed step size h, we define rmax as
rmax = max
max
|r|,
??[?,L] r?roots(??h (z))
where ??h is defined in (8). If rmax < 1 and its multiplicity is equal to m, then the speed of
convergence of the sequence xk produced by the linear multistep method to the equilibrium x? of the
differential equation is given by
k
kxk ? x? k = O(k m?1 rmax
).
(9)
We can now use these properties to analyze and design multistep methods.
3
Analysis and design of multi-step methods
As shown previously, we want to integrate (Gradient Flow) and Proposition 1.1 gives a rate of
convergence in the continuous case. If the method tracks x(t) with sufficient accuracy, then the rate
of the method will be close to the rate of convergence of x(kh). So, larger values of h yield faster
convergence of x(t) to the equilibrium x? . However h cannot be too large, as the method may be
too inaccurate and/or unstable as h increases. Convergence rates of optimization algorithms are thus
controlled by our ability to discretize the gradient flow equation using large step sizes. We recall the
different conditions that proper linear multistep methods should satisfy.
? Monic polynomial (Section 2.1). Without loss of generality (dividing both sides of the
difference equation of the multistep method (2) by ?s does not change the method).
? Explicit method (Section 2.1). We assume that the scheme is explicit in order to avoid
solving a non-linear system at each step.
5
? Consistency (Section 2.2). If the method is not consistent, then the local error does not
converge when the step size goes to zero.
? Zero-stability (Section 2.2). Zero-stability ensures convergence of the global error (Section
2.3) when the method is also consistent.
? Region of absolute stability (Section 2.4). If ?h is not inside the region of absolute stability
for any ? ? [?, L], then the method is divergent when tmax increases.
Using the remaining degrees of freedom, we can tune the algorithm to improve the convergence
rate on (Linear ODE), which corresponds to the optimization of a quadratic function. Indeed, as
showed in Proposition 2.8, the largest root of ??h (z) gives us the rate of convergence on quadratic
functions (when ? ? [?, L]). Since smooth and strongly convex functions are close to quadratic (being
sandwiched between two quadratics), this will also give us a good idea of the rate of convergence
on these functions. We do not derive a proof of convergence of the sequence for general smooth
and (strongly) convex function (but convergence is proved by Nesterov [2013] or using Lyapunov
techniques by Wilson et al. [2016]). Still our results provide intuition on why accelerated methods
converge faster.
3.1
Analysis of two-step methods
We now analyze convergence of two-step methods (an analysis of Euler?s method is provided in the
Supplementary Material). We first translate the conditions multistep method, listed at the beginning
of this section, into constraints on the coefficients:
?2 = 1
?2 = 0
?0 + ?1 + ?2 = 0
?0 + ?1 + ?2 = ?1 + 2?2
|Roots(?)| ? 1
(Monic polynomial)
(Explicit method)
(Consistency)
(Consistency)
(Zero-stability).
Equality contraints yield three linear constraints, defining the set L such that
L = {?0 , ?1 , ?0 , ?1 :
?1 = ?(1 + ?0 ),
?1 = 1 ? ?0 ? ?0 ,
|?0 | < 1}.
(10)
We now seek conditions on the remaining parameters to produce a stable method. Absolute stability
requires that all roots of the polynomial ??h (z) in (8) are inside the unit circle, which translates into
condition on the roots of second order equations here. The following proposition gives the values of
the roots of ??h (z) as a function of the parameters ?i and ?i .
Proposition 3.1 Given constants 0 < ? ? L, a step size h > 0 and a linear two-step method defined
by (?, ?), under the conditions
(?1 + ?h?1 )2 ? 4(?0 + ?h?0 ),
(?1 + Lh?1 )2 ? 4(?0 + Lh?0 ),
(11)
the roots r? (?) of ??h , defined in (8), are complex conjugate for any ? ? [?, L]. Moreover, the
largest root modulus is equal to
max |r? (?)|2 = max {?0 + ?h?0 , ?0 + Lh?0 } .
??[?,L]
(12)
The proof can be found in the Supplementary Material. The next step is to minimize the largest
modulus (12) in the coefficients ?i and ?i to get the best rate of convergence, assuming the roots are
complex (the case were the roots are real leads to weaker results).
3.2
Design of a family of two-step methods for quadratics
We now have all ingredients to build a two-step method for which the sequence xk converges quickly
to x? for quadratic functions. Optimizing the convergence rate means solving the following problem,
min max {?0 + ?h?0 , ?0 + Lh?0 }
s.t. (?0 , ?1 , ?0 , ?1 ) ? L
(?1 + ?h?1 )2 ? 4(?0 + ?h?0 )
(?1 + Lh?1 )2 ? 4(?0 + Lh?0 ),
6
in the variables ?0 , ?1 , ?0 , ?1 , h > 0, where L is defined in (10). If we use the equality constraints
in (10) and make the following change of variables,
? = h(1 ? ?0 ), c? = ?0 + ?h?0 , cL = ?0 + Lh?0 ,
h
(13)
? in the variables c? , cL . In that case, the optimal solution is
the problem can be solved, for fixed h,
given by
q
p
? 2 , c? = (1 ? Lh)
? 2,
c? = (1 ? ?h)
(14)
?
L
2
? ?]0, (1+?/L) [. Now if we fix h
? we can recover
obtained by tightening the two first inequalities, for h
L
a two step linear method defined by
step size h by using the equations in (13). We define
p (?, ?) and a p
the following quantity ? = (1 ? ?/L)/(1 + ?/L).
? = 1/L for example, the parameters of the correspondA suboptimal two-step method. Setting h
ing two-step method, called method M1 , are
1
M1 = ?(z) = ? ? (1 + ?)z + z 2 , ?(z) = ??(1 ? ?) + (1 ? ? 2 )z, h =
L(1 ? ?)
(15)
and its largest modulus root (12) is given by
q
p
?
rate(M1 ) = max{c? , cL } = c? = 1 ? ?/L.
? which minimizes the
Optimal two-step method for quadratics. We can compute the optimal h
maximum of the two roots c?? and c?L defined in (14). The solution simply balances the two terms in
? ? = (1 + ?)2 /L. This choice of h
? leads to the method M2 , described by
the maximum, with h
1
M2 = ?(z) = ? 2 ? (1 + ? 2 )z + z 2 , ?(z) = (1 ? ? 2 )z, h = ?
(16)
?L
with convergence rate
p
p
?
?
rate(M2 ) = c? = cL = ? = (1 ? ?/L)/(1 + ?/L) < rate(M1 ).
We will now see that methods M1 and M2 are actually related to Nesterov?s accelerated method and
Polyak?s heavy ball algorithms.
4
On the link between integration and optimization
? We
In the previous section, we derived a family of linear multistep methods, parametrized by h.
will now compare these methods to common optimization algorithms used to minimize L-smooth,
?-strongly convex functions.
4.1
Polyak?s heavy ball method
The heavy ball method was proposed by Polyak [1964]. It adds a momentum term to the gradient step
xk+2 = xk+1 ? c1 ?f (xk+1 ) + c2 (xk+1 ? xk ),
?
? 2
where c1 = 4/( L + ?) and c2 = ? 2 . We can organize the terms in the sequence to match the
general structure of linear multistep methods, to get
? 2 xk ? (1 + ? 2 )xk+1 + xk+2 = c1 (??f (xk+1 )) .
We easily identify ?(z) = ? 2 ?(1+? 2 )z +z 2 and h?(z) = c1 z. To extract h, we will assume that the
method is consistent (see conditions (5)). All computations done, we can identify the corresponding
linear multistep method as
1
2
2
2
MPolyak = ?(z) = ? ? (1 + ? )z + 1, ?(z) = (1 ? ? )z, h = ?
.
(17)
?L
This shows that MPolyak = M2 . In fact, this result was expected since Polyak?s method is known
to be optimal for quadratic functions. However, it is also known that Polyak?s algorithm does not
converge for a general smooth and strongly convex function [Lessard et al., 2016].
7
4.2
Nesterov?s accelerated gradient
Nesterov?s accelerated method in its simplest form is described by two sequences xk and yk , with
yk+1
=
xk+1
=
1
?f (xk ),
L
yk+1 + ?(yk+1 ? yk ).
xk ?
As above, we will write Nesterov?s accelerated gradient as a linear multistep method by expanding
yk in the definition of xk , to get
1
(??(??f (xk )) + (1 + ?)(??f (xk+1 ))) .
L
Again, assuming as above that the method is consistent to extract h, we identify the linear multistep
method associated to Nesterov?s algorithm. After identification,
1
2
2
MNest = ?(z) = ? ? (1 + ?)z + z , ?(z) = ??(1 ? ?) + (1 ? ? )z, h =
,
L(1 ? ?)
?xk ? (1 + ?)xk+1 + xk+2 =
which means that M1 = MNest .
4.3
The convergence rate of Nesterov?s method
Pushing the analysis a little bit further, we have a simple intuitive argument that explains why
Nesterov?s algorithm is faster than the gradient method. There is of course a complete proof of its
rate of convergence [Nesterov, 2013], even using differential equations arguments [Wibisono et al.,
2016; Wilson et al., 2016], but we take a simpler approach here. The key parameter is the step size h.
If we compare it with the step
p size in the classical gradient method, Nesterov?s method uses a step
size which is (1 ? ?)?1 ? L/? times larger.
Recall that, in continuous time, the rate of convergence of x(t) to x? is given by
f (x(t)) ? f (x? ) ? e?2?t (f (x0 ) ? f (x? )).
The gradient method tries to approximate x(t) using an Euler scheme with step size h = 1/L, which
(grad)
means xk
? x(k/L), so
(grad)
f (xk
?
) ? f (x? ) ? f (x(k/L)) ? f (x? ) ? (f (x0 ) ? f (x? ))e?2k L .
However, Nesterov?s method has a step size equal to
p
p
1 + ?/L
1
1
?
hNest =
=
4?L
.
??
which means xnest
?
x
k/
k
L(1 ? ?)
2 ?L
4?L
while maintaining stability. In that case, the estimated rate of convergence becomes
?
?
?
f (xnest
? f (x? ) ? (f (x0 ) ? f (x? ))e?k ?/L ,
k ) ? f (x ) ? f x k/ 4?L
which is approximatively the rate
p of convergence of Nesterov?s algorithm in discrete time and we
recover the accelerated rate in ?/L versus ?/L for gradient descent.
Overall, the accelerated method is more efficient because it integrates the gradient flow faster than
simple gradient descent, making longer steps. A numerical simulation in Figure 1 makes this argument
more visual.
5
Generalization and Future Work
We showed that accelerated optimization methods can be seen as multistep integration schemes
applied to the basic gradient flow equation. Our results give a natural interpretation of acceleration:
multistep schemes allow for larger steps, which speeds up convergence. In the Supplementary
Material, we detail further links between integration methods and other well-known optimization
algorithms such as proximal point methods, mirror gradient decent, proximal gradient descent, and
8
9
9
x0
xstar
Exact
Euler
Nesterov
Polyak
8.5
8
x0
xstar
Exact
Euler
Nesterov
Polyak
8.5
8
7.5
7.5
7
7
6.5
6.5
6
6
5
5.5
6
6.5
7
5
5.5
6
6.5
Figure 1: Integration of a linear ODE with optimal (left) and small (right) step sizes.
discuss the weakly convex case. The extra-gradient algorithm and its recent accelerated version Diakonikolas and Orecchia [2017] can also be linked to another family of integration methods called
Runge-Kutta which include notably predictor-corrector methods.
Our stability analysis is limited to the quadratic case, the definition of A-stability being too restrictive
for the class of smooth and strongly convex functions. A more appropriate condition would be Gstability, which extends A-stability to non-linear ODEs, but this condition requires strict monotonicity
of the error (which is not the case with accelerated algorithms). Stability may also be tackled by
recent advances in lower bound theory provided by Taylor [2017] but these yield numerical rather
than analytical convergence bounds. Our next objective is thus to derive a new stability condition in
between A-stability and G-stability.
Acknowledgments
The authors would like to acknowledge support from a starting grant from the European Research
Council (ERC project SIPA), from the European Union?s Seventh Framework Programme (FP7PEOPLE-2013-ITN) under grant agreement number 607290 SpaRTaN, an AMX fellowship, as well as
support from the chaire ?conomie des nouvelles donn?es with the data science joint research initiative
with the fonds AXA pour la recherche and a gift from Soci?t? G?n?rale Cross Asset Quantitative
Research.
9
References
Allen Zhu, Z. and Orecchia, L. [2017], Linear coupling: An ultimate unification of gradient and
mirror descent, in ?Proceedings of the 8th Innovations in Theoretical Computer Science?, ITCS 17.
Beck, A. and Teboulle, M. [2003], ?Mirror descent and nonlinear projected subgradient methods for
convex optimization?, Operations Research Letters 31(3), 167?175.
Ben-Tal, A. and Nemirovski, A. [2001], Lectures on modern convex optimization: analysis, algorithms, and engineering applications, SIAM.
Bubeck, S., Tat Lee, Y. and Singh, M. [2015], ?A geometric alternative to nesterov?s accelerated
gradient descent?, ArXiv e-prints .
Diakonikolas, J. and Orecchia, L. [2017], ?Accelerated extra-gradient descent: A novel accelerated
first-order method?, arXiv preprint arXiv:1706.04680 .
Duchi, J. C., Shalev-Shwartz, S., Singer, Y. and Tewari, A. [2010], Composite objective mirror
descent., in ?COLT?, pp. 14?26.
Gautschi, W. [2011], Numerical analysis, Springer Science & Business Media.
Krichene, W., Bayen, A. and Bartlett, P. L. [2015], Accelerated mirror descent in continuous and
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Lessard, L., Recht, B. and Packard, A. [2016], ?Analysis and design of optimization algorithms via
integral quadratic constraints?, SIAM Journal on Optimization 26(1), 57?95.
Nesterov, Y. [1983], A method of solving a convex programming problem with convergence rate o
(1/k2), in ?Soviet Mathematics Doklady?, Vol. 27, pp. 372?376.
Nesterov, Y. [2007], ?Gradient methods for minimizing composite objective function?.
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Science & Business Media.
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Programming 152(1-2), 381?404.
Polyak, B. T. [1964], ?Some methods of speeding up the convergence of iteration methods?, USSR
Computational Mathematics and Mathematical Physics 4(5), 1?17.
Su, W., Boyd, S. and Candes, E. [2014], A differential equation for modeling nesterov?s accelerated
gradient method: Theory and insights, in ?Advances in Neural Information Processing Systems?,
pp. 2510?2518.
S?li, E. and Mayers, D. F. [2003], An introduction to numerical analysis, Cambridge University
Press.
Taylor, A. [2017], Convex Interpolation and Performance Estimation of First-order Methods for
Convex Optimization, PhD thesis, Universit? catholique de Louvain.
Wibisono, A., Wilson, A. C. and Jordan, M. I. [2016], ?A variational perspective on accelerated
methods in optimization?, Proceedings of the National Academy of Sciences p. 201614734.
Wilson, A. C., Recht, B. and Jordan, M. I. [2016], ?A lyapunov analysis of momentum methods in
optimization?, arXiv preprint arXiv:1611.02635 .
10
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accelerated:19 wibisono:3 phenomenon:2 |
6,315 | 6,712 | Sharpness, Restart and Acceleration
Vincent Roulet
INRIA, ENS
Paris France
[email protected]
Alexandre d?Aspremont
CNRS, ENS
Paris France
[email protected]
Abstract
The ?ojasiewicz inequality shows that sharpness bounds on the minimum of convex
optimization problems hold almost generically. Sharpness directly controls the
performance of restart schemes, as observed by Nemirovskii and Nesterov [1985].
The constants quantifying error bounds are of course unobservable, but we show
that optimal restart strategies are robust, and searching for the best scheme only
increases the complexity by a logarithmic factor compared to the optimal bound.
Overall then, restart schemes generically accelerate accelerated methods.
Introduction
We study convex optimization problems of the form
minimize f (x)
(P)
where f is a convex function defined on Rn . The complexity of these problems using first order
methods is generically controlled by smoothness assumptions on f such as Lipschitz continuity of its
gradient. Additional assumptions such as strong convexity or uniform convexity provide respectively
linear [Nesterov, 2013b] and faster polynomial [Juditski and Nesterov, 2014] rates of convergence.
However, these assumptions are often too restrictive to be applied. Here, we make a much weaker and
generic assumption that describes the sharpness of the function around its minimizers by constants
? ? 0 and r ? 1 such that
?
d(x, X ? )r ? f (x) ? f ? , for every x ? K,
(Sharp)
r
where f ? is the minimum of f , K ? Rn is a compact set, d(x, X ? ) = miny?X ? kx ? yk is the
distance from x to the set X ? ? K of minimizers of f 1 for the Euclidean norm k ? k. This defines a
lower bound on the function around its minimizers: for r = 1, f shows a kink around its minimizers
and the larger is r the flatter is the function around its minimizers. We tackle this property by restart
schemes of classical convex optimization algorithms.
Sharpness assumption (Sharp) is better known as a H?lderian error bound on the distance to the set
of minimizers. Hoffman [Hoffman, 1952] first introduced error bounds to study system of linear
inequalities. Natural extensions were then developed for convex optimization [Robinson, 1975;
Mangasarian, 1985; Auslender and Crouzeix, 1988], notably through the concept of sharp minima
[Polyak, 1979; Burke and Ferris, 1993; Burke and Deng, 2002]. But the most striking discovery was
made by ?ojasiewicz [?ojasiewicz, 1963, 1993] who proved inequality (Sharp) for real analytic and
subanalytic functions. It has then been extended to non-smooth subanalytic convex functions by
Bolte et al. [2007]. Overall, since (Sharp) essentially measures the sharpness of minimizers, it holds
somewhat generically. On the other hand, this inequality is purely descriptive as we have no hope of
ever observing either r or ?, and deriving adaptive schemes is crucial to ensure practical relevance.
1
We assume the problem feasible, i.e. X ? 6= ?.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
?ojasiewicz inequalities either in the form of (Sharp) or as gradient dominated properties [Polyak,
1979] led to new simple convergence results [Karimi et al., 2016], in particular for alternating and
splitting methods [Attouch et al., 2010; Frankel et al., 2015], even in the non-convex case [Bolte et al.,
2014]. Here we focus on H?lderian error bounds as they offer simple explanation of accelerated rates
of restart schemes.
Restart schemes were already studied for strongly or uniformly convex functions [Nemirovskii and
Nesterov, 1985; Nesterov, 2013a; Juditski and Nesterov, 2014; Lin and Xiao, 2014]. In particular,
Nemirovskii and Nesterov [1985] link a ?strict minimum? condition akin to (Sharp) with faster
convergence rates using restart schemes which form the basis of our results, but do not study the
cost of adaptation and do not tackle the non-smooth case. In a similar spirit, weaker versions of this
strict minimum condition were used more recently to study the performance of restart schemes in
[Renegar, 2014; Freund and Lu, 2015; Roulet et al., 2015]. The fundamental question of a restart
scheme is naturally to know when must an algorithm be stopped and relaunched. Several heuristics
[O?Donoghue and Candes, 2015; Su et al., 2014; Giselsson and Boyd, 2014] studied adaptive restart
schemes to speed up convergence of optimal methods. The robustness of restart schemes was then
theoretically studied by Fercoq and Qu [2016] for quadratic error bounds, i.e. (Sharp) with r = 2,
that LASSO problem satisfies for example. Fercoq and Qu [2017] extended recently their work to
produce adaptive restarts with theoretical guarantees of optimal performance, still for quadratic error
bounds. Previous references focus on smooth problems, but error bounds appear also for non-smooth
ones, Gilpin et al. [2012] prove for example linear converge of restart schemes in bilinear matrix
games where the minimum is sharp, i.e. (Sharp) with r = 1.
Our contribution here is to derive optimal scheduled restart schemes for general convex optimization
problems for smooth, non-smooth or H?lder smooth functions satisfying the sharpness assumption.
We then show that for smooth functions these schemes can be made adaptive with nearly optimal
complexity (up to a squared log term) for a wide array of sharpness assumptions. We also analyze
restart criterion based on a sufficient decrease of the gap to the minimum value of the problem, when
this latter is known in advance. In that case, restart schemes are shown ot be optimal without requiring
any additional information on the function.
1
1.1
Problem assumptions
Smoothness
Convex optimization problems (P) are generally divided in two classes: smooth problems, for which
f has Lipschitz continuous gradients, and non-smooth problems for which f is not differentiable.
Nesterov [2015] proposed to unify point of views by assuming generally that there exist constants
1 ? s ? 2 and L > 0 such that
k?f (x) ? ?f (y)k ? Lkx ? yks?1 ,
for all x, y ? Rn
(Smooth)
where ?f (x) is any sub-gradient of f at x if s = 1 (otherwise this implies differentiability of f ).
For s = 2, we retrieve the classical definition of smoothness [Nesterov, 2013b]. For s = 1 we get a
classical assumption made in non-smooth convex optimization, i.e., that sub-gradients of the function
are bounded. For 1 < s < 2, this assumes gradient of f to be H?lder Lipschitz continuous. In a
first step, we will analyze restart schemes for smooth convex optimization problems, then generalize
to general smoothness assumption (Smooth) using appropriate accelerated algorithms developed by
Nesterov [2015].
1.2
Error bounds
In general, an error bound is an inequality of the form
d(x, X ? ) ? ?(f (x) ? f ? ),
where ? is an increasing function at 0, called the residual function, and x may evolve either in the
whole space or in a bounded set, see Bolte et al. [2015] for more details. We focus on H?lderian
Error Bounds (Sharp) as they are the most common in practice. They are notably satisfied by a
analytic and subanalytic functions but the proof (see e.g. Bierstone and Milman [1988, Theorem
6.4]) is shown using topological arguments that are far from constructive. Hence, outside of some
2
particular cases (e.g. strong convexity), we cannot assume that the constants in (Sharp) are known,
even approximately.
Error bounds can generically be linked to ?ojasiewicz inequality that upper bounds magnitude of the
gradient by values of the function [Bolte et al., 2015]. Such property paved the way to many recent
results in optimization [Attouch et al., 2010; Frankel et al., 2015; Bolte et al., 2014]. Here we will
see that (Sharp) is sufficient to acceleration of convex optimization algorithms by their restart. Note
finally that in most cases, error bounds are local properties hence the convergence results that follow
will generally be local.
1.3
Sharpness and smoothness
Let f be a convex function on Rn satisfying (Smooth) with parameters (s, L). This property ensures
that, f (x) ? f ? + Ls kx ? yks , for given x ? Rn and y ? X ? . Setting y to be the projection of x
onto X ? , this yields the following upper bound on suboptimality
f (x) ? f ? ?
L
d(x, X ? )s .
s
(1)
Now, assume that f satisfies the error bound (Sharp) on a set K with parameters (r, ?). Combining (1)
and (Sharp) this leads for every x ? K,
s?
? d(x, X ? )s?r .
rL
This means that necessarily s ? r by taking x ? X ? . Moreover if s < r, this last inequality can
only be valid on a bounded set, i.e. either smoothness or error bound or both are valid only on a
bounded set. In the following, we write
2
2
s
? , L s /? r
(2)
and
? ,1?
r
respectively a generalized condition number for the function f and a condition number based on the
ratio of powers in inequalities (Smooth) and (Sharp). If r = s = 2, ? matches the classical condition
number of the function.
2
Scheduled restarts for smooth convex problems
In this section, we seek to solve (P) assuming that the function f is smooth, i.e. satisfies (Smooth)
with s = 2 and L > 0. Without further assumptions on f , an optimal algorithm to solve the smooth
convex optimization problem (P) is Nesterov?s accelerated gradient method [Nesterov, 1983]. Given
an initial point x0 , this algorithm outputs, after t iterations, a point x = A(x0 , t) such that
f (x) ? f ? ?
cL
d(x0 , X ? )2 ,
t2
(3)
where c > 0 denotes a universal constant (whose value will be allowed to vary in what follows, with
c = 4 here). We assume without loss of generality that f (x) ? f (x0 ). More details about Nesterov?s
algorithm are given in Supplementary Material.
In what follows, we will also assume that f satisfies (Sharp) with parameters (r, ?) on a set K ? X ? ,
which means
?
d(x, X ? )r ? f (x) ? f ? , for every x ? K.
(Sharp)
r
As mentioned before if r > s = 2, this property is necessarily local, i.e. K is bounded. We assume
then that given a starting point x0 ? Rn , sharpness is satisfied on the sublevel set {x| f (x) ? f (x0 )}.
Remark that if this property is valid on an open set K ? X ? , it will also be valid on any compact set
K 0 ? K with the same exponent r but a potentially lower constant ?. The scheduled restart schemes
we present here rely on a global sharpness hypothesis on the sublevel set defined by the initial point
and are not adaptive to constant ? on smaller sublevel sets. On the other hand, restarts on criterion
that we present in Section 4, assuming that f ? is known, adapt to the value of ?. We now describe a
restart scheme exploiting this extra regularity assumption to improve the computational complexity
of solving problem (P) using accelerated methods.
3
2.1
Scheduled restarts
Here, we schedule the number of iterations tk made by Nesterov?s algorithm between restarts, with
tk the number of (inner) iterations at the k th algorithm run (outer iteration). Our scheme is described
in Algorithm 1 below.
Algorithm 1 Scheduled restarts for smooth convex minimization
Inputs : x0 ? Rn and a sequence tk for k = 1, . . . , R.
for k = 1, . . . , R do
xk := A(xk?1 , tk )
end for
Output : x
? := xR
The analysis of this scheme and the following ones relies on two steps. We first choose schedules that
ensure linear convergence in the iterates xk at a given rate. We then adjust this linear rate to minimize
the complexity in terms of the total number of iterations.
We begin with a technical lemma which assumes linear convergence holds, and connects the growth
of tk , the precision reached and the total number of inner iterations N .
Lemma 2.1. Let xk be a sequence whose k th iterate is generated from the previous one by an
PR
algorithm that runs tk iterations and write N = k=1 tk the total number of iterations to output a
point xR . Suppose setting tk = Ce?k , k = 1, . . . , R for some C > 0 and ? ? 0 ensures that outer
iterations satisfy
f (xk ) ? f ? ? ?e??k ,
(4)
for all k ? 0 with ? ? 0 and ? ? 0. Then precision at the output is given by,
f (xR ) ? f ? ? ? exp(??N/C),
and
f (xR ) ? f ? ?
when ? = 0,
?
?
(?e?? C ?1 N + 1) ?
,
when ? > 0.
Proof. When ? = 0, N = RC, and inserting this in (4) at the last point xR yields the desired
PR
? e?R ?1
result. On the other hand, when ? > 0, we have N =
k=1 tk = Ce e? ?1 , which gives
?
R = log ee??1
C N + 1 /?. Inserting this in (4) at the last point, we get
?
?
f (xR ) ? f ? ? ? exp ? ?? log ee??1
?
? ,
C N +1
?? ?1
(?e
C
N +1) ?
where we used ex ? 1 ? x. This yields the second part of the result.
The last approximation in the case ? > 0 simplifies the analysis that follows without significantly
affecting the bounds. We also show in Supplementary Material that using t?k = dtk e does not
significantly affect the bounds above. Remark that convergence bounds are generally linear or
polynomial such that we can extract a subsequence that converges linearly. Therefore our approach
does not restrict the analysis of our scheme. It simplifies it and can be used for other algorithms like
the gradient descent as detailed in Supplementary Material.
We now analyze restart schedules tk that ensure linear convergence. Our choice of tk will heavily
depend on the ratio between r and s (with s = 2 for smooth functions here), incorporated in the
parameter ? = 1 ? s/r defined in (2). Below, we show that if ? = 0, a constant schedule is sufficient
to ensure linear convergence. When ? > 0, we need a geometrically increasing number of iterations
for each cycle.
Proposition 2.2. Let f be a smooth convex function satisfying (Smooth) with parameters (2, L)
and (Sharp) with parameters (r, ?) on a set K. Assume that we are given x0 ? Rn such that
?
{x| f (x) ? f (x0 )} ? K. Run Algorithm 1 from x0 with iteration schedule tk = C?,?
e? k , for
k = 1, . . . , R, where
1
?
?
C?,?
, e1?? (c?) 2 (f (x0 ) ? f ? )? 2 ,
(5)
4
with ? and ? defined in (2) and c = 4e2/e here. The precision reached at the last point x
? is given by,
1
1
f (?
x) ? f ? ? exp ?2e?1 (c?)? 2 N (f (x0 ) ? f ? ) = O exp(??? 2 N ) , when ? = 0, (6)
while,
f (?
x) ? f ? ?
where N =
PR
k=1 tk
f (x0 ) ? f ?
? e?1 (f (x
0)
?
?
1
f ? ) 2 (c?)? 2 N
+1
?2
?2 = O N ? ,
when ? > 0,
(7)
is the total number of iterations.
Proof. Our strategy is to choose tk such that the objective is linearly decreasing, i.e.
f (xk ) ? f ? ? e??k (f (x0 ) ? f ? ),
(8)
for some ? ? 0 depending on the choice of tk . This directly holds for k = 0 and any ? ? 0.
Combining (Sharp) with the complexity bound in (3), we get
f (xk ) ? f ? ?
c?
(f
t2k
2
(xk?1 ) ? f ? ) r ,
where c = 4e2/e using that r2/r ? e2/e . Assuming recursively that (8) is satisfied at iteration k ? 1
for a given ?, we have
f (xk ) ? f ? ?
2
c?e?? r (k?1)
(f (x0 )
t2k
2
? f ?) r ,
and to ensure (8) at iteration k, we impose
2
c?e?? r (k?1)
(f (x0 )
t2k
2
? f ? ) r ? e??k (f (x0 ) ? f ? ).
Rearranging terms in this last inequality, using ? defined in (2), we get
tk ? e
?(1?? )
2
1
?
(c?) 2 (f (x0 ) ? f ? )? 2 e
??
2
k
.
(9)
For a given ? ? 0, we can set tk = Ce?k where
C=e
?(1?? )
2
1
?
(c?) 2 (f (x0 ) ? f ? )? 2
and
? = ? ?/2,
(10)
and Lemma 2.1 then yields,
?
1
f (?
x) ? f ? ? exp ??e? 2 (c?)? 2 N (f (x0 ) ? f ? ),
when ? = 0, while
f (?
x) ? f ? ?
(f (x0 )?f ? )
?
?
?
2
2 ?e
2
1
?
(c?)? 2 (f (x0 )?f ? ) 2 N +1 ?
,
when ? > 0. These bounds are minimal for ? = 2, which yields the desired result.
When ? = 0, bound (6) matches the classical complexity bound for smooth strongly convex functions [Nesterov, 2013b]. When ? > 0 on the other hand, bound (7) highlights a much faster
convergence rate than accelerated gradient methods. The sharper the function (i.e. the smaller r), the
faster the convergence. This matches the lower bounds for optimizing smooth and sharp functions
functions [Arjevani and Shamir, 2016; Nemirovskii and Nesterov, 1985, Page 6] up to constant
?
factors. Also, setting tk = C?,?
e? k yields continuous bounds on precision, i.e. when ? ? 0, bound
(7) converges to bound (6), which also shows that for ? near zero, constant restart schemes are almost
optimal.
5
2.2
Adaptive scheduled restart
The previous restart schedules depend on the sharpness parameters (r, ?) in (Sharp). In general of
course, these values are neither observed nor known a priori. Making our restart scheme adaptive is
thus crucial to its practical performance. Fortunately, we show below that a simple logarithmic grid
search strategy on these parameters is enough to guarantee nearly optimal performance.
We run several schemes with a fixed number of inner iterations N to perform a log-scale grid search
on ? and ?. We define these schemes
as follows.
Si,0 : Algorithm 1 with tk = Ci ,
(11)
Si,j : Algorithm 1 with tk = Ci e?j k ,
where Ci = 2i and ?j = 2?j . We stop these schemes when the total number of inner algorithm
PR
iterations has exceed N , i.e. at the smallest R such that k=1 tk ? N . The size of the grid search in
Ci is naturally bounded as we cannot restart the algorithm after more than N total inner iterations,
so i ? [1, . . . , blog2 N c]. We will also show that when ? is smaller than 1/N , a constant schedule
performs as well as the optimal geometrically increasing schedule, which crucially means we can
also choose j ? [1, . . . , dlog2 N e] and limits the cost of grid search. The following result details the
convergence of this method, its notations are the same as in Proposition 2.2 and its technical proof
can be found in Supplementary Material.
Proposition 2.3. Let f be a smooth convex function satisfying (Smooth) with parameters (2, L)
and (Sharp) with parameters (r, ?) on a set K. Assume that we are given x0 ? Rn such that
{x| f (x) ? f (x0 )} ? K and denote N a given number of iterations. Run schemes Si,j defined in
(11) to solve (P) for i ? [1, . . . , blog2 N c] and j ? [0, . . . , dlog2 N e], stopping each time after N
PR
total inner algorithm iterations i.e. for R such that k=1 tk ? N .
?
Assume N is large enough, so N ? 2C?,?
, and if
1
N
?
> 1.
> ? > 0, C?,?
If ? = 0, there exists i ? [1, . . . , blog2 N c] such that scheme Si,0 achieves a precision given by
1
f (?
x) ? f ? ? exp ?e?1 (c?)? 2 N (f (x0 ) ? f ? ).
If ? > 0, there exist i ? [1, . . . , blog2 N c] and j ? [1, . . . , dlog2 N e] such that scheme Si,j achieves
a precision given by
f (?
x) ? f ? ?
f (x0 )?f ?
1
? e?1 (c?)? 2
2
?
(f (x0 )?f ? ) 2 (N ?1)/4+1 ?
.
Overall, running the logarithmic grid search has a complexity (log2 N )2 times higher than running
N iterations using the optimal (oracle) scheme.
As showed in Supplementary Material, scheduled restart schemes are theoretically efficient only if
the algorithm itself makes a sufficient number of iterations to decrease the objective value. Therefore
we need N large enough to ensure the efficiency of the adaptive method. If ? = 0, we naturally
?
?
?
?
have C?,0
? 1, therefore if N1 > ? > 0 and N is large, assuming C?,?
? C?,0
, we get C?,?
? 1.
This adaptive bound is similar to the one of Nesterov [2013b] to optimize smooth strongly convex
functions in the sense that we lose approximately a log factor of the condition number of the function.
However our assumptions are weaker and we are able to tackle all regimes of the sharpness property,
i.e. any exponent r ? [2, +?], not just the strongly convex case.
In the supplementary material we also analyze the simple gradient descent method under the sharpness
(Sharp) assumption. It shows that simple gradient descent achieves a O(?? ) complexity for a given
accuracy . Therefore restarting accelerated gradient methods reduces complexity to O(?? /2 )
compared to simple gradient descent. This result is similar to the acceleration of gradient descent. We
extend now this restart scheme to solve non-smooth or H?lder smooth convex optimization problem
under the sharpness assumption.
3
Universal scheduled restarts for convex problems
In this section, we use the framework introduced by Nesterov [2015] to describe smoothness of a
convex function f , namely, we assume that there exist s ? [1, 2] and L > 0 on a set J ? Rn , i.e.
k?f (x) ? ?f (y)k ? Lkx ? yks?1 , for every x, y ? J.
6
Without further assumptions on f , the optimal rate of convergence for this class of functions is
bounded as O(1/N ? ), where N is the total number of iterations and
? = 3s/2 ? 1,
(12)
which gives ? = 2 for smooth functions and ? = 1/2 for non-smooth functions. The universal
fast gradient method [Nesterov, 2015] achieves this rate by requiring only a target accuracy and a
starting point x0 . It outputs after t iterations a point x , U(x0 , , t), such that
2
cL s d(x0 , X ? )2
+
,
(13)
2 2?
2
2
s t s
4s?2
where c is a constant (c = 2 s ). More details about the universal fast gradient method are given in
Supplementary Material.
f (x) ? f ? ?
We will again assume that f is sharp with parameters (r, ?) on a set K ? X ? , i.e.
?
d(x, X ? )r ? f (x) ? f ? , for every x ? K.
(Sharp)
r
As mentioned in Section 1.2, if r > s, smoothness or sharpness are local properties, i.e. either J or
K or both are bounded, our analysis is therefore local. In the following we assume for simplicity,
given an initial point x0 , that smoothness and sharpness are satisfied simultaneously on the sublevel
set {x| f (x) ? f (x0 )}. The key difference with the smooth case described in the previous section is
that here we schedule both the target accuracy k used by the algorithm and the number of iterations
tk made at the k th run of the algorithm. Our scheme is described in Algorithm 2.
Algorithm 2 Universal scheduled restarts for convex minimization
Inputs : x0 ? Rn , 0 ? f (x0 ) ? f ? , ? ? 0 and a sequence tk for k = 1, . . . , R.
for k = 1, . . . , R do
k := e?? k?1 ,
xk := U(xk?1 , k , tk )
end for
Output : x
? := xR
Our strategy is to choose a sequence tk that ensures
f (xk ) ? f ? ? k ,
for the geometrically decreasing sequence k . The overall complexity of our method will then depend
on the growth of tk as described in Lemma 2.1. The proof is similar to the smooth case and can be
found in Supplementary Material.
Proposition 3.1. Let f be a convex function satisfying (Smooth) with parameters (s, L) on a set J and
(Sharp) with parameters (r, ?) on a set K. Given x0 ? Rn assume that {x|f (x) ? f (x0 )} ? J ? K.
Run Algorithm 2 from x0 for a given 0 ? f (x0 ) ? f ? with
? = ?,
?
tk = C?,?,?
e? k ,
s
? ??
?
where C?,?,?
, e1?? (c?) 2? 0
where ? is defined in (12), ? and ? are defined in (2) and c = 8e2/e here. The precision reached at
the last point x
? is given by,
s
s
f (?
x) ? f ? ? exp ??e?1 (c?)? 2? N 0 = O exp(??? 2? N ) , when ? = 0,
while,
f (?
x) ? f ? ?
where N =
PR
k=1 tk
0
?
?
s
? e?1 (c?)? 2? 0
N +1
s
??
? ?? = O ? 2? N ? ,
when ? > 0,
is total number of iterations.
This bound matches the lower bounds for optimizing smooth and sharp functions [Nemirovskii and
Nesterov, 1985, Page 6] up to constant factors. Notice that, compared to Nemirovskii and Nesterov
[1985], we can tackle non-smooth convex optimization by using the universal fast gradient algorithm
of Nesterov [2015]. The rate of convergence in Proposition 3.1 is controlled by the ratio between ?
and ?. If these are unknown, a log-scale grid search won?t be able to reach the optimal rate, even if ?
is known since we will miss the optimal rate by a constant factor. If both are known, in the case of
non-smooth strongly convex functions for example, a grid-search on C recovers nearly the optimal
bound. Now we will see that if f ? is known, restart produces adaptive optimal rates.
7
4
Restart with termination criterion
Here, we assume that we know the optimum f ? of (P), or have an exact termination criterion. This is
the case for example in zero-sum matrix games problems or non-degenerate least-squares without
regularization. We assume again that f satisfies (Smooth) with parameters (s, L) on a set J and
(Sharp) with parameters (r, ?) on a set K. Given an initial point x0 we assume that smoothness and
sharpness are satisfied simultaneously on the sublevel set {x| f (x) ? f (x0 )}. We use again the
universal gradient method U. Here however, we can stop the algorithm when it reaches the target
accuracy as we know the optimum f ? , i.e. we stop after t inner iterations such that x = U(x0 , , t )
satisfies f (x) ? f ? ? , and write x , C(x0 , ) the output of this method.
Here we simply restart this method and decrease the target accuracy by a constant factor after each
restart. Our scheme is described in Algorithm 3.
Algorithm 3 Restart on criterion
Inputs : x0 ? Rn , f ? , ? ? 0, 0 = f (x0 ) ? f ?
for k = 1, . . . , R do
k := e?? k?1 ,
end for
Output : x
? := xR
xk := C(xk?1 , k )
The following result describes the convergence of this method. It relies on the idea that it cannot do
more iterations than the best scheduled restart to achieve the target accuracy at each iteration. Its
proof can be found in Supplementary Material.
Proposition 4.1. Let f be a convex function satisfying (Smooth) with parameters (s, L) on a set J
and (Sharp) with parameters (r, ?) on a set K. Given x0 ? Rn assume that {x, f (x) ? f (x0 )} ?
J ? K. Run Algorithm 3 from x0 with parameter ? = ?. The precision reached at the last point x
? is
given by,
s
s
f (?
x) ? f ? ? exp ??e?1 (c?)? 2? N (f (x0 ) ? f ? ) = O exp(??? 2? N ) , when ? = 0,
while,
f (?
x) ? f ? ?
f (x0 ) ? f ?
s
? e?1 (c?)? 2? (f (x
0)
?
?
f ?) ? N
+1
s
??
?? = O ? 2? N ? ,
when ? > 0,
where N is the total number of iterations, ? is defined in (12), ? and ? are defined in (2) and c = 8e2/e
here.
Therefore if f ? is known, this method is adaptive, contrary to the general case in Proposition 3.1.
It can even adapt to the local values of L or ? as we use a criterion instead of a preset schedule.
Here, stopping using f (xk ) ? f ? implicitly yields optimal choices of C and ? . A closer look at
the proof shows that the dependency in ? of this restart scheme is a factor h(?) = ?e??/? of
the number of iterations. Taking ? = 1, leads then to a suboptimal constant factor of at most
h(?)/h(1) ? e/2 ? 1.3 for ? ? [1/2, 2], so running this scheme with ? = 1 makes it parameter-free
while getting nearly optimal bounds.
5
Numerical Results
We illustrate our results by testing our adaptive restart methods, denoted Adap and Crit, introduced
respectively in Sections 2.2 and 4 on several problems and compare them against simple gradient
descent (Grad), accelerated gradient methods (Acc), and the restart heuristic enforcing monotonicity
(Mono in [O?Donoghue and Candes, 2015]). For Adap we plot the convergence of the best method
found by grid search to compare with the restart heuristic. This implicitly assumes that the grid
search is run in parallel with enough servers. For Crit we use the optimal f ? found by another solver.
This gives an overview of its performance in order to potentially approximate it along the iterations
8
in a future work as done with Polyak steps [Polyak, 1987]. All restart schemes were done using the
accelerated gradient with backtracking line search detailed in the Supplementary Material, with large
dots representing restart iterations.
The results focused on unconstrained problems but our approach can directly be extended to composite
problems by using the proximal variant of the gradient, accelerated gradient and universal fast gradient
methods [Nesterov, 2015] as detailed in the Supplementary Material. This includes constrained
optimization as a particular case by adding the indicator function of the constraint set to the objective
(as in the SVM example below).
In Figure 1, we solve classification problems with various losses on the UCI Sonar data set [Asuncion
and Newman, 2007]. For least square loss on sonar data set, we observe much faster convergence
of the restart schemes compared to the accelerated method. These results were already observed by
O?Donoghue and Candes [2015]. For logistic loss, we observe that restart does not provide much
improvement. The backtracking line search on the Lipschitz constant may be sufficient to capture
the geometry of the problem. For hinge loss, we regularized by a squared norm and optimize the
dual, which means solving a quadratic problem with box constraints. We observe here that the
scheduled restart scheme convergences much faster, while restart heuristics may be activated too
late. We observe similar results for the LASSO problem. In general Crit ensures the theoretical
accelerated rate but Adap exhibits more consistent behavior. This highlights the benefits of a sharpness
assumption for these last two problems. Precisely quantifying sharpness from data/problem structure
is a key open problem.
10 -10
0
200
400
600
800
Iterations
10 -1
10 -2
f(x)-f
Grad
Acc
Mono
Adap
Crit
Grad
Acc
Mono
Adap
Crit
0
500
1000
Iterations
*
10 0
*
10 0
*
10 -5
f(x)-f
f(x)-f
*
10 0
10 -5
10 -10
f(x)-f
10 0
Grad
Acc
Mono
Adap
Crit
0
500
Iterations
1000
10 -5
10 -10
Grad
Acc
Mono
Adap
Crit
0
500
1000
Iterations
Figure 1: From left to right: least square loss, logistic loss, dual SVM problem and LASSO. We use
adaptive restarts (Adap), gradient descent (Grad), accelerated gradient (Acc) and restart heuristic
enforcing monotonicity (Mono). Large dots represent the restart iterations. Regularization parameters
for dual SVM and LASSO were set to one.
Acknowledgments
The authors would like to acknowledge support from the chaire ?conomie des nouvelles donn?es with
the data science joint research initiative with the fonds AXA pour la recherche, a gift from Soci?t?
G?n?rale Cross Asset Quantitative Research and an AMX fellowship. The authors are affiliated to
PSL Research University, Paris, France.
9
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11
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6,316 | 6,713 | Learning Koopman Invariant Subspaces
for Dynamic Mode Decomposition
Naoya Takeishi? , Yoshinobu Kawahara?,? , Takehisa Yairi?
Department of Aeronautics and Astronautics, The University of Tokyo
?
The Institute of Scientific and Industrial Research, Osaka University
?
RIKEN Center for Advanced Intelligence Project
{takeishi,yairi}@ailab.t.u-tokyo.ac.jp, [email protected]
?
Abstract
Spectral decomposition of the Koopman operator is attracting attention as a tool
for the analysis of nonlinear dynamical systems. Dynamic mode decomposition
is a popular numerical algorithm for Koopman spectral analysis; however, we
often need to prepare nonlinear observables manually according to the underlying
dynamics, which is not always possible since we may not have any a priori
knowledge about them. In this paper, we propose a fully data-driven method for
Koopman spectral analysis based on the principle of learning Koopman invariant
subspaces from observed data. To this end, we propose minimization of the residual
sum of squares of linear least-squares regression to estimate a set of functions that
transforms data into a form in which the linear regression fits well. We introduce
an implementation with neural networks and evaluate performance empirically
using nonlinear dynamical systems and applications.
1
Introduction
A variety of time-series data are generated from nonlinear dynamical systems, in which a state evolves
according to a nonlinear map or differential equation. In summarization, regression, or classification
of such time-series data, precise analysis of the underlying dynamical systems provides valuable
information to generate appropriate features and to select an appropriate computation method. In
applied mathematics and physics, the analysis of nonlinear dynamical systems has received significant
interest because a wide range of complex phenomena, such as fluid flows and neural signals, can
be described in terms of nonlinear dynamics. A classical but popular view of dynamical systems
is based on state space models, wherein the behavior of the trajectories of a vector in state space is
discussed (see, e.g., [1]). Time-series modeling based on a state space is also common in machine
learning. However, when the dynamics are highly nonlinear, analysis based on state space models
becomes challenging compared to the case of linear dynamics.
Recently, there is growing interest in operator-theoretic approaches for the analysis of dynamical
systems. Operator-theoretic approaches are based on the Perron?Frobenius operator [2] or its adjoint,
i.e., the Koopman operator (composition operator) [3], [4]. The Koopman operator defines the
evolution of observation functions (observables) in a function space rather than state vectors in a state
space. Based on the Koopman operator, the analysis of nonlinear dynamical systems can be lifted
to a linear (but infinite-dimensional) regime. Consequently, we can consider modal decomposition,
with which the global characteristics of nonlinear dynamics can be inspected [4], [5]. Such modal
decomposition has been intensively used for scientific purposes to understand complex phenomena
(e.g., [6]?[9]) and also for engineering tasks, such as signal processing and machine learning. In fact,
modal decomposition based on the Koopman operator has been utilized in various engineering tasks,
including robotic control [10], image processing [11], and nonlinear system identification [12].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
One of the most popular algorithms for modal decomposition based on the Koopman operator is
dynamic mode decomposition (DMD) [6], [7], [13]. An important premise of DMD is that the target
dataset is generated from a set of observables that spans a function space invariant to the Koopman
operator (referred to as Koopman invariant subspace). However, when only the original state vectors
are available as the dataset, we must prepare appropriate observables manually according to the
underlying nonlinear dynamics. Several methods have been proposed to utilize such observables,
including the use of basis functions [14] and reproducing kernels [15]. Note that these methods work
well only if appropriate basis functions or kernels are prepared; however, it is not always possible to
prepare such functions if we have no a priori knowledge about the underlying dynamics.
In this paper, we propose a fully data-driven method for modal decomposition via the Koopman
operator based on the principle of learning Koopman invariant subspaces (LKIS) from scratch using
observed data. To this end, we estimate a set of parametric functions by minimizing the residual sum
of squares (RSS) of linear least-squares regression, so that the estimated set of functions transforms
the original data into a form in which the linear regression fits well. In addition to the principle of
LKIS, an implementation using neural networks is described. Moreover, we introduce empirical
performance of DMD based on the LKIS framework with several nonlinear dynamical systems and
applications, which proves the feasibility of LKIS-based DMD as a fully data-driven method for
modal decomposition via the Koopman operator.
2
2.1
Background
Koopman spectral analysis
We focus on a (possibly nonlinear) discrete-time autonomous dynamical system
xt+1 = f (xt ), x ? M, t ? T = {0} ? N,
(1)
where M denotes the state space and (M, ?, ?) represents the associated probability space. In
dynamical system (1), Koopman operator K [4], [5] is defined as an infinite-dimensional linear
operator that acts on observables g : M ? R (or C), i.e.,
Kg(x) = g(f (x)),
(2)
with which the analysis of nonlinear dynamics (1) can be lifted to a linear (but infinite-dimensional)
regime. Since K is linear, let us consider a set of eigenfunctions {?1 , ?2 , . . . } of K with eigenvalues
{?1 , ?2 , . . . }, i.e., K?i = ?i ?i for i ? N, where ? : M ? C and ? ? C. Further, suppose
that g can
be expressed as a linear combination of those infinite number of eigenfunctions, i.e.,
P?
g(x) = i=1 ?i (x)ci with a set of coefficients {c1 , c2 , . . . }. By repeatedly applying K to both sides
of this equation, we obtain the following modal decomposition:
?
X
g(xt ) =
?ti ?i (x0 )ci .
(3)
i=1
Here, the value of g is decomposed into a sum of Koopman modes wi = ?i (x0 )ci , each of which
evolves over time with its frequency and decay rate respectively given by ??i and |?i |, since ?i
is a complex value. The Koopman modes and their eigenvalues can be investigated to understand
the dominant characteristics of complex phenomena that follow nonlinear dynamics. The above
discussion can also be applied straightforwardly to continuous-time dynamical systems [4], [5].
Modal decomposition based on K, often referred to as Koopman spectral analysis, has been receiving
attention in nonlinear physics and applied mathematics. In addition, it is a useful tool for engineering
tasks including machine learning and pattern recognition; the spectra (eigenvalues) of K can be used
as features of dynamical systems, the eigenfunctions are a useful representation of time-series for
various tasks, such as regression and visualization, and K itself can be used for prediction and optimal
control. Several methods have been proposed to compute modal decomposition based on K, such
as generalized Laplace analysis [5], [16], the Ulam?Galerkin method [17], and DMD [6], [7], [13].
DMD, which is reviewed in more detail in the next subsection, has received significant attention and
been utilized in various data analysis scenarios (e.g., [6]?[9]).
Note that the Koopman operator and modal decomposition based on it can be extended to random
dynamical systems actuated by process noise [4], [14], [18]. In addition, Proctor et al. [19], [20]
discussed Koopman analysis of systems with control signals. In this paper, we primarily target
autonomous deterministic dynamics (e.g., Eq. (1)) for the sake of presentation clarity.
2
2.2
Dynamic mode decomposition and Koopman invariant subspace
Let us review DMD, an algorithm for Koopman spectral analysis (further details are in the suppleT
mentary). Consider a set of observables {g1 , . . . , gn } and let g = [g1 ? ? ? gn ] be a vector-valued
n?m
observable. In addition, define two matrices Y0 , Y1 ? R
generated by x0 , f and g, i.e.,
Y0 = [g(x0 ) ? ? ? g(xm?1 )] and Y1 = [g(f (x0 )) ? ? ? g(f (xm?1 ))] ,
(4)
where m + 1 is the number of snapshots in the dataset. The core functionality of DMD algorithms
is computing the eigendecomposition of matrix A = Y1 Y0? [13], [21], where Y0? is the Moore?
Penrose pseudoinverse of Y0 . The eigenvectors of A are referred to as dynamic modes, and they
coincide with the Koopman modes if the corresponding eigenfunctions of K are in span{g1 , . . . , gn }
[21]. Alternatively (but nearly equivalently), the condition under which DMD works as a numerical
realization of Koopman spectral analysis can be described as follows.
Rather than calculating the infinite-dimensional K directly, we can consider the restriction of K to
a finite-dimensional subspace. Assume the observables are elements of L2 (M, ?). The Koopman
invariant subspace is defined as G ? L2 (M, ?) s.t. ?g ? G, Kg ? G. If G is spanned by a finite
number of functions, then the restriction of K to G, which we denote K, becomes a finite-dimensional
linear operator. In the sequel, we assume the existence of such G. If {g1 , . . . , gn } spans G, then
DMD?s matrix A = Y1 Y0? coincides with K ? Rn?n asymptotically, wherein K is the realization of
K with regard to the frame (or basis) {g1 , . . . , gn }. For modal decomposition (3), the (vector-valued)
Koopman modes are given by w and the values of the eigenfunctions are obtained by ? = z H g,
where w and z are the right- and left-eigenvectors of K normalized such that wiH zj = ?i,j [14],
[21], and z H denotes the conjugate transpose of z.
Here, an important problem in the practice of DMD arises, i.e., we often have no access to g that
spans a Koopman invariant subspace G. In this case, for nonlinear dynamics, we must manually
prepare adequate observables. Several researchers have addressed this issue; Williams et al. [14]
leveraged a dictionary of predefined basis functions to transform original data, and Kawahara [15]
defined Koopman spectral analysis in a reproducing kernel Hilbert space. Brunton et al. [22] proposed
the use of observables selected in a data-driven manner [23] from a function dictionary. Note that, for
these methods, we must select an appropriate function dictionary or kernel function according to the
target dynamics. However, if we have no a priori knowledge about them, which is often the case,
such existing methods do not have to be applied successfully to nonlinear dynamics.
3
3.1
Learning Koopman invariant subspaces
Minimizing residual sum of squares of linear least-squares regression
In this paper, we propose a method to learn a set of observables {g1 , . . . , gn } that spans a Koopman
invariant subspace G, given a sequence of measurements as the dataset. In the following, we
summarize desirable properties for such observables, upon which the proposed method is constructed.
Theorem 1. Consider a set of square-integrable observables {g1 , . . . , gn }, and define a vectorT
valued observable g = [g1 ? ? ? gn ] . In addition, define a linear operator G whose matrix form
?
R
R
is given as G = M (g ? f )g H d?
gg H d? . Then, ?x ? M, g(f (x)) = Gg(x) if and only
M
if {g1 , . . . , gn } spans a Koopman invariant subspace.
Pn
Proof. If ?x ? M, g(f (x)) = Gg(x), then for any g? =
i=1 ai gi ? span{g1 , . . . , gn },
!
n
n
n
X
X
X
K?
g=
ai gi (f (x)) =
ai Gi,j gj (x) ? span{g1 , . . . , gn },
i=1
j=1
i=1
where Gi,j denotes the (i, j)-element of G; thus, span{g1 , . . . , gn } is a Koopman invariant subspace.
On the other hand, if {g1 , . . . , gn } spans a Koopman
R invariant subspace,Rthere exists a linear operator
K such that ?x ? M, g(f (x)) = Kg(x); thus, M (g ? f )g H d? = M Kgg H d?. Therefore, an
instance of the matrix form of K is obtained in the form of G.
According to Theorem 1, we should obtain g that makes g ? f ? Gg zero. However, such problems
cannot be solved with finite data because g is a function. Thus, we give the corresponding empirical
3
risk minimization problem based on the assumption of ergodicity of f and the convergence property
of the empirical matrix as follows.
Assumption 1. For dynamical system (1), the time-average and space-average of a function g :
M ? R (or C) coincide in m ? ? for almost all x0 ? M, i.e.,
Z
m?1
1 X
lim
g(xj ) =
g(x)d?(x), for almost all x0 ? M.
m?? m
M
j=0
Theorem 2. Define Y0 and Y1 by Eq. (4) and suppose that Assumption 1 holds. If all modes are
sufficiently excited in the data (i.e., rank(Y0 ) = n), then matrix A = Y1 Y0? almost surely converges
to the matrix form of linear operator G in m ? ?.
R
1
1
H
H
to M (g ? f )g H d? and
RProof. HFrom Assumption 1, m Y1 Y0 and m Y0 Y0 respectively converge
1
gg d? for almost all x0 ? M. In addition, since the rank of Y0 Y0H is always n, ( m
Y0 Y H )? conM
10
R
1
H
?
H
H ?
verges to ( M gg d?) in m ? ? [24]. Consequently, in m ? ?, A = m Y1 Y0
m Y0 Y0
almost surely converges to G, which is the matrix form of linear operator G.
Since A = Y1 Y0? is the minimum-norm solution of the linear least-squares regression from the
columns of Y0 to those of Y1 , we constitute the learning problem to estimate a set of function that
transforms the original data into a form in which the linear least-squares regression fits well. In
particular, we minimize RSS, which measures the discrepancy between the data and the estimated
regression model (i.e., linear least-squares in this case). We define the RSS loss as follows:
2
LRSS (g; (x0 , . . . , xm )) =
Y1 ? (Y1 Y0? )Y0
,
(5)
F
which becomes zero when g spans a Koopman invariant subspace. If we implement a smooth
parametric model on g, the local minima of LRSS can be found using gradient descent. We adopt g
that achieves a local minimum of LRSS as a set of observables that spans (approximately) a Koopman
invariant subspace.
3.2
Linear delay embedder for state space reconstruction
In the previous subsection, we have presented an important part of the principle of LKIS, i.e.,
minimization of the RSS of linear least-squares regression. Note that, to define RSS loss (5), we need
access to a sequence of the original states, i.e., (x0 , . . . , xm ) ? Mm+1 , as a dataset. In practice,
however, we cannot necessarily observe full states x due to limited memory and sensor capabilities.
In this case, only transformed (and possibly degenerated) measurements are available, which we
denote y = ?(x) with a measurement function ? : M ? Rr . To define RSS loss (5) given only
degenerated measurements, we must reconstruct the original states x from the actual observations y.
Here, we utilize delay-coordinate embedding, which has been widely used for state space reconstruction in the analysis of nonlinear dynamics. Consider a univariate time-series (. . . , yt?1 , yt , yt+1 , . . . ),
which is a sequence of degenerated measurements yt = ?(xt ). According to the well-known
Taken?s theorem [25], [26], a faithful representation of xt that preserves the structure of the state
T
? t = yt yt?? ? ? ? yt?(d?1)? with some lag parameter ? and
space can be obtained by x
embedding dimension d if d is greater than 2 dim(x). For a multivariate time-series, embedding
with non-uniform lags provides better reconstruction [27]. For example, when we have a twoT
dimensional time-series yt = [y1,t y2,t ] , an embedding with non-uniform lags is similar to
T
? t = y1,t y1,t??11 ? ? ? y1,t??1d1 y2,t y2,t??21 ? ? ? y2,t??2d2 with each value of ? and
x
d. Several methods have been proposed for selection of ? and d [27]?[29]; however, appropriate
values may depend on the given application (attractor inspection, prediction, etc.).
In this paper, we propose to surrogate the parameter selection of the delay-coordinate embedding by
learning a linear delay embedder from data. Formally, we learn embedder ? such that
T
(k)
T
T
? t = ?(yt ) = W? ytT yt?1
? ? ? yt?k+1
x
, W? ? Rp?kr ,
(6)
? r = dim(y), and k is a hyperparameter of maximum lag. We estimate weight
where p = dim(x),
? instead
W? as well as the parameters of g by minimizing RSS loss (5), which is now defined using x
of x. Learning ? from data yields an embedding that is suitable for learning a Koopman invariant
subspace. Moreover, we can impose L1 regularization on weight W? to make it highly interpretable
if necessary according to the given application.
4
g
?t
x
. . . , yt
?t)
g(x
k+1 , yt k+2 , . . . , yt , yt+1 , . . .
original time-series
h
Lrec
yt
h
Lrec
yt+1
LRSS
g
? t+1
x
? t+1 )
g(x
Figure 1: An instance of LKIS framework, in which g and h are implemented by MLPs.
3.3
Reconstruction of original measurements
Simple minimization of LRSS may yield trivial g, such as constant values. We should impose some
constraints to prevent such trivial solutions. In the proposed framework, modal decomposition is
first obtained in terms of learned observables g; thus, the values of g must be back-projected to the
space of the original measurements y to obtain a physically meaningful representation of the dynamic
modes. Therefore, we modify the loss function by employing an additional term such that the original
?
measurements y can be reconstructed from the values of g by a reconstructor h, i.e., y ? h(g(x)).
Such term is given as follows:
m
X
?0, . . . , x
? m )) =
? j ))k2 ,
Lrec (h, g; (x
kyj ? h(g(x
(7)
j=0
and, if h is a smooth parametric model, this term can also be reduced using gradient descent. Finally,
the objective function to be minimized becomes
? k?1 , . . . , x
? m )) + ?Lrec (h, g; (x
? k?1 , . . . , x
? m )), (8)
L(?, g, h; (y0 , . . . , ym )) = LRSS (g, ?; (x
where ? is a parameter that controls the balance between LRSS and Lrec .
3.4
Implementation using neural networks
In Sections 3.1?3.3, we introduced the main concepts for the LKIS framework, i.e., RSS loss
minimization, learning the linear delay embedder, and reconstruction of the original measurements.
Here, we demonstrate an implementation of the LKIS framework using neural networks.
Figure 1 shows a schematic diagram of the implementation of the framework. We model g and h
using multi-layer perceptrons (MLPs) with a parametric ReLU activation function [30]. Here, the
sizes of the hidden layer of MLPs are defined by the arithmetic means of the sizes of the input and
output layers of the MLPs. Thus, the remaining tunable hyperparameters are k (maximum delay
? and n (dimensionality of g). To obtain g with dimensionality much
of ?), p (dimensionality of x),
greater than that of the original measurements, we found that it was useful to set k > 1 even when
full-state measurements (e.g., y = x) were available.
After estimating the parameters of ?, g, and h, DMD can be performed normally by using the values
of the learned g, defining the data matrices in Eq. (4), and computing the eigendecomposition of
A = Y1 Y0? ; the dynamic modes are obtained by w, and the values of the eigenfunctions are obtained
by ? = z H g, where w and z are the right- and left-eigenvectors of A. See Section 2.2 for details.
In the numerical experiments described in Sections 5 and 6, we performed optimization using firstorder gradient descent. To stabilize optimization, batch normalization [31] was imposed on the
inputs of hidden layers. Note that, since RSS loss function (5) is not decomposable with regard to
data points, convergence of stochastic gradient descent (SGD) cannot be shown straightforwardly.
However, we empirically found that the non-decomposable RSS loss was often reduced successfully,
even with mini-batch SGD. Let us show an example; the full-batch RSS loss (denoted L?RSS ) under the
updates of the mini-batch SGD are plotted in the rightmost panel of Figure 4. Here, L?RSS decreases
rapidly and remains small. For SGD on non-decomposable losses, Kar et al. [32] provided guarantees
for some cases; however, examining the behavior of more general non-decomposable losses under
mini-batch updates remains an open problem.
4
Related work
The proposed framework is motivated by the operator-theoretic view of nonlinear dynamical systems.
In contrast, learning a generative (state-space) model for nonlinear dynamical systems directly has
5
12
12
8
LKIS
linear Hankel
basis exp.
truth
0.2
Im(6)
6
4
2
0.3
noisy x1
noisy x2
10
8
LKIS
linear Hankel
basis exp.
truth
0.2
6
0.1
Im(6)
0.3
x1
x2
10
4
2
0
0
0.1
0
0
-0.1
-2
-4
20
40
60
80
100
-0.1
-2
-0.2
-0.6
-4
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
20
40
60
80
100
-0.2
-0.6
-0.4
-0.2
0
Re(6)
0.2
0.4
0.6
0.8
1
Re(6)
Figure 2: (left) Data generated from system (9)
and (right) the estimated Koopman eigenvalues.
While linear Hankel DMD produces an inconsistent eigenvalue, LKIS-DMD successfully identifies ?, ?, ?2 , and ?0 ?0 = 1.
Figure 3: (left) Data generated from system (9)
and white Gaussian observation noise and (right)
the estimated Koopman eigenvalues. LKIS-DMD
successfully identifies the eigenvalues even with
the observation noise.
been actively studied in machine learning and optimal control communities, on which we mention a
few examples. A classical but popular method for learning nonlinear dynamical systems is using an
expectation-maximization algorithm with Bayesian filtering/smoothing (see, e.g., [33]). Recently,
using approximate Bayesian inference with the variational autoencoder (VAE) technique [34] to learn
generative dynamical models has been actively researched. Chung et al. [35] proposed a recurrent
neural network with random latent variables, Gao et al. [36] utilized VAE-based inference for neural
population models, and Johnson et al. [37] and Krishnan et al. [38] developed inference methods for
structured models based on inference with a VAE. In addition, Karl et al. [39] proposed a method to
obtain a more consistent estimation of nonlinear state space models. Moreover, Watter et al. [40]
proposed a similar approach in the context of optimal control. Since generative models are intrinsically
aware of process and observation noises, incorporating methodologies developed in such studies to
the operator-theoretic perspective is an important open challenge to explicitly deal with uncertainty.
5
Numerical examples
In this section, we provide numerical examples of DMD based on the LKIS framework (LKIS-DMD)
implemented using neural networks. We conducted experiments on three typical nonlinear dynamical
systems: a fixed-point attractor, a limit-cycle attractor, and a system with multiple basins of attraction.
We show the results of comparisons with other recent DMD algorithms, i.e., Hankel DMD [41], [42],
extended DMD [14], and DMD with reproducing kernels [15]. The detailed setups of the experiments
discussed in this section and the next section are described in the supplementary.
Fixed-point attractor Consider a two-dimensional nonlinear map on xt = [x1,t
x1,t+1 = ?x1,t ,
x2,t+1 = ?x2,t + (?2 ? ?)x21,t ,
T
x2,t ] :
(9)
which has a stable equilibrium at the origin if ?, ? < 1. The Koopman eigenvalues of system (9)
include ? and ?, and the corresponding eigenfunctions are ?? (x) = x1 and ?? (x) = x2 ? x21 ,
respectively. ?i ?j is also an eigenvalue with corresponding eigenfunction ?i? ?j? . A minimal
Koopman invariant subspace of system (9) is span{x1 , x2 , x21 }, and the eigenvalues of the Koopman
operator restricted to such subspace include ?, ? and ?2 . We generated a dataset using system (9)
with ? = 0.9 and ? = 0.5 and applied LKIS-DMD (n = 4), linear Hankel DMD [41], [42] (delay 2),
and DMD with basis expansion by {x1 , x2 , x21 }, which corresponds to extended DMD [14] with a
right and minimal observable dictionary. The estimated Koopman eigenvalues are shown in Figure 2,
wherein LKIS-DMD successfully identifies the eigenvalues of the target invariant subspace. In
Figure 3, we show eigenvalues estimated using data contaminated with white Gaussian observation
noise (? = 0.1). The eigenvalues estimated by LKIS-DMD coincide with the true values even with
the observation noise, whereas the results of DMD with basis expansion (i.e., extended DMD) are
directly affected by the observation noise.
Limit-cycle attractor
We generated data from the limit cycle of the FitzHugh?Nagumo equation
x?1 = x31 /3 + x1 ? x2 + I,
x?2 = c(x1 ? bx2 + a),
(10)
where a = 0.7, b = 0.8, c = 0.08, and I = 0.8. Since trajectories in a limit-cycle are periodic, the
(discrete-time) Koopman eigenvalues should lie near the unit circle. Figure 4 shows the eigenvalues
estimated by LKIS-DMD (n = 16), linear Hankel DMD [41], [42] (delay 8), and DMDs with
6
104
RBF
polynomial
Im(6)
Im(6)
Im(6)
linear Hankel
log(L?RSS )
log(,L?rec )
102
100
Im(6)
LKIS
10-2
10-4
10-6
Re(6)
Re(6)
Re(6)
Re(6)
0
50
100
150
iterations
Figure 4: The left four panels show the estimated Koopman eigenvalues on the limit-cycle of the
FitzHugh-Nagumo equation by LKIS-DMD, linear Hankel DMD, and kernel DMDs with polynomial
and RBF kernels. The hyperparameters of each DMD are set to produce 16 eigenvalues. The
rightmost plot shows the full-batch (size 2,000) loss under mini-batch (size 200) SGD updates along
iterations. Non-decomposable part L?RSS decreases rapidly and remains small, even by SGD.
5
0
x_
x_
Im(log(6)=/t)
10
-5
-10
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
x
Re(log(6)=/t)
x
Figure 5: (left) The continuous-time Koopman eigenvalues estimated by LKIS-DMD on the Duffing
equation. (center) The true basins of attraction of the Duffing equation, wherein points in the blue
region evolve toward (1, 0) and points in the red region evolve toward (?1, 0). Note that the stable
manifold of the saddle point is not drawn precisely. (right) The values of the Koopman eigenfunction
with a nearly zero eigenvalue computed by LKIS-DMD, whose level sets should correspond to the
basins of attraction. There is rough agreement between the true boundary of the basins of attraction
and the numerically computed boundary. The right two plots are best viewed in color.
reproducing kernels [15] (polynomial kernel of degree 4 and RBF kernel of width 1). The eigenvalues
produced by LKIS-DMD agree well with those produced by kernel DMDs, whereas linear Hankel
DMD produces eigenvalues that would correspond to rapidly decaying modes.
Multiple basins of attraction Consider the unforced Duffing equation
x
? = ?? x? ? x(? + ?x2 ),
x = [x
T
x]
? ,
(11)
T
T
where ? = 1, ? = ?1, and ? = 0.5. States x following (11) evolve toward [1 0] or [?1 0]
depending on which basin of attraction the initial value belongs to unless the initial state is on
the stable manifold of the saddle. Generally, a Koopman eigenfunction whose continuous-time
eigenvalue is zero takes a constant value in each basin of attraction [14]; thus, the contour plot of
such an eigenfunction shows the boundary of the basins of attraction. We generated 1,000 episodes
of time-series starting at different initial values uniformly sampled from [?2, 2]2 . The left plot in
Figure 5 shows the continuous-time Koopman eigenvalues estimated by LKIS-DMD (n = 100), all
of which correspond to decaying modes (i.e., negative real parts) and agree with the property of the
data. The center plot in Figure 5 shows the true basins of attraction of (11), and the right plot shows
the estimated values of the eigenfunction corresponding to the eigenvalue of the smallest magnitude.
The surface of the estimated eigenfunction agrees qualitatively with the true boundary of the basins
of attractions, which indicates that LKIS-DMD successfully identifies the Koopman eigenfunction.
6
Applications
The numerical experiments in the previous section demonstrated the feasibility of the proposed
method as a fully data-driven method for Koopman spectral analysis. Here, we introduce practical
applications of LKIS-DMD.
Chaotic time-series prediction Prediction of a chaotic time-series has received significant interest
in nonlinear physics. We would like to perform the prediction of a chaotic time-series using DMD,
since
as follows. Since g(xt ) is decomposed as
Pn DMD can be naturally utilized for prediction
H
of K, the next
i=1 ?i (xt )ci and ? is obtained by ?i (xt ) = zi g(xt ) where zi is a left-eigenvalue
Pn
step of g can be described in terms of the current step, i.e., g(xt+1 ) = i=1 ?i (ziH g(xt ))ci . In
addition, in the case of LKIS-DMD, the values of g must be back-projected to y using the learned h.
7
20
LKIS
LSTM
linear Hankel
3
RMS error
raw
15
2.5
10
5
2
0
1.5
-5
1
-10
0.5
-15
0
10
20
30
2.2
30-step prediction
truth
LKIS
-20
10
2
OC-SVM
1.8
5
RMS error
1.6
1.4
0
1.2
1
0.8
-5
0.6
0.4
RuLSIF
-10
0.2
0
10
20
30
Figure 6: The left plot shows RMS errors from
1- to 30-step predictions, and the right plot
shows a part of the 30-step prediction obtained
by LKIS-DMD on (upper) the Lorenz-x series
and (lower) the Rossler-x series.
Figure 7: The top plot shows the raw time-series
obtained by a far-infrared laser [45]. The other plots
show the results of unstable phenomena detection,
wherein the peaks should correspond to the occurrences of unstable phenomena.
We generated two types of univariate time-series by extracting the {x} series of the Lorenz attractor
[43] and the Rossler attractor [44]. We simulated 25,000 steps for each attractor and used the first
10,000 steps for training, the next 5,000 steps for validation, and the last 10,000 steps for testing
prediction accuracy. We examined the prediction accuracy of LKIS-DMD, a simple LSTM network,
and linear Hankel DMD [41], [42], all of whose hyperparameters were tuned using the validation set.
The prediction accuracy of every method and an example of the predicted series on the test set by
LKIS-DMD are shown in Figure 6. As can be seen, the proposed LKIS-DMD achieves the smallest
root-mean-square (RMS) errors in the 30-step prediction.
Unstable phenomena detection One of the most popular applications of DMD is the investigation
of the global characteristics of dynamics by inspecting the spatial distribution of the dynamic modes.
In addition to the spatial distribution, we can investigate the temporal profiles of mode activations by
examining the values of corresponding eigenfunctions. For example, assume there is an eigenfunction
?? 1 that corresponds to a discrete-time eigenvalue ? whose magnitude is considerably smaller
than one. Such a small eigenvalue indicates a rapidly decaying (i.e., unstable) mode; thus, we can
detect occurrences of unstable phenomena by observing the values of ?? 1 . We applied LKIS-DMD
(n = 10) to a time-series generated by a far-infrared laser, which was obtained from the Santa Fe
Time Series Competition Data [45]. We investigated the values of eigenfunction ?? 1 corresponding
to the eigenvalue of the smallest magnitude. The original time-series and values of ?? 1 obtained
by LKIS-DMD are shown in Figure 7. As can be seen, the activations of ?? 1 coincide with
sudden decays of the pulsation amplitudes. For comparison, we applied the novelty/change-point
detection technique using one-class support vector machine (OC-SVM) [46] and direct density-ratio
estimation by relative unconstrained least-squares importance fitting (RuLSIF) [47]. We computed
AUC, defining the sudden decays of the amplitudes as the points to be detected, which were 0.924,
0.799, and 0.803 for LKIS, OC-SVM, and RuLSIF, respectively.
7
Conclusion
In this paper, we have proposed a framework for learning Koopman invariant subspaces, which
is a fully data-driven numerical algorithm for Koopman spectral analysis. In contrast to existing
approaches, the proposed method learns (approximately) a Koopman invariant subspace entirely
from the available data based on the minimization of RSS loss. We have shown empirical results for
several typical nonlinear dynamics and application examples.
We have also introduced an implementation using multi-layer perceptrons; however, one possible
drawback of such an implementation is the local optima of the objective function, which makes
it difficult to assess the adequacy of the obtained results. Rather than using neural networks, the
observables to be learned could be modeled by a sparse combination of basis functions as in [23] but
still utilizing optimization based on RSS loss. Another possible future research direction could be
incorporating approximate Bayesian inference methods, such as VAE [34]. The proposed framework
is based on a discriminative viewpoint, but inference methodologies for generative models could be
used to modify the proposed framework to explicitly consider uncertainty in data.
8
Acknowledgments
This work was supported by JSPS KAKENHI Grant No. JP15J09172, JP26280086, JP16H01548,
and JP26289320.
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11
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6,317 | 6,714 | Soft-to-Hard Vector Quantization for End-to-End
Learning Compressible Representations
Eirikur Agustsson
ETH Zurich
Fabian Mentzer
ETH Zurich
Michael Tschannen
ETH Zurich
[email protected]
[email protected]
[email protected]
Lukas Cavigelli
ETH Zurich
Radu Timofte
ETH Zurich & Merantix
Luca Benini
ETH Zurich
[email protected]
[email protected]
[email protected]
Luc Van Gool
KU Leuven & ETH Zurich
[email protected]
Abstract
We present a new approach to learn compressible representations in deep architectures with an end-to-end training strategy. Our method is based on a soft
(continuous) relaxation of quantization and entropy, which we anneal to their
discrete counterparts throughout training. We showcase this method for two challenging applications: Image compression and neural network compression. While
these tasks have typically been approached with different methods, our soft-to-hard
quantization approach gives results competitive with the state-of-the-art for both.
1
Introduction
In recent years, deep neural networks (DNNs) have led to many breakthrough results in machine
learning and computer vision [20, 28, 10], and are now widely deployed in industry. Modern DNN
models often have millions or tens of millions of parameters, leading to highly redundant structures,
both in the intermediate feature representations they generate and in the model itself. Although
overparametrization of DNN models can have a favorable effect on training, in practice it is often
desirable to compress DNN models for inference, e.g., when deploying them on mobile or embedded
devices with limited memory. The ability to learn compressible feature representations, on the other
hand, has a large potential for the development of (data-adaptive) compression algorithms for various
data types such as images, audio, video, and text, for all of which various DNN architectures are now
available.
DNN model compression and lossy image compression using DNNs have both independently attracted
a lot of attention lately. In order to compress a set of continuous model parameters or features, we
need to approximate each parameter or feature by one representative from a set of quantization
levels (or vectors, in the multi-dimensional case), each associated with a symbol, and then store the
assignments (symbols) of the parameters or features, as well as the quantization levels. Representing
each parameter of a DNN model or each feature in a feature representation by the corresponding
quantization level will come at the cost of a distortion D, i.e., a loss in performance (e.g., in
classification accuracy for a classification DNN with quantized model parameters, or in reconstruction
error in the context of autoencoders with quantized intermediate feature representations). The rate
R, i.e., the entropy of the symbol stream, determines the cost of encoding the model or features in a
bitstream.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
To learn a compressible DNN model or feature representation we need to minimize D + ?R, where
? > 0 controls the rate-distortion trade-off. Including the entropy into the learning cost function can
be seen as adding a regularizer that promotes a compressible representation of the network or feature
representation. However, two major challenges arise when minimizing D + ?R for DNNs: i) coping
with the non-differentiability (due to quantization operations) of the cost function D + ?R, and ii)
obtaining an accurate and differentiable estimate of the entropy (i.e., R). To tackle i), various methods
have been proposed. Among the most popular ones are stochastic approximations [39, 19, 7, 32, 5]
and rounding with a smooth derivative approximation [15, 30]. To address ii) a common approach
is to assume the symbol stream to be i.i.d. and to model the marginal symbol distribution with a
parametric model, such as a Gaussian mixture model [30, 34], a piecewise linear model [5], or a
Bernoulli distribution [33] (in the case of binary symbols).
DNN model compression
In this paper, we propose a unified end-to-end learning framework for learning compressible representations, jointly optimizing the model parameters, the quantization levels, and
the entropy of the resulting symbol stream to compress either a subset of feature representations in the network or the
model itself (see inset figure). We address both challenges i)
and ii) above with methods that are novel in the context DNN
model and feature compression. Our main contributions are:
x
F1 ( ? ; w1 )
x(1)
x(K?1)
FK ( ? ; wK )
x(K)
z = [w1 , w2 , . . . , wK ]
data compression
x
Fb ? ... ? F1
z = x(b)
FK ? ... ? Fb+1
x(K)
z: vector to be compressed
? We provide the first unified view on end-to-end learned compression of feature representations and
DNN models. These two problems have been studied largely independently in the literature so far.
? Our method is simple and intuitively appealing, relying on soft assignments of a given scalar
or vector to be quantized to quantization levels. A parameter controls the ?hardness? of the
assignments and allows to gradually transition from soft to hard assignments during training. In
contrast to rounding-based or stochastic quantization schemes, our coding scheme is directly
differentiable, thus trainable end-to-end.
? Our method does not force the network to adapt to specific (given) quantization outputs (e.g.,
integers) but learns the quantization levels jointly with the weights, enabling application to a wider
set of problems. In particular, we explore vector quantization for the first time in the context of
learned compression and demonstrate its benefits over scalar quantization.
? Unlike essentially all previous works, we make no assumption on the marginal distribution of
the features or model parameters to be quantized by relying on a histogram of the assignment
probabilities rather than the parametric models commonly used in the literature.
? We apply our method to DNN model compression for a 32-layer ResNet model [13] and fullresolution image compression using a variant of the compressive autoencoder proposed recently
in [30]. In both cases, we obtain performance competitive with the state-of-the-art, while making
fewer model assumptions and significantly simplifying the training procedure compared to the
original works [30, 6].
The remainder of the paper is organized as follows. Section 2 reviews related work, before our
soft-to-hard vector quantization method is introduced in Section 3. Then we apply it to a compressive autoencoder for image compression and to ResNet for DNN compression in Section 4 and 5,
respectively. Section 6 concludes the paper.
2
Related Work
There has been a surge of interest in DNN models for full-resolution image compression, most
notably [32, 33, 4, 5, 30], all of which outperform JPEG [35] and some even JPEG 2000 [29]
The pioneering work [32, 33] showed that progressive image compression can be learned with
convolutional recurrent neural networks (RNNs), employing a stochastic quantization method during
training. [4, 30] both rely on convolutional autoencoder architectures. These works are discussed in
more detail in Section 4.
In the context of DNN model compression, the line of works [12, 11, 6] adopts a multi-step procedure
in which the weights of a pretrained DNN are first pruned and the remaining parameters are quantized
using a k-means like algorithm, the DNN is then retrained, and finally the quantized DNN model
is encoded using entropy coding. A notable different approach is taken by [34], where the DNN
2
compression task is tackled using the minimum description length principle, which has a solid
information-theoretic foundation.
It is worth noting that many recent works target quantization of the DNN model parameters and
possibly the feature representation to speed up DNN evaluation on hardware with low-precision
arithmetic, see, e.g., [15, 23, 38, 43]. However, most of these works do not specifically train the DNN
such that the quantized parameters are compressible in an information-theoretic sense.
Gradually moving from an easy (convex or differentiable) problem to the actual harder problem during
optimization, as done in our soft-to-hard quantization framework, has been studied in various contexts
and falls under the umbrella of continuation methods (see [3] for an overview). Formally related but
motivated from a probabilistic perspective are deterministic annealing methods for maximum entropy
clustering/vector quantization, see, e.g., [24, 42]. Arguably most related to our approach is [41],
which also employs continuation for nearest neighbor assignments, but in the context of learning a
supervised prototype classifier. To the best of our knowledge, continuation methods have not been
employed before in an end-to-end learning framework for neural network-based image compression
or DNN compression.
3
3.1
Proposed Soft-to-Hard Vector Quantization
Problem Formulation
Preliminaries and Notations. We consider the standard model for DNNs, where we have an
architecture F : Rd1 7? RdK+1 composed of K layers F = FK ? ? ? ? ? F1 , where layer Fi
maps Rdi ? Rdi+1 , and has parameters wi ? Rmi . We refer to W = [w1 , ? ? ? , wK ] as the
parameters of the network and we denote the intermediate layer outputs of the network as x(0) :=
x
and
x(i) := Fi (x(i?1) ), such that F (x) = x(K) and x(i) is the feature vector produced
by layer Fi .
The parameters of the network are learned w.r.t. training data X = {x1 , ? ? ? , xN } ? Rd1 and labels
Y = {y1 , ? ? ? , yN } ? RdK+1 , by minimizing a real-valued loss L(X , Y; F ). Typically, the loss can
be decomposed as a sum over the training data plus a regularization term,
N
1 X
L(X , Y; F ) =
`(F (xi ), yi ) + ?R(W),
(1)
N i=1
where `(F (x), y)
Pis the sample loss, ? > 0 sets the regularization strength, and R(W) is a regularizer
(e.g., R(W) = i kwi k2 for l2 regularization). In this case, the parameters of the network can be
learned using stochastic gradient descent over mini-batches. Assuming that the data X , Y on which
the network is trained is drawn from some distribution PX,Y , the loss (1) can be thought of as an
estimator of the expected loss E[`(F (X), Y) + ?R(W)]. In the context of image classification, Rd1
would correspond to the input image space and RdK+1 to the classification probabilities, and ` would
be the categorical cross entropy.
We say that the deep architecture is an autoencoder when the network maps back into the input space,
with the goal of reproducing the input. In this case, d1 = dK+1 and F (x) is trained to approximate x,
e.g., with a mean squared error loss `(F (x), y) = kF (x) ? yk2 . Autoencoders typically condense
the dimensionality of the input into some smaller dimensionality inside the network, i.e., the layer
with the smallest output dimension, x(b) ? Rdb , has db d1 , which we refer to as the ?bottleneck?.
Compressible representations. We say that a weight parameter wi or a feature x(i) has a compressible representation if it can be serialized to a binary stream using few bits. For DNN compression, we
want the entire network parameters W to be compressible. For image compression via an autoencoder,
we just need the features in the bottleneck, x(b) , to be compressible.
Suppose we want to compress a feature representation z ? Rd in our network (e.g., x(b) of an
autoencoder) given an input x. Assuming that the data X , Y is drawn from some distribution PX,Y , z
will be a sample from a continuous random variable Z.
To store z with a finite number of bits, we need to map it to a discrete space. Specifically, we map
z to a sequence of m symbols using a (symbol) encoder E : Rd 7? [L]m , where each symbol is an
index ranging from 1 to L, i.e., [L] := {1, . . . , L}. The reconstruction of z is then produced by a
? = D(E(z)) ? Rd . Since z is
(symbol) decoder D : [L]m 7? Rd , which maps the symbols back to z
3
a sample from Z, the symbol stream E(z) is drawn from the discrete probability distribution PE(Z) .
Thus, given the encoder E, according to Shannon?s source coding theorem [8], the correct metric for
compressibility is the entropy of E(Z):
X
H(E(Z)) = ?
P (E(Z) = e) log(P (E(Z) = e)).
(2)
e?[L]m
Our generic goal is hence to optimize the rate distortion trade-off between the expected loss and the
entropy of E(Z):
min EX,Y [`(F? (X), Y) + ?R(W)] + ?H(E(Z)),
(3)
E,D,W
?, and ? > 0 controls the trade-off
where F? is the architecture where z has been replaced with z
between compressibility of z and the distortion it imposes on F? .
However, we cannot optimize (3) directly. First, we do not know the distribution of X and Y.
Second, the distribution of Z depends in a complex manner on the network parameters W and the
distribution of X. Third, the encoder E is a discrete mapping and thus not differentiable. For our first
approximation we consider the sample entropy instead of H(E(Z)). That is, given the data X and
some fixed network parameters W, we can estimate the probabilities P (E(Z) = e) for e ? [L]m
via a histogram. For this estimate to be accurate, we however would need |X | Lm . If z is the
bottleneck of an autoencoder, this would correspond to trying to learn a single histogram for the
entire discretized data space. We relax this by assuming the entries of E(Z) are i.i.d. such that we
can instead compute the histogram over the L distinct values. More
Qm precisely, we assume that for
e = (e1 , ? ? ? , em ) ? [L]m we can approximate P (E(Z) = e) ? l=1 pel , where pj is the histogram
estimate
|{el (zi )|l ? [m], i ? [N ], el (zi ) = j}|
pj :=
,
(4)
mN
where we denote the entries of E(z) = (e1 (z), ? ? ? , em (z)) and zi is the output feature z for training
data point xi ? X . We then obtain an estimate of the entropy of Z by substituting the approximation
(3.1) into (2),
!
!
m
m
L
X
Y
Y
X
pel = ?m
pj log pj = mH(p),
(5)
H(E(Z)) ? ?
pel log
e?[L]m
l=1
l=1
j=1
where the first (exact) equality is due to [8], Thm. 2.6.6, and H(p) := ?
entropy for the (i.i.d., by assumption) components of E(Z) 1 .
PL
j=1
pj log pj is the sample
We now can simplify the ideal objective of (3), by replacing the expected loss with the sample mean
over ` and the entropy using the sample entropy H(p), obtaining
N
1 X
`(F (xi ), yi ) + ?R(W) + ?mH(p).
(6)
N i=1
We note that so far we have assumed that z is a feature output in F , i.e., z = x(k) for some k ? [K].
However, the above treatment would stay the same if z is the concatenation of multiple feature
outputs. One can also obtain a separate sample entropy term for separate feature outputs and add
them to the objective in (6).
In case z is composed of one or more parameter vectors, such as in DNN compression where z = W,
? cease to be random variables, since W is a parameter of the model. That is, opposed to the
z and z
? which we want to be compressible,
case where we have a source X that produces another source Z
we want the discretization of a single parameter vector W to be compressible. This is analogous to
compressing a single document, instead of learning a model that can compress a stream of documents.
In this case, (3) is not the appropriate objective, but our simplified objective in (6) remains appropriate.
This is because a standard technique in compression is to build a statistical model of the (finite) data,
which has a small sample entropy. The only difference is that now the histogram probabilities in (4)
are taken over W instead of the dataset X , i.e., N = 1 and zi = W in (4), and they count towards
storage as well as the encoder E and decoder D.
1
In fact, from [8], Thm. 2.6.6, it follows that if the histogram estimates pj are exact, (5) is an upper bound
for the true H(E(Z)) (i.e., without the i.i.d. assumption).
4
Challenges. Eq. (6) gives us a unified objective that can well describe the trade-off between compressible representations in a deep architecture and the original training objective of the architecture.
However, the problem of finding a good encoder E, a corresponding decoder D, and parameters W
that minimize the objective remains. First, we need to impose a form for the encoder and decoder,
and second we need an approach that can optimize (6) w.r.t. the parameters W. Independently of the
choice of E, (6) is challenging since E is a mapping to a finite set and, therefore, not differentiable.
This implies that neither H(p) is differentiable nor F? is differentiable w.r.t. the parameters of z and
layers that feed into z. For example, if F? is an autoencoder and z = x(b) , the output of the network
will not be differentiable w.r.t. w1 , ? ? ? , wb and x(0) , ? ? ? , x(b?1) .
These challenges motivate the design decisions of our soft-to-hard annealing approach, described in
the next section.
3.2
Our Method
Encoder and decoder form. For the encoder E : Rd 7? [L]m we assume that we have L centers
vectors C = {c1 , ? ? ? , cL } ? Rd/m . The encoding of z ? Rd is then performed by reshaping it
?(m) ] ? R(d/m)?m , and assigning each column z
?(l) to the index of its
into a matrix Z = [?
z(1) , ? ? ? , z
d
nearest neighbor in C. That is, we assume the feature z ? R can be modeled as a sequence of m
points in Rd/m , which we partition into the Voronoi tessellation over the centers C. The decoder
? ? R(d/m)?m from a symbol sequence (e1 , ? ? ? , em ) by
D : [L]m 7? Rd then simply constructs Z
? = [ce , ? ? ? , ce ], from which z
? back
? is formed by reshaping Z
picking the corresponding centers Z
1
m
? = D(E(Z)).
? = D(E(z)) and Z
into Rd . We will interchangeably write z
The idea is then to relax E and D into continuous mappings via soft assignments instead of the hard
nearest neighbor assignment of E.
? ? Rd/m to C as
Soft assignments. We define the soft assignment of z
2
?(?
z) := softmax(??[k?
z ? c1 k , . . . , k?
z ? cL k2 ]) ? RL ,
(7)
yj
e
where softmax(y1 , ? ? ? , yL )j := ey1 +???+eyL is the standard softmax operator, such that ?(?
z) has
positive entries and k?(?
z)k1 = 1. We denote the j-th entry of ?(?
z) with ?j (?
z) and note that
(
1 if j = arg minj 0 ?[L] k?
z ? cj 0 k
lim ?j (?
z) =
???
0 otherwise
? z) := lim??? ?(?
? in C. We
such that ?(?
z) converges to a one-hot encoding of the nearest center to z
? z) as the hard assignment of z
? to C and the parameter ? > 0 as the hardness of
therefore refer to ?(?
the soft assignment ?(?
z).
? as
Using soft assignment, we define the soft quantization of z
L
X
? z) :=
Q(?
cj ?i (?
z) = C?(?
z),
j=1
where we write the centers as a matrix C = [c1 , ? ? ? , cL ] ? Rd/m?L . The corresponding hard
? z) := lim??? Q(?
? z) = ce(?z) , where e(?
?.
assignment is taken with Q(?
z) is the center in C nearest to z
Therefore, we can now write:
? z(1) ), ? ? ? , ?(?
? z(m) )].
? = D(E(Z)) = [Q(?
? z(1) ), ? ? ? , Q(?
? z(m) )] = C[?(?
Z
? via hard nearest neighbor assignments, we can approximate it with
Now, instead of computing Z
?
a smooth relaxation Z := C[?(?
z(1) ), ? ? ? , ?(?
z(m) )] by using the soft assignments instead of the
?, this gives us a differentiable
hard assignments. Denoting the corresponding vector form by z
? in the network with z
?.
approximation F? of the quantized architecture F? , by replacing z
Entropy estimation. Using the soft assignments, we can similarly define a soft histogram, by
summing up the partial assignments to each center instead of counting as in (4):
N m
1 XX
(l)
qj :=
?j (?
zi ).
mN i=1
l=1
5
This gives us a valid probability mass function q = (q1 , ? ? ? , qL ), which is differentiable but converges
to p = (p1 , ? ? ? , pL ) as ? ? ?.
We can now define the ?soft entropy? as the cross entropy between p and q:
L
X
?
H(?)
:= H(p, q) = ?
pj log qj = H(p) + DKL (p||q)
j=1
P
where DKL (p||q) =
Since
j pj log(pj /qj ) denotes the Kullback?Leibler divergence.
?
DKL (p||q) ? 0, this establishes H(?)
as an upper bound for H(p), where equality is obtained
when p = q.
We have therefore obtained a differentiable ?soft entropy? loss (w.r.t. q), which is an upper bound on
?
the sample entropy H(p). Hence, we can indirectly minimize H(p) by minimizing H(?),
treating
the histogram probabilities of p as constants for gradient computation. However, we note that while
qj is additive over the training data and the symbol sequence, log(qj ) is not. This prevents the use
?
of mini-batch gradient descent on H(?),
which can be an issue for large scale learning problems.
?
?
In this case, we can instead re-define the soft entropy H(?)
as H(q, p). As before, H(?)
? H(p)
?
?
as ? ? ?, but H(?)
ceases to be an upper bound for H(p). The benefit is that now H(?)
can be
decomposed as
L
N X
m X
L
X
X
1
(l)
?
H(?)
:= H(q, p) = ?
qj log pj = ?
?j (?
zi ) log pj ,
(8)
mN
j=1
i=1
j=1
l=1
such that we get an additive loss over the samples xi ? X and the components l ? [m].
Soft-to-hard deterministic annealing. Our soft assignment scheme gives us differentiable ap?
proximations F? and H(?)
of the discretized network F? and the sample entropy H(p), respectively.
However, our objective is to learn network parameters W that minimize (6) when using the encoder
and decoder with hard assignments, such that we obtain a compressible symbol stream E(z) which
we can compress using, e.g., arithmetic coding [40].
To this end, we anneal ? from some initial value ?0 to infinity during training, such that the soft
approximation gradually becomes a better approximation of the final hard quantization we will use.
Choosing the annealing schedule is crucial as annealing too slowly may allow the network to invert
the soft assignments (resulting in large weights), and annealing too fast leads to vanishing gradients
too early, thereby preventing learning. In practice, one can either parametrize ? as a function of the
iteration, or tie it to an auxiliary target such as the difference between the network losses incurred by
soft quantization and hard quantization (see Section 4 for details).
For a simple initialization of ?0 and the centers C, we can sample the centers from the set Z :=
P
(l)
? z)k2
z ? Q(?
{?
zi |i ? [N ], l ? [m]} and then cluster Z by minimizing the cluster energy z??Z k?
using SGD.
4
Image Compression
We now show how we can use our framework to realize a simple image compression system. For
the architecture, we use a variant of the convolutional autoencoder proposed recently in [30] (see
Appendix A.1 for details). We note that while we use the architecture of [30], we train it using our
soft-to-hard entropy minimization method, which differs significantly from their approach, see below.
Our goal is to learn a compressible representation of the features in the bottleneck of the autoencoder.
Because we do not expect the features from different bottleneck channels to be identically distributed,
we model each channel?s distribution with a different histogram and entropy loss, adding each entropy
term to the total loss using the same ? parameter. To encode a channel into symbols, we separate the
channel matrix into a sequence of pw ? ph -dimensional patches. These patches (vectorized) form the
columns of Z ? Rd/m?m , where m = d/(pw ph ), such that Z contains m (pw ph )-dimensional points.
Having ph or pw greater than one allows symbols to capture local correlations in the bottleneck,
which is desirable since we model the symbols as i.i.d. random variables for entropy coding. At test
time, the symbol encoder E then determines the symbols in the channel by performing a nearest
? as described above. During
neighbor assignment over a set of L centers C ? Rpw ph , resulting in Z,
?
training we instead use the soft quantized Z, also w.r.t. the centers C.
6
1.00
MS-SSIM
ImageNET100
1.00
B100
MS-SSIM
1.00
MS-SSIM
Urban100
1.00
0.98
0.98
0.98
0.98
0.96
0.96
0.96
0.96
0.94
0.94
0.94
0.94
0.92
0.92
0.92
0.92
0.90
0.90
0.90
0.90
0.88
0.88
0.88
0.88
0.86
0.86
0.86
0.2
0.4
rate [bpp]
0.6
0.20bpp / 0.91 / 0.69 / 23.88dB
SHVQ (ours)
0.2
0.4
rate [bpp]
0.6
0.20bpp / 0.90 / 0.67 / 24.19dB
BPG
SHVQ (ours)
BPG
JPEG 2000
JPEG
0.86
0.2
0.4
rate [bpp]
0.6
0.20bpp / 0.88 / 0.63 / 23.01dB
JPEG 2000
Kodak
MS-SSIM
0.2
0.4
rate [bpp]
0.6
0.22bpp / 0.77 / 0.48 / 19.77dB
JPEG
Figure 1: Top: MS-SSIM as a function of rate for SHVQ (Ours), BPG, JPEG 2000, JPEG, for each
data set. Bottom: A visual example from the Kodak data set along with rate / MS-SSIM / SSIM /
PSNR.
We trained different models using Adam [17], see Appendix A.2. Our training set is composed
similarly to that described in [4]. We used a subset of 90,000 images from ImageNET [9], which
we downsampled by a factor 0.7 and trained on crops of 128 ? 128 pixels, with a batch size of 15.
To estimate the probability distribution p for optimizing (8), we maintain a histogram over 5,000
images, which we update every 10 iterations with the images from the current batch. Details about
other hyperparameters can be found in Appendix A.2.
The training of our autoencoder network takes place in two stages, where we move from an identity
function in the bottleneck to hard quantization. In the first stage, we train the autoencoder without any
quantization. Similar to [30] we gradually unfreeze the channels in the bottleneck during training (this
gives a slight improvement over learning all channels jointly from the start). This yields an efficient
weight initialization and enables us to then initialize ?0 and C as described above. In the second stage,
we minimize (6), jointly learning network weights and quantization levels. We anneal ? by letting the
gap between soft and hard quantization error go to zero as the number of iterations t goes to infinity.
Let eS = kF? (x)?xk2 be the soft error, eH = kF? (x)?xk2 be the hard error. With gap(t) = eH ?eS
we can denote the error between the actual the desired gap with eG (t) = gap(t) ? T /(T + t) gap(0),
such that the gap is halved after T iterations. We update ? according to ?(t + 1) = ?(t) + KG eG (t),
where ?(t) denotes ? at iteration t. Fig. 3 in Appendix A.4 shows the evolution of the gap, soft and
hard loss as sigma grows during training. We observed that both vector quantization and entropy loss
lead to higher compression rates at a given reconstruction MSE compared to scalar quantization and
training without entropy loss, respectively (see Appendix A.3 for details).
Evaluation. To evaluate the image compression performance of our Soft-to-Hard Vector Quantization Autoencoder (SHVQ) method we use four datasets, namely Kodak [2], B100 [31], Urban100 [14],
ImageNET100 (100 randomly selected images from ImageNET [25]) and three standard quality
measures, namely peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) [37], and
multi-scale SSIM (MS-SSIM), see Appendix A.5 for details. We compare our SHVQ with the
standard JPEG, JPEG 2000, and BPG [1], focusing on compression rates < 1 bits per pixel (bpp) (i.e.,
the regime where traditional integral transform-based compression algorithms are most challenged).
As shown in Fig. 1, for high compression rates (< 0.4 bpp), our SHVQ outperforms JPEG and JPEG
2000 in terms of MS-SSIM and is competitive with BPG. A similar trend can be observed for SSIM
(see Fig. 4 in Appendix A.6 for plots of SSIM and PSNR as a function of bpp). SHVQ performs
best on ImageNET100 and is most challenged on Kodak when compared with JPEG 2000. Visually,
SHVQ-compressed images have fewer artifacts than those compressed by JPEG 2000 (see Fig. 1,
and Fig. 5?12 in Appendix A.7).
Related methods and discussion. JPEG 2000 [29] uses wavelet-based transformations and adaptive EBCOT coding. BPG [1], based on a subset of the HEVC video compression standard, is the
7
M ETHOD
O RIGINAL MODEL
P RUNING + FT. + INDEX CODING + H. C ODING [12]
P RUNING + FT. + K - MEANS + FT. + I.C. + H.C. [11]
P RUNING + FT. + H ESSIAN - WEIGHTED K - MEANS + FT. + I.C. + H.C.
P RUNING + FT. + U NIFORM QUANTIZATION + FT. + I.C. + H.C.
P RUNING + FT. + I TERATIVE ECSQ + FT. + I.C. + H.C.
S OFT- TO -H ARD A NNEALING + FT. + H. C ODING ( OURS )
S OFT- TO -H ARD A NNEALING + FT. + A. C ODING ( OURS )
ACC
[%]
92.6
92.6
92.6
92.7
92.7
92.7
92.1
92.1
C OMP.
RATIO
1.00
4.52
18.25
20.51
22.17
21.01
19.15
20.15
Table 1: Accuracies and compression factors for different DNN compression techniques, using a
32-layer ResNet on CIFAR-10. FT. denotes fine-tuning, IC. denotes index coding and H.C. and A.C.
denote Huffman and arithmetic coding, respectively. The pruning based results are from [6].
current state-of-the art for image compression. It uses context-adaptive binary arithmetic coding
(CABAC) [21].
SHVQ (ours)
Theis et al. [30]
The recent works of [30, 5]
Quantization
vector
quantization
rounding
to integers
also showed competitive performance with JPEG 2000. While Backpropagation grad. of soft relaxation grad. of identity mapping
Gaussian scale mixtures
we use the architecture of [30], Entropy estimation (soft) histogram
Training material ImageNET
high quality Flickr images
there are stark differences beOperating points
single model
ensemble
tween the works, summarized
in the inset table. The work of [5] build a deep model using multiple generalized divisive normalization (GDN) layers and their inverses (IGDN), which are specialized layers designed to capture local
joint statistics of natural images. Furthermore, they model marginals for entropy estimation using
linear splines and also use CABAC[21] coding. Concurrent to our work, the method of [16] builds on
the architecture proposed in [33], and shows that impressive performance in terms of the MS-SSIM
metric can be obtained by incorporating it into the optimization (instead of just minimizing the MSE).
In contrast to the domain-specific techniques adopted by these state-of-the-art methods, our framework
for learning compressible representation can realize a competitive image compression system, only
using a convolutional autoencoder and simple entropy coding.
5
DNN Compression
For DNN compression, we investigate the ResNet [13] architecture for image classification. We adopt
the same setting as [6] and consider a 32-layer architecture trained for CIFAR-10 [18]. As in [6], our
goal is to learn a compressible representation for all 464,154 trainable parameters of the model.
We concatenate the parameters into a vector W ? R464,154 and employ scalar quantization (m = d),
such that ZT = z = W. We started from the pre-trained original model, which obtains a 92.6%
accuracy on the test set. We implemented the entropy minimization by using L = 75 centers and
chose ? = 0.1 such that the converged entropy would give a compression factor ? 20, i.e., giving
? 32/20 = 1.6 bits per weight. The training was performed with the same learning parameters as
the original model was trained with (SGD with momentum 0.9). The annealing schedule used was a
simple exponential one, ?(t + 1) = 1.001 ? ?(t) with ?(0) = 0.4. After 4 epochs of training, when
?(t) has increased by a factor ? 20, we switched to hard assignments and continued fine-tuning at
a 10? lower learning rate. 2 Adhering to the benchmark of [6, 12, 11], we obtain the compression
factor by dividing the bit cost of storing the uncompressed weights as floats (464, 154 ? 32 bits) with
the total encoding cost of compressed weights (i.e., L ? 32 bits for the centers plus the size of the
compressed index stream).
Our compressible model achieves a comparable test accuracy of 92.1% while compressing the DNN
by a factor 19.15 with Huffman and 20.15 using arithmetic coding. Table 1 compares our results with
state-of-the-art approaches reported by [6]. We note that while the top methods from the literature
also achieve accuracies above 92% and compression factors above 20?, they employ a considerable
amount of hand-designed steps, such as pruning, retraining, various types of weight clustering, special
encoding of the sparse weight matrices into an index-difference based format and then finally use
2
We switch to hard assignments since we can get large gradients for weights that are equally close to two
? converges to hard nearest neighbor assignments. One could also employ simple gradient clipping.
centers as Q
8
entropy coding. In contrast, we directly minimize the entropy of the weights in the training, obtaining
a highly compressible representation using standard entropy coding.
In Fig. 13 in Appendix A.8, we show how the sample entropy H(p) decays and the index histograms
develop during training, as the network learns to condense most of the weights to a couple of centers
when optimizing (6). In contrast, the methods of [12, 11, 6] manually impose 0 as the most frequent
center by pruning ? 80% of the network weights. We note that the recent works by [34] also manages
to tackle the problem in a single training procedure, using the minimum description length principle.
In contrast to our framework, they take a Bayesian perspective and rely on a parametric assumption
on the symbol distribution.
6
Conclusions
In this paper we proposed a unified framework for end-to-end learning of compressed representations
for deep architectures. By training with a soft-to-hard annealing scheme, gradually transferring
from a soft relaxation of the sample entropy and network discretization process to the actual nondifferentiable quantization process, we manage to optimize the rate distortion trade-off between the
original network loss and the entropy. Our framework can elegantly capture diverse compression
tasks, obtaining results competitive with state-of-the-art for both image compression as well as DNN
compression. The simplicity of our approach opens up various directions for future work, since our
framework can be easily adapted for other tasks where a compressible representation is desired.
Acknowledgments
This work was supported by EUs Horizon 2020 programme under grant agreement No 687757 ?
REPLICATE, by NVIDIA Corporation through the Academic Hardware Grant, by ETH Zurich, and
by Armasuisse.
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11
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6,318 | 6,715 | Learning spatiotemporal piecewise-geodesic
trajectories from longitudinal manifold-valued data
Juliette Chevallier
CMAP, ?cole polytechnique
[email protected]
Pr St?phane Oudard
Oncology Department
USPC, AP-HP, HEGP
St?phanie Allassonni?re
CRC, Universit? Paris Descartes
[email protected]
Abstract
We introduce a hierarchical model which allows to estimate a group-average
piecewise-geodesic trajectory in the Riemannian space of measurements and individual variability. This model falls into the well defined mixed-effect models.
The subject-specific trajectories are defined through spatial and temporal transformations of the group-average piecewise-geodesic path, component by component. Thus we can apply our model to a wide variety of situations. Due to the
non-linearity of the model, we use the Stochastic Approximation ExpectationMaximization algorithm to estimate the model parameters. Experiments on synthetic data validate this choice. The model is then applied to the metastatic renal
cancer chemotherapy monitoring: we run estimations on RECIST scores of treated
patients and estimate the time they escape from the treatment. Experiments highlight the role of the different parameters on the response to treatment.
1
Introduction
During the past few years, the way we treat renal metastatic cancer was profoundly changed: a new
class of anti-angiogenic therapies targeting the tumor vessels instead of the tumor cells has emerged
and drastically improved survival by a factor of three (Escudier et al., 2016). These new drugs,
however, do not cure the cancer, and only succeed in delaying the tumor growth, requiring the use of
successive therapies which must be continued or interrupted at the appropriate moment according to
the patient?s response. The new medicine process has also created a new scientific challenge: how
to choose the more efficient drug therapy. This means that one has to properly understand how the
patient reacts to the possible treatments. Actually, there are several strategies and taking the right
decision is a contested issue (Rothermundt et al., 2015, 2017).
To achieve that goal, physicians took an interest in mathematical modeling. Mathematics has already
demonstrated its efficiency and played a role in the change of stop-criteria for a given treatment
(Burotto et al., 2014). However, to the best of our knowledge, there only exists one model which
was designed by medical practitioners. Although, very basic mathematically, it seems to show that
this point of view may produce interesting results. Introduced by Stein et al. in 2008, the model
performs a non-linear regression using the least squares method to fit an increasing or/and decreasing
exponential curve. This model is still used but suffers limitations. First, as the profile are fitted
individual-by-individual independently, the model cannot explain a global dynamic. Then, the choice
of exponential growth avoids the emergence of plateau effects which are often observed in practice.
This opens the way to new models which would explain both a population and each individual with
other constraints on the shape of the response.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Learning models of disease progression from such databases raises great methodological challenges.
We propose here a very generic model which can be adapted to a large number of situations. For a
given population, our model amounts to estimating an average trajectory in the set of measurements
and individual variability. Then we can define continuous subject-specific trajectories in view of the
population progression. Trajectories need to be registered in space and time, to allow anatomical
variability (as different tumor sizes), different paces of progression and sensitivity to treatments. The
framework of mixed-effects models is well suited to deal with this hierarchical problem. Mixedeffects models for longitudinal measurements were introduced in the seminal paper of Laird and Ware
(1982) and have been widely developed since then. The recent generic approach of Schiratti et al.
(2015) to align patients is even more suitable. First, anatomical data are naturally modeled as points on
Riemannian manifolds while the usual mixed-effects models are defined for Euclidean data. Secondly,
the model was built with the aim of granting individual temporal and spatial variability through
individual variations of a common time-line grant and parallel shifting of the average trajectory.
However, Schiratti et al. (2015) have made a strong hypothesis to build their model as they consider
that the mean evolution is a geodesic. This would mean in our targeted situation that the cancer
would either go on evolving or is always sensitive to the treatment. Unfortunately, the anti-angiogenic
treatments may be inefficient, efficient or temporarily efficient, leading to a re-progression of the
metastasis. Therefore, we want to relax this assumption on the model.
In this paper, we propose a generic statistical framework for the definition and estimation of spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data. Riemannian geometry
allows us to derive a method that makes few assumptions about the data and applications dealt with.
We first introduce our model in its most generic formulation and then make it explicit for RECIST
(Therasse et al., 2000) score monitoring, i.e. for one-dimension manifolds. Experimental results
on those scores are given in section 4.2. The introduction of a more general model is a deliberate
choice as we are expecting to apply our model to the corresponding medical images. Because of
the non-linearity of the model, we have to use a stochastic version of the Expectation-Maximization
algorithm (Dempster et al., 1977), namely the MCMC-SAEM algorithm, for which theoretical results
regarding the convergence have been proved in Delyon et al. (1999) and Allassonni?re et al. (2010)
and numerical efficiency has been demonstrated for these types of models (Schiratti et al. (2015),
M ONOLIX ? MOd?les NOn LIn?aires ? effets miXtes).
2
Mixed-effects model for piecewise-geodesically distributed data
We consider a longitudinal dataset obtained by repeating measurements of n ? N? individuals,
where each individual i ? J1, nK is observed ki ? N? times, at the time points ti = (ti,j )16j6ki
and where yiP
= (yi,j )16j6ki denotes the sequence of observations for this individual. We also
n
denote k =
i=1 ki the total numbers of observations. We assume that each observation yi,j
is a point on a d-dimensional geodesically complete Riemannian manifold (M, g), so that y =
(yi,j )16i6n, 16j6ki ? M k .
We generalize the idea of Schiratti et al. (2015) and build our model in a hierarchical way. We see
our data points as samples along trajectories and suppose that each individual trajectory derives
from a group-average scenario through spatiotemporal transformations. Key to our model is that the
group-average trajectory in no longer assumed to be geodesic but piecewise-geodesic.
2.1
Generic piecewise-geodesic curves model
Let m ? N? and tR = ?? < t1R < . . . < tm?1
< +? a subdivision of R, called thebreakingR
up times sequence. Let M0 a d-dimensional geodesically complete manifold and ??0` 16`6m a
family of geodesics on M0 . To completely define our average trajectory, we introduce m isometries
?`0 : M0 ? M0` := ?`0 (M0 ). This defines m new geodesics ? on the corresponding space M0` ? by
setting down ?0` = ?`0 ? ??0 ` . The isometric nature of the mapping ?`0 ensures that the manifolds
M0` remain Riemannian and that the curves ?0` remain geodesic. In particular, each ?0` remains
parametrizable (Gallot et al., 2004). We define the average trajectory by
?t ? R,
?0 (t) =
?01 (t) 1]??,t1R ] (t)
+
m?1
X
?0` (t) 1]t`?1 ,t` ] (t) + ?0m (t) 1]tm?1 ,+?[ (t) .
R
`=2
2
R
R
In this framework, M0 may be understood as a manifold-template of the geodesic components of the
curve ?0 .
Because of the piecewise nature of our average-trajectory ?0 , constraints have to be formulated on
each interval of the subdivision tR . Following the formulation of the local existence and uniqueness
theorem (Gallot et al., 2004), constraints on geodesics are generally formulated by forcing a value
and a tangent vector at a given time-point. However, such an approach cannot ensure the curve ?0
to be at least continuous. That is why we re-formulate these constraints in our model as boundary
m+1
conditions. Let a sequence A? = (A?0 , . . . , A?m ) ? (M0 )
, an initial time t0 ? R and a final time
1
1
t1 ? R. We impose that for all ` ? J1, m ? 1K, ??0 (t0 ) = A?0 , ??0` (t`R ) = A?` , ??0`+1 (t`R ) = A?` and
??0m (t1 ) = A?m . Notably, the 2m constraints are defined step by step. In one dimension (cf section
2.2), the geodesics could be written explicitly and such constraints do not complicate the model so
much. In higher dimension, we have to use shooting or matching methods to enforce this constraint.
In practice, the choice of the isometries ?`0 and the geodesics ??0` have to be done with the aim to be
"as regular as possible" (at least continuous as said above) at the rupture points t`R . In one dimension
for instance, we build trajectories that are continuous, not differentiable but with a very similar slope
on each side of the breaking-points.
We want the individual trajectories to represent a wide variety of behaviors and to derive from the
group average path by spatiotemporal transformations. To do that, we define for each component ` of
the piecewise-geodesic curve ?0 a couple of transformations (?`i , ?i` ). These transformations, namely
the diffeomorphic component deformations and the time component reparametrizations, characterize
respectively the spatial and the temporal variability of propagation among the population. Thus,
individual trajectories may write in the form of
?t ? R,
?i (t) = ?i1 (t) 1]??,t1R,i ] (t) +
m?1
X
?i` (t) 1]t`?1 ,t`
R,i
R,i ]
(t) + ?im (t) 1]tm?1 ,+?[ (t) (?)
R,i
`=2
where the functions ?i` are obtained from ?0` through the applications of the two transformations
?`i and ?i` described below. Note
that, in particular, each individual possesses his own sequence
of rupture times tR,i = t`R,i 16`<m . Moreover, we require the fewest constraints possible in the
construction: at least continuity and control of the slopes at these breaking-up points.
For compactness, we will now abusively denote t0R for t0 and tm
R for t1 .
To allow different paces in the progression and different breaking-up times for each individual,
we introduce some temporal transformations ?i` , called time-warp, that are defined for the subject
`?1
`
i ? J1, nK and for the geodesic component ` ? J1, mK by ?i` (t) = ?i` (t ? t`?1
R ? ?i ) + tR . The
`
parameters ?i correspond to the time-shift between the mean and the individual progression onset and
the ?i` are the acceleration factors that describe the pace of individuals, being faster or slower than
the average. To ensure good adjunction at the rupture points, we demand the individual breaking-up
`?1
times t`R,i and the time-warps to satisfy ?i` (t`R,i ) = t`R and ?i` (t`?1
R,i ) = tR . Hence the subdivision
tR,i is constrained by the time reparametrizations, which are also constrained. Only the acceleration
factors ?i` and the first time shift ?i1 are free: for all ` ? J1, mK, the constraints rewrite step by step as
`
t`R,i = t`?1
R + ?i +
t`R ?t`?1
R
?`i
`?1
`?1
and ?i` = tR,i
? tR
.
Concerning the space variability, we introduce m diffeomorphic deformations ?`i which enable the
different components of the individual trajectories to vary more irrespectively of each other. We just
enforce the adjunction to be at least continuous and therefore the diffeomorphisms ?`i have to satisfy
?`i ? ?0` (t`R ) = ?`+1
? ?0`+1 (t`R ). Note that the mappings ?`i do not need to be isometric anymore, as
i
the individual trajectories are no longer required to be geodesic.
Finally, for all i ? J1, nK and ` ? J1, mK, we set ?i` = ?`i ? ?0` ? ?i` and define ?i as in (?). The
observations yi = (yi,j ) are assumed to be distributed along the curve ?i and perturbed by an additive
Gaussian noise ?i ? N (0, ? 2 Iki ) :
?(i, j) ? J1, nK ? J1, ki K,
1
yi,j = ?i (ti,j ) + ?i,j
where ?i,j ? N (0, ? 2 ) .
By defining A` = ?`0 (A?` ) for each ` we can apply the constraints on ?0` instead of ??0` .
3
The choice of the isometries ?`0 and the diffeomorphisms ?`i will induce a large panel of piecewisegeodesic models. For example, if m = 1, ?0 = Id and if ?1i denotes the application that maps the
curve ?0 onto its parallel curve for a given non-zero tangent vector wi , we feature the model proposed
by Schiratti et al. (2015). In the following paragraph we propose another specific model which can be
used for chemotherapy monitoring for instance (see section 4.2).
2.2
Piecewise-logistic curve model
We focus in the following on the case of piecewise-logistic model, which presents a real interest
regarding to our target application (cf section 4.2). We assume that m = 2 and d = 1 and we set
M0 = ]0, 1[ equipped with the
logistic metric. Given three real numbers
?0init , ?0escap and ?0fin we set
escap
escap
escap
1
init
2
fin
down ?0 : x 7? ?0 ? ?0
x + ?0
and ?0 : x 7? ?0 ? ?0
x + ?0escap . Thus, we can map
escap
escap
M0 onto the intervals ]?0 , ?0init [ and ]?0 , ?0fin [ respectively: if ??0 refers to the sigmoid function,
?10 ? ??0 will be a logistic curve, growing from ?0escap to ?0init .
In this way, there is essentially a single breaking-up time and we will denote it tR at the population
level and tiR at the individual one. Moreover, due to our target applications, we force the first logistic
to be decreasing and the second one increasing (this condition may be relaxed). Logistics are defined
on open intervals, with asymptotic constraints. We want to formulate our constraints on some noninfinite time-points, as explained in the previous paragraph, we set a positive threshold ? close to zero
and demand the logistics ?01 and ?02 to be ?-near from their corresponding asymptotes. More precisely,
we impose the average trajectory ?0 to be of the form ?0 = ?01 1]??,tR ] + ?02 1]tR ,+?[ where
( escap
?01 : R ? ]?0escap , ?0init [
?02 : R ? ]?0escap , ?0fin [
?0 + 2? 6 ?0init
?0init + ?0escap e(at+b)
?0fin + ?0escap e?(ct+d)
?0escap + 2? 6 ?0fin
t
?
7
(at+b)
?(ct+d)
1+e
1+e
and a, b, c and d are some positive numbers given by the following constraints
t 7?
?01 (t0 ) = ?0init ? ? ,
?01 (tR ) = ?02 (tR ) = ?0escap + ?
and
?02 (t1 ) = ?0fin ? ? .
In our context, the initial time of the process is known: it is the beginning of the treatment. So,
we assume that the average initial time t0 is equal to zero. Especially t0 is no longer a variable.
1
1
1
Moreover, for each individual i ? J1, nK, the time-warps
write ?i (t) = ?i (t ? t0 ? ?i ) + t0 and
1??1i
?1i
?i2 (t) = ?i2 (t ? tR ? ?i2 ) + tR where ?i2 = ?i1 +
(tR ? t0 ). From now on, we note ?i for ?i1 .
RECIST score (dimentionless)
In the same way as the time-warp, the diffeomorphisms ?1i and ?2i are chosen to allow different
amplitudes and rupture values: for each subject i ? J1, nK, given the two scaling factors ri1 and ri2
and the space-shift ?i , we define ?`i (x) = ri` (x ? ?0 (tR )) + ?0 (tR ) + ?i , ` ? {1, 2}. Other choices
are conceivable but in the context of our target applications, this one is appropriate. Mathematically,
any regular and injective function defined on ]?0escap , ?0init [ (respectively ]?0escap , ?0fin [) is suited.
?i
?0
?1
?2
?3
?4
?5
?6
?7
400
200
?0init ??
ri1
t0 t1
tR
0
?
0
ri2
?0
?0fin ??
1,000
2,000
Times (in days)
?i
?i
escap
+?
0
t0
tiR
ti1 tR
t1
(b) From average to individual trajectory.
(a) Diversity of individual trajectories.
Figure 1: Model description. Figure 1a represents a typical average trajectory and several individual
ones, for different vectors Pi . The rupture times are represented by diamonds and the initial/final
times by stars. Figure 1b illustrates the non-standard constraints for ?0 and the transition from the
average trajectory to an individual one: the trajectory ?i is subject to a temporal and a spacial warp.
In other "words", ?i = ?1i ? ?01 ? ?i1 1]??,tiR ] + ?2i ? ?02 ? ?i2 1]tiR ,+?[ .
4
To sum up, each individual trajectory
?i depends on the average curve ?0 through
fixed effects
zpop = ?0init , ?0escap , ?0fin , tR , t1 and random effects zi = ?i1 , ?i2 , ?i , ri1 , ri2 , ?i . This leads to a
non-linear mixed-effects model. More precisely, for all (i, j) ? J1, nK ? J1, ki K,
yi,j = ri1 ?i1 (ti,j ) ? ?0 (tR ) + ?0 (tR ) + ?i 1]??,tiR ] (ti,j )
+ ri2 ?i2 (ti,j ) ? ?0 (tR ) + ?0 (tR ) + ?i 1]tiR ,+?[ (ti,j ) + ?i,j
0
where ?i1 = ?1i ? ?01 , ?i2 = ?2i ? ?02 and tiR = t0 + ?i1 + tR??t
. Figure 1 provides an illustration of the
1
i
model. On each subfigure, the bold black curve represents the average trajectory ?0 and the colour
curves several individual trajectories.
The acceleration and the scaling parameters have to be positive and equal to one on average while the
`
time and space shifts are of any signs and must be zero on average. For these reasons, we set ?i` = e?i
`
and ri` = e?i for ` ? {1, 2} leading to Pi = t ?i1 ?i2 ?i ?1i ?2i ?i . We assume that Pi ? N (0, ?)
where ? ? Sp R, p = 6. This assumption is important in view of the applications. Usually, the
random effects are studied independently. Here, we are interested in correlations between the two
phases of patient?s response to treatment (see section 4.2).
3
Parameters estimation with the MCMC-SAEM algorithm
In this section, we explain how to use a stochastic version of the EM algorithm to produce maximum
a posteriori estimates of the parameters.
3.1
Statistical analysis of the piecewise-logistic curves model
We want to estimate (zpop , ?, ?). The theoretical convergence of the EM algorithm, and a fortiori
of the SAEM algorithm (Delyon et al., 1999), is proved only if the model belongs to the curved
exponential family. Moreover, for numerical performances this framework is important. Without
further hypothesis, the piecewise-logistic model does not satisfy this constraint. We proceed as in
Kuhn and Lavielle (2005): we assume that zpop is the realization of independent Gaussian random
variables with fixed small variances andestimate the means of those
variables. So, the parameters we
escap
init
fin
want to estimate are from now on ? = ?0 , ?0 , ?0 , tR , t1 , ?, ? .
The fixed and random effects z = (zpop , zi )16i6n are considered as latent variables. Our model write
in a hierarchical way as
?
ki
n O
?
O
?
?
?
y | z, ? ?
N ?i (ti,j ), ? 2
?
?
i=1 j=1
n
?
O
?
?
escap
2
2
init
fin , ? 2 ) ? N (t , ? 2 ) ? N (t , ? 2 )
?
z
|
?
?
N
(?
N (0, ?)
,
?
)
?
N
(?
,
?
)
?
N
(?
?
R
1
init
escap
fin
R
1
0
0
0
?
i=1
where ?init , ?escap , ?fin , ?R and ?1 are hyperparameters of the model. The product measures ? mean
that the corresponding entries are considered to be independent in our model. Of course, it is not the
case for the observations which are obtained by repeating measurements for several individuals but
this assumption leads us to a more computationally tractable algorithm.
In this context, the EM algorithm is very efficient to compute the maximum likelihood estimate of ?.
Due to the non-linearity of our model, a stochastic version of the EM algorithm is adopted, namely
the Stochastic Approximation Expectation-Maximization (SAEM) algorithm. As the conditional
distribution q(z|y, ?) is unknown, the Expectation step is replaced using a Monte-Carlo Markov
Chain (MCMC) sampling algorithm, leading to consider the MCMC-SAEM algorithm introduced in
Kuhn and Lavielle (2005) and Allassonni?re et al. (2010). It alternates between a simulation step,
a stochastic approximation step and a maximization step until convergence. The simulation step is
achieved using a symmetric random walk Hasting-Metropolis within Gibbs sampler (Robert and
Casella, 1999). See the supplementary material for details about algorithmics.
To ensure the existence of the maximum a posteriori (theorem 1), we use a "partial" Bayesian
formalism, i.e. we assume the following prior
(?, ?) ? W ?1 (V, m? ) ? W ?1 (v, m? ) where V ? Sp R, v, m? , m? ? R
5
and W ?1 (V, m? ) denotes the inverse Wishart distribution with scale matrix V and degrees of
freedom m? . In order for the inverse Wishart to be non-degenerate, the degrees m? and m? must
satisfy m? > 2p and m? > 2. In practice, we yet use degenerate priors but with correct posteriors
.To be consistent with the one-dimension inverse Wishart distribution, we define the density function
of distribution of higher dimension as
p
!m?
|V |
1
1
?1
fW ?1 (V,m? ) (?) =
exp ? tr V ?
pp
2
?p m2?
2 2 |?|
where ?p is the multivariate gamma function. The maximization step is straightforward given the
sufficient statistics of our exponential model: we update the parameters by taking a barycenter
between the corresponding sufficient statistic and the prior. See the supplementary material for
explicit equations.
3.2
Existence of the Maximum a Posteriori
The next theorem ensures us that the model is well-posed and that the maximum we are looking for
through the MCMC-SAEM algorithm exists. Let ? the space of admissible parameters :
n
o
? =
?0init , ?0escap , ?0fin , tR , t1 , ?, ? ? R5 ? Sp R ? R+ ? positive-definite .
Theorem 1 (Existence of the MAP). Given the piecewise-logistic model and the choice of
probability distributions for the parameters and latent variables of the model, for any dataset
(ti,j , yi,j )i?J1,nK, j?J1,ki K , there exist ?bM AP ? argmax q(?|y).
???
A detailed proof is postponed to the supplementary material.
4
Experimental results
The piecewise-logistic model has been designed for chemotherapy monitoring. More specifically,
we have met radiologists of the H?pital Europ?en Georges-Pompidou (HEGP ? Georges Pompidou
European Hospital) to design our model. In practice, patients suffer from the metastatic kidney cancer
and take a drug each day. Regularly, they come to the HEGP to check the tumor evolution. The
response to a given treatment has generally two distinct phases: first, tumor?s size reduces; then, the
tumor grows again. A practical question is to quantify the correlation between both phases and to
determine as accurately as possible the individual rupture times tiR which are related to an escape of
the patient?s response to treatment.
4.1
Synthetic data
In order to validate our model and numerical scheme, we first run experiments on synthetic data.
We well understood that the covariance matrix ? gives a lot of information on the health status of a
patient: pace and amplitude of tumor progression, individual rupture times. . . Therefore, we have to
pay special attention to the estimation of ? in this paragraph.
An important point was to allow a lot of different individual behaviors. In our synthetic example,
Figure 1a illustrates this variability. From a single average trajectory (?0 in bold plain line), we can
generate individuals who are cured at the end (dot-dashed lines: ?3 and ?4 ), some whose response to
the treatment is bad (dashed lines: ?5 and ?6 ), some who only escape (no positive response to the
treatments ? dotted lines: ?7 ). Likewise, we can generate "patients" with only positive responses or
no response at all. The case of individual 4 is interesting in practice: the tumor still grows but so
slowly that the growth is negligible, at least in the short-run.
Figure 2 illustrates the qualitative performance of the estimation. We are notably able to understand
various behaviors and fit subjects which are far from the average path, such as the orange and the
green curves. We represent only five individuals but 200 subjects have been used to perform the
estimation.
To measure the influence of the sample size on our model/algorithm, we generate synthetic datasets
of various size and perform the estimation 50 times for each dataset. Means and standard deviations
6
RECIST score (dimentionless)
RECIST score (dimentionless)
200
100
0
200
100
0
0
500
1,000
Times (in days)
0
1,500
500
1,000
Times (in days)
1,500
(b) After 600 iterations.
(a) Initialisation.
Figure 2: Initialisation and "results". On both figures, the estimated trajectories are in plain lines and
the target curves in dashed lines. The (noisy) observations are represented by crosses. The average
path is in bold black line, the individuals in color. Figure 2a: Population parameters zpop and latent
variables zpop are initialized at the empirical mean of the observations; individual trajectories are
initialized on the average trajectory (P = 0, ? = 0.1Ip , ? = 1). Figure 2b: After 600 iterations,
sometime less, the estimated curves fit very well the observations. As the algorithm is stochastic,
estimated curves ? and effectively the individuals ? still wave around the target curves.
Table 1: Mean (standard deviation) of relative error (expressed as a percentage) for the population
parameters zpop and the residual standard deviation ? for 50 runs according to the sample size n.
Sample
size n
?0init
?0escap
?0fin
tR
t1
?
50
100
150
1.63 (1.46)
2.42 (1.50)
2.14 (1.17)
9.45 (5.40)
9.07 (5.19)
11.40 (5.72)
6.23 (2.25)
7.82 (2.43)
5.82 (2.55)
11.58 (1.64)
13.62 (1.31)
9.24 (1.63)
4.41 (0.75)
5.27 (0.60)
3.42 (0.71)
25.24 (12.84)
10.35 (3.96)
2.83 (2.31)
of the relative errors for the real parameters, namely ?0init , ?0escap , ?0fin , tR , t1 and ?, are compiled
in Table 1. To compare things which are comparable, we have generated a dataset of size 200 and
shortened them to the desired size. Moreover, to put the algorithm on a more realistic situation, the
synthetic individual times are non-periodically spaced, individual sizes vary between 12 and 18 and
the observed values are noisy (? = 3).
We remark that our algorithm is stable and that the bigger the sample size, the better we learn the
residual standard deviation ?. The parameters tR and ?0escap are quite difficult to learn as they occur
on the flat section of the trajectory. However, the error we made is not crippling as the most important
for clinicians is the dynamic along both phases. As the algorithm enables to estimate both the mean
trajectory and the individual dynamic, it succeeds in estimating the inter-individual variability. This
ends in a good estimate of the covariance matrix ? (see figure 4).
4.2
Chemotherapy monitoring: RECIST score of treated patients
We now run our estimation algorithm on real data from HEGP.
The RECIST (Response Evaluation Criteria In Solid Tumors) score (Therasse et al., 2000) measures
the tumoral growth and is a key indicator of the patient survival. We have performed the estimation
over a drove of 176 patients of the HEGP. There is an average of 7 visits per subjects (min: 3, max:
22), with an average duration of 90 days between consecutive visits.
We have run the algorithm several times, with different proposal laws for the sampler (a Symmetric
Random Walk Hasting-Metropolis within Gibbs one) and different priors. We present here a run with
a low residual standard variation in respect to the amplitude of the trajectories and complexity of the
dataset: ? = 14.50 versus max(?0init , ?0fin ) ? ?0escap = 452.4. Figure 3a illustrates the performance of
the model on the first eight patients. Although we cannot explain all the paths of progression, the
algorithm succeeds in fitting various types of curves: from the yellow curve ?3 which is rather flat
and only escape to the red ?7 which is spiky. From Figure 3b, it seems that the rupture times occur
early in the progression in average. Nevertheless , this result is to be considered with some reserve:
the rupture time generally occurs on a stable phase of the disease and the estimation may be difficult.
7
RECIST score (dimentionless)
400
?0
?1
?2
?3
?4
?5
?6
?7
?8
200
0
0
100
200
300
Times (in days)
400
40
20
0
500
0
1,000
2,000
3,000
4,000
Individual rupture times (in days)
5,000
(b) Individual rupture times tiR .
(a) After 600 iterations.
Individual
rupture times tiR
(in days)
High score?
10
4,000
Fast response?
?Low step
High step?
ste r ? 2
p?
fa
cto
de
pl
itu
5
Lo
w
0
0
ste
p
?10
0
1st amplitude factor ?1
gh
?Slow response
?5
4 5
Hi
?2
0
2
1st acceleration factor ? 1
0
2 nd
am
0
2,000
?
?5
?10
?4
Space shift ?
0
?Low score
500
? 2 nd
Fa
st acc
pr el
og er
re at
ss
io
n
Sl fa
ow ct
pr or ?
2
og
re
ss
?
Last onset?
?Early onset
Time shift ?
Figure 3: RECIST score. We keep conventions of the previous figures. Figure 3a is the result of a 600
iterations run. We represent here only the first 8 patients among the 176. Figure 3b is the histogram of
the rupture times tiR for this run. In black bold line, the estimated average rupture time tR is a good
estimate of the average of the individual rupture times although there exists a large range of escape.
(b) The space warp.
(a) The time warp.
Figure 4: Individual random effects. Figure 4a: log-acceleration factors ?i1 and ?i2 against times
shifts ?i . Figure 4b: log-amplitude factors ?1i and ?2i against space shifts ?i . In both figure, the color
corresponds to the individual rupture time tiR . These estimations hold for the same run as Figure 3.
In Figure 4, we plot the individual estimates of the random effects (obtained from the last iteration) in
comparison to the individual rupture times. Even though the parameters which lead the space warp,
i.e. ?1i , ?2i and ?i are correlated, the correlation with the rupture time is not clear. In other words, the
volume of the tumors seems to not be relevant to evaluate the escape of a patient. On the contrary,
which is logical, the time warp strongly impacts the rupture time.
4.3
Discussion and perspective
We propose here a generic spatiotemporal model to analyze longitudinal manifold-valued measurements. Contrary to Schiratti et al. (2015), the average trajectory is not assumed to be geodesic
anymore. This allows us to apply our model to more complex situations: in chemotherapy monitoring
for example, where the patients are treated and tumors may respond, stabilize or progress during
the treatment, with different conducts for each phase. At the age of personalized medicine, to give
physicians decision support systems is really important. Therefore learning correlations between both
phases is crucial. This has been taken into account here.
For purpose of working with more complicated data, medical images for instance, we have first
presented our model in a very generic version. Then we made it explicit for RECIST scores monitoring
and performed experiments on data from the HEGP. However, we have studied that dataset as if all
patients behave similarly, which is not true in practice. We believe that a possible amelioration of our
model is to put it into a mixture model.
Lastly, the SAEM algorithm is really sensitive to initial conditions. This phenomenon is emphasized
by the non-independence between the individual variables: actually, the average trajectory ?0 is not
exactly the trajectory of the average parameters. Fortunately, the more the sample size n increases,
the more ?0 draws closer to the trajectory of the average parameters.
8
Acknowledgments
Ce travail b?n?ficie d?un financement public Investissement d?avenir, reference ANR-11-LABX0056-LMH. This work was supported by a public grant as part of the Investissement d?avenir, project
reference ANR-11-LABX-0056-LMH.
Travail r?alis? dans le cadre d?un projet financ? par la Fondation de la Recherche M?dicale,
"DBI20131228564". Work performed as a part of a project funded by the Fondation of Medical
Research, grant number "DBI20131228564".
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9
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6,319 | 6,716 | Improving Regret Bounds for Combinatorial
Semi-Bandits with Probabilistically Triggered Arms
and Its Applications
Qinshi Wang
Princeton University
Princeton, NJ 08544
[email protected]
Wei Chen
Microsoft Research
Beijing, China
[email protected]
Abstract
We study combinatorial multi-armed bandit with probabilistically triggered arms
and semi-bandit feedback (CMAB-T). We resolve a serious issue in the prior
CMAB-T studies where the regret bounds contain a possibly exponentially large
factor of 1/p? , where p? is the minimum positive probability that an arm is triggered by any action. We address this issue by introducing a triggering probability
modulated (TPM) bounded smoothness condition into the general CMAB-T framework, and show that many applications such as influence maximization bandit and
combinatorial cascading bandit satisfy this TPM condition. As a result, we completely remove the factor of 1/p? from the regret bounds, achieving significantly
better regret bounds for influence maximization and cascading bandits than before.
Finally, we provide lower bound results showing that the factor 1/p? is unavoidable
for general CMAB-T problems, suggesting that the TPM condition is crucial in
removing this factor.
1
Introduction
Stochastic multi-armed bandit (MAB) is a classical online learning framework modeled as a game
between a player and the environment with m arms. In each round, the player selects one arm and
the environment generates a reward of the arm from a distribution unknown to the player. The player
observes the reward, and use it as the feedback to the player?s algorithm (or policy) to select arms in
future rounds. The goal of the player is to cumulate as much reward as possible over time. MAB
models the classical dilemma between exploration and exploitation: whether the player should keep
exploring arms in search for a better arm, or should stick to the best arm observed so far to collect
rewards. The standard performance measure of the player?s algorithm is the (expected) regret, which
is the difference in expected cumulative reward between always playing the best arm in expectation
and playing according to the player?s algorithm.
In recent years, stochastic combinatorial multi-armed bandit (CMAB) receives many attention (e.g.
[9, 7, 6, 10, 13, 15, 14, 16, 8]), because it has wide applications in wireless networking, online
advertising and recommendation, viral marketing in social networks, etc. In the typical setting of
CMAB, the player selects a combinatorial action to play in each round, which would trigger the
play of a set of arms, and the outcomes of these triggered arms are observed as the feedback (called
semi-bandit feedback). Besides the exploration and exploitation tradeoff, CMAB also needs to deal
with the exponential explosion of the possible actions that makes exploring all actions infeasible.
One class of the above CMAB problems involves probabilistically triggered arms [7, 14, 16], in
which actions may trigger arms probabilistically. We denote it as CMAB-T in this paper. Chen et al.
[7] provide such a general model and apply it to the influence maximization bandit, which models
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
stochastic influence diffusion in social networks and sequentially selecting seed sets to maximize
the cumulative influence spread over time. Kveton et al. [14, 16] study cascading bandits, in which
arms are probabilistically triggered following a sequential order selected by the player as the action.
However, in both studies, the regret bounds contain an undesirable factor of 1/p? , where p? is the
minimum positive probability that any arm can be triggered by any action,1 and this factor could be
exponentially large for both influence maximization and cascading bandits.
In this paper, we adapt the general CMAB framework of [7] in a systematic way to completely remove
the factor of 1/p? for a large class of CMAB-T problems including both influence maximization and
combinatorial cascading bandits. The key observation is that for these problems, a harder-to-trigger
arm has less impact to the expected reward and thus we do not need to observe it as often. We turn
this key observation into a triggering probability modulated (TPM) bounded smoothness condition,
adapted from the original bounded smoothness condition in [7]. We eliminates the 1/p? factor
in the regret bounds for all CMAB-T problems with the TPM condition, and show that influence
maximization bandit and the conjunctive/disjunctive cascading bandits all satisfy the TPM condition.
Moreover, for general CMAB-T without the TPM condition, we show a lower bound result that 1/p?
is unavoidable, because the hard-to-trigger arms are crucial in determining the best arm and have to
be observed enough times.
Besides removing the exponential factor, our analysis is also tighter in other regret factors or constants
comparing to the existing influence maximization bandit results [7, 25], combinatorial cascading
bandit [16], and linear bandits without probabilistically triggered arms [15]. Both the regret analysis
based on the TPM condition and the proof that influence maximization bandit satisfies the TPM
condition are technically involved and nontrivial, but due to the space constraint, we have to move
the complete proofs to the supplementary material. Instead we introduce the key techniques used in
the main text.
Related Work. Multi-armed bandit problem is originally formated by Robbins [20], and has been
extensively studied in the literature [cf. 3, 21, 4]. Our study belongs to the stochastic bandit research,
while there is another line of research on adversarial bandits [2], for which we refer to a survey
like [4] for further information. For stochastic MABs, an important approach is Upper Confidence
Bound (UCB) approach [1], on which most CMAB studies are based upon.
As already mentioned in the introduction, stochastic CMAB has received many attention in recent
years. Among the studies, we improve (a) the general framework with probabilistically triggered arms
of [7], (b) the influence maximization bandit results in [7] and [25], (c) the combinatorial cascading
bandit results in [16], and (d) the linear bandit results in [15]. We defer the technical comparison
with these studies to Section 4.3. Other CMAB studies do not deal with probabilistically triggered
arms. Among them, [9] is the first study on linear stochastic bandit, but its regret bound has since
been improved by Chen et al. [7], Kveton et al. [15]. Combes et al. [8] improve the regret bound of
[15] for linear bandits in a special case where arms are mutually independent. Most studies above
are based on the UCB-style CUCB algorithm or its minor variant, and differ on the assumptions and
regret analysis. Gopalan et al. [10] study Thompson sampling for complex actions, which is based on
the Thompson sample approach [22] and can be applied to CMAB, but their regret bound has a large
exponential constant term.
Influence maximization is first formulated as a discrete optimization problem by Kempe et al. [12],
and has been extensively studied since (cf. [5]). Variants of influence maximization bandit have also
been studied [18, 23, 24]. Lei et al. [18] use a different objective of maximizing the expected size of
the union of the influenced nodes over time. Vaswani et al. [23] discuss how to transfer node level
feedback to the edge level feedback, and then apply the result of [7]. Vaswani et al. [24] replace
the original maximization objective of influence spread with a heuristic surrogate function, avoiding
the issue of probabilistically triggered arms. But their regret is defined against a weaker benchmark
relaxed by the approximation ratio of the surrogate function, and thus their theoretical result is weaker
than ours.
1
The factor of 1/f ? used for the combinatorial disjunctive cascading bandits in [16] is essentially 1/p? .
2
2
General Framework
In this section we present the general framework of combinatorial multi-armed bandit with probabilistically triggered arms originally proposed in [7] with a slight adaptation, and denote it as CMAB-T.
We illustrate that the influence maximization bandit [7] and combinatorial cascading bandits [14, 16]
are example instances of CMAB-T.
CMAB-T is described as a learning game between a learning agent (or player) and the environment.
The environment consists of m random variables X1 , . . . , Xm called base arms (or arms) following
a joint distribution D over [0, 1]m . Distribution D is picked by the environment from a class of
distributions D before the game starts. The player knows D but not the actual distribution D.
The learning process proceeds in discrete rounds. In round t ? 1, the player selects an action St from
an action space S based on the feedback history from the previous rounds, and the environment draws
(t)
(t)
from the joint distribution D an independent sample X (t) = (X1 , . . . , Xm ). When action St is
played on the environment outcome X (t) , a random subset of arms ?t ? [m] are triggered, and the
(t)
outcomes of Xi for all i ? ?t are observed as the feedback to the player. The player also obtains a
nonnegative reward R(St , X (t) , ?t ) fully determined by St , X (t) , and ?t . A learning algorithm aims
at properly selecting actions St ?s over time based on the past feedback to cumulate as much reward
as possible. Different from [7], we allow the action space S to be infinite. In the supplementary
material, we discuss an example of continuous influence maximization [26] that uses continuous and
infinite action space while the number of base arms is still finite.
We now describe the triggered set ?t in more detail, which is not explicit in [7]. In general, ?t may
have additional randomness beyond the randomness of X (t) . Let Dtrig (S, X) denote a distribution
of the triggered subset of [m] for a given action S and an environment outcome X. We assume that ?t
is drawn independently from Dtrig (St , X (t) ). We refer Dtrig as the probabilistic triggering function.
To summarize, a CMAB-T problem instance is a tuple ([m], S, D, Dtrig , R), with elements already
described above. These elements are known to the player, and hence establishing the problem input
to the player. In contrast, the environment instance is the actual distribution D ? D picked by the
environment, and is unknown to the player. The problem instance and the environment instance
together form the (learning) game instance, in which the learning process would unfold. In this paper,
we fix the environment instance D, unless we need to refer to more than one environment instances.
For each arm i, let ?i = EX?D [Xi ]. Let vector ? = (?1 , . . . , ?m ) denote the expectation vector of
arms. Note that vector ? is determined by D. Same as in [7], we assume that the expected reward
E[R(S, X, ? )], where the expectation is taken over X ? D and ? ? Dtrig (S, X), is a function of
action S and the expectation vector ? of the arms. Henceforth, we denote rS (?) , E[R(S, X, ? )].
We remark that Chen et al. [6] relax the above assumption and consider the case where the entire
distribution D, not just the mean of D, is needed to determine the expected reward. However, they
need to assume that arm outcomes are mutually independent, and they do not consider probabilistically
triggered arms. It might be interesting to incorporate probabilistically triggered arms into their setting,
but this is out of the scope of the current paper.
The performance of a learning algorithm A is measured by its (expected) regret, which is the difference
in expected cumulative reward between always playing the best action and playing actions selected
by algorithm A. Formally, let opt? = supS?S rS (?), where ? = EX?D [X], and we assume
that opt? is finite. Same as in [7], we assume that the learning algorithm has access to an offline
(?, ?)-approximation oracle O, which takes ? = (?1 , . . . , ?m ) as input and outputs an action S O
such that Pr{r? (S O ) ? ? ? opt? } ? ?, where ? is the approximation ratio and ? is the success
probability. Under the (?, ?)-approximation oracle, the benchmark cumulative reward should be the
?? fraction of the optimal reward, and thus we use the following (?, ?)-approximation regret:
Definition 1 ((?, ?)-approximation Regret). The T -round (?, ?)-approximation regret of a learning algorithm A (using an (?, ?)-approximation oracle) for a CMAB-T game instance
([m], S, D, Dtrig , R, D) with ? = EX?D [X] is
"
A
Reg?,?,?
(T ) = T ? ? ? ? ? opt? ? E
T
X
#
R(StA , X (t) , ?t ) = T ? ? ? ? ? opt? ? E
i=1
"
T
X
i=1
3
#
rStA (?) ,
where StA is the action A selects in round t, and the expectation is taken over the randomness of
the environment outcomes X (1) , . . . , X (T ) , the triggered sets ?1 , . . . , ?T , as well as the possible
randomness of algorithm A itself.
We remark that because probabilistically triggered arms may strongly impact the determination of
the best action, but they may be hard to trigger and observe, the regret could be worse and the regret
analysis is in general harder than CMAB without probabilistically triggered arms.
The above framework essentially follows [7], but we decouple actions from subsets of arms, allow
action space to be infinite, and explicitly model triggered set distribution, which makes the framework
more powerful in modeling certain applications (see supplementary material for more discussions).
2.1
Examples of CMAB-T: Influence Maximization and Cascading Bandits
In social influence maximization [12], we are given a weighted directed graph G = (V, E, p), where
V and E are sets of vertices and edges respectively, and each edge (u, v) is associated with a
probability p(u, v). Starting from a seed set S ? V , influence propagates in G as follows: nodes
in S are activated at time 0, and at time t ? 1, a node u activated in step t ? 1 has one chance to
activate its inactive out-neighbor v with an independent probability p(u, v). The influence spread of
seed set S, ?(S), is the expected number of activated nodes after the propagation ends. The offline
problem of influence maximization is to find at most k seed nodes in G such that the influence spread
is maximized. Kempe et al. [12] provide a greedy algorithm with approximation ratio 1 ? 1/e ? ?
and success probability 1 ? 1/|V |, for any ? > 0.
For the online influence maximization bandit [7], the edge probabilities p(u, v)?s are unknown and
need to be learned over time through repeated influence maximization tasks: in each round t, k seed
nodes St are selected, the influence propagation from St is observed, the reward is the number of
nodes activated in this round, and one wants to repeat this process to cumulate as much reward as
possible. Putting it into the CMAB-T framework, the set of edges E is the set of arms [m], and their
outcome distribution D is the joint distribution of m independent Bernoulli distributions with means
p(u, v) for all (u, v) ? E. Any seed set S ? V with at most k nodes is an action. The triggered
arm set ?t is the set of edges (u, v) reached by the propagation, that is, u can be reached from St
(t)
by passing through only edges e ? E with Xe = 1. In this case, the distribution Dtrig (St , X (t) )
degenerates to a deterministic triggered set. The reward R(St , X (t) , ?t ) equals to the number of nodes
(t)
in V that is reached from S through only edges e ? E with Xe = 1, and the expected reward is
exactly the influence spread ?(St ). The offline oracle is a (1 ? 1/e ? ?, 1/|V |)-approximation greedy
algorithm. We remark that the general triggered set distribution Dtrig (St , X (t) ) (together with infinite
action space) can be used to model extended versions of influence maximization, such as randomly
selected seed sets in general marketing actions [12] and continuous influence maximization [26] (see
supplementary material).
Now let us consider combinatorial cascading bandits [14, 16]. In this case, we have m independent
Bernoulli random variables X1 , . . . , Xm as base arms. An action is to select an ordered sequence
from a subset of these arms satisfying certain constraint. Playing this action means that the player
reveals the outcomes of the arms one by one following the sequence order until certain stopping
condition is satisfied. The feedback is the outcomes of revealed arms and the reward is a function
form of these arms. In particular, in the disjunctive form the player stops when the first 1 is revealed
and she gains reward of 1, or she reaches the end and gains reward 0. In the conjunctive form, the
player stops when the first 0 is revealed (and receives reward 0) or she reaches the end with all 1
outcomes (and receives reward 1). Cascading bandits can be used to model online recommendation
and advertising (in the disjunctive form with outcome 1 as a click) or network routing reliability (in
the conjunctive form with outcome 0 as the routing edge being broken). It is straightforward to see
that cascading bandits fit into the CMAB-T framework: m variables are base arms, ordered sequences
are actions, and the triggered set is the prefix set of arms until the stopping condition holds.
3
Triggering-Probability Modulated Conditions
Chen et al. [7] use two conditions to guarantee the theoretical regret bounds. The first one is
monotonicity, which we also use in this paper, and is restated below.
4
Algorithm 1 CUCB with computation oracle.
Input: m, Oracle
1: For each arm i, Ti ? 0 {maintain the total number of times arm i is played so far}
2: For each arm i, ?
?i ? 1 {maintain the empirical mean of Xi }
3: for t = 1, 2, 3, . . . do
q
ln t
4:
For each arm i ? [m], ?i ? 32T
{the confidence radius, ?i = +? if Ti = 0}
i
5:
For each arm i ? [m], ?
?i = min {?
?i + ?i , 1} {the upper confidence bound}
6:
S ? Oracle(?
?1 , . . . , ?
?m )
(t)
7:
Play action S, which triggers a set ? ? [m] of base arms with feedback Xi ?s, i ? ?
(t)
8:
For every i ? ? , update Ti and ?
?i : Ti = Ti + 1, ?
?i = ?
?i + (Xi ? ?
?i )/Ti
9: end for
Condition 1 (Monotonicity). We say that a CMAB-T problem instance satisfies monotonicity, if for
any action S ? S, for any two distributions D, D0 ? D with expectation vectors ? = (?1 , . . . , ?m )
and ?0 = (?01 , . . . , ?0m ), we have rS (?) ? rS (?0 ) if ?i ? ?0i for all i ? [m].
The second condition is bounded smoothness. One key contribution of our paper is to properly
strengthen the original bounded smoothness condition in [7] so that we can both get rid of the
undesired 1/p? term in the regret bound and guarantee that many CMAB problems still satisfy the
conditions. Our important change is to use triggering probabilities to modulate the condition, and
thus we call such conditions triggering probability modulated (TPM) conditions. The key point of
TPM conditions is including the triggering probability in the condition. We use pD,S
to denote the
i
probability that action S triggers arm i when the environment instance is D. With this definition,
we can also technically define p? as p? = inf i?[m],S?S,pD,S >0 pD,S
. In this section, we further use
i
i
1-norm based conditions instead of the infinity-norm based condition in [7], since they lead to better
regret bounds for the influence maximization and cascading bandits.
Condition 2 (1-Norm TPM Bounded Smoothness). We say that a CMAB-T problem instance satisfies 1-norm TPM bounded smoothness, if there exists B ? R+ (referred as the bounded smoothness
constant) such that, for any two distributions D, D0 ? D with expectation vectors ? and ?0 , and any
P
action S, we have |rS (?) ? rS (?0 )| ? B i?[m] pD,S
|?i ? ?0i |.
i
Note that the corresponding non-TPM version of the above condition would remove pD,S
in the
i
above condition, which is a generalization of the linear condition used in linear bandits [15]. Thus, the
TPM version is clearly stronger than the non-TPM version (when the bounded smoothness constants
are the same). The intuition of incorporating the triggering probability pD,S
to modulate the 1-norm
i
condition is that, when an arm i is unlikely triggered by action S (small pD,S
), the importance of
i
arm i also diminishes in that a large change in ?i only causes a small change in the expected reward
rS (?). This property sounds natural in many applications, and it is important for bandit learning ?
although an arm i may be difficult to observe when playing S, it is also not important to the expected
reward of S and thus does not need to be learned as accurately as others more easily triggered by S.
4
CUCB Algorithm and Regret Bound with TPM Bounded Smoothness
We use the same CUCB algorithm as in [7] (Algorithm 1). The algorithm maintains the empirical
estimate ?
?i for the true mean ?i , and feed the upper confidence bound ?
?i to the offline oracle to
obtain the next action S to play. The upper confidence bound ?
?i is large if arm i is not triggered often
(Ti is small), providing optimistic estimates for less observed arms. We next provide its regret bound.
Definition 2 (Gap). Fix a distribution D and its expectation vector ?. For each action S, we define
the gap ?S = max(0, ? ? opt? ? rS (?)). For each arm i, we define
?imin =
inf
S?S:pD,S
>0,?S >0
i
?imax =
?S ,
sup
?S .
S?S:pD,S
>0,?S >0
i
As a convention, if there is no action S such that pD,S
> 0 and ?S > 0, we define ?imin = +?,
i
i
i
?max = 0. We define ?min = mini?[m] ?min , and ?max = maxi?[m] ?imax .
5
?
Let S? = {i ? [m] | p?,S
> 0} be the set of arms that could be triggered by S. Let K = maxS?S |S|.
i
For convenience, we use dxe0 to denote max{dxe, 0} for any real number x.
Theorem 1. For the CUCB algorithm on a CMAB-T problem instance that satisfies monotonicity
(Condition 1) and 1-norm TPM bounded smoothness (Condition 2) with bounded smoothness constant
B, (1) if ?min > 0, we have distribution-dependent bound
2
X
X 576B 2 K ln T
2BK
?
+
log2 i
+2 ?
Reg?,?,? (T ) ?
? ?max + 4Bm; (1)
6
?imin
?min 0
i?[m]
i?[m]
(2) we have distribution-independent bound
?
log2
Reg?,?,? (T ) ? 12B mKT ln T +
T
18 ln T
+2 ?m?
0
?2
? ?max + 2Bm.
6
(2)
?
For the above theorem, we remark that the regret bounds are tight (up to a O( log T ) factor in the
case of distribution-independent bound) base on a lower bound result in [15]. More specifically,
Kveton et al. [15] show that for linear bandits (a special class of CMAB-T without probabilistic triggering), the distribution-dependent regret is lower bounded by ?( (m?K)K
log T ), and the
?
?
distribution-independent regret is lower bounded by ?( mKT ) when T ? m/K, for some instance where ?imin = ? for all i ? [m] and ?imin < ?. Comparing with our regret upper
bound in the above theorem, (a) for distribution-dependent bound, we have the regret upper bound
O( (m?K)K
log T ) since for that instance B = 1 and there are K arms with ?imin = ?, so tight with
?
the?
lower bound in [15]; and (b) for distribution-independent
bound, we have the regret upper bound
?
O( mKT log T ), tight to the lower bound up to a O( log T ) factor, same as the upper bound for
the linear bandits in [15]. This indicates that parameters m and K appeared in the above regret
bounds are all needed. As for parameter B, we can view it simply as a scaling parameter. If we
scale the reward of an instance to B times larger than before, certainly, the regret is B times larger.
Looking at the distribution-dependent regret bound (Eq. (1)), ?imin would also be scaled by a factor
of B, canceling one B factor from B 2 , and ?max is also scaled by a factor of B, and thus the regret
bound in Eq. (1) is also scaled by a factor of B. In the distribution-independent regret bound (Eq. (2)),
the scaling of B is more direct. Therefore, we can see that all parameters m, K, and B appearing in
the above regret bounds are needed.
4.1
Novel Ideas in the Regret Analysis
Due to the space limit, the full proof of Theorem 1 is moved to the supplementary material. Here
we briefly explain the novel aspects of our analysis that allow us to achieve new regret bounds and
differentiate us from previous analyses such as the ones in [7] and [16, 15].
We first give an intuitive explanation on how to incorporate the TPM bounded smoothness condition
to remove the factor 1/p? in the regret bound. Consider a simple illustrative example of two actions
S0 and S, where S0 has a fixed reward r0 as a reference action, and S has a stochastic reward
depending on the outcomes of its triggered base arms. Let S? be the set of arms that can be triggered
? suppose i can be triggered by action S with probability pS , and its true mean is ?i
by S. For i ? S,
i
and its empirical mean at the end of round t is ?
?i,t . The analysis in [7] would need a property that, if
for all i ? S? |?
?i,t ? ?i | ? ?i for some properly defined ?i , then S no longer generates regrets. The
analysis would conclude that arm i needs to be triggered ?(log T /?i2 ) times for the above condition
to happen. Since arm i is only triggered with probability pSi , it means action S may need to be played
?(log T /(pSi ?i2 )) times. This is the essential reason why the factor 1/p? appears in the regret bound.
Now with the TPM bounded smoothness, we know that the impact of |?
?i,t ? ?i | ? ?i to the
difference in the expected reward is only pSi ?i , or equivalently, we could relax the requirement to
|?
?i,t ? ?i | ? ?i /pSi to achieve the same effect as in the previous analysis. This translates to the result
that action S would generate regret in at most O(log T /(pSi (?i /pSi )2 )) = O(pSi log T /?i2 ) rounds.
We then need to handle
P the case when we have multiple actions that could trigger arm i. The
simple addition of S:pS >0 pSi log T /?i2 is not feasible since we may have exponentially or even
i
infinitely many such actions. Instead, we introduce the key idea of triggering probability groups,
such that the above actions are divided into groups by putting their triggering probabilities pSi
6
into geometrically separated bins: (1/2, 1], (1/4, 1/2] . . . , (2?j , 2?j+1 ], . . . The actions in the same
group would generate regret in at most O(2?j+1 log T /?i2 ) rounds
with a similar argument, and
P
summing up together, they could generate regret in at most O( j 2?j+1 log T /?i2 ) = O(log T /?i2 )
rounds. Therefore, the factor of 1/pSi or 1/p? is completely removed from the regret bound.
Next, we briefly explain our idea to achieve the improved bound over the linear bandit result
in [15]. The key step is to bound regret ?St generated in round t. By a derivation similar to
[15, 7] together with the 1-norm TPM bounded smoothness condition, we would obtain that ?St ?
P
t
B i?S?t pD,S
(?
?i,t ? ?i ) with high probability. The analysis in [15] would analyze the errors
i
|?
?i,t ? ?i | by a cascade of infinitely many sub-cases of whether there are xj arms with errors larger
than yj with decreasing yj , but it may still be loose. Instead we directly work on the above summation.
Naive bounding the about error summation would not give a O(log T ) bound because there could
be too many arms with small errors. Our trick is to use a reverse amortization: we cumulate small
errors on many sufficiently sampled arms and treat them as errors of insufficiently sample arms, such
that an arm sampled O(log T ) times would not contribute toward the regret. This trick tightens our
analysis and leads to significantly improved constant factors.
4.2
Applications to Influence Maximization and Combinatorial Cascading Bandits
The following two lemmas show that both the cascading bandits and the influence maximization
bandit satisfy the TPM condition.
Lemma 1. For both disjunctive and conjunctive cascading bandit problem instances, 1-norm TPM
bounded smoothness (Condition 2) holds with bounded smoothness constant B = 1.
Lemma 2. For the influence maximization bandit problem instances, 1-norm TPM bounded smooth? where C? is the largest number
ness (Condition 2) holds with bounded smoothness constant B = C,
of nodes any node can reach in the directed graph G = (V, E).
The proof of Lemma 1 involves a technique called bottom-up modification. Each action in cascading
bandits can be viewed as a chain from top to bottom. When changing the means of arms below, the
triggering probability of arms above is not changed. Thus, if we change ? to ?0 backwards, the
triggering probability of each arm is unaffected before its expectation is changed, and when changing
the mean of an arm i, the expected reward of the action is at most changed by pD,S
|?0i ? ?i |.
i
The proof of Lemma 2 is more complex, since the bottom-up modification does not work directly
on graphs with cycles. To circumvent this problem, we develop an influence tree decomposition
technique as follows. First, we order all influence paths from the seed set S to a target v. Second,
each edge is independently sampled based on its edge probability to form a random live-edge graph.
Third, we divide the reward portion of activating v among all paths from S to v: for each live-edge
graph L in which v is reachable from S, assign the probability of L to the first path from S to v in L
according to the path total order. Finally, we compose all the paths from S to v into a tree with S
as the root and copies of v as the leaves, so that we can do bottom-up modification on this tree and
properly trace the reward changes based on the reward division we made among the paths.
4.3
Discussions and Comparisons
We now discuss the implications of Theorem 1 together with Lemmas 1 and 2 by comparing them
with several existing results.
Comparison with [7] and CMAB with ?-norm bounded smoothness conditions. Our work is
a direct adaption of the study in [7]. Comparing with [7], we see that the regret bounds in Theorem 1
are not dependent on the inverse of triggering probabilities, which is the main issue in [7]. When
applied to influence maximization bandit, our result is strictly stronger than that of [7] in two aspects:
?
(a)
p we remove the factor of 1/p by using the TPM condition; (b) we reduce a factor of |E| and
|E| in the dominant terms of distribution-dependent and -independent bounds, respectively, due
to our use of 1-norm instead of ?-norm conditions used in Chen et al. [7]. In the supplementary
material, we further provide the corresponding ?-norm TPM bounded smoothness conditions and
the regret bound results, since in general the two sets of results do not imply each other.
Comparison with [25] on influence maximization bandits. Let G = (V, E) be the social graph
we consider. By Lemma 2, our Theorem 1 can be applied to the influence maximization bandit
7
?
? which gives concrete O(log T ) distribution-dependent and O( T log T ) distributionwith B = C,
independent bounds for the influence maximization bandit. Wen et al. [25] also study the influence maximization bandit aiming at eliminating the exponential factor 1/p? , but they only provide
distribution-independent regret bounds when the graph is a forest. Therefore, our result on influence
maximization bandit is much more general than theirs. Even limiting our result to the forest case,
our result is still better, as we now explain. When the graph is a bidirectional forest, C? is the
size of the largest connected component in the forest.
? Then we can apply Theorem 1 to obtain
? T log T ). In contrast, their regret bound is
the distribution-independent
regret
bound
as
O(|E|
C
?
O(|E|C? T log T ), where C? is a parameter with complicated dependency
on the graph topology
?
and edge influence probabilities. Clearly, our regret bound is O( log T ) better than theirs on pa3
2
rameter T . When comparing C? with C? , we have C? ? |V | and
pC? ? |V | , thus our worst-case
bound is better than their worst-case bound by an additional O( |V |) factor. They do not provide
simple analytical properties on C? , but on the three simple graph examples they gave, our parameter
C? is either comparable or better than C? . Wen et al. [25] also study a generalization of linear
transformation of edge probabilities, which is orthogonal to our current discussion, and could be
potentially incorporated into the general CMAB-T framework.
Comparison with [16] on combinatorial cascading bandits By Lemma 1, we can apply Theorem 1 to combinatorial conjunctive
? smoothness
P and disjunctive cascading bandits with bounded
constant B = 1, achieving O( ?i1 K log T ) distribution-dependent, and O( mKT log T )
min
distribution-independent regret. In contrast,
besides having exactly these terms, the results
Q
Q in [16] have
?
?
an extra factor of 1/f , where f = i?S ? p(i) for conjunctive cascades, and f ? = i?S ? (1 ? p(i))
for disjunctive cascades, with S ? being the optimal solution and p(i) being the probability of success
for item (arm) i. For conjunctive cascades, f ? could be reasonably close to 1 in practice as argued in
[16], but for disjunctive cascades, f ? could be exponentially small since items in optimal solutions
typically have large p(i) values. Therefore, our result completely removes the dependency on 1/f ?
and is better than their result. Moreover, we also have much smaller constant factors owing to the
new reverse amortization method described in Section 4.1.
Comparison with [15] on linear bandits.
When there is no probabilistically triggered arms
(i.e. p? = 1), Theorem 1 would have tighter bounds since some analysis dealing with probabilistic
triggering is not needed. In particular, in Eq. (1) the leading constant 624 would be reduced to 48, the
dlog2 xe0 term is gone, and 6Bm becomes 2Bm; in Eq. (2) the leading constant 50 is reduced to 14,
and the other changes are the same as above (see the supplementary material). The result itself is also
a new contribution, since it generalizes the linear bandit of [15] to general 1-norm conditions with
matching regret bounds, while significantly reducing the leading constants (their constants are 534
and 47 for distribution-dependent and independent bounds, respectively). This improvement comes
from the new reversed amortization method described in Section 4.1.
5
Lower Bound of the General CMAB-T Model
In this section, we show that there exists some CMAB-T problem instance such that
p the regret
bound in [7] is tight, i.e. the factor 1/p? in the distribution-dependent bound and 1/p? in the
distribution-independent bound are unavoidable, where p? is the minimum positive probability that
any base arm i is triggered by any action S. It also implies that the TPM bounded smoothness may
not be applied to all CMAB-T instances.
For our purpose, we only need a simplified version of the bounded smoothness condition of [7] as
below: There exists a bounded smoothness constant B such that, for every action S and every pair of
mean outcome vectors ? and ?0 , we have |rS (?) ? rS (?0 )| ? B maxi?S? |?i ? ?0i |, where S? is the
set of arms that could possibly be triggered by S.
We prove the lower bounds using the following CMAB-T problem instance ([m], S, D, Dtrig , R). For
each base arm i ? [m], we define an action Si , with the set of actions S = {S1 , . . . , Sm }. The family
of distributions D consists of distributions generated by every ? ? [0, 1]m such that the arms are
independent Bernoulli variables. When playing action Si in round t, with a fixed probability p, arm i
(t)
(t)
is triggered and its outcome Xi is observed, and the reward of playing Si is p?1 Xi ; otherwise
with probability 1 ? p no arm is triggered, no feedback is observed and the reward is 0. Following the
8
CMAB-T framework, this means that Dtrig (Si , X), as a distribution on the subsets of [m], is either
{i} with probability p or ? with probability 1 ? p, and the reward R(Si , X, ? ) = p?1 Xi ? I{? = {i}}.
The expected reward rSi (?) = ?i . So this instance satisfies the above bounded smoothness with
constant B = 1. We denote the above instance as FTP(p), standing for fixed triggering probability
instance. This instance is similar with position-based model [17] with only one position, while the
feedback is different. For the FTP(p) instance, we have p? = p and rSi (?)P= p ? p?1 ?i = ?i .
Then applying the result in [7], we have distributed-dependent upper bound O( i p?1i log T ) and
min
p
distribution-independent upper bound O( p?1 mT log T ).
We first provide the distribution-independent lower bound result.
Theorem 2. Let p be a real number with 0 < p < 1. Then for any CMAB-T algorithm A, if
T ? 6p?1 , there exists a CMAB-T environment instance D with mean ? such that on instance
FTP(p),
s
1
mT
A
Reg? (T ) ?
.
170
p
The proof of the above and the next theorem
p are all based on the results for the classical MAB
problems. Comparing to the upper bound O( p?1 mT log T ). obtained
from [7], Theorem 2 implies
?
that the regret upper bound of CUCB in [7] is tight up to a O( log T ) factor. This means that the
1/p? factor in the regret bound of [7] cannot be avoided in the general class of CMAB-T problems.
Next we give the distribution-dependent lower bound. For a learning algorithm, we say that it is
consistent if, for every ?, every non-optimal arm is played o(T a ) times in expectation, for any real
number a > 0. Then we have the following distribution-dependent lower bound.
Theorem 3. For any consistent algorithm A running on instance FTP(p) and ?i < 1 for every arm
i, we have
A
X
Reg?
(T )
p?1 ?i
lim inf
?
,
T ?+?
ln T
kl(?i , ?? )
i:? <??
i
?
?
where ? = maxi ?i , ?i = ? ? ?i , and kl(?, ?) is the Kullback-Leibler divergence function.
Again we see that the distribution-dependent upper bound obtained from [7] asymptotically match the
lower bound above. Finally, we remark that even if we rescale the reward from [1, 1/p] back to [0, 1],
the corresponding scaling factor B would become
? p, and thus we would still obtain the conclusion
that the regret bounds in [7] is tight (up to a O( log T ) factor), and thus 1/p? is in general needed in
those bounds.
6
Conclusion and Future Work
In this paper, we propose the TPM bounded smoothness condition, which conveys the intuition that
an arm difficult to trigger is also less important in determining the optimal solution. We show that this
condition is essential to guarantee low regret, and prove that important applications, such as influence
maximization bandits and combinatorial cascading bandits all satisfy this condition.
There are several directions one may further pursue. One is to improve the regret bound for some
specific problems. For example, for the influence maximization bandit, can we give a better algorithm
or analysis to achieve a better regret bound than the one provided by the general TPM condition?
Another direction is to look into other applications with probabilistically triggered arms that may not
satisfy the TPM condition or need other conditions to guarantee low regret. Combining the current
CMAB-T framework with the linear generalization as in [25] to achieve scalable learning result is
also an interesting direction.
9
Acknowledgment
Wei Chen is partially supported by the National Natural Science Foundation of China (Grant No.
61433014).
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avoiding:1 ex:3 |
6,320 | 6,717 | Predictive-State Decoders:
Encoding the Future into Recurrent Networks
Arun Venkatraman1? , Nicholas Rhinehart1?, Wen Sun1 ,
Lerrel Pinto , Martial Hebert1 , Byron Boots2 , Kris M. Kitani1 , J. Andrew Bagnell1
1
The Robotics Institute, Carnegie-Mellon University, Pittsburgh, PA
2
School of Interactive Computing, Georgia Institute of Technology, Atlanta, GA
1
Abstract
Recurrent neural networks (RNNs) are a vital modeling technique that rely on
internal states learned indirectly by optimization of a supervised, unsupervised, or
reinforcement training loss. RNNs are used to model dynamic processes that are
characterized by underlying latent states whose form is often unknown, precluding
its analytic representation inside an RNN. In the Predictive-State Representation
(PSR) literature, latent state processes are modeled by an internal state representation that directly models the distribution of future observations, and most recent
work in this area has relied on explicitly representing and targeting sufficient statistics of this probability distribution. We seek to combine the advantages of RNNs
and PSRs by augmenting existing state-of-the-art recurrent neural networks with
P REDICTIVE -S TATE D ECODERS (P SDs), which add supervision to the network?s
internal state representation to target predicting future observations. P SDs are
simple to implement and easily incorporated into existing training pipelines via
additional loss regularization. We demonstrate the effectiveness of P SDs with
experimental results in three different domains: probabilistic filtering, Imitation
Learning, and Reinforcement Learning. In each, our method improves statistical
performance of state-of-the-art recurrent baselines and does so with fewer iterations
and less data.
1
Introduction
Despite their wide success in a variety of domains, recurrent neural networks (RNNs) are often
inhibited by the difficulty of learning an internal state representation. Internal state is a unifying
characteristic of RNNs, as it serves as an RNN?s memory. Learning these internal states is challenging
because optimization is guided by the indirect signal of the RNN?s target task, such as maximizing
the cost-to-go for reinforcement learning or maximizing the likelihood of a sequence of words. These
target tasks have a latent state sequence that characterizes the underlying sequential data-generating
process. Unfortunately, most settings do not afford a parametric model of latent state that is available
to the learner.
However, recent work has shown that in certain settings, latent states can be characterized by
observations alone [8, 24, 26] ? which are almost always available to recurrent models. In such
partially-observable problems (e.g. Fig. 1a), a single observation is not guaranteed to contain enough
information to fully represent the system?s latent state. For example, a single image of a robot is
insufficient to characterize its latent velocity and acceleration. While a latent state parametrization
may be known in some domains ? e.g. a simple pendulum can be sufficiently modeled by its angle
? ? data from most domains cannot be explicitly parametrized.
and angular velocity (?, ?)
?
Contributed equally to this work. Direct correspondence to: {arunvenk,nrhineha}@cs.cmu.edu
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(a) The process generating sequential data has latent
state st which generates the next latent state st+1 . st
is usually unknown but generates the observations xt
which are used to learn a model for the system.
(b) An overview of our approach for modelling the process from Fig. 1a. We attach a decoder to the internal
state of an RNN to predict statistics of future observations xt to xt+k observed at training time.
Figure 1: Data generation process and proposed model
In lieu of ground truth access to latent states, recurrent neural networks [32, 47] employ internal states
to summarize previous data, serving as a learner?s memory. We avoid the terminology ?hidden state"
as it refers to the internal state in the RNN literature but refers to the latent state in the HMM, PSR,
and related literature. Internal states are modified towards minimizing the target application?s loss,
e.g., minimizing observation loss in filtering or cumulative reward in reinforcement learning. The
target application?s loss is not directly defined over the internal states: they are updated via the chain
rule (backpropagation) through the global loss. Although this modeling is indirect, recurrent networks
nonetheless can achieve state-of-the-art results on many robotics [18, 23], vision [34, 50], and natural
language tasks [15, 20, 38] when training succeeds. However, recurrent model optimization is
hampered by two main difficulties: 1) non-convexity, and 2) the loss does not directly encourage
the internal state to model the latent state. A poor internal state representation can yield poor task
performance, but rarely does the task objective directly measure the quality of the internal state.
Predictive-State Representations (PSRs) [8, 24, 26] offer an alternative internal state representation
to that of RNNs in terms of the available observations. Spectral learning methods for PSRs provide
theoretical guarantees on discovering the global optimum for the model and internal state parameters
under the assumptions of infinite training data and realizability. However, in the non-realizable setting
? i.e. model mismatch (e.g., using learned parameters of a linear system model for a non-linear system)
? these algorithms lose any performance guarantees on using the learned model for the target inference
tasks. Extensions to handle nonlinear systems rely on RKHS embeddings [43], which themselves
can be computationally infeasible to use with large datasets. Nevertheless, when these models are
trainable, they often achieve strong performance [24, 45]; the structure they impose significantly
simplifies the learning problem.
We leverage ideas from the both RNN and PSR paradigms, resulting in a marriage of two orthogonal
sequential modeling approaches. When training an RNN, P REDICTIVE -S TATE D ECODERS (Fig. 1b)
provide direct supervision on the internal state, aiding the training problem. The proposed method
can be viewed as an instance of Multi-Task Learning (MTL) [13] and self-supervision [27], using the
inputs to the learner to form a secondary unsupervised objective. Our contribution is a general method
that improves performance of learning RNNs for sequential prediction problems. The approach is
easy to implement as a regularizer on traditional RNN loss functions with little overhead and can
thus be incorporated into a variety of existing recurrent models.
In our experiments, we examine three domains where recurrent models are used to model temporal
dependencies: probabilistic filtering, where we predict the future observation given past observations;
Imitation Learning, where the learner attempts to mimic an expert?s actions; and Reinforcement
Learning, where a policy is trained to maximize cumulative reward. We observe that our method
improves loss convergence rates and results in higher-quality final objectives in these domains.
2
Latent State Space Models
To model sequential prediction problems, it is common to cast the problem into the Markov Process
framework. Predictive distributions in this framework satisfy the Markov property:
P (st+1 |st , st?1 , . . . , s0 ) = P (st+1 |st )
2
(1)
Figure 2: Learning recurrent models consists of learning a function f that updates the internal state
ht given the latest observation xt . The internal state may also be used to predict targets yt , such as
control actions for imitation and reinforcement learning. These are then inputs to a loss function `
which accumulate as the multi-step loss L over all timesteps.
where st is the latent state of the system at timestep t. Intuitively, this property tells us that the
future st+1 is only dependent on the current state2 st and does not depend on any previous state
s0 , . . . , st?1 . As st is latent, the learner only has access to observations xt , which are produced by
st . For example, in robotics, xt may be joint angles from sensors or a scene observed as an image. A
common graphical model representation is shown in Fig. 1a.
The machine learning problem is to find a model f that uses the latest observation xt to recursively
update an internal state, denoted ht , illustrated in Fig. 2. Note that ht is distinct from st . ht is the
learner?s internal state, and st is the underlying configuration of the data-generating Markov
Process. For example, the internal state in the Bayesian filtering/POMDP setup is represented as a
belief state [49], a ?memory" unit in neural networks, or as a distribution over observations for PSRs.
Unlike traditional supervised machine learning problems, learning models for latent state problems
must be accomplished without ground-truth supervision of the internal states themselves. Two distinct
paradigms for latent state modeling exist. The first are discriminative approaches based on RNNs, and
the second is a set of theoretically well-studied approaches based on Predictive-State Representations.
In the following sections we provide a brief overview of each class of approach.
2.1
Recurrent Models and RNNs
A classical supervised machine learning approach for learning internal models involves choosing
an explicit parametrization for the internal states and assuming ground-truth access to these states
and observations at training time [17, 29, 33, 37]. These models focus on learning only the recursive
model f in Fig. 2, assuming access to the st (Fig. 1a) at training time. Another class of approaches
drop the assumption of access to ground truth but still assume a parametrization of the internal state.
These models set up a multi-step prediction error and use expectation maximization to alternate
between optimizing over the model?s parameters and the internal state values [2, 19, 16].
While imposing a fixed representation on the internal state adds structure to the learning problem, it
can limit performance. For many problems such as speech recognition [20] or text generation [48], it
is difficult to fully represent a latent state inside the model?s internal state. Instead, typical machine
learning solutions rely on the Recurrent Neural Network architecture. The RNN model (Fig. 2) uses
the internal state to make predictions yt = f (ht , xt ) and is trained by minimizing a series of loss
functions `t over each prediction, as shown in the following optimization problem:
X
min L = min
`t (f (ht , xt ))
(2)
f
f
t
The loss functions `t are usually application- and domain-specific. For example, in a probabilistic
filtering problem, the objective may be to minimize the negative log-likelihood of the observations [4, 52] or the prediction of the next observation [34]. For imitation learning, this objective
function will penalize deviation of the prediction from the expert?s action [39], and for policygradient reinforcement learning methods, the objective includes the log-likelihood of choosing
actions weighted by their observed returns. In general, the task objective optimized by the network
does not directly specify a loss directly over the values of the internal state ht .
2
In Markov Decision Processes (MDPs), P (st+1 |st ) may depend on an action taken at st .
3
The general difficulty with the objective in Eq. (2) is that the recurrence with f results in a highly
non-convex and difficult optimization [2].
RNN models are thus often trained with backpropagation-through-time (BPTT) [55]. BPTT allows
future losses incurred at timestep t to be back-propogated and affect the parameter updates to f .
These updates to f then change the distribution of internal states computed during the next forward
pass through time. The difficulty is then that small updates to f can drastically change the distribution
of ht , sometimes resulting in error exponential in the time horizon [53]. This ?diffusion problem"
can yield an unstable training procedure with exponentially exploding or vanishing gradients [7].
While techniques such as truncated gradients [47] or gradient-clipping [35] can alleviate some of
these problems, each of these techniques yields stability by discarding information about how future
observations and predictions should backpropagate through the current internal state. A significant
innovation in training internal states with long-term dependence was the LSTM [25]. Many variants
on LSTMs exist (e.g. GRUs [14]), yet in the domains evaluated by Greff et al. [21], none consistently
exhibit statistically significant improvements over LSTMs.
In the next section, we discuss a different paradigm for learning temporal models. In contrast with the
open-ended internal-state learned by RNNs, Predictive-State methods do not parameterize a specific
representation of the internal state but use certain assumptions to construct a mathematical structure
in terms of the observations to find a globally optimal representation.
2.2
Predictive-State Models
Predictive-State Representations (PSRs) address the problem of finding an internal state by formulating the representation directly in terms of observable quantities. Instead of targeting a prediction loss as with RNNs, PSRs define a belief over the distribution of k future observations,
gt = [xTt , ..., xTt+k?1 ]T ? Rkn given all the past observations pt = [x0 , . . . xt?1 ] [10]. In the case of
linear systems, this k is similar to the rank of the observability matrix [6]. The key assumption in
PSRs is that the definition of state is equivalent to having sufficient information to predict everything
about gt at time-step t [42], i.e. there is a bijective function that maps P (st |pt?1 ) ? the distribution
of latent state given the past ? to P (gt |pt?1 ) ? the belief over future observations.
Spectral learning approaches were developed to find an globally optimal internal state representation and the transition model f for these Predictive-State models. In the controls literature, these
approaches were developed as subspace identification [51], and in the ML literature as spectral
approaches for partially-observed systems [9, 8, 26, 56]. A significant improvement in model
learning was developed by Boots et al. [10], Hefny et al. [24], where sufficient feature functions
? (e.g., moments) map distributions
P (gt |pt ) to points in feature space E [?(gt )|pt ]. For example,
E [?(gt )|pt ] = E gt , gt gtT |pt are the sufficient statistics for a Gaussian distribution. With this
representation, learning latent state prediction models can be reduced to supervised learning.
Hefny et al. [24] used this along with Instrumental Variable Regression [11] to develop a procedure
that, in the limit of infinite data, and under a linear-system realiziablity assumption, would converge
to the globally optimal solution. Sun et al. [45] extended this setup to create a practical algorithm,
Predictive-State Inference Machines (PSIMs) [44, 45, 54], based on the concept of inference machines [31, 40]. Unlike in Hefny et al. [24], which attempted to find a generative observation model
and transition model, PSIMs directly learned the filter function, an operator f , that can deterministically pass the predictive states forward in time conditioned on the latest observation, by minimizing
the following loss over f :
`p =
X
2
k?(gt+1 ) ? f (ht , xt )k ,
ht+1 = f (ht , xt )
(3)
t
This loss function, which we call the predictive-state loss, forms the basis of our P REDICTIVE -S TATE
D ECODERS. By minimizing this supervised loss function, PSIM assigns statistical meaning to internal
states: it forces the internal state ht to match sufficient statistics of future observations E [?(gt )|pt ] at
every timestep t. We observe an empirical sample of the future gt = [xt , . . . , xt+k ] at each timestep
by looking into the future in the training dataset or by waiting for streaming future observations.
Whereas [45] primarily studied algorithms for minimizing the predictive-state loss, we adapt it to
augment general recurrent models such as LSTMs and for a wider variety of applications such as
imitation and reinforcement learning.
4
Figure 3: Predictive-State Decoders Architecture. We augment the RNN from Fig. 2 with an
additional objective function R which targets decoding of the internal state through F at each time
step to the predictive-state which is represented as statistics over the future observations.
3
Predictive-State Decoders
Our P REDICTIVE -S TATE D ECODERS architecture extends the Predictive-State Representation idea
to general recurrent architectures. We hypothesize that by encouraging the internal states to encode
information sufficient for reconstructing the predictive state, the resulting internal states better capture
the underlying dynamics and learning can be improved. The result is a simple-to-implement objective
function which is coupled with the existing RNN loss. To represent arbitrary sizes and values of
PSRs with a fixed-size internal state in the recurrent network, we attach a decoding module F (?) to
the internal states to produce the resulting PSR estimates. Figure 3 illustrates our approach.
Our P SD objective R is the predictive-state loss:
X
2
R=
kF (ht ) ? ?([xt+1 , xt+2 , . . .])k2 ,
ht = f (ht?1 , xt?1 ),
(4)
t
where F is a decoder that maps from the internal state ht to an empirical sample of the predictivestate, computed from a sequence of observed future observations available at training. The network
is optimized by minimizing the weighted total loss function L + ?R where ? is the weighting on
the predictive-state objective R. This penalty encourages the internal states to encode information
sufficient for directly predicting sufficient future observations. Unlike more standard regularization
techniques, R does not regularize the parameters of the network but instead regularizes the output
variables, the internal states predicted by the network.
Our method may be interpreted as an instance of Multi-Task Learning (MTL) [13]. MTL has found
use in recent deep neural networks [5, 27, 30]. The idea of MTL is to employ a shared representation
to perform complementary or similar tasks. When the learner exhibits good performance on one
task, some of its understanding can be transferred to a related task. In our case, forcing RNNs to be
able to more explicitly reason about the future they will encounter is an intuitive and general method.
Endowing RNNs with a theoretically-motivated representation of the future better enables them
to serve their purpose of making sequential predictions, resulting in more effective learning. This
difference is pronounced in applications such as imitation and reinforcement learning (Sections 4.2
and 4.3) where the primary objective is to find a control policy to maximize accumulated future
reward while receiving only observations from the system. MTL with P SDs supervises the network
to predict the future and implicitly the consequences of the learned policy. Finally, our P SD objective
can be considered an instance of self-supervision [27] as it uses the inputs to the learner to form a
secondary unsupervised objective.
As discussed in Section 2.1, the purpose of the internal state in recurrent network models (RNNs,
LSTMs, deep, or otherwise) is to capture a quantity similar to that of state. Ideally, the learner
would be able to back-propagate through the primary objective function L and discover the best
representation of the latent state of the system towards minimizing the objective. However, as this
problem highly non-convex, BPTT often yields a locally-optimal solution in a basin determined by
the initialization of the parameters and the dataset. By introducing R, the space of feasible models is
reduced. We observe next how this objective leads our method to find better models.
5
Helicopter GRU Network
101
100
Hopper GRU Network
Baseline
k = 2, ? = 1.0
k = 2, ? = 10.0
k = 5, ? = 1.0
k = 5, ? = 10.0
101
9 ? 100
Observation Loss
Baseline
k = 2, ? = 1.0
k = 2, ? = 10.0
k = 5, ? = 1.0
k = 5, ? = 10.0
Observation Loss
Observation Loss
Pendulum GRU Network
8 ? 100
100
200
300
Iteration
400
500
0
Baseline
k = 2, ? = 1.0
k = 2, ? = 10.0
k = 5, ? = 1.0
k = 5, ? = 10.0
101
100
200
300
Iteration
400
500
100
200
300
Iteration
400
(a) Pendulum
3 ? 101
2 ? 101
0
Baseline
k = 2, ? = 1.0
k = 2, ? = 10.0
k = 5, ? = 1.0
k = 5, ? = 10.0
101
500
100
200
300
Iteration
400
500
Hopper LSTM Network
Baseline
k = 2, ? = 10.0
k = 5, ? = 0.5
k = 5, ? = 10.0
k = 10, ? = 10.0
6 ? 101
9 ? 100
0
4 ? 101
Helicopter LSTM Network
Observation Loss
Observation Loss
Pendulum LSTM Network
100
Observation Loss
0
Baseline
k = 2, ? = 1.0
k = 5, ? = 5.0
k = 5, ? = 10.0
k = 10, ? = 5.0
6 ? 101
4 ? 101
3 ? 101
2 ? 101
0
100
200
300
Iteration
(b) Helicopter
400
500
0
100
200
300
Iteration
400
500
(c) Hopper
Figure 4: Loss over predicting future observations during filtering. For both RNNs with GRU cells
(top) and with with LSTM cells (bottom), adding P SDs to the RNN networks can often improve
performance and convergence rate.
4
Experiments
We present results on problems of increasing complexity for recurrent models: probabilistic filtering,
Imitation Learning (IL), and Reinforcement Learning (RL). The first is easiest, as the goal is to predict
the next future observation given the current observation and internal state. For imitation learning, the
recurrent model is given training-time expert guidance with the goal of choosing actions to maximize
the sequence of future rewards. Finally, we analyze the challenging domain of reinforcement learning,
where the goal is the same as imitation learning but expert guidance is unavailable.
P REDICTIVE -S TATE D ECODERS require two hyperparameters: k, the number of observations to
characterize the predictive state and ?, the regularization trade-off factor. In most cases, we primarily
tune ?, and set k to one of {2, . . . , 10}. For each domain, for each k, there were ? values for which
the performance was worse than the baseline. However, for many sets of hyperparameters, the
performance exceeded the baselines. Most notably, for many experiments, the convergence rate was
significantly better using P SDs, implying that P SDs allows for more efficient data utilization for
learning recurrent models.
P SDs also require a specification of two other parameters in the architecture: the featurization function
? and decoding module F . For simplicity, we use an affine function as the decoder F in Eq. (4). The
results presented below use an identity featurization ? for the presented results but include a short
discussion of second order featurization. We find that in each domain, we are able to improve the
performance of the state-of-the-art baselines. We observe improvements with both GRU and LSTM
cells across a range of k and ?. In IL with P SDs, we come significantly closer and occasionally
eclipse the expert?s performance, whereas the baselines never do. In our RL experiments, our method
achieves statistically significant improvements over the state-of-the-art approach of [18, 41] on the 5
different settings we tested.
4.1
Probabilistic Filtering
In the probabilistic filtering problem, the goal is to predict the future from the current internal state.
Recurrent models for filtering use a multi-step objective function that maximizes the likelihood of the
future observations over the internal states and dynamics model f ?s parameters. Under a Gaussian
assumption (e.g. like a Kalman filter [22]), the equivalent objective that minimizes the negative
P
2
log-likelihood is given as L = t kxt+1 ? f (xt , ht )k .
While traditional methods would explicitly solve for parametric internal states ht using an EM
style approach, we use BPTT to implicitly find an non-parametric internal state. We optimize the
6
Figure 5: Cumulative rewards for AggreVaTeD and AggreVaTeD+P REDICTIVE -S TATE D ECODERS
on partially observable Acrobot and CartPole with both LSTM cells and GRU cells averaged over 15
runs with different random seeds.
end-to-end filtering performance through the P SD joint objective minf,F L + ?R. Our experimental
results are shown in Fig. 4. The experiments were run with ? as the identity, capturing statistics
representing the first moment. We tested ? as second-order statistics and found while the performance
improved over the baseline, it was outperformed by the first moment. In all environments, a dataset
was collected using a preset control policy. In the Pendulum experiments, we predict the pendulum?s
angle ?. The LQR controlled Helicopter experiments [3] use a noisy state as the observation, and
the Hopper dataset was generated using the OpenAI simulation [12] with robust policy optimization
algorithm [36] as the controller.
We test each environment with Tensorflow?s built-in GRU and LSTM cells [1]. We sweep over
various k and ? hyperparameters and present the average results and standard deviations from runs
with different random seeds. Fig. 4 baselines are recurrent models equivalent to P SDs with ? = 0.
4.2
Imitation Learning
We experiment with the partially observable CartPole and Acrobot domains3 from OpenAI Gym [12].
We applied the method of AggreVaTeD [46], a policy-gradient method, to train our expert models.
AggreVaTeD uses access to a cost-to-go oracle in order to train a policy that is sensitive to the value
of the expert?s actions, providing an advantage over behavior cloning IL approaches. The experts
have access to the full state of the robots, unlike the learned recurrent policies.
We tune the parameters of LSTM and GRU agents (e.g., learning rate, number of internal units) and
afterwards only tune ? for P SDs. In Fig. 5, we observe that P SDs improve performance for both GRUand LSTM-based agents and increasing the predictive-state horizon k yields better results. Notably,
P SDs achieves 73% relative improvement over baseline LSTM and 42% over GRU on Cartpole.
Difference random seeds were used. The cumulative reward of the current best policy is shown.
4.3
Reinforcement Learning
Reinforcement learning (RL) increases the problem complexity from imitation learning by removing
expert guidance. The latent state of the system is heavily influenced by the RL agent itself and
changes as the policy improves. We use [18]?s implementation of TRPO [41], a Natural Policy
3
The observation function only provides positional information (including joint angles), excluding velocities.
7
TRPO
TRPO+PSD
Figure 6: Walker Cumulative Rewards and Sorted Percentiles. N = 15, 5e4 TRPO steps per iteration.
Table 1: Top: Mean Average Returns ? one standard deviation, with N = 15 for Walker2d? and
N = 30 otherwise. Bottom: Relative improvement of on the means. ? indicates p < 0.05 and
??
indicates p < 0.005 on Wilcoxon?s signed-rank test for significance of improvement. All runs
computed with 5e3 transitions per iteration, except Walker2d? , with 5e4.
Swimmer
HalfCheetah
Hopper
Walker2d
Walker2d?
[41]
[41]+P SDs
91.3 ? 25.5
97.0 ? 19.4
330 ? 158
372 ? 143
1103 ? 264
1195 ? 272
383 ? 96
416 ? 88
1396 ? 396
1611 ? 436
Rel. ?
6.30%?
13.0%?
9.06%?
8.59%?
15.4%??
Gradient method [28]. Although [41] defines a KL-constraint on policy parameters that affect actions,
our implementation of P SDs introduces parameters (those of the decoder) that are unaffected by the
constraint, as the decoder does not directly govern the agent?s actions.
In these experiments, results are highly stochastic due to both environment randomness and nondeterministic parallelization of rllab [18]. We therefore repeat each experiment at least 15 times
with paired random seeds. We use
(k = 4 for Hopper), the identity
k = 2 for most experiments
featurization for ?, and vary ? in 101 , 100 , . . . , 10?6 , and employ the LSTM cell and other default
parameters of TRPO. We report the same metric as [18]: per-TRPO batch average return across
learning iterations. Additionally, we report per-run performance by plotting the sorted average TRPO
batch returns (each item is a number representing a method?s performance for a single seed).
Figs. 6 and 7 demonstrate that our method generally produces higher-quality results than the baseline.
These results are further summarized by their means and stds. in Table 1. In Figure 6, 40% of our
method?s models are better than the best baseline model. In Figure 7c, 25% of our method?s models
are better than the second-best (98th percentile) baseline model. We compare various RNN cells in
Table 2, and find our method can improve Basic (linear + tanh nonlinearity), GRU, and LSTM RNNs,
and usually reduces the performance variance. We used Tensorflow [1] and passed both the ?hidden"
and ?cell" components of an LSTM?s internal state to the decoder. We also conducted preliminary
additional experiments with second order featurization (?(x) = [x, vec(xxT )]). Corresponding
to Tab. 2, column 1 for the inverted pendulum, second order features yielded 861 ? 41, a 4.9%
improvement in the mean and a large reduction in variance.
5
Conclusion
We introduced a theoretically-motivated method for improving the training of RNNs. Our method
stems from previous literature that assigns statistical meaning to a learner?s internal state for modelling
latent state of the data-generating processes. Our approach uses the objective in PSIMs and applies it to
more complicated recurrent models such as LSTMs and GRUs and to objectives beyond probabilistic
filtering such as imitation and reinforcement learning. We show that our straightforward method
improves performance across all domains with which we experimented.
8
TRPO
(a) Swimmer, N=30
TRPO+PSD
(b) HalfCheetah, N=30
(c) Hopper, N=40
Figure 7: Top: Per-iteration average returns for TRPO and TRPO+P REDICTIVE -S TATE D ECODERS
vs. batch iteration, with 5e3 steps per iteration. Bottom: Sorted per-run mean (across iterations)
average returns. Our method generally produces better models.
Table 2: Variations of RNN units. Mean Average Returns ? one standard deviation, with N = 20.
1e3 transitions per iteration are used. Our method can improve each recurrent unit we tested.
InvertedPendulum
Swimmer
Basic
GRU
LSTM
Basic
GRU
LSTM
[41]
[41]+P SDs
820 ? 139
820 ? 118
673 ? 268
782 ? 183
640 ? 265
784 ? 215
66.0 ? 21.4
71.4 ? 26.9
64.6 ? 55.3
75.1 ? 28.8
56.5 ? 23.8
61.0 ? 23.8
Rel. ?
?0.08%
20.4%
22.6%
8.21%
16.1%
7.94%
Acknowledgements
This material is based upon work supported in part by: Office of Naval Research (ONR) contract
N000141512365, and National Science Foundation NRI award number 1637758.
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12
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6,321 | 6,718 | Optimistic posterior sampling for reinforcement
learning: worst-case regret bounds
Shipra Agrawal
Columbia University
[email protected]
Randy Jia
Columbia University
[email protected]
Abstract
We present an algorithm based on posterior sampling (aka Thompson sampling)
that achieves near-optimal worst-case regret bounds when the underlying Markov
Decision Process (MDP) is communicating with a finite, though unknown,
diameter.
?
?
Our main result is a high probability regret upper bound of O(D
SAT ) for any
communicating MDP with S states, A actions and diameter D, when T ? S 5 A.
Here, regret compares the total reward achieved by the algorithm to the total
expected reward of an optimal infinite-horizon undiscounted average reward policy,
in time horizon ?
T . This result improves over the best previously known upper
?
bound of O(DS
AT ) achieved by any algorithm in this
? setting, and matches the
dependence on S in the established lower bound of ?( DSAT ) for this problem.
Our techniques involve proving some novel results about the anti-concentration of
Dirichlet distribution, which may be of independent interest.
1
Introduction
Reinforcement Learning (RL) refers to the problem of learning and planning in sequential decision
making systems when the underlying system dynamics are unknown, and may need to be learned by
trying out different options and observing their outcomes. A typical model for the sequential decision
making problem is a Markov Decision Process (MDP), which proceeds in discrete time steps. At each
time step, the system is in some state s, and the decision maker may take any available action a to
obtain a (possibly stochastic) reward. The system then transitions to the next state according to a fixed
state transition distribution. The reward and the next state depend on the current state s and the action
a, but are independent of all the previous states and actions. In the reinforcement learning problem,
the underlying state transition distributions and/or reward distributions are unknown, and need to be
learned using the observed rewards and state transitions, while aiming to maximize the cumulative
reward. This requires the algorithm to manage the tradeoff between exploration vs. exploitation, i.e.,
exploring different actions in different states in order to learn the model more accurately vs. taking
actions that currently seem to be reward maximizing.
Exploration-exploitation tradeoff has been studied extensively in the context of stochastic multiarmed bandit (MAB) problems, which are essentially MDPs with a single state. The performance of
MAB algorithms is typically measured through regret, which compares the total reward obtained by
the algorithm to the total expected reward of an optimal action. Optimal regret bounds have been
established for many variations of MAB (see Bubeck et al. [2012] for a survey), with a large majority
of results obtained using the Upper Confidence Bound (UCB) algorithm, or more generally, the
optimism in the face of uncertainty principle. Under this principle, the learning algorithm maintains
tight over-estimates (or optimistic estimates) of the expected rewards for individual actions, and
at any given step, picks the action with the highest optimistic estimate. More recently, posterior
sampling, aka Thompson Sampling [Thompson, 1933], has emerged as another popular algorithm
design principle in MAB, owing its popularity to a simple and extendible algorithmic structure, an
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
attractive empirical performance [Chapelle and Li, 2011, Kaufmann et al., 2012], as well as provably
optimal performance bounds that have been recently obtained for many variations of MAB [Agrawal
and Goyal, 2012, 2013b,a, Russo and Van Roy, 2015, 2014, Bubeck and Liu, 2013]. In this approach,
the algorithm maintains a Bayesian posterior distribution for the expected reward of every action;
then at any given step, it generates an independent sample from each of these posteriors and takes the
action with the highest sample value.
We consider the reinforcement learning problem with finite states S and finite actions A in a similar
regret based framework, where the total reward of the reinforcement learning algorithm is compared
to the total expected reward achieved by a single benchmark policy over a time horizon T . In our
setting, the benchmark policy is the infinite-horizon undiscounted average reward optimal policy for
the underlying MDP, under the assumption that the MDP is communicating with (unknown) finite
diameter D. The diameter D is an upper bound on the time it takes to move from any state s to any
other state s0 using an appropriate policy, for each pair s, s0 . A finite diameter is understood to be
necessary for interesting bounds on the regret of any algorithm in this setting [Jaksch et al., 2010].
The UCRL2 algorithm of Jaksch et al. [2010],?
which is based on the optimism principle, achieved the
?
best previously known upper bound of O(DS
AT ) for this problem. A similar bound was achieved
by Bartlett and Tewari [2009], though assuming the knowledge
of the diameter D. Jaksch et al.
?
[2010] also established a worst-case lower bound of ?( DSAT ) on the regret of any algorithm for
this problem.
Our main contribution is a?
posterior sampling based algorithm with a high
? probability worst-case
?
?
regret upper bound of O(D
SAT + DS 7/4 A3/4 T 1/4 ), which is O(D
SAT ) when T ?? S 5 A.
This improves the previously best known upper bound for this problem by a factor of S, and
matches the dependence on S in the lower bound, for large enough T .
Our algorithm uses an ?optimistic version? of the posterior sampling heuristic, while utilizing several
ideas from the algorithm design structure in Jaksch et al. [2010], such as an epoch based execution
and the extended MDP construction. The algorithm proceeds in epochs, where in the beginning of
?
every epoch, it generates ? = O(S)
sample transition probability vectors from a posterior distribution
for every state and action, and solves an extended MDP with ?A actions and S states formed using
these samples. The optimal policy computed for this extended MDP is used throughout the epoch.
Posterior Sampling for Reinforcement Learning (PSRL) approach has been used previously in Osband
et al. [2013], Abbasi-Yadkori and Szepesvari [2014], Osband and Van Roy [2016], but in a Bayesian
regret framework. Bayesian regret is defined as the expected regret over a known
prior on the
?
?
transition probability matrix. Osband and Van Roy [2016] demonstrate an O(H
SAT ) bound on
the expected Bayesian regret for PSRL in finite-horizon episodic Markov decision processes, when
the episode length is H. In this paper, we consider the stronger notion of worst-case regret, aka
minimax regret, which requires bounding the maximum regret for any instance of the problem. 1
Further, ?
we consider a non-episodic communicating MDP setting, and produce a comparable bound
?
of O(D
SAT ) for large T , where D is the unknown diameter of the communicating MDP. In
comparison to a single sample from the posterior in PSRL, our algorithm is slightly inefficient as
?
it uses multiple (O(S))
samples. It is not entirely clear if the extra samples are only an artifact
of the analysis. In an empirical study of a multiple sample version of posterior sampling for RL,
Fonteneau et al. [2013] show that multiple samples can potentially improve the performance of
posterior sampling in terms of probability of taking the optimal decision. Our analysis utilizes some
ideas from the Bayesian regret analysis, most importantly the technique of stochastic optimism from
Osband et al. [2014] for deriving tighter deviation bounds. However, bounding the worst-case regret
requires several new technical ideas, in particular, for proving ?optimism? of the gain of the sampled
MDP. Further discussion is provided in Section 4.
We should also compare our result with the very recent result of Azar et al. [2017], which provides
an optimistic version of value-iteration algorithm with a minimax (i.e., worst-case) regret bound of
1
Worst-case regret is a strictly stronger notion of regret in case the reward distribution function is known
and only the transition probability distribution is unknown, as we will assume here for the most part. In case of
unknown reward distribution, extending our worst-case regret bounds would require an assumption of bounded
rewards, where as the Bayesian regret bounds in the above-mentioned literature allow more general (known)
priors on the reward distributions with possibly unbounded support. Bayesian regret bounds in those more
general settings are incomparable to the worst-case regret bounds presented here.
2
?
? HSAT ) when T ? H 3 S 3 A. However, the setting considered in Azar et al. [2017] is that of an
O(
episodic MDP, where the learning agent interacts with the system in episodes of fixed and known
length H. The initial state of each episode can be arbitrary, but importantly, the sequence of these
initial states is shared by the algorithm and any benchmark policy. In contrast, in the non-episodic
setting considered in this paper, the state trajectory of the benchmark policy over T time steps can be
completely different from the algorithm?s trajectory. To the best of our understanding, the shared
sequence of initial states and the fixed known length H of episodes seem to form crucial components
of the analysis in Azar et al. [2017], making it difficult to extend their analysis to the non-episodic
communicating MDP setting considered in this paper.
Among other related work, Burnetas and Katehakis [1997] and Tewari and Bartlett
[2008] present optimistic linear programming approaches that achieve logarithmic regret
bounds with problem dependent constants. Strong PAC bounds have been provided in
Kearns and Singh [1999], Brafman and Tennenholtz [2002], Kakade et al. [2003], Asmuth et al.
[2009], Dann and Brunskill [2015]. There, the aim is to bound the performance of the policy
learned at the end of the learning horizon, and not the performance during learning as quantified by
regret. Strehl and Littman [2005], Strehl and Littman [2008] provide an optimistic algorithm for
bounding regret in a discounted reward setting, but the definition of regret is slightly different in that
it measures the difference between the rewards of an optimal policy and the rewards of the learning
algorithm along the trajectory taken by the learning algorithm.
2
2.1
Preliminaries and Problem Definition
Markov Decision Process (MDP)
We consider a Markov Decision Process M defined by tuple {S, A, P, r, s1 }, where S is a finite
state-space of size S, A is a finite action-space of size A, P : S ? A ? ?S is the transition model,
r : S ? A ? [0, 1] is the reward function, and s1 is the starting state. When an action a ? A is taken
0
in a state s ? S, a reward rs,a
P is generated and the system transitions to the next state s ? S with
probability Ps,a (s0 ), where s0 ?S Ps,a (s0 ) = 1.
We consider ?communicating? MDPs with finite ?diameter? (see Bartlett and Tewari [2009]
for an in-depth discussion). Below we define communicating MDPs, and recall some useful known
results for such MDPs.
Definition 1 (Policy). A deterministic policy ? : S ? A is a mapping from state space to action
space.
Definition 2 (Diameter D(M)). Diameter D(M) of an MDP M is defined as the minimum time
required to go from one state to another in the MDP using some deterministic policy:
D(M) =
max
?
min Ts?s
0,
s6=s0 ,s,s0 ?S ?:S?A
?
0
where Ts?s
0 is the expected number of steps it takes to reach state s when starting from state s and
using policy ?.
Definition 3 (Communicating MDP). An MDP M is communicating if and only if it has a finite
diameter. That is, for any two states s 6= s0 , there exists a policy ? such that the expected number of
?
steps to reach s0 from s, Ts?s
0 , is at most D, for some finite D ? 0.
Definition 4 (Gain of a policy). The gain of a policy ?, from starting state s1 = s, is defined as the
infinite horizon undiscounted average reward, given by
T
1X
? (s) = E[ lim
rst ,?(st ) |s1 = s].
T ?? T
i=1
?
where st is the state reached at time t.
Lemma 2.1 (Optimal gain for communicating MDPs). For a communicating MDP M with diameter
D:
(a) (Puterman [2014] Theorem 8.1.2, Theorem 8.3.2) The optimal (maximum) gain ?? is state
independent and is achieved by a deterministic stationary policy ? ? , i.e., there exists a
3
deterministic policy ? ? such that
?
?? := max
max ?? (s0 ) = ?? (s), ?s ? S.
0
?
s ?S
Here, ? ? is referred to as an optimal policy for MDP M.
(b) (Tewari and Bartlett [2008], Theorem 4) The optimal gain ?? satisfies the following equations,
T
T
?? = min max rs,a + Ps,a
h ? hs = max rs,a + Ps,a
h? ? h?s , ?s
(1)
h?RS s,a
a
?
where h , referred to as the bias vector of MDP M, satisfies:
max h?s ? min h?s ? D.
s
s
Given the above definitions and results, we can now define the reinforcement learning problem
studied in this paper.
2.2
The reinforcement learning problem
The reinforcement learning problem proceeds in rounds t = 1, . . . , T . The learning agent starts from
a state s1 at round t = 1. In the beginning of every round t, the agent takes an action at ? A and
observes the reward rst ,at as well as the next state st+1 ? Pst ,at , where r and P are the reward
function and the transition model, respectively, for a communicating MDP M with diameter D.
The learning agent knows the state-space S, the action space A, as well as the rewards rs,a , ?s ?
S, a ? A, for the underlying MDP, but not the transition model P or the diameter D. (The assumption
of known and deterministic rewards has been made here only for simplicity of exposition, since the
unknown transition model is the main source of difficulty in this problem. Our algorithm and results
can be extended to bounded stochastic rewards with unknown distributions using standard Thompson
Sampling for MAB, e.g., using the techniques in Agrawal and Goyal [2013b].)
The agent can use the past observations to learn the underlying MDP model and decide future actions.
PT
The goal is to maximize the total reward t=1 rst ,at , or equivalently, minimize the total regret over
a time horizon T , defined as
PT
R(T, M) := T ?? ? t=1 rst ,at
(2)
where ?? is the optimal gain of MDP M.
We present an algorithm for the learning agent with a near-optimal upper bound on the regret
R(T, M) for any communicating MDP M with diameter D, thus bounding the worst-case regret
over this class of MDPs.
3
Algorithm Description
Our algorithm combines the ideas of Posterior sampling (aka Thompson Sampling) with the extended
MDP construction used in Jaksch et al. [2010]. Below we describe the main components of our
algorithm.
t
Some notations: Ns,a
denotes the total number of times the algorithm visited state s and played
t
t
action a until before time t, and Ns,a
(i) denotes the number of time steps among these Ns,a
steps
where the next
state
was
i,
i.e.,
a
transition
from
state
s
to
i
was
observed.
We
index
the
states
from
1
PS
t
t
(i) = Ns,a
to S, so that i=1 Ns,a
for any t. We use the symbol 1 to denote the vector of all 1s, and
1i to denote the vector with 1 at the ith coordinate and 0 elsewhere.
Doubling epochs: Our algorithm uses the epoch based execution framework of Jaksch et al. [2010].
An epoch is a group of consecutive rounds. The rounds t = 1, . . . , T are broken into consecutive
epochs as follows: the k th epoch begins at the round ?k immediately after the end of (k ? 1)th epoch
?
?k
and ends at the first round ? such that for some state-action pair s, a, Ns,a
? 2Ns,a
. The algorithm
computes a new policy ?
?k at the beginning of every epoch k, and uses that policy through all the
rounds in that epoch. It is easy to observe that irrespective of how the policy ?
?k is computed, the
number of epochs in T rounds is bounded by SA log(T ).
4
Posterior Sampling: We use posterior sampling to compute the policy ?
?k in the beginning of
every epoch. Dirichlet distribution is a convenient choice maintaining posteriors for the transition
probability vectors Ps,a for every s ? S, a ? A, as they satisfy the following useful property: given
a prior Dirichlet(?1 , . . . , ?S ) on Ps,a , after observing a transition from state s to i (with underlying
probability Ps,a (i)), the posterior distribution is given by Dirichlet(?1 , . . . , ?i + 1, . . . , ?S ). By this
property, for any s ? S, a ? A, on starting from prior Dirichlet(1) for Ps,a , the posterior at time t is
t
Dirichlet({Ns,a
(i) + 1}i=1,...,S ).
Our algorithm uses a modified, optimistic version of this approach. At the beginning of every epoch
k, for every s ? S, a ? A such that Ns,a ? ?, it generates multiple samples for Ps,a from a ?boosted?
posterior. Specifically, it generates ? = O(S log(SA/?)) independent sample probability vectors
?,k
Q1,k
s,a , . . . , Qs,a as
?k
Qj,k
s,a ? Dirichlet(Ms,a ),
t
where Mts,a denotes the vector [Ms,a
(i)]i=1,...,S , with
t
t
Ms,a
(i) := ?1 (Ns,a
(i) + ?), for i = 1, . . . , S.
(3)
q
Here, ? = O(log(T /?)), ? = O(log(T /?)), ? = TAS + 12?S 2 , and ? ? (0, 1) is a parameter of
the algorithm. In the regret analysis, we derive sufficiently large constants that can be used in the
definition of ?, ?, ? to guarantee the bounds. However, no attempt has been made to optimize those
constants, and it is likely that much smaller constants suffice.
For every remaining s, a, i.e., those with small Ns,a (Ns,a < ?) the algorithm use a simple optimistic
sampling described in Algorithm 1. This special sampling for s, a with small Ns,a has been introduced
to handle a technical difficulty in analyzing the anti-concentration of Dirichlet posteriors when the
parameters are very small. We suspect that with an improved analysis, this may not be required.
Extended MDP: The policy ?
?k to be used in epoch k is computed as the optimal policy of an
? k defined by the sampled transition probability vectors, using the construction of
extended MDP M
Jaksch et al. [2010]. Given sampled vectors Qj,k
s,a , j = 1, . . . , ?, for every state-action pair s, a, we
k
?
define extended MDP M by extending the original action space as follows: for every s, a, create
? actions for every action a ? A, denoting by aj the action corresponding to action a and sample
? k , on taking action aj in state s, reward is rs,a but transitions to the next state
j; then, in MDP M
follows the transition probability vector Qj,k
s,a .
? k to take actions in the action
Note that the algorithm uses the optimal policy ?
?k of extended MDP M
? k , where the policy ?
space A which is technically different from the action space of MDP M
?k is
defined. We slightly abuse the notation to say that the algorithm takes action at = ?
? (st ) to mean that
the algorithm takes action at = a ? A when ?
?k (st ) = aj for some j.
Our algorithm is summarized as Algorithm 1.
4
Regret Bounds
We prove the following bound on the regret of Algorithm 1 for the reinforcement learning problem.
Theorem 1. For any communicating MDP M with S states, A actions, and diameter D, with
probability 1 ? ?. the regret of Algorithm 1 in time T ? CDA log2 (T /?) is bounded as:
?
? D SAT + DS 7/4 A3/4 T 1/4 + DS 5/2 A
R(T, M) ? O
where C is an absolute constant. For T ? S 5 A, this implies a regret bound of
?
? D SAT .
R(T, M) ? O
? hides logarithmic factors in S, A, T, ? and absolute constants.
Here O
The rest of this section is devoted to proving the above theorem. Here, we provide a sketch of the
proof and discuss some of the key lemmas, all missing details are provided in the supplementary
material.
5
Algorithm 1 A posterior sampling based algorithm for the reinforcement learning problem
Inputs: State space S, Action space A, starting state s1 , reward function r, time horizon
T , parameters ? ? (0, 1], ? = O(S log(SA/?)), ? = O(log(T /?)), ? = O(log(T /?)), ? =
q
TS
A
+ 12?S 2 .
1
Initialize: ? 1 := 1, M?s,a
= ?1.
for all epochs k = 1, 2, . . . , do
Sample transition probability vectors: For each s, a, generate ? independent sample probability
vectors Qj,k
s,a , j = 1, . . . , ?, as follows:
?k
? (Posterior sampling): For s, a such that Ns,a
? ?, use samples from the Dirichlet
distribution:
?k
Qj,k
s,a ? Dirichlet(Ms,a ),
?
?k
< ?, use the following
(Simple optimistic sampling): For remaining s, a, with Ns,a
simple optimistic sampling: let
?
Ps,a
= P?s,a ? ?,
r
?k
Ns,a
(i)
3P?s,a (i) log(4S)
?
where Ps,a (i) = N ?k , and ?i = min
+
?
N k
s,a
s,a
3 log(4S) ?
, Ps,a (i)
?k
Ns,a
, and
let z be a random vector picked uniformly at random from {11 , . . . , 1S }; set
PS
?
?
Qj,k
s,a = Ps,a + (1 ?
i=1 Ps,a (i))z.
? k constructed using sample
Compute policy ?
? k : as the optimal gain policy for extended MDP M
j,k
set {Qs,a , j = 1, . . . , ?, s ? S, a ? A}.
Execute policy ?
?k:
for all time steps t = ?k , ?k + 1, . . . , until break epoch do
Play action at = ?
?k (st ).
Observe the transition to the next state st+1 .
t+1
t+1
(i) for all a ? A, s, i ? S as defined (refer to Equation (3)).
(i), Ms,a
Set Ns,a
?k
t+1
If Nst ,at ? 2Nst ,at , then set ?k+1 = t + 1 and break epoch.
end for
end for
4.1
Proof of Theorem 1
PT
As defined in Section 2, regret R(T, M) is given by R(T, M) = T ?? ? t=1 rst ,at , where ?? is
the optimal gain of MDP M, at is the action taken and st is the state reached by the algorithm at
time t. Algorithm 1 proceeds in epochs k = 1, 2, . . . , K, where K ? SA log(T ). To bound its regret
in time T , we first analyze the regret in each epoch k, namely,
P?k+1 ?1
Rk := (?k+1 ? ?k )?? ? t=?
rst ,at ,
k
and bound Rk by roughly
?k+1
X Ns,a
? N ?k
p ? s,a
D
k
Ns,a
s,a
?
k+1
?k
where, by definition, for every s, a, (Ns,a
? Ns,a
) is the number of times this state-action pair is
visited in epoch k. The proof of this bound has two main components:
(a) Optimism: The policy ?
?k used by the algorithm in epoch k is computed as an optimal gain policy
? k . The first part of the proof is to show that with high probability, the
of the extended MDP M
k
?
extended MDP M is (i) a communicating MDP with diameter at most 2D, and (ii) optimistic,
i.e., has optimal gain at least (close to) ?? . Part (i) is stated as Lemma 4.1, with a proof provided
? k be the optimal gain of the extended MDP M
? k . In
in the supplementary material. Now, let ?
6
Lemma 4.2, which forms one of the main novel technical components of our proof, we show that
with probability 1 ? ?,
q
SA
? k ? ?? ? O(D
?
?
).
T
We first show that above holds if for every s, a, there exists a sample transition probability vector
whose projection on a fixed unknown vector (h? ) is optimistic. Then, in Lemma 4.3 we prove this
optimism by deriving a fundamental new result on the anti-concentration of any fixed projection
of a Dirichlet random vector (Proposition A.1 in the supplementary material).
Substituting this upper bound on ?? , we have the following bound on Rk with probability 1 ? ?:
q
P?k+1 ?1 ?
SA
?
Rk ? t=?
?
?
r
+
O(D
)
.
(4)
k
s
,a
t t
T
k
? k for MDP
(b) Deviation bounds: Optimism guarantees that with high probability, the optimal gain ?
?
k
?
?
? k.
?k , ?k is the gain of the chosen policy ?
?k for MDP M
M is at least ? . And, by definition of ?
However, the algorithm executes this policy on the true MDP M. The only difference between
? k , the next
the two is the transition model: on taking an action aj := ?
?k (s) in state s in MDP M
state follows the sampled distribution
P?s,a := Qj,k
s,a ,
(5)
where as on taking the corresponding action a in MDP M, the next state follows the distribution
? k and the average reward obtained by the
Ps,a . The next step is to bound the difference between ?
?
algorithm by bounding the deviation (Ps,a ? Ps,a ). This line of argument bears similarities to the
analysis of UCRL2 in Jaksch et al. [2010], but with tighter deviation bounds that we are able to
guarantee due to the use of posterior sampling instead of deterministic optimistic bias used in
? k , the bias vector h,
? and
UCRL2. Now, since at = ?
?k (st ), using the relation between the gain ?
k
?
reward vector of optimal policy ?
?k for communicating MDP M (refer to Lemma 2.1)
P?k+1 ?1 ?
P?k+1 ?1 ?
?
? ? rst ,at
=
(Pst ,at ? 1st )T h
t=?k
t=?k
P?k+1 ?1 ?
?
=
(Pst ,at ? Pst ,at + Pst ,at ? 1st )T h
(6)
t=?k
? ? RS , the bias vector of MDP M
? k satisfies
where with high probability, h
? s ? mins h
? s ? D(M
? k ) ? 2D (refer to Lemma 4.1).
maxs h
? for all s, a, to bound the first term in above.
Next, we bound the deviation (P?s,a ? Ps,a )T h
?
Note that h is random and can be arbitrarily correlated with P? , therefore, we need to bound
? ? [0, 2D]S ).
maxh?[0,2D]S (P?s,a ? Ps,a )T h. (For the above term, w.l.o.g. we can assume h
?k
For s, a such that Ns,a
> ?, P?s,a = Qj,k
s,a is a sample from the Dirichlet posterior. In Lemma 4.4,
we show that with high probability,
k
? pD + DS
(7)
max (P?s,a
? Ps,a )T h ? O(
?k ).
?k
Ns,a
Ns,a
?
This bound is an improvement by a S factor over the corresponding deviation bound obtainable
for the optimistic estimates of Ps,a in UCRL2. The derivation of this bound utilizes and extends
?k
the stochastic optimism technique from Osband et al. [2014]. For s, a with Ns,a
? ?, P?s,a = Qj,k
s,a
is a sample from the simple optimistic sampling, where we can only show the following weaker
?k
bound, but since this is used only while Ns,a
is small, the total contribution of this deviation will
be small:
s
!
S
DS
k
T
?
?
max (Ps,a ? Ps,a ) h ? O D
.
(8)
?k +
?k
Ns,a
Ns,a
h?[0,2D]S
h?[0,2D]S
? ?k , h,
? st ] = P T h
?
Finally, to bound the second term in (6), we observe that E[1Tst+1 h|?
st ,at and use
?
Azuma-Hoeffding inequality to obtain with probability (1 ? SA ):
p
P?k+1 ?1
? ? O( (?k+1 ? ?k ) log(SA/?)).
(Pst ,at ? 1st )T h
(9)
t=?k
7
Combining the above observations (equations (4), (6), (7), (8), (9)), we obtain the following bound
on Rk within logarithmic factors:
r
?k+1
X Ns,a
?
p
? N ?k
SA
?k+1
?k+1
p ? s,a 1(Ns,a
>
?)
+
D(?k+1 ??k )
+D
S
1
(N
?
?)
+D ?k+1 ? ?k .
s,a
k
T
Ns,a
s,a
(10)
We can finish the proof by observing that (by definition of an epoch) the number of visits of any
state-action pair can at most double in an epoch,
?k+1
?k
?k
Ns,a
? Ns,a
? Ns,a
,
and therefore, substituting this observation in (10), we can bound (within logarithmic factors) the
PK
total regret R(T ) = k=1 Rk as:
!
q
K
p ?
p
P
P
P
?
?k
SA
k
D(?k+1 ? ?k ) T + D
Ns,a + D
SNs,a + D ?k+1 ? ?k
?
k=1
?
k >?
s,a:Ns,a
k <?
s,a:Ns,a
?
?
?K
Ns,a
) + D log(K)(SA S?) + D KT
P
?k+1
?k
where we used Ns,a
? 2Ns,a
and k (?k+1 ? ?k ) = T . Now, we use that K ? SA log(T ),
q
?
TS
2
and SA S? = O(S 7/4 A3/4 T 1/4 + S 5/2 A log(T /?)) (using ? =
A + 12?S ). Also, since
p
?
P
P
?K
?K
Ns,a
? SAT , and we obtain,
s,a Ns,a ? T , by simple worst scenario analysis,
s,a
?
?
SAT + DS 7/4 A3/4 T 1/4 + DS 5/2 A).
R(T, M) ? O(D
?
P
? D SAT + D log(K)( s,a
4.2
p
Main lemmas
Following lemma form the main technical components of our proof. All the missing proofs are
provided in the supplementary material.
Lemma 4.1. Assume T ? CDA log2 (T /?) for a large enough constant C. Then, with probability
? k is bounded by 2D.
1 ? ?, for every epoch k, the diameter of MDP M
? k of the extended MDP
Lemma 4.2. With probability 1 ? ?, for every epoch k, the optimal gain ?
k
?
M satisfies:
q
? k ? ?? ? O D log2 (T /?) SA ,
?
T
?
where ? the optimal gain of MDP M and D is the diameter.
Proof. Let h? be the bias vector for an optimal policy ? ? of MDP M (refer to Lemma 2.1 in the
preliminaries section). Since h? is a fixed (though unknown) vector with |hi ? hj | ? D, we can
apply Lemma 4.3 to obtain that with probability 1 ? ?, for all s, a, there exists a sample vector Qj,k
s,a
for some j ? {1, . . . , ?} such that
T ?
T
?
(Qj,k
s,a ) h ? Ps,a h ? ?
q
? k which for any s, takes
where ? = O D log2 (T /?) SA
. Now, consider the policy ? for MDP M
T
action aj , with a = ? ? (s) and j being a sample satisfying above inequality. Let Q? be the transition
?
matrix for this policy, whose rows are formed by the vectors Qj,k
s,? ? (s) , and P? be the transition
matrix whose rows are formed by the vectors Ps,?? (s) . Above implies Q? h? ? P?? h? ? ?1.
We use this inequality along with the known relations between the gain and the bias of optimal
?
? k satisfies
policy in communicating MDPs to obtain that the gain ?(?)
of policy in ? for MDP M
?
?
?(?)
? ? ? ? (details provided in the supplementary material), which proves the lemma statement
? k ? ?(?).
?
since by optimality ?
8
Lemma 4.3. (Optimistic Sampling) Fix any vector h ? RS such that |hi ? hi0 | ? D for any i, i0 ,
?
and any epoch k. Then, for every s, a, with probability 1 ? SA
there exists at least one j such that
q
2
SA
T
T
(Qj,k
.
s,a ) h ? Ps,a h ? O D log (T /?)
T
Lemma 4.4. (Deviation bound) With probability 1 ? ?, for all epochs k, sample j, all s, a
s
?
!
?
S
log(SAT
/?)
log(SAT
/?)
?
?k
? O D
>?
+D
, Ns,a
?
?k
?k
?
N
N
s,a
s,a
j,k
T
s
!
max (Qs,a ? Ps,a ) h ?
?
h?[0,2D]S
?
S log(S)
S log(SAT /?)
?
?k
?
+D
,
Ns,a
??
? O D
N ?k
N ?k
s,a
5
s,a
Conclusions
We presented an algorithm inspired by posterior sampling that achieves near-optimal worst-case
regret bounds for the reinforcement learning problem with communicating MDPs in a non-episodic,
undiscounted average reward setting. Our algorithm may be viewed as a more efficient randomized
version of the UCRL2 algorithm
of Jaksch et al. [2010], with randomization via posterior sampling
?
forming the key to the S factor improvement in the regret bound provided by our algorithm. Our
analysis demonstrates that posterior sampling provides the right amount of uncertainty in the samples,
so that an optimistic policy can be obtained without excess over-estimation.
While our work surmounts some important technical difficulties in obtaining worst-case regret bounds
for posterior sampling based algorithms for communicating MDPs, the provided bound is tight in
its dependence on S and A only for large T (specifically, for T ? S 5 A). Other related results on
tight
? worst-case regret bounds have a similar requirement of large T (Azar et al. [2017] produce an
? HSAT ) bound when T ? H 3 S 3 A). Obtaining a cleaner worst-case regret bound that does
O(
not require such a condition remains an open question. Other important directions of future work
?
include reducing the number of posterior samples required in every epoch from O(S)
to constant or
logarithmic in S, and extensions to contextual and continuous state MDPs.
9
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11
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6,322 | 6,719 | Mean teachers are better role models:
Weight-averaged consistency targets improve
semi-supervised deep learning results
Antti Tarvainen
The Curious AI Company
[email protected]
Harri Valpola
The Curious AI Company
[email protected]
Abstract
The recently proposed Temporal Ensembling has achieved state-of-the-art results in
several semi-supervised learning benchmarks. It maintains an exponential moving
average of label predictions on each training example, and penalizes predictions
that are inconsistent with this target. However, because the targets change only once
per epoch, Temporal Ensembling becomes unwieldy when learning large datasets.
To overcome this problem, we propose Mean Teacher, a method that averages
model weights instead of label predictions. As an additional benefit, Mean Teacher
improves test accuracy and enables training with fewer labels than Temporal
Ensembling. Using the same network architecture, Mean Teacher achieves error
rate 4.35% on SVHN with 250 labels, better than Temporal Ensembling does with
1000 labels. We show that Mean Teacher is compatible with residual networks, and
improve state of the art on CIFAR-10 with 4000 labels from 10.55% to 6.28%. Our
preliminary experiments also suggest a large improvement over state of the art on
semi-supervised ImageNet 2012.
1
Introduction
Deep learning has seen tremendous success in areas such as image and speech recognition. In order
to learn useful abstractions, deep learning models require a large number of parameters, thus making
them prone to over-fitting (Figure 1a). Moreover, adding high-quality labels to training data manually
is often expensive. Therefore, it is desirable to use regularization methods that exploit unlabeled data
effectively to reduce over-fitting in semi-supervised learning.
When a percept is changed slightly, a human typically still considers it to be the same object. Correspondingly, a classification model should favor functions that give consistent output for similar data
points. One approach for achieving this is to add noise to the input of the model. To enable the model
to learn more abstract invariances, the noise may be added to intermediate representations, an insight
that has motivated many regularization techniques, such as Dropout [27]. Rather than minimizing
the classification cost at the zero-dimensional data points of the input space, the regularized model
minimizes the cost on a manifold around each data point, thus pushing decision boundaries away
from the labeled data points (Figure 1b).
Since the classification cost is undefined for unlabeled examples, the noise regularization by itself
does not aid in semi-supervised learning. To overcome this, the model [20] evaluates each data
point with and without noise, and then apply a consistency cost between the two predictions. In this
case, the model assumes a dual role as a teacher and a student. As a student, it learns as before; as a
teacher, it generates targets, which are then used by itself as a student for learning. Since the model
itself generates targets, they may very well be incorrect. If too much weight is given to the generated
targets, the cost of inconsistency outweighs that of misclassification, preventing the learning of new
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: A sketch of a binary classification task with two labeled examples (large blue dots) and
one unlabeled example, demonstrating how the choice of the unlabeled target (black circle) affects
the fitted function (gray curve). (a) A model with no regularization is free to fit any function that
predicts the labeled training examples well. (b) A model trained with noisy labeled data (small dots)
learns to give consistent predictions around labeled data points. (c) Consistency to noise around
unlabeled examples provides additional smoothing. For the clarity of illustration, the teacher model
(gray curve) is first fitted to the labeled examples, and then left unchanged during the training of the
student model. Also for clarity, we will omit the small dots in figures d and e. (d) Noise on the teacher
model reduces the bias of the targets without additional training. The expected direction of stochastic
gradient descent is towards the mean (large blue circle) of individual noisy targets (small blue circles).
(e) An ensemble of models gives an even better expected target. Both Temporal Ensembling and the
Mean Teacher method use this approach.
information. In effect, the model suffers from confirmation bias (Figure 1c), a hazard that can be
mitigated by improving the quality of targets.
There are at least two ways to improve the target quality. One approach is to choose the perturbation
of the representations carefully instead of barely applying additive or multiplicative noise. Another
approach is to choose the teacher model carefully instead of barely replicating the student model.
Concurrently to our research, Miyato et al. [15] have taken the first approach and shown that Virtual
Adversarial Training can yield impressive results. We take the second approach and will show that it
too provides significant benefits. To our understanding, these two approaches are compatible, and
their combination may produce even better outcomes. However, the analysis of their combined effects
is outside the scope of this paper.
Our goal, then, is to form a better teacher model from the student model without additional training.
As the first step, consider that the softmax output of a model does not usually provide accurate
predictions outside training data. This can be partly alleviated by adding noise to the model at
inference time [4], and consequently a noisy teacher can yield more accurate targets (Figure 1d). This
approach was used in Pseudo-Ensemble Agreement [2] and has lately been shown to work well on
semi-supervised image classification [13, 22]. Laine & Aila [13] named the method the ? model; we
will use this name for it and their version of it as the basis of our experiments.
The ? model can be further improved by Temporal Ensembling [13], which maintains an exponential
moving average (EMA) prediction for each of the training examples. At each training step, all
the EMA predictions of the examples in that minibatch are updated based on the new predictions.
Consequently, the EMA prediction of each example is formed by an ensemble of the model?s current
version and those earlier versions that evaluated the same example. This ensembling improves the
quality of the predictions, and using them as the teacher predictions improves results. However, since
each target is updated only once per epoch, the learned information is incorporated into the training
process at a slow pace. The larger the dataset, the longer the span of the updates, and in the case of
on-line learning, it is unclear how Temporal Ensembling can be used at all. (One could evaluate all
the targets periodically more than once per epoch, but keeping the evaluation span constant would
require O(n2 ) evaluations per epoch where n is the number of training examples.)
2
Mean Teacher
To overcome the limitations of Temporal Ensembling, we propose averaging model weights instead
of predictions. Since the teacher model is an average of consecutive student models, we call this the
Mean Teacher method (Figure 2). Averaging model weights over training steps tends to produce a
2
prediction
prediction
3
3
classification
cost
consistency
cost
?
??
? exponential ??
moving
average
3
label
input
student model
teacher model
Figure 2: The Mean Teacher method. The figure depicts a training batch with a single labeled
example. Both the student and the teacher model evaluate the input applying noise (?, ? 0 ) within
their computation. The softmax output of the student model is compared with the one-hot label
using classification cost and with the teacher output using consistency cost. After the weights of the
student model have been updated with gradient descent, the teacher model weights are updated as an
exponential moving average of the student weights. Both model outputs can be used for prediction,
but at the end of the training the teacher prediction is more likely to be correct. A training step with
an unlabeled example would be similar, except no classification cost would be applied.
more accurate model than using the final weights directly [18]. We can take advantage of this during
training to construct better targets. Instead of sharing the weights with the student model, the teacher
model uses the EMA weights of the student model. Now it can aggregate information after every
step instead of every epoch. In addition, since the weight averages improve all layer outputs, not just
the top output, the target model has better intermediate representations. These aspects lead to two
practical advantages over Temporal Ensembling: First, the more accurate target labels lead to a faster
feedback loop between the student and the teacher models, resulting in better test accuracy. Second,
the approach scales to large datasets and on-line learning.
More formally, we define the consistency cost J as the expected distance between the prediction of
the student model (with weights ? and noise ?) and the expected prediction of the teacher model
(with weights ?0 and noise ? 0 ).
h
J(?) = Ex,?0 ,? kf (x, ?0 , ? 0 )
f (x, ?, ?)k
2
i
The difference between the ? model, Temporal Ensembling, and Mean teacher is how the teacher
predictions are generated. Whereas the ? model uses ?0 = ?, and Temporal Ensembling approximates
f (x, ?0 , ? 0 ) with a weighted average of successive predictions, we define ?t0 at training step t as the
EMA of successive ? weights:
?t0 = ??t0
1
+ (1
?)?t
where ? is a smoothing coefficient hyperparameter. An additional difference between the three
algorithms is that the ? model applies training to ?0 whereas Temporal Ensembling and Mean Teacher
treat it as a constant with regards to optimization.
We can approximate the consistency cost function J by sampling noise ?, ? 0 at each training step
with stochastic gradient descent. Following Laine & Aila [13], we use mean squared error (MSE) as
our consistency cost function. In the experiments section, we will also explore the use of other cost
functions.
3
Table 1: Error rate percentage on SVHN over 10 runs (4 runs when using all labels). We use
exponential moving average weights in the evaluation of all our models. All the methods use a similar
13-layer CNN architecture. See Table 5 in the Appendix for results without input augmentation.
250 labels
73257 images
GAN [24]
? model [13]
Temporal Ensembling [13]
VAT+EntMin [15]
Supervised-only
27.77 ? 3.18
? model
9.69 ? 0.92
Mean Teacher
4.35 ? 0.50
500 labels
73257 images
18.44 ? 4.8
6.65 ? 0.53
5.12 ? 0.13
16.88 ? 1.30
6.83 ? 0.66
4.18 ? 0.27
1000 labels
73257 images
8.11 ? 1.3
4.82 ? 0.17
4.42 ? 0.16
3.86
12.32 ? 0.95
4.95 ? 0.26
3.95 ? 0.19
73257 labels
73257 images
2.54 ? 0.04
2.74 ? 0.06
2.75 ? 0.10
2.50 ? 0.07
2.50 ? 0.05
Table 2: Error rate percentage on CIFAR-10 over 10 runs (4 runs when using all labels).
1000 labels
50000 images
GAN [24]
? model [13]
Temporal Ensembling [13]
VAT+EntMin [15]
Supervised-only
46.43 ? 1.21
? model
27.36 ? 1.20
Mean Teacher
21.55 ? 1.48
3
2000 labels
50000 images
33.94 ? 0.73
18.02 ? 0.60
15.73 ? 0.31
4000 labels
50000 images
18.63 ? 2.32
12.36 ? 0.31
12.16 ? 0.31
10.55
20.66 ? 0.57
13.20 ? 0.27
12.31 ? 0.28
50000 labels
50000 images
5.56 ? 0.10
5.60 ? 0.10
5.82 ? 0.15
6.06 ? 0.11
5.94 ? 0.15
Experiments
To test our hypotheses, we first replicated the ? model [13] in TensorFlow [1] as our baseline. We
then modified the baseline model to use weight-averaged consistency targets. The model architecture
is a 13-layer convolutional neural network (CNN) with three types of noise: random translations and
horizontal flips of the input images, Gaussian noise on the input layer, and dropout applied within the
network. We use mean squared error as the consistency cost and ramp up its weight from 0 to its final
value during the first 80 epochs. The details of the model and the training procedure are described in
Appendix B.1.
3.1
Comparison to other methods on SVHN and CIFAR-10
We ran experiments using the Street View House Numbers (SVHN) and CIFAR-10 benchmarks [16].
Both datasets contain 32x32 pixel RGB images belonging to ten different classes. In SVHN, each
example is a close-up of a house number, and the class represents the identity of the digit at the center
of the image. In CIFAR-10, each example is a natural image belonging to a class such as horses, cats,
cars and airplanes. SVHN contains of 73257 training samples and 26032 test samples. CIFAR-10
consists of 50000 training samples and 10000 test samples.
Tables 1 and 2 compare the results against recent state-of-the-art methods. All the methods in the
comparison use a similar 13-layer CNN architecture. Mean Teacher improves test accuracy over the
? model and Temporal Ensembling on semi-supervised SVHN tasks. Mean Teacher also improves
results on CIFAR-10 over our baseline ? model.
The recently published version of Virtual Adversarial Training by Miyato et al. [15] performs even
better than Mean Teacher on the 1000-label SVHN and the 4000-label CIFAR-10. As discussed in the
introduction, VAT and Mean Teacher are complimentary approaches. Their combination may yield
better accuracy than either of them alone, but that investigation is beyond the scope of this paper.
4
Table 3: Error percentage over 10 runs on SVHN with extra unlabeled training data.
? model (ours)
Mean Teacher
500 labels
73257 images
6.83 ? 0.66
4.18 ? 0.27
500 labels
173257 images
4.49 ? 0.27
3.02 ? 0.16
500 labels
573257 images
3.26 ? 0.14
2.46 ? 0.06
Figure 3: Smoothened classification cost (top) and classification error (bottom) of Mean Teacher and
our baseline ? model on SVHN over the first 100000 training steps. In the upper row, the training
classification costs are measured using only labeled data.
3.2
SVHN with extra unlabeled data
Above, we suggested that Mean Teacher scales well to large datasets and on-line learning. In addition,
the SVHN and CIFAR-10 results indicate that it uses unlabeled examples efficiently. Therefore, we
wanted to test whether we have reached the limits of our approach.
Besides the primary training data, SVHN includes also an extra dataset of 531131 examples. We
picked 500 samples from the primary training as our labeled training examples. We used the rest of
the primary training set together with the extra training set as unlabeled examples. We ran experiments
with Mean Teacher and our baseline ? model, and used either 0, 100000 or 500000 extra examples.
Table 3 shows the results.
3.3
Analysis of the training curves
The training curves on Figure 3 help us understand the effects of using Mean Teacher. As expected, the
EMA-weighted models (blue and dark gray curves in the bottom row) give more accurate predictions
than the bare student models (orange and light gray) after an initial period.
Using the EMA-weighted model as the teacher improves results in the semi-supervised settings. There
appears to be a virtuous feedback cycle of the teacher (blue curve) improving the student (orange) via
consistency cost, and the student improving the teacher via exponential moving averaging. If this
feedback cycle is detached, the learning is slower, and the model starts to overfit earlier (dark gray
and light gray).
Mean Teacher helps when labels are scarce. When using 500 labels (middle column) Mean Teacher
learns faster, and continues training after ? model stops improving. On the other hand, in the
all-labeled case (left column), Mean Teacher and the ? model behave virtually identically.
5
Figure 4: Validation error on 250-label SVHN over four runs per hyperparameter setting and
their means. In each experiment, we varied one hyperparameter, and used the evaluation run
hyperparameters of Table 1 for the rest. The hyperparameter settings used in the evaluation runs are
marked with the bolded font weight. See the text for details.
Mean Teacher uses unlabeled training data more efficiently than the ? model, as seen in the middle
column. On the other hand, with 500k extra unlabeled examples (right column), ? model keeps
improving for longer. Mean Teacher learns faster, and eventually converges to a better result, but the
sheer amount of data appears to offset ? model?s worse predictions.
3.4
Ablation experiments
To assess the importance of various aspects of the model, we ran experiments on SVHN with 250
labels, varying one (or a few) hyperparameters at a time while keeping the others fixed.
Removal of noise (Figures 4(a) and 4(b)). In the introduction and Figure 1, we presented the
hypothesis that the ? model produces better predictions by adding noise to the model on both sides.
But after the addition of Mean Teacher, is noise still needed? Yes. We can see that either input
augmentation or dropout is necessary for passable performance. On the other hand, input noise does
not help when augmentation is in use. Dropout on the teacher side provides only a marginal benefit
over just having it on the student side, at least when input augmentation is in use.
Sensitivity to EMA decay and consistency weight (Figures 4(c) and 4(d)). The essential hyperparameters of the Mean Teacher algorithm are the consistency cost weight and the EMA decay ?. How
sensitive is the algorithm to their values? We can see that in each case the good values span roughly
an order of magnitude and outside these ranges the performance degrades quickly. Note that EMA
decay ? = 0 makes the model a variation of the ? model, although somewhat inefficient one because
the gradients are propagated through only the student path. Note also that in the evaluation runs we
used EMA decay ? = 0.99 during the ramp-up phase, and ? = 0.999 for the rest of the training. We
chose this strategy because the student improves quickly early in the training, and thus the teacher
should forget the old, inaccurate, student weights quickly. Later the student improvement slows, and
the teacher benefits from a longer memory.
Decoupling classification and consistency (Figure 4(e)). The consistency to teacher predictions
may not necessarily be a good proxy for the classification task, especially early in the training. So
far our model has strongly coupled these two tasks by using the same output for both. How would
decoupling the tasks change the performance of the algorithm? To investigate, we changed the model
to have two top layers and produce two outputs. We then trained one of the outputs for classification
and the other for consistency. We also added a mean squared error cost between the output logits, and
then varied the weight of this cost, allowing us to control the strength of the coupling. Looking at the
results (reported using the EMA version of the classification output), we can see that the strongly
coupled version performs well and the too loosely coupled versions do not. On the other hand, a
moderate decoupling seems to have the benefit of making the consistency ramp-up redundant.
6
Table 4: Error rate percentage of ResNet Mean Teacher. We report the test results from 10 runs for
CIFAR-10 and validation results from 2 runs for ImageNet.
CIFAR-10
1000 labels
50000 images
State of the art
CNN Mean Teacher
ResNet Mean Teacher
21.55 ? 1.48
10.08 ? 0.41
CIFAR-10
4000 labels
50000 images
10.55 [15]
12.31 ? 0.28
6.28 ? 0.15
ImageNet 2012
128000 labels
1280000 images
35.24 ? 0.90 [19]
19.76 ? 0.05
Changing from MSE to KL-divergence (Figure 4(f)) Following Laine & Aila [13], we use mean
squared error (MSE) as our consistency cost function, but KL-divergence would seem a more natural
choice. Which one works better? We can see that MSE works better than KL-divergence or their
intermediate versions. See Appendix C for details of the cost function family we used for this
experiment and for our intuition about why MSE may perform so well.
3.5
Mean Teacher with residual networks on CIFAR-10 and ImageNet
In the experiments above, we used a traditional 13-layer convolutional architecture, which has
the benefit of making comparisons to earlier work easy. In order to explore the effect of the
model architecture, we ran experiments using a 12-block Residual Network [8] with Shake-Shake
regularization [5] on CIFAR-10. The details of the model and the training procedure are described
in Appendix B.2. As shown in Table 4, the results improve remarkably with the better network
architecture.
To test scaling of the method to realistic images, we ran experiments on Imagenet 2012 dataset [21]
with 10% of labels. We used a residual network with squeeze-and-excitation blocks [10] and ShakeShake regularization, and saw a clear improvement over the state of the art. However, the ImageNet
results come with caveats: As the test set is not publicly available, we measured the results using
the validation set. We also used a very small 8-block architecture, and probably did not use the best
hyperparameters. Regardless, the results suggest that Mean Teacher is useful also on large natural
images.
4
Related work
Noise regularization of neural networks was proposed by Sietsma & Dow [25]. More recently, several
types of perturbations have been shown to regularize intermediate representations effectively in
deep learning. Adversarial Training [6] changes the input slightly to give predictions that are as
different as possible from the original predictions. Dropout [27] zeroes random dimensions of layer
outputs. Dropconnect [30] generalizes Dropout by zeroing individual weights instead of activations.
Stochastic Depth [11] drops entire layers of residual networks, and Swapout [26] generalizes Dropout
and Stochastic Depth. Shake-shake regularization [5] duplicates residual paths and samples a linear
combination of their outputs independently during forward and backward passes.
Several semi-supervised methods are based on training the model predictions to be consistent to
perturbation. The Denoising Source Separation framework (DSS) [28] uses denoising of latent
variables to learn their likelihood estimate. The variant of Ladder Network [20] implements DSS
with a deep learning model for classification tasks. It produces a noisy student predictions and clean
teacher predictions, and applies a denoising layer to predict teacher predictions from the student
predictions. The ? model [13] improves the model by removing the explicit denoising layer and
applying noise also to the teacher predictions. Similar methods had been proposed already earlier for
linear models [29] and deep learning [2]. Virtual Adversarial Training [15] is similar to the ? model
but uses adversarial perturbation instead of independent noise.
The idea of a teacher model training a student is related to model compression [3] and distillation [9].
The knowledge of a complicated model can be transferred to a simpler model by training the
simpler model with the softmax outputs of the complicated model. The softmax outputs contain
7
more information about the task than the one-hot outputs, and the requirement of representing this
knowledge regularizes the simpler model. Besides its use in model compression, distillation can be
used to harden trained models against adversarial attacks [17]. The difference between distillation
and consistency regularization is that distillation is performed after training whereas consistency
regularization is performed on training time.
Consistency regularization can be seen as a form of label propagation [32]. Training samples that
resemble each other are more likely to belong to the same class. Label propagation takes advantage
of this assumption by pushing label information from each example to examples that are near it
according to some metric. Label propagation can also be applied to deep learning models [31].
However, ordinary label propagation requires a predefined distance metric in the input space. In
contrast, consistency targets employ a learned distance metric implied by the abstract representations
of the model. As the model learns new features, the distance metric changes to accommodate these
features. Therefore, consistency targets guide learning in two ways. On the one hand they spread the
labels according to the current distance metric, and on the other hand, they aid the network learn a
better distance metric.
5
Conclusion
Temporal Ensembling, Virtual Adversarial Training and other forms of consistency regularization
have recently shown their strength in semi-supervised learning. In this paper, we propose Mean
Teacher, a method that averages model weights to form a target-generating teacher model. Unlike
Temporal Ensembling, Mean Teacher works with large datasets and on-line learning. Our experiments
suggest that it improves the speed of learning and the classification accuracy of the trained network.
In addition, it scales well to state-of-the-art architectures and large image sizes.
The success of consistency regularization depends on the quality of teacher-generated targets. If the
targets can be improved, they should be. Mean Teacher and Virtual Adversarial Training represent
two ways of exploiting this principle. Their combination may yield even better targets. There are
probably additional methods to be uncovered that improve targets and trained models even further.
Acknowledgements
We thank Samuli Laine and Timo Aila for fruitful discussions about their work, and Phil Bachman
and Colin Raffel for corrections to the pre-print version of this paper. We also thank everyone at The
Curious AI Company for their help, encouragement, and ideas.
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6,323 | 672 | Object-Based Analog VLSI Vision
Circuits
Christof Koch
Computation and Neural Systems
California Institute of Technology
Pasadena, CA
Bimal Mathur, Shih-Chii Liu
Rockwell International Science Center
Thousand Oaks, CA
John G. Harris
MIT Artificial Intelligence Laboratory
Cambridge, MA
Jin Luo, Massimo Sivilotti
Tanner Research, Inc.
Pasadena, CA
Abstract
We describe two successfully working, analog VLSI vision circuits
that move beyond pixel-based early vision algorithms. One circuit,
implementing the dynamic wires model, provides for dedicated lines
of communication among groups of pixels that share a common
property. The chip uses the dynamic wires model to compute the
arclength of visual contours. Another circuit labels all points inside
a given contour with one voltage and all other with another voltage. Its behavior is very robust, since small breaks in contours are
automatically sealed, providing for Figure-Ground segregation in
a noisy environment. Both chips are implemented using networks
of resistors and switches and represent a step towards object level
processing since a single voltage value encodes the property of an
ensemble of pixels.
1
CONTOUR-LENGTH CHIP
Contour length computation is useful for further processing such as structural
saliency (Shaashua and Ullman, 1988), which is thought to be an important stage
before object recognition. This computation is impossible on an analog chip if we
828
Object-Based Analog VLSI Vision Circuits
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contours scanned into the contour length chip. The voltage is a linear function of
contour length.
are restricted to pure pixel- or image-based operations. The dynamic wire methodology provides dedicated lines of communication among groups of pixels of an image
which share common properties (Liu and Harris, 1992). In simple applications, object regions can be grouped together to compute the area or the center of mass of
each object. Alternatively, object boundaries may be used to compute curvature
or contour length. These ideas are not limited to sets of simple electrical wires;
resistive networks can also be configured on the fly. The problem of smoothing
object contours using resistive dynamic wires has been previously studied (Liu and
Harris, 1992).
In the contour-length application, pixels along image contours are electrically connected by a reconfigurable dynamic wire. The first step of processing requires that
each contour choose an arbitrary but unique leader pixel. The top of Fig. 2 shows
several examples of contours and indicates which pixels where chosen as leaders by
the chip. The leader is responsible for connecting a shunting resistor between the
shared dynamic wire and ground. If each pixel on the contour supplies a constant
amount of current to the dynamic wire, all of the current must flow through the
shunting resistor. Therefore, the voltage on the wire will encode the contour length.
Fig. 1 shows the linear relationship between the measured voltage and the contour
length. The bottom half of Fig. 2 shows the length of several example contours using
an intensity coding. The brighter contours indicate a higher voltage and therefore a
longer contour. The contour length chip was fabricated through MOSIS using 2J.Lm
CMOS technology. The prototype 2x2 mm 2 chip contains an a 7x7 pixel array.
829
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Figure 2: Four binary contour images were scanned into the contour-length chip and
are shown in the top figure. The highlighted pixel in each contour was chosen by the
chip to be the leader. The bottom figure shows the measured voltages (indicated
by intensity) from the contour-length chip for the four images are shown. Since the
intensity of each pixel encodes its length, the longer contours are brighter.
Object-Based Analog VLSI Vision Circuits
The most challenging aspect of the design of the contour-length chip is the circuitry
to uniquely select a pixel from each contour to be the leader. The leader is selected
by entering all the pixels along each contour in a competition. The winner of this
competition will be the leader. This competition requires each node to charge up its
own capacitor once a reset line has been triggered. The first node that charges its
capacitor above the trip point of a digital inverter will pull down a global precharge
wire which connects all the pixels along the contour. This wire will in turn latch the
states of the winner and losers. One of the pixels will normally toggle first because
of the inherent offsets and component mismatches in silicon.
2
FIGURE-GROUND CHIP
Ullman (1984) proposed that a visual routine is used in human vision to determine
if a specified point in the visual field is inside or outside of one (or more) closed
visual contours. We describe such a chip that labels all points inside a givenpossibly incomplete and broken-contour. We assume that the presence of an edge
in the image causes switches at the corresponding grid point within a rectangular
resistive network to open (Fig. 3). A closed edge contour will then correspond to a
series of open switches on this grid. We assume that the visual contour will always
encompass the central grid point in the array. At this point, the resistive grid is
connected to the battery V/ig, while the periphery of the array is grounded to Vgnd.
If the voltage at all other grid points is left floating and the contour is complete,
that is, the central grid point is completely isolated from the periphery of the chip
by a series of open grid points, the voltage at all points inside the contour rises to
V/ig, while the voltage at grid points outside the contour will settle to V gnd . Thus,
the figure will be labeled by one voltage level and ground by another.
Contours in real images are frequently incomplete, but instead have broken segments
of one or more pixels. This will enable the current to flow through these holes in
the contour, smearing out the voltage level between inside and outside. We exploit
a property of Mead's (1989) Hres circuit, used to implement the resistances, to
achieve contour completion. While the current flowing through Hres is linear in the
voltage gradient for small voltage differences, it saturates for large voltage gradients.
At those locations where the contour is broken, the saturating resistances limit the
current flow, preventing smoothing of the voltage profile to occur.
Figure 4 shows the responses of the Figure-Ground chip to different input patterns
collected with a fixed bias: V/ ig 3.5 V and Vgnd 2 V. The two-dimensional data
is presented as pairs of images. The input patterns are located on the left while the
corresponding voltage outputs are presented next to the input on the right. The
black-white patterns are used to represent the binary input data encoding object
boundaries. Thus, at all locations marked in black, the associated switches shown in
Fig. 1a are opened. The gray-scale on the right denotes output voltage levels, where
the darkest value corresponds to V/ig and the brightest to Vgnd. The center pixel
of the view field is always set to V/ig. Notice that at every node where a boundary
input signal (in black) appears and the switches are opened, the output voltage at
that node is tied to Vgnd . This can be seen best in (e; white outline). To evaluate
the ability of our circuit to perform Figure-Ground segregation in the presence of
breaks in the contour, more and wider breaks are introduced into a simple square
=
=
831
832
Koch, Mathur, Liu, Harris, Luo, and Sivilotti
+~+
,.,
t+.?.
.-!"
Voltage
+
vr
;:' .
-+
+
Vgnd
(a)
-.&
(b)
Figure 3: (a) The Figure-Ground network is made up of resistors and switches. The
input to the chip is a binary edge map. At every grid point in the rectangular array
where edges have been found, four switches are opened, isolating that node from its
four neighbors (the shaded edge contour corresponds to a series of isolated nodes).
We assume that the central point in the array is always enclosed by the contour.
This point is connected to a voltage source V/ig, while the periphery is connected to
the voltage Vgnd. If the contour is unbroken, the voltage at each interior point will
then rise to V/ig, while all outside grid points will settle to Vgnd. Thus, the object
is rapidly segregated from the background. If the contour is not complete, the
saturating resistors (indicated with simple resistors) will limit the current flowing
through these holes in the contour and partially seal off the boundary. (b) represents
a conceptual view of how an object (figure) is segregated from the background in
the two-dimensional view field, in terms of two distinct voltage levels (V/ig labels
the object and Vgnd labels the background). The circuit has 48 by 48 nodes on a 4.6
by 6.8 mm 2 die size and was implemented using MOSIS 2 pm CMOS technology.
Object-Based Analog VLSI Vision Circuits
(a) The input consists of a
completely enclosed box. The
network is therefore broken into
two isolated segments, the inside
and the outside of the box and
labeled by two very different
voltage values, Vfig and Vgnd.
(b) The object boundary has a
break equal to one pixel at the
center of the left and right edges.
Due to the large Voltage
difference across these two leaks,
the
saturated
horizontal
resistances,
HRes,
saturate,
thereby helping to "seal" off these
breaks using a very simple
algorithm.
n
LJ
I
I
(C) The width of the breaks in the
contour increases to three pixels
each. Yet HRes still acts to
effectively seal the two holes and
the "Figure" is segregated from
the "Surround".
I
I
(d) The width of the breaks
I
I
increases to five pixels each. Due
to the much smaller voltage
gradient across this wider gap in
the contour, the voltage spreads
outside the figure.
I
I
(e) A total of four breaks, each
fi ve pixels wide, prevents the
''Figure'' from being segregated.
The system can't decide whether a
single object with wide breaks at
its side or four separate objects
are present.
II
L-.J
Figure 4
833
834
Koch, Mathur, Liu, Harris, Luo, and Sivilotti
?
?
?
??
--_....__......_-_.
( a )
(
b
)
(c)
Figure 5: The Figure-Ground response to a noisy and incomplete contour outlining
a hand (the binary image shown in (a) is scanned in from off-chip). The output
voltage is shown as intensity in (b) and as a 3-D plot in (c) . The center node is
tied to 3.5 V and marked as black in (c). The shaded area labels all pixels whose
voltage is above 2.4 V. Notice the voltage decay along the little finger, due to an
incomplete contour at the finger tip.
Object-Based Analog VLSI Vision Circuits
contour. The box and break points on the sides are center-row symmetrical and the
breaks are respectively one, three and five pixels wide. In (e), two additional, five
pixel wide breaks have been included. For small enough breaks, our circuit has an
excellent boundary-completion capabilities. This is important for machine vision,
since real images rarely have complete boundaries.
The performance of the chip is illustrated in Fig. 4. If the contour is unbroken,
the voltage inside the figure rises to V/ ig , segregating it from the surround. If a
small gap appears in the contour, it can be partially sealed off by the action of
the saturating resistance Hres, which limits the current flowing through this gap,
inhibiting full voltage equalization from occurring. As the break in the contour
becomes larger, the voltage gradient across the illusionary contour between the
upper and the lower part of the figure becomes smaller and smaller. If Hres is set
to a low conductance, the gradient becomes larger again (Fig. 5c); now, however,
the chip fails to discriminate between very small and large gaps. Note that inside
and outside are strictly defined only for a closed contour. Thus, it is somewhat
arbitrary at what distance two edges are considered to be part of the same or
separate contours (e.g., Fig. 5). If the output voltage is thresholded at 3.0 V (in
the case of Fig. 5b), the contour with one or two pixel breaks would be considered
a single Figure, while the two larger breaks would not be.
3
CONCLUSION
Most analog vision chips are restricted to work either at the local, pixel-level or
the global, image-level. The dynamic wire and figure-ground chips discussed in
this paper allow data-dependent neighborhoods to form. With these configured
neighborhoods, analog chips can now perform object-level processing.
Acknowledgements
This Work is supported by the National Science Foundation, the Office of Naval
Research and Rockwell International Science Center. We thank MOSIS for all chip
fabrication. JGH is supported by an NSF postdoctoral fellowship.
References
Liu, S. and Harris, J .G. (1992), Dynamic wires: an analog VLSI model for object
processing, Internat. Journal of Compo Vision. 8: pp. 231-239 .
Luo, J., Koch, C. and Mathur, B. (1992), Figure-Ground segregation using an
analog VLSI Chip, IEEE Micro, Vol. 12 46-57, 1992.
Shaashua, A. and Ullman, S. (1988), Structural saliency: The detection of globally
salient structures using a locally connected network. In Proceedings of the IEEE
Computer Vision and Pattern Recognition Conference.
Ullman, S. (1984), Visual routines, Cognition, Vol. 18, pp. 97-159, 1984.
835
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6,324 | 6,720 | Matching neural paths: transfer from recognition to
correspondence search
Nikolay Savinov1
1
Lubor Ladicky1
Marc Pollefeys1,2
Department of Computer Science at ETH Zurich, 2 Microsoft
{nikolay.savinov,lubor.ladicky,marc.pollefeys}@inf.ethz.ch
Abstract
Many machine learning tasks require finding per-part correspondences between
objects. In this work we focus on low-level correspondences ? a highly ambiguous
matching problem. We propose to use a hierarchical semantic representation of
the objects, coming from a convolutional neural network, to solve this ambiguity.
Training it for low-level correspondence prediction directly might not be an option
in some domains where the ground-truth correspondences are hard to obtain. We
show how transfer from recognition can be used to avoid such training. Our idea is
to mark parts as ?matching? if their features are close to each other at all the levels
of convolutional feature hierarchy (neural paths). Although the overall number
of such paths is exponential in the number of layers, we propose a polynomial
algorithm for aggregating all of them in a single backward pass. The empirical
validation is done on the task of stereo correspondence and demonstrates that we
achieve competitive results among the methods which do not use labeled target
domain data.
1
Introduction
Finding per-part correspondences between objects is a long-standing problem in machine learning.
The level at which correspondences are established can go as low as pixels for images or millisecond
timestamps for sound signals. Typically, it is highly ambiguous to match at such a low level: a pixel
or a timestamp just does not contain enough information to be discriminative and many false positives
will follow. A hierarchical semantic representation could help to solve the ambiguity: we could
choose the low-level match which also matches at the higher levels. For example, a car contains a
wheel which contains a bolt. If we want to check if this bolt matches the bolt in another view of the
car, we should check if the wheel and the car match as well.
One possible hierarchical semantic representation could be computed by a convolutional neural
network. The features in such a network are composed in a hierarchical manner: the lower-level
features are used to compute higher-level features by applying convolutions, max-poolings and
non-linear activation functions on them. Nevertheless, training such a convolutional neural network
for correspondence prediction directly (e.g., [25], [2]) might not be an option in some domains
where the ground-truth correspondences are hard and expensive to obtain. This raises the question of
scalability of such approaches and motivates the search for methods which do not require training
correspondence data.
To address the training data problem, we could transfer the knowledge from the source domain where
the labels are present to the target domain where no labels or few labeled data are present. The most
common form of transfer is from classification tasks. Its promise is two-fold. First, classification
labels are one of the easiest to obtain as it is a natural task for humans. This allows to create huge
recognition datasets like Imagenet [18]. Second, the features from the low to mid-levels have been
shown to transfer well to a variety of tasks [22], [3], [15].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Although there has been a huge progress in transfer from classification to detection [7], [17], [19],
[16], segmentation [12], [1] and other semantic reasoning tasks like single-image depth prediction
[4], the transfer to correspondence search has been limited [13], [10], [8].
We propose a general solution to unsupervised transfer from recognition to correspondence search at
the lowest level (pixels, sound millisecond timestamps). Our approach is to match paths of activations
coming from a convolutional neural network, applied on two objects to be matched. More precisely,
to establish matching on the lowest level, we require the features to match at all different levels of
convolutional feature hierarchy. Those different-level features form paths. One such path would
consist of neural activations reachable from the lowest-level feature to the highest-level feature in the
network topology (in other words, the lowest level feature lies in the receptive field of the highest
level). Since every lowest-level feature belongs to many paths, we do voting based on all of them.
Although the overall number of such paths is exponential in the number of layers and thus infeasible
to compute naively, we prove that the voting is possible in polynomial time in a single backward
pass through the network. The algorithm is based on dynamic programming and is similar to the
backward pass for gradient computation in the neural network.
Empirical validation is done on the task of stereo correspondence on two datasets: KITTI 2012 [6]
and KITTI 2015 [14]. We quantitatively show that our method is competitive among the methods
which do not require labeled target domain data. We also qualitatively show that even dramatic
changes in low-level structure can be handled reasonably by our method due to the robustness of the
recognition hierarchy: we apply different style transfers [5] to corresponding images in KITTI 2015
and still successfully find correspondences.
2
Notation
Our method is generally applicable to the cases where the input data has a multi-dimensional grid
topology layout. We will assume input objects o to be from the set of B-dimensional grids ? ? RB
and run convolutional neural networks on those grids. The per-layer activations from those networks
will be contained in the set of (B + 1)-dimensional grids ? ? RB+1 . Both the input data and the
activations will be indexed by a (B + 1)-dimensional vector x = (x, y, . . . , c) ? NB+1 , where x is a
column index, y is a row index, etc., and c ? {1, . . . , C} is the channel index (we will assume C = 1
for the input data, which is a non-restrictive assumption as we will explain later).
We will search for correspondences between those grids, thus our goal will be to estimate shifts
d ? D ? ZB+1 for all elements in the grid. The choice of the shift set D is task-dependent. For
example, for sound B = 1 and only 1D shifts can be considered. For images, B = 2 and D could be
a set of 1D shifts (usually called a stereo task) or a set of 2D shifts (usually called an optical flow
task).
In this work, we will be dealing with convolutional neural network architectures, consisting of
convolutions, max-poolings and non-linear activation functions (one example of such an architecture
is a VGG-net [20], if we omit softmax which we will not use for the transfer). We assume every
convolutional layer to be followed by a non-linear activation function throughout the paper and will
not specify those functions explicitly.
The computational graph of these architectures is a directed acyclic graph G = {A, E}, where
A = {a1 , . . . , a|A| } is a set of nodes, corresponding to neuron activations (|A| denotes the size of
this set), and E = {e1 , . . . , e|E| } is a set of arcs, corresponding to computational dependencies (|E|
denotes the size of this set). Each arc is represented as a tuple (ai , aj ), where ai is the input (origin),
SL
aj is the output (endpoint). The node set consists of disjoint layers A = `=0 A` . The arcs are only
allowed to go from the previous layer to the next one.
We will use the notation A` (x) for the node in `-th layer at position x; in(x` ) for the set of
origins x`?1 of arcs, entering layer ` at position x` of the reference object; x`+1 ? out(x` ) for
the set of endpoints of arcs, exiting layer ` at position x` of the reference object. Let f` ? F =
{maxpool, conv} be the mathematic operator which corresponds to forward computation in layer
` as a ? f` (in(a)), a ? A` (with a slight abuse of notation, we use a for both the nodes in the
computational graph and the activation values which are computed in those nodes).
2
Searched G?
Reference G
Shifts
Max-pooling
k4 (d) = 0 k4 (d) = 0 k4 (d) = 0
Convolution
k3 (d) = 1 k3 (d) = 1 k3 (d) = 1
Max-pooling
k2 (d) = 1 k2 (d) = 1 k2 (d) = 1
Convolution
k1 (d) = 3 k1 (d) = 3 k1 (d) = 2
Input
k0 (d) = 3 k0 (d) = 3 k0 (d) = 2
Figure 1: Four siamese paths are shown. Two of them (red) have the same origin and support the
hypothesis of the shift d = 3 for this origin. The other two (green and pink) have different origins
and support hypotheses d = 3 and d = 2 for their respective origins.
3
Correspondence via path matching
We will consider two objects, reference o ? ? and searched o0 ? ?, for which we want to find
correspondences. After applying a CNN on them, we get graphs G and G0 of activations. The goal is
to establish correspondences between the input-data layers A0 and A00 . That is, every cell A0 (x) in
the reference object o ? ? has a certain shift d ? D in the searched object o0 ? ?, and we want to
estimate d.
Here comes the cornerstone idea of our method: we establish the matching of A0 (x) with A00 (x ? d)
for a shift d if there is a pair of ?parallel? paths (we call this pair a siamese path), originating at
those nodes and ending at the last layers AL , A0L , which match. This pair of paths must have the
same spatial shift with respect to each other at all layers, up to subsampling, and go through the
same feature channels with respect to each other. We take the subsampling into account by per-layer
functions
$ %
?
? = d ,
k` (d) = ?` (k`?1 (d)), ` = 1, . . . , L,
?` (d)
k0 (d) = d,
(1)
q`
where k` (d) is how the zero-layer shift d transforms at layer `, q` is the `-th layer spatial subsampling
factor (note that rounding and division on vectors is done element-wise). Then a siamese path P can
be represented as
P = (p, p0 ),
P
p = (A0 (xP
0 ), . . . , AL (xL )),
0
P
p0 = (A00 (xP
0 ? k0 (d)), . . . , AL (xL ? kL (d)))
(2)
P
where xP
0 = x and x` denotes the position at which the path P intersects layer ` of the reference
activation graph. Such paths are illustrated in Fig. 1. The logic is simple: matching in a siamese
path means that the recognition hierarchy detects the same features at different perception levels with
the same shifts (up to subsampling) with respect to the currently estimated position x, which allows
for a confident prediction of match. The fact that a siamese path is ?matched? can be established by
computing the matching function (high if it matches, low if not)
M (P ) =
L
K
0
P
m` (A` (xP
` ), A` (x` ? k` (d)))
(3)
`=0
where m` (?, ?) is a matching function for individual neurons (prefers them both to be similar and
non-zero at the same time) and is a logical-and-like operator. Both will be discussed later.
Since we want to estimate the shift for a node A0 (x), we will consider all possible shifts and vote for
each of them. Let us denote a set of siamese paths, starting at A` (x) and A0` (x ? d) and ending at
the last layer, as P` (x, d).
For every shift d ? D we introduce U (x, d) as the log-likelihood of the event that d is the correct
shift, i.e. A0 (x) matches A00 (x ? d). To collect the evidence from all possible paths, we ?sum up?
3
the matching functions for all individual paths, leading to
M
U (x, d) =
M (P ) =
P ?P0 (x,d)
M
L
K
0
P
m` (A` (xP
` ), A` (x` ? k` (d)))
(4)
P ?P0 (x,d) `=0
where the sum-like operator ? will be discussed later.
The distribution U (x, d) can be used to either obtain the solution as d? (x) = arg maxd?D U (x, d)
or to post-process the distribution with any kind of spatial smoothing optimization and then again
take the best-cost solution.
The obvious obstacle to using the distribution U (x, d) is that
Observation 1. If K is the minimal number of activation channels in all the layers of the network
and L is the number of layers, the number of paths, considered in the computation of U (x, d) for a
single originating node, is ?(K L ) ? at least exponential in the number of layers.
In practice, it is infeasible to compute U (x, d) naively. In this work, we prove that it is possible
to compute U (x, d) in O(|A| + |E|) ? thus linear in the number of layers ? using the algorithm
which will be introduced in the next section.
4
Linear-time backward algorithm
Theorem 1. For any m` (?, ?) and any pair of operators h?, i such that is left-distributive over
?, i.e. a (b ? c) = a b ? a c, we can compute U (x, d) for all x and d in O(|A| + |E|).
Proof Since there is distributivity, we can use a dynamic programming approach similar to the one
developed for gradient backpropagation.
`
First, let us introduce subsampling functions ks` (d) = ?s (ks?1
(d)), k`` (d) = d, s ? `. Note that
0
ks = ks as introduced in Eq. 1.
Then, let us introduce auxiliary variables U` (x` , d) for each layer ` = 0, . . . , L, which have the same
definition as U (x, d) except for the fact that the paths, considered in them, start from the later layer `:
M
U` (x` , d) =
M (P ) =
P ?P` (x` ,d)
M
L
K
0
P
`
ms (As (xP
s ), As (xs ? ks (d))).
(5)
P ?P` (x` ,d) s=`
Note that U (x, d) = U0 (x, d). The idea is to iteratively recompute U` (x` , d) based on known
U`+1 (x`+1 , ?` (d)) for all x`+1 . Eventually, we will get to the desired U0 (x, d).
The first step is to notice that all the paths share the same prefix and write it out explicitly:
U` (x` , d) =
M
L
K
0
P
`
ms (As (xP
s ), As (xs ? ks (d)))
P ?P` (x` ,d) s=`
"
=
M
m` (A` (x` ), A0` (x` ? d))
P ?P` (x` ,d)
L
K
#
0
P
`
ms (As (xP
s ), As (xs ? ks (d))) .
s=`+1
(6)
Now, we want to pull the prefix m` (A` (x` ), A0` (x` ? d)) out of the ?sum?. For that purpose, we
will need the set of endpoints out(x` ), introduced in the notation in Section 2. The ?sum? can be
re-written in terms of those endpoints as
" L
#
M
K
0
P
`
U` (x` , d) =
m` (A` (x` ), A0` (x` ? d))
ms (As (xP
s ), As (xs ? ks (d))) .
x`+1 ?out(x` )
P ?P`+1 (x`+1 ,?`+1 (d))
s=`+1
(7)
4
Algorithm 1 Backward pass
1: procedure BACKWARD(G, G0 )
2:
for AL (xL ) ? AL do
3:
for d ? kL (D) do
4:
UL (xL , d) ? mL (AL (xL ), A0L (xL ? d)),
. Initialize the last layer.
5:
end for
6:
end for
7:
for ` = L-1, ..., 0 do
8:
for A` (x` ) ? A` do
9:
for d ? k` (D) do
10:
S ? 0,
11:
for x`+1 ? out(x` ) do
12:
S ? S ? U`+1 (x`+1 , ?`+1 (d)),
13:
end for
14:
U` (x` , d) ? m` (A` (x` ), A0` (x` ? d)) S,
15:
end for
16:
end for
17:
end for
18:
return U0
. Return the distribution for the first layer.
19: end procedure
The last step is to use the left-distributivity of over ? to pull the prefix out of the ?sum?:
U` (x` , d) = m` (A` (x` ), A0` (x` ? d))
M
L
K
0
P
`
ms (As (xP
s ), As (xs ? ks (d)))
s=`+1
x`+1 ?out(x` )
P ?P`+1 (x`+1 ,?`+1 (d))
= m` (A` (x` ), A0` (x` ? d))
M
U`+1 (x`+1 , ?`+1 (d)).
(8)
x`+1 ?out(x` )
The detailed procedure is listed in Algorithm 1. We use the notation k` (D) for the set of subsampled
shifts which is the result of applying function k` to every element of the set of initial shifts D.
5
Choice of neuron matching function m and operators ?,
For the convolutional layers, we use the matching function
(
0
if w = 0, v = 0,
mconv (w, v) = min(w,v)
otherwise.
max(w,v)
(9)
For the max-pooling layers, the computational graph can be truncated to just one active connection
(as only one element influences higher-level features). Moreover, max-pooling does not create any
additional features, only passes/subsamples the existing ones. Thus it does not make sense to take into
account the pre-activations for those layers as they are the same as activations (up to subsampling).
For these reasons, we use
mmaxpool (w, v) = ?(w = arg max Nw ) ? ?(v = arg max Nv ),
(10)
where Nw is the neighborhood of max-pooling covering node w, ?(?) is the indicator function (1 if
the condition holds, 0 otherwise).
In this paper, we use sum as ? and product as . Another possible choice would be max for ?
and min or product for ? theoretically, those combinations satisfy the conditions in Theorem 1.
Nevertheless, we found sum/product combination working better than others. This could be explained
by the fact that max as ? would be taken over a huge set of paths which is not robust in practice.
6
Experiments
We validate our approach in the field of computer vision as our method requires a convolutional
neural network trained on a large recognition dataset. Out of the vision correspondence tasks, we
5
Table 1: Summary of the convolutional neural network VGG-16. We only show the part up to the 8-th
layer as we do not use higher activations (they are not pixel-related enough). In the layer type row, c
stands for 3x3 convolution with stride 1 followed by the ReLU non-linear activation function [11] and
p for 2x2 max-pooling with stride 2. The input to convolution is padded with the ?same as boundary?
rule.
Layer index
1
2
3
4
5
6
7
8
Layer type
Output channels
c
64
c
64
p
64
c
128
c
128
p
128
c
256
c
256
chose stereo matching to validate our method. For this task, the input data dimensionality is B = 2
and the shift set is represented by horizontal shifts D = {(0, 0, 0), . . . , (Dmax , 0, 0)}. We always
convert images to grayscale before running CNNs, following the observation by [25] that color does
not help.
For pre-trained recognition CNN, we chose the VGG-16 network [20]. This network is summarized
in Table 1. We will further refer to layer indexes from this table. It is important to mention that we
have not used the whole range of layers in our experiments. In particular, we usually started from
layer 2 and finished at layer 8. As such, it is still necessary to consider multi-channel input. To
extend our algorithm to this case, we create a virtual input layer with C = 1 and virtual per-pixel
arcs to all the real input channels. While starting from a later layer is an empirical observation which
improves the results for our method, the advantage of finishing at an earlier layer was discovered by
other researchers as well [5] (starting from some layer, the network activations stop being related to
individual pixels). We will thus abbreviate our methods as ?ours(s, t)? where ?s? is the starting layer
and ?t? is the last layer.
6.1
Experimental setup
For the stereo matching, we chose the largest available datasets KITTI 2012 and KITTI 2015. All
image pairs in these datasets are rectified, so correspondences can be searched in the same row.
For each training pair, the ground-truth shift is measured densely per-pixel. This ground truth was
obtained by projecting the point cloud from LIDAR on the reference image. The quality measure is
the percentage Errt of pixels whose predicted shift error is bigger than a threshold of t pixels. We
considered a range of thresholds t = 1, . . . , 5, while the main benchmark measure is Err3 . This
measure is only computed for the pixels which are visible in both images from the stereo pair.
For comparison with the baselines, we used the setup proposed in [25] ? the seminal work which
introduced deep learning for stereo matching and which currently stays one of the best methods on the
KITTI datasets. [24] is an extensive study which has a representative comparison of learning-based
and non-learning-based methods under the same setup and open-source code [24] for this setup. The
whole pipeline works as follows. First, we obtain the raw scores U (x, d) from Algorithm 1 for the
shifts up to Dmax = 228. Then we normalize the scores U (x, ?) per-pixel by dividing them over
the maximal score, thus turning them into the range [0, 1], suitable for running the post-processing
code [24]. Finally, we run the post-processing code with exactly the same parameters as the original
method [25] and measure the quality on the same 40 validation images.
6.2
Baselines
We have two kinds of baselines in our evaluation: those coming from [25] and our simpler versions
of deep feature transfer similar to [13], which do not consider paths.
The first group of baselines from [25] are the following: the sum of absolute differences ?sad?,
the census transform ?cens? [23], the normalized cross-correlation ?ncc?. We also included the
learning-based methods ?fst? and ?acrt? [25] for completeness, although they use training data to
learn features while our method does not.
For the second group of baselines, we stack up the activation volumes for the given layer range
and up-sample the layer volumes if they have reduced resolution. Then we compute normalized
cross-correlation of the stacked features. Those baselines are denoted ?corr(s, t)? where ?s? is the
6
Table 2: This table shows the percentages of erroneous pixels Errt for thresholds t = 1, . . . , 5
on the KITTI 2012 validation set from [25]. Our method is denoted ?ours(2, 8)?. The two rightmost columns ?fst? and ?acrt? correspond to learning-based methods from [25]. We give them for
completeness, as all the other methods, including ours, do not use learning.
Methods
Threshold
sad
cens
ncc
corr(1, 2)
corr(2, 2)
corr(2, 8)
ours(2, 8)
fst
acrt
1
2
3
4
5
8.16
-
4.90
-
8.93
-
20.6
10.5
7.58
6.19
5.40
20.4
10.4
7.52
6.13
5.36
20.7
8.14
5.23
4.02
3.42
17.4
6.40
3.94
2.99
2.49
3.02
-
2.61
-
Table 3: KITTI 2012 ablation study.
Methods
Threshold
ours(2, 2)
ours(2, 3)
central(2, 8)
ours(2, 8)
1
2
3
4
5
17.7
7.90
5.28
4.08
3.41
18.4
8.16
5.41
4.05
3.32
17.3
6.58
4.02
3.04
2.53
17.4
6.40
3.94
2.99
2.49
starting layer, ?t? is the last layer. Note that we correlate the features before applying ReLU following
what [25] does for the last layer. Thus we use the input to the ReLU inside the layers.
All the methods, including ours, undergo the same post-processing pipeline. This pipeline consists of
semi-global matching [9], left-right consistency check, sub-pixel enhancement by fitting a quadratic
curve, median and bilateral filtering. We refer the reader to [25] for the full description. While the
first group of baselines was tuned by [25] and we take the results from that paper, we had to tune the
post-processing hyper-parameters of the second group of baselines to obtain the best results.
6.3
KITTI 2012
The dataset consists of 194 training image pairs and 195 test image pairs. The reflective surfaces like
windshields were excluded from the ground truth.
The results in Table 2 show that our method ?ours(2, 8)? performs better compared to the baselines.
At the same time, its performance is lower than learning-based methods from [25]. The main promise
of our method is scalability: while we test it on a task where huge effort was invested into collecting
the training data, there are other important tasks without such extensive datasets.
6.4
Ablation study on KITTI 2012
The goal of this section is to understand how important is the deep hierarchy of features versus one or
few layers. We compared the following setups: ?ours(2, 2)? uses only the second layer, ?ours(2, 3)?
uses only the range from layer 2 to layer 3, ?central(2, 8)? considers the full range of layers but only
with central arcs in the convolutions (connecting same pixel positions between activations) taken into
account in the backward pass, ?ours(2, 8)? is the full method. The result in Table 3 shows that it is
profitable to use the full hierarchy both in terms of depth and coverage of the receptive field.
6.5
KITTI 2015
The stereo dataset consists of 200 training image pairs and 200 test image pairs. The main difference
to KITTI 2012 is that the images are colored and the reflective surfaces are present in the evaluation.
Similar conclusions to KITTI 2012 can be drawn from experimental results: our method provides a
reasonable transfer, being inferior only to learning-based methods ? see Table 4. We show our depth
map results in Fig. 2.
7
Table 4: This table shows the percentages of erroneous pixels Errt for thresholds t = 1, . . . , 5
on the KITTI 2015 validation set from [25]. Our method is denoted ?ours(2, 8)?. The two rightmost columns ?fst? and ?acrt? correspond to learning-based methods from [25]. We give them for
completeness, as all the other methods, including ours, do not use learning.
Methods
Threshold
sad
cens
ncc
corr(1, 2)
corr(2, 2)
corr(2, 8)
ours(2, 8)
fst
acrt
1
2
3
4
5
9.44
-
5.03
-
8.89
-
26.6
10.9
6.68
5.05
4.22
26.5
10.8
6.63
5.03
4.20
29.6
11.2
6.16
4.42
3.60
26.2
9.27
4.78
3.36
2.72
3.99
-
3.25
-
Figure 2: Results on KITTI 2015. Top to bottom: reference image, searched image, our depth result.
The depth is visualized in the standard KITTI color coding (from close to far: yellow, green, purple,
red, blue).
6.6
Style transfer experiment on KITTI 2015
The goal of this experiment is to show the robustness of recognition hierarchy for the transfer to
correspondence search ? something we advocated in the introduction as the advantage of our
approach. We apply the style transfer method [5], implemented in the Prisma app. We ran different
style transfers on the left and right images. While now very different at the pixel level, the higher
level descriptions of the images remain the same which allows to successfully run our method. The
qualitative results show the robustness of our path-based method in Fig. 3 (see also Fig. 2 for visual
comparison to normal data).
Figure 3: Results for the style transfer on KITTI 2015. Top to bottom: reference image, searched
image, our depth result. The depth is visualized in the standard KITTI color coding (from close to
far: yellow, green, purple, red, blue).
8
7
Conclusion
In this work, we have presented a method for transfer from recognition to correspondence search at
the lowest level. For that, we re-use activation paths from deep convolutional neural networks and
propose an efficient polynomial algorithm to aggregate an exponential number of such paths. The
empirical results on the stereo matching task show that our method is competitive among methods
which do not use labeled data from the target domain. It would be interesting to apply this technique
to sound, which should become possible once a high-quality deep convolutional model becomes
accessible to the public (e.g., [21]).
Acknowledgements
We would like to thank Dmitry Laptev, Alina Kuznetsova and Andrea Cohen for their comments
about the manuscript. We also thank Valery Vishnevskiy for running our code while our own cluster
was down. This work is partially funded by the Swiss NSF project 163910 ?Efficient Object-Centric
Detection?.
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6,325 | 6,721 | Linearly constrained Gaussian processes
Carl Jidling
Department of Information Technology
Uppsala University, Sweden
[email protected]
Niklas Wahlstr?m
Department of Information Technology
Uppsala University, Sweden
[email protected]
Adrian Wills
School of Engineering
University of Newcastle, Australia
[email protected]
Thomas B. Sch?n
Department of Information Technology
Uppsala University, Sweden
[email protected]
Abstract
We consider a modification of the covariance function in Gaussian processes to
correctly account for known linear operator constraints. By modeling the target
function as a transformation of an underlying function, the constraints are explicitly
incorporated in the model such that they are guaranteed to be fulfilled by any
sample drawn or prediction made. We also propose a constructive procedure for
designing the transformation operator and illustrate the result on both simulated
and real-data examples.
Introduction
Bayesian non-parametric modeling has had a
profound impact in machine learning due, in
no small part, to the flexibility of these model
structures in combination with the ability to encode prior knowledge in a principled manner [6].
These properties have been exploited within the
class of Bayesian non-parametric models known
as Gaussian Processes (GPs), which have received significant research attention and have
demonstrated utility across a very large range of
real-world applications [16].
0.7
0.8
0.9
1
1.1
1.2
Predicted
magnetic field
strength [a.u.]
1.3
2
1.5
x3 [m]
1
1
0.5
0
1
Abstracting from the myriad number of these applications, it has been observed that the efficacy
of GPs modeling is often intimately dependent
on the appropriate choice of mean and covariance functions, and the appropriate tuning of
their associated hyper-parameters. Often, the
most appropriate mean and covariance functions
are connected to prior knowledge of the underlying problem. For example, [10] uses functional
expectation constraints to consider the problem
of gene-disease association, and [13] employs
a multivariate generalized von Mises distribution to produce a GP-like regression that handles
circular variable problems.
3
0
2
1
-1
0
-2
x2 [m]
-3
-1
x1 [m]
Figure 1: Predicted strength of a magnetic field at
three heights, given measured data sampled from
the trajectory shown (blue curve). The three components (x1 , x2 , x3 ) denote the Cartesian coordinates, where the x3 -coordinate is the height above
the floor. The magnetic field is curl-free, which
can be formulated in terms of three linear constraints. The method proposed in this paper can
exploit these constraints to improve the predictions.
See Section 5.2 for details.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
At the same time, it is not always obvious how one might construct a GP model that obeys underlying
principles, such as equilibrium conditions and conservation "laws". One straightforward approach
to this problem is to add fictitious measurements that observe the constraints at a finite number of
points of interest. This has the benefit of being relatively straightforward to implement, but has
the sometimes significant drawback of increasing the problem dimension and at the same time not
enforcing the constraints between the points of interest.
A different approach to constraining the GP model is to construct mean and covariance functions that
obey the constraints. For example, curl and divergence free covariance functions are used in [22] to
improve the accuracy for regression problems. The main benefit of this approach is that the problem
dimension does not grow, and the constraints are enforced everywhere, not pointwise. However, it is
not obvious how these approaches can be scaled for an arbitrary set of linear operator constraints.
The contribution of this paper is a new way to include constraints into multivariate GPs. In particular,
we develop a method that transforms a given GP into a new, derived, one that satisfies the constraints.
The procedure relies upon the fact that GPs are closed under linear operators, and we propose an
algorithm capable of constructing the required transformation. We will demonstrate the utility of
this new method on both simulated examples and on a real-world application, the latter in form of
predicting the components of a magnetic field, as illustrated in Figure 1.
To make these ideas more concrete, we present a simple example that will serve as a focal point
several times throughout the paper. To that end, assume that we have a two-dimensional function
f (x) : R2 7? R2 on which we put a GP prior f (x) ? GP (?(x), K(x, x0 )) . We further know that
f (x) should obey the differential equation
?f1
?f2
+
= 0.
?x1
?x2
(1)
In this paper we show how to modify K(x, x0 ) and ?(x) such that any sample from the new GP is
guaranteed to obey the constraints like (1), considering any kind of linear operator constraint.
2
Problem formulation
Assume that we are given a data set of N observations {xk , yk }N
k=1 where xk denotes the input and
yk the output. Both the input and output are potentially vector-valued, where xk ? RD and yk ? RK .
We consider the regression problem where the data can be described by a non-parametric model
yk = f (xk ) + ek , where ek is zero-mean white noise representing the measurement uncertainty. In
this work, we place a vector-valued GP prior on f
f (x) ? GP (?(x), K(x, x0 )) ,
(2)
with the mean function and the covariance function
?(?) : RD 7? RK ,
K(?, ?) : RD ? RD 7? RK ? RK .
(3)
{xk , yk }N
k=1 ,
Based on the data
we would now like to find a posterior over the function f (x). In
addition to the data, we know that the function f should fulfill certain constraints
Fx [f ] = 0,
(4)
where Fx is an operator mapping the function f (x) to another
h function ig(x) as Fx [f ] = g(x). We
further require Fx to be a linear operator meaning that Fx ?1 f 1 + ?2 f 2 = ?1 Fx [f 1 ] + ?2 Fx [f 2 ],
where ?1 , ?2 ? R. The operator Fx can for example be a linear transform Fx [f ] = Cf (x) which
together with the constraint (4) forces a certain linear combination of the outputs to be linearly
dependent.
The operator Fx could also include other linear operations on the function f (x). For example, we
might know that the function f (x) : R2 ? R2 should obey a certain partial differential equation
?f1
?f2
Fx [f ] = ?x
+ ?x
. A few more linear operators are listed in Section 1 of the Supplementary
1
2
material, including integration as one the most well-known.
The constraints (4) can either come from known physical laws or other prior knowledge of the process
generating the data. Our objective is to encode these constraints in the mean and covariance functions
(3) such that any sample from the corresponding GP prior (2) always obeys the constraint (4).
2
3
3.1
Building a constrained Gaussian process
Approach based on artificial observations
Just as Gaussian distributions are closed under linear transformations, so are GPs closed under linear
operations (see Section 2 in the Supplementary material). This can be used for a straightforward way
of embedding linear operator constraints of the form (4) into GP regression. The idea is to treat the
?
? . The
? k }N
? k = 0 for all k = 1 . . . N
constraints as noise-free artificial observations {?
xk , y
k=1 with y
? k = Fx? k [f ], where x
? k are input points in the domain of
regression is then performed on the model y
? k coincide with the points of prediction.
interest. For example, one could let these artificial inputs x
An advantage of this approach is that it allows constraints of the type (4) with a non-zero right hand
side. Furthermore, there is no theoretical limit on how many constraints we can include (i.e. number
of rows in Fx ) ? although in practice, of course, there is.
However, this is problematic mainly for two reasons. First of all, it makes the problem size grow.
This increases memory requirements and execution time, and the numerical stability is worsen due to
an increased condition number. This is especially clear from the fact that we want these observations
to be noise-free, since the noise usually has a regularizing effect. Secondly, the constraints are only
enforced point-wise, so a sample drawn from the posterior fulfills the constraint only in our chosen
points. The obvious way of compensating for this is by increasing the number of points in which the
constraints are observed ? but that exacerbates the first problem. Clearly, the challenge grows quickly
with the dimension of the inferred function.
Embedding the constraints in the covariance function removes these issues ? it makes the enforcement
continuous while the problem size is left unchanged. We will now address the question of how to
design such a covariance function.
3.2
A new construction
We want to find a GP prior (2) such that any sample f (x) from that prior obeys the constraints (4). In
turn, this leads to constraints on the mean and covariance functions (3) of that prior. However, instead
of posing these constraints on the mean and covariance functions directly, we consider f (x) to be
related to another function g(x) via some operator Gx
f (x) = Gx [g].
(5)
Fx [ Gx [g]] = 0.
(6)
The constraints (4) then amounts to
We would like this relation to be true for any function g(x). To do that, we will interpret Fx and
Gx as matrices and use a similar procedure to that of solving systems of linear equations. Since Fx
and Gx are linear operators, we can think of Fx [f ] and Gx [g] as matrix-vector multiplications where
P
Fx [f ] = Fx f , with ( Fx f )i = K
j=1 ( Fx )ij fj where each element ( Fx )ij in the operator matrix
Fx is a scalar operator. With this notation, (6) can be written as
Fx Gx = 0.
(7)
This reformulation imposes constraints on the operator Gx rather than on the GP prior for f (x)
directly. We can now proceed by designing a GP prior for g(x) and transform it using the mapping
(5). We further know that GPs are closed under linear operations. More specifically, if g(x) is
modeled as a GP with mean ?g (x) and covariance Kg (x, x0 ), then f (x) is also a GP with
f (x) = Gx g ? GP Gx ?g , Gx Kg GxT0 .
(8)
We use ( Gx Kg GxT0 )ij to denote that ( Gx Kg GxT0 )ij = ( Gx )ik ( Gx0 )jl (Kg )kl , where Gx and Gx0 act
on the first and second argument of Kg (x, x0 ), respectively. See Section 2 in the Supplementary
material for further details on linear operations on GPs.
The procedure to find the desired GP prior for f can now be divided into the following three steps
1. Find an operator Gx that fulfills the condition (6).
3
2. Choose a mean and covariance function for g(x).
3. Find the mean and covariance functions for f (x) according to (8).
In addition to being resistant to the disadvantages of the approach described in Section 3.1, there are
some additional strengths worth pointing out with this method. First of all, we have separated the task
of encoding the constraints and encoding other desired properties of the kernel. The constraints are
encoded in Fx and the remaining properties are determined by the prior for g(x), such as smoothness
assumptions. Hence, satisfying the constraints does not sacrifice any desired behavior of the target
function.
Secondly, K(x, x0 ) is guaranteed to be a valid covariance function provided that Kg (x, x0 ) is, since
GPs are closed under linear functional transformations. From (8), it is clear that each column of K
must fulfill all constraints encoded in Fx . Possibly K could be constructed only with this knowledge,
assuming a general form and solving the resulting equation system. However, a solution may not just
be hard to find, but one must also make sure that it is indeed a valid covariance function.
Furthermore, this approach provides a simple and straightforward way of constructing the covariance
function even if the constraints have a complicated form. It makes no difference if the linear operators
relate the components of the target function explicitly or implicitly ? the procedure remains the same.
3.3
Illustrating example
We will now illustrate the method using the example (1) introduced already in the introduction.
?f2
?f1
+ ?x
= 0, where x = [x1 , x2 ]T and
Consider a function f (x) : R2 7? R2 satisfying ?x
1
2
T
f (x) = [f1 (x), f2 (x)] . This equation describes all two-dimensional divergence-free vector fields.
?
?
The constraint can be written as a linear constraint on the form (4) where Fx = [ ?x
] and
1 ?x2
T
f (x) = [f1 (x) f2 (x)] . Modeling this function with a GP and building the covariance structure as
described above, we first need to find the transformation Gx such that (7) is fulfilled. For example,
we could pick
T
(9)
Gx = ? ?x? 2 ?x? 1 .
2
0
If the underlying function g(x) : R 7? R isgiven by g(x) ? GP 0, kg (x, x ) , then we can make
use of (8) to obtain f (x) ? GP 0, K(x, x0 ) where
?
?
2
?2
? ?x?2 x0
0
?x
x
2
2
1?
kg (x, x0 ).
K(x, x0 ) = Gx kg (x, x0 ) GxT = ?
2
?2
? ?x?1 x0
0
?x1 x
2
1
Using a covariance function with the following structure, we know that the constraint will be fulfilled
by any function generated from the corresponding GP.
4
Finding the operator Gx
In a general setting it might be hard to find an operator Gx that fulfills the constraint (7). Ultimately,
we want an algorithm that can construct Gx from a given Fx . In more formal terms, the function
Gx g forms the nullspace of Fx . The concept of nullspaces for linear operators is well-established
[11], and does in many ways relate to real-number linear algebra.
However, an important difference is illustrated by considering a one-dimensional function f (x)
?
subject to the constraint Fx f = 0 where Fx = ?x
. The solution to this differential equation can not
be expressed in terms of an arbitrary underlying function, but it requires f (x) to be constant. Hence,
?
the nullspace of ?x
consists of the set of horizontal lines. Compare this with the real number equation
ab = 0, a 6= 0, which is true only if b = 0. Since the nullspace differs between operators, we must be
careful when discussing the properties of Fx and Gx based on knowledge from real-number algebra.
T
Let us denote the rows in Fx as fT
1 , . . . , fL . We now want to find all solutions g such that
Fx g = 0
?
fTi g = 0,
? i = 1, . . . , L.
(10)
The solutions g1 , . . . , gP to (10) will then be the columns of Gx . Each row vector fj can be written
as fi = ?i ?f where ?i ? RK?Mf and ?f = [?1 , . . . , ?Mf ]T is a vector of Mf scalar operators
4
Algorithm 1 Constructing Gx
Input: Operator matrix Fx
Output: Operator matrix Gx where Fx Gx = 0
Step 1: Make an ansatz g = ?? g for the columns in Gx .
Step 2: Expand Fx ?? g and collect terms.
Step 3: Construct A ? vec(?) = 0 and find the vectors ?1 . . . ?P spanning its nullspace.
Step 4: If P = 0, go back to Step 1 and make a new ansatz, i.e. extend the set of operators.
Step 5: Construct Gx = [?1 ? g , . . . , ?P ? g ].
included in Fx . We now assume that g also can be written in a similar form g = ?? g where
? ? RK?M g and ? g = [?1 , . . . , ?M g ]T is a vector of M g scalar operators. One may make the
assumption that the same set of operators that are used to describe fi also can be used to describe g,
i.e., ? g = ?f. However, this assumption might need to be relaxed. The constraints (10) can then be
written as
(?f)T ?i ?? g = 0,
? i = 1, . . . , L.
(11)
We perform the multiplication and collect the terms in ?f and ? g . The condition (11) then results in
conditions on the parameters in ? resulting a in a homogeneous system of linear equations
A ? vec(?) = 0.
(12)
The vectors vec(?1 ), . . . , vec(?P ) spanning the nullspace of A in (12) are then used to compute the
columns in Gx = [ g1 , . . . gP ] where gp = ?p ? g . If it turns out that the nullspace of A is empty,
one should start over with a new ansatz and extend the set of operators in ? g .
The outline of the procedure as described above is summarized in Algorithm 1. The algorithm is
based upon a parametric ansatz rather than directly upon the theory for linear operators. Not only
is it more intuitive, but it does also remove any conceptual challenges that theory may provide. A
problem with this is that one may have to iterate before having found the appropriate set of operators
in Gx . It might be of interest to examine possible alternatives to this algorithm that does not use a
parametric approach. Let us now illustrate the method with an example.
4.1
Divergence-free example revisited
Let us return to the example discussed in Section 3.3, and show how the solution found by visual
inspection also can be found with the algorithm described above. Since Fx only contains first-order
derivative operators, we assume that a column in Gx does so as well. Hence, let us propose the
following ansatz (step 1)
" ? #
?11 ?12 ?x1
g= ?
= ?? g .
(13)
?
21 ?22
?x2
Applying the constraint, expanding and collecting terms (step 2) we find
" ? #
?2
?2
?2
?
?
Fx ?? g = ?x? 1 ?x? 2 ?11 ?12 ?x? 1 = ?11 2 + (?12 + ?21 )
+ ?22 2 ,
21
22
?x1
?x1 ?x2
?x2
?x2
where we have used the fact that
expression (14) equals zero if
"
1
0
0
?2
?xi ?xj
0
1
0
0
1
0
=
?2
?xj ?xi
(14)
assuming continuous second derivatives. The
? ?
# ?11
0
?? ?
0 ? 12 ? = A ? vec(?) = 0.
?21
1
?22
(15)
The nullspace is spanned by a single vector (step 3) [?11 ?12 ?21 ?22 ]T = ?[0 ? 1 1 0]T , ? ? R.
?
? T
Choosing ? = 1, we get Gx = ? ?x
(step 5), which is the same as in (9).
?x
2
1
5
4.2
Generalization
Although there are no conceptual problems with the algorithm introduced above, the procedure of
expanding and collecting terms appears a bit informal. In a general form, the algorithm is reformulated
such that the operators are completely left out from the solution process. The drawback of this is a
more cumbersome notation, and we have therefore limited the presentation to this simplified version.
However, the general algorithm is found in the Supplementary material of this paper.
5
5.1
Experimental results
Simulated divergence-free function
f1 (x1 , x2 ) = e
?ax1 x2
f2 (x1 , x2 ) = e?ax1 x2
?f1
?x1
+
ax1 sin(x1 x2 ) ? x1 cos(x1 x2 ) ,
x2 cos(x1 x2 ) ? ax2 sin(x1 x2 ) ,
Consider the example in Section 3.3. An example of a function fulfilling
?f2
?x2
= 0 is
(16)
where a denotes a constant. We will now study how the regression of this function differs when
using the covariance function found in Section 3.3 as compared to a diagonal covariance function
K(x, x0 ) = k(x, x0 )I. The measurements generated are corrupted with Gaussian noise such that
yk = f (x
ek , where ek ? N (0, ? 2 I). The squared exponential covariance function k(x, x0 ) =
k1) +
2
?2
?f exp ? 2 l kx ? x0 k2 has been used for kg and k with hyperparameters chosen by maximizing
the marginal likelihood. We have used the value a = 0.01 in (16).
We have used 50 measurements randomly picked over the domain [0 4] ? [0 4], generated with the
noise level ? = 10?4 . The points for prediction corresponds to a discretization using 20 uniformly
distributed points in each direction, and hence a total of NP = 202 = 400. We have included the
approach described is Section 3.1 for comparison. The number of artificial observations have been
chosen as random subsets of the prediction points, up to and including the full set.
q
1 ?T ?
The comparison is made with regard to the root mean squared error erms =
NP f ? f ? , where
?f ? = ?
?f ? ?f and ?f is a concatenated vector storing the true function values in all prediction points and
?
?f denotes the reconstructed equivalent. To decrease the impact of randomness, each error value has
been formed as an average over 50 reconstructions given different sets of measurements.
An example of the true field, measured values and reconstruction errors using the different methods
is seen in Figure 2. The result from the experiment is seen in Figure 3a. Note that the error from the
approach with artificial observations is decreasing as the number of observations is increased, but
only to a certain point. Have in mind, however, that the Gram matrix is growing, making the problem
larger and worse conditioned. The result from our approach is clearly better, while the problem size
is kept small and numerical problems are therefore avoided.
Figure 2: Left: Example of field plots illustrating the measurements (red arrows) and the true field
(gray arrows). Remaining three plots: reconstructed fields subtracted from the true field. The artificial
observations of the constraint have been made in the same points as the predictions are made.
5.2
Real data experiment
Magnetic fields can mathematically be considered as a vector field mapping a 3D position to a 3D
magnetic field strength. Based on the magnetostatic equations, this can be modeled as a curl-free
6
0.9
0.038
Our approach
Diagonal
Artificial obs
erms
erms
Our approach
Diagonal
0.7
Artificial obs
0.036
0.5
25
100
0.034
10 1
400
Nc
10 2
10 3
Nc
(a) Simulated experiment
(b) Real-data experiment
Figure 3: Accuracy of the different approaches as the number of artificial observations Nc is increased.
vector field. Following Section 3.1 in the Supplementary material, our method can be used to encode
the constraints in the following covariance function (which also has been presented elsewhere [22])
!
0
0 T
kx?x0 k2
x
?
x
x
?
x
Kcurl (x, x0 ) = ?f2 e? 2l2
I3 ?
.
(17)
l
l
With a magnetic sensor and an optical positioning system, both position and magnetic field data have
been collected in a magnetically distorted indoor environment, see the Supplementary material for
details about the experimental details. In Figure 1 the predicted magnitude of the magnetic field over
a two-dimensional domain for three different heights above the floor is displayed. The predictions
have been made based on 500 measurements sampled from the trajectory given by the blue curve.
Similar to the simulated experiment in Section 5.1, we compare the predictions of the curl-free covariance function (17) with the diagonal covariance function and the diagonal covariance function using
artificial observations. The results have been formed by averaging the error over 50 reconstructions.
In each iteration, training data and test data were randomly selected from the data set collected in the
experiment. 500 train data points and 1 000 test data points were used.
The result is seen in Figure 3b. We recognize the same behavior as we saw for the simulated
experiment in Figure 3a. Note that the accuracy of the artificial observation approach gets very close
to our approach for a large number of artificial observations. However, in the last step of increasing
the artificial observations, the accuracy decreases. This is probably caused by the numerical errors
that follows from an ill-conditioned Gram matrix.
6
Related work
Many problems in which GPs are used contain some kind of constraint that could be well exploited
to improve the quality of the solution. Since there are a variety of ways in which constraints
may appear and take form, there is also a variety of methods to deal with them. The treatment
of inequality constraints in GP regression have been considered for instance in [1] and [5], based
on local representations in a limited set of points. The paper [12] proposes a finite-dimensional
GP-approximation to allow for inequality constraints in the entire domain.
It has been shown that linear constraints satisfied by the training data will be satisfied by the GP
prediction as well [19]. The same paper shows how this result can be extended to quadratic forms
through a parametric reformulation and minimization of the Frobenious norm, with application
demonstrated for pose estimation. Another approach on capturing human body features is described
in [18], where a face-shape model is included in the GP framework to imply anatomic correctness.
A rigorous theoretical analysis of degeneracy and invariance properties of Gaussian random fields
is found in [7], including application examples for one-dimensional GP problems. The concept of
learning the covariance function with respect to algebraic invariances is explored in [9].
Although constraints in most situations are formulated on the outputs of the GP, there are also
situations in which they are acting on the inputs. An example of this is given in [21], describing a
method of benefit from ordering constraints on the input to reduce the negative impact of input noise.
Applications within medicine include gene-disease association through functional expectation constraints [10] and lung disease sub-type identification using a mixture of GPs and constraints encoded
with Markov random fields [17]. Another way of viewing constraints is as modified prior distributions.
By making use of the so-called multivariate generalized von Mises distribution, [13] ends up in a
version of GP regression customized for circular variable problems. Other fields of interest include
using GPs in approximately solving one-dimensional partial differential equations [8, 14, 15].
7
Generally speaking, the papers mentioned above consider problems in which the constraints are dealt
with using some kind of external enforcement ? that is, they are not explicitly incorporated into the
model, but rely on approximations or finite representations. Therefore, the constraints may just be
approximately satisfied and not necessarily in a continuous manner, which differs from the method
proposed in this paper. Of course, comparisons can not be done directly between methods that have
been developed for different kinds of constraints. The interest in this paper is multivariate problems
where the constraints are linear combinations of the outputs that are known to equal zero.
For multivariate problems, constructing the covariance function is particularly challenging due to
the correlation between the output components. We refer to [2] for a very useful review. The basic
idea behind the so-called separable kernels is to separate the process of modeling the covariance
function for each component and the process of modeling the correlation between them. The final
covariance function is chosen for example according to some method of regularization. Another
class of covariance functions is the invariant kernels. Here, the correlation is inherited from a
known mathematical relation. The curl- and divergence free covariance functions are such examples
where the structure follows directly from the underlying physics, and has been shown to improve
the accuracy notably for regression problems [22]. Another example is the method proposed in
[4], where the Taylor expansion is used to construct a covariance model given a known relationship
between the outputs. A very useful property on linear transformations is given in [20], based on the
GPs natural inheritance of features imposed by linear operators. This fact has for example been used
in developing a method for monitoring infectious diseases [3].
The method proposed in this work is exploiting the transformation property to build a covariance
function of the invariant kind for a multivariate GP. We show how this property can be exploited to
incorporate knowledge of linear constraints into the covariance function. Moreover, we present an
algorithm of constructing the required transformation. This way, the constraints are built into the
prior and are guaranteed to be fulfilled in the entire domain.
7
Conclusion and future work
We have presented a method for designing the covariance function of a multivariate Gaussian process
subject to known linear operator constraints on the target function. The method will by construction
guarantee that any sample drawn from the resulting process will obey the constraints in all points.
Numerical simulations show the benefits of this method as compared to alternative approaches.
Furthermore, it has been demonstrated to improve the performance on real data as well.
As mentioned in Section 4, it would be desirable to describe the requirements on Gx more rigorously.
That might allow us to reformulate the construction algorithm for Gx in a way that allows for a more
straightforward approach as compared to the parametric ansatz that we have proposed. In particular,
our method relies upon the requirement that the target function can be expressed in terms of an
underlying potential function g. This leads to the intriguing and nontrivial question: Is it possible to
mathematically guarantee the existence of such a potential? If the answer to this question is yes, the
next question will of course be what it look like and how it relates to the target function.
Another possible topic of further research is the extension to constraints including nonlinear operators,
which for example might rely upon a linearization in the domain of interest. Furthermore, it may be
of potential interest to study the extension to a non-zero right-hand side of (4).
Acknowledgements
This research is financially supported by the Swedish Foundation for Strategic Research (SSF) via
the project ASSEMBLE (Contract number: RIT 15-0012). The work is also supported by the Swedish
Research Council (VR) via the project Probabilistic modeling of dynamical systems (Contract number:
621-2013-5524). We are grateful for the help and equipment provided by the UAS Technologies
Lab, Artificial Intelligence and Integrated Computer Systems Division (AIICS) at the Department of
Computer and Information Science (IDA), Link?ping University, Sweden. The real data set used in
this paper has been collected by some of the authors together with Manon Kok, Arno Solin, and Simo
S?rkk?. We thank them for allowing us to use this data. We also thank Manon Kok for supporting us
with the data processing. Furthermore, we would like to thank Carl Rasmussen and Marc Deisenroth
for fruitful discussions on constrained GPs.
8
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MIT press, Cambridge, MA, 2006.
[17] James Ross and Jennifer Dy. Nonparametric mixture of Gaussian processes with constraints. In
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pages 1346?1354. JMLR Workshop and Conference Proceedings, 2013.
[18] Ognjen Rudovic and Maja Pantic. Shape-constrained gaussian process regression for facialpoint-based head-pose normalization?. In Proceedings of the International Conference on
Computer Vision (ICCV), 2011.
[19] Mathieu Salzmann and Raquel Urtasun. Implicitly constrained Gaussian process regression for
monocular non-rigid pose estimation. In Neural Information Processing Systems (NIPS), 2010.
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9
[21] Cuong Tran, Vladimir Pavlovic, and Robert Kopp. Gaussian process for noisy inputs with
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10
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6,326 | 6,722 | Fixed-Rank Approximation of a
Positive-Semidefinite Matrix from Streaming Data
Joel A. Tropp
Caltech
Alp Yurtsever
EPFL
Madeleine Udell
Cornell
Volkan Cevher
EPFL
[email protected] [email protected] [email protected] [email protected]
Abstract
Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a
sequence of linear updates. Because of storage limitations, it may only be possible
to retain a sketch of the psd matrix. This paper develops a new algorithm for
fixed-rank psd approximation from a sketch. The approach combines the Nystr?m
approximation with a novel mechanism for rank truncation. Theoretical analysis
establishes that the proposed method can achieve any prescribed relative error in
the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques
for fixed-rank psd matrix approximation across a wide range of examples.
1
Motivation
In recent years, researchers have studied many applications where a large positive-semidefinite (psd)
matrix is presented as a series of linear updates. A recurring theme is that we only have space to store
a small summary of the psd matrix, and we must use this information to construct an accurate psd
approximation with specified rank. Here are two important cases where this problem arises.
Streaming Covariance Estimation. Suppose that we receive a stream h1 , h2 , h3 , ? ? ? ? Rn of
high-dimensional vectors. The psd sample covariance matrix of these vectors has the linear dynamics
A(0) ? 0 and
A(i) ? (1 ? i?1 )A(i?1) + i?1 hi h?i .
When the dimension n and the number of vectors are both large, it is not possible to store the vectors
or the sample covariance matrix. Instead, we wish to maintain a small summary that allows us to
compute the rank-r psd approximation of the sample covariance matrix A(i) at a specified instant i.
This problem and its variants are often called streaming PCA [3, 12, 14, 15, 25, 32].
Convex Low-Rank Matrix Optimization with Optimal Storage. A primary application of
semidefinite programming (SDP) is to search for a rank-r psd matrix that satisfies additional constraints. Because of storage costs, SDPs are difficult to solve when the matrix variable is large.
Recently, Yurtsever et al. [44] exhibited the first provable algorithm, called SketchyCGM, that
produces a rank-r approximate solution to an SDP using optimal storage.
Implicitly, SketchyCGM forms a sequence of approximate psd solutions to the SDP via the iteration
A(0) ? 0
and A(i) ? (1 ? ?i )A(i?1) + ?i hi h?i .
The step size ?i = 2/(i + 2), and the vectors hi do not depend on the matrices A(i) . In fact,
SketchyCGM only maintains a small summary of the evolving solution A(i) . When the iteration
terminates, SketchyCGM computes a rank-r psd approximation of the final iterate using the method
described by Tropp et al. [37, Alg. 9].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1.1
Notation and Background
The scalar field F = R or F = C. Define ?(R) = 1 and ?(C) = 0. The asterisk ? is the (conjugate)
transpose, and the dagger ? denotes the Moore?Penrose pseudoinverse. The notation A1/2 refers to
the unique psd square root of a psd matrix A. For p ? [1, ?], the Schatten p-norm k ? kp returns the
`p norm of the singular values of a matrix. As usual, ?r refers to the rth largest singular value.
For a nonnegative integer r, the phrase ?rank-r? and its variants mean ?rank at most r.? For a
matrix M , the symbol JM Kr denotes a (simultaneous) best rank-r approximation of the matrix
M with respect to any Schatten p-norm. We can take JM Kr to be any r-truncated singular value
decomposition (SVD) of M [24, Sec. 6]. Every best rank-r approximation of a psd matrix is psd.
2
Sketching and Fixed-Rank PSD Approximation
We begin with a streaming data model for a psd matrix that evolves via a sequence of general
linear updates, and it describes a randomized linear sketch for tracking the psd matrix. To compute
a fixed-rank psd approximation, we develop an algorithm based on the Nystr?m method [40], a
technique from the literature on kernel methods. In contrast to previous approaches, our algorithm
uses a distinct mechanism to truncate the rank of the approximation.
The Streaming Model. Fix a rank parameter r in the range 1 ? r ? n. Initially, the psd matrix
A ? Fn?n equals a known psd matrix Ainit ? Fn?n . Then A evolves via a series of linear updates:
A ? ?1 A + ?2 H
H ? Fn?n is (conjugate) symmetric.
where ?i ? R,
(2.1)
In many applications, the innovation H is low-rank and/or sparse. We assume that the evolving matrix
A always remains psd. At one given instant, we must produce an accurate rank-r approximation of
the psd matrix A induced by the stream of linear updates.
The Sketch. Fix a sketch size parameter k in the range r ? k ? n. Independent from A, we draw
and fix a random test matrix
? ? Fn?k .
(2.2)
See Sec. 3 for a discussion of possible distributions. The sketch of the matrix A takes the form
Y = A? ? Fn?k .
(2.3)
The sketch (2.3) supports updates of the form (2.1):
Y ? ?1 Y + ?2 H?.
(2.4)
To find a good rank-r approximation, we must set the sketch size k larger than r. But storage costs
and computation also increase with k. One of our main contributions is to clarify the role of k.
Under the model (2.1), it is more or less necessary to use a randomized linear sketch to track A [28].
For psd matrices, sketches of the form (2.2)?(2.3) appear explicitly in Gittens?s work [16, 17, 19].
Tropp et al. [37] relies on a more complicated sketch developed in [7, 42].
The Nystr?m Approximation. The Nystr?m method is a general technique for low-rank psd matrix
approximation. Various instantiations appear in the papers [5, 11, 13, 16, 17, 19, 22, 27, 34, 40].
Here is the application to the present situation. Given the test matrix ? and the sketch Y = A?, the
Nystr?m method constructs a rank-k psd approximation of the psd matrix A via the formula
?nys = Y (?? Y )? Y ? .
A
(2.5)
In most work on the Nystr?m method, the test matrix ? depends adaptively on A, so these approaches
are not valid in the streaming setting. Gittens?s framework [16, 17, 19] covers the streaming case.
Fixed-Rank Nystr?m Approximation: Prior Art. To construct a Nystr?m approximation with
exact rank r from a sketch of size k, the standard approach is to truncate the center matrix to rank r:
?nysfix = Y (J?? Y Kr )? Y ? .
A
r
(2.6)
The truncated Nystr?m approximation (2.6) appears in the many papers, including [5, 11, 18, 34].
We have found (Sec. 5) that the truncation method (2.6) performs poorly in the present setting. This
observation motivated us to search for more effective techniques.
2
Fixed-Rank Nystr?m Approximation: Proposal. The purpose of this paper is to develop, analyze,
and evaluate a new approach for fixed-rank approximation of a psd matrix under the streaming model.
We propose a more intuitive rank-r approximation:
? r = JA
?nys Kr .
A
(2.7)
That is, we report a best rank-r approximation of the full Nystr?m approximation (2.5).
This ?matrix nearness? approach to fixed-rank approximation appears in the papers [21, 22, 37]. The
combination with the Nystr?m method (2.5) is totally natural. Let us emphasize that the approach (2.7)
also applies to Nystr?m approximations outside the streaming setting.
Summary of Contributions. This paper contains a number of advances over the prior art:
1. We propose a new technique (2.7) for truncating the Nystr?m approximation to rank r. This
formulation differs from the published literature on fixed-rank Nystr?m approximations.
2. We present a stable numerical implementation of (2.7) based on the best practices outlined
in the paper [27]. This approach is essential for achieving high precision! (Sec. 3)
3. We establish informative error bounds for the method (2.7). In particular, we prove that it
attains (1 + ?)-relative error in the Schatten 1-norm when k = ?(r/?). (Sec. 4)
4. We document numerical experiments on real and synthetic data to demonstrate that our
method dominates existing techniques [18, 37] for fixed-rank psd approximation. (Sec. 5)
Psd matrix approximation is a ubiquitous problem, so we expect these results to have a broad impact.
Related Work. Randomized algorithms for low-rank matrix approximation were proposed in
the late 1990s and developed into a technology in the 2000s; see [22, 30, 41]. In the absence of
constraints, such as streaming, we recommend the general-purpose methods from [22, 23, 27].
Algorithms for low-rank matrix approximation in the important streaming data setting are discussed
in [4, 7, 8, 15, 22, 37, 41, 42]. Few of these methods are designed for psd matrices.
Nystr?m methods for low-rank psd matrix approximation appear in [11, 13, 16, 17, 19, 22, 26, 34,
37, 40, 43]. These works mostly concern kernel matrices; they do not focus on the streaming model.
We are only aware of a few papers [16, 17, 19, 37] on algorithms for psd matrix approximation
that operate under the streaming model (2.1). These papers form the comparison group.
After this paper was submitted, we learned about two contemporary works [35, 39] that propose the
fixed-rank approximation (2.7) in the context of kernel methods. Our research is distinctive because
we focus on the streaming setting, we obtain precise error bounds, we address numerical stability,
and we include an exhaustive empirical evaluation.
Finally, let us mention two very recent theoretical papers [6, 33] that present existential results on
algorithms for fixed-rank psd matrix approximation. The approach in [6] is only appropriate for
sparse input matrices, while the work [33] is not valid in the streaming setting.
3
Implementation
Distributions for the Test Matrix. To ensure that the sketch is informative, we must draw the
test matrix (2.2) at random from a suitable distribution. The choice of distribution determines the
computational requirements for the sketch (2.3), the linear updates (2.4), and the matrix approximation (2.7). It also affects the quality of the approximation (2.7). Let us outline some of the most useful
distributions. A full discussion is outside the scope of our work, but see [17, 19, 22, 29, 30, 37, 41].
Isotropic Models. Mathematically, the most natural model is to construct a test matrix ? ? Fn?k
whose range is a uniformly random k-dimensional subspace in Fn . There are two approaches:
1. Gaussian. Draw each entry of the matrix ? ? Fn?k independently at random from the
standard normal distribution on F.
2. Orthonormal. Draw a Gaussian matrix G ? Fn?k , as above. Compute a thin orthogonal?
triangular factorization G = ?R to obtain the test matrix ? ? Fn?k . Discard R.
Gaussian and orthonormal test matrices both require storage of kn floating-point numbers in F for
the test matrix ? and another kn floating-point numbers for the sketch Y . In both cases, the cost of
multiplying a vector in Fn into ? is ?(kn) floating-point operations.
3
Algorithm 1 Sketch Initialization. Implements (2.2)?(2.3) with a random orthonormal test matrix.
Input: Positive-semidefinite input matrix A ? Fn?n ; sketch size parameter k
Output: Constructs test matrix ? ? Fn?k and sketch Y = A? ? Fn?k
local: ?, Y
function N YSTROM S KETCH(A; k)
if F = R then
? ? randn(n, k)
if F = C then
? ? randn(n, k) + i ? randn(n, k)
? ? orth(?)
Y ? A?
1
2
3
4
5
6
7
8
. Internal variables for N YSTROM S KETCH
. Constructor
. Improve numerical stability
Algorithm 2 Linear Update. Implements (2.4).
Input: Scalars ?1 , ?2 ? R and conjugate symmetric H ? Fn?n
Output: Updates sketch to reflect linear innovation A ? ?1 A + ?2 H
local: ?, Y
function L INEARU PDATE(?1 , ?2 , H)
Y ? ?1 Y + ?2 H?
1
2
3
. Internal variables for N YSTROM S KETCH
For isotropic models, we can analyze the approximation (2.7) in detail. In exact arithmetic, Gaussian
and isotropic test matrices yield identical Nystr?m approximations (Supplement). In floating-point
arithmetic, orthonormal matrices are more stable for large k, but we can generate Gaussian matrices
with less arithmetic and communication. References for isotropic test matrices include [21, 22, 31].
Subsampled Scrambled Fourier Transform (SSFT). One shortcoming of the isotropic models is
the cost of storing the test matrix and the cost of multiplying a vector into the test matrix. We can
often reduce these costs using an SSFT test matrix. An SSFT takes the form
? = ?1 F ?2 F R ? Fn?k .
(3.1)
The ?i ? Fn?n are independent, signed permutation matrices,1 chosen uniformly at random. The
matrix F ? Fn?n is a discrete Fourier transform (F = C) or a discrete cosine transform (F = R).
The matrix R ? Fn?k is a restriction to k coordinates, chosen uniformly at random.
An SSFT ? requires only ?(n) storage, but the sketch Y still requires storage of kn numbers.
We can multiply a vector in Fn into ? using ?(n log n) arithmetic operations via an FFT or FCT
algorithm. Thus, for most choices of sketch size k, the SSFT improves over the isotropic models.
In practice, the SSFT yields matrix approximations whose quality is identical to those we obtain with
an isotropic test matrix (Sec. 5). Although the analysis for SSFTs is less complete, the empirical
evidence confirms that the theory for isotropic models also offers excellent guidance for SSFTs.
References for SSFTs and related test matrices include [1, 2, 9, 22, 29, 36, 42].
Numerically Stable Implementation. It requires care to compute the fixed-rank approximation (2.7).
The supplement shows that a poor implementation may produce an approximation with 100% error!
Let us outline a numerically stable and very accurate implementation of (2.7), based on an idea
from [27, 38]. Fix a small parameter ? > 0. Instead of approximating the psd matrix A directly, we
approximate the shifted matrix A? = A + ?I and then remove the shift. Here are the steps:
1.
2.
3.
4.
5.
6.
1
Construct the shifted sketch Y? = Y + ??.
Form the matrix B = ?? Y? .
Compute a Cholesky decomposition B = CC ? .
Compute E = Y? C ?1 by back-substitution.
Compute the (thin) singular value decomposition E = U ?V ? .
?r = U J?2 ? ?IKr U ? .
Form A
A signed permutation has exactly one nonzero entry in each row and column; the nonzero has modulus one.
4
Algorithm 3 Fixed-Rank PSD Approximation. Implements (2.7).
Input: Matrix A in sketch must be psd; rank parameter 1 ? r ? k
Output: Returns factors U ? Fn?r with orthonormal columns and nonnegative, diagonal ? ? Fr?r
?r = U ?U ? of the sketched matrix A
that form a rank-r psd approximation A
1
2
3
4
5
6
7
8
9
10
local: ?, Y
. Internal variables for N YSTROM S KETCH
function F IXED R ANK PSDA PPROX(r)
? ? ? norm(Y )
. ? = 2.2 ? 10?16 in double precision
Y ? Y + ??
. Sketch of shifted matrix A + ?I
B ? ?? Y
C ? chol((B + B ? )/2)
. Force symmetry
(U , ?, ?) ? svd(Y /C, ?econ?)
. Solve least squares problem; form thin SVD
U ? U (:, 1:r) and ? ? ?(1:r, 1:r)
. Truncate to rank r
? ? max{0, ?2 ? ?I}
. Square to get eigenvalues; remove shift
return (U , ?)
The pseudocode addresses some additional implementation details. Related, but distinct, methods
were proposed by Williams & Seeger [40] and analyzed in Gittens?s thesis [17].
Pseudocode. We present detailed pseudocode for the sketch (2.2)?(2.4) and the implementation of
the fixed-rank psd approximation (2.7) described above. For simplicity, we only elaborate the case of
a random orthonormal test matrix; we have also developed an SSFT implementation for empirical
testing. The pseudocode uses both mathematical notation and M ATLAB 2017 A functions.
Algorithms and Computational Costs. Algorithm 1 constructs a random orthonormal test matrix,
and computes the sketch (2.3) of an input matrix. The test matrix and sketch require the storage of
2kn floating-point numbers. Owing to the orthogonalization step, the construction of the test matrix
requires ?(k 2 n) floating-point operations. For a general input matrix, the sketch requires ?(kn2 )
floating-point operations; this cost can be removed by initializing the input matrix to zero.
Algorithm 2 implements the linear update (2.4) to the sketch. Nominally, the computation requires
?(kn2 ) arithmetic operations, but this cost can be reduced when H has structure (e.g., low rank).
Using the SSFT test matrix (3.1) also reduces this cost.
Algorithm 3 computes the rank-r psd approximation (2.7). This method requires additional storage of
?(kn). The arithmetic cost is ?(k 2 n) operations, which is dominated by the SVD of the matrix E.
4
Theoretical Results
Relative Error Bound. Our first result is an accurate bound for the expected Schatten 1-norm error
in the fixed-rank psd approximation (2.7).
Theorem 4.1 (Fixed-Rank Nystr?m: Relative Error). Assume 1 ? r < k ? n. Let A ? Fn?n be a
psd matrix. Draw a test matrix ? ? Fn?k from the Gaussian or orthonormal distribution, and form
?r given by (2.5) and (2.7) satisfies
the sketch Y = A?. Then the approximation A
r
?
E kA ? Ar k1 ? 1 +
? kA ? JAKr k1 ;
(4.1)
k?r??
r
?r k? ? kA ? JAKr k? +
E kA ? A
? kA ? JAKr k1 .
(4.2)
k?r??
The quantities ?(R) = 1 and ?(C) = 0. Similar results hold with high probability.
The proof appears in the supplement.
In contrast to all previous analyses of randomized Nystr?m methods, Theorem 4.1 yields explicit,
sharp constants. (The contemporary work [39, Thm. 1] contains only a less precise variant of (4.1).)
As a consequence, the formulae (4.1)?(4.2) offer an a priori mechanism for selecting the sketch size
k to achieve a desired error bound. In particular, for each ? > 0,
k = (1 + ??1 )r + ?
implies
?r k1 ? (1 + ?) ? kA ? JAKr k1 .
E kA ? A
5
Thus, we can attain an arbitrarily small relative error in the Schatten 1-norm. In the streaming setting,
the scaling k = ?(r/?) is optimal for this result [14, Thm. 4.2]. Furthermore, it is impossible [41,
Sec. 6.2] to obtain ?pure? relative error bounds in the Schatten ?-norm unless k = ?(n).
The Role of Spectral Decay. To circumvent these limitations, it is necessary to develop a different
kind of error bound. Our second result shows that the fixed-rank psd approximation (2.7) automatically
exploits decay in the spectrum of the input matrix.
Theorem 4.2 (Fixed-Rank Nystr?m: Spectral Decay). Instate the notation and assumptions of
Theorem 4.1. Then
%
?r k1 ? kA ? JAKr k1 + 2 min
? kA ? JAK% k1 ;
(4.3)
E kA ? A
1+
%<k??
k?%??
%
?r k? ? kA ? JAKr k? + 2 min
E kA ? A
1+
? kA ? JAK% k1 .
(4.4)
%<k??
k?%??
The index % ranges over the natural numbers.
The proof of Theorem 4.2 appears in the supplement.
Here is one way to understand this result. As the index % increases, the quantity %/(k?%??) increases
while the rank-% approximation error decreases. Theorem 4.2 states that the approximation (2.7)
automatically achieves the best tradeoff between these two terms. When the spectrum of A decays,
the rank-% approximation error may be far smaller than the rank-r approximation error. In this case,
Theorem 4.2 is tighter than Theorem 4.1, although the prediction is more qualitative.
Additional Results. The proofs can be extended to obtain high-probability bounds, as well as results
for other Schatten norms or for other test matrices (Supplement).
5
Numerical Performance
Experimental Setup. In many streaming applications, such as [44], it is essential that the sketch
uses as little memory as possible and that the psd approximation achieves the best possible error. For
the methods we consider, the arithmetic costs of linear updates and psd approximation are roughly
comparable. Therefore, we only assess storage and accuracy.
For the numerical experiments, the field F = C except when noted explicitly. Choose a psd input
matrix A ? Fn?n and a target rank r. Then fix a sketch size parameter k with r ? k ? n. For each
trial, draw the test matrix ? from the orthonormal or the SSFT distribution, and form the sketch
?r defined
Y = A? of the input matrix. Using Algorithm 3, compute the rank-r psd approximation A
in (2.7). We evaluate the performance using the relative error metric:
? r kp
kA ? A
Schatten p-norm relative error =
? 1.
(5.1)
kA ? JAKr kp
We perform 20 independent trials and report the average error.
We compare our method (2.7) with the standard truncated Nystr?m approximation (2.6); the best
reference for this type of approach is [18, Sec. 2.2]. The approximation (2.6) is constructed from the
same sketch as (2.7), so the experimental procedure is identical.
We also consider the sketching method and psd approximation algorithm [37, Alg. 9] based on earlier
work from [7, 22, 42]. We implemented this sketch with orthonormal matrices and also with SSFT
matrices. The sketch has two different parameters (k, `), so we select the parameters that result in the
minimum relative error. Otherwise, the experimental procedure is the same.
We apply the methods to representative input matrices; see the Supplement for plots of the spectra.
Synthetic Examples. The synthetic examples are diagonal with dimension n = 103 ; results for
larger and non-diagonal matrices are similar. These matrices are parameterized by an effective rank
parameter R, which takes values in {5, 10, 20}. We compute approximations with rank r = 10.
1. Low-Rank + PSD Noise. These matrices take the form
A = diag(1, . . . , 1, 0, . . . , 0) + ?n?1 W ? Fn?n .
| {z }
R
6
Relative Error (S1 )
10 0
10 -2
10 -2
10 -4
10 -4
10 -6
10 -6
10 -8
10 -8
1
2
4
8
16
32
64
128
6
12
Storage (T )
( A ) PhaseRetrieval (r = 1)
Relative Error (S1 )
24
48
96
192
Storage (T )
( B ) PhaseRetrieval (r = 5)
[TYUC17, Alg. 9]
Standard (2.6)
Proposed (2.7)
10 2
10 -2
10 1
10 -4
10 0
10
-6
10
-8
10 -1
1
2
4
8
16
32
64
128
16
32
64
128
256
Storage (T )
Storage (T )
( C ) MaxCut (r = 1)
( D ) MaxCut (r = 14)
F IGURE 5.1: Application Examples, Approximation Rank r, Schatten 1-Norm Error. The data
series show the performance of three algorithms for rank-r psd approximation. Solid lines are
generated from the Gaussian sketch; dashed lines are from the SSFT sketch. Each panel displays the
Schatten 1-norm relative error (5.1) as a function of storage cost T . See Sec. 5 for details.
The matrix W ? Fn?n has the W ISHART(n, n; F) distribution; that is, W = GG? where
G ? Fn?n is standard normal. The parameter ? controls the signal-to-noise ratio. We
consider three examples: LowRankLowNoise (? = 10?4 ), LowRankMedNoise (? = 10?2 ),
LowRankHiNoise (? = 10?1 ).
2. Polynomial Decay. These matrices take the form
A = diag(1, . . . , 1, 2?p , 3?p , . . . , (n ? R + 1)?p ) ? Fn?n .
| {z }
R
The parameter p > 0 controls the rate of polynomial decay. We consider three examples:
PolyDecaySlow (p = 0.5), PolyDecayMed (p = 1), PolyDecayFast (p = 2).
3. Exponential Decay. These matrices take the form
A = diag(1, . . . , 1, 10?q , 10?2q , . . . , 10?(n?R)q ) ? Fn?n .
| {z }
R
The parameter q > 0 controls the rate of exponential decay. We consider three examples:
ExpDecaySlow (q = 0.1), ExpDecayMed (q = 0.25), ExpDecayFast (q = 1).
Application Examples. We also consider non-diagonal matrices inspired by the SDP algorithm [44].
1. MaxCut: This is a real-valued psd matrix with dimension n = 2 000, and its effective rank
R = 14. We form approximations with rank r ? {1, 14}. The matrix is an approximate
solution to the M AX C UT SDP [20] for the sparse graph G40 [10].
2. PhaseRetrieval: This is a psd matrix with dimension n = 25 921. It has exact rank 250,
but its effective rank R = 5. We form approximations with rank r ? {1, 5}. The matrix is
an approximate solution to a phase retrieval SDP; the data is drawn from our paper [44].
Experimental Results. Figures 5.1?5.2 display the performance of the three fixed-rank psd approximation methods for a subcollection of the input matrices. The vertical axis is the Schatten 1-norm
7
Relative Error (S1 )
[TYUC17, Alg. 9]
Standard (2.6)
Proposed (2.7)
10
10 -1
0
10 -1
10 -1
10 -2
10 -2
12
24
48
96
192
12
24
48
96
192
12
24
Storage (T )
Storage (T )
( A ) LowRankLowNoise
48
96
192
Storage (T )
( B ) LowRankMedNoise
( C ) LowRankHiNoise
Relative Error (S1 )
10 0
10 -1
10 -1
10 -1
10 -2
10 -3
10 -2
10 -2
12
24
48
96
192
12
24
Storage (T )
48
96
192
12
24
Storage (T )
( D ) PolyDecayFast
48
96
192
Storage (T )
( E ) PolyDecayMed
( F ) PolyDecaySlow
10 0
10 0
Relative Error (S1 )
10 0
10 -2
10 -2
10 -4
10 -4
10 -6
10 -6
10 -6
10 -8
10 -8
10 -8
10 -2
10 -4
12
24
48
96
Storage (T )
( G ) ExpDecayFast
192
12
24
48
96
Storage (T )
( H ) ExpDecayMed
192
12
24
48
96
192
Storage (T )
( I ) ExpDecaySlow
F IGURE 5.2: Synthetic Examples with Effective Rank R = 10, Approximation Rank r = 10,
Schatten 1-Norm Error. The data series show the performance of three algorithms for rank-r psd
approximation with r = 10. Solid lines are generated from the Gaussian sketch; dashed lines are
from the SSFT sketch. Each panel displays the Schatten 1-norm relative error (5.1) as a function of
storage cost T .
relative error (5.1). The variable T on the horizontal axis is proportional to the storage required for
the sketch only. For the Nystr?m-based approximations (2.6)?(2.7), we have the correspondence
T = k. For the approximation [37, Alg. 9], we set T = k + `.
The experiments demonstrate that the proposed method (2.7) has a significant benefit over the
alternatives for input matrices that admit a good low-rank approximation. It equals or improves on the
competitors for almost all other examples and storage budgets. The supplement contains additional
numerical results; these experiments only reinforce the message of Figures 5.1?5.2.
Conclusions. This paper makes the case for using the proposed fixed-rank psd approximation (2.7)
in lieu of the alternatives (2.6) or [37, Alg. 9]. Theorem 4.1 shows that the proposed fixed-rank psd
approximation (2.7) can attain any prescribed relative error, and Theorem 4.2 shows that it can exploit
spectral decay. Furthermore, our numerical work demonstrates that the proposed approximation
improves (almost) uniformly over the competitors for a range of examples. These results are timely
because of the recent arrival of compelling applications, such as [44], for sketching psd matrices.
8
Acknowledgments. The authors wish to thank Mark Tygert and Alex Gittens for helpful feedback
on preliminary versions of this work. JAT gratefully acknowledges partial support from ONR Award
N00014-17-1-2146 and the Gordon & Betty Moore Foundation. VC and AY were supported in
part by the European Commission under Grant ERC Future Proof, SNF 200021-146750, and SNF
CRSII2-147633. MU was supported in part by DARPA Award FA8750-17-2-0101.
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6,327 | 6,723 | Multi-Modal Imitation Learning from Unstructured
Demonstrations using Generative Adversarial Nets
Karol Hausman?? , Yevgen Chebotar??? , Stefan Schaal?? , Gaurav Sukhatme? , Joseph J. Lim?
?
University of Southern California, Los Angeles, CA, USA
Max-Planck-Institute for Intelligent Systems, T?bingen, Germany
{hausman, ychebota, sschaal, gaurav, limjj}@usc.edu
?
Abstract
Imitation learning has traditionally been applied to learn a single task from demonstrations thereof. The requirement of structured and isolated demonstrations
limits the scalability of imitation learning approaches as they are difficult to
apply to real-world scenarios, where robots have to be able to execute a multitude of tasks. In this paper, we propose a multi-modal imitation learning
framework that is able to segment and imitate skills from unlabelled and unstructured demonstrations by learning skill segmentation and imitation learning
jointly. The extensive simulation results indicate that our method can efficiently
separate the demonstrations into individual skills and learn to imitate them using a single multi-modal policy. The video of our experiments is available at
http://sites.google.com/view/nips17intentiongan.
1
Introduction
One of the key factors to enable deployment of robots in unstructured real-world environments is
their ability to learn from data. In recent years, there have been multiple examples of robot learning
frameworks that present promising results. These include: reinforcement learning [31] - where a
robot learns a skill based on its interaction with the environment and imitation learning [2, 5] - where
a robot is presented with a demonstration of a skill that it should imitate. In this work, we focus on
the latter learning setup.
Traditionally, imitation learning has focused on using isolated demonstrations of a particular skill [29].
The demonstration is usually provided in the form of kinesthetic teaching, which requires the user to
spend sufficient time to provide the right training data. This constrained setup for imitation learning
is difficult to scale to real world scenarios, where robots have to be able to execute a combination
of different skills. To learn these skills, the robots would require a large number of robot-tailored
demonstrations, since at least one isolated demonstration has to be provided for every individual skill.
In order to improve the scalability of imitation learning, we propose a framework that can learn to
imitate skills from a set of unstructured and unlabeled demonstrations of various tasks.
As a motivating example, consider a highly unstructured data source, e.g. a video of a person cooking
a meal. A complex activity, such as cooking, involves a set of simpler skills such as grasping,
reaching, cutting, pouring, etc. In order to learn from such data, three components are required: i) the
ability to map the image stream to state-action pairs that can be executed by a robot, ii) the ability to
segment the data into simple skills, and iii) the ability to imitate each of the segmented skills. In this
work, we tackle the latter two components, leaving the first one for future work. We believe that the
capability proposed here of learning from unstructured, unlabeled demonstrations is an important
step towards scalable robot learning systems.
?
Equal contribution
In this paper, we present a novel imitation learning method that learns a multi-modal stochastic
policy, which is able to imitate a number of automatically segmented tasks using a set of unstructured
and unlabeled demonstrations. Our results indicate that the presented technique can separate the
demonstrations into sensible individual skills and imitate these skills using a learned multi-modal
policy. We show applications of the presented method to the tasks of skill segmentation, hierarchical
reinforcement learning and multi-modal policy learning.
2
Related Work
Imitation learning is concerned with learning skills from demonstrations. Approaches that are suitable
for this setting can be split into two categories: i) behavioral cloning [27], and ii) inverse reinforcement
learning (IRL) [24]. While behavioral cloning aims at replicating the demonstrations exactly, it
suffers from the covariance shift [28]. IRL alleviates this problem by learning a reward function that
explains the behavior shown in the demonstrations. The majority of IRL works [16, 35, 1, 12, 20]
introduce algorithms that can imitate a single skill from demonstrations thereof but they do not readily
generalize to learning a multi-task policy from a set of unstructured demonstrations of various tasks.
More recently, there has been work that tackles a problem similar to the one presented in this paper,
where the authors consider a setting where there is a large set of tasks with many instantiations [10].
In their work, the authors assume a way of communicating a new task through a single demonstration.
We follow the idea of segmenting and learning different skills jointly so that learning of one skill
can accelerate learning to imitate the next skill. In our case, however, the goal is to separate the
mix of expert demonstrations into single skills and learn a policy that can imitate all of them, which
eliminates the need of new demonstrations at test time.
The method presented here belongs to the field of multi-task inverse reinforcement learning. Examples
from this field include [9] and [4]. In [9], the authors present a Bayesian approach to the problem,
while the method in [4] is based on an EM approach that clusters observed demonstrations. Both of
these methods show promising results on relatively low-dimensional problems, whereas our approach
scales well to higher dimensional domains due to the representational power of neural networks.
There has also been a separate line of work on learning from demonstration, which is then iteratively
improved through reinforcement learning [17, 6, 23]. In contrast, we do not assume access to the
expert reward function, which is required to perform reinforcement learning in the later stages of the
above algorithms.
There has been much work on the problem of skill segmentation and option discovery for hierarchical
tasks. Examples include [25, 19, 14, 33, 13]. In this work, we consider a possibility to discover
different skills that can all start from the same initial state, as opposed to hierarchical reinforcement
learning where the goal is to segment a task into a set of consecutive subtasks. We demonstrate,
however, that our method may be used to discover the hierarchical structure of a task similarly to the
hierarchical reinforcement learning approaches. In [13], the authors explore similar ideas to discover
useful skills. In this work, we apply some of these ideas to the imitation learning setup as opposed to
the reinforcement learning scenario.
Generative Adversarial Networks (GANs) [15] have enjoyed success in various domains including
image generation [8], image-image translation [34, 18] and video prediction [22]. More recently, there
have been works connecting GANs and other reinforcement learning and IRL methods [26, 11, 16].
In this work, we expand on some of the ideas presented in these works and provide a novel framework
that exploits this connection.
The works that are most closely related to this paper are [16], [7] and [21]. In [7], the authors show
a method that is able to learn disentangled representations and apply it to the problem of image
generation. In this work, we provide an alternative derivation of our method that extends their
work and applies it to multi-modal policies. In [16], the authors present an imitation learning GAN
approach that serves as a basis for the development of our method. We provide an extensive evaluation
of the hereby presented approach compared to the work in [16], which shows that our method, as
opposed to [16], can handle unstructured demonstrations of different skills. A concurrent work [21]
introduces a method similar to ours and applies it to detecting driving styles from unlabelled human
data.
2
3
Preliminaries
Let M = (S, A, P, R, p0 , ?, T ) be a finite-horizon Markov Decision Process (MDP), where S and A
are state and action spaces, P : S ? A ? S ? R+ is a state-transition probability function or system
dynamics, R : S ? A ? R a reward function, p0 : S ? R+ an initial state distribution, ? a reward
discount factor, and T a horizon. Let ? = (s0 , a0 , . . . , sT , aT ) be a trajectory of states and actions and
PT
R(? ) = t=0 ? t R(st , at ) the trajectory reward. The goal of reinforcement learning methods is to
find parameters ? of a policy ?? (a|s) that maximizes the expected discounted reward over trajectories
induced by the policy: E?? [R(? )] where s0 ? p0 , st+1 ? P (st+1 |st , at ) and at ? ?? (at |st ).
In an imitation learning scenario, the reward function is unknown. However, we are given a set of
demonstrated trajectories, which presumably originate from some optimal expert policy distribution
?E1 that optimizes an unknown reward function RE1 . Thus, by trying to estimate the reward function
RE1 and optimizing the policy ?? with respect to it, we can recover the expert policy. This approach
is known as inverse reinforcement learning (IRL) [1]. In order to model a variety of behaviors, it is
beneficial to find a policy with the highest possible entropy that optimizes RE1 . We will refer to this
approach as the maximum-entropy IRL [35] with the optimization objective
min max H(?? ) + E?? R(s, a) ? E?E1 R(s, a),
(1)
R
??
where H(?? ) is the entropy of the policy ?? .
Ho and Ermon [16] showed that it is possible to redefine the maximum-entropy IRL problem with
multiple demonstrations sampled from a single expert policy ?E1 as an optimization of GANs [15]. In
this framework, the policy ?? (a|s) plays the role of a generator, whose goal is to make it difficult for a
discriminator network Dw (s, a) (parameterized by w) to differentiate between imitated samples from
?? (labeled 0) and demonstrated samples from ?E1 (labeled 1). Accordingly, the joint optimization
goal can be defined as
max min E(s,a)??? [log(Dw (s, a))] + E(s,a)??E1 [log(1 ? Dw (s, a))] + ?H H(?? ).
?
w
(2)
The discriminator and the generator policy are both represented as neural networks and optimized by
repeatedly performing alternating gradient updates. The discriminator is trained on the mixed set of
expert and generator samples and outputs probabilities that a particular sample has originated from
the generator or the expert policies. This serves as a reward signal for the generator policy that tries
to maximize the probability of the discriminator confusing it with an expert policy. The generator can
be trained using the trust region policy optimization (TRPO) algorithm [30] with the cost function
log(Dw (s, a)). At each iteration, TRPO takes the following gradient step:
E(s,a)??? [?? log ?? (a|s) log(Dw (s, a))] + ?H ?? H(?? ),
(3)
which corresponds to minimizing the objective in Eq. (2) with respect to the policy ?? .
4
Multi-modal Imitation Learning
The traditional imitation learning scenario described in Sec. 3 considers a problem of learning to
imitate one skill from demonstrations. The demonstrations represent samples from a single expert
policy ?E 1 . In this work, we focus on an imitation learning setup where we learn from unstructured
and unlabelled demonstrations of various tasks. In this case, the demonstrations come from a
set of expert policies ?E1 , ?E2 , . . . , ?Ek , where k can be unknown, that optimize different reward
functions/tasks. We will refer to this set of unstructured expert policies as a mixture of policies ?E .
We aim to segment the demonstrations of these policies into separate tasks and learn a multi-modal
policy that will be able to imitate all of the segmented tasks.
In order to be able to learn multi-modal policy distributions, we augment the policy input with a
latent intention i distributed by a categorical or uniform distribution p(i), similar to [7]. The goal of
the intention variable is to select a specific mode of the policy, which corresponds to one of the skills
presented in the demonstrations. The resulting policy can be expressed as:
?(a|s, i) = p(i|s, a)
3
?(a|s)
.
p(i)
(4)
We augment the trajectory to include the latent intention as ?i = (s0 , a0 , i0 , ...sT , aT , iT ). The
PT
resulting reward of the trajectory with the latent intention is R(?i ) = t=0 ? t R(st , at , it ). R(a, s, i)
is a reward function that depends on the latent intention i as we have multiple demonstrations that
optimize different
expected discounted reward is equal to:
R reward functions for different tasks. The
QT ?1
E?? [R(?i )] = R(?i )?? (?i )d?i where ?? (?i ) = p0 (s0 ) t=0 P (st+1 |st , at )?? (at |st , it )p(it ).
Here, we show an extension of the derivation presented in [16] (Eqs. (1, 2)) for a policy ?(a|s, i)
augmented with the latent intention variable i, which uses demonstrations from a set of expert policies
?E , rather than a single expert policy ?E1 . We are aiming at maximum entropy policies that can be
determined from the latent intention variable i. Accordingly, we transform the original IRL problem
to reflect this goal:
min max H(?(a|s)) ? H(?(a|s, i)) + E? R(s, a, i) ? E?E R(s, a, i),
(5)
?
R
where ?(a|s) =
P
?(a|s, i)p(i), which results in the policy averaged over intentions (since the
i
p(i) is constant). This goal reflects our objective: we aim to obtain a multi-modal policy that has a
high entropy without any given intention, but it collapses to a particular task when the intention is
specified. Analogously to the solution for a single expert policy, this optimization objective results in
the optimization goal of the generative adversarial imitation learning network, with the exception that
the state-action pairs (s, a) are sampled from a set of expert policies ?E :
max min Ei?p(i),(s,a)??? [log(Dw (s, a))] + E(s,a)??E [1 ? log(Dw (s, a))]
w
?
(6)
+?H H(?? (a|s)) ? ?I H(?? (a|s, i)),
where ?I , ?H correspond to the weighting parameters on the respective objectives. The resulting
entropy H(?? (a|s, i)) term can be expressed as:
H(?? (a|s, i)) = Ei?p(i),(s,a)??? (? log(?? (a|s, i))
(7)
?? (a|s)
= ?Ei?p(i),(s,a)??? log p(i|s, a)
p(i)
= ?Ei?p(i),(s,a)??? log(p(i|s, a)) ? Ei?p(i),(s,a)??? log ?? (a|s) + Ei?p(i) log p(i)
= ?Ei?p(i),(s,a)??? log(p(i|s, a)) + H(?? (a|s)) ? H(i),
which results in the final objective:
max min Ei?p(i),(s,a)??? [log(Dw (s, a))] + E(s,a)??E [1 ? log(Dw (s, a))]
?
w
(8)
+(?H ? ?I )H(?? (a|s)) + ?I Ei?p(i),(s,a)??? log(p(i|s, a)) + ?I H(i),
where H(i) is a constant that does not influence the optimization. This results in the same
optimization objective as for the single expert policy (see Eq. (2)) with an additional term
?I Ei?p(i),(s,a)??? log(p(i|s, a)) responsible for rewarding state-action pairs that make the latent
intention inference easier. We refer to this cost as the latent intention cost and represent p(i|s, a) with
a neural network. The final reward function for the generator is:
Ei?p(i),(s,a)??? [log(Dw (s, a))] + ?I Ei?p(i),(s,a)??? log(p(i|s, a)) + ?H 0 H(?? (a|s)).
4.1
(9)
Relation to InfoGAN
In this section, we provide an alternative derivation of the optimization goal in Eq. (8) by extending
the InfoGAN approach presented in [7]. Following [7], we introduce the latent variable c as a means
to capture the semantic features of the data distribution. In this case, however, the latent variables
are used in the imitation learning scenario, rather than the traditional GAN setup, which prevents us
from using additional noise variables (z in the InfoGAN approach) that are used as noise samples to
generate the data from.
Similarly to [7], to prevent collapsing to a single mode, the policy optimization objective is augmented
with mutual information I(c; G(??c , c)) between the latent variable and the state-action pairs generator
G dependent on the policy distribution ??c . This encourages the policy to produce behaviors that are
4
interpretable from the latent code, and given a larger number of possible latent code values leads to an
increase in the diversity of policy behaviors. The corresponding generator goal can be expressed as:
Ec?p(c),(s,a)???c [log(Dw (s, a))] + ?I I(c; G(??c , c)) + ?H H(??c )
(10)
In order to compute I(c; G(??c , c)), we follow the derivation from [7] that introduces a lower bound:
I(c; G(??c , c)) = H(c) ? H(c|G(??c , c))
(11)
0
= E(s,a)?G(??c ,c) [Ec0 ?P (c|s,a) [log P (c |s, a)]] + H(c)
= E(s,a)?G(??c ,c) [DKL (P (?|s, a)||Q(?|s, a)) + Ec0 ?P (c|s,a) [log Q(c0 |s, a)]] + H(c)
? E(s,a)?G(??c ,c) [Ec0 ?P (c|s,a) [log Q(c0 |s, a)]] + H(c)
= Ec?P (c),(s,a)?G(??c ,c) [log Q(c|s, a)] + H(c)
By maximizing this lower bound we maximize I(c; G(??c , c)). The auxiliary distribution Q(c|s, a)
can be parametrized by a neural network.
The resulting optimization goal is
max min Ec?p(c),(s,a)???c [log(Dw (s, a))] + E(s,a)??E [1 ? log(Dw (s, a))]
?
w
+ ?I Ec?P (c),(s,a)?G(??c ,c) [log Q(c|s, a)] +
(12)
?H H(??c )
which results in the generator reward function:
Ec?p(c),(s,a)???c [log(Dw (s, a))] + ?I Ec?P (c),(s,a)?G(??c ,c) [log Q(c|s, a)] + ?H H(??c ).
(13)
This corresponds to the same objective that was derived in Section 4. The auxiliary distribution over
the latent variables Q(c|s, a) is analogous to the intention distribution p(i|s, a).
5
Implementation
In this section, we discuss implementation details that can alleviate instability of the training procedure
of our model. The first indicator that the training has become unstable is a high classification accuracy
of the discriminator. In this case, it is difficult for the generator to produce a meaningful policy as
the reward signal from the discriminator is flat and the TRPO gradient of the generator vanishes.
In an extreme case, the discriminator assigns all the generator samples to the same class and it is
impossible for TRPO to provide a useful gradient as all generator samples receive the same reward.
Previous work suggests several ways to avoid this behavior. These include leveraging the Wasserstein
distance metric to improve the convergence behavior [3] and adding instance noise to the inputs of
the discriminator to avoid degenerate generative distributions [32]. We find that adding the Gaussian
noise helped us the most to control the performance of the discriminator and to produce a smooth
reward signal for the generator policy. During our experiments, we anneal the noise similar to [32],
as the generator policy improves towards the end of the training.
An important indicator that the generator policy distribution has collapsed to a uni-modal policy is a
high or increasing loss of the intention-prediction network p(i|s, a). This means that the prediction
of the latent variable i is difficult and consequently, the policy behavior can not be categorized into
separate skills. Hence, the policy executes the same skill for different values of the latent variable. To
prevent this, one can increase the weight of the latent intention cost ?I in the generator loss or add
more instance noise to the discriminator, which makes its reward signal relatively weaker.
In this work, we employ both categorical and continuous latent variables to represent the latent
intention. The advantage of using a continuous variable is that we do not have to specify the number
of possible values in advance as with the categorical variable and it leaves more room for interpolation
between different skills. We use a softmax layer to represent categorical latent variables, and use a
uniform distribution for continuous latent variables as proposed in [7].
6
Experiments
Our experiments aim to answer the following questions: (1) Can we segment unstructured and
unlabelled demonstrations into skills and learn a multi-modal policy that imitates them? (2) What
5
Figure 1: Left: Walker-2D running forwards, running backwards, jumping. Right: Humanoid running
forwards, running backwards, balancing.
Figure 2: Left: Reacher with 2 targets: random initial state, reaching one target, reaching another
target. Right: Gripper-pusher: random initial state, grasping policy, pushing (when grasped) policy.
is the influence of the introduced intention-prediction cost on the resulting policies? (3) Can we
autonomously discover the number of skills presented in the demonstrations, and even accomplish
them in different ways? (4) Does the presented method scale to high-dimensional policies? (5)
Can we use the proposed method for learning hierarchical policies? We evaluate our method on
a series of challenging simulated robotics tasks described below. We would like to emphasize
that the demonstrations consist of shuffled state-action pairs such that no temporal information
or segmentation is used during learning. The performance of our method can be seen in our
supplementary video2 .
6.1
Task setup
Reacher The Reacher environment is depicted in Fig. 2 (left). The actuator is a 2-DoF arm attached
at the center of the scene. There are several targets placed at random positions throughout the
environment. The goal of the task is, given a data set of reaching motions to random targets, to
discover the dependency of the target selection on the intention and learn a policy that is capable of
reaching different targets based on the specified intention input. We evaluate the performance of our
framework on environments with 1, 2 and 4 targets.
Walker-2D The Walker-2D (Fig. 1 left) is a 6-DoF bipedal robot consisting of two legs and feet
attached to a common base. The goal of this task is to learn a policy that can switch between three
different behaviors dependent on the discovered intentions: running forward, running backward and
jumping. We use TRPO to train single expert policies and create a combined data set of all three
behaviors that is used to train a multi-modal policy using our imitation framework.
Humanoid Humanoid (Fig. 1 right) is a high-dimensional robot with 17 degrees of freedom. Similar
to Walker-2D the goal of the task is to be able to discover three different policies: running forward,
running backward and balancing, from the combined expert demonstrations of all of them.
Gripper-pusher This task involves controlling a 4-DoF arm with an actuated gripper to push a sliding
block to a specified goal area (Fig. 2 right). We provide separate expert demonstrations of grasping
the object, and pushing it towards the goal starting from the object already being inside the hand. The
initial positions of the arm, block and the goal area are randomly sampled at the beginning of each
episode. The goal of our framework is to discover both intentions and the hierarchical structure of the
task from a combined set of demonstrations.
6.2
Multi-Target Imitation Learning
Our goal here is to analyze the ability of our method to segment and imitate policies that perform
the same task for different targets. To this end, we first evaluate the influence of the latent intention
cost on the Reacher task with 2 and 4 targets. For both experiments, we use either a categorical
intention distribution with the number of categories equal to the number of targets or a continuous,
2
http://sites.google.com/view/nips17intentiongan
6
Figure 3: Results of the imitation GAN with (top row) and without (bottom row) the latent intention
cost. Left: Reacher with 2 targets(crosses): final positions of the reacher (circles) for categorical (1)
and continuous (2) latent intention variable. Right: Reacher with 4 targets(crosses): final positions of
the reacher (circles) for categorical (3) and continuous (4) latent intention variable.
Figure 4: Left: Rewards of different Reacher policies for 2 targets for different intention values over
the training iterations with (1) and without (2) the latent intention cost. Right: Two examples of a
heatmap for 1 target Reacher using two latent intentions each.
uniformly-distributed intention variable, which means that the network has to discover the number of
intentions autonomously. Fig. 3 top shows the results of the reaching tasks using the latent intention
cost for 2 and 4 targets with different latent intention distributions. For the continuous latent variable,
we show a span of different intentions between -1 and 1 in the 0.2 intervals. The colors indicate
the intention ?value?. In the categorical distribution case, we are able to learn a multi-modal policy
that can reach all the targets dependent on the given latent intention (Fig. 3-1 and Fig. 3-3 top). The
continuous latent intention is able to discover two modes in case of two targets (Fig. 3-2 top) but
it collapses to only two modes in the four targets case (Fig. 3-4 top) as this is a significantly more
difficult task.
As a baseline, we present the results of the Reacher task achieved by the standard GAN imitation
learning presented in [16] without the latent intention cost. The obtained results are presented in
Fig. 3 bottom. Since the network is not encouraged to discover different skills through the intention
learning cost, it collapses to a single target for 2 targets in both the continuous and discrete latent
intention variables. In the case of 4 targets, the network collapses to 2 modes, which can be explained
by the fact that even without the latent intention cost the imitation network tries to imitate most of
the presented demonstrations. Since the demonstration set is very diverse in this case, the network
learned two modes without the explicit instruction (latent intention cost) to do so.
To demonstrate the development of different intentions, in Fig. 4 (left) we present the Reacher
rewards over training iterations for different intention variables. When the latent intention cost is
included, (Fig. 4-1), the separation of different skills for different intentions starts to emerge around
the 1000-th iteration and leads to a multi-modal policy that, given the intention value, consistently
reaches the target associated with that intention. In the case of the standard imitation learning GAN
setup (Fig. 4-2), the network learns how to imitate reaching only one of the targets for both intention
values.
In order to analyze the ability to discover different ways to accomplish the same task, we use our
framework with the categorical latent intention in the Reacher environment with a single target.
7
Figure 5: Top: Rewards of Walker-2D policies for different intention values over the training iterations
with (left) and without (right) the latent intention cost. Bottom: Rewards of Humanoid policies for
different intention values over the training iterations with (left) and without (right) the latent intention
cost.
Since we only have a single set of expert trajectories that reach the goal in one, consistent manner,
we subsample the expert state-action pairs to ease the intention learning process for the generator.
Fig. 4 (right) shows two examples of a heatmap of the visited end-effector states accumulated for
two different values of the intention variable. For both cases, the task is executed correctly, the robot
reaches the target, but it achieves it using different trajectories. These trajectories naturally emerged
through the latent intention cost as it encourages different behaviors for different latent intentions. It
is worth noting that the presented behavior can be also replicated for multiple targets if the number of
categories in the categorical distribution of the latent intention exceeds the number of targets.
6.3
Multi-Task Imitation Learning
We also seek to further understand whether our model extends to segmenting and imitating policies
that perform different tasks. In particular, we evaluate whether our framework is able to learn a
multi-modal policy on the Walker-2D task. We mix three different policies ? running backwards,
running forwards, and jumping ? into one expert policy ?E and try to recover all of them through
our method. The results are depicted in Fig. 5 (top). The additional latent intention cost results in
a policy that is able to autonomously segment and mimic all three behaviors and achieve a similar
performance to the expert policies (Fig. 5 top-left). Different intention variable values correspond to
different expert policies: 0 - running forwards, 1 - jumping, and 2 - running backwards. The imitation
learning GAN method is shown as a baseline in Fig. 5 (top-right). The results show that the policy
collapses to a single mode, where all different intention variable values correspond to the jumping
behavior, ignoring the demonstrations of the other two skills.
To test if our multi-modal imitation learning framework scales to high-dimensional tasks, we evaluate
it in the Humanoid environment. The expert policy is constructed using three expert policies: running
backwards, running forwards, and balancing while standing upright. Fig. 5 (bottom) shows the
rewards obtained for different values of the intention variable. Similarly to Walker-2D, the latent
8
Figure 6: Time-lapse of the learned Gripper-pusher policy. The intention variable is changed manually
in the fifth screenshot, once the grasping policy has grasped the block.
intention cost enables the neural network to segment the tasks and learn a multi-modal imitation
policy. In this case, however, due to the high dimensionality of the task, the resulting policy is able
to mimic running forwards and balancing policies almost as well as the experts, but it achieves a
suboptimal performance on the running backwards task (Fig. 5 bottom-left). The imitation learning
GAN baseline collapses to a uni-modal policy that maps all the intention values to a balancing
behavior (Fig. 5 bottom-right).
Finally, we evaluate the ability of our method to discover options in hierarchical IRL tasks. In order
to test this, we collect expert policies in the Gripper-pusher environment that consist of grasping and
pushing when the object is grasped demonstrations. The goal of this task is to check whether our
method will be able to segment the mix of expert policies into separate grasping and pushing-whengrasped skills. Since the two sub-tasks start from different initial conditions, we cannot present the
results in the same form as for the previous tasks. Instead, we present a time-lapse of the learned
multi-modal policy (see Fig. 6) that presents the ability to change in the intention during the execution.
The categorical intention variable is manually changed after the block is grasped. The intention
change results in switching to a pushing policy that brings the block into the goal region. We present
this setup as an example of extracting different options from the expert policies that can be further
used in an hierarchical reinforcement learning task to learn the best switching strategy.
7
Conclusions
We present a novel imitation learning method that learns a multi-modal stochastic policy, which is
able to imitate a number of automatically segmented tasks using a set of unstructured and unlabeled
demonstrations. The presented approach learns the notion of intention and is able to perform different
tasks based on the policy intention input. We evaluated our method on a set of simulation scenarios
where we show that it is able to segment the demonstrations into different tasks and to learn a
multi-modal policy that imitates all of the segmented skills. We also compared our method to a
baseline approach that performs imitation learning without explicitly separating the tasks.
In the future work, we plan to focus on autonomous discovery of the number of tasks in the given
pool of demonstrations as well as evaluating this method on real robots. We also plan to learn an
additional hierarchical policy over the discovered intentions as an extension of this work.
Acknowledgements
This research was supported in part by National Science Foundation grants IIS-1205249, IIS-1017134,
EECS-0926052, the Office of Naval Research, the Okawa Foundation, and the Max-Planck-Society.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of
the author(s) and do not necessarily reflect the views of the funding organizations.
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11
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6,328 | 6,724 | Learning to Inpaint for Image Compression
Mohammad Haris Baig?
Department of Computer Science
Dartmouth College
Hanover, NH
Vladlen Koltun
Intel Labs
Santa Clara, CA
Lorenzo Torresani
Dartmouth College
Hanover, NH
Abstract
We study the design of deep architectures for lossy image compression. We present
two architectural recipes in the context of multi-stage progressive encoders and empirically demonstrate their importance on compression performance. Specifically,
we show that: (a) predicting the original image data from residuals in a multi-stage
progressive architecture facilitates learning and leads to improved performance at
approximating the original content and (b) learning to inpaint (from neighboring
image pixels) before performing compression reduces the amount of information
that must be stored to achieve a high-quality approximation. Incorporating these
design choices in a baseline progressive encoder yields an average reduction of over
60% in file size with similar quality compared to the original residual encoder.
1
Introduction
Visual data constitutes most of the total information created and shared on the Web every day and it
forms a bulk of the demand for storage and network bandwidth [13]. It is customary to compress
image data as much as possible as long as there is no perceptible loss in content. In recent years
deep learning has made it possible to design deep models for learning compact representations for
image data [2, 16, 18, 19, 20]. Deep learning based approaches, such as the work of Rippel and
Bourdev [16], significantly outperform traditional methods of lossy image compression. In this paper,
we show how to improve the performance of deep models trained for lossy image compression.
We focus on the design of models that produce progressive codes. Progressive codes are a sequence
of representations that can be transmitted to improve the quality of an existing estimate (from a
previously sent code) by adding missing detail. This is in contrast to non-progressive codes whereby
the entire data for a certain quality approximation must be transmitted before the image can be viewed.
Progressive codes improve the user?s browsing experience by reducing loading time of pages that are
rich in images. Our main contributions in this paper are two-fold.
1. While traditional progressive encoders are optimized to compress residual errors in each
stage of their architecture (residual-in, residual-out), instead we propose a model that is
trained to predict at each stage the original image data from the residual of the previous
stage (residual-in, image-out). We demonstrate that this leads to an easier optimization
resulting in better image compression. The resulting architecture reduces the amount of
information that must be stored for reproducing images at similar quality by 18% compared
to a traditional residual encoder.
2. Existing deep architectures do not exploit the high degree of spatial coherence exhibited by
neighboring patches. We show how to design and train a model that can exploit dependences
between adjacent regions by learning to inpaint from the available content. We introduce
multi-scale convolutions that sample content at multiple scales to assist with inpainting.
?
http://www.cs.dartmouth.edu/ haris/compression
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
We jointly train our proposed inpainting and compression models and show that inpainting
reduces the amount of information that must be stored by an additional 42%.
2
Approach
We begin by reviewing the architecture and the learning objective of a progressive multi-stage encoderdecoder with S stages. We adopt the convolutional-deconvolutional residual encoder proposed by
Toderici et al. [19] as our reference model. The model extracts a compact binary representation B
from an image patch P . This binary representation, used to reconstruct an approximation of the
original patch, consists of the sequence of representations extracted by the S stages of the model,
B = [B1 , B2 , . . . BS ].
The first stage of the model extracts a binary code B1 from the input patch P . Each of the subsequent stages learns to extract representations Bs , to model the compressions residuals Rs?1 from
the previous stage. The compression residuals Rs are defined as Rs = Rs?1 ? Ms (Rs?1 |?s ),
where Ms (Rs?1 |?s ) represents the reconstruction obtained by the stage s when modelling the
residuals Rs?1 . The model at each stage is split into an encoder Bs = Es (Rs?1 |?E
s ) and a decoder
E
D
E
D
Ds (Bs |?D
)
such
that
M
(R
|?
)
=
D
(E
(R
|?
)|?
)
and
?
=
{?
,
?
s
s?1
s
s s
s?1
s
s
s
s
s
s }. The paramth
eters for the s stage of the model are denoted by ?s . The residual encoder-decoder is trained on a
dataset P, consisting of N image patches, according to the following objective:
?
L(P;
?1:S ) =
S X
N
X
(i)
(i)
kRs?1 ? Ms (Rs?1 |?s )k22.
(1)
s=1 i=1
(i)
(i)
Rs represents the compression residual for the ith patch P (i) after stage s and R0 = P (i) .
Residual encoders are difficult to optimize as gradients have to traverse long paths from later stages to
affect change in the previous stages. When moving along longer paths, gradients tend to decrease in
magnitude as they get to earlier stages. We address this shortcoming of residual encoders by studying
a class of architectures we refer to as ?Residual-to-Image? (R2I).
2.1
Residual-to-Image (R2I)
To address the issue of vanishing gradients we add connections between subsequent stages and restate
the loss to predict the original data at the end of each stage, thus performing residual-to-image
prediction. This leads to the new objective shown below:
L(P; ?1:S ) =
S X
N
X
(i)
kP (i) ? Ms (Rs?1 |?s )k22.
(2)
s=1 i=1
Stage s of this model takes as input the compression residuals Rs?1 computed with respect to the
original data, Rs?1 = P ? Ms?1 (Rs?2 |?s?1 ), and Ms?1 (Rs?2 |?s?1 ) now approximates the
reconstruction of the original data P at stage s ? 1. To allow complete image reconstructions to
be produced at each stage while only feeding in residuals, we introduce connections between the
layers of adjacent stages. These connections allow for later stages to incorporate information that has
been recovered by earlier stages into their estimate of the original image data. Consequently, these
connections (between subsequent stages) allow for better optimization of the model.
In addition to assisting with modeling the original image, these connections play two key roles. Firstly,
these connections create residual blocks [10] which encourage explicit learning of how to reproduce
information which could not be generated by the previous stage. Secondly, these connections reduce
the length of the path along which information has to travel from later stages to impact the earlier
stages, leading to a better joint optimization.
This leads us to the question of where should such connections be introduced and how should
information be propagated? We consider two types of connections to propagate information between
successive stages. 1) Prediction connections are analogous to the identity shortcuts introduced by
He et al. [10] for residual learning. They act as parameter-free additive connections: the output of
2
Residuals
Encoder layer
Data
Residual Prediction
Decoder layer
Data Prediction
10
Binarizer
Information flow
Additive connection
Parametrized connection
+
10101000
10101001
01010001
a) Residual encoder
10101000
01010001
10101000
c) Full connections
b) Prediction connections
01010001
10101000
d) Decoding connections
Residual to Image (R2I) Models
Figure 1: Multiple approaches for introducing connections between successive stages. These designs
for progressive architectures allow for varying degrees of information to be shared. Architecture
(b-d) do not reconstruct residuals, but the original data at every stage. We call these architectures
?residual-to-image? (R2I).
each stage is produced by simply adding together the residual predictions of the current stage and all
preceding stages (see Figure 1(b)) before applying a final non-linearity.2) Parametric connections
are referred to as projection shortcuts by He et al. [10]. Here we use them to connect corresponding
layers in two consecutive stages of the compression model. The features of each layer from the
previous stage are convolved with learned filters before being added to the features of the same
layer in the current stage. A non-linearity is then applied on top. The prediction connections only
yield the benefit of creating residual blocks, albeit very large and thus difficult to optimize. In
contrast, parametric connections allow for the intermediate representations from previous stages to
be passed to the subsequent stages. They also create a denser connectivity pattern with gradients
now moving along corresponding layers in adjacent stages. We consider two variants of parametric
connections: ?full? which use parametric connections between all the layers in two successive stages
(see Figure 1(c)), and ?decoding? connections which link only corresponding decoding layers (i.e.,
there are no connections between encoding layers of adjacent stages). We note that the LSTM-based
model of Toderici et al. [20] represents a particular instance of R2I network with full connections.
In Section 3 we demonstrate that R2I models with decoding connections outperform those with full
connections and provide an intuitive explanation for this result.
2.2
Inpainting Network
Image compression architectures learn to encode and decode an image patch-by-patch. Encoding all
patches independently assumes that the regions contain truly independent content. This assumption
generally does not hold true when the patches being encoded are contiguous. We observe that the
content of adjacent image patches is not independent. We propose a new module for the compression
model designed to exploit the spatial coherence between neighboring patches. We achieve this goal
by training a model with the objective of predicting the content of each patch from information
available in the neighboring regions.
Deep models for inpainting, such as the one proposed by Pathak et al. [14], are trained to predict
? from a context region C? (as shown in Figure 2). As there
the values of pixels in the region W
is data present all around the region to be inpainted this imposes strong constraints on what the
inpainted region should look like. We consider the scenario where images are encoded and decoded
block-by-block moving from left to right and going from top to bottom (similar to how traditional
codecs process images [1, 21]). Now, at decoding time only content above and to the left of each
patch will have been reconstructed (see Figure 2(a)). This gives rise to the problem of ?partial-context
inpainting?. We propose a model that, given an input region C, attempts to predict the content of
the current patch P . We denote by P? the dataset which contains all the patches from the dataset P
3
Content available
for inpainting
C?
Region to inpaint
C
P?
P
Full-context Inpainting
Partial-context Inpainting
(b) Multi-scale convolutional layer
(a) Variations of the inpainting problem
Figure 2: (a) The two kinds of inpainting problems. (b) A multi-scale convolutional layer with 3
dilation factors. The colored boxes represent pixels from which the content is sampled.
and the respective context regions C for each patch. The loss function used to train our inpainting
network is:
? ?I ) =
Linp (P;
N
X
kP (i) ? MI (C (i) |?I )k22.
(3)
i=1
The output of the inpainting network is denoted by MI (C (i) |?I ), where ?I refers to the parameters
of the inpainting network.
2.2.1
Architecture of the Partial-Context Inpainting Network
Our inpainting network has a feed-forward architecture which propagates information from the
context region C to the region being inpainted, P . To improve the ability of our model at predicting
content, we use a multi-scale convolutional layer as the basic building block of our inpainting network.
We make use of the dilated convolutions described by Yu and Koltun [23] to allow for sampling
at various scales. Each multi-scale convolutional layer is composed of k filters for each dilation
factor being considered. Varying the dilation factor of filters gives us the ability to analyze content
at various scales. This structure of filters provides two benefits. First, it allows for a substantially
denser and more diverse sampling of data from context and second it allows for better propagation of
content at different spatial scales. A similarly designed layer was also used by Chen et al. [5] for
sampling content at multiple scales for semantic segmentation. Figure 2(b) shows the structure of a
multi-scale convolutional layer.
The multi-scale convolutional layer also gives us the freedom to propagate content at full resolution
(no striding or pooling) as only a few multi-scale layers suffice to cover the entire region. This allows
us to train a relatively shallow yet highly expressive architecture which can propagate fine-grained
information that might otherwise be lost due to sub-sampling. This light-weight and efficient design
is needed to allow for joint training with a multi-stage compression model.
2.2.2
Connecting the Inpainting Network with the R2I Compression model
Next, we describe how to use the prediction of the inpainting network for assisting with compression.
Whereas the inpainting network learns to predict the data as accurately as possible, we note that
this is not sufficient to achieve good performance on compression, where it is also necessary that
the ?inpainting residuals? be easy to compress. We describe the inpainting residuals as R0 =
P ? MI (C|?I ), where MI (C|?I ) denotes the inpainting estimate. As we wanted to train our
model to always predict the data, we add the inpainting estimate to the final prediction of each stage
of our compression model. This allows us to (a) produce the original content at each stage and (b) to
4
discover an inpainting that is beneficial for all stages of the model because of joint training. We now
train our complete model as
? ?I , ?1:S ) = Linp (P;
? ?I ) +
LC (P;
N X
S
X
(i)
kP (i) ? [Ms (Rs?1 |?s ) + MI (C (i) |?I )] k22.
(4)
i=1 s=1
(i)
In this new objective LC , the first term Linp corresponds to the original inpainting loss, R0
corresponds to the inpainting residual for example i. We note that each stage of this inpainting-based
progressive coder directly affects what is learned by the inpainting network. We refer to the model
trained with this joint objective as ?Inpainting for Residual-to-Image Compression? (IR2I).
Whereas we train our model to perform inpainting from the original image content, we use a lossy
approximation of the context region C when encoding images with IR2I. This is done because at
decoding time our model does not have access to the original image data. We use the approximation
from stage 2 of our model for performing inpainting at encoding and decoding time, and transmit
the binary codes for the first two stages as a larger first code. This strategy allows us to leverage
inpainting while performing progressive image compression.
2.3
Implementation Details
Our models were trained on 6,507 images from the ImageNet dataset [7], as proposed by Ball? et
al. [2] to train their single-stage encoder-decoder architectures. A full description of the R2I models
and the inpainting network is provided in the supplementary material. We use the Caffe library [11]
to train our models. The residual encoder and R2I models were trained for 60,000 iterations whereas
the joint inpainting network was trained for 110,000 iterations. We used the Adam optimizer [12]
for training our models and the MSRA initialization [9] for initializing all stages. We used initial
learning rates of 0.001 and the learning rate was dropped after 30K and 45K for the R2I models.
For the IR2I model, the learning rate was dropped after 30K, 65K, and 90K iterations by a factor of
10 each time. All of our models were trained to reproduce the content of 32 ? 32 image patches.
Each of our models has 8 stages, with each stage contributing 0.125 bits-per-pixel (bpp) to the total
representation of a patch. Our models handle binary optimization by employing the biased estimators
approach proposed by Raiko et al. [15] as was done by Toderici et al. [19, 20].
Our inpainting network has 8 multi-scale convolutional layers for content propagation and one
standard convolutional layer for performing the final prediction. Each multi-scale convolutional
layer consists of 24 filters each for dilation factors 1, 2, 4, 8. Our inpainting network takes as input a
context region C of size 64 ? 64, where the bottom right 32 ? 32 region is zeroed out and represents
the region to be inpainted.
3
Results
We investigate the improvement brought about by the presented techniques. We are interested in
studying the reduction in bit-rate, for varying quality of reconstruction, achieved after adaptation
from the residual encoder proposed by Toderici et al. [19]. To evaluate performance, we perform
compression with our models on images from the Kodak dataset [8]. The dataset consists of 24
uncompressed color images of size 512 ? 768. The quality is measured according to the MSSSIM [22] metric (higher values indicate better quality). We use the Bjontegaard-Delta metric [4] to
compute the average reduction in bit-rate across all quality settings.
3.1
R2I - Design and Performance
The table in Figure 3(a) shows the percentage reduction in bit-rate achieved by the three variations of
the Residual-to-Image models. As can be seen, adding side-connections and training for the more
desirable objective (i.e., approximating the original data) at each stage helps each of our models.
That said, having connections in the decoder only helps more compared to using a ?full? connection
approach or only sharing the final prediction.
5
18
16
14
Approach
R2I Prediction
R2I Full
R2I Decoding
SSIM
MS-SSIM
4.483
10.015
20.002
5.177
7.652
17.951
MS?SSIM (dB)
Rate Savings (%)
12
10
Residual Encoder
(a)
R2I Prediction connection
8
R2I Full connections
R2I Decoding connections
6
0.1
0.2
0.3
0.4
0.5
0.6
BitRate
0.7
0.8
0.9
1
(b)
Figure 3: (a) Average rate savings for each of the three R2I variants compared to the residual encoder
proposed by Toderici et al. [19]. (b) Figure shows the quality of images produced by each of the three
R2I variants across a range of bit-rates.
50
200
50
Enc?Dec
Dec?only
Enc?Dec
Dec?only
Enc?Dec
Dec?only
R2I full connections
40
R2I decoding connections
40
100
Training Loss (MSE)
Training Loss (MSE)
Training Loss (MSE)
150
30
20
30
20
50
10
0
0.5
1
1.5
2
2.5
3
3.5
Iterations
Iterations
a) Stage 1
4
4.5
5
5.5
6
0
10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0
Iterations
Iterations
(x104)
2.5 3 3.5
Iterations
Iterations
b) Stage 4
0.5
1
1.5
2
c) Stage 8
4
4.5
5
5.5
6
Figure 4: The R2I training loss from 3 different stages (start, middle, end) viewed as a function
of iterations for the ?full? and the ?decoding? connections models. We note that the decoding
connections model converges faster, to a lower value, and shows less variance.
The model which shares only the prediction between stages performs poorly in comparison to the
other two designs as it does not allow for features from earlier stages to be altered as efficiently as
done by the full or decoding connections architectures.
The model with decoding connections does better than the architecture with full connections because
for the model with connections at decoding only the binarization layer in each stage extracts a
representation from the relevant information only (the residuals with respect to the data). In contrast,
when connections are established in both the encoder and the decoder, the binary representation may
include information that has been captured by a previous stage, thereby adding burden on each stage
in identifying information pertinent to improving reconstruction, leading to a tougher optimization.
Figure 4 shows that the model with full connections struggles to minimize the training error compared
to the model with decoding connections. This difference in training error points to the fact that
connections in the encoder make it harder for the model to do well at training time. This difficulty of
optimization amplifies with the increase in stages as can be seen by the difference between the full
and decoding architecture performance (shown in Figure 3(b)) because the residuals become harder
to compress.
We note that R2I models significantly improve the quality of reconstruction at higher bit rates but do
not improve the estimates at lower bit-rates as much (see Figure 3(b)). This tells us that the overall
performance can be improved by focusing on approaches that yield a significant improvement at
lower bit-rates, such as inpainting, which is analyzed next.
6
18
17
16
Rate Savings (%)
SSIM
MS-SSIM
R2I Decoding
R2I Decoding Sep-Inp
R2I Decoding Joint-Inp
20.002
27.379
63.353
17.951
27.794
60.446
MS?SSIM (dB)
15
Approach
14
13
12
(a) Impact of inpainting on the performance
at compression. All bit-rate savings are reported with respect to the residual encoder
by Toderici et al. [19]
11
Residual Encoder
R2I?Decoding Sep?Inp
10
R2I?Decoding Joint?Inp
9
0.2
0.3
0.4
0.5
0.6
BitRate
0.7
0.8
0.9
1
(b)
Figure 5: (a) Average rate savings with varying forms of inpainting. (b) The quality of images with
each of our proposed approaches at varying bit-rates.
3.2
Impact of Inpainting
We begin analyzing the performance of the inpainting network and other approaches on partial-context
inpainting. We compare the performance of the inpainting network with both traditional approaches
as well as a learning-based baseline. Table 1 shows the average SSIM achieved by each approach for
inpainting all non-overlapping patches in the Kodak dataset.
Approach
SSIM
PDE-based
Exemplar-based
[3]
[6]
Vanilla network
Learning-based
Inpainting network
0.4574
0.4611
0.4545
0.5165
Table 1: Average SSIM for partial-context inpainting on the Kodak dataset [8]. The vanilla model is
a feed-forward CNN with no multi-scale convolutions.
The vanilla network corresponds to a 32-layer (4 times as deep as the inpainting network) model that
does not use multi-scale convolutions (all filters have a dilation factor of 1), has the same number of
parameters, and also operates at full resolution (as our inpainting network). This points to the fact that
the improvement in performance of the inpainting network over the vanilla model is a consequence of
using multi-scale convolutions. The inpainting network improves over traditional approaches because
our model learns the best strategy for propagating content as opposed to using hand-engineered
principles of content propagation. The low performance of the vanilla network shows that learning
by itself is not superior to traditional approaches and multi-scale convolutions play a key role in
achieving better performance.
Whereas inpainting provides an initial estimate of the content within the region it by no means
generates a perfect reconstruction. This leads us to the question of whether this initial estimate
is better than not having an estimate? The table in Figure 5(a) shows the performance on the
compression task with and without inpainting. These results show that the greatest reduction in file
size is achieved when the inpainting network is jointly trained with the R2I model. We note (from
Figure 5(b)) that inpainting greatly improves the quality of results obtained at lower and at higher bit
rates.
The baseline where the inpainting network is trained separately from the compression network
is presented here to emphasize the role of joint training. Traditional codecs [1] use simple non
learning-based inpainting approaches and their predefined methods of representing data are unable to
compactly encode the inpainting residuals. Learning to inpaint separately improves the performance
7
as the inpainted estimate is better than not having any estimate. But given that the compression model
has not been trained to optimize the compression residuals the reduction in bit-rate for achieving high
quality levels is low. We show that with joint training, we can not only train a model that does better
inpainting but also ensure that the inpainting residuals can be represented compactly.
3.3
Comparison with Existing Approaches
Table 2 shows a comparison of the performance of various approaches compared to JPEG [21] in the
0.125 to 1 bits-per-pixel (bpp) range. We select this range as images from our models towards the
end of this range show no perceptible artifacts of compression.
The first part of the table evaluates the performance of learning-based progressive approaches. We
note that our proposed model outperforms the multi-stage residual encoder proposed by Toderici
et al. [19] (trained on the same 6.5K dataset) by 17.9% and IR2I outperforms the residual encoder
by reducing file-sizes by 60.4%. The residual-GRU, while similar in architecture to our ?full?
connections model, does not do better even when trained on a dataset that is 1000 times bigger and
for 10 times more training time. The results shown here do not make use of entropy coding as the
goal of this work is to study how to improve the performance of deep networks for progressive image
compression and entropy coding makes it harder to understand where the performance improvements
are coming from. As various approaches use different entropy coding methods, this further obfuscates
the source of the improvements.
The second part of the table shows the performance of existing codecs. Existing codecs use entropy
coding and rate-distortion optimization. We note that even without using either of these powerful
post processing techniques, our final ?IR2I? model is competitive with traditional methods for
compression, which use both of these techniques. A comparison with recent non-progressive
approaches [2, 18], which also use these post-processing techniques for image compression, is
provided in the supplementary material.
Approach
Number of
Training Images
Progressive
Rate Savings
(%)
Residual Encoder [19]
Residual-GRU [20]
R2I (Decoding connections)
IR2I
6.5K
6M
6.5K
6.5K
Yes
Yes
Yes
Yes
2.56
33.26
18.53
51.25
JPEG-2000 [17]
WebP [1]
N/A
N/A
No
No
63.01
64.98
Table 2: Average rate savings compared to JPEG [21]. The savings are computed on the Kodak [8]
dataset with rate-distortion profiles measuring MS-SSIM in the 0-1 bpp range.
We observe that a naive implementation of IR2I creates a linear dependence in content (as all regions
used as context have to be decoded before being used for inpainting) and thus may be substantially
slower. In practice, this slowdown would be negligible as one can use a diagonal scan pattern (similar
to traditional codecs) for ensuring high parallelism thereby reducing run times. Furthermore, we
perform inpainting using predictions from the first step only. Therefore, the dependence only exists
when generating the first progressive code. For all subsequent stages, there is no dependence in
content, and our approach is comparable in run time to similar approaches.
4
Conclusion and Future Work
We study a class of ?Residual to Image? models and show that within this class, architectures which
have decoding connections perform better at approximating image data compared to designs with
other forms of connectivity. We observe that our R2I decoding connections model struggles at low
bit-rates and we show how to exploit spatial coherence between content of adjacent patches via
inpainting to improve performance at approximating image content at low bit-rates. We design a new
8
model for partial-context inpainting using multi-scale convolutions and show that the best way to
leverage inpainting is by jointly training the inpainting network with our R2I Decoding model.
One interesting extension of this work would be to incorporate entropy coding within our progressive
compression framework to train models that produce binary codes which have low-entropy and can
be represented even more compactly. Another possible direction would be to extend our proposed
framework to video data, where the gains from our discovery of recipes for improving compression
may be even greater.
5
Acknowledgements
This work was funded in part by Intel Labs and NSF award CNS-120552. We gratefully acknowledge
NVIDIA and Facebook for the donation of GPUs used for portions of this work. We would like to
thank George Toderici, Nick Johnston, Johannes Balle for providing us with information needed for
accurate assessment. We are grateful to members of the Visual Computing Lab at Intel Labs, and
members of the Visual Learning Group at Dartmouth College for their feedback.
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6,329 | 6,725 | Adaptive Bayesian Sampling with Monte Carlo EM
Anirban Roychowdhury, Srinivasan Parthasarathy
Department of Computer Science and Engineering
The Ohio State University
[email protected], [email protected]
Abstract
We present a novel technique for learning the mass matrices in samplers obtained
from discretized dynamics that preserve some energy function. Existing adaptive
samplers use Riemannian preconditioning techniques, where the mass matrices are
functions of the parameters being sampled. This leads to significant complexities in
the energy reformulations and resultant dynamics, often leading to implicit systems
of equations and requiring inversion of high-dimensional matrices in the leapfrog
steps. Our approach provides a simpler alternative, by using existing dynamics in
the sampling step of a Monte Carlo EM framework, and learning the mass matrices
in the M step with a novel online technique. We also propose a way to adaptively
set the number of samples gathered in the E step, using sampling error estimates
from the leapfrog dynamics. Along with a novel stochastic sampler based on
Nos?-Poincar? dynamics, we use this framework with standard Hamiltonian Monte
Carlo (HMC) as well as newer stochastic algorithms such as SGHMC and SGNHT,
and show strong performance on synthetic and real high-dimensional sampling
scenarios; we achieve sampling accuracies comparable to Riemannian samplers
while being significantly faster.
1
Introduction
Markov Chain Monte Carlo sampling is a well-known set of techniques for learning complex Bayesian
probabilistic models that arise in machine learning. Typically used in cases where computing the
posterior distributions of parameters in closed form is not feasible, MCMC techniques that converge
reliably to the target distributions offer a provably correct way (in an asymptotic sense) to draw
samples of target parameters from arbitrarily complex probability distributions. A recently proposed
method in this domain is Hamiltonian Monte Carlo (HMC) [1, 2], that formulates the target density
as an ?energy function? augmented with auxiliary ?momentum? parameters, and uses discretized
Hamiltonian dynamics to sample the parameters while preserving the energy function. The resulting
samplers perform noticeably better than random walk-based methods in terms of sampling efficiency
and accuracy [1, 3]. For use in stochastic settings, where one uses random minibatches of the data
to calculate the gradients of likelihoods for better scalability, researchers have used Fokker-Planck
correction steps to preserve the energy in the face of stochastic noise [4], as well as used auxiliary
?thermostat? variables to control the effect of this noise on the momentum terms [5, 6]. As with the
batch setting, these methods have exploited energy-preserving dynamics to sample more efficiently
than random walk-based stochastic samplers [4, 7, 8].
A primary (hyper-)parameter of interest in these augmented energy function-based samplers in the
?mass? matrix of the kinetic energy term; as noted by various researchers [1, 3, 6, 8, 9], this matrix
plays an important role in the trajectories taken by the samplers in the parameter space of interest,
thereby affecting the overall efficiency. While prior efforts have set this to the identity matrix or some
other pre-calculated value [4, 5, 7], recent work has shown that there are significant gains to be had in
efficiency as well as convergent accuracy by reformulating the mass in terms of the target parameters
to be sampled [3, 6, 8], thereby making the sampler sensitive to the underlying geometry. This is
done by imposing a positive definite constraint on the adaptive mass, and using it as the metric of
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
the Riemannian manifold of probability distributions parametrized by the target parameters. This
constraint also satisfies the condition that the momenta be sampled from a Gaussian with the mass as
the covariance. Often called Riemannian preconditioning, this idea has been applied in both batch [3]
as well as stochastic settings [6, 8] to derive HMC-based samplers that adaptively learn the critically
important mass matrix from the data.
Although robust, these reformulations often lead to significant complexities in the resultant dynamics;
one can end up solving an implicit system of equations in each half-step of the leapfrog dynamics
[3, 6], along with inverting large O(D2 ) matrices. This is sometimes sidestepped by performing
fixed point updates at the cost of additional error, or restricting oneself to simpler formulations that
honor the symmetric positive definite constraint, such as a diagonal matrix [8]. While this latter
choice ameliorates a lot of the added complexity, it is clearly suboptimal in the context of adapting
to the underlying geometry of the parameter space. Thus we would ideally need a mechanism to
robustly learn this critical mass hyperparameter from the data without significantly adding to the
computational burden.
We address this issue in this work with the Monte Carlo EM (MCEM) [10, 11, 12, 13] framework. An
alternative to the venerable EM technique, MCEM is used to locally optimize maximum likelihood
problems where the posterior probabilities required in the E step of EM cannot be computed in closed
form. In this work, we perform existing dynamics derived from energy functions in the Monte Carlo
E step while holding the mass fixed, and use the stored samples of the momentum term to learn the
mass in the M step. We address the important issue of selecting appropriate E-step sampling iterations,
using error estimates to gradually increase the sample sizes as the Markov chain progresses towards
convergence. Combined with an online method to update the mass using sample covariance estimates
in the M step, this gives a clean and scalable adaptive sampling algorithm that performs favorably
compared to the Riemannian samplers. In both our synthetic experiments and a high dimensional
topic modeling problem with a complex Bayesian nonparametric construction [14], our samplers
match or beat the Riemannian variants in sampling efficiency and accuracy, while being close to an
order of magnitude faster.
2
2.1
Preliminaries
MCMC with Energy-Preserving Dynamics
In Hamiltonian Monte Carlo, the energy function is written as
1
H(?, p) = ?L(?) + pT M ?1 p.
(1)
2
Here X is the observed data, and ? denotes the model parameters. L(?) = log p(X|?) + log p(?)
denotes the log likelihood of the data given the parameters along with the Bayesian prior, and
p denotes the auxiliary ?momentum? mentioned above. Note that the second term in the energy
function, the kinetic energy, is simply the kernel of a Gaussian with the mass matrix M acting as
covariance. Hamilton?s equations of motions are then applied to this energy function to derive the
following differential equations, with the dot accent denoting a time derivative:
?? = M ?1 p, p? = ?L(?).
These are discretized using the generalized leapfrog algorithm [1, 15] to create a sampler that is both
symplectic and time-reversible, upto a discretization error that is quadratic in the stepsize.
Machine learning applications typically see the use of very large datasets for which computing the
gradients of the likelihoods in every leapfrog step followed by a Metropolis-Hastings correction
ratio is prohibitively expensive. To address this, one uses random ?minibatches? of the dataset
in each iteration [16], allowing some stochastic noise for improved scalability, and removes the
Metropolis-Hastings (M-H) correction steps [4, 7]. To preserve the system energy in this context one
has to additionally apply Fokker-Planck corrections to the dynamics [17]. The stochastic sampler in
[4] uses these techniques to preserve the canonical Gibbs energy above (1). Researchers have also
used the notion of ?thermostats? from the molecular dynamics literature [9, 18, 19, 20] to further
control the behavior of the momentum terms in the face of stochastic noise; the resulting algorithm
[5] preserves an energy of its own [21] as well.
2.2
Adaptive MCMC using Riemannian Manifolds
As mentioned above, learning the mass matrices in these MCMC systems is an important challenge.
Researchers have traditionally used Riemannian manifold refomulations to address this, and integrate
2
the updating of the mass into the sampling steps. In [3] the authors use this approach to derive
adaptive variants of first-order Langevin dynamics as well as HMC. For the latter the reformulated
energy function can be written as:
1
1
Hgc (?, p) = ?L(?) + pT G(?)?1 p + log (2?)D |G(?)| ,
(2)
2
2
where D is the dimensionality of the parameter space. Note that the momentum variable p can
be integrated out to recover the desired marginal density of ?, in spite of the covariance being a
function of ?. In the machine learning literature, the authors of [8] used a diagonal G(?) to produce
an adaptive variant of the algorithm in [7], whereas the authors in [6] derived deterministic and
stochastic algorithms from a Riemannian variant of the Nos?-Poincar? energy [9], with the resulting
adaptive samplers preserving symplecticness as well as canonical system temperature.
2.3
Monte Carlo EM
The EM algorithm [22] is widely used to learn maximum likelihood parameter estimates for complex
probabilistic models. In cases where the expectations of the likelihoods required in the E step are
not tractable, one can use Monte Carlo simulations of the posterior instead. The resulting Monte
Carlo EM (MCEM) framework [10] has been widely studied in the statistics literature, with various
techniques developed to efficiently draw samples and estimate Monte Carlo errors in the E step
[11, 12, 13]. For instance, the expected log-likelihood is usually replaced with the following Monte
m
P
1
Carlo approximation: Q(?|? t ) = m
log p(X, utl |?), where u represents the latent augmentation
l=1
variables used in EM, and m is the number of samples taken in the E step. While applying this
framework, one typically has to carefully tune the number of samples gathered in the E step, since
the potential distance from the stationary distribution in the early phases would necessitate drawing
relatively fewer samples, and progressively more as the sampler nears convergence.
In this work we leverage this MCEM framework to learn M in (1) and similar energies using samples
of p; the discretized dynamics constitute the E step of the MCEM framework, with suitable updates
to M performed in the corresponding M step. We also use a novel mechanism to dynamically adjust
the sample count by using sampling errors estimated from the gathered samples, as described next.
3
3.1
Mass-Adaptive Sampling with Monte Carlo EM
The Basic Framework
Riemannian samplers start off by reformulating the energy function, making the mass a function
of ? and adding suitable terms to ensure constancy of the marginal distributions. Our approach is
fundamentally different: we cast the task of learning the mass as a maximum likelihood problem
over the space of symmetric positive definite matrices. For instance, we can construct the following
problem for standard HMC:
1
1
max L(?) ? pT M ?1 p ? log |M |.
(3)
M 0
2
2
Recall that the joint likelihood is p(?, p) ? exp(?H(?, p)), H(?, ?) being the energy from (1). Then,
we use correct samplers that preserve the desired densities in the E step of a Monte Carlo EM
(MCEM) framework, and use the obtained samples of p in the corresponding M step to perform
suitable updates for the mass M . Specifically, to wrap the standard HMC sampler in our framework,
we perform the generalized leapfrog steps [1, 15] to obtain proposal updates for ?, p followed by
Metropolis-Hastings corrections in the E step, and use the obtained p values in the M step. The
resultant adaptive sampling method is shown in Alg. 1.
Note that this framework can also be applied to stochastic samplers that preserve the energy, upto
standard discretization errors. We can wrap the SGHMC sampler [4] in our framework as well,
since it uses Fokker-Planck corrections to approximately preserve the energy (1) in the presence
of stochastic noise. We call the resulting method SGHMC-EM, and specify it in Alg. 3 in the
supplementary.
As another example, the SGNHT sampler [5] is known to preserve a modified Gibbs energy [21];
therefore we can propose the following max-likelihood problem for learning the mass:
1
1
? 2 /2,
max L(?) ? pT M ?1 p ? log |M | + ?(? ? ?)
(4)
M 0
2
2
3
where ? is the thermostat variable, and ?, ?? are constants chosen to preserve correct marginals. The
SGNHT dynamics can used in the E step to maintain the above energy, and we can use the collected
p samples in the M step as before. We call the resultant method SGNHT-EM, as shown in Alg. 2.
Note that, unlike standard HMC above, we do not perform Metropolis-Hastings corrections steps on
the gathered samples for these cases. As shown in the algorithms, we collect one set of momenta
samples per epoch, after the leapfrog iterations. We use S_count to denote the number of such
samples collected before running an M-step update.
The advantage of this MCEM approach over the parameter-dependent Riemannian variants is twofold:
1. The existing Riemannian adaptive algorithms in the literature [3, 6, 8] all start by modifying the
energy function, whereas our framework does not have any such requirement. As long as one uses a
sampling mechanism that preserves some energy with correct marginals for ?, in a stochastic sense
or otherwise, it can be used in the E step of our framework.
2. The primary disadvantage of the Riemannian algorithms is the added complexity in the dynamics
derived from the modified energy functions. One typically ends up using generalized leapfrog
dynamics [3, 6], which can lead to implicit systems of equations; to solve these one either has to use
standard solvers that have complexity at least cubic in the dimensionality [23, 24], with scalability
issues in high dimensional datasets, or use fixed point updates with worsened error guarantees. An
alternative approach is to use diagonal covariance matrices, as mentioned earlier, which ignores the
coordinate correlations. Our MCEM approach sidesteps all these issues by keeping the existing
dynamics of the desired E step sampler unchanged. As shown in the experiments, we can match or
beat the Riemannian samplers in accuracy and efficiency by using suitable sample sizes and M step
updates, with significantly improved sampling complexities and runtimes.
3.2
Dynamic Updates for the E-step Sample Size
We now turn our attention to the task of
learning the sample size in the E step from
the data. The nontriviality of this issue
is due to the following reasons: first, we
cannot let the sampling dynamics run to
convergence in each E step without making
the whole process prohibitively slow; second, we have to account for the correlation
among successive samples, especially early
on in the process when the Markov chain is
far from convergence, possibly with ?thinning? techniques; and third, we may want
to increase the sample count as the chain
matures and gets closer to the stationary
distribution, and use relatively fewer samples early on.
Algorithm 1 HMC-EM
Input: ? (0) , , LP _S, S_count
? Initialize M ;
repeat
? Sample p(t) ? N (0, M );
for i = 1 to LP _S do
? p(i) ? p(i+?1) , ? (i) ? ? (i+?1) ;
? p(i+/2) ? p(i) ? 2 ?? H(? (i) , p(i) );
? ? (i+) ? ? (i) + 2 ?p H(? (i) , p(i+/2) );
? p(i+) ? p(i+/2) ? 2 ?? H(? (i+) , p(i+/2) );
end for
? Set ? (t+1) , p(t+1) from ? LP _S+ , pLP _S+
using Metropolis-Hastings
? Store MC-EM sample p(t+1) ;
if (t + 1) mod S_count = 0 then
To this end, we leverage techniques de? Update M using MC-EM samples;
rived from the MCEM literature in statisend if
tics [11, 13, 25] to first evaluate a suitable
? Update S_count as described in the text;
?test? function of the target parameters at
until forever
certain subsampled steps, using the gathered samples and current M step estimates.
We then use confidence intervals created around these evaluations to gauge the relative effect of
successive MCEM estimates over the Monte Carlo error. If the updated values of these functions
using newer M-step estimates lie in these intervals, we increase the number of samples collected in
the next MCEM loop.
Specifically,
similar to [13],
we start off with the following test function for HMC-EM (Alg. 1):
q(?) = M ?1 p, ?L(?) . We then subsample some timesteps as mentioned below, evaluate q at
those steps, and create confidence intervals using sample means and variances:
mS =
S
1X
q ,
S s=1 s
vS =
S
1X 2
q ? m2S ,
S s=1 s
4
CS := mS ? z1??/2 vS ,
where S denotes the subsample count, z1??/2 is the (1 ? ?) critical value of a standard Gaussian, and
CS the confidence
(Alg. 2), we use the following test
interval mentioned earlier. For SGNHT-EM
function: q(?) = M ?1 p, ?L(?) + ?M ?1 p, pT M ?1 p , derived from the SGNHT dynamics.
One can adopt the following method described in [25]: choose the
Pssubsampling
(0)
offsets
{t
.
.
.
t
}
as
t
=
1
S
s
Input: ? , , A, LP _S, S_count
i=1 xi , where
xi ? 1 ? Poisson(?id ), with suitably cho? Initialize ? (0) , p(0) and M ;
sen ? ? 1 and d > 0. We found both this
repeat
and a fixed set of S offsets to work well in
for i = 1 to LP _S do
(i+1)
(i)
?
? p?
? p(i) ? ? (i) M ?1 p(i) ? ?L(?
)+ our experiments.
2AN (0, );
With the subsamples collected using this
? ? (i+1) ? ? (i) + M ?1 p(i+1) ;
mechanism, we calculate the confidence
1 (i+1)T
? ? (i+1) ? ? (i) + D
p
M ?1 p(i+1) ? 1 ; intervals as described earlier. The assumpend for
tion is that this interval provides an esti
? Set ? (t+1) , p(t+1) , ? (t+1) =
mate of the spread of q due to the Monte
Carlo error. We then perform the M-step,
? (LP _S+1) , p(LP _S+1) , ? (LP _S+1) ;
and evaluate q using the updated M-step es(t+1)
? Store MC-EM sample p
;
timates. If this value lies in the previously
if (t + 1) mod S_count = 0 then
calculated confidence bound, we increase
? Update M using MC-EM samples;
S as S = S + S/SI in the following iterend if
ation to overcome the Monte Carlo noise.
? Update S_count as described in the text;
See [11, 13] for details on these procedures.
until forever
Values for the constants ?, ?, d, SI , as well
as initial estimates for S are given in the
supplementary. Running values for S are denoted S_count hereafter.
Algorithm 2 SGNHT-EM
3.3
An Online Update for the M-Step
Next we turn our attention to the task of updating the mass matrices using the collected momenta
samples. As shown in the energy functions above, the momenta are sampled from zero-mean
normal distributions, enabling us to use standard covariance estimation techniques from the literature.
However, since we are using discretized MCMC to obtain these samples, we have to address the
variance arising from the Monte Carlo error, especially during the burn-in phase. To that end, we
found a running average of the updates to work well in our experiments; in particular, we updated the
inverse mass matrix, denoted as MI , at the k th M-step as:
(k)
MI
(k?1)
= (1 ? ?(k) )MI
(k,est)
+ ?(k) MI
,
(5)
(k,est)
where
is a suitable estimate computed from the gathered samples in the k th M-step, and
(k) MI
?
is a step sequence satisfying some standard assumptions, as described below. Note that the
MI s correspond to the precision matrix of the Gaussian distribution of the momenta; updating this
during the M-step also removes the need to invert the mass matrices during the leapfrog iterations.
(k,est)
Curiously, we found the inverse of the empirical covariance matrix to work quite well as MI
in
our experiments.
These updates also induce a fresh perspective on the convergence of the overall MCEM procedure.
Existing convergence analyses in the statistics literature fall into three broad categories: a) the almost
sure convergence presented in [26] as t ? ? with increasing sample sizes, b) the asymptotic angle
presented in [27], where the sequence of MCEM updates are analyzed as an approximation to the
standard EM sequence as the sample size, referred to as S_count above, tends to infinity, and c)
the asymptotic consistency results obtained from multiple Gibbs chains in [28], by letting the chain
counts and iterations tend to ?. Our analysis differs from all of these, by focusing on the maximum
likelihood situations noted above as convex optimization problems, and using SGD convergence
(k)
techniques [29] for the sequence of iterates MI .
(k,est)
Proposition 1. Assume the MI
?s provide an unbiased estimate of ?J, and have
bounded
eigenvalues. Let inf kMI ?MI? k2 > ?J(MI ) > 0 ? > 0. Further, let the sequence ?(k) satisfy
n
o
2
P (k)
P
(k)
= ?, k ?(k) < ?. Then the sequence MI
converges to the MLE of the precision
k?
almost surely.
5
Recall that the (negative) precision is a natural parameter of the normal distribution written in
exponential family notation, and that the log-likelihood is a concave function of the natural parameters
for this family; this makes max-likelihood a convex optimization problem over the precision, even in
the presence of linear constraints [30, 31]. Therefore, this implies that the problems (3), (4) have a
unique maximum, denoted by MI? above. Also note that the update (5) corresponds to a first order
update on the iterates with an L2-regularized objective, with unit regularization parameter; this is
denoted by J(MI ) in the proposition. That is, J is the energy preserved by our sampler(s), as a
function of the mass (precision), augmented with an L2 regularization term. The resultant strongly
convex optimization problem can be analyzed using SGD techniques under the assumptions noted
above; we provide a proof in the supplementary for completeness.
We should note here that the ?stochasticity? in the proof does not refer to the stochastic gradients
of L(?) used in the leapfrog dynamics of Algorithms 2 through 5; instead we think of the collected
momenta samples as a stochastic minibatch used to compute the gradient of the regularized energy,
as a function of the covariance (mass), allowing us to deal with the Monte Carlo error indirectly. Also
(k,est)
note that our assumption on the unbiasedness of the MI
estimates is similar to [26], and distinct
from assuming that the MCEM samples of ? are unbiased; indeed, it would be difficult to make this
latter claim, since stochastic samplers in general are known to have a convergent bias.
3.4
Nos?-Poincar? Variants
We next develop a stochastic version of the dynamics derived from the Nos?-Poincar? Hamiltonian,
followed by an MCEM variant. This allows for a direct comparison of the Riemann manifold formulation and our MCEM framework for learning the kinetic masses, in a stochastic setting with thermostat
controls on the momentum terms and desired properties like reversibility and symplecticness provided
by generalized leapfrog discretizations. The Nos?-Poincar? energy function can be written as [6, 9]:
1 p ?1 p
q2
HN P = s ?L(?) +
+ gkT log s ? H0 ,
M
+
(6)
2 s
s
2Q
where L(?) is the joint log-likelihood, s is the thermostat control, p and q the momentum terms
corresponding to ? and s respectively, and M and Q the respective mass terms. See [6, 9] for
descriptions of the other constants. Our goal is to learn both M and Q using the MCEM framework,
as opposed to [6], where both were formulated in terms of ?. To that end, we propose the following
system of equations for the stochastic scenario:
t+/2 2
?
B(?)
A(?)s t+/2
pt+ /2 = p +
s?L(?) ? ? M ?1 pt+ /2 ,
(q
) + 1+
q
2
4Q
2Q
s
t+/2
1 pt+/2
p
?
?gkT (1 + log s) +
M ?1
+ L(?)
+ H0 = 0,
? q+
2
2
s
s
t+/2
1
1
q
t+/2
t+
?1
t+
s+s
+
, ?
= ? + M p
,
s
=s+
Q
s st+
t+ ?
B(? t+ ) ?1 t+/2
t+
t+/2
t+
t+
t+/2
? t+ )
, q
p
=p
+
s ?L(? ) ? ?
M p
=q
+ H0 + L(?
2
2
st+
2
t+/2
q t+/2
1 pt+/2
p
A(?)st+ t+/2
?1
t+
? gkT (1 + log s ) +
M
?
q
?
,
2 st+
st+
2Q
2Q
(7)
where t + /2 denotes the half-step dynamics, ? signifies noisy stochastic estimates, and A(?) and
B(?) denote the stochastic noise terms, necessary for the Fokker-Planck corrections [6]. Note that
we only have to solve a quadratic equation for q t+/2 with the other updates also being closed-form,
as opposed to the implicit system of equations in [6].
Proposition 2. The dynamics (7) preserve the Nos?-Poincar? energy (6).
The proof is a straightforward application of the Fokker-Planck corrections for stochastic noise
to the Hamiltonian dynamics derived from (6), and is provided in the supplementary. With these
dynamics, we first develop the SG-NPHMC algorithm (Alg. 4 in the supplementary) as a counterpart
to SGHMC and SGNHT, and wrap it in our MCEM framework to create SG-NPHMC-EM (Alg. 5
in the supplementary). As we shall demonstrate shortly, this EM variant performs comparably to
SGR-NPHMC from [6], while being significantly faster.
6
4
Experiments
In this section we compare the performance of the MCEM-augmented variants of HMC, SGHMC
as well as SGNHT with their standard counterparts, where the mass matrices are set to the identity
matrix. We call these augmented versions HMC-EM, SGHMC-EM, and SGNHT-EM respectively.
As baselines for the synthetic experiments, in addition to the standard samplers mentioned above, we
also evaluate RHMC [3] and SGR-NPHMC [6], two recent algorithms based on dynamic Riemann
manifold formulations for learning the mass matrices. In the topic modeling experiment, for scalability
reasons we evaluate only the stochastic algorithms, including the recently proposed SGR-NPHMC,
and omit HMC, HMC-EM and RHMC. Since we restrict the discussions in this paper to samplers
with second-order dynamics, we do not compare our methods with SGLD [7] or SGRLD [8].
4.1
Parameter Estimation of a 1D Standard Normal Distribution
In this experiment we aim to learn the parameters of a unidimensional standard normal distribution in
both batch and stochastic settings, using 5, 000 data points generated from N (0, 1), analyzing the
impact of our MC-EM framework on the way. We compare all the algorithms mentioned so far: HMC,
HMC-EM, SGHMC, SGHMC-EM, SGNHT, SGNHT-EM, SG-NPHMC, SG-NPHMC-EM along
with RHMC and SGR-NPHMC. The generative model consists of normal-Wishart priors on the mean
? and precision ? , with posterior distribution p(?, ? |X) ? N (X|?, ? )W(? |1, 1), where W denotes
the Wishart distribution. We run all the algorithms for the same number of iterations, discarding
the first 5, 000 as ?burn-in?. Batch sizes were fixed to 100 for all the stochastic algorithms, along
with 10 leapfrog iterations across the board. For SGR-NPHMC and RHMC, we used the observed
Fisher information plus the negative Hessian of the prior as the tensor, with one fixed point iteration
on the implicit system of equations arising from the dynamics of both. For HMC we used a fairly
high learning rate of 1e ? 2. For SGHMC and SGNHT we used A = 10 and A = 1 respectively. For
SGR-NPHMC we used A, B = 0.01.
We show the RMSE numbers colM ETHOD
RMSE (?) RMSE (? )
T IME
lected from post-burn-in samples as
HMC
0.0196
0.0197
0.417 MS
well as per-iteration runtimes in TaHMC-EM
0.0115
0.0104
0.423 MS
ble 1. An ?iteration? here refers to a
RHMC
0.0111
0.0089
5.748 MS
complete E step, with the full quota
SGHMC
0.1590
0.1646
0.133 MS
SGHMC-EM
0.0713
0.2243
0.132 MS
of leapfrog jumps. The improvements
SG-NPHMC
0.0326
0.0433
0.514 MS
afforded by our MCEM framework
SG-NPHMC-EM
0.0274
0.0354
0.498 MS
are immediately noticeable; HMCSGR-NPHMC
0.0240
0.0308
3.145 MS
EM matches the errors obtained from
SGNHT
0.0344
0.0335
0.148 MS
RHMC, in effect matching the samSGNHT-EM
0.0317
0.0289
0.148 MS
ple distribution, while being much
faster (an order of magnitude) per iteration. The stochastic MCEM algo- Table 1: RMSE of the sampled means, precisions and perrithms show markedly better perfor- iteration runtimes (in milliseconds) from runs on synthetic
mance as well; SGNHT-EM in partic- Gaussian data.
ular beats SGR-NPHMC in RMSE-? while being significantly faster due to simpler updates for the
mass matrices. Accuracy improvements are particularly noticeable for the high learning rate regimes
for HMC, SGHMC and SG-NPHMC.
4.2
Parameter Estimation in 2D Bayesian Logistic Regression
Next we present some results obtained from a Bayesian logistic regression experiment, using both
synthetic and real datasets. For the synthetic case, we used the same methodology as [6]; we generated
2, 000 observations from a mixture of two normal distributions with means at [1, ?1] and [?1, 1], with
mixing weights set to (0.5, 0.5) and the covariance set to I. We then classify these points using a linear
classifier with weights {W0 , W1 } = [1, ?1], and attempt to learn these weights using our samplers.
We put N (0, 10I) priors on the weights, and used the metric tensor described in ?7 of [3] for the
Riemannian samplers. In the (generalized) leapfrog steps of the Riemannian samplers, we opted to
use 2 or 3 fixed point iterations to approximate the solutions to the implicit equations. Along with this
synthetic setup, we also fit a Bayesian LR model to the Australian Credit and Heart regression datasets
from the UCI database, for additional runtime comparisons. The Australian credit dataset contains
690 datapoints of dimensionality 14, and the Heart dataset has 270 13-dimensional datapoints.
7
For the synthetic case, we discard the first
10, 000 samples as burn-in, and calculate RMSE
values from the remaining samples. Learning
rates were chosen from {1e ? 2, 1e ? 4, 1e ? 6},
and values of the stochastic noise terms were selected from {0.001, 0.01, 0.1, 1, 10}. Leapfrog
steps were chosen from {10, 20, 30}. For the
stochastic algorithms we used a batchsize of
100.
M ETHOD
HMC
HMC-EM
RHMC
SGHMC
SGHMC-EM
SG-NPHMC
SG-NPHMC-EM
SGR-NPHMC
SGNHT
SGNHT-EM
RMSE (W0 )
0.0456
0.0145
0.0091
0.2812
0.2804
0.4945
0.0990
0.1901
0.2035
0.1983
RMSE (W1 )
0.1290
0.0851
0.0574
0.2717
0.2583
0.4263
0.4229
0.1925
0.1921
0.1729
The RMSE numbers for the synthetic dataset
are shown in Table 2, and the per-iteration runtimes for all the datasets are shown in Table 3. Table 2: RMSE of the two regression parameters,
We used initialized S_count to 300 for HMC- for the synthetic Bayesian logistic regression exEM, SGHMC-EM, and SGNHT-EM, and 200 periment. See text for details.
for SG-NPHMC-EM. The MCEM framework
noticeably improves the accuracy in almost all cases, with no computational overhead. Note the
improvement for SG-NPHMC in terms of RMSE for W0 . For the runtime calculations, we set all
samplers to 10 leapfrog steps, and fixed S_count to the values mentioned above.
The comparisons with the
M ETHOD
T IME ( SYNTH ) T IME (AUS ) T IME (H EART )
Riemannian algorithms tell
HMC
1.435 MS
0.987 MS
0.791 MS
a clear story: though we
HMC-EM
1.428 MS
0.970 MS
0.799 MS
do get somewhat better
RHMC
1550 MS
367 MS
209 MS
accuracy with these samSGHMC
0.200 MS
0.136 MS
0.112 MS
SGHMC-EM
0.203 MS
0.141 MS
0.131 MS
plers, they are orders of
SG-NPHMC
0.731 MS
0.512 MS
0.403 MS
magnitude slower.
In
SG-NPHMC-EM
0.803 MS
0.525 MS
0.426 MS
our synthetic case, for inSGR-NPHMC
6.720
MS
4.568
MS
3.676 MS
stance, each iteration of
SGNHT
0.302 MS
0.270 MS
0.166 MS
RHMC (consisting of all
SGNHT-EM
0.306 MS
0.251 MS
0.175 MS
the leapfrog steps and the
M-H ratio calculation) takes
more than a second, us- Table 3: Per-iteration runtimes (in milliseconds) for Bayesian logistic
ing 10 leapfrog steps and 2 regression experiments, on both synthetic and real datasets.
fixed point iterations for the
implicit leapfrog equations, whereas both HMC and HMC-EM are simpler and much faster. Also
note that the M-step calculations for our MCEM framework involve a single-step closed form update
for the precision matrix, using the collected samples of p once every S_count sampling steps; thus
we can amortize the cost of the M-step over the previous S_count iterations, leading to negligible
changes to the per-sample runtimes.
4.3
Topic Modeling using a Nonparametric Gamma Process Construction
Next we turn our attention to a high-dimensional topic modeling experiment using a nonparametric
Gamma process construction. We elect to follow the experimental setup described in [6]. Specifically,
we use the Poisson factor analysis framework of [32]. Denoting the vocabulary as V , and the
documents in the corpus as D, we model the observed counts of the vocabulary terms as DV ?N =
Poi(??), where ?K?N models the counts of K latent topics in the documents, and ?V ?K denotes
the factor load matrix, that encodes the relative importance of the vocabulary terms in the latent topics.
Following standard Bayesian convention, we put model the columns of ? as ??,k ? Dirichlet(?),
using normalized Gamma variables: ?v,k = P?v?v , with ?v ? ?(?, 1). Then we have ?n,k ?
v
pj
?(rk , 1?p
); we put ?(a0 , b0 ) priors on the document-specific mixing probabilities pj . We then set
j
the rk s to the atom weights generated by the constructive Gamma process definition of [14]; we refer
the reader to that paper for the details of the formulation. It leads to a rich nonparametric construction
of this Poisson factor analysis model for which closed-form Gibbs updates are infeasible, thereby
providing a testing application area for the stochastic MCMC algorithms. We omit the Metropolis
Hastings correction-based HMC and RHMC samplers in this evaluation due to poor scalability.
8
(a)
(b)
Figure 1: Test perplexities plotted against (a) post-burnin iterations and (b) wall-clock time for the
20-Newsgroups dataset. See text for experimental details.
We use count matrices from the 20-Newsgroups and Reuters Corpus Volume 1 corpora [33]. The
former has 2, 000 words and 18, 845 documents, while the second has a vocabulary of size 10, 000
over 804, 414 documents. We used a chronological 60?40 train-test split for both datasets. Following
standard convention for stochastic algorithms, following each minibatch we learn document-specific
parameters from 80% of the test set, and calculate test perplexities on the remaining 20%. Test
perplexity, a commonly used measure for such evaluations, is detailed in the supplementary.
As noted in [14], the atom weights have three sets of components: the Ek s, Tk s and the hyperparameters ?, ? and c. As in [6], we ran three parallel chains for these parameters, collecting samples of the
momenta from the Tk and hyperparameter chains for the MCEM mass updates. We kept the mass of
the Ek chain fixed to IK , and chose K = 100 as number of latent topics. We initialized S_count,
the E-step sample size in our algorithms, to 50 for NPHMC-EM and 100 for the rest. Increasing
S_count over time yielded fairly minor improvements, hence we kept it fixed to the values above
for simplicity. Additional details on batch sizes, learning rates, stochastic noise estimates, leapfrog
iterations etc are provided in the supplementary. For the 20-Newsgroups dataset we ran all algorithms
for 1, 500 burn-in iterations, and collected samples for the next 1, 500 steps thereafter, with a stride
of 100, for perplexity calculations. For the Reuters dataset we used 2, 500 burn-in iterations. Note
that for all these algorithms, an ?iteration? corresponds to a full E-step with a stochastic minibatch.
The numbers obtained at the end
M ETHOD
20-N EWS R EUTERS T IME (20-N EWS )
of the runs are shown in Table 2,
SGHMC
759
996
0.047 S
along with per-iteration runtimes.
SGHMC-EM
738
972
0.047 S
The post-burnin perplexity-vsSGNHT
757
979
0.045 S
SGNHT-EM
719
968
0.045 S
iteration plots from the 20SGR-NPHMC
723
952
0.410 S
Newsgroups dataset are shown in
SG-NPHMC
714
958
0.049 S
Figure 1. We can see significant
SG-NPHMC-EM
712
947
0.049 S
improvements from the MCEM
framework for all samplers, with
that of SGNHT being highly pro- Table 4: Test perplexities and per-iteration runtimes on 20nounced (719 vs 757); indeed, Newsgroups and Reuters datasets.
the SG-NPHMC samplers have
lower perplexities (712) than those obtained by SGR-NPHMC (723), while being close to an order
of magnitude faster per iteration for 20-Newsgroups even when the latter used diagonalized metric
tensors, ostensibly by avoiding implicit systems of equations in the leapfrog steps to learn the kinetic
masses. The framework yields nontrivial improvements for the Reuters dataset as well.
5
Conclusion
We propose a new theoretically grounded approach to learning the mass matrices in Hamiltonianbased samplers, including both standard HMC and stochastic variants, using a Monte Carlo EM
framework. In addition to a newly proposed stochastic sampler, we augment certain existing samplers
with this technique to devise a set of new algorithms that learn the kinetic masses dynamically from
the data in a flexible and scalable fashion. Experiments conducted on synthetic and real datasets
demonstrate the efficacy and efficiency of our framework, when compared to existing Riemannian
manifold-based samplers.
9
Acknowledgments
We thank the anonymous reviewers for their insightful comments and suggestions. This material is
based upon work supported by the National Science Foundation under Grant No. DMS-1418265.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of
the author(s) and do not necessarily reflect the views of the National Science Foundation.
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6,330 | 6,726 | ADMM without a Fixed Penalty Parameter:
Faster Convergence with New Adaptive Penalization
Yi Xu? , Mingrui Liu? , Qihang Lin? , Tianbao Yang?
Department of Computer Science, The University of Iowa, Iowa City, IA 52242, USA
?
Department of Management Sciences, The University of Iowa, Iowa City, IA 52242, USA
{yi-xu, mingrui-liu, qihang-lin, tianbao-yang}@uiowa.edu
?
Abstract
Alternating direction method of multipliers (ADMM) has received tremendous
interest for solving numerous problems in machine learning, statistics and signal
processing. However, it is known that the performance of ADMM and many of its
variants is very sensitive to the penalty parameter of a quadratic penalty applied
to the equality constraints. Although several approaches have been proposed for
dynamically changing this parameter during the course of optimization, they do not
yield theoretical improvement in the convergence rate and are not directly applicable to stochastic ADMM. In this paper, we develop a new ADMM and its linearized
variant with a new adaptive scheme to update the penalty parameter. Our methods
can be applied under both deterministic and stochastic optimization settings for
structured non-smooth objective function. The novelty of the proposed scheme lies
at that it is adaptive to a local sharpness property of the objective function, which
marks the key difference from previous adaptive scheme that adjusts the penalty
parameter per-iteration based on certain conditions on iterates. On theoretical side,
given the local sharpness characterized by an exponent ? ? (0, 1], we show that the
1?? 1
e
) in the
proposed ADMM enjoys an improved iteration complexity of O(1/
2(1??)
e
deterministic setting and an iteration complexity of O(1/
) in the stochastic
setting without smoothness and strong convexity assumptions. The complexity in
either setting improves that of the standard ADMM which only uses a fixed penalty
parameter. On the practical side, we demonstrate that the proposed algorithms
converge comparably to, if not much faster than, ADMM with a fine-tuned fixed
penalty parameter.
1
Introduction
Our problem of interest is the following convex optimization problem that commonly arises in
machine learning, statistics and signal processing:
min
x??
F (x) , f (x) + ?(Ax)
(1)
where ? ? Rd is a closed convex set, f : Rd ? R and ? : Rm ? R are proper lower-semicontinuous
convex functions, and A ? Rm?d is a matrix. In this paper, we consider solving (1) by alternating
direction method of multipliers (ADMM) in two paradigms, namely deterministic optimization and
stochastic optimization. In both paradigms, ADMM has been employed widely for solving the
regularized statistical learning problems like (1) due to its capability of tackling the sophisticated
structured regularization term ?(Ax) in (1) (e.g., the generalized lasso kAxk1 ), which is often an
1
e suppresses a logarithmic factor.
O()
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
obstacle for applying other methods such as proximal gradient method. As follows, we describe
the standard ADMM and its variants for solving (1) in different optimization paradigms. It is worth
mentioning that all algorithms presented in this paper can be easily extended to handle a more general
term ?(A(x) + c), where A is a linear mapping.
To apply ADMM, the original problem (1) is first cast into an equivalent constrained optimization
problem via decoupling:
min
x??,y?Rm
f (x) + ?(y),
s.t. y = Ax.
(2)
An augmented Lagrangian function for (2) is defined as
?
kAx ? yk22 ,
(3)
2
where ? is a constant called penalty parameter and ? ? Rm is a dual variable. Then, the standard
ADMM solves problem (1) by executing the following three steps in each iteration:
2
?
1
x? +1 = arg min L(x, y? , ?? ) = arg min f (x) +
(Ax ? y? ) ? ??
,
(4)
x??
x??
2
?
2
2
1
?
,
(Ax
?
y)
?
(5)
?
y? +1 = arg min L(x? +1 , y, ?? ) = arg minm ?(y) +
? +1
?
x??
y?R
2
?
2
?? +1 = ?? ? ?(Ax? +1 ? y? +1 ).
(6)
L(x, y, ?) = f (x) + ?(y) ? ?> (Ax ? y) +
When A is not an identity matrix, solving the subproblem (4) above for x? +1 might be difficult. To
alleviate the issue, linearized ADMM [33, 34, 8] has been proposed, which solves the following
problem instead of (4):
2
1
?
1
+ kx ? x? k2G ,
x? +1 = arg min f (x) +
(Ax ? y? ) ? ??
(7)
x??
2
?
2
2
?
where kxkG = x> Gx and G ? Rd?d is a positive semi-definite matrix. By setting G =
?I ? ?A> A 0, the term x> A> Ax in (7) vanishes. It has been established that both standard
ADMM and linearized ADMM have an O(1/t) convergence rate for solving (2) [8] , where t is the
number of iterations. Under a minor condition, this result implies an O(1/) iteration complexity for
solving the original problem (1) (see Corollary 1).
In addition, we consider ADMM for solving (1) in stochastic optimization with
f (x) = E? [f (x; ?)]
(8)
where ? is a random variable. This formulation captures many risk minimization problems in
machine learning where ? denotes a data point sampled from a distribution and f (x; ?) denotes a
loss function
the model x on the data ?. It also covers as a special case the empirical loss where
Pof
n
f (x) = n1 i=1 f (x; ?i ) with n is the number of samples. For these problems, computing f (x)
itself might be prohibitive (e.g., when n is very large) or even impossible. To address this issue,
one usually considers the stochastic optimization paradigm, where it is assumed that f (x; ?) and
its subgradient ?f (x; ?) can be efficiently computed. To solve the stochastic optimization problem,
stochastic ADMM algorithms have been proposed [21, 23], which update y? +1 and ?? +1 the same
to (5) and (6), respectively, but update x? +1 as
2
kx ? x? k2G?
?
1
x? +1 = arg min f (x? ; ?? )+?f (x? ; ?? )> (x?x? )+
(Ax
?
y
)
?
?
(9)
?
?
+
x??
2
?
??
2
where ?? is a random sample, ?? is a stepsize and G? = ?I ? ??? A> A I [23] or G? = I [21].
Other stochastic variants of ADMM for
? general convex optimization were also proposed in [23, 35].
These work have established an O(1/ t) convergence rate of stochastic ADMM for solving (2) with
f (x) being (8). Under a minor condition, we can also show that these stochastic ADMM algorithms
suffer from a higher iteration complexity of O(1/2 ) for finding an -optimal solution to the original
problem (1) (see Corollary 3).
Although the variants of ADMM with fast convergence rates have been developed under smoothness,
strong convexity and other regularity conditions (e.g., the matrix A has full rank), the best iteration
2
complexities of deterministic ADMM and stochastic ADMM for general convex optimization remain
O(1/) and O(1/2 ), respectively. On the other hand, many studies have reported that the performance of ADMM is very sensitive to the penalty parameter ?. How to address or alleviate this issue
has attracted many studies and remains an active topic. In particular, it remains an open question
how to quantify the improvement in ADMM?s theoretical convergence by using adaptive penalty
parameters. Of course, the answer to this question depends on the adaptive scheme being used.
Almost all previous works focus on using self-adaptive schemes that update the penalty parameter
during the course of optimization according to the historical iterates (e.g., by balancing the primal
residual and dual residual). However, there is hitherto no quantifiable improvement in terms of
convergence rate or iteration complexity for these self-adaptive schemes.
In this paper, we focus on the design of adaptive penalization for both deterministic and stochastic
ADMM and show that, with the proposed adaptive updating scheme on the penalty parameter, the
theoretical convergence properties of ADMM can be improved without imposing any smoothness and
strong convexity assumptions on the objective function. The key difference between the proposed
adaptive scheme and previous self-adaptive schemes is that the proposed penalty parameter is adaptive
to an local sharpness property of the objective function, namely the local error bound (see Definition 1).
Given the exponent constant ? ? (0, 1] that characterizes this local sharpness property, we show that
1?? 2
e
the proposed deterministic ADMM enjoys an improved iteration complexity of O(1/
) and the
2(1??)
e
proposed stochastic ADMM enjoys an iteration complexity of O(1/
), both of which improve
the complexity of their standard counterparts which only use a fixed penalty parameter. To the best of
our knowledge, this is the first evidence that an adaptive penalty parameter used in ADMM can lead
to provably lower iteration complexities. We call the proposed ADMM algorithms locally adaptive
ADMM because of its adaptivity to the problem?s local property.
2
Related Work
Since there is a tremendous amount of studies on ADMM, the review below mainly focuses on the
ADMMs with a variable penalty parameter. A convergence rate of O(1/t) was first shown for both the
standard and linearized variants of ADMM [8, 19, 9] on general non-smooth and non-strongly convex
problems. Later, smoothness and strong convexity assumptions are introduced to develop faster
convergence rates of ADMMs
[22, 3, 11, 6]. Stochastic ADMM was considered in [21, 23] with a
?
e
convergence rate of O(1/ t) for general convex problems and O(1/t)
for strongly convex problems.
Recently, many variance reduction techniques have been borrowed into stochastic ADMMPto achieve
n
improved convergence rates for finite-sum optimization problems where f (x) = n1 i=1 fi (x)
under the smoothness and strong convexity assumptions [37, 36, 24]. Nevertheless, most of these
aforementioned works focus on using a constant penalty parameter.
He et al. [10] analyzed ADMM with self-adaptive penalty parameters. The motivation for their
self-adaptive penalty is to balance the order of the primal residual and the dual residual. However,
the convergence of ADMM with self-adaptive penalty is not guaranteed unless the adaptive scheme
is turned off after a number of iterations. Additionally, their self-adaptive rule requires computing
the deterministic subgradient of f (x) so that is not appropriate for stochastic optimization. Tian
& Yuan [25] proposed a variant of ADMM with variable penalty parameters. Their analysis and
algorithm require the smoothness assumption of ?(Ax) and full column rank of the A matrix. Zhou
et al. [15] focused on solving low-rank representation by linearized ADMM and also proposed a
non-decreasing self-adaptive penalty scheme. However, their scheme is only applicable to an equality
constraint Ax + By = c with c 6= 0. Recently, Xu et al. [31] proposed a self-adaptive penalty
scheme for ADMM based on the Barzilai and Borwein gradient methods. The convergence of their
ADMM relies on the analysis in He et al. [10] and thus requires the penalty parameter to be fixed after
a number of iterations. In contrast, our adaptive scheme fpr the penalty parameter is different from
the previous methods in the following aspects: (i) it is adaptive to the local sharpness property of the
problem; (ii) it allows the penalty parameter to increase to infinity as the algorithm proceeds; (iii) it
can be employed for both deterministic and stochastic ADMMs as well as their linearized versions.
It is also notable that the presented algorithms and their convergence theory share many similarities
with the recent developments leveraging the local error bound condition [32, 30, 29], where similar
iteration complexities have been established. However, we would like to emphasize that the newly
2
e suppresses a logarithmic factor.
O()
3
proposed ADMM algorithms are more effective to tackle problems with structured regularizers (e.g.,
generalized lasso) than the methods in [32, 30, 29], and have an additional unique feature of using
adaptive penalty parameter.
3
Preliminaries
Recall that the problem of our interest:
min F (x) , f (x) + ?(Ax),
(10)
x??
where ? ? Rd is a closed convex set, f : Rd ? (??, +?] and ? : Rm ? (??, +?] are proper
lower-semicontinuous convex functions, and A ? Rm?d is a matrix. Let ?? and F? denote the
optimal set of (10) and the optimal value, respectively. We present some assumptions that will be
used in the paper.
Assumption 1. For the convex optimization problem (10), we assume (a) there exist known x0 ? ?
and 0 ? 0 such that F (x0 ) ? F? ? 0 ; (b) ?? is a non-empty convex compact set; (c) there exists a
constant ? such that k??(y)k2 ? ? for all y; (d) ? is defined everywhere.
?
For a positive semi-definite matrix G, the G-norm is defined as kxkG = x> Gx. Let B(x, r) =
{u ? Rd : ku ? xk2 ? r} denote the Euclidean ball centered x with a radius r. We denote by
dist(x, ?? ) the distance between x and the set ?? , i.e., dist(x, ?? ) = minv??? kx ? vk2 . We
denote by S the -sublevel set of F (x), respectively, i.e., S = {x ? ? : F (x) ? F? + }.
Local Sharpness. Below, we introduce a condition, namely local error bound condition, to characterize the local sharpness property of the objective function.
Definition 1 (Local error bound (LEB)). A function F (x) is said to satisfy a local error bound
condition on the -sublevel set if there exist ? ? (0, 1] and c > 0 such that for any x ? S
dist(x, ?? ) ? c(F (x) ? F? )? .
(11)
Remark: We will refer to ? as the local sharpness parameter. A recent study [1] has shown that the
local error bound condition is equivalent to the famous Kurdyka - ?ojasiewicz (KL) property [13],
which characterizes that under a transformation of ?(s) = cs? , the function F (x) can be made sharp
around the optimal solutions, i.e, the norm of subgradient of the transformed function ?(F (x) ? F? )
is lowered bounded by a constant 1. Note that by allowing ? = 0 in the above condition we can
capture a full spectrum of functions. However, a broad family of functions can have a sharper upper
bound, i.e., with a non-zero constant ? in the above condition. For example, for functions that are
semi-algebraic and continuous, the above inequality is known to hold on any compact set (c.f. [1] and
references therein). The value of ? has been revealed for many functions (c.f. [18, 14, 20, 1, 32]).
4
Locally Adaptive ADMM for Deterministic Optimization
Since the proposed locally adaptive ADMM algorithm builds upon the standard ADMM, we first
present the detailed steps of ADMM in Algorithm 1. Note that if we set G = 0 ? Rd?d , it gives
the standard ADMM; and if we use G = ?I ? ?A> A 0, it gives the linearized variant, which
can make the computation of x? +1 easier. To ensure G 0, the minimum valid value for ? in the
linearized variant is ?kAk22 . To present the convergence result of ADMM (Algorithm 1), we first
introduce some notations.
?
?
!
x
?A> ?
?,
y , F(u) = ?
u=
?
?
Ax ? y
t
bt =
u
1X
u? ,
t ? =1
t
bt =
x
t
1X
x? ,
t ? =1
bt =
y
1X
y? ,
t ? =1
t
X
bt = 1
?
?? .
t ? =1
We recall the convergence result of [8] for the equality constrained problem (2), which does not
assume any smoothness, strong convexity and other regularity conditions.
4
Algorithm 1 ADMM(x0 , ?, t)
Algorithm 2 LA-ADMM (x0 , ?1 , K, t)
1: Input: x0 ? ?, the penalty parameter ?, the number 1: Input: x0 ? ?, the number of stages
K, and the number of iterations t per
of iterations t
stage, initial value of penalization pa2: Initialize: x1 = x0 , y1 = Ax1 , ?1 = 0, ? = ?kAk22
rameter ?1
and G = ?I ? ?A> A or G = 0.
2: for k = 1, . . . , K do
3: for ? = 1, . . . , t do
3:
Let xk = ADMM(xk?1 , ?k , t)
4:
Update x? +1 by (7), y? +1 by (5),
4:
Update ?k+1 = 2?k
5:
Update ?? +1 by (6)
5: end for
6: end for
P
6: Output: xK
bt = t? =1 x? /t
7: Output: x
Proposition 1 (Theorem 4.1 in [8]). For any x ? ?, y ? Rm and ? ? Rm , we have
f (b
xt ) + ?(b
yt ) ? [f (x) + ?(y)] + (b
ut ? u)> F(u) ?
?ky ? y1 k22
k? ? ?1 k22
kx ? x1 k2G
+
+
.
2t
2t
2?t
Remark: The above result establishes a convergence rate for the variational inequality pertained
bt ) converges to the optimal solutions of (2) in a rate of O(1/t).
to (2). When t ? ?, (b
xt , y
Since our goal is to solve the problem (1), next we present a corollary exhibiting the convergence of
ADMM for solving the original problem (1). All omitted proofs can be found in the supplement.
bt be the output of ADMM. For any x ? ?,
Corollary 1. Suppose Assumption 1.c and 1.d hold. Let x
we have
kx ? x0 k2G
?kAk22 kx ? x0 k22
?2
F (b
xt ) ? F (x) ?
+
+
.
2t
2t
2?t
Remark: For the standard ADMM with G = 0 the first term in the R.H.S vanishes. For the linearized
ADMM with G = ?I ? ?A> A 0, we can bound kx ? x0 k2G ? ?kx ? x0 k22 . One can also derive
a theoretically optimal value of ? by setting x = x? ? ?? and minimizing the upper bound, which
?
results in ? = kAk2 kx?? ?x0 k2 for the standard ADMM or ? = ?2kAk kx
for the linearized
2
? ?x0 k2
ADMM. Finally, the above result implies that the iteration
complexity
of standard and linearized
0 k2
ADMM for finding an -optimal solution of (1) is O ?kAk2 kx?x
.
Next, we present our locally adaptive ADMM and our main result in this section regarding its iteration
complexity. The proposed algorithm is described in Algorithm 2, which is referred to as LA-ADMM.
The algorithm runs with multiple stages by calling ADMM at each stage with a warm start and a
constant number of iterations t. The penalty parameter ?k is increased by a constant factor larger
than 1 (e.g., 2) after each stage and has an initial value dependent on ?, kAk2 , 0 , ? and the targeted
accuracy . The convergence result of LA-ADMM employing G = ?I ? ?A> A is established below.
A slightly better result in terms of a constant factor can be established for employing G = 0.
a local errormbound condition on the Theorem 2. Suppose Assumption 1 holds and F (x) obeys
l
2?1??
kAk2 0 ,
K = dlog2 (0 /)e and t =
8?kAk2 max(1,c2 )
1??
, we have F (xK ) ? F? ?
1??
e
2. The iteration complexity of LA-ADMM for achieving an 2-optimal solution is O(1/
).
sublevel. Let ?1 =
Remark: There are two levels of adaptivity to the local sharpness of the penalty parameter. First,
the initial value ?1 in Algorithm 3 depends on the local sharpness parameter ?. Second, the time
interval to increase the penalty parameter is determined by the value of t which is also dependent on
?. Compared to the iteration complexity O(1/) of vanilla ADMM, LA-ADMM can enjoy a lower
iteration complexity.
5
Locally Adaptive ADMM for Stochastic Optimization
In this section, we consider stochastic optimization problem as the following:
min F (x) , E? [f (x; ?)] + ?(Ax),
x??
(12)
where ? is a random variable and f (x; ?) : Rd ? (??, +?] is a proper lower-semicontinuous
convex function for each realization of ?. For this problem, in addition to Assumption 1, we make
5
Algorithm 3 SADMM(x0 , ?, ?, t, ? )
Algorithm 4 LA-SADMM (x0 , ?1 , ?1 , D1 , K, t)
d
1: Input: x0 ? R , a step size ?, penalty 1: Input: x0 ? Rd , the number of stages K, the numparameter ?, the number of iterations t
ber of iterations t per stage, the initial step size ?1 ,
and a domain ?.
the initial parameter ?1 and the initial radius D1 .
2: Initialize: x1 = x0 , y1 = Ax1 , ?1 = 0 2: for k = 1, . . . , K do
3: for ? = 1, . . . , t do
3:
Let xk = SADMM(xk?1 , ?k , ?k , t, Bk ? ?)
4:
Update x? +1 by (9) and y? +1 by (5) 4: Update ?k+1 = ?k /2 and ?k+1 = 2?k , Dk+1 =
Dk /2.
5:
Update ?? +1 by (6)
5:
end
for
6: end for
P
bt = t? =1 x? /t
6: Output: xK
7: Output: x
the following assumption for our development, which is a standard assumption for many previous
stochastic gradient methods.
Assumption 2. For the stochastic optimization problem (12), we assume that there exists a constant
R such that k?f (x; ?)k2 ? R almost surely for any x ? ?.
We present a framework of stochastic ADMM (SADMM) in Algorithm 3. The convergence results
for solving the equivalent constrained optimization problem of stochastic ADMM with different
choices of G? have been established in [21, 23, 35].
Below, we will focus on G? = ?I ? ??A> A I because it leads to computationally more efficient
update for x? +1 than other two choices for high-dimensional problems. Using G? = I will yield a
similar convergence bound except for a constant term and using the idea of AdaGrad for computing
G? will lead to the same order of convergence in the worst-case, which we will postpone to future
work for exploration. The corollary below will be used in our analysis.
Corollary 3. Suppose Assumption 1.c, 1.d and Assumption 2 hold. Let G? = ?I ? ??A> A I in
Algorithm 3. For any x ? ?,
?R2
?kx1 ? xk22
?kAk22 kx1 ? xk22
?2
?kAk2 kx1 ? xt+1 k2
F (b
xt ) ? F (x) ?
+
+
+
+
2
2?t
2t
2?t
t
t
X
1
+
(E[g? ] ? g? )> (x? ? x).
t ? =1
Remark: Taking expectation on both sides will yield the expectational convergence bound. We can
also use an analysis of large deviation to bound
with high
?
? the last term to obtain the convergence
probability. In particular, by setting ? ? 1/ ? , the above result implies an O(1/ t) convergence
rate, i.e., O(1/2 ) iteration complexity of stochastic ADMM.
Next, we discuss our locally adaptive stochastic ADMM (LA-SADMM) algorithm in Algorithm 4.
The key idea is similar to LA-ADMM, i.e., calling SADMM in multiple stages with warm start. The
step size ?k in each call of SADMM is fixed and decreases by a certain fraction after one stage. The
penalty parameter is updated similarly to that in LA-ADMM but with a different initial value. A key
difference from LA-ADMM is that we employ a domain shrinking approach to modify the domain
of the solutions x? +1 at each stage. For the k-th stage, the domain for x is the intersection of ?
and Bk = B(xk?1 , Dk ), where the latter is a ball with a radius of Dk centered at xk?1 (the initial
solution of the k-th stage). The radius Dk will decrease geometrically between stages. The purpose
of using the domain shrinking approach is to tackle the last term of the upper bound in Corollary 3 so
that it can decrease geometrically as the stage number increases. A similar idea has been adopted
in [29, 7, 5]. Note that during each SADMM, we can use the three choices of G? as mentioned before.
Below we only present the convergence result of the variant with G? = ?I ? ?k ?k A> A.
Theorem 4. Suppose Assumptions 1 and 2 hold and F (x) obeys the local error bound condition
0
c0
6R2
on S . Given ? ? (0, 1), let ?? = ?/K, K = dlog2 ( 0 )e, ?1 = 6R
2 , ?1 = kAk2 , D1 ? 1?? ,
0
2
t be the smallest integer such that t ? max{
>
? 2 12?kAk2 D1 ?2 kAk2
6912R2 log(1/?)D
1
,
, R2 2 }
0
20
and G? =
2I ? ?1 ?1 A A I. Then LA-SADMM guarantees that, with a probability 1 ? ?, we have F (xK ) ?
F? ? 2. The iteration complexity of LA-SADMM for achieving an 2-optimal solution with a high
c0
2(1??)
e
probability 1 ? ? is O(log(1/?)/
), provided D1 = O( (1??)
).
6
Algorithm 5 LA-ADMM with Restarting
Algorithm 6 LA-SADMM with Restarting
(1)
t1 , ?1
1: Input:
2: Initialization: x(0)
3: for s = 1, 2, . . . , do
(s)
4:
x(s) =LA-ADMM(x(s?1) , ?1 , K, ts )
1??
5:
ts+1 = ts 2
6: end for
7: Output: x(S)
,
(s+1)
?1
=
(s)
?1 /21??
(1)
and ? 0 /2
0
2: Initialization: x(0) , ?1 = 6R
2 , ?1 =
3: for s = 1, 2, . . . , do
1: Input: t1 , D1
6R2
kAk22 0
(s)
4:
x(s) =LA-SADMM(x(s?1) , ?1 , ?1 , D1 , K, ts )
(s+1)
(s)
5:
ts+1 = ts 22(1??) , D1
= D1 21??
6: end for
7: Output: x(S)
Remark: Interestingly, unlike that in LA-ADMM, the initial value ?1 does not depend on ?. The
adaptivity of the penalty parameters lies on the time interval t which determines when the value of ?
is increased. The difference comes from the first two terms in Corollary 3.
Before ending this section, we discuss two points. First, both Theorem 2 and Theorem 4 exhibit the
dependence of the two algorithms on the c parameter (e.g., t in Algorithm 2 and D1 in Algorithm 4)
that is usually unknown. Nevertheless, this issue can be easily addressed by using another level of
restarting and increasing sequence of t and D1 similar to the practical variants in [29, 32]. Due to
the limit of space, we only present the algorithms in Algorithm 5 and Algorithm 6 with their formal
guarantee presented in supplement. The conclusion is that under mild conditions as long as ? (1)
(1)
in Algorithm 5 is sufficiently small, t1 and D1 in Algorithm 6 are sufficiently large, the iteration
1??
2(1??)
e
e
complexities remain O(1/
) and O(1/
) when ? in LEB condition is known. Second, these
variants can be even employed when the local sharpness parameter ? is unknown by simply setting it
to 0, and still enjoy reduced iteration complexities in terms of a multiplicative factor compared to
vanilla ADMMs. Detailed results are included in the supplement.
6
Applications and Experiments
In this section, we present some experimental results of the proposed algorithms for solving three
tasks, namely generalized LASSO, robust regression with a low-rank regularizer (RR-LR) and
learning low-rank representation. For generalized lasso, our experiment focuses on comparing the
proposed LA-SADMM with SADMM. For the latter tasks, we focus on comparing the proposed
LA-ADMM with previous linearized ADMM with and without self-adaptive penalty parameters.
We first consider generalized LASSO, which can find applications in many problems in statistics and
machine learning [28]. The objective of generalized LASSO can be expressed as:
n
min F (x) =
x?Rd
1X
`(x> ai , bi ) + ?kAxk1
n i=1
(13)
where (ai , bi ) is a set of pairs of training data, i = 1, . . . , n, ? ? 0 is a regularization parameter,
A ? Rm?d is a specified matrix, and `(z, b) is a convex loss function in terms of z. The above
formulation include many formulations as special cases, e.g., the standard LASSO where A = I ?
Rd?d [26], fused LASSO that penalizes the `1 norm of both the coefficients and their successive
differences [27], graph-guided fused LASSO (GGLASSO) where A = F ? Rm?d encodes some
graph information about features [12], and sparse graph-guided fused LASSO (S-GGLASSO) where
kAxk1 = ?2 kxk1 + ?1 kF xk1 [21].
Let us first discuss the local sharpness parameter of generalized lasso with different loss
functions. For the loss function, let us first consider piecewise linear loss function such as
hinge loss `(z, b) = max(0, 1 ? bz), absolute loss `(z, b) = |z ? b| and -insensitive loss
`(z, b) = max(|z ? b| ? , 0). Then the objective is a polyhedral function. According to the
results in [32], the local sharpness parameter is ? = 1. It then implies that both LA-ADMM and
LA-SADMM enjoy linear convergence results for solving the problem (13) with a piecewise linear
loss function. To the best of our knowledge, these are the first linear convergence results of ADMM
without smoothness and strong convexity conditions. One can also consider piecewise quadratic loss
such as square loss `(z, b) = (z ? b)2 for b ? R and squared hinge loss `(z, b) = max(0, 1 ? bz)2
for b ? {1, ?1}. According to [14], the problem with convex piecewise quadratic loss has a local
e ?) and O(1/)
e
sharpness parameter ? = 1/2, implying O(1/
for LA-ADMM and LA-SADMM.
7
gisette
w8a
0.16
0.55
SADMM
LA-SADMM
0.15
0.145
0.14
0.135
0.13
0.45
0.4
0.35
0.3
0
0.5
1
1.5
2
2.5
0.25
3
?10 6
# of iterations
0
2
4
6
8
11
10
0
20
0.155
0.15
100
0.45
0.4
0.35
0.145
ADMM-best
ADMM-worst
ADMM-AP
ADMM-RB
LA-ADMM
7
log(objective)
objective
0.165
0.16
80
shape
SADMM
LA-SADMM
0.5
60
(c) RR + LR
gisette
0.55
SADMM
LA-SADMM
40
# of iterations
(b) SVM + GGLASSO
w8a
objective
11.5
10
?10 5
# of iterations
0.175
6.5
6
5.5
0.3
0.14
0.135
12
10.5
(a) SVM + GGLASSO
0.17
ADMM-best
ADMM-worst
ADMM-AP
ADMM-RB
LA-ADMM
12.5
log(objective)
0.5
objective
objective
0.155
synthetic data
13
SADMM
LA-SADMM
5
0
1
2
# of iterations
3
4
?10
(d) SVM + S-GGLASSO
6
0.25
0
2
4
6
# of iterations
8
10
5
?10
(e) SVM + S-GGLASSO
0
200
400
600
800
1000
# of iterations
(f) LRR
Figure 1: Comparison of different algorithms for solving different tasks. RR + LR represents robust
regression with a low rank regularizer. LRR represents low-rank representation.
For more examples with different values of ?, we refer readers to [32, 30, 29, 17].
SVM Classification with GGLASSO and S-GGLASSO Regularizers To generate the A
matrix, we first need to construct a dependency graph of features. We follow [21] to generate a
dependency graph by sparse inverse covariance selection [4]. Specifically, we get the estimator
? ?1 via sparse inverse covariance estimation with
of the inverse covariance matrix denoted by ?
? ?1 , where i, j ? {1, . . . , d}, i 6= j, an edge
the graphical lasso [4]. For each nonzero entry ?
ij
between i and j is created. If we denote by G ? {V, E} the resulting graph, where V is a set of
d vertices, which correspond to d features in the data, and E = {e1 , . . . , em } denotes the set of
m edges between elements of V, where ei consists of a tuple of two elements, then the k-th row
of A has two non-zero elements corresponding to the k-th edge ek = (i, j) ? E with Ak,i = 1
and Ak,j = ?1. We choose two medium-scale data sets from libsvm website, namely w8a data
(n = 49749, d = 300) and gisette data (n = 6000, d = 5000), to conduct the experiment. In
the process of estimating inverse covariance matrix, we choose a penalty parameter to be 0.01
that renders the percentage of non-zero elements of the A matrix to be around 3% for w8a
data and 1% for gisette data. We compare the performance of the LA-SADMM algorithm
? with
SADMM [23], where in SADMM we use G? = ?I ? ??? A> A I with ?? ? ?1 / ? . For
fairness, we set the same initial solution with all zero entries. We fix the value of regularization
parameters (? in GGLASSO and ?1 , ?2 in S-GGLASSO) to be n1 , where n is the number of
samples. For SADMM, we tune both ?1 and ? from {10?5:1:5 } . For LA-SADMM, we set the
initial step size and penalty parameter to their theoretical value in Theorem 4, and select D1 from
{100, 1000}. The values of t in LA-SADMM is set to 105 and 5 ? 104 for w8a and gisette, respectively. The results of comparing the objective values versus the number of iterations are presented
in Figure 1 (a,b,d,e). We can see that LA-SADMM exhibits a much faster convergence than SADMM.
Robust Regression with a Low-rank Regularizer The objective function is F (X) =
?kXk? + kAX ? Ck1 . We can form an equality constraint Y = AX ? C and solve the problem
by linearized ADMM. The value of the local sharpness parameter of this problem is still an open
problem. We compare the proposed LA-ADMM, the vanilla linearized ADMM with a fixed
penalty parameter (ADMM), the linearized ADMM with self-adaptive penalty proposed in [15]
(ADMM-AP), and the linearized ADMM with residual balancing in [10, 2] (ADMM-RB). We
construct a synthetic data where A ? R1000?100 is generated following a Gaussian distribution with
mean 0 and standard deviation 1. To construct C ? R1000?50 , we first generate X ? R100?50 and
8
? and then let C = AX
? + ?, where ? is a Gaussian
retain only its top 20 components denoted by X
noise matrix with mean zero and standard deviation 0.01. We set ? = 100. For the vanilla linearized
ADMM, we try different penalty parameters from {10?3:1:3 } and report the best performance
(using ? = 0.01) and worst performance (using ? = 0.001). To demonstrate the capability of
adaptive ADMM, we choose ? = 0.001 as the initial step size for LA-ADMM and ADMM-AP.
Other parameters of ADMM-AP is the same as suggested in the original paper. For LA-ADMM, we
implement its restarting variant (Algorithm 5), and start with the number of inner iterations t = 2 and
increase its value by a factor 2 after 10 stages, and also increase the value of ? by 10 times after each
stage. The results are reported in Figure 1 (c), from which we can see that LA-ADMM performs
comparably with ADMM with the best penalty parameter and also better than ADMM-AP. We also
include the results in terms of running time in the supplement.
Low-rank Representation [16] The objective function is F (X) = ?kXk? + kAX ? Ak2,1 , where
A ? Rn?d is a data matrix. We used the shape image 3 and set ? = 10. For the vanilla linearized
ADMM, we try different penalty parameters from {10?3:1:3 } and report the best performance (using
? = 0.1) and worst performance (using ? = 0.01). To demonstrate the capability of adaptive
ADMM, we choose ? = 0.01 as the initial step size for LA-ADMM and ADMM-AP. Other
parameters of ADMM-AP is the same as suggested in the original paper. For LA-ADMM, we
start with the number of inner iterations t = 20 and increase its value by a factor 2 after 2 stages,
and also increase the value of ? by 2 times after each stage. The results are reported in Figure 1
(f), from which we can see that LA-ADMM performs comparably with ADMM with the best
penalty parameter and also better than ADMM-AP. We can see from the figure that the results of
ADMM-worst and ADMM-AP are quite similar. We also include the results in terms of running time
in the supplement.
7
Conclusion
In this paper, we have presented a new theory of (linearized) ADMM for both deterministic and
stochastic optimization with adaptive penalty parameters. The new adaptive scheme is different
from previous self-adaptive schemes and is adaptive to the local sharpness of the problem. We
have established faster convergence of the proposed algorithms of ADMM with penalty parameters
adaptive to the local sharpness parameter. Experimental results have demonstrated the superior
performance of the proposed stochastic and deterministic adaptive ADMM.
Acknowlegements
We thank the anonymous reviewers for their helpful comments. Y. Xu, M. Liu and T. Yang are
partially supported by National Science Foundation (IIS-1463988, IIS-1545995). Y. Xu would like to
thank Yan Yan for useful discussions on the low-rank representation experiments.
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6,331 | 6,727 | Shape and Material from Sound
Zhoutong Zhang
MIT
Jiajun Wu
MIT
Qiujia Li
University of Cambridge
Zhengjia Huang
ShanghaiTech University
Joshua B. Tenenbaum
MIT
William T. Freeman
MIT, Google Research
Abstract
What can we infer from hearing an object falling onto the ground? Based on
knowledge of the physical world, humans are able to infer rich information from
such limited data: the rough shape of the object, its material, the height of the fall,
etc. In this paper, we aim to approximate such competency. We first mimic the
human knowledge about the physical world using a fast physics-based generative
model. Then, we present an analysis-by-synthesis approach to infer properties
of the falling object. We further approximate human past experience by directly
mapping audio to object properties using deep learning with self-supervision.
We evaluate our method through behavioral studies, where we compare human
predictions with ours on inferring object shape, material and initial height of
falling. Results show that our method achieves near-human performance, without
any annotations. We further test our model using real world data, illustrating its
potential in inference from real world data.
1
Introduction
Given a short audio clip of interacting objects, humans, even young children, can recover rich
information about materials, surface smoothness, and the quantity of objects involved [Zwicker and
Fastl, 2013, Kunkler-Peck and Turvey, 2000, Siegel et al., 2014]. How does our cognitive system
recover so much content from the audio clip? What is the role of past experience in understanding
auditory inputs?
For physical scene understanding from visual input, recent behavioral and computational studies
suggest that human judgments can be well explained as approximate, probabilistic simulations of a
mental physics engine [Battaglia et al., 2013, Sanborn et al., 2013]. These studies suggest that the
brain encodes rich but noisy knowledge of physical properties of objects and basic laws of physical
interactions between objects. To understand, reason, and predict about a physical scene, humans
seem to rely on simulations from this mental physics engine.
In this paper, we develop a computational system to interpret audio clips of falling objects, inspired
by the idea that humans may use a physics engine as part of a generative model to understand the
physical world. The first component of our generative model is the representation of a rigid object,
which includes its 3D geometric shape, position in space, and its physical properties including mass,
Young?s modulus, Rayleigh damping coefficients, and restitution. All of these object attributes are
treated as latent variables in our model, which we aim to infer from auditory inputs.
The second part includes an efficient physics-based audio synthesis engine. Given the initial conditions
and properties of an object, which serves as the hypothesis for our generative model, the engine first
simulates the rigid body motion of the object and generate the object?s trajectory with corresponding
collision profile under rigid body physics. Then the object?s trajectory and collision profile, along
with its pre-computed sound statistics, are used to generate the sound of this entire physical process.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Given an audio of a single object falling, we utilize our generative model to infer latent
variables that could best reproduce the sound.
With such efficient forward models, it is possible for us to infer the prescribed latent variables in
an analysis-by-synthesis style. In short, given an audio clip, we aim to find a set of latent variables
that best reproduce it. One challenge in this goal is to design a likelihood function that measures
the perceptual distance between two sounds. To address this challenge, we utilize the observation
that a simple feature space, such as spectrogram, can be effective if the degrees of freedom for latent
variables are restricted. This observation allows us to infer latent variables via methods like Gibbs
sampling, where we can only focus on approximating the conditional probability of a single variable
given the others.
To further accelerate our inference procedure, we incorporate past experience via a supervised learning
system that uses unlabeled data with inferred labels. To this end, we propose a self-supervised learning
algorithm inspired by the wake/sleep phases in Helmholtz machines [Dayan et al., 1995]. A deep
neural network is trained as the recognition model (i.e., sleep cycle), where the labels are generated
using our inference algorithm. Then, for any future audio clip, the output of the recognition model
can be used as a good initialization, accelerating the inference procedure.
We evaluate our models on a range of perception tasks: inferring object shape, material and initial
height from the sound. We also collect human responses for each of these tasks and compare them
with model estimates. Our results indicate that humans are quite successful in these tasks; our model
not only closely matches human successes, but also makes similar errors as humans do. Due to the
hardness of acquiring quality audio data with rich labels, we use synthetic data to evaluate our models
in above tasks. To prove our model is capable of such inference tasks given real world audio, we
tested our model using real captured data under constraint settings.
Our work makes three contributions. First, we propose a novel model for estimating physical
properties of objects from auditory inputs by incorporating the feedback of a physics engine and an
audio engine into the inference process. Second, we train a deep learning based recognition model
that leads to efficient inference in the generative model. Third, we test our model and compare it to
humans on a variety of judgment tasks, and demonstrate the correlation between human responses
and model estimates.
2
Related Work
Human visual and auditory perception In the field of auditory perception or psychoacoustics,
researchers have explored how humans can infer object properties including shape, material, size
from audio in the past decades [Zwicker and Fastl, 2013, Kunkler-Peck and Turvey, 2000, Rocchesso
and Fontana, 2003, Klatzky et al., 2000, Siegel et al., 2014]. Recently, McDermott et al. [2013]
proposed compact sound representations that capture semantic information and are informative of
human auditory perception.
2
iteration
Generative Model
shape
rotation
Gibbs
Sampling
height
Likelihood Function
material
Target Audio
iteration
Figure 2: Our inference pipeline. We use Gibbs sampling over the latent variables. The conditional
probability is approximated using the likelihood between reconstructed sound and the input sound.
Sound simulation Our sound synthesis engine builds upon and extends existing sound simulation
systems in computer graphics and computer vision [O?Brien et al., 2001, 2002, James et al., 2006,
Bonneel et al., 2008, Van den Doel and Pai, 1998, Zhang et al., 2017]. Van den Doel and Pai [1998]
simulated object vibration using the finite element method and approximated the vibrating object as
a single point source. O?Brien et al. [2001, 2002] used the Rayleigh method to approximate wave
equation solutions for better synthesis quality. James et al. [2006] proposed to solve Helmholtz
equations using the Boundary Element Method, where each object?s vibration mode is approximated
by a set of vibrating points. Recently, Zhang et al. [2017] built a framework for synthesizing largescale audio-visual data. In this paper, we accelerate the framework by Zhang et al. [2017] to achieve
near real-time rendering, and explore learning object representations from sound with the synthesis
engine in the loop.
Physical Object Perception There has been a growing interest in understanding physical object
properties like masses and frictions from visual input or scene dynamics [Chang et al., 2017, Battaglia
et al., 2016, Wu et al., 2015, 2016, 2017]. Most of the existing research has been focusing on object
properties from visual data. Recently, researchers started to explore learning object representations
from sound. Owens et al. [2016a] attempted to infer material properties from audio, focusing on
the scenario of hitting objects with a drumstick. Owens et al. [2016b] further demonstrated audio
signals can be used as supervision on learning object concepts from visual data, and Aytar et al.
[2016] proposed to learn sound representations from corresponding video frames. Zhang et al. [2017]
discussed the complementary role of auditory and visual data in recovering both geometric and
physical object properties. In this paper, we propose to learn physical object representations through
a combination of powerful deep recognition models and analysis-by-synthesis inference methods.
Analysis-by-synthesis Our framework also relates to the field of analysis-by-synthesis, or generative models with data-driven proposals [Yuille and Kersten, 2006, Zhu and Mumford, 2007, Wu et al.,
2015], as we are incorporating a graphics engine as a black-box synthesizer. Unlike earlier methods
that focus mostly on explaining visual data, our work aims to infer latent parameters from auditory
data. Please refer to Bever and Poeppel [2010] for a review of analysis-by-synthesis methods.
3
An Efficient, Physics-Based Audio Engine
At the core of our inference pipeline is an efficient audio synthesis engine. In this section, we first
give a brief overview of existing synthesis engines, and then present our technical innovations on
accelerating existing systems to real-time rendering for our inference algorithm.
3.1
Audio Synthesis Engine
Audio synthesis engines generate realistic sound by following fundamental physical laws. First, the
interaction between an object and the environment is generated using rigid body simulation, where
3
Settings
Waveform
Audio Wave
Original algorithm
Amplitude cutoff
Principal modes
Multi-threading
SoundNet-8
conv1 pool1 ??
conv7 pool7 fc
Figure 3: Our 1D deep convolutional network. Its
architecture follows that in Aytar et al. [2016], where
raw audio waves are forwarded through consecutive
conv-pool layers, and then passed to a fully connected
layer to produce output.
All
Time (s)
30.4
24.5
12.7
1.5
0.8
Table 1: Acceleration break down of each
technique we adopted. Timing is evaluated
by synthesizing an audio with 200 collisions.
The last row reports the final timing after
adopting all techniques.
Newton?s physics laws dictate the object?s motion and collisions over time. According to vibration
analysis, each collision causes the object to vibrate in certain patterns, changing the air pressure
around its surface. Such turbulence then propagates in air to the recording position and creates the
sound of this physical process.
Rigid Body Simulation Given an object?s 3D position and orientation and its mass and restitution,
a physics engine can simulate the physical processes, and output the object?s position, orientation,
and collision information over time. In our implementation, an open-source physics engine, Bullet [Coumans, 2010], is used to simulate this process. In order to achieve accurate results, we use a
time step of 1/300 second for the simulation. Specifically, we record the 3D pose and position of the
object over time, as well as the collision locations, magnitudes, and directions. Object sound can then
be approximated by accumulating sounds caused by those discrete impulse collisions on its surface.
Audio Synthesis The audio synthesis procedure is built upon previous work on simulating realistic
sounds [James et al., 2006, Bonneel et al., 2008, O?Brien et al., 2001]. To facilitate fast synthesis,
this process is decomposed into two parts: online and offline. The offline part first uses finite element
method (FEM) to obtain the object?s vibration modes, which depends on object?s shape and its
Young?s modulus. The object vibration modes are then used as Neumann boundary conditions of the
Helmholtz equation, which can be solved using boundary element method (BEM). The solution is
then approximated using techniques reported by James et al. [2006], where the resulting pressure field
is approximated by a sparse set of vibrating points. Note that the computation above only depends
on object?s shape and Young?s modulus, but not on the physical process that it undergoes. This
allows us to pre-compute a number of shape-modulus configurations before simulation; only minimal
computation is needed at the simulation phase.
The online part of the audio engine loads pre-computed approximation and decomposes impulses on
the surface mesh of the object onto its modal bases. Summing up pressure changes at the observation
point induced by vibrations in each mode produces the desired sound. An evaluation of its authenticity
can be found in Zhang et al. [2017].
3.2
Accelerating Audio Synthesis
The prerequisite for performing efficient inference via analysis-by-synthesis is fast audio synthesis.
Unfortunately, the simulation procedure described above is expensive to compute. We therefore
present ways for accelerating the computation to near real-time.
First, we pick out the most significant modes excited by each impulse. By setting a threshold
at 90% energy cutoff, we shorten the computing time by ignoring sound components generated
by insignificant modes, which are about one-half of total modes on average. Secondly, we stop
synthesizing as the amplitude of the sound is damped below a small threshold, where humans can
hardly perceive. Thirdly, we parallelize the synthesis process by treating each collision independently,
which can be computed on an independent thread. We then join the completed threads into a shared
buffer according to their time-stamps. The effect of acceleration is shown in Table 1. Online sound
synthesis only contains variables that are fully decoupled from the offline stage, which enables us to
freely manipulate other variables with little computational cost during simulation time.
4
Variable
Range
C/T
Variable
Range
C/T
6
Primitive shape (s)
Height (z)
Rotation axis (i, j, k)
14 classes
[1, 2]
S2
D
C
C
Specific modulus (E/?)
Restitution (e)
Rotation angle (w)
[1, 30] ? 10
[0.6, 0.9]
[??, ?)
D
C
C
Rayleigh damping (?)
10[?8,?5]
C
Rayleigh damping (?)
2[0,5]
C
Table 2: Variables in our generative model, where the C/T column indicates whether sampling takes
place in continuous (C) or discrete (D) domain, and values inside parentheses are the range we
uniformly sampled from. Rotation is defined in quaternions.
3.3
Generating Stimuli
Because real audio recordings with rich labels are hard to acquire, we synthesize random audio
clips using our physics-based simulation in order to test and evaluate our models. Specifically, we
constrain ourselves only focusing on primitive objects falling onto the ground. We first construct a
sound statistic data set that includes 14 primitives (partly shown in Table 2), each with 10 different
specific moduli (defined as Young?s modulus over density). With this pre-computed data, we are
able to generate synthetic audio clips in a near-real-time fashion. Since the process of objects falling
onto the ground is relatively fast, we set the total simulation time of each scenario to 3 seconds. An
detailed synthesis setup can be found in Table 2.
4
Inference
In this section, we investigate four models for inferring object properties, each corresponding to a
different scenario. We start from an unsupervised model where the input is only one single test case
with no annotation. Inspired by how humans can infer scene information using a mental physics
engine [Battaglia et al., 2013, Sanborn et al., 2013], we adopt Gibbs sampling over latent variables to
find the combination that best reproduces the given audio. We develop this model further with the
help from past experience, where a deep neural network is trained using data with inferred labels as
supervision. Such self-supervised scheme is able to approximate the most probable configurations
faster. We further investigate the case when labels can be acquired but are extremely coarse. We first
train a recognition model with weak labels, then randomly pick candidates from those labels as an
initialization for our analysis-by-synthesis inference. Lastly, in order to understand the limit for this
inference task, we train a deep neural network with fully labeled data that yields the upper-bound
performance.
4.1
Models
Unsupervised Given an audio clip S, we would like to recover all latent variables x, so that the
reproduced sound g(x) is most similar to S. Suppose L(?, ?) is a likelihood function that measure
the perceptual distance between two sounds, then the goal is to maximize L(g(x), S). We denote
L(g(x), S) as p(x) for brevity. In order to find x that maximizes p(x), p(x) can be treated as an
distribution p?(x) scaled with an unknown partition function Z. Since we do not have an exact form for
p(?) nor p?(x), Gibbs sampling is applied to draw samples from p(x) using conditional probabilities.
Specifically, at sweep round t, we update each variable xi by drawing samples from
t?1
p?(xi |xt1 , xt2 , ...xti?1 , xt?1
i+1 , ...xn ).
(1)
Such conditional probabilities, however, are much easier to approximate. For example, to sample
Young?s modulus conditioned on other variables, we could simply use the spectrogram as features,
and measure the l2 distance between two sounds, since Young?s modulus would only affect the
frequency at each collision. Under such observation, we use the spectrogram as features for all
variables except height. Since the height can be inferred from the time of the first collision, a simple
likelihood function can be designed as measuring the time difference between the first impact in two
sounds. Note that this is only an approximate measure: object?s shape and orientation also affects
5
the time of the first impact. Nonetheless, since we are only concerned with conditional probabilities,
such measure can be very effective.
To sample from the conditional probabilities, we adopt the Metropolis?Hastings algorithm, where
samples are drawn from a Gaussian distribution and are accepted by flipping a biased coin according
to its likelihood compared with the previous sample. Specifically, we calculate the l2 distance dt
in feature space between g(xt ) and S. For a new sample xt+1 , we also calculate the l2 distance
dt+1 in feature space between g(xt+1 ) and S. The new sample is accepted if dt+1 is smaller than
dt ; otherwise, xt+1 is accepted with probability exp(?(dt+1 ? dt )/T ), where T is a time varying
function inspired by simulated annealing algorithm. In our implementation, T is set as a quadratic
function of the current MCMC sweep number t.
Self-supervised Learning To accelerate the above sampling process we propose a self-supervised
model, which is analogous to a Helmholtz machine trained by the wake-sleep algorithm. We first train
a deep neural network, whose labels are generated by the unsupervised inference model suggested
above for a limited number of iterations. For a new audio clip, our self-supervised model uses the
result from neural network as an initialization, and then runs our analysis-by-synthesis algorithm to
make further inference. By making use of past experience, the sampling process is expected to start
off from a better position, and takes much fewer iterations to converge than the unsupervised model.
Weakly-supervised Learning We further investigate the case where weak supervision might be
helpful for accelerating the inference process. Since the latent variables we aim to recover are hard to
obtain in real world settings, it is more realistic to assume that we could acquire very coarse labels,
such as the type of material, rough attributes of the object?s shape, the height of the fall, etc. Based
on such assumptions, we coarsen ground truth labels for all variables. For our primitive shapes,
three attributes are defined, namely ?with edge,? ?with curved surface,? and ?pointy.? For material
parameters, i.e., specific modulus, Rayleigh damping coefficients and restitution, they are mapped to
steel, ceramic, polystyrene and wood by finding the nearest neighbor to those real material parameters.
Height is divided into ?low? and ?high? categories. A deep convolutional neural network is trained
on our synthesized dataset with coarse labels. As shown in Figure 4, even trained using coarse labels,
our network learns features very similar to the ones learned by the fully supervised network. To go
beyond coarse labels, the unsupervised model is applied using the initialization suggested by the
neural network.
Fully-supervised Learning To investigate the limit of this inference task, we train an oracle model
with ground truth labels. To visualize what kind of abstraction and characteristic features are learned
by oracle model, we plot inputs that maximally activate some hidden units in the last layer of the
network. Figure 4 illustrates some of the most interesting waveforms. A selection of them learned to
recognize specific temporal patterns, and others are most sensitive to specific frequencies. Similar
patterns are found between weakly and fully supervised models.
4.2
Contrasting Model Performance
We would like to evaluate how well our model performs under different settings, especially on how
past experience or coarse labeling could improve over the unsupervised model. We first present the
implementation details of all four models, then compare their quantitative results on all inference
tasks.
Sampling Setup We performed 80 sweeps of MCMC sampling over all the 7 latent variables; for
every sweep, each variable is sampled twice. Shape, specific modulus and rotation are sampled by
uniform distributions across their corresponding dimensions. For other continuous variables, we
define an auxiliary Gaussian variable xi ? N (?i , ?i2 ) for sampling, where the mean ?i is based on
current state. To evaluate the likelihood function between the input and the sampled audio (both with
sample rate of 44.1k), we compute the spectrogram of the signal using Tuckey window of length
5,000 with a 2,000 sample overlap. For each window, a 10,000 point Fourier transform is applied.
Deep Learning Setup Our fully supervised and self-supervised recognition models inherit the
architecture of SoundNet-8 [Aytar et al., 2016] as Figure 3, which takes an arbitrarily long raw audio
wave as an input, and produces a 1024-dim feature vector. We append a fully connected layer at the
6
Time Domain
small damping
multiple collisions
low frequency
Frequency Domain
mid frequency
high frequency
Weakly
Supervised
Fully
Supervised
Figure 4: Visualization of top two sound waves that activate the hidden unit most significantly, in
temporal and spectral domain. Their common characteristics can reflect the values of some latent
variables, e.g. Rayleigh damping, restitution and specific modulus from left to right. Both weakly
and fully supervised models capture similar features.
Latent Variables
Inference Model
shape
mod.
height
?
?
Unsupervised
init
infer
8%
54%
10%
56%
0.179
0.003
0.144
0.069
0.161
0.173
Self-supervised
init
infer
14%
52%
16%
62%
0.060
0.005
0.092
0.061
0.096
0.117
Weakly supervised
init
infer
18%
62%
12%
66%
0.018
0.005
0.077
0.055
0.095
0.153
Fully supervised
infer
98%
100%
0.001
0.001
0.051
Table 3: Initial and final classification accuracies and label MSE errors of three different inference
models after 80 iterations of MCMC, upper bounded by the fully supervised model.
end to produce a 28-dim vector as the final output of the neural network. The former 14 dimensions
are the one-hot encoding of primitive shapes, the following 10 dimensions are encodings of the
specific modulus. The last 4 dimensions regress the initial height, two Rayleigh damping coefficients
and the restitution respectively. All the regression dimensions are normalized to a [?1, 1] range. The
weakly supervised model preserves the structure of fully supervised one, but with an 8-dim final
output: 3 for shape attributes, 1 for height, and 4 for materials. We used stochastic gradient descent
for training, with a learning rate of 0.001, a momentum of 0.9 and a batch size of 16. Mean Square
Error(MSE) loss is used for back-propagation. We implemented our framework in Torch7 [Collobert
et al., 2011], and trained all models from scratch.
Results Results for the four inference models proposed above are shown in Table 3. For shapes and
specific modulus, we evaluate the results as classification accuracies; for height, Rayleigh damping
coefficients, and restitution, results are evaluated as MSE. Before calculating MSE, we normalize
values of each latent variable to [?1, 1] interval, so that the MSE score is comparable across variables.
From Table 3, we can conclude that self-supervised and weakly supervised models provide better
initialization for our analysis-by-synthesis algorithm, especially on the last four continuous latent
variables. One can also observe that final inference accuracies and MSEs are marginally better than
the unsupervised case. To illustrate the rate of convergence, we plot the likelihood value, exp(?kd)
where d is the distance of sound features, along iterations of MCMC in Figure 5. The mean curve of
self-supervised model meets our expectation, i.e., it converges much faster than the unsupervised
model, and reaches a slightly higher likelihood at the end of 30 iterations. The fully supervised model,
which is trained on 200,000 audios with the full set of ground-truth labels, yields near-perfect results
for all latent variables.
7
0.8
steel
1
1.0
0.8
0.9
ceramic
0.6
ceramic
0.6
poly
0.4
poly
0.4
0.2
wood
0
steel ceramic poly wood
0.2
wood
steel ceramic poly wood
Likelihood
1
steel
0.7
random
self-supervised
0.6
0
0.5 0
Human
Unsupervised model
0.8
5
10
15
20
25
Number of MCMC Sweeps
30
Figure 5: Left and middle: confusion matrix of material classification performed by human and our
unsupervised model. Right: mean likelihood curve over MCMC iterations.
HasEdge
Edge
With
Edge
Has
Has
Curved
Surface
With
Has
Curved
Surface
Pointy
Pointy
IsIs
Pointy
Height
Height
1.001.00
Accuracy
0.75
0.75
1.00
1.00
0.75
0.75
0.830.83
0.670.67
0.50
0.50
0.50
0.50
0.500.50
0.330.33
0.25
0.25
3030
5050
I Iterations
8080
0.89
0.89
0.50
0.50
0.78
0.78
0.33
0.33
0.67
0.67
0.25
0.25
0.170.17
1010
0.67
0.67
Material
human
0.17
0.17
0.56
0.56
10 10 30 30 50 50 80 80
Iterations
10
10
30
30
50
50
Iterations
Iterations
80
80
10
10
30
30
50
50
Iterations
80
80
10
10
30
30
50
50
Iterations
80
80
Figure 6: Human performance and unsupervised performance comparison. The horizontal line
represents human performance for each task. Our algorithm closely matches human performance.
5
Evaluations
We first evaluate the performance of our inference procedure by comparing its performance with
humans. The evaluation is conducted using synthetic audio with their ground truth labels. Then,
we investigate whether our inference algorithm performs well on real-world recordings. Given an
audio recorded under experiment settings, our algorithm is able to distinguish its shape among other
candidates.
5.1
Human Studies
We try to understand how capable our inference model is relative to human performances. We
designed three tasks on inferring object?s shape, material and height of the fall, the most intuitive
attributes when hearing object falling. Those tasks are designed to be classification problems, where
the labels are in accordance with coarse labels used by our weakly-supervised model. The study was
conducted on Amazon Mechanical Turk. For each experiment (shape, material, height), we randomly
selected 52 test cases. Before answering test questions, the subject is shown 4 training examples
with ground truths as familiarization of the setup. We collected 192 responses for the experiment on
inferring shape, 566 for material, and 492 for height, resulting in a total of 1,250 responses.
Inferring Shapes After being familiarized with the experiment, participants are asked to make
three binary judgments about the shape by listening to our synthesized audio clip. Prior examples are
given for people to understand the distinctions of ?with edge,? ?with curved surface,? and ?pointy?
attributes. Due to material variations, humans mostly make those decisions based on temporal
information rather than spectral information, i.e. the time sequence of collisions. As shown in
Figure 6, humans are relatively good at recognizing shape attributes from sound and are around the
same level of competency when the unsupervised algorithm runs for 10?30 iterations.
Inferring Materials We sampled audio clips whose physical properties ? density, Young?s modulus
and damping coefficients ? are in the vicinity of true parameters of steel, ceramic, polystyrene and
wood. Participants are required to choose one out of four possible materials. However, it can still be
challenging to distinguish between materials, especially when sampled ones have similar damping
8
Likelihood
1
0.8
0.6
0.4
0
(a) Real shape and sound
(b) Inferred shape and sound
5
10 15 20 25 30 35 40 45 50 55
Iteration
(c) Normalized likelihood over iterations
Figure 7: Results of inference on real world data. The test recording is made by dropping the metal
dice in (a). Our inferred shape and reproduced sound is shown in (b). Likelihood over iteration is plot
ted in (c).
and specific modulus. Our algorithm confuses steel with ceramic and ceramic with polystyrene
occasionally, which is in accordance with human performance, as shown in Figure 5.
Inferring Heights In this task, we ask participants to choose whether the object is dropped from a
high position or a low one. We provided example videos and audios to help people anchor reference
height. Under our scene setup, the touchdown times of the two extremes of the height range differ
by 0.2s. To address the potential bias that algorithms may be better at exploiting falling time, we
explicitly told humans that the silence at the beginning is informative. Second, we make sure that
the anchoring example is always available during the test, which participants can always compare
and refer to. Third, the participant has to play each test clip manually, and therefore has control over
when the audio begins. Last, we tested on different object shapes. Because the time of first impact is
shape-dependent, differently shaped objects dropped from the same height would have first impacts
at different times, making it harder for the machine to exploit the cue.
5.2
Transferring to Real Scenes
In addition to the synthetic data, we designed real world experiments to test our unsupervised model.
We select three candidate shapes: tetrahedron, octahedron, and dodecahedron. We record the sound
a metal octahedron dropping on a table and used our unsupervised model to recover the latent
variables. Because the real world scenarios may introduce highly complex factors that cannot be
exactly modeled in our simulation, a more robust feature and a metric are needed. For every audio
clip, we use its temporal energy distribution as the feature, which is derived from spectrogram. A
window of 2,000 samples with a 1,500 sample overlap is used to calculate the energy distribution.
Then, we use the earth mover?s distance (EMD) [Rubner et al., 2000] as the metric, which is a natural
choice for measuring distances between distributions.
The inference result is illustrated in Figure 7. Using the energy distribution with EMD distance
measure, our generated sound aligns its energy at major collision events with the real audio, which
greatly reduces ambiguities among the three candidate shapes. We also provide our normalized
likelihood function overtime to show our sampling has converged to produce highly probable samples.
6
Conclusion
In this paper, we propose a novel model for estimating physical properties of objects from auditory inputs, by incorporating the feedback of an efficient audio synthesis engine in the loop. We
demonstrate the possibility of accelerating inference with fast recognition models. We compare
our model predictions with human responses on a variety of judgment tasks and demonstrate the
correlation between human responses and model estimates. We also show that our model generalizes
to constrained real data.
Acknowledgements
The authors would like to thank Changxi Zheng, Eitan Grinspun, and Josh H. McDermott for helpful
discussions. This work is supported by NSF #1212849 and #1447476, ONR MURI N00014-16-12007, Toyota Research Institute, Samsung, Shell, and the Center for Brain, Minds and Machines
(NSF STC award CCF-1231216).
9
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11
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6,332 | 6,728 | Flexible statistical inference for mechanistic models of
neural dynamics
Jan-Matthis Lueckmann? 1 , Pedro J. Gon?alves? 1 , Giacomo Bassetto1 ,
Kaan ?cal1,2 , Marcel Nonnenmacher1 , Jakob H. Macke?1
1
research center caesar, an associate of the Max Planck Society, Bonn, Germany
2
Mathematical Institute, University of Bonn, Bonn, Germany
{jan-matthis.lueckmann, pedro.goncalves, giacomo.bassetto,
kaan.oecal, marcel.nonnenmacher, jakob.macke}@caesar.de
Abstract
Mechanistic models of single-neuron dynamics have been extensively studied in
computational neuroscience. However, identifying which models can quantitatively
reproduce empirically measured data has been challenging. We propose to overcome this limitation by using likelihood-free inference approaches (also known
as Approximate Bayesian Computation, ABC) to perform full Bayesian inference
on single-neuron models. Our approach builds on recent advances in ABC by
learning a neural network which maps features of the observed data to the posterior distribution over parameters. We learn a Bayesian mixture-density network
approximating the posterior over multiple rounds of adaptively chosen simulations.
Furthermore, we propose an efficient approach for handling missing features and
parameter settings for which the simulator fails, as well as a strategy for automatically learning relevant features using recurrent neural networks. On synthetic data,
our approach efficiently estimates posterior distributions and recovers ground-truth
parameters. On in-vitro recordings of membrane voltages, we recover multivariate
posteriors over biophysical parameters, which yield model-predicted voltage traces
that accurately match empirical data. Our approach will enable neuroscientists to
perform Bayesian inference on complex neuron models without having to design
model-specific algorithms, closing the gap between mechanistic and statistical
approaches to single-neuron modelling.
1
Introduction
Biophysical models of neuronal dynamics are of central importance for understanding the mechanisms
by which neural circuits process information and control behaviour. However, identifying which
models of neural dynamics can (or cannot) reproduce electrophysiological or imaging measurements
of neural activity has been a major challenge [1]. In particular, many models of interest ? such as
multi-compartment biophysical models [2], networks of spiking neurons [3] or detailed simulations
of brain activity [4] ? have intractable or computationally expensive likelihoods, and statistical
inference has only been possible in selected cases and using model-specific algorithms [5, 6, 7]. Many
models are defined implicitly through simulators, i.e. a set of dynamical equations and possibly a
description of sources of stochasticity [1]. In addition, it is often of interest to identify models which
can reproduce particular features in the data, e.g. a firing rate or response latency, rather than the full
temporal structure of a neural recording.
?
?
Equal contribution
Current primary affiliation: Centre for Cognitive Science, Technical University Darmstadt
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
A
B
C
?1
h1
?
h2
h3
?
?
?
fI(s)
?
?K
?
f1(s)
f2(s)
hH
?1
Means
?1
?
Precision
factors
?K
xo
x
proposal prior
prior
Mixture
weights
?K
true posterior
posterior
forward pass
feature 1
Figure 1: Flexible likelihood-free inference for models of neural dynamics. A. We want to
flexibly and efficiently infer the posterior over model parameters given observed data, on a wide
range of models of neural dynamics. B. Our method approximates the true posterior on ? around the
observed data xo by performing density estimation on data simulated using a proposal prior. C. We
train a Bayesian mixture-density network (MDN) for posterior density estimation.
In the absence of likelihoods, the standard approach in neuroscience has been to use heuristic
parameter-fitting methods [2, 8, 9]: distance measures are defined on multiple features of interest,
and brute-force search [10, 11] or evolutionary algorithms [2, 9, 12, 13] (neither of which scales to
high-dimensional parameter spaces) are used to minimise the distances between observed and modelderived features. As it is difficult to trade off distances between different features, the state-of-the-art
methods optimise multiple objectives and leave the final choice of a model to the user [2, 9]. As
this approach is not based on statistical inference, it does not provide estimates of the full posterior
distribution ? thus, while this approach has been of great importance for identifying ?best fitting?
parameters, it does not allow one to identify the full space of parameters that are consistent with data
and prior knowledge, or to incrementally refine and reject models.
Bayesian inference for likelihood-free simulator models, also known as Approximate Bayesian
Computation [14, 15, 16], provides an attractive framework for overcoming these limitations: like
parameter-fitting approaches in neuroscience [2, 8, 9], it is based on comparing summary features
between simulated and empirical data. However, unlike them, it provides a principled framework for
full Bayesian inference and can be used to determine how to trade off goodness-of-fit across summary
statistics. However, to the best of our knowledge, this potential has not been realised yet, and ABC
approaches are not used for linking mechanistic models of neural dynamics with experimental data
(for an exception, see [17]). Here, we propose to use ABC methods for statistical inference of
mechanistic models of single neurons. We argue that ABC approaches based on conditional density
estimation [18, 19] are particularly suited for neuroscience applications.
We present a novel method (Sequential Neural Posterior Estimation, SNPE) in which we sequentially
train a mixture-density network across multiple rounds of adaptively chosen simulations1 . Our
approach is directly inspired by prior work [18, 19], but overcomes critical limitations: first, a
flexible mixture-density network trained with an importance-weighted loss function enables us to
use complex proposal distributions and approximate complex posteriors. Second, we represent a full
posterior over network parameters of the density estimator (i.e. a ?posterior on posterior-parameters?)
which allows us to take uncertainty into account when adjusting weights. This enables us to perform
?continual learning?, i.e. to effectively utilise all simulations without explicitly having to store them.
Third, we introduce an approach for efficiently dealing with simulations that return missing values,
or which break altogether ? a common situation in neuroscience and many other applications of
simulator-based models ? by learning a model that predicts which parameters are likely to lead to
breaking simulations, and using this knowledge to modify the proposal distribution. We demonstrate
the practical effectiveness and importance of these innovations on biophysical models of single
neurons, on simulated and neurophysiological data. Finally, we show how recurrent neural networks
can be used to directly learn relevant features from time-series data.
1
Code available at https://github.com/mackelab/delfi
2
1.1
Related work using likelihood-free inference for simulator models
Given experimental data xo (e.g. intracellular voltage measurements of a single neuron, or extracellular recordings from a neural population), a model p(x|?) parameterised by ? (e.g. biophysical
parameters, or connectivity strengths in a network simulation) and a prior distribution p(?), our goal
is to perform statistical inference, i.e. to find the posterior distribution p?(?|x = xo ). We assume that
the model p(x|?) is only defined through a simulator [14, 15]: we can generate samples xn ? x|?
from it, but not evaluate p(x|?) (or its gradients) explicitly. In neural modelling, many models are
defined through specification of a dynamical system with external or intrinsic noise sources or even
through a black-box simulator (e.g. using the NEURON software [20]).
In addition, and in line with parameter-fitting approaches in neuroscience and most ABC techniques
[14, 15, 21], we are often interested in capturing summary statistics of the experimental data (e.g.
firing rate, spike-latency, resting potential of a neuron). Therefore, we can think of x as resulting
from applying a feature function f to the raw simulator output s, x = f (s), with dim(x) dim(s).
Classical ABC algorithms simulate from multiple parameters, and reject parameter sets which yield
data that are not within a specified distance from the empirically observed features. In their basic
form, proposals are drawn from the prior (?rejection-ABC? [22]). More efficient variants make
use of a Markov-Chain Monte-Carlo [23, 24] or Sequential Monte-Carlo (SMC) samplers [25, 26].
Sampling-based ABC approaches require the design of a distance metric on summary features, as
well as a rejection criterion (?), and are exact only in the limit of small ? (i.e. many rejections) [27],
implying strong trade-offs between accuracy and scalability. In SMC-ABC, importance sampling is
used to sequentially sample from more accurate posteriors while ? is gradually decreased.
Synthetic-likelihood methods [28, 21, 29] approximate the likelihood p(x|?) using multivariate
Gaussians fitted to repeated simulations given ? (see [30, 31] for generalisations). While the
Gaussianity assumption is often motivated by the central limit theorem, distributions over features can
in practice be complex and highly non-Gaussian [32]. For example, neural simulations sometimes
result in systematically missing features (e.g. spike latency is undefined if there are no spikes), or
diverging firing rates.
Finally, methods originating from regression correction [33, 18, 19] simulate multiple data xn from
different ? n sampled from a proposal distribution p?(?), and construct a conditional density estimate
q(?|x) by performing a regression from simulated data xn to ? n . Evaluating this density model at
the observed data xo , q(?|xo ) yields an estimate of the posterior distribution. These approaches do
not require parametric assumptions on likelihoods or the choice of a distance function and a tolerance
(?) on features. Two approaches are used for correcting the mismatch between prior and proposal
distributions: Blum and Fran?ois [18] proposed the importance weights p(?)/?
p(?), but restricted
themselves to proposals which were truncated priors (i.e. all importance weights were 0 or 1), and did
not sequentially optimise proposals over multiple rounds. Papamakarios and Murray [19] recently
used stochastic variational inference to optimise the parameters of a mixture-density network, and a
post-hoc division step to correct for the effect of the proposal distribution. While highly effective in
some cases, this closed-form correction step can be numerically unstable and is restricted to Gaussian
and uniform proposals, limiting both the robustness and flexibility of this approach. SNPE builds on
these approaches, but overcomes their limitations by introducing four innovations: a highly flexible
proposal distribution parameterised as a mixture-density network, a Bayesian approach for continual
learning from multiple rounds of simulations, and a classifier for predicting which parameters will
result in aborted simulations or missing features. Fourth, we show how this approach, when applied
to time-series data of single-neuron activity, can automatically learn summary features from data.
2
2.1
Methods
Sequential Neural Posterior Estimation for likelihood-free inference
In SNPE, our goal is to learn the parameters ? of a posterior model q? (?|x = f (s)) which, when
evaluated at xo , approximates the true posterior p(?|xo ) ? q? (?|x = xo ). Given a prior p(?), a
proposal prior p?(?), pairs of samples (? n , xn ) generated from the proposal prior and the simulator,
and a calibration kernel K? , the posterior model can be trained by minimising the importanceweighted log-loss
1 X p(? n )
L(?) = ?
K? (xn , xo ) log q? (? n |xn ),
(1)
N n p?(? n )
3
as is shown by extending the argument in [19] with importance-weights p(? n )/?
p(? n ) and a kernel
K? in Appendix A.
Sampling from a proposal prior can be much more effective than sampling from the prior. By
including the importance weights in the loss, the analytical correction step of [19] (i.e. division by
the proposal prior) becomes unnecessary: SNPE directly estimates the posterior density rather than
a conditional density that is reweighted post-hoc. The analytical step of [19] has the advantage of
side-stepping the additional variance brought about by importance-weights, but has the disadvantages
of (1) being restricted to Gaussian proposals, and (2) the division being unstable if the proposal prior
has higher precision than the estimated conditional density.
The calibration kernel K? (x, xo ) can be used to calibrate the loss function by focusing it on simulated
data points x which are close to xo [18]. Calibration kernels K? (x, xo ) are to be chosen such that
K? (xo , xo ) = 1 and that K? decreases with increasing distance kx ? xo k, given a bandwidth ? 2 .
Here, we only used calibration kernels to exclude bad simulations by assigning them kernel value
zero. An additional use of calibration kernels would be to limit the accuracy of the posterior density
estimation to a region near xo . Choice of the bandwidth implies a bias-variance trade-off [18]. For
the problems we consider here, we assumed our posterior model q? (?|x) based on a multi-layer
neural network to be sufficiently flexible, such that limiting bandwidth was not necessary.
P
We sequentially optimise the density estimator q? (?|x) = k ?k N (?|?k , ?k ) by training a mixturedensity network (MDN) [19] with parameters ? over multiple ?rounds? r with adaptively chosen
proposal priors p?(r) (?) (see Fig. 1). We initialise the proposal prior at the prior, p?(1) (?) = p(?),
and subsequently take the posterior of the previous round as the next proposal prior (Appendix B).
Our approach is not limited to Gaussian proposals, and in particular can utilise multi-modal and
heavy-tailed proposal distributions.
2.2
Training the posterior model with stochastic variational inference
To make efficient use of simulation time, we want the posterior network q? (?|x) to use all simulations,
including ones from previous rounds. For computational and memory efficiency, it is desirable to
avoid having to store all old samples, or having to train a new model at each round. To achieve this
goal, we perform Bayesian inference on the weights w of the MDN across rounds. We approximate
the distribution over weights as independent Gaussians [34, 35]. Note that the parameters ? of this
Bayesian MDN are are means and standard deviations per each weight, i.e., ? = {?m , ?s }. As an
extension to the approach of [19], rather than assuming a zero-centred prior over weights, we use
the posterior over weights of the previous round, ??(r?1) (w), as a prior for the next round. Using
stochastic variational inference, in each round, we optimise the modified loss
1 X p(? n )
L(?(r) ) = ?
K? (xn , xo ) log qw (? n |xn ) ? (w)
(r)
N n p? (? n )
?(r)
(2)
1
+ DKL ??(r) (w)||??(r?1) (w) .
N
Here, the distributions ?(w) are approximated by multivariate normals with diagonal covariance. The
continuity penalty ensures that MDN parameters that are already well constrained by previous rounds
are less likely to be updated than parameters with large uncertainty (see Appendix C). In practice,
gradients of the expectation over networks are approximated using the local reparameterisation trick
[36].
2.3
Dealing with bad simulations and bad features, and learning features from time series
Bad simulations: Simulator-based models, and single-neuron models in particular, frequently
generate nonsensical data (which we name ?bad simulations?), especially in early rounds in which the
relevant region of parameter space has not yet been found. For example, models of neural dynamics
can easily run into self-excitation loops with diverging firing rates [37] (Fig. 4A). We introduce
a feature b(s) = 1 to indicate that s and x correspond to a bad simulation. We set K(xn , xo ) = 0
2
While we did not investigate this here, an attractive idea would be to base the kernel of the distance between xn and xo on the divergence between the associated posteriors, e.g. K? (xn , xo ) =
exp(?1/? DKL (q (r?1) (?|xn )||q (r?1) (?|xo ))) ? in this case, two data would be regarded as similar if the
current estimation of the density network assigns similar posterior distributions to them, which is a natural
measure of similarity in this context.
4
whenever b(xn ) = 1 since the density estimator should not spend resources on approximating the
posterior for bad data. With this choice of calibration kernel, bad simulations are ignored when
updating the posterior model ? however, this results in inefficient use of simulations.
We propose to learn a model g? : ? ? [0, 1] to predict the probability that a simulation from ? will
break. While any probabilistic classifier could be used, we train a binary-output neural network with
log-loss on (? n , b(sn )). For each proposed ?, we reject ? with probability g?(?), and do not carry out
the expensive simulation3 . The rejections could be incorporated into the importance weights (which
would require estimating the corresponding partition function, or assuming it to be constant across
rounds), but as these rejections do not depend on the data xo , we interpret them as modifying the
prior: from an initially specified prior p(?), we obtain a modified prior excluding those parameters
which likely will lead to nonsensical simulations. Therefore, the predictive model g?(?) does not only
lead to more efficient inference (especially in strongly under-constrained scenarios), but is also useful
in identifying an effective prior ? the space of parameters deemed plausible a priori intersected with
the space of parameters for which the simulator is well-behaved.
Bad features: It is frequently observed that individual features of interest for fitting single-neuron
models cannot be evaluated: for example, the spike latency cannot be evaluated if a simulation
does not generate spikes, but the fact that this feature is missing might provide valuable information
(Fig. 4C). SNPE can be extended to handle ?bad features? by using a carefully designed posterior
network. For each feature fi (s), we introduce a binary feature mi (s) which indicates whether fi
is missing. We parameterise the input layer of the posterior network with multiplicative terms of
the form hi (s) = fi (s) ? (1 ? mi (s)) + ci ? mi (s) where the term ci is to be learned. This approach
effectively learns an imputation value ci for each missing feature. For a more expressive model, one
could also include terms which learn interactions across different missing-feature indicators and/or
features, but we did not explore this here.
Learning features: Finally, we point out that using a neural network for posterior estimation yields
a straightforward way of learning relevant features from data [38, 39, 40]. Rather than feeding
summary features f (s) into the network, we directly feed time-series recordings of neural activity
into the network. The first layer of the MDN becomes a recurrent layer instead of a fully-connected
one. By minimising the variational objective (Eq.2), the network learns informative summary features
about posterior densities.
3
Results
While SNPE is in principle applicable to any simulator-based model, we designed it for performing
inference on models of neural dynamics. In our applications, we concentrate on single-neuron models.
We demonstrate the ability of SNPE to recover ground-truth posteriors in Gaussian Mixtures and
Generalised Linear Models (GLMs) [41], and apply SNPE to a Hodgkin-Huxley neuron model and
an autapse model, which can have parameter regimes of unstable behaviour and missing features.
3.1
Statistical inference on simple models
Gaussian mixtures: We first demonstrate the effectiveness of SNPE for inferring the posterior of
mixtures of two Gaussians, for which we can analytically compute true posteriors. We are interested
in the numerical stability of the method (?robustness?) and the ?flexibility? to approximate multi-modal
posteriors. To illustrate the robustness of SNPE, we apply SNPE and the method proposed by [19]
(which we refer to by Conditional Density Estimation for Likelihood-free Inference, CDE-LFI) to
infer the common mean of a mixture of two Gaussians, given samples from the mixture distribution
(Fig. 2A; details in Appendix D.1). Whereas SNPE works robustly across multiple algorithmic
rounds, CDE-LFI can become unstable: its analytical correction requires a division by a Gaussian
which becomes unstable if the precision of the Gaussian does not increase monotonically across
rounds (see 2.1). Constraining the precision-matrix to be non-decreasing fixes the numerical issue,
but leads to biased estimates of the posterior. Second, we apply both SNPE and CDE-LFI to infer
the two means of a mixture of two Gaussians, given samples x from the mixture distribution (Fig.
2B; Appendix D.1). While SNPE can use bi-modal proposals, CDE-LFI cannot, implying reduced
efficiency of proposals on strongly non-Gaussian or multi-modal problems.
3
An alternative approach would be to first learn p(?|b(s) = 0) by applying SNPE to a single feature,
f1 (s) = b(s), and to subsequently run SNPE on the full feature-set, but using p(?|b(s) = 0) as prior ? however,
this would ?waste? simulations for learning p(?|b(s) = 1).
5
0
?2
2
10
0
2
3
4
5
# of rounds
6
?2
0
?
-10
2
0.1
D
true value
SNPE
PG-MCMC
?2
5
parameter
value
?10
0
?
10
?10
0
?
10
E
...
-3.0
-0.5
b0
...
10
-0.7
1.5
h1
...
-0.3
2.4
h2
...
SNPE covariance
true value
CDE-LFI
PG-MCMC
5
parameter
8
0.1
0
1
xo
-0.0
10
2
?2
0
x
0
F
SNPE
CDE-LFI
0
2
1
CDE-LFI
density
SNPE
B
PG-MCMC covariance
value
C
0
?
50
~
p(?)(2)
~(?)(6)
p
density
1
100
?
2
density
% completed runs
p *(?|x = xo)
A
-0.0
Figure 2: Inference on simple statistical models. A. Robustness of posterior inference on 1-D
Gaussian Mixtures (GMs). Left: true posterior given observation at xo = 0. Middle: percentage
of completed runs as a function of number of rounds; SNPE is robust. Right: Gaussian proposal
priors tend to underestimate tails of posterior (red). B. Flexibility of posterior inference. Left: True
posterior for 1-D bimodal GM and observation xo . Middle and right: First round proposal priors
(dotted), second round proposal priors (dashed) and estimated posteriors (solid) for CDE-LFI and
SNPE respectively (true posterior red). SNPE allows multi-modal proposals. C, F. Application to
GLM. Posterior means and variances are recovered well by both CDE-LFI and SNPE. For reference,
we approximate the posterior using likelihood-based PG-MCMC. D. Covariance matrices for SNPE
and PG-MCMC. E. Partial view of the posterior for 3 out of 10 parameters (all 10 parameters in
Appendix G). Ground-truth parameters in red. 2-D marginals for SNPE (lines) and PG-MCMC
(histograms). White and yellow contour lines correspond to 68% and 95% of the mass, respectively.
Generalised linear models: Generalised linear models (GLM) are commonly used to model
neural responses to sensory stimuli. For these models, several techniques are available to estimate the
posterior distribution over parameters, making them ideally suited to test SNPE in a single-neuron
model. We evaluated the posterior distribution over the parameters of a GLM using a P?lya-Gamma
sampler (PG-MCMC, [42, 43]) and compared it to the posterior distributions estimated by SNPE
(Appendix D.2 for details). We found a good agreement of the posterior means and variances (Fig.
2C), covariances (Fig. 2D), as well as pairwise marginals (Fig. 2E). We note that, since GLMs have
close-to-Gaussian posteriors, the CDE-LFI method works extremely well on this problem (Fig. 2F).
In summary, SNPE leads to accurate and robust estimation of the posterior in simple models. It works
effectively even on multi-modal posteriors on which CDE-LFI exhibits worse performance. On a
GLM-example with an (almost) Gaussian posterior, the CDE-LFI method works extremely well,
but SNPE yields very similar posterior estimates (see Appendix F for additional comparison with
SMC-ABC).
3.2
Statistical inference on Hodgkin-Huxley neuron models
Simulated data: The Hodgkin-Huxley equations [44] describe the dynamics of a neuron?s membrane potential and ion channels given biophysical parameters (e.g. concentration of sodium and
potassium channels) and an injected input current (Fig. 3A, see Appendix D.3). We applied SNPE
to a Hodgkin-Huxley model with channel kinetics as in [45] and inferred the posterior over 12
biophysical parameters, given 20 voltage features of the simulated data. The true parameter values are
close to the mode of the inferred posterior (Fig. 3B, D), and in a region of high posterior probability.
Samples from the posterior lead to voltage traces that are similar to the original data, supporting the
correctness of the approach (Fig. 3C).
6
4.3
?80
0.55
0.00
...
ln(gNa)
0.9
2.0
60
time (ms)
40
?20
...
ln(gK)
0
C
voltage (mV)
3.2
?20
120
-3.0
?80
-1.9
ln(gl)
2.3
0
60
time (ms)
120
SNPE mean
best IBEA
1.2
E
40
voltage (mV)
0.0
?20
gNa
gK
ENa
kbn1
noise
VT
gM
?El
F
...
3.2
4.3
...
ln(gNa)
?80
0.19
0.00
0.9
2.0
625
time (ms)
1250
tmax
G
gl
kbn2
?EK
40
?20
...
ln(gK)
0
?80
...
input (nA)
...
voltage (mV)
||? - ? * || / ??
D
B
40
...
input (nA)
voltage (mV)
A
-3.0
-1.9
ln(gl)
0
625
time (ms)
1250
Figure 3: Application to Hodgkin-Huxley model: A. Simulation of Hodgkin-Huxley model with
current injection. B. Posterior over 3 out of 12 parameters inferred with SNPE (12 parameters in
Appendix G). True parameters have high posterior probabilities (red). C. Traces for the mode (cyan)
of and samples (orange) from the inferred posterior match the original data (blue). D. Comparison
between SNPE and a standard parameter-fitting procedure based on a genetic algorithm, IBEA:
difference between the mode of SNPE or IBEA best parameter set, and the ground-truth parameters,
normalised by the standard deviations obtained by SNPE. E-G. Application to real data from Allen
Cell Type Database. Inference over 12 parameters for cell 464212183. Results presented as in A-C.
Biophysical neuron models are typically fit to data with genetic algorithms applied to the distance
between simulated and measured data-features [2, 8, 9, 46]. We compared the performance of SNPE
with a commonly used genetic algorithm (Indicator Based Evolutionary Algorithm, IBEA, from the
BluePyOpt package [9]), given the same number of model simulations (Fig. 3D). SNPE is comparable
to IBEA in approximating the ground-truth parameters ? note that defining an objective measure to
compare the two approaches is difficult, as they both minimise different criteria. However, unlike
IBEA, SNPE also returns a full posterior distribution, i.e. the space of all parameters consistent with
the data, rather than just a ?best fit?.
In-vitro recordings: We also applied the approach to in vitro recordings from the mouse visual
cortex (see Appendix D.4, Fig. 3E-G). The posterior mode over 12 parameters of a Hodgkin-Huxley
model leads to a voltage trace which is similar to the data, and the posterior distribution shows the
space of parameters for which the output of the model is preserved. These posteriors could be used to
motivate further experiments for constraining parameters, or to study invariances in the model.
3.3
Dealing with bad simulations and features
Bad simulations: We demonstrate our approach (see Section 2.3) for dealing with ?bad simulations?
(e.g. for which firing rates diverge) using a simple, two-parameter ?autapse? model for which the region
of stability is known. During SNPE, we concurrently train a classifier to predict ?bad simulations? and
update the prior accordingly. This approach does not only lead to a more efficient use of simulations,
but also identifies the parameter space for which the simulator is well-defined, information that could
be used for further model analysis (Fig. 4A, B).
Bad features: Many features of interest in neural models, e.g. the latency to first spike after the
injection of a current input, are only well defined in the presence of other features, e.g. the presence
of spikes (Fig. 4C). Given that large parts of the parameter space can lead to non-spiking behaviour,
missing features occur frequently and cannot simply be ignored. We enriched our MDN with an extra
layer which imputes values to the absent features, values which are optimised alongside the rest of
the parameters of the network (Fig. 4D; Appendix E). Such imputation has marginal computational
7
0.0
2.5
C
102
101
g^ (?)
?
rate (Hz)
103
voltage (mV)
B
104
D
40
m1(s)
c1
m2(s)
?
?20
+
100
0
observed data
bad simulation
10?1
0
50
time (ms)
100
-1
0.0
1
J
2.0
1.0
h1
?
h2
h3
1-m1(s)
input (mA)
A
?80
3.6
?
?
f1(s)
?
f2(s)
0.0
0
60
time (ms)
120
?
hH
?
Figure 4: Inference on neural dynamics has to deal with diverging simulations and missing
features. A. Firing rate of a model neuron connected to itself (autapse). If the strength of the selfconnection (parameter J) is bigger than 1, the dynamics are unstable (orange line - bad simulation).
B. Portion of parameter space leading to diverging simulations learned by the classifier (yellow: low
probability of bad simulation, blue: high probability), and comparison with analytically computed
boundaries (white, see Appendix D.5). C. Illustration of a model neuron in two parameter regimes,
spiking (grey trace) and non-spiking (blue). When the neuron does not spike, features that depend on
the presence of spiking, such as the latency to first spike, are not defined. D. Our MDN is augmented
with a multiplicative layer which imputes values for missing features.
cost and grants us the convenience of not having to hand-tune imputation values, or to reject all
simulations for which any individual feature might be missing.
Learning features with recurrent neural networks (RNNs): In neural modelling, it is often
of interest to work with hand-designed features that are thought to be particularly important or
informative for particular analysis questions [2]. For instance, the shape of the action potential is
intimately related to the dynamics of sodium and potassium channels in the Hodgkin-Huxley model.
However, the space of possible features is immense, and given the highly non-linear nature of many of
the neural models in question, it can sometimes be of interest to simply perform statistical inference
without having to hand-design features. Our approach provides a straightforward means of doing that:
we augment the MDN with a RNN which runs along the recorded voltage trace (and stimulus, here a
coloured-noise input) to learn appropriate features to constrain the model parameters. As illustrated in
figure 5B, the first layer of the network, which previously received pre-computed summary statistics
as inputs, is replaced by a recurrent layer that receives full voltage and current traces as inputs. In
order to capture long-term dependencies in the sequence input, we use gated-recurrent units (GRUs)
for the RNN [47]. Since we are using 25 GRU units and only keep the final output of the unrolled
RNN (many-to-one), we introduce a bottleneck. The RNN thus transforms the voltage trace and
stimulus into a set of 25 features, which allow SNPE to recover the posterior over the 12 parameters
(Fig. 5C). As expected, the presence of spikes in the observed data leads to a tighter posterior for
parameters associated to the main ion channels involved in spike generation, ENa , EK , gNa and gK .
4
Discussion
Quantitatively linking models of neural dynamics to data is a central problem in computational
neuroscience. We showed that likelihood-free inference is at least as general and efficient as ?blackbox? parameter fitting approaches in neuroscience, but provides full statistical inference, suggesting
it to be the method of choice for inference on single-neuron models. We argued that ABC approaches
based on density estimation are particularly useful for neuroscience, and introduced a novel algorithm
(SNPE) for estimating posterior distributions. We can flexibly and robustly estimate posterior
distributions, even when large regions of the parameter space correspond to unstable model behaviour,
or when features of choice are missing. Furthermore, we have extended our approach with RNNs to
automatically define features, thus increasing the potential for better capturing salient aspects of the
data with highly non-linear models. SNPE is therefore equipped to estimate posterior distributions
under common constraints in neural models.
Our approach directly builds on a recent approach for density estimation ABC (CDE-LFI, [19]).
While we found CDE-LFI to work well on problems with unimodal, close-to-Gaussian posteriors and
stable simulators, our approach extends the range of possible applications, and these extensions are
critical for the application to neuron models. A key component of SNPE is the proposal prior, which
guides the sampling on each round of the algorithm. Here, we used the posterior on the previous
round as the proposal for the next one, as in CDE-LFI and in many Sequential-MC approaches. Our
8
voltage (mV)
A
B
40
60
240
?20
?80
C
v1
i1
v2
i2
v3
i3
?
?
vT
iT
GRUs
GRUs
GRUs
?
GRUs
input (nA)
Features
2.55
0.00
f1 f2 ? fN
60
240
gNa
gK
gl
ENa
?EK
?El
gM
tmax
kbn1
kbn2
VT
noise
Mixture Density Network
0
120
time (ms)
240
Figure 5: We can learn informative features using a recurrent mixture-density network (RMDN). A. We consider a neuron driven by a colored-noise input current. B. Rather than engineering
summary features to reduce the dimensionality of observations, we provide the complete voltage
trace and input current as input to an R-MDN. The unrolled forward pass is illustrated, where a
many-to-one recurrent network reduces the dimensionality of the inputs (T time steps long) to a
feature vector of dimensionality N . C. Our goal is to infer the posterior density for two different
observations: (1) the full 240ms trace shown in panel A; and (2) the initial 60ms of its duration, which
do not show any spike. We show the obtained marginal posterior densities for the two observations,
using a 25-dimensional feature vector learned by the RNN. In the presence of spikes, the posterior
uncertainty gets tighter around the true parameters related to spiking.
method could be extended by alternative approaches to designing proposal priors [48, 49], e.g. by
exploiting the fact that we also represent a posterior over MDN parameters: for example, one could
design proposals that guide sampling towards regions of the parameter space where the uncertainty
about the parameters of the posterior model is highest. We note that, while here we concentrated
on models of single neurons, ABC methods and our approach will also be applicable to models of
populations of neurons. Our approach will enable neuroscientists to perform Bayesian inference on
complex neuron models without having to design model-specific algorithms, closing the gap between
mechanistic and statistical models, and enabling theory-driven data-analysis [50].
Acknowledgements
We thank Maneesh Sahani, David Greenberg and Balaji Lakshminarayanan for useful comments
on the manuscript. This work was supported by SFB 1089 (University of Bonn) and SFB 1233
(University of T?bingen) of the German Research Foundation (DFG) to JHM and by the caesar
foundation.
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11
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6,333 | 6,729 | Online Prediction with Selfish Experts
Tim Roughgarden
Department of Computer Science
Stanford University
Stanford, CA 94305
[email protected]
Okke Schrijvers
Department of Computer Science
Stanford University
Stanford, CA 94305
[email protected]
Abstract
We consider the problem of binary prediction with expert advice in settings where
experts have agency and seek to maximize their credibility. This paper makes
three main contributions. First, it defines a model to reason formally about settings
with selfish experts, and demonstrates that ?incentive compatible? (IC) algorithms
are closely related to the design of proper scoring rules. Second, we design
IC algorithms with good performance guarantees for the absolute loss function.
Third, we give a formal separation between the power of online prediction with
selfish versus honest experts by proving lower bounds for both IC and non-IC
algorithms. In particular, with selfish experts and the absolute loss function,
there is no (randomized) algorithm for online prediction?IC or otherwise?with
asymptotically vanishing regret.
1
Introduction
In the months leading up to elections and referendums, a plethora of pollsters try to figure out
how the electorate is going to vote. Different pollsters use different methodologies, reach different
people, and may have sources of random errors, so generally the polls don?t fully agree with each
other. Aggregators such as Nate Silver?s FiveThirtyEight, and The Upshot by the New York Times
consolidate these different reports into a single prediction, and hopefully reduce random errors.1
FiveThirtyEight in particular has a solid track record for their predictions, and as they are transparent
about their methodology we use them as a motivating example. To a first-order approximation, they
operate as follows: first they take the predictions of all the different pollsters, then they assign a
weight to each of the pollsters based on past performance (and other factors), and finally they use the
weighted average of the pollsters to run simulations and make their own prediction.2
But could the presence of an institution that rates pollsters inadvertently create perverse incentives
for pollsters? The FiveThirtyEight pollster ratings are publicly available.3 They can be interpreted
as a reputation, and a low rating can negatively impact future revenue opportunities for a pollster.
Moreover, it has been demonstrated in practice that experts do not always report their true beliefs
about future events. For example, in weather forecasting there is a known ?wet bias,? where consumerfacing weather forecasters deliberately overestimate low chances of rain (e.g. a 5% chance of rain is
reported as a 25% chance of rain) because people don?t like to be surprised by rain [Bickel and Kim,
2008].
1
https://fivethirtyeight.com/, https://www.nytimes.com/section/upshot.
This is of course a simplification. FiveThirtyEight also uses features like the change in a poll over time,
the state of the economy, and correlations between states. See https://fivethirtyeight.com/features/
how-fivethirtyeight-calculates-pollster-ratings/ for details. Our goal in this paper is not to
accurately model all of the fine details of FiveThirtyEight (which are anyways changing all the time). Rather, it
is to formulate a general model of prediction with experts that clearly illustrates why incentives matter.
3
https://projects.fivethirtyeight.com/pollster-ratings/
2
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
These examples motivate the development of models of aggregating predictions that endow agency to
the data sources.4 While there are multiple models in which we can investigate this issue, a natural
candidate is the problem of prediction with expert advice. By focusing on a standard model, we
abstract away from the fine details of FiveThirtyEight (which are anyways changing all the time),
which allows us to formulate a general model of prediction with experts that clearly illustrates why
incentives matter. In the classical model [Littlestone and Warmuth, 1994, Freund and Schapire, 1997],
at each time step, several experts make predictions about an unknown event. An online prediction
algorithm aggregates experts? opinions and makes its own prediction at each time step. After this
prediction, the event at this time step is realized and the algorithm incurs a loss as a function of its
prediction and the realization. To compare its performance against individual experts, for each expert
the algorithm calculates what its loss would have been had it always followed the expert?s prediction.
While the problems introduced in this paper are relevant for general online prediction, to focus on
the most interesting issues we concentrate on the case of binary events, and real-valued predictions
in [0, 1]. For different applications, different notions of loss are appropriate, so we parameterize the
model by a loss function `. Thus our formal model is: at each time step t = 1, 2, . . . , T :
(t)
1. Each expert i makes a prediction pi 2 [0, 1], representing advocacy for event ?1.?
2. The online algorithm commits to a probability q (t) 2 [0, 1] as a prediction for event ?1.?
3. The outcome r(t) 2 {0, 1} is realized.
(t)
4. The algorithm incurs expected loss `(q (t) , r(t) ), each expert i is assigned loss `(pi , r(t) ).
The standard goal in this problem is to design an online prediction algorithm that is guaranteed to
have expected loss not much larger than that incurred by the best expert in hindsight. The classical
solutions maintain a weight for each expert and make a prediction according to which outcome has
more expert weight behind it. An expert?s weight can be interpreted as a measure of its credibility in
light of its past performance. The (deterministic) Weighted Majority (WM) algorithm always chooses
the outcome with more expert weight. The Randomized Weighted Majority (RWM) algorithm
randomizes between the two outcomes with probability proportional to their total expert weights.
(t)
The most common method of updating experts? weights is via multiplication by 1 ?`(pi , r(t) )
after each time step t, where ? is the learning rate. We call this the ?standard? or ?classical? version
of the WM and RWM algorithms.
The classical model instills no agency in the experts. To account for this, in this paper we replace
Step 1 of the classical model by:
(t)
1a. Each expert i formulates a belief bi 2 [0, 1].
(t)
1b. Each expert i reports a prediction pi 2 [0, 1] to the algorithm.
(t)
Each expert now has two types of loss at each time step ? the reported loss `(pi , r(t) ) with respect
(t)
to the reported prediction and the true loss `(bi , r(t) ) with respect to her true beliefs.5
When experts care about the weight that they are assigned, and with it their reputation and influence
in the algorithm, different loss functions can lead to different expert behaviors. For example, for
the quadratic loss function, in the standard WM and RWM algorithms, experts have no reason to
misreport their beliefs (see Proposition 8). This is not the case for other loss functions, such as the
absolute loss function.6 The standard algorithm with the absolute loss function incentivizes extremal
(t)
1
reporting, i.e. an expert reports 1 whenever bi
2 and 0 otherwise. This follows from a simple
4
More generally, one can investigate how the presence of machine learning algorithms affects data-generating
processes, either during learning or deployment. We discuss some of this work in the related work section.
5
When we speak of the best expert in hindsight, we are always referring to the true losses. Guarantees with
respect to reported losses follow from standard results [Littlestone and Warmuth, 1994, Freund and Schapire,
1997, Cesa-Bianchi et al., 2007], but are not immediately meaningful.
6
The loss function is often tied to the particular application. For example, in the current FiveThirtyEight pollster rankings, the performance of a pollster is primarily measured according to an absolute
loss function and also whether the candidate with the highest polling numbers ended up winning (see
https://github.com/fivethirtyeight/data/tree/master/pollster-ratings). However, in 2008
FiveThirtyEight used the notion of ?pollster introduced error? or PIE, which is the square root of a difference
of squares, as the most important feature in calculating the weights, see https://fivethirtyeight.com/
features/pollster-ratings-v31/.
2
derivation or alternatively from results in the property elicitation literature.7 This shows that for the
absolute loss function the standard WM algorithm is not ?incentive-compatible? in a sense that we
formalize in Section 2. There are similar examples for the other commonly studied weight update
rules and for the RWM algorithm. We might care about truthful reporting for its own sake, but
additionally the worry is that non-truthful reports will impede our ability to get good regret guarantees
(with respect to experts? true losses).
We study several fundamental questions about online prediction with selfish experts:
1. What is the design space of ?incentive-compatible? online prediction algorithms, where
every expert is incentivized to report her true beliefs?
2. Given a loss function like absolute loss, are there incentive-compatible algorithms with good
regret guarantees?
3. Is online prediction with selfish experts strictly harder than in the classical model with
honest experts?
Our Results. The first contribution of this paper is the development of a model for reasoning
formally about the design and analysis of weight-based online prediction algorithms when experts are
selfish (Section 2), and the definition of an ?incentive-compatible? (IC) such algorithm. Intuitively, an
IC algorithm is such that each expert wants to report its true belief at each time step. We demonstrate
that the design of IC online prediction algorithms is closely related to the design of strictly proper
scoring rules. Using this, we show that for the quadratic loss function, the standard WM and RWM
algorithms are IC, whereas these algorithms are not generally IC for other loss functions.
Our second contribution is the design of IC prediction algorithms for the absolute loss function with
non-trivial performance guarantees. For example, our best result for deterministic algorithms is: the
WM algorithm, with experts? weights evolving
according to the spherical proper scoring rule (see
p
Section 3), is IC and has loss at most 2 + 2 times the loss of best expert in hindsight (in the limit as
T ! 1). A variant of the RWM algorithm with the Brier scoring rule is IC and has expected loss at
most 2.62 times that of the best expert in hindsight (also in the limit, see Section 5).
Our third and most technical contribution is a formal separation between online prediction with
selfish experts and the traditional setting with honest experts. Recall that with honest experts, the
classical (deterministic) WM algorithm has loss at most twice that of the best expert in hindsight (as
T ! 1) [Littlestone and Warmuth, 1994]. We prove in Section 4 that the worst-case loss of every
(deterministic) IC algorithm, and every (deterministic) non-IC algorithm satisfying mild technical
conditions, is bounded away from twice that of the best expert in hindsight (even as T ! 1).
A consequence of our lower bound is that, with selfish experts, there is no natural (randomized)
algorithm for online prediction?IC or otherwise?with asymptotically vanishing regret.
Finally, in Section 6 we show simulations that indicate that different IC methods show similar regret
behavior, and that their regret is substantially better than that of the non-IC standard algorithms,
suggesting that the worst-case characterization we prove holds more generally.
Related Work. We believe that our model of online prediction over time with selfish experts is
novel. We next survey the multiple other ways in which online learning and incentive issues have
been blended, and the other efforts to model incentive issues in machine learning.
There is a large literature on prediction and decision markets (e.g. Chen and Pennock [2010], Horn
et al. [2014]), which also aim to aggregate information over time from multiple parties and make use
of proper scoring rules to do it. However, prediction markets provide incentives through payments,
rather than influence, and lack the feedback mechanism to select among experts. While there are
strong mathematical connections between cost function-based prediction markets and regularizationbased online learning algorithms in the standard (non-IC) model [Abernethy et al., 2013], there does
not appear to be any interesting implications for online prediction with selfish experts.
There is also an emerging literature on ?incentivizing exploration? in partial feedback models such as
the bandit model (e.g. Frazier et al. [2014], Mansour et al. [2016]). Here, the incentive issues concern
the learning algorithm itself, rather than the experts (or ?arms?) that it makes use of.
7
The absolute loss function is known to elicit the median [Bonin, 1976][Thomson, 1979], and since we have
binary realizations, the median is either 0 or 1.
3
The question of how an expert should report beliefs has been studied before in the literature on strictly
proper scoring rules [Brier, 1950, McCarthy, 1956, Savage, 1971, Gneiting and Raftery, 2007], but
this literature typically considers the evaluation of a single prediction, rather than low-regret learning.
Bayarri and DeGroot [1989] look at correlated settings where strictly proper scoring rules don?t
suffice, though they also do not consider how an aggregator can achieve low regret.
Finally, there are many works that fall under the broader umbrella of incentives in machine learning.
Roughly, work in this area can be divided into two genres: incentives during the learning stage, e.g.
[Cai et al., 2015, Shah and Zhou, 2015, Liu and Chen, 2016, Dekel et al., 2010], or incentives during
the deployment stage, e.g. Br?ckner and Scheffer [2011], Hardt et al. [2016]. Finally, Babaioff et al.
[2010] consider the problem of no-regret learning with selfish experts in an ad auction setting, where
the incentives come from the allocations and payments of the auction, rather than from weights as in
our case.
2
Preliminaries and Model
Standard Model. At each time step t 2 1, ..., T we want to predict a binary realization r(t) 2
{0, 1}. To help in the prediction, we have access to n experts that for each time step report a prediction
(t)
pi 2 [0, 1] about the realization. The realizations are determined by an oblivious adversary, and
the predictions of the experts may or may not be accurate. The goal is to use the predictions of the
experts in such a way that the algorithm performs nearly as well as the best expert in hindsight. Most
of the algorithms proposed for this problem fall into the following framework.
Definition 1 (Weight-update Online Prediction Algorithm). A weight-update online prediction algoPn
(t)
(t) (t)
rithm maintains a weight wi for each expert and makes its prediction q (t) based on i=1 wi pi
Pn (t)
(t)
and i wi (1 pi ). After the algorithm makes its prediction, the realization r(t) is revealed, and
the algorithm updates the weights of experts using the rule
?
?
(t+1)
(t)
(t)
wi
= f pi , r(t) ? wi ,
(1)
where f : [0, 1] ? {0, 1} ! R+ is a positive function on its domain.
(t)
(t)
The standard WM algorithm has f (pi , r(t) ) = 1 ?`(pi , r(t) ) where ? 2 (0, 12 ) is the learning
Pn (t) (t) Pn (t)
(t)
rate, and predicts q (t) = 1 if and only if i wi pi
i wi (1 pi ). Let the total loss of the alP
PT
(T )
(t)
T
gorithm be M (T ) = t=1 `(q (t) , r(t) ) and let the total loss of expert i be mi = t=1 `(pi , r(t) ).
(T )
n
The MW algorithm has the property that M (T ) ? 2(1 + ?)mi + 2 ln
for each expert i, and
?
Pn (t) (t)
RWM ?where the algorithm picks 1 with probability proportional to i wi pi ? satisfies
(T )
M (T ) ? (1 + ?)mi + ln?n for each expert i [Littlestone and Warmuth, 1994][Freund and Schapire,
1997]. The notion of ?no ?-regret? [Kakade et al., 2009] captures the idea that the per time-step loss
of an algorithm is ? times that of the best expert in hindsight, plus a term that goes to 0 as T grows:
(T )
Definition 2 (?-regret). An algorithm is said to have no ?-regret if M (T ) ? ? mini mi + o(T ).
p
By taking ? = O(1/ T ), MW is a no 2-regret algorithm, and RWM is a no 1-regret algorithm.
Selfish Model. We consider a model in which experts have agency about the prediction they report,
and care about the weight that they are assigned. In the selfish model, at time t the expert formulates
(t)
(t)
a private belief bi about the realization, but she is free to report any prediction pi to the algorithm.
Let Bern(p) be a Bernoulli random variable with parameter p. For any non-negative weight update
function f ,
?
?
(t+1)
(t)
(t)
?
?
?
?
max Eb(t) [wi
] = max Er?Bern b(t) [f (p, r) wi ] = wi ? max Er?Bern b(t) [f (p, r)] .
p
i
p
So expert i will report whichever
p
i
(t)
pi
i
will maximize the expectation of the weight update function.
Performance of an algorithm with respect to the reported loss of experts follows from the standard
analysis [Littlestone and Warmuth, 1994]. However, the true loss may be worse (in Section 3 we
4
show this for the standard update rule, Section 4 shows it more generally). Unless explicitly stated
PT
(T )
(t)
otherwise, in the remainder of this paper mi = t=1 `(bi , r(t) ) refers to the true loss of expert i.
(t)
(t)
For now this motivates restricting the weight update rule f to functions where reporting pi = bi
maximizes the expected weight of experts. We call these weight-update rules Incentive Compatible
(IC).
Definition 3 (Incentive Compatibility). A weight-update function f is incentive compatible (IC) if
(t)
reporting the true belief bi is always a best response for every expert at every time step. It is strictly
(t)
(t)
IC when pi = bi is the only best response.
By a ?best response,? we mean an expected utility-maximizing report, where the expectation is with
respect to the expert?s beliefs.
Collusion. The definition of IC does not rule out the possibility that experts can collude to jointly
misreport to improve their weights. We therefore also consider a stronger notion of incentive
compatibility for groups with transferable utility.8
Definition 4 (IC for Groups with Transferable Utility). A weight-update function f is IC for groups
with transferable utility (TU-GIC) if for every subset S of players, the total expected weight of the
P
(t+1)
(t)
group i2S Eb(t) [wi
] is maximized by each reporting their private belief bi .
i
Proper Scoring Rules. Incentivizing truthful reporting of beliefs has been studied extensively, and
the set of functions that do this is called the set of proper scoring rules. Since we focus on predicting
a binary event, we restrict our attention to this class of functions.
Definition 5 (Binary Proper Scoring Rule, [Schervish, 1989]). A function f : [0, 1] ? {0, 1} !
R [ {?1} is a binary proper scoring rule if it is finite except possibly on its boundary and whenever
for p 2 [0, 1] it holds that p 2 maxq2[0,1] p ? f (q, 1) + (1 p) ? f (q, 0).
A function f is a strictly proper scoring rule if p is the only value that maximizes the expectation.
The first and perhaps most well-known proper scoring rule is the Brier scoring rule.
Example 6 (Brier Scoring Rule, [Brier, 1950]). The Brier score is Br(p, r) = 2pr
where pr = pr + (1 p)(1 r) is the report for the event that materialized.
(p2 + (1
p)2 )
We will use the Brier scoring rule in Section 5 to construct an incentive-compatible randomized
algorithm with good guarantees. The following proposition follows directly from Definitions 3 and 5.
Proposition 7. Weight-update rule f is (strictly) IC if and only if f is a (strictly) proper scoring rule.
Surprisingly, this result remains true even when experts can collude. While the realizations are
obviously correlated, linearity of expectation causes the sum to be maximized exactly when each
expert maximizes their own expected weight.
Proposition 8. A weight-update rule f is (strictly) incentive compatible for groups with transferable
utility if and only if f is a (strictly) proper scoring rule.
Thus, for online prediction with selfish experts, we get TU-GIC ?for free.? It is quite uncommon for
problems in non-cooperate game theory to admit good TU-GIC solutions. For example, results for
auctions (either for revenue or welfare) break down once bidders collude, see e.g. [Goldberg and
Hartline, 2005]. In the remainder of the paper we will simply use IC to refer to IC and TU-GIC, as
strictly proper scoring rules yield algorithms that satisfy both definitions.
Thus, for IC algorithms we are restricted to considering (bounded) proper scoring rules as weightupdate rules. Conversely, any bounded scoring rule can be used, possibly after an affine transformation
(which preserve proper-ness). Are there any proper scoring rules that give an online prediction
algorithm with a good performance guarantee? The standard algorithm for quadratic losses yields a
weight-update function that is equivalent to the Brier strictly proper scoring rule, and thus is IC. The
standard algorithm with absolute losses is not IC, so in the remainder of this paper we discuss this
setting in more detail.
8
Note that TU-GIC is a strictly stronger concept than IC and group IC with nontransferable utility (NTU-GIC)
[Moulin, 1999][Jain and Mahdian, 2007].
5
3
Deterministic Algorithms for Selfish Experts
This section studies the question if there are good online prediction algorithms with selfish experts
for the absolute loss function. We restrict our attention here to deterministic algorithms; Section 5
gives a randomized algorithm with good guarantees.
Proposition 7 tells us that for selfish experts to have a strict incentive to report truthfully, the weightupdate rule must be a strictly proper scoring rule. This section
p gives a deterministic algorithm based
on the spherical strictly proper scoring rule that has no (2 + 2)-regret (Theorem 10). Additionally,
we consider the question if the non-truthful reports from experts in using the standard (non-IC) WM
algorithm are harmful. We show that this is the case by proving it is not a no (4 O(1))-regret
algorithm for any constant smaller than 4 (Proposition 11). This shows that, when experts are selfish,
the IC online prediction algorithm with the spherical rule outperforms the standard WM algorithm
(in the worst case).
Online Prediction using a Spherical Rule. We next give an algorithm that uses a strictly proper
scoring rule that is based on the spherical rule scoring rule.9 Consider the following weight-update
rule:
?
?
?
?
?
? q
(t) (t)
(t)
(t)
(t)
(t)
(t)
(t)
fsp pi , r
=1 ? 1
1 |pi
r | / pi ? pi + (1 pi ) ? (1 pi ) . (2)
The following proposition establishes that this is in fact a strictly proper scoring rule. Due to space
constraints, all proofs appear in Appendix A of the supplementary material.
Proposition 9. The spherical weight-update rule in (2) is a strictly proper scoring rule.
In addition to incentivizing truthful reporting, the WM algorithm with the update rule fsp does not do
much worse than the best expert in hindsight.
p
p
Theorem 10. WM with weight-update rule (2) for ? = O(1/ T ) < 12 has no (2 + 2)-regret.
True Loss of the Non-IC Standard Rule. It is instructive to compare the guarantee in Theorem 10
with the performance of the standard (non-IC) WM algorithm. WM with the standard weight update
(t)
(t)
function f (pi , r(t) ) = 1 ?|pi
r(t) | for ? 2 (0, 12 ) has no 2-regret with respect to the reported
loss of experts. However, this algorithm incentivizes extremal reports (for details see Appendix B in
the supplementary material), and in the worst case, this algorithm?s loss can be as bad as 4 times the
true loss of the best expert in hindsight. Theorem 10 shows that a suitable IC algorithm obtains a
superior worst-case guarantee.
?
?
(t)
(t)
Proposition 11. The standard WM algorithm with weight-update rule f pi , r(t) = 1 ?|pi
r(t) | results in a total worst-case loss no better than M (T )
4
(T )
4 ? mini mi
o(1).
The Cost of Selfish Experts
We now address the third fundamental question: whether or not online prediction with selfish experts
is strictly harder than with honest experts. As there exists a deterministic algorithm for honest experts
with no 2-regret, showing a separation between honest and selfish experts boils down to proving that
there exists a constant > 0 such that best possible no ?-regret algorithm has ? = 2 + . In this
section we show that such a exists, and that it is independent of the learning rate. Hence the lower
bound also holds for algorithms that, like the classical prediction algorithms, use a time-varying
learning rate. Due to space considerations, this section only states the main results, for details
and proofs refer to the supplementary materials where in Appendix D we give the results for IC
algorithms, and in Appendix E we give the results for the non-IC algorithms. We extend these results
to randomized algorithms in Section 5, where we rule out the existence of a (possibly randomized)
no-regret algorithm for selfish experts.
9
In Appendix G in the supplementary materials we give an intuition for why this rule yields better results
than other natural candidates, such as the Brier scoring rule.
6
IC Algorithms. To prove the lower bound, we have to be specific about which set of algorithms
we consider. To cover algorithms that have a decreasing learning parameter, we first show that any
positive proper scoring rule can be interpreted as having a learning parameter ?.
Proposition 12. Let f be any strictly proper scoring rule. We can write f as f (p, r) = a + bf 0 (p, r)
with a 2 R, b 2 R+ and f 0 a strictly proper scoring rule with min(f 0 (0, 1), f 0 (1, 0)) = 0 and
max(f 0 (0, 0), f 0 (1, 1)) = 1.
We call f 0 : [0, 1] ? {0, 1} ! [0, 1] a normalized scoring rule. Using normalized scoring rules, we
can define a family of scoring rules with different learning rates ?. Define F as the following family
of proper scoring rules generated by normalized strictly proper scoring rule f :
F = {f 0 (p, r) = a (1 + ?(f (p, r)
1)) : a > 0 and ? 2 (0, 1)}
By Proposition 12 the union of families generated by normalized strictly proper scoring rules cover
all strictly proper scoring rules. Using this we can now formulate the class of deterministic algorithms
that are incentive compatible.
Definition 13 (Deterministic IC Algorithms). Let Ad be the set of deterministic algorithms that
(t+1)
(t)
(t)
update weights by wi
= a(1 + ?(f (pi , r(t) ) 1))wi , for a normalized strictly proper scoring
Pn
(t) (t) Pn
(t)
rule f and ? 2 (0, 12 ) with ? possibly decreasing over time. For q = i=1 wi pi / i=1 wi , A
picks q (t) = 0 if q < 12 , q (t) = 1 if q > 12 and uses any deterministic tie breaking rule for q = 12 .
Using this definition we can now state our main lower bound result for IC algorithms:
Theorem 14. For the absolute loss function, there does not exists a deterministic and incentivecompatible algorithm A 2 Ad with no 2-regret.
Of particular interest are symmetric scoring rules, which occur often in practice, and which have a
relevant parameter that drives the lower bound results:
Definition 15 (Scoring Rule Gap). The scoring rule gap of family F with generator f is =
f ( 12 ) 12 (f (0) + f (1)) = f ( 12 ) 12 .
By definition, the scoring rule gap for strictly proper scoring rules is strictly positive, and it drives the
lower bound for symmetric functions:
Lemma 16. Let F be a family of scoring rules generated by a symmetric strictly proper scoring rule
0
f , and let be the scoring rule gap of F. In the
? worst case,
? MW with any scoring rule f from F
(T
)
with ? 2 (0, 12 ) can do no better than M (T )
2 + d 1 1 e ? mi .
As a consequence of Lemma 16, we can calculate lower bounds for specific strictly proper scoring
rules. For example, the spherical
rule used in Section 3 is a symmetric strictly proper scoring rule
p
2
1
1
with a gap parameter = 2
e = 15 .
2 , and hence 1/d
Non-IC Algorithms. What about non-incentive-compatible algorithms? Could it be that, even
with experts reporting strategically instead of honestly, there is a deterministic algorithm with loss at
most twice that of the best expert in hindsight (or a randomized algorithm with vanishing regret), to
match the classical results for honest experts? Under mild technical conditions, the answer is no. The
following definition captures how players are incentivized to report differently from their beliefs.
Definition 17 (Rationality Function). For a weight update function f , let ?f : [0, 1] ! [0, 1] be the
function from beliefs to predictions, such that reporting ?f (b) is rational for an expert with belief b.
Under mild technical conditions on the rationality function, we show our main lower bound for
(potentially non-IC) algorithms.
Theorem 18. For a weight update function f with continuous or non-strictly increasing rationality
function ?f , there is no deterministic no 2-regret algorithm.
Note that Theorem 18 covers the standard algorithm, as well as other common update rules such
(t)
(t)
(t)
as the Hedge update rule fHedge (pi , r(t) ) = e ?|pi r | [Freund and Schapire, 1997], and all IC
methods, since they have the identity rationality function (though the bounds in Thm 14 are stronger).
7
5
Randomized Algorithms: Upper and Lower Bounds
Impossibility of Vanishing Regret. We now consider randomized online learning algorithms,
which can typically achieve better worst-case guarantees than deterministic algoritms. For example,
with honest experts, there are randomized algorithms no 1-regret. Unfortunately, the lower bounds in
Section 4 imply that no such result is possible for randomized algorithms (more details in Appendix F).
Corollary 19. Any incentive compatible randomized weight-update algorithm or non-IC randomized
algorithm with continuous or non-strictly increasing rationality function cannot be no 1-regret.
An IC Randomized Algorithm. While we cannot hope to achieve a no-regret algorithm for online
prediction with selfish experts, we can do better than the deterministic algorithm from Section 3.
Consider the following class of randomized algorithms:
Definition 20 (?-randomized weighted majority). Let Ar be the class of algorithms that maintains
(t)
Pn
w
(t)
expert weights as in Definition 1. Let b(t) = i=1 Pn i (t) ? pi be the weighted predictions. For
w
j=1
j
8
if b(t) ? ?
<0
parameter ? 2 [0, 12 ] the algorithm chooses 1 with probability p(t) = b(t) if ? < b(t) ? 1 ? .
:
1
otherwise
(t)
(t)
We call algorithms in Ar ?-RWM algorithms. We?ll use the Brier rule fBr (pi , r(t) ) = 1 ?((pi )2 +
(t)
(t)
(t)
(t)
(1 pi )2 + 1)/2 (1 si )) with si = |pi
r(t) |.
Theorem 21. Let A 2 Ar be
p a ?-RWM algorithm with the Brier weight update rule fBr and
? = 0.382 and with ? = O(1/ T ) 2 (0, 12 ). A has no 2.62-regret.
6
Simulations
The theoretical results presented so far indicate that when faced with selfish experts, one should
use an IC weight update rule, and ones with smaller scoring rule gap are better. Two objections to
these conclusions are: first, the presented results are worst-case, and may not represent behavior on
a typical input. It is of particular interest to see if on non-worst-case inputs, the non-IC standard
weight-update rule does better or worse than the IC methods proposed in this paper. Second, there
is a gap between our upper and lower bounds for IC rules, so it?s interesting to see what numerical
regret is obtained.
Results. In our first simulation, experts are represented by a simple two-state hidden Markov model
(HMM) with a ?good? state and a ?bad? state. Realization r(t) is given by a fair coin. For r(t) = 0
(t)
(otherwise beliefs are reversed), in the good state expert i believes bi ? min{Exp(1)/5, 1}, in the
(t)
1
bad state bi ? U[ 12 , 1]. The probability to exit a state is 10
for both states. This data generating
process models that experts that have information about the event are more accurate than experts
who lack the information. Figure 1a shows the regret as a function of time for the standard (nonIC) algorithm, and IC scoring rules including one from the Beta family [Buja et al., 2005] with
(t)
(t)
(t)
? = = 12 . For the IC methods, experts report pi = bi , for the standard algorithm pi = 1 if
(t)
(t)
1
bi
2 and pi = 0 otherwise. The y axis is the ratio of the total loss of each of the algorithms to
the performance of the best expert at that time. The plot is for 10 experts, T = 10, 000, ? = 10 2 ,
and the randomized10 versions of the algorithms, averaged over 30 runs. Varying model parameters
and the deterministic version show similar results.
Each of the IC methods does significantly better than the standard weight-update algorithm, and
even at T = 200, 000 (not shown in the graph), the IC methods have a regret factor of about 1.003,
whereas the standard algorithm still has 1.14. This gives credence to the notion that failing to account
for incentive issues is problematic beyond the worst-case bounds presented earlier. Moreover, while
there is a worst-case lower bound that rules out no-regret, for natural synthetic data, the loss of all the
IC algorithms approaches that of the best expert in hindsight, while the standard algorithm fails to do
10
Here we use the regular RWM algorithm, so in the notation of Section 5, we have ? = 0.
8
(a) The HMM data-generating process.
(b) The greedy lower bound instance.
Figure 1: Regret for different data-generating processes.
Table 1: Comparison of lower bound results with simulation. The simulation is run for T =
10, 000, ? = 10 4 and we report the average of 30 runs. For the lower bounds, the first number is the
lower bound from Lemma 16, i.e. 2 + d 1 1 e , the second number (in parentheses) is 2 + .
Greedy LB
LB Sim
Lem 16 LB
Beta .1
2.3708
2.4414
2.33 (2.441)
Beta .5
2.2983
2.3186
2.25 (2.318)
Beta .7
2.2758
2.2847
2.25 (2.285)
Beta .9
2.2584
2.2599
2.25 (2.260)
Brier
2.2507
2.2502
2.25
Spherical
2.2071
2.2070
2.2 (2.207)
this. This seems to indicate that eliciting the truthful beliefs of the experts is more important than the
exact weight-update rule.
Comparison of LB Instances. We consider both the lower bound instance described the proof of
(t)
Lemma 16, and a greedy version that punishes the algorithm every time w0 is ?sufficiently? large.11
Figure 1b shows the regret for different algorithms on the greedy lower bound instance. Table 1 shows
that it very closely traces 2 + , as do the numerical results for the lower bound from Lemma 16. In
fact, for the analysis, we needed to use d 1 e when determining the first phase of the instance. When
we use instead numerically, the regret seems to trace 2 + quite closely, rather than the weaker
proven lower bound of 2 + d 1 1 e . Table 1 shows that the analysis of Lemma 16 is essentially tight
(up to the rounding of ). Closing the gap between the lower and upper bound requires finding a
different lower bound instance, or a better analysis for the upper bound.
7
Open Problems
There area number of interesting questions that this work raises. First of all, our utility model
effectively causes experts to optimize their weight independently of other experts. Bayarri and
DeGroot [1989] discuss different objective functions for experts, including optimizing relative weight
among experts under different informational assumptions. These would impose different constraints
as to which algorithms would lead to truthful reporting, and it would be interesting to see if no-regret
learning is possible in this setting.
It also remains an open problem to close the gap between the best known upper and lower bounds that
we presented in this paper. The simulations showed that the analysis for the lower bound instances is
almost tight, so this requires a novel upper bound and/or a different lower bound instance.
Finally, strictly proper scoring rules are also well-defined beyond binary outcomes. It would be
interesting to see what bounds can be proved for predictions over more than two outcomes.
11
(t)
(t)
(t)
When w0 is sufficiently large we make e0 (and thus the algorithm) wrong twice: b0 = 0, b1 = 1,
(t+1)
(t)
(t)
= 12 , r(t) = 1, and b0
= 0, b1 = 12 , b0 = 1, r(t) = 1. ?Sufficiently? here means that weight of e0 is
high enough for the algorithm to follow its advice during both steps.
(t)
b2
9
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6,334 | 673 | A Hybrid Linear/Nonlinear Approach to Channel
Equalization Problems
Wei-Tsih Lee
John Pearson
David Sarnoff Research Center
CN5300
Princeton, NJ 08543
Abstract
Channel equalization problem is an important problem in high-speed
communications. The sequences of symbols transmitted are distorted by
neighboring symbols. Traditionally, the channel equalization problem is
considered as a channel-inversion operation. One problem of this
approach is that there is no direct correspondence between error probability and residual error produced by the channel inversion operation. In
this paper, the optimal equalizer design is formulated as a classification
problem. The optimal classifier can be constructed by Bayes decision
rule. In general it is nonlinear. An efficient hybrid linear/nonlinear
equalizer approach has been proposed to train the equalizer. The error
probability of new linear/nonlinear equalizer has been shown to be better than a linear equalizer in an experimental channel.
1 INTRODUCTION
In a typical communication system, a sequence of symbols {ld are transmitted though a
linear time-dispersive channel h(t). Let x(t) be the received signal, it can be written as
x (t) =
L/jh (t-n1) + W (I)
j
(1)
where h(t) denotes the elementary pulse waveform, and wet) represents the random noise
with iid Gaussian distribution. In a Quadrature Amplitude Modulation (QAM), symbols
(Id are represented by complex numbers. During the transmission, interferences from
neighboring symbols may distort the received signals. It is called Intersymbol Interference
(lSI). It mainly because following reasons: nonideal channel which introduces phase or
amplitude distortions, phase jitter, and impulse noise. Thus, equalization techniques are
used to reduce the lSI.
674
A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems
2 ADAPTIVE LINEAR/RADIAL BASIS FUNCTION APPROACH
TO EQUALIZER DESIGN
Traditionally, the channel equalization problem is considered as a channel-inversion operation. The idea is that an equalizer is constructed as to undo the interference from neighboring symbols as they passing through a linear dispersive channel. It can be used to
explain different equalizer structures (Zero-forcing, Least mean square, and decision feedback) and their performance [Proakis, 1989]. One problem of this approach is that in general there is no direct correspondence between error probability and residual error
produced by the channel inversion operation. In [Gibson, etal, 1991], authors proposed a
classification viewpoint for the equalizer design. They suggested that the optimal equalizer should be a classifier whose decision boundary is constructed according to Bayes
decision rule. Compared with the channel inversion approach, the outputs of receiver are
used as features for a classifier. The decision is made solely based on the classifier output,
hence, on feature distribution. As it is well-known in [Fukunaga, 1978], the optimal decision boundaries can rapidly be computed if the features are Gaussian distributed. However, there is no idea about the structure of the optimal equalizer (classifier)for timedispersive channel outputs. In next section, we prove that for a linear channel, the optimal
equalizer is nonlinear.
2.1 THE OPTIMAL EQUALIZER OF A LINEAR TIME-DISPERSIVE CHANNEL
Let us first consider a two-value equalization problem. Symbols with two possible values
( -1, I) are transmitted. Let the channel be represented in a discrete form as a FIR of (hi),
i=O,N-l. The output Xi can be written as
X?
,
=
N-l
~ I . .h .+w .
L
'-I I
,
(2)
j=o
The optimal equalizer design is equivalent to the following Bayes decision problem.
Given (xi}, decide Ii by
1
/. = {
,
-1
if
P(lj=-II X j,xj+l'?????,xi+N-l) >P(lj=
lI X j,xj+l'?????,xj+N-l)
(3)
where P (lj = 11 xi,xi + 1, ??.?. ,xi+N-l) is the posterior probability of the transmitted symbol
Ii being 1 given channel output (xd.
By Bayes theorem, expression(3) can be expanded to the following form:
(4)
i+N-l
L
IT
j =
j
kl'
k;z?.... ,kj _ N + I e
P(xjl Ii
= 1,???Jj _ N + 1 = ki - N + 1) P(lj = 1,... J j - N + 1 = k j _ N + 1)
{I. -1 }
(5)
Since conditional probability P (xii I j = 1,... J j -N + 1 = k j _ N + 1) in (5) is a Gaussian distribution, the numerator in (5) is a mixture of Gaussian distribution. Plugging (5) into (3),
Bayes decision rule determines the optimal decision boundary as the solution of eqUality.
Since denominator is the same on both sides, it can be ignored. Rearranging the equation,
675
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Lee and Pearson
it can be written as summation of exponential functions. The solution of this equation is
nonlinear function of {xil. In general, no analytical form can be found. However, it can be
solved by numerical methods. Thus, the optimal decision boundary can be determined.
The result can be extended to multi-class problems.
Based on the result established above, we provide a theoretical justification of a nonlinear
equalizer approach to linear time-dispersive channel. The theoretical comparison of performances of linear and optimal equalizers can be found in [Gibson, et.al, 1991]. They
concluded that performance of linear equalizers can not be improved by increasing tap
length. This also suggests that a nonlinear equalizer approach is necessary. Another reason
for nonlinear equalization approach is due to channels with spectrum hulls [Proakis,
1989]. In this case, the linear equalizer can not achieve the desired performance due to
"noise enhancement".
2.2 NONLINEAR EQUALIZER DESIGN PROBLEM
There are several approaches to nonlinear equalizer design. To reduce the Least Mean
Square (LMS) error, Voterra-series approach uses high-order product terms of input as
new features. The tree-structured linear equalizer method [Gelfand, et.al., 1991] partitions
the feature-space, and makes a piecewise linear approximation to the optimal nonlinear
equalizer. As reported in [Gelfand, et.al., 1991], the tree-structured linear equalizer
approach provides reasonable fast convergence and lower error probability as compared
with linear and Voterra series approaches. The problem of this approach is that a lot of
training samples are needed to achieve good performance. A neural network approach,
MultiLayer Perceptron(MLP) [Gibson, et.al, 1991], trains 3 or 4 layers interconnected
Perceptrons to form the nonlinear decision boundary. It is observed in [Gibson, et.al,
1991] that the performance of a MLP equalizer is close to optimal Bayes classifier. However, the training time is long and a fine-turing procedure is used. A nonlinear equalizer
approach using radial basis functions is also reponed in [Chen, et.al., 1991].
To put equalizers into use, the long training time is unpractical, and a fine-adjusting procedure is not allowed. Hence, it is desired to have an efficient, automatic procedure for nonlinear equalize design. To achieve this goal, we propose a hybrid linear and radial basis
functions approach for automatic nonlinear equalizer design.
Although the optimal equalizer should be nonlinear, all these nonlinear design methods
require long training time or large amount of training samples. Linear equalizers are not
optimal, but with following advantages: easy training, fast convergence. It is also reported
that the linear equalizer is relatively robust [Fukunaga, 1978]. Hence, it is desirable to
combine the advantages of both linear and nonlinear equalizers. However, the hybrid
structure should provide desired properties: fast convergence, automatic training procedure, and low error rate. To satisfy these constraints, we propose a feature-space partitioning approach to hybrid equalizer design.
2.3 FEATURE?SPACE PARTITIONING APPROACH TO HYBRID EQUALIZER
DESIGN
To design a hybrid linear/nonlinear equalizer, we adopt the feature-space partitioning concept. The idea is similar to the one developed in [Gelfand, et.al, 1991]. Here, we consider
a partitioning method based on geometrical reasoning for eqUalization problems. The idea
is based on the fact that linear equalizers can recover distorted signals, except the cases
when strong noise push samples into boundaries where two classes overlaid with each
A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems
other. We consider the "confused" samples as these samples near decision boundaries. The
separation of "confused" samples can be accomplished based on the output values of linear equalizers. If the distance between output value and the closest point in signal constellation [Proakis, 1989] is greater than a threshold, then we consider current sample is
"confused". This means that the sample is the one close to decision boundary. To achieve
an accurate classification, we classify it by a nonlinear equalizer, which is constructed for
separating the samples near Bayes decision boundary.
The hybrid structure consisted of a linear equalizer, followed by a radial basis function
(RBS) network, as shown in Fig. 1. A RBS network (Fig.2) is a two-layered network with
radial_basis_function nodes in first layer, and a weighted linear combination of outputs of
these nodes.
Each feature vector consisted of a collection of consecutive data from the channel. It is
assumed that these data are properly time and carrier synchronized [Proakis, 1989]. For
the QAM, a complex-valued linear equalizer is adopted. The distance between output
value of linear equalizer and the closest point is then computed and compared with threshold as described before. The "confused" samples are classified by a nonlinear RBS equalizer. The output of a RBS network can be written as weighted summation of outputs of
nodes as follows:
(6)
where f(x) is the output of network. The oUfut value of each node is computed according
to the bell-shaped function centered at ci' (J is the width of a node. Wi is the weight associated with ith node. In our experiments, the width of all nodes are fixed. The first N training samples are assigned to the centers of N -nodes network. The weights are adjusted
according to stochastic gradient decent rule:
(7)
where 11 is the learning rate. d k is the desired output of network
To train a hybrid LE/RBS equalizer, a collection of training samples is used to adjust the
parameters of linear equalizer. The training samples for RBS network are collected
according to the distance rule described above. They are used to adjust weights of RBS
network only.
The classification of a unknown sample follows a similar rule. The output value of the linear equalizer is computed. If the distance between output value and the closest point in
signal constellation is smaller than the threshold, then the closest point is considered as the
recovered signal. If not, the output of the RBS network is used to classify the sample. The
closest point in signal constellation from output of RBS network is then used for sample
class. Note, however, that there is a different interpretation for the output of linear and
RBS equalizer. The function of linear equalizers can be considered as an approximation of
channel inversion. Hence, it is similar to a deconvolution computation [Proakis, 1989].
However, for a RBS network, the output is the summation of weighted local Gaussian
functions. For closely located points, the network is asked to give same output by then
training procedure. Thus, it is more a classification approach than a deconvolution
method.
The approach provides a design method for hybrid LE/RBS equalizer. The linear equaliz-
677
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Lee and Pearson
{Xn }
line:lI' Equalizer
yes
Radial_basis _ function
network
Fig. 1 System Diagram of Hybrid Linear/Nonlinear Equalizer
I(x)
Fig. 2 A RadiaCBasis_Function Network
A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems
ers perfonn the channel inversion or partitioning of the feature space, depending on the
output value. More complicated tree-structured equalizer [Gelfand, et.al., 1991] can be
adopted for this proposes. The nonlinear RBS networks are used for classifying "confused" samples. They can be replaced by MLPs. Hence, the approach provides a general
method for designing hybrid structure eqalizers. However, the trade-off between complexity and efficiency of these combinations has to to be considered. For example, a multilayer
tree-structured equalizer can divide the space into smaller regions for finer classification.
However, the small amount of training samples in practice can be a problem for this
method. A MLP network can be used for nonlinear classifier. Nevertheless, convergence
time will be a major concern.
3 EXPERIMENT
We have applied our hybrid design method to a 4-QAM system. The channel is modeled
by
Xj _
where I j
1
= 0.406/ +0.814/
j
j _
1 +0.407/j
_
2 +w j
(8)
= {- 1 - j,- 1 + j,l- j,1 + j} .
A 7-tapped complex linear equalizer is used for classifying the input. Threshold for nonlinear equalizer is 0.1. We use 4,000training samples and 5,000 testing samples. A 400
nodes RBS network is used for nonlinear equalizer. The first 400 "confused" training samples are used for the center of network. The network is trained according to (6). Learning
coefficient T) is chosen to be 0.01. The width of a RBS node is 1.0.
Fig. 3 shows the symbol error probability vs. SNR. The error probability is evaluated over
5,000 testing samples. The hybrid LE/RBS network produces nearly 10% reduction of
error rate compared with linear equalizer. This shows that a hybrid linear/RBS network
equalizer can reduce the error rate by classifying "confused" samples near decision
boundaries. No comparison with Bayes classifier has been made. In our experiments, it is
observed that the error rate can be reduced further by increasing the number of RBS
nodes. This seems imply that a large-size RBS network will in general produce better classification result. However, since there is always a limitation of the computation resources:
computation time and memory storage, the perfonnance of the hybrid linear/RBS network
is limited, especially in high signal constellation case discussed below.
Equalization in high signal constellation, 16 and 64-QAM, have been tried. The result
shows no significant improvement This can be explained by the increasing of complexity.
Recall that the RBS network is to separate the samples near the boundary. To deal with the
increasing of number of classes due to high signal constellation, the number of nodes of
network must increase proportionally. Since the increasing rate is exponential in terms of
number of classes, it implies a straight-forward implementation of RBS network method
can not be used for high signal constellation. In [Chen, et.al., 1991], authors suggest a
dynamical RBS network with adjustable center location and width. The algorithm runs in
batch mode. It is reported that the size of network can be reduced dramatically by the
dynamical RBS network method. However, for equalizer application, on-line version of
the algorithm is needed.
679
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Lee and Pearson
??
?
:<
:
?
_CUlt
1.
om
SNR
Fig. 3 Error Probability of Hybrid Linear/Radial_Basis_Function Network Equalizer for a
Linear Channel xi - 1 = 0.406Ii + 0.814/i _ 1 + 0.407li _ 2 +wi with 4-QAM.
4 CONCLUSION AND DISCUSSION FOR HYBRID LE/RBS
EQUALIZER DESIGN
By combining feature-space partitioning and nonlinear equalizers, we have developed a
hybrid linear/nonlinear equalization approach. The major contribution of this research is
to provide a theoretical justification of nonlinear equalization approach for linear time-dispersive channels. A feature-space partitioning method by linear equalizer is proposed.
RBS networks for nonlinear equalizers are integrated into the design to separate the samples near decision boundary. The experiments for 4-QAM equalization have demonstrated
the feasibility of the approach. For high signal constellation modulation, a dynamical RBS
network method [Chen, etal., 1991] has been suggested to overcome the problem of
increasing complexity.
The hybrid Linear/nonlinear equalization approach combines the strength of linear and
nonlinear equalizers. It offers a framework to integrate the deconvolution and classification methods. The approach can be generalized to include complicated partitioning
A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems
scheme and other nonlinear networks, such as :MLP, as well.
More researches need to be conducted to make this approach practical for general use. The
relationship between the performance of hybrid equalizer and taps length of linear equalizers, the width and the number ofRBS nodes need to be investigated. The on-line version
of dynamical RBS network [Chen, et.al., 1991] need to be developed.
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Proakis, J. G., Digital Communications, McGrwa-Hill company, New York, 1989.
Gibson, GJ., Siu, S., Cowan, C.F.N., "The Application of Nonlinear Structures to Reconstruction of Binary Signals," IEEE. Trans. on Signal Processing, vol. 39, No.8, Aug.,pp.
1877-1884, 1991.
Fukunaga, K., Introduction to Statistical Pattern Recognition, Academic Press, New York,
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Gelfand, S.B., Ravishankar, C.S., and Delp, EJ., "Tree-structured Piecewise Linear Adaptive Equalization," ICC91, 001383, 1386.
Chen, S., Gilbson, GJ., Cowan, C.F.N., and Grant, P.M., "Reconstruction of binary signals using an adaptive radial-bas is-function equalizer," Signal Processing, 22, pp. 77-93,
1991.
Chen, S., Cowan, C.F.N., and Grant, P.M., "Orthogonal Least Squares Learning Algorithm
for Radial Basis Function Networks," IEEE. Trans. on Neural Networks, vol. 2, no., 2,
March, pp.302-309, 1991.
681
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6,335 | 6,730 | Tensor Biclustering
Soheil Feizi
Stanford University
[email protected]
Hamid Javadi
Stanford University
[email protected]
David Tse
Stanford University
[email protected]
Abstract
Consider a dataset where data is collected on multiple features of multiple individuals over multiple times. This type of data can be represented as a three dimensional
individual/feature/time tensor and has become increasingly prominent in various
areas of science. The tensor biclustering problem computes a subset of individuals
and a subset of features whose signal trajectories over time lie in a low-dimensional
subspace, modeling similarity among the signal trajectories while allowing different scalings across different individuals or different features. We study the
information-theoretic limit of this problem under a generative model. Moreover,
we propose an efficient spectral algorithm to solve the tensor biclustering problem
and analyze its achievability bound in an asymptotic regime. Finally, we show the
efficiency of our proposed method in several synthetic and real datasets.
1
Introduction
Let T ? Rn1 ?n2 be a data matrix whose rows and columns represent individuals and features,
respectively. Given T, the matrix biclustering problem aims to find a subset of individuals (i.e.,
J1 ? {1, 2, ..., n1 }) which exhibit similar values across a subset of features (i.e., J2 ? {1, 2, ..., n2 })
(Figure 1-a). The matrix biclustering problem has been studied extensively in machine learning and
statistics and is closely related to problems of sub-matrix localization, planted clique and community
detection [1, 2, 3].
In modern datasets, however, instead of collecting data on every individual-feature pair at a single
time, we may collect data at multiple times. One can visualize a trajectory over time for each
individual-feature pair. This type of datasets has become increasingly prominent in different areas of
science. For example, the roadmap epigenomics dataset [4] provides multiple histon modification
marks for genome-tissue pairs, the genotype-tissue expression dataset [5] provides expression data
on multiple genes for individual-tissue pairs, while there have been recent efforts to collect various
omics data in individuals at different times [6].
Suppose we have n1 individuals, n2 features, and we collect data for every individual-feature pair
at m different times. This data can be represented as a three dimensional tensor T ? Rn1 ?n2 ?m
(Figure 1-b). The tensor biclustering problem aims to compute a subset of individuals and a subset
of features whose trajectories are highly similar. Similarity is modeled as the trajectories as lying
in a low-dimensional (say one-dimensional) subspace (Figure 1-d). This definition allows different
scalings across different individuals or different features, and is important in many applications
such as in omics datasets [6] because individual-feature trajectories often have their own intrinsic
scalings. In particular, at each time the individual-feature data matrix may not exhibit a matrix
bicluster separately. This means that repeated applications of matrix biclustering cannot solve the
tensor biclustering problem. Moreover, owing to the same reason, trajectories in a bicluster can
have large distances among themselves (Figure 1-d). Thus, a distance-based clustering of signal
trajectories is likely to fail as well.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Our Model
(b)
(c)
times
Tensor Biclustering
featu
res
features
(d)
featu
als
individu
2
Tensor
Biclustering
1
0
-1
2
0
-2
-4 -4
-2
als
individu
res
Trajectories in the Bicluster
All Trajectories
3
-2
4
Tensor Triclustering
times
Matrix Biclustering
individuals
(a)
0
2
4
3
2
1
0
-1
-2
-3
4
2
0
-2
-4 -4
-2
0
2
4
Figure 1: (a) The matrix biclustering problem. (b) The tensor biclustering problem. (c) The tensor
triclustering problem. (d) A visualization of a bicluster in a three dimensional tensor. Trajectories in
the bicluster (red points) form a low dimensional subspace.
This problem formulation has two main differences with tensor triclustering, which is a natural
generalization of matrix biclustering to a three dimensional tensor (Figure 1-c). Firstly, unlike tensor
triclustering, tensor biclustering has an asymmetric structure along tensor dimensions inspired by
aforementioned applications. That is, since a tensor bicluster is defined as a subset of individuals
and a subset of features with similar trajectories, the third dimension of the tensor (i.e., the time
dimension) plays a different role compared to the other two dimensions. This is in contrast with
tensor triclustering where there is not such a difference between roles of tensor dimensions in defining
the cluster. Secondly, in tensor biclustering, the notion of a cluster is defined regarding to trajectories
lying in a low-dimensional subspace while in tensor triclustering, a cluster is defined as a sub-cube
with similar entries.
Finding statistically significant patterns in multi-dimensional data tensors has been studied in dimensionality reduction [7, 8, 9, 10, 11, 12, 13, 14], topic modeling [15, 16, 17], among others. One
related model is the spiked tensor model [7]. Unlike the tensor biclustering model that is asymmetric
along tensor dimensions, the spiked tensor model has a symmetric structure. Computational and
statistical limits for the spiked tensor model have been studied in [8, 9, 10, 14], among others. For
more details, see Supplementary Materials (SM) Section 1.3.
In this paper, we study information-theoretic and computational limits for the tensor biclustering
problem under a statistical model described in Section 2. From a computational perspective, we
present four polynomial time methods and analyze their asymptotic achievability bounds. In particular,
one of our proposed methods, namely tensor folding+spectral, outperforms other methods both
theoretically (under realistic model parameters) and numerically in several synthetic and real data
experiments. Moreover, we characterize a fundamental limit under which no algorithm can solve the
tensor biclustering problem reliably in a minimax sense. We show that above this limit, a maximum
likelihood estimator (MLE) which has an exponential computational complexity can solve this
problem with vanishing error probability.
1.1
Notation
We use T , X , and Z to represent input, signal, and noise tensors, respectively. For any set J, |J|
denotes its cardinality. [n] represents the set {1, 2, ..., n}. J? = [n] ? J. kxk2 = (xt x)1/2 is the
second norm of the vector x. x ? y is the Kronecker product of two vectors x and y. The asymptotic
notation a(n) = O(b(n)) means that, there exists a universal constant c such that for sufficiently
2
large n, we have |a(n)| < cb(n). If there exists c > 0 such that a(n) = O(b(n) log(n)c ), we use
?
?
the notation a(n) = O(b(n)).
The asymptotic notation a(n) = ?(b(n)) and a(n) = ?(b(n))
is the
?
same as b(n) = O(a(n)) and b(n) = O(a(n)),
respectively. Moreover, we write a(n) = ?(b(n))
?
?
iff a(n) = ?(b(n)) and b(n) = ?(a(n)). Similarly, we write a(n) = ?(b(n))
iff a(n) = ?(b(n))
?
and b(n) = ?(a(n)).
2
Problem Formulation
Let T = X + Z where X is the signal tensor and Z is the noise tensor. Consider
q
X
1)
T =X +Z =
?r u(J
? wr(J2 ) ? vr + Z,
r
(1)
r=1
(J )
(J )
where ur 1 and wr 2 have zero entries outside of J1 and J2 index sets, respectively. We assume
?1 ? ?2 ? ... ? ?q > 0. Under this model, trajectories X (J1 , J2 , :) form an at most q dimensional
subspace. We assume q min(m, |J1 | ? |J2 |).
Definition 1 (Tensor Biclustering). The problem of tensor biclustering aims to compute bicluster
index sets (J1 , J2 ) given T according to (1).
In this paper, we make the following simplifying assumptions: we assume q = 1, n = |n1 | = |n2 |,
(J )
and k = |J1 | = |J2 |. To simplify notation, we drop superscripts (J1 ) and (J2 ) from u1 1 and
(J )
w1 2 , respectively. Without loss of generality, we normalize signal vectors such that ku1 k =
kw1 k = kv1 k = 1. Moreover, we assume that for every (j1 , j2 ) ? J1 ? J2 , ? ? u1 (j1 ) ? c?
and ? ? w1 (j2 ) ? c?, where c is a constant. Under these assumptions, a signal trajectory can be
written as X (j1 , j2 , :) = u1 (j1 )w1 (j2 )v1 . The scaling of this trajectory depends on row and column
specific parameters u1 (j1 ) and w1 (j2 ). Note that our analysis can be extended naturally to a more
general setup of having multiple embedded biclusters with q > 1. We discuss this in Section 7.
Next we describe the noise model. If (j1 , j2 ) ?
/ J1 ? J2 , we assume that entries of the noise trajectory
Z(j1 , j2 , :) are i.i.d. and each entry has a standard normal distribution. If (j1 , j2 ) ? J1 ? J2 , we
assume that entries of Z(j1 , j2 , :) are i.i.d. and each entry has a Gaussian distribution with zero mean
and ?z2 variance. We analyze the tensor biclustering problem under two noise models for ?z2 :
- Noise Model I: In this model, we assume ?z2 = 1, i.e., the variance of the noise within and outside
of the bicluster is assumed to be the same. This is the noise model often considered in analysis
of sub-matrix localization [2, 3] and tensor PCA [7, 8, 9, 10, 11, 12, 14]. Although this model
simplifies the analysis, it has the following drawback: under this noise model, for every value
of ?1 , the average trajectory lengths in the bicluster is larger than the average trajectory lengths
outside of the bicluster. See SM Section 1.2 for more details.
?2
- Noise Model II: In this model, we assume ?z2 = max(0, 1 ? mk12 ), i.e., ?z2 is modeled to minimize
the difference between the average trajectory lengths within and outside of the bicluster. If
?12 < mk 2 , noise is added to make the average trajectory lengths within and outside of the bicluster
comparable. See SM Section 1.2 for more details.
3
3.1
Computational Limits of the Tensor Biclustering Problem
Tensor Folding+Spectral
Recall the formulation of the tensor biclustering problem (1). Let
T(j1 ,1) , T (j1 , :, :) and
T(j2 ,2) , T (:, j2 , :),
(2)
be horizontal (the first mode) and lateral (the second mode) matrix slices of the tensor T , respectively.
One way to learn the embedded bicluster in the tensor is to compute row and column indices whose
trajectories are highly correlated with each other. To do that, we compute
n
n
X
X
C1 ,
Tt(j2 ,2) T(j2 ,2) and C2 ,
Tt(j1 ,1) T(j1 ,1) .
(3)
j2 =1
j1 =1
3
Input Tensor
T
T(1, : , : )
T(k, : , : )
Matrix Slices m
m
n2
n1
n2
...
t
Spectral
Decomposition n
2
Combined
Covariance
...
t
T(1, : , : ) T(1, : , : )
Bicluster Index Set
(J2)
T(n1, : , : )
T(k+1, : , : )
T(n1, : , : ) T(n1, : , : )
n2
n2
n2
...
...
Figure 2: A visualization of the tensor folding+spectral algorithm 1 to compute the bicluster index
set J2 . The bicluster index set J1 can be computed similarly.
Algorithm 1 Tensor Folding+Spectral
Input: T , k
? 1 , the top eigenvector of C1
Compute u
? 1 , the top eigenvector of C2
Compute w
? 1|
Compute J?1 , indices of the k largest values of |w
Compute J?2 , indices of the k largest values of |?
u1 |
Output: J?1 and J?2
C1 represents a combined covariance matrix along the tensor columns (Figure 2). We refer to it as
the folded tensor over the columns. If there was no noise, this matrix would be equal to ?12 u1 ut1 .
Thus, its eigenvector corresponding to the largest eigenvalue would be equal to u1 . On the other
hand, we have u1 (j1 ) = 0 if j1 ?
/ J1 and |u1 (j1 )| > ?, otherwise. Therefore, selecting k indices of
the top eigenvector with largest magnitudes would recover the index set J1 . However, with added
noise, the top eigenvector of the folded tensor would be a perturbed version of u1 . Nevertheless one
can estimate J1 similarly (Algorithm 1). A similar argument holds for C2 .
? 1 and w
? 1 be top eigenvectors of C1 and C2 , respectively. Under both noise models
Theorem 1. Let u
I and II,
? ?n), if ? 2 = ?(n),
?
- for m < O(
1
?
? n), if ? 2 = ?(
? ?n max(n, m)),
- for m = ?(
1
? 1 (j2 )| > |w
? 1 (j20 )| for every
as n ? ?, with high probability, we have |?
u1 (j1 )| > |?
u1 (j10 )| and |w
j1 ? J1 , j10 ? J?1 , j2 ? J2 and j20 ? J?2 .
In the proof of Theorem 1, following the result of [18] for a Wigner noise matrix, we have proved an
l? version of the Davis-Kahan Lemma for a Wishart noise matrix. This lemma can be of independent
interest for the readers.
3.2
Tensor Unfolding+Spectral
2
Let Tunf olded ? Rm?n be the unfolded tensor T such that Tunf olded (:, (j1 ? 1)n + j2 ) =
T (j1 , j2 , :) for 1 ? j1 , j2 ? n. Without noise, the right singular vector of this matrix is u1 ? w1
which corresponds to the singular value ?1 . Therefore, selecting k 2 indices of this singular vector
with largest magnitudes would recover the index set J1 ? J2 . With added noise, however, the top
singular vector of the unfolded tensor will be perturbed. Nevertheless one can estimate J1 ? J2
similarly (SM Section 2).
4
? be the top right singular vector of Tunf olded . Under both noise models I and II,
Theorem 2. Let x
2
?
if ?12 = ?(max(n
, m)), as n ? ?, with high probability, we have |?
x(j 0 )| < |?
x(j)| for every j in
0
the bicluster and j outside of the bicluster.
3.3
Thresholding Sum of Squared and Individual Trajectory Lengths
If the average trajectory lengths in the bicluster is larger than the one outside of the bicluster, methods
based on trajectory length statistics can be successful in solving the tensor biclustering problem. One
such method is thresholding individual trajectory lengths. In this method, we select k 2 indices (j1 , j2 )
with the largest trajectory length kT (j1 , j2 , :)k (SM Section 2).
Theorem 3. As n ? ?, with high probability, J?1 = J1 and J?2 = J2
? ?mk 2 ), under noise model I.
- if ?12 = ?(
2
?
- if ?12 = ?(mk
), under noise model II.
Another method to solve the tensor biclustering problem is thresholding sum of squared trajectory
lengths. In this method, we select k row indices with the largest sum of squared trajectory lengths
along the columns as an estimation of J1 . We estimate J2 similarly (SM Section 2).
Theorem 4. As n ? ?, with high probability, J?1 = J1 and J?2 = J2
? ?nm), under noise model I.
- if ?12 = ?(k
?
2
?
- if ?12 = ?(mk
+ k nm), under noise model II.
4
4.1
Statistical (Information-Theoretic) Limits of the Tensor Biclustering
Problem
Coherent Case
In this section, we study a statistical (information theoretic) boundary for
? the tensor biclustering
problem under the following
statistical
model:
We
assume
u
(j
)
=
1/
k for j1 ? J1 . Similarly,
1 1
?
we assume w1 (j2 ) = 1/ k for j2 ? J2 . Moreover, we assume v1 is a fixed given vector with
kv1 k = 1. In the next section, we consider a non-coherent model where v1 is random and unknown.
Let T be an observed tensor from the tensor biclustering model (J1 , J2 ). Let Jall be the set of
2
all possible (J1 , J2 ). Thus, |Jall | = nk . A maximum likelihood estimator (MLE) for the tensor
biclustering problem can be written as:
max
? all
J?J
v1t
X
(j1 ,j2 )?J?1 ?J?2
T (j1 , j2 , :) ?
k(1 ? ?z2 )
2?1
X
2
kT (j1 , j2 , :)k
(4)
(j1 ,j2 )?J?1 ?J?2
(J?1 , J?2 ) ? Jall .
Note that under the noise model I, the second term is zero. To solve this optimization, one needs
2
to compute the likelihood function for nk possible bicluster indices. Thus, the computational
complexity of the MLE is exponential in n.
?
as n ? ?, with high probability, (J1 , J2 ) is
Theorem 5. Under noise model I, if ?12 = ?(k),
the optimal solution of optimization (4). A similar result holds under noise model II if mk =
?(log(n/k)).
Next, we establish an upper bound on ?12 under which no computational method can solve the tensor
biclustering problem with vanishing probability of error. This upper bound indeed matches with the
MLE achievability bound of Theorem 5 indicating its tightness.
Theorem 6. Let T be an observed tensor from the tensor biclustering model with bicluster indices
(J1 , J2 ). Let A be an algorithm that uses T and computes (J?1 , J?2 ). Under noise model I, for any
5
fixed 0 < ? < 1, if ?12 < c? k log(n/k), as n ? ?, we have
h
i
inf
sup
P J?1 6= J1 or J?2 6= J2 > 1 ? ? ?
A?AllAlg (J1 ,J2 )?Jall
log(2)
.
2k log(ne/k)
(5)
A similar result holds under noise model II if mk = ?(log(n/k)).
4.2
Non-coherent Case
In this section we consider a similar setup to the one of Section 4.1 with the difference that v1 is
assumed to be uniformly distributed over a unit sphere. For simplicity, in this section we only consider
noise model I. The ML optimization in this setup can be written as follows:
X
max k
T (j1 , j2 , :)k2
(6)
? all
J?J
(j1 ,j2 )?J?1 ?J?2
(J?1 , J?2 ) ? Jall .
?
?
Theorem 7. Under noise model I, if ?12 = ?(max(k,
km)), as n ? ?, with high probability,
(J1 , J2 ) is the optimal solution of optimization (6).
If k > ?(m), the achievability bound of Theorem 7 simplifies to the one of Theorem 5. In this case,
using the result of?
Theorem 6, this bound is tight. If k < O(m), the achievability bound of Theorem
? mk) which is larger than the one of Theorem 5 (this is the price we pay for not
7 simplifies to ?(
knowing v1 ). In the following, we show that this bound is also tight.
To show the converse of Theorem 7, we consider the detection task which is presumably easier than
the estimation task. Consider two probability distributions: (1) P?1 under which the observed tensor
is T = ?1 u1 ? w1 ? v1 + Z where J1 and J2 have uniform distributions over k subsets of [n] and
v1 is uniform over a unit sphere. (2) P0 under which the observed tensor is T = Z. Noise entries are
i.i.d. normal. We need the following definition of contiguous distributions ([8]):
Definition 2. For every n ? N, let P0,n and P1,n be two probability measures on the same measure
space. We say that the sequence (P1,n ) is contiguous with respect to (P0,n ), if, for any sequence of
events An , we have
lim P0,n (An ) = 0 ? lim P1,n (An ) = 0.
?
n??
n??
(7)
? mk), P? is contiguous with respect to P0 .
Theorem 8. If ?12 < O(
1
This theorem with Lemma 2 of [8] establishes the converse of Theorem 7. The proof is based on
bounding the second moment of the Radon-Nikodym derivative of P?1 with respect to P0 (SM
Section 4.9).
5
Summary of Asymptotic Results
?
Table 1 summarizes asymptotic bounds for the case of ? = 1/ k and m = ?(n). For the MLE we
consider the coherent model of Section 4.1. Also in Table 1 we summarize computational complexity
of different tensor biclustering methods. We discuss analytical and empirical running time of these
methods in SM Section 2.2.
Table 1: Comparative analysis of
? tensor biclustering methods. Results have been simplified for the
case of m = ?(n) and ? = 1/ k.
Methods
Tensor Folding+Spectral
Tensor Unfolding+Spectral
Th. Sum of Squared Trajectory Lengths
Th. Individual Trajectory Lengths
Maximum Likelihood
Statistical Lower Bound
?12 , noise model I
? 3/2 )
?(n
? 2)
?(n
?
?(nk)
? 2 ?n)
?(k
?
?(k)
?
O(k)
6
?12 , noise model II
? 3/2 )
?(n
? 2)
?(n
2
?
?(nk
)
?
?(nk 2 )
?
?(k)
?
O(k)
Comp. Complexity
O(n4 )
O(n3 )
O(n3 )
O(n3 )
exp(n)
-
(a)
(b)
noise model I
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
noise model II
1
0.9
Bicluster Recovery Rate
Bicluster Recovery Rate
1
50
100
150
200
250
300
350
400
Signal Strength
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
400
Signal Strength
Tensor folding+Spectral
Th. sum of squared trajectory lengths
Tensor unfolding+Spectral
Th. individual trajectory lengths
Figure 3: Performance of different tensor biclustering methods in various values of ?1 (i.e., the signal
strength), under both noise models I and II. We consider n = 200, m = 50, k = 40. Experiments
have been repeated 10 times for each point.
In both noise models, the maximum likelihood estimator which has an exponential computational
complexity leads to the best achievability bound compared to other methods. Below this bound, the
inference is statistically impossible. Tensor folding+spectral
method outperforms other methods
?
with polynomial computational complexity if k > n under noise model I, and k > n1/4 under
noise model II. For smaller values of k, thresholding individual trajectory lengths lead to a better
achievability bound. This case is a part of the high-SNR regime where the average trajectory lengths
within the bicluster is significantly larger than the one outside of the bicluster. Unlike thresholding
individual trajectory lengths, other methods use the entire tensor to solve the tensor biclustering
problem. Thus, when k is very small, the accumulated noise can dominate the signal strength.
Moreover, the performance of the tensor unfolding method is always worst than the one of the tensor
folding method. The reason is that, the tensor unfolding method merely infers a low dimensional
subspace of trajectories, ignoring the block structure that true low dimensional trajectories form.
6
6.1
Numerical Results
Synthetic Data
In this section we evaluate the performance of different tensor biclustering methods in synthetic
datasets. We use the statistical model described in Section 4.1 to generate the input tensor T . Let
(J?1 , J?2 ) be estimated bicluster indices (J1 , J2 ) where |J?1 | = |J?2 | = k. To evaluate the inference
quality we compute the fraction of correctly recovered bicluster indices (SM Section 3.1).
In our simulations we consider n = 200, m = 50, k = 40. Figure 3 shows the performance of
four tensor biclustering methods in different values of ?1 (i.e., the signal strength), under both noise
models I and II. Tensor folding+spectral algorithm outperforms other methods in both noise models.
The gain is larger in the setup of noise model II compared to the one of noise model I.
6.2
Real Data
In this section we apply tensor biclustering methods to the roadmap epigenomics dataset [4] which
provides histon mark signal strengths in different segments of human genome in various tissues and
cell types. In this dataset, finding a subset of genome segments and a subset of tissues (cell-types)
with highly correlated histon mark values can provide insight on tissue-specific functional roles
of genome segments [4]. After pre-processing the data (SM Section 3.2), we obtain a data tensor
T ? Rn1 ?n2 ?m where n1 = 49 is the number of tissues (cell-types), n2 = 1457 is the number of
7
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folding
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squared TL
TL
random
Figure 4: An application of tensor biclustering methods to the the roadmap epigenomics data.
genome segments, and m = 7 is the number of histon marks. Note that although in our analytical
results for simplicity we assume n1 = n2 , our proposed methods can be used in a more general case
such as the one considered in this section.
We form two combined covariance matrices C1 ? Rn1 ?n1 and C2 ? Rn2 ?n2 according to (3).
Figure 4-(a,b) shows largest eigenvalues of C1 and C2 , respectively. As illustrated in these figures,
spectral gaps (i.e., ?1 ? ?2 ) of these matrices are large, indicating the existence of a low dimensional
signal tensor in the input tensor. We also form an unfolded tensor Tunf olded ? Rm?n1 n2 . Similarly,
there is a large gap between the first and second largest singular values of Tunf olded (Figure 4-c).
We use the tensor folding+spectral algorithm 1 with |J1 | = 10 and |J2 | = 400 (we consider other
values for the bicluster size in SM Section 3.2). The output of the algorithm (J?1 , J?2 ) is illustrated in
Figure 4-d (note that for visualization purposes, we re-order rows and columns to have the bicluster
appear in the corner). Figure 4-e illustrates the unfolded subspace {T (j1 , j2 , :) : (j1 , j2 ) ? J?1 ? J?2 }.
In this inferred bicluster, Histon marks H3K4me3, H3K9ac, and H3K27ac have relatively high values.
Reference [4] shows that these histon marks indicate a promoter region with an increased activation
in the genome.
To evaluate the quality of the inferred bicluster, we compute total absolute pairwise correlations
among vectors in the inferred bicluster. As illustrated in Figure 4-f, the quality of inferred bicluster
by tensor folding+spectral algorithm is larger than the one of other methods. Next, we compute the
bicluster quality by choosing bicluster indices uniformly at random with the same cardinality. We
repeat this experiment 100 times. There is a significant gap between the quality of these random
biclusters and the ones inferred by tensor biclustering methods indicating the significance of our
inferred biclusters. For more details on these experiment, see SM Section 3.2.
7
Discussion
In this paper, we introduced and analyzed the tensor biclustering problem. The goal is to compute a
subset of tensor rows and columns whose corresponding trajectories form a low dimensional subspace.
To solve this problem, we proposed a method called tensor folding+spectral which demonstrated
improved analytical and empirical performance compared to other considered methods. Moreover, we
characterized computational and statistical (information theoretic) limits for the tensor biclustering
problem in an asymptotic regime, under both coherent and non-coherent statistical models.
Our results consider the case when the rank of the subspace is equal to one (i.e., q = 1). When q > 1,
in both tensor folding+spectral and tensor unfolding+spectral methods, the embedded subspace in
the signal matrix will have a rank of q > 1, with singular values ?1 ? ?2 ? ... ? ?q > 0. In this
8
setup, we need the spectral radius of the noise matrix to be smaller than ?q in order to guarantee
the recovery of the subspace. The procedure to characterize asymptotic achievability bounds would
follow from similar steps of the rank one case with some technical differences. For example, we
would need to extend Lemma 6 to the case where the signal matrix has rank q > 1. Moreover, in our
problem setup, we assumed that the size of the bicluster k and the rank of its subspace q are know
parameters. In practice, these parameters can be learned approximately from the data. For example,
in the tensor folding+spectral method, a good choice for the q parameter would be the index where
eigenvalues of the folded matrix decrease significantly. Knowing q, one can determine the size of
the bicluster similarly as the number of indices in top eigenvectors with significantly larger absolute
values. Another practical approach to estimate model parameters would be trial and error plus cross
validations.
Some of the developed proof techniques may be of independent interest as well. For example, we
proved an l? version of the Davis-Kahan lemma for a Wishart noise matrix. Solving the tensor
biclustering problem for the case of having multiple overlapping biclusters, for the case of having
incomplete tensor, and for the case of a priori unknown bicluster sizes are among future directions.
8
Code
We provide code for tensor biclustering methods in the following link: https://github.com/
SoheilFeizi/Tensor-Biclustering.
9
Acknowledgment
We thank Prof. Ofer Zeitouni for the helpful discussion on detectably proof techniques of probability
measures.
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10
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6,336 | 6,731 | DPSCREEN: Dynamic Personalized Screening
Kartik Ahuja
Electrical and Computer Engineering Department
University of California, Los Angeles
[email protected]
William R. Zame
Economics Department
University of California, Los Angeles
[email protected]
Mihaela van der Schaar
Engineering Science Department, University of Oxford
Electrical and Computer Engineering Department, University of California, Los Angeles
[email protected]
Abstract
Screening is important for the diagnosis and treatment of a wide variety of diseases.
A good screening policy should be personalized to the features of the patient
and to the dynamic history of the patient (including the history of screening).
The growth of electronic health records data has led to the development of many
models to predict the onset and progression of different diseases. However, there
has been limited work to address the personalized screening for these different
diseases. In this work, we develop the first framework to construct screening
policies for a large class of disease models. The disease is modeled as a finite state
stochastic process with an absorbing disease state. The patient observes an external
information process (for instance, self-examinations, discovering comorbidities,
etc.) which can trigger the patient to arrive at the clinician earlier than scheduled
screenings. The clinician carries out the tests; based on the test results and the
external information it schedules the next arrival. Computing the exactly optimal
screening policy that balances the delay in the detection against the frequency of
screenings is computationally intractable; this paper provides a computationally
tractable construction of an approximately optimal policy. As an illustration, we
make use of a large breast cancer data set. The constructed policy screens patients
more or less often according to their initial risk ? it is personalized to the features
of the patient ? and according to the results of previous screens ? it is personalized
to the history of the patient. In comparison with existing clinical policies, the
constructed policy leads to large reductions (28-68%) in the number of screens
performed while achieving the same expected delays in disease detection.
1
Introduction
Screening plays an important role in the diagnosis and treatment of a wide variety of diseases,
including cancer, cardiovascular disease, HIV, diabetes and many others by leading to early detection
of disease [1]-[3]. For some diseases (e.g., breast cancer, pancreatic cancer), the benefit of early
detection is enormous [4] [5]. Because screening ? especially screening that requires invasive
procedures such as mammograms, CT scans, biopsies, angiograms, etc. ? imposes financial and
health costs on the patient and resource costs on society, good screening policies should trade off
benefit and cost [6]. The best screening policies should take into account that the trade-off between
benefit and cost should be different for different diseases ? but also for different patients ? patients
whose features suggest that they are at high risk should be screened more often; patients whose
features suggest that they are at low risk should be screened less often ? and even different for the
same individual at different points in time, as the perceived risk for that patient changes. Thus the
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
best screening policies should account for the disease type and be personalized to the features of the
patient and to the history of the patient (including the history of screening) [32]. This paper develops
the first such personalized screening policies in a very general setting.
A screening policy prescribes what tests should/should not be done and when. Developing personalized screening policies that optimally balance the frequency of testing against the delay in the
detection of the disease is extremely difficult for a number of reasons. (1) The onset and progression
of different diseases varies significantly across the diseases. For instance, in [7] the development of
breast cancer is modeled as a stationary Markov process, in [36] the development of HIV is modeled
using a non-stationary survival process and, in [46] the development of colon cancer is modeled as a
Semi-Markov process. The test outcomes observed over time may follow a non-stationary stochastic
process that depends on the disease process upto that time and the features of the patient [35][36].
Existing works on screening [7] [9] are restricted to Markov disease processes and stationary Markov
test outcome models, while this is not the case for many diseases and their test outcomes [10][35]-[37].
(2) The cost of not screening is the delay in detection of disease, which is not known. Hence the
decision maker must act on the basis of beliefs about future disease states in addition to beliefs about
the current disease state. (3) Patients can arrive at the scheduled time but may also arrive earlier on
the basis of external information so the decision maker?s beliefs must take this external information
into account. For instance, external information can be the development of lumps on breasts [25][26],
or the development of a comorbidity [33][41]. (4) Given models of the progression of the disease and
of the external information, solving for that policy is computationally intractable in general.
This paper addresses all of these problems. We provide a computationally effective procedure that
solves for an approximately optimal policy and we provide bounds for the approximation error (loss
in performance) that arises from using the approximately optimal policy rather than the exactly
optimal policy. Our procedure is applicable to many disease models such as dynamic survival models
[11]-[13][36]-[37], first hitting time models [7][9][14]-[17].
Evaluating a proposed personalized screening policy using observational data is challenging. Observational data does not contain the counterfactuals: we cannot know what would have happened if a
patient had been screened more often or an additional test had been performed. Instead, we follow an
alternative route that has become standard in the literature [7]-[10]: we learn the disease progression
model from the observational data and then evaluate the screening policy on the basis of the learned
model. We also account for the fact that the disease model may be incorrectly estimated. We show
that if the estimation error and the approximation error are small, then the policy we construct is very
close to the policy for the correctly estimated model.
In this work, use of a large breast cancer data set to illustrate the proposed personalized screening
policy. We show that high risk patients are screened more often than low risk patients (personalization
to the features of the patient) and that patients with bad test results are screened more often than
patients with good test results (personalization to the dynamic history of the patient). The effect of
these personalizations is that, in comparison with existing clinical policies, the policy we construct
leads to large reductions (28-68%) in screening while achieving the same expected delays in disease
detection. To illustrate the impact of the disease on the policy, we carry out a synthetic exercise across
diseases, one for which the delay cost is linear and one for which the delay cost is quadratic. We
show that the regime of operation (frequency of tests vs expected delay in detection) for the policies
for the two costs are significantly different, thus highlighting the importance of choice of costs.
2
Model and Problem Formulation
Time Time is discrete and the time horizon is finite; we write T = {1, ..., T } for the set of time slots.
Patient Features Patients are distinguished by a (fixed) feature x. We assume that the features of a
patient (age, sex, family history, etc.) are observable and that the set X of all patient features is finite.
Disease Model We model the disease in terms of the (true physiological) state, where the state space
is S. The disease follows a finite state stochastic process; S T is the space of state trajectories. The
probability distribution over trajectories depends on the patient?s features; for ~s ? S T , x ? X we
write P r(~s|x) for the probability that the state trajectory is ~s given that the patient?s features are
x. We distinguish one state D ? S as the disease state; the disease state D is absorbing.1 Hence
1
The restriction to a single absorbing disease state is only for expositional convenience.
2
P r(s(t) = D, s(t0 ) 6= D) = 0 for every time t and every time t0 > t. The true state is hidden/not
observed.2
Our stochastic process model of disease encompasses many of the disease models in the literature,
including discrete time survival models. The (discrete time) Cox Proportional Odds model [11], for
instance, is the particular case of our model in which there are two states (Healthy H and Disease
D) and the probability distribution over state trajectories is determined from the hazard rates. To
be precise: if ~s is the state trajectory for which the disease state first occurs at time t0 , so that
s(t) = H for t < t0 and s(t) = D for t ? t0 , ?(t|x) is the hazard at time t conditional on x, then
P r(~s|x) = [1 ? ?(1|x)] ? ? ? [1 ? ?(t0 ? 1|x)][?(t0 |x)] and P r(~s|x) = 0 for all trajectories not having
this form. Similar constructions show that other dynamic survival models [14]-[17] [10][37] also fit
in the rubric of the general model presented here.3
External Information The clinician performs tests that are informative about the patient?s true
state; in addition, external information may also arrive (for instance, patient self-examines breasts for
lumps, patient discovers comorbidities, etc.). The patient observes an external information process
modeled by a finite state stochastic process with state space Y; the information at time t is Y (t) ? Y
(for instance, Y = {Lump, No Lump}). If the patient visits clinician at time t, then this external
information Y (t) arrives to the clinician. Y (t) may be correlated with the patient?s state trajectory
through time t and the patient?s features; we write P r(Y (t) = y|~s(t), x) for the probability that
the external information at time t is y ? Y, conditional on the state trajectory through time t and
features x. We assume that at each time t the external information Y (t) is independent of the past
observations conditional on the state trajectory through time t, ~s(t), and features x.
Arrival The patient visits the clinician at time t if either (a) the information process Y (t) exceeds
some threshold y? or (b) t is the time for the next recommended screening (determined in the Screening
Policies described below). The first visit of the patient to the hospital depends on the screening
policy and the patient?s features (See the description below). If the patient visits the clinician at time
t, the clinician performs a sequence of tests and observes the results. For simplicity of exposition,
we assume that the clinician performs only a single test, with a finite set Z of outcomes. We write
P r(Z(t) = z|~s(t), x) as the probability that test performed at time t yields the result z, conditional
on the (unobserved) state trajectory and the patient?s features. We assume that the current test result
is independent of past test results, conditional on the state trajectory and patient features. We also
assume that current test result is independent of the external information conditional on the state
trajectory through time t and the patient features. These assumptions are standard [7] [36]. We
adopt the convention that z(t) = ? if the patient does not visit the clinician at time t so that no test
is performed. If the test outcome z ? Z + ? Z, then the patient is diagnosed to have the disease.
We assume that there are no false positives. If a patient is diagnosed to be in the disease state, then
screening ends and treatment begins.
Screening Policies The history of a patient through time t consists of the trajectories of external
information, test results and screening recommendations through time t. Write H(t) for the set of
ST
histories through time t and H = t=0 H(t) for the set of all histories. By convention H(0) consists
only of the empty history. A screening policy is a map ? : X ? H ? {1, . . . , T } ? {D} that specifies,
for each feature x and history h either the next screening time t+ or the decision that the patient is in
the disease state D and so treatment should begin. A screening policy ? begins at time 0, when the
history is empty, so ?(x, ?) specifies the first screening time for a patient with features x. (For riskier
patients, screening should begin earlier.) Write ? for the space of all screening policies.
Screening Cost We normalize so that the cost of each screening is 1. (We can easily generalize to
the more general setting in which the clinician decides from multiple tests [50], and different tests
have different costs.) The cost of screening is a proxy for some combination of the monetary cost, the
resource cost and the health cost to the patient. We discount screening P
costs over time so if Ts is the
set of times at which the patient is screened then the screening cost is t?Ts ? t , where ? ? (0, 1).
2
For many diseases, it seems natural to identify states intermediate between Healthy and Disease. For
instance, because breast lumps [26] or colon polyps [9] that are found to be benign may become malignant, it
seems natural to distinguish at least one Risky state, intermediate between the Healthy and Disease states.
3
We can encompass the possibility of competing risks (e.g., different kinds of heart failure) [13] simply by
allowing for multiple absorbing states.
3
Delay Cost If disease first occurs at time tD (the incidence time) but is detected only at time
td > tD (the detection time) then the patient incurs a delay cost C(td ? tD ; tD ). If the disease
is never detected the delay cost is C(T ? tD ; tD ). We assume that the delay cost function C :
{1, . . . , T } ? {1, . . . , T ? 1} ? (0, ?) is increasing in the first argument (the lag in detection)
and decreasing in the second argument (the incidence time). The cost of delay is 0 if disease never
occurs or occurs only at time t = T . Note that as soon as the disease is detected screening ends
and treatment begins; in particular, there is a single unique time of incidence and a single unique
time of detection. We allow for general delay costs because the impact of early/late detection on the
probability of survival/successful treatment is different for different diseases.
Expected Costs If the patient features are x ? X then every screening policy ? ? ? induces a
probability distribution P r(?|x, ?) on the space H(T ) of all histories through time T and in particular
induces probability distributions ? = P r(?|x, ?) on the families Ts ? 2{1,...,T ?1} of screening times
and ? = P r((?, ?)|x,?)
detection time. The expected
P on thet pairs (tD , td ) of incidence time and
screening cost is E?
?
and
the
expected
delay
cost
is
E
? C(td ? tD , tD ) . We provide a
t?Ts
graphical model for the entire setup in the Appendix B of the Supplementary Materials.
Optimal Screening Policy The objective of the screening policy is to minimize a weighted sum of
the screening cost and the delay cost; i.e. the optimal screening policy is defined by
n
hX i
o
arg min (1 ? w) E?
? t + wE? [C(td ? tD , tD )]
(1)
???
t?Ts
The weight w reflects social/medical policy; for instance, w might be chosen to minimize cost subject
to some accepted tolerance in delay (Further discussion on this is in Section 4).
Comment The standard decision theory methods [18]-[21] used in screening [7][9] cannot be used
to solve the above problem. In standard POMDPs, the interval between two decision epochs (in this
case, screening times) is fixed exogenously; in standard POSMDPs, the time between two decision
epochs is the sojourn time for the underlying core-state process. In our setting, the time between two
decision epochs depends on the action (follow-up date), the external information process, and the
state trajectory. In standard POMDPs (POSMDPs) the cost incurred in a decision epoch depend on
the current state, while in the above problem the delay cost depends on the state trajectory. Moreover,
in our setting the disease state trajectory is not restricted to a Markovian or Semi-Markovian process.
3
Proposed Approach
Beliefs By a belief b we mean a probability distribution over the pairs consisting of state trajectories
and a label l for the diagnosis: l = 1 if the patient has been diagnosed with
Pthe disease, l = 0 otherwise.
By definition, a belief is a function b : S T ? {0, 1} ? [0, 1] such that ~s,l b(~s, l) = 1 but it is often
convenient to view a belief as a vector. Beliefs are updated using Bayesian updating every time there
is a new observation (test outcomes, patient arrival, external information). Knowledge of beliefs
will be sufficient to solve the optimization problem (1); see the Appendix C in the Supplementary
Materials. We write B for the space of all beliefs.
Bellman Equations To solve (1) we will formulate and solve the Bellman equations. To this end,
we begin by defining the various components of the Bellman equations. Fix a time t. The cost C?
incurred at time t depends on what happens at that time: i) if the patient (with diagnosis status l = 0
before the test) is tested and found to have acquired the disease, the cost is the sum of the cost of
testing and the cost of delay, ii) if the patient has the disease and is not detected, then the cost of delay
is incurred in the time slot T , and iii) if the patient does not have the disease, then the cost incurred in
time slot t depends on whether a test was done in time slot t or not. We write these cases below.
?
t
t ? T, l = 0, z ? Z +
?wC(t ? tD ; tD ) + (1 ? w)? I(z 6= ?)
?
C(~s, t, z, l) = wC(T ? tD ; tD )
(2)
t = T, l = 0
?
(1 ? w)? t I(z 6= ?)
otherwise
A recommendation plan ? : Z ? T maps the observation z at the end of time slot t to the next
scheduled follow-up time. Note that the recommendation plan is defined for a time t and is different
than the policy. Denote the probability distribution over the observations (test outcome z, duration to
the next arrival ??, and the external information at the next arrival
time y) conditional on the current
belief b and the current recommendation plan ? by P r(z, y, ?? b, ? , x). The belief b is updated to
4
?
b in the next arrival time ?? based on the observations, current recommended plan and the current
beliefs using Bayesian updating as ?b(~s, l) = P r(~s, l|b, ? , y, z, ??, x).
The optimal values for the objective in (2) starting from different initial beliefs can be expressed in
terms of a value function V : B ? {1, ..., T + 1} ? R. The value function at time t when the patient
is screened solves the Bellman equation:
hX
X
i
? t + ?? (3)
? s, t, z, l) +
V (b, t) = max
?b(~s, l)P r(z|~s, x) C(~
P r(z, y, ?? b, ? , x)V b,
?
z,?
? ,y
~
s,l,z
We define V (b, T + 1) = 0 for all beliefs. Note that the computation of the first term in the RHS
of (3) has a worst case computation time of |S|T . Therefore, solving for exact V (b, T ) that satisfies
(3) is computationally intractable when T is large. Next, we derive a useful property of the value
function. (The proof of this and all other results are in the Appendix D-F of the Supplementary
Material.).
Lemma 1 For every t, the value function V (b, t) is the maximum of a finite family of functions that
are linear in the beliefs b. In particular, the value function is convex and piecewise linear.
The above property was shown for POMDPs in [39], we use the same ideas to extend it to our setup.
3.1
Constructing the Exactly Optimal Policy
Every linear function of beliefs is of the form ?? b for some vector ?. (We view ?, b as column
vectors and write ?? for the transpose.) Hence Lemma 1 tells us that there is a finite set of vectors
?(t) such that V (b, t) = max???(t) ?? b. We refer to ?(t) as the set of alpha vectors. In view of
Lemma 1, to determine the value functions we need only determine the sets of alpha vectors. If we
substitute the expression V (b, t) = max???(t) ?? b into (3), then we obtain a recursive expression
for ?(t) in terms of ?(t + 1). By definition, the value function at time T + 1 is identically 0 so
?(T + 1) = {0}, where 0 is the |S T ? {0, 1}| dimensional zero vector, so we have an explicit starting
point for this recursive procedure. There is an optimal action associated with each alpha vector.
The action corresponding to the optimal alpha vector at a certain belief is the output of the optimal
action given that belief, and so constructing the sets of alpha vectors yields the optimal policy; the
details of the algorithm are in the Algorithm 3 in the Appendix A of the Supplementary Materials.
Unfortunately, the algorithm to compute the sets of alpha vectors is computationally intractable (as
expected). We therefore propose an algorithm that is tractable to compute an approximately optimal
policy.
3.2
Constructing the Approximately Optimal Policy
Point-Based Value Iteration (PBVI) approximation algorithms are known to work well for standard
POMDPs [18]. These algorithms rely on choosing a finite set of belief vectors and constructing alpha
vectors for these belief vectors and their success depends very much on the efficient construction of
the set of belief vectors. The standard approaches [18] for belief construction are not designed to
cope with settings like ours when beliefs lie in a very high dimensional space; in our setup belief has
|S T ? {0, 1}| dimensions. In Algorithm 1 (pseudo-code in the Appendix A of the Supplementary
Materials), we first construct a lower dimensional belief space by sampling trajectories that are more
likely to occur for the disease and then sampling the set of beliefs in the lower dimensional space that
are likely to occur over the course of various screening policies. The key steps for Algorithm 1 are
1. Sample typical physiological state trajectories Sample a set S? ? S T of K physiological
trajectories from the distribution P r(~s|x).
2. Construct the set of reachable belief vectors Say that a belief vector b2 is reachable from
the belief vector b1 if it can be derived by Bayesian updating on the basis of some underlying
screening policy. We construct the sets of belief vectors that can be reached under different screening
policies. For the first time slot, we start with a belief vector that lies in the space S? ? {0, 1} given as
?
? l = 0. For subsequent times, we select the beliefs that are encountered
P r(~s|x)/P r(S|x),
?~s ? S,
under random exploration of the actions (recommendation of future test dates). In addition to using
random exploration, we can choose actions determined from a set of policies such as the clinical
policies used in practice [27] [28] [47] to construct the set of reachable belief vectors.
5
? and the set of all such beliefs as
Denote the set of belief vectors constructed at time t by B[t]
?
?
? (see Algorithm
B = {B[t], ?t}. We carry out point-based value backups on these beliefs B
2 in the Appendix A of the Supplementary Materials), to construct the alpha vectors and thus
the approximately optimal policy. Henceforth, we refer to our approach (Algorithm 1 and 2) as
DPSCREEN.
Computational Complexity
The worst case computation of the policy requires
?
O T (B)2 T 2 K|Y||Z| steps, where B = maxt |B[t]|
is the maximum over the number of
points sampled by the Algorithm 1 for any time slot t. The complexity can be reduced by restricting
the space of actions; e.g. by bounding the amount of time allowed between successive screenings.
Moreover, the proposed algorithms can be easily parallelized (many operations carried inside the
iterations in Algorithm 2 can be done parallel), thus significantly reducing computation time.
Approximation Error Because we only sample a finite number of trajectories, the policy we
construct is not optimal but we can bound the loss of performance in comparison to the exactly optimal
policy and hence justify the term ?approximately optimal policy.? Define the approximation error to
be the difference between the value achieved by the exact optimal policy (solution to (1)) and the value
achieved by the approximately optimal policy (output from Algorithm 2). As a measure of the density
0
? = ? maxt?T maxB minb?B[t]
of sampling of the belief simplex we set ?(B)
? ||b ? b ||1 , where ? is
a constant that measures the maximum expected loss that can occur in one time slot. We make a few
assumptions for the proposition to follow. The cost for delay is C(td ? tD ; tD ) = c(td ? tD )? tD ,
where c(d) is a convex function of d. The test outcome is accurate, i.e. no false positives and no
false negatives. The maximum screening interval is bounded by W < T . The time horizon T is
sufficiently large. We show that the loss of performance is bounded by the sampling density.
?
Proposition 1 The approximation error is bounded above by ?(B).
3.3 Robustness
Estimation Error To this point, it has been assumed that the model parameters are known. In practice, the model parameters need to be estimated using the observational data. In the next section, we
will give a concrete example of how we estimate these parameters using observational data for breast
cancer. Here we discuss the effect of error in estimation. Suppose that the model being estimated
0
(true model) is m ? M , where M is the space of all the possible models (model parametrizations)
under consideration. (We assume that the probability distribution of the physiological state transition,
the patient?s self-observation outcomes, and the clinician?s observation outcomes are continuous on
M .) Write L = M ? B for the joint space of models and beliefs. Let the estimate of the model be
? Let us assume that for every model in M the solution to (1) is unique. Therefore, we can define
m.
a mapping ? ? : L ? Z ? T ? T |Z| , where ? ? (l, z, t) is the optimal recommended screening time
at l, at time t following z. For a fixed model m, ? ? ((m, b), z, t) is the maximizer in (3).
Theorem 1. There is a closed lower dimensional set E ? L such that the function ? ? is locally
constant on the complement of E.
? and the true model m0 are
Theorem 1 implies that, with probability 1, if the model estimate m
sufficiently close, then the actions recommended by the exactly optimal policies for both models are
identical. Therefore, the impact of estimation error on the exactly optimal policy is minimal. However,
we construct approximately optimal policies. We can combine these conditions with Proposition 1 to
? goes to zero, then the approximately optimal policy (for m)
say that if the approximation error ?(B)
?
0
will also converge to the exactly optimal policy for true model m .
Personalization: Figure 1 provides a graphical representation of the way in which DPSCREEN
is personalized to the patients. We consider three Patients. The disease model for each patient
is given by the ex ante survival curve (the probability of not becoming diseased by a given time).
As shown in the graphs, the survival curves for Patients 1, 2 are the same; the survival curve for
Patient 3 begins below the survival curve for Patients 1, 2 but is flatter and so eventually crosses
the survival curve for Patients 1, 2. All three patients are screened at date 1; for all three the test
outcome is z = Low. Hence the belief (risk assessment) for all three patients decreases. As a result,
Patients 1, 2 are scheduled for next screening at date 4 but Patient 3, who has a lower ex ante survival
probability, is scheduled for next screening at date 3. Thus, the policy is personalized to the ex ante
risk. However, at date 2, all three patients experience an external information shock which causes
them to be screened early. The test outcome for Patient 1 is z = Medium so Patient 1 is assessed
to be at higher risk and is scheduled for next screening at date 3; the test outcome for Patient 2 is
6
Belief Disease
?
?
Patient 1 and 2: Personalization through histories
Same features, different histories ? different screening
? = ??????
? = ???
? ' = 3
? ' = 4
?
Patient 2
Belief Disease
1
?
?
2
?
?
?
4
5
6 Time
? ' = 4 ? ' = 5
1
Patient 3
3
Patient 2 and 3: Personalization through features
Same history, different hazard rates ? different screening
? = ???
? = ???
?
Belief Disease
Survival probability Survival probability
Survival probability
Patient 1
2
3
4
5
6 Time
? = ???
'
? = ???
? = 3
? ' = 6
1
2
3
4
? ' : prescribed next arrival time
5
6 Time
?: test outcomes
Figure 1: Illustration of dynamic personalization
z = Low so Patient 2 is assessed to be at lower risk and is scheduled for next screening at date 5.
Thus the policy is personalized to the dynamic history. The test outcome for Patient 3 is z = Low
and Patient 3?s ex ante survival probability is higher so Patient 3?s risk is assessed to be very low, and
Patient 3 is scheduled for next screening at date 6. Thus the policy adjusts to time-varying model
parameters.
4
Illustrative Experiments
Here we demonstrate the effectiveness of our policy in a real setting: screening for breast cancer.
Description of the dataset: We use a de-identified dataset (from Athena Health Network [22])
of 45, 000 patients aged 60-65 who underwent screening for breast cancer. For most individuals
we have the following associated features: age, the number of family members with breast cancer,
weight, etc. Each patient had at least one mammogram; some had several. (In total, there are 84,000
mammograms in the dataset.) If the patients had a positive mammogram, a biopsy is carried out.
Further description of mammogram output is in the Appendix G of the Supplementary Materials.
Model description We model the disease progression using a two-state Markov model: S = {H, D}
(H = Healthy, D = Disease/Cancer). Given patient features x, the initial probability of cancer is
pin (x) and the probability of transition from the H to D is ptr (x). The external information Y
is the size (perhaps 0) of a breast lump, based on the patient?s own self-examination. In view
of the universal growth law for tumor described in [23], we model Y (t) = g(t) + (t), where
g(t) = (1 ? e??(t?ts ) )I(t > ts ) is the size of the tumor and ts is the time at which patient actually
develops cancer (the lump exists), (t) is a zero mean white noise process with variance ? 2 and I() is
the indicator function. If the lump size Y exceeds the threshold y?, then the patient visits the clinician,
where tests are carried out. The set of test outcomes is Z = {?, 1, 2, 3}, where z = ? when no test is
done, z = 1 when the mammogram is negative and no biopsy is done, z = 2 when the mammogram
is positive and the biopsy is negative, z = 3 when both mammogram and biopsy is positive.
Model Estimation We use the specificity and sensitivity for the mammogram from [7]. Each patient
has a different (initial) risk for developing cancer; we compute the risk scores using the Gail model
[24], which we use as the feature x. We assumed pin (x) and ptran (x) are logistic functions of x. We
use standard Markov Chain Monte Carlo methods to estimate these functions pin (x) and ptran (x)
(further details in the Appendix G of the Supplementary Materials). We assume that each woman has
one self-examination per month [25] [26]. We use the value ? = 0.9 as stated in [23]. We estimate
the parameters for the self-examinations ? = 0.43 and y? = 1 on the basis of the values of sensitivity
and specificity for the self-examination from the literature [43]. In the comparisons to follow, we
7
will also analyze the setting when there are no self-examinations. We divide the population into two
risk groups; the Low risk group consists of patients whose prior estimated risk of developing cancer
within five years is less than 5%; the High risk group consists of patients whose prior estimated risk
exceeds 5%.
Performance Metrics, Objective and Benchmarks: Our objective is to minimize the number
of screenings subject to a constraint on expected delay cost. We assume the delay cost is linear:
C(td ? tD , tD ) = td ? tD . To derive the solution to this constrained problem from construction,
which minimizes the weighted sum of screening cost and delay cost, we solve the weighted problem
for some weight w, and then tune w to select the policy that minimizes the number of screenings
subject to a constraint on expected delay cost. For comparison purposes, we take the constraint
on expected delay cost to be the expected delay that arises from current clinical practice (annual
screening in the US [27][28], biennial screening in some other countries [29]). (Because our objective
is to minimize the number of screenings, we take the cost of each screening to be 1, whether or not a
biopsy is performed.)
Comment At this point, we remind that existing frameworks [7][9][10] cannot be used to solve for
the optimal screening policy in the above setup because: i) the costs incurred (delay) depends on
the state trajectory and not just the current state, and ii) the lump growth model and the patient?s
self-examination of the lump is not easy to incorporate in these works.
Comparisons with clinical screening policies: We compare our constructed policies (for the two
groups), with and without self-examination, in terms of three metrics: i) E[N |R]: the expected
number of tests per year, conditional on the risk group; ii) E[?|R]: the expected delay, conditional
on the risk group; iii) E[?|R, D]: the expected delay, conditional on the risk group and the patient
actually developing cancer. Because E[?|R] is the expected unconditional delay, it accounts for
patients who do not develop cancer as well as for patients who do have cancer; because most patients
do not develop cancer, E[?|R] is small. We show the comparisons with the annual policies in Table 1;
we show the comparisons with biennial screening in the Appendix G of the Supplementary Materials.
In Table 1 we compare the performance of DPSCREEN (with and without self-examination) for Low
and High risk groups against the current clinical policy of annual screening. For both risk groups,
the proposed policy achieves approximately the same expected delay as the benchmark policy while
doing many fewer tests (in expectation). With self-examinations, the expected reduction in number of
screens is 57-68% (depending on risk group); even without self-examinations, the expected reduction
in number of screens is 28-45% percent (depending on risk group).
In Table 2 we contrast the difference in DPSCREEN across the two risk groups. To keep the
comparison fair, we fix the tolerance in the delay to a fixed value. The proposed policy is personalized
as it recommends significantly fewer tests to the low risk patients in contrast to the high risk patients.
E[N jR; Cost]
3
Impact of the type of disease: We have so
High risk, Linear cost
w=0.9 w=0.9
far considered breast cancer as an example
High risk, Quadratic cost
2.5
Low risk, Linear cost
and assumed linear delay costs. For some disLow risk, Quadratic cost
2
eases (such as Pancreatic cancer [30][5]) the
w=0.5
w=0.5
survival probability decreases very quickly with
1.5
w=0.3
the delay in detection and therefore it might
w=0.3
w=0.9
1
be reasonable to assume a cost of delay that
is strictly convex (such as quadratic costs) in
w=0.9
w=0.5
0.5
w=0.3
delay time for some disease. In Figure 2, we
w=0.5w=0.3
0
show that for a fixed risk group and for the same
0
0.2
0.4
0.6
0.8
1
E["jR; Cost] months
weights the policy constructed using quadratic
Figure 2: Impact of the type of disease
costs is much more aggressive in testing. Moreover, the regime of operation of the policy (the
points achieved by the policy in the 2-D plane
E[N |R, Cost] vs E[?|R, Cost]) can vary a lot depending on the choice of cost function even though
the same weights are used. Therefore, the cost should be chosen based on the disease.
5
Related Works
In Section 2, following the equation (1), we compared our methods with frameworks to some general
frameworks in decision theory [18]-[21]. Next, we compare with other relevant works.
8
Table 1: Comparison of the proposed policy with annual screening for both high and low risk group.
Risk
Group
Low
High
Metrics
E[N |R], E[?|R], E[?|R, D]
E[N |R], E[?|R], E[?|R, D]
DPSCREEN with
self-examination
0.32, 0.23, 9.2
0.43, 0.50, 6.7
DPSCREEN w/o
self-examination
0.55, 0.23, 9.2
0.72, 0.52,7.07
Annual
1, 0.24, 9.6
1, 0.52, 7.07
Table 2: Comparison of the proposed policies across different risk groups
Risk
Group
Low
High
DPSCREEN with self-examination
DPSCREEN w/o self-examination
E[N |R], E[?|R], E[?|R, D]
0.12, 0.33, 13.7
0.80, 0.35, 4.73
E[N |R], E[?|R], E[?|R, D]
0.32, 0.33, 13.7
1.09, 0.35, 4.73
Screening frameworks for different diseases in operations research: Many works have focused
on optimizing population-based screening schedules, which are not personalized (See [42] and
references therein). In [7] [9] the authors develop personalized POMDP based screening models.
The underlying disease evolution (breast and colon cancer) is assumed to follow a Markov process.
External information process such as self-exams and the test outcomes over time are assumed to
follow a stationary i.i.d process given the disease process. In [10] authors develop personalized
screening models based on principles of Bayesian design for maximizing information gain (based
on [40]). The underlying disease model (cardiac disease) is a dynamic (two-state) survival model
and the cost of misdetection is a constant and does not depend on the delay. The test outcomes are
modeled using generalized linear mixed effects models, and there is no external information process.
To summarize, all of the above methods rely on very specific models for their disease, test outcomes,
and external information, while our method imposes much less restrictions on the same.
Screening frameworks for different diseases in medical literature: The Medical research literature on screening (e.g., Cancer Intervention and Surveillance Modelling Network, US preventive
services task force, etc.) relies on stochastic simulation based methods: fix a disease model and a
set of screening policies to be compared; for each policy in the set, simulate outcome paths from
the model; compare across the set of policies [44]-[48]. The clinical guidelines for screening issued
by the US preventive services task force [47][49] for colon cancer cancers are created based on the
MISCAN-COLON [46] model for colon cancer. Simulations were carried out to compare different
screening policies suggested by experts for that specific disease model- MISCAN-COLON. This
approach allows more realistically complex models but it only compares a fixed set of policies, all of
which may be far from optimal.
Controlled Sensing: In controlled sensing [21][34][38] the problem of sensor scheduling requires
deciding which sensor to use and when; this problem is similar the personalized screening problem
studied here. In these works [21][34][38], the main focus is to exploit (or derive) structural properties
of the process being sensed and the cost functions such that the exactly optimal sensing schedule is
easy to characterize and compute. The structural assumptions such as the process that is sampled is
stationary and Markov make these works less suited for personalized screening.
6
Conclusion
In this work, we develop a novel methodology for constructing personalized screening policies that
balance the cost of screening against the cost of delay in detection of disease. The disease is modeled
as an arbitrary finite state stochastic process with an absorbing disease state. Our method incorporates
the possibility of external information, such as self-examination or discovery of co-morbidities, that
may trigger arrival of the patient to the clinician in advance of a scheduled screening appointment.
We use breast cancer data to develop the disease model. In comparison with current clinical policies,
our personalized screening policies reduce the number of screenings performed while maintaining
the same delay in detection of disease.
9
7
Acknowledgements
This work was supported by the Office of Naval Research (ONR) and the National Science Foundation
(NSF) (Grant number: 1533983 and Grant number: 1407712).
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12
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6,337 | 6,732 | Learning Unknown Markov Decision Processes:
A Thompson Sampling Approach
Yi Ouyang
University of California, Berkeley
[email protected]
Mukul Gagrani
University of Southern California
[email protected]
Ashutosh Nayyar
University of Southern California
[email protected]
Rahul Jain
University of Southern California
[email protected]
Abstract
We consider the problem of learning an unknown Markov Decision Process (MDP)
that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes
(TSDE). At the beginning of each episode, the algorithm generates a sample from
the posterior distribution over the unknown model parameters. It then follows the
optimal stationary policy for the sampled model for the rest of the episode. The
duration of each episode is dynamically determined by two stopping criteria. The
first stopping criterion controls the growth rate of episode length. The second
stopping criterion happens when
? the number of visits to any state-action pair is
?
doubled. We establish O(HS
AT ) bounds on expected regret under a Bayesian
setting, where S and A are the sizes of the state and action spaces, T is time, and
H is the bound of the span. This regret bound matches the best available bound for
weakly communicating MDPs. Numerical results show it to perform better than
existing algorithms for infinite horizon MDPs.
1
Introduction
We consider the problem of reinforcement learning by an agent interacting with an environment while
trying to minimize the total cost accumulated over time. The environment is modeled by an infinite
horizon Markov Decision Process (MDP) with finite state and action spaces. When the environment
is perfectly known, the agent can determine optimal actions by solving a dynamic program for the
MDP [1]. In reinforcement learning, however, the agent is uncertain about the true dynamics of the
MDP. A naive approach to an unknown model is the certainty equivalence principle. The idea is to
estimate the unknown MDP parameters from available information and then choose actions as if the
estimates are the true parameters. But it is well-known in adaptive control theory that the certainty
equivalence principle may lead to suboptimal performance due to the lack of exploration [2]. This
issue actually comes from the fundamental exploitation-exploration trade-off: the agent wants to
exploit available information to minimize cost, but it also needs to explore the environment to learn
system dynamics.
One common way to handle the exploitation-exploration trade-off is to use the optimism in the
face of uncertainty (OFU) principle [3]. Under this principle, the agent constructs confidence sets
for the system parameters at each time, find the optimistic parameters that are associated with the
minimum cost, and then selects an action based on the optimistic parameters. The optimism procedure
encourages exploration for rarely visited states and actions. Several optimistic algorithms are proved
to possess strong theoretical performance guarantees [4?10].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
An alternative way to incentivize exploration is the Thompson Sampling (TS) or Posterior Sampling
method. The idea of TS was first proposed by Thompson in [11] for stochastic bandit problems. It has
been applied to MDP environments [12?17] where the agent computes the posterior distribution of
unknown parameters using observed information and a prior distribution. A TS algorithm generally
proceeds in episodes: at the beginning of each episode a set of MDP parameters is randomly
sampled from the posterior distribution, then actions are selected based on the sampled model during
the episode. TS algorithms have the following advantages over optimistic algorithms. First, TS
algorithms can easily incorporate problem structures through the prior distribution. Second, they are
more computationally efficient since a TS algorithm only needs to solve the sampled MDP, while an
optimistic algorithm requires solving all MDPs that lie within the confident sets. Third, empirical
studies suggest that TS algorithms outperform optimistic algorithms in bandit problems [18, 19] as
well as in MDP environments [13, 16, 17].
Due to the above advantages, we focus on TS algorithms for the MDP learning problem. The main
challenge in the design of a TS algorithm is the lengths of the episodes. For finite horizon MDPs
under the episodic setting, the length of each episode can be set as the time horizon [13]. When there
exists a recurrent state under any stationary policy, the TS algorithm of [15] starts a new episode
whenever the system enters the recurrent state. However, the above methods to end an episode can
not be applied to MDPs without the special features. The work of [16] proposed a dynamic episode
schedule based on the doubling trick used in [7], but a mistake in their proof of regret bound was
pointed out by [20]. In view of the mistake in [16], there is no TS algorithm with strong performance
guarantees for general MDPs to the best of our knowledge.
We consider the most general subclass of weakly communicating MDPs in which meaningful finite
time regret guarantees can be analyzed. We propose the Thompson Sampling with Dynamic Episodes
(TSDE) learning algorithm. In TSDE, there are two stopping criteria for an episode to end. The
first stopping criterion controls the growth rate of episode length. The second stopping criterion
is the doubling trick similar to the one in [7?10, 16] that stops when the number of visits to any
state-action pair is doubled. Under a Bayesian framework,
we show that the expected regret of TSDE
?
?
? hides logarithmic factors. Here S
accumulated up to time T is bounded by O(HS
AT ) where O
and A are the sizes of the state and action spaces, T is time, and H is the bound of the span. This
regret bound matches the best available bound for weakly communicating MDPs [7], and it matches
the theoretical lower bound in order of T except for logarithmic factors. We present numerical results
that show that TSDE actually outperforms current algorithms with known regret bounds that have the
same order in T for a benchmark MDP problem as well as randomly generated MDPs.
2
Problem Formulation
2.1
Preliminaries
An infinite horizon Markov Decision Process (MDP) is described by (S, A, c, ?). Here S is the state
space, A is the action space, c : S ? A ? [0, 1]1 is the cost function, and ? : S 2 ? A ? [0, 1]
represents the transition probabilities such that ?(s0 |s, a) = P(st+1 = s0 |st = s, at = a) where
st ? S and at ? A are the state and the action at t = 1, 2, 3 . . . . We assume that S and A are finite
spaces with sizes S ? 2 and A ? 2, and the initial state s1 is a known and fixed state. A stationary
policy is a deterministic map ? : S ? A that maps a state to an action. The average cost per stage of
a stationary policy is defined as
T
J? (?) = lim sup
T ??
i
1 hX
E
c(st , at ) .
T
t=1
Here we use J? (?) to explicitly show the dependency of the average cost on ?.
To have meaningful finite time regret bounds, we consider the subclass of weakly communicating
MDPs defined as follows.
Definition 1. An MDP is weakly communicating (or weak accessible) if its states can be partitioned
into two subsets: in the first subset all states are transient under every stationary policy, and every
two states in the second subset can be reached from each other under some stationary policy.
1
Since S and A are finite, we can normalize the cost function to [0, 1] without loss of generality.
2
From MDP theory [1], we know that if the MDP is weakly communicating, the optimal average cost
per stage J(?) = min? J? (?) satisfies the Bellman equation
n
o
X
J(?) + v(s, ?) = min c(s, a) +
?(s0 |s, a)v(s0 , ?)
(1)
a?A
s0 ?S
for all s ? S. The corresponding optimal stationary policy ? ? is the minimizer of the above
optimization given by
a = ? ? (s, ?).
(2)
Since the cost function c(s, a) ? [0, 1], J(?) ? [0, 1] for all ?. If v satisfies the Bellman equation, v
plus any constant also satisfies the Bellman equation. Without loss of generality, let mins?S v(s, ?) =
0 and define the span of the MDP as sp(?) = maxs?S v(s, ?). 2
We define ?? to be the set of all ? such that the MDP with transition probabilities ? is weakly
communicating, and there exists a number H such that sp(?) ? H. We will focus on MDPs with
transition probabilities in the set ?? .
2.2
Reinforcement Learning for Weakly Communicating MDPs
We consider the reinforcement learning problem of an agent interacting with a random weakly
communicating MDP (S, A, c, ?? ). We assume that S, A and the cost function c are completely
known to the agent. The actual transition probabilities ?? is randomly generated at the beginning
before the MDP interacts with the agent. The value of ?? is then fixed but unknown to the agent. The
complete knowledge of the cost is typical as in [7, 15]. Algorithms can generally be extended to the
unknown costs/rewards case at the expense of some constant factor for the regret bound.
At each time t, the agent selects an action according to at = ?t (ht ) where ht =
(s1 , s2 , . . . , st , a1 , a2 , . . . , at?1 ) is the history of states and actions. The collection ? = (?1 , ?2 . . . )
is called a learning algorithm. The functions ?t allow for the possibility of randomization over actions
at each time.
We focus on a Bayesian framework for the unknown parameter ?? . Let ?1 be the prior distribution
for ?? , i.e., for any set ?, P(?? ? ?) = ?1 (?). We make the following assumptions on ?1 .
Assumption 1. The support of the prior distribution ?1 is a subset of ?? . That is, the MDP is weakly
communicating and sp(?? ) ? H.
In this Bayesian framework, we define the expected regret (also called Bayesian regret or Bayes risk)
of a learning algorithm ? up to time T as
R(T, ?) = E
T h
hX
ii
c(st , at ) ? J(?? )
(3)
t=1
where st , at , t = 1, . . . , T are generated by ? and J(?? ) is the optimal per stage cost of the MDP.
The above expectation is with respect to the prior distribution ?1 for ?? , the randomness in state
transitions, and the randomized algorithm. The expected regret is an important metric to quantify the
performance of a learning algorithm.
3
Thompson Sampling with Dynamic Episodes
In this section, we propose the Thompson Sampling with Dynamic Episodes (TSDE) learning
algorithm. The input of TSDE is the prior distribution ?1 . At each time t, given the history ht , the
agent can compute the posterior distribution ?t given by ?t (?) = P(?? ? ?|ht ) for any set ?. Upon
applying the action at and observing the new state st+1 , the posterior distribution at t + 1 can be
updated according to Bayes? rule as
?t+1 (d?) = R
?(st+1 |st , at )?t (d?)
.
?0 (st+1 |st , at )?t (d?0 )
2
(4)
See [7]for a discussion on the connection of the span with other parameters such as the diameter appearing
in the lower bound on regret.
3
Let Nt (s, a) be the number of visits to any state-action pair (s, a) before time t. That is,
Nt (s, a) = |{? < t : (s? , a? ) = (s, a)}|.
(5)
With these notations, TSDE is described as follows.
Algorithm 1 Thompson Sampling with Dynamic Episodes (TSDE)
Input: ?1
Initialization: t ? 1, tk ? 0
for episodes k = 1, 2, ... do
Tk?1 ? t ? tk
tk ? t
Generate ?k ? ?tk and compute ?k (?) = ? ? (?, ?k ) from (1)-(2)
while t ? tk + Tk?1 and Nt (s, a) ? 2Ntk (s, a) for all (s, a) ? S ? A do
Apply action at = ?k (st )
Observe new state st+1
Update ?t+1 according to (4)
t?t+1
end while
end for
The TSDE algorithm operates in episodes. Let tk be start time of the kth episode and Tk = tk+1 ? tk
be the length of the episode with the convention T0 = 1. From the description of the algorithm,
t1 = 1 and tk+1 , k ? 1, is given by
tk+1 = min{t > tk : t > tk + Tk?1 or Nt (s, a) > 2Ntk (s, a) for some (s, a)}.
(6)
At the beginning of episode k, a parameter ?k is sampled from the posterior distribution ?tk . During
each episode k, actions are generated from the optimal stationary policy ?k for the sampled parameter
?k . One important feature of TSDE is that its episode lengths are not fixed. The length Tk of each
episode is dynamically determined according to two stopping criteria: (i) t > tk + Tk?1 , and (ii)
Nt (s, a) > 2Ntk (s, a) for some state-action pair (s, a). The first stopping criterion provides that the
episode length grows at a linear rate without triggering the second criterion. The second stopping
criterion ensures that the number of visits to any state-action pair (s, a) during an episode should not
be more than the number visits to the pair before this episode.
Remark 1. Note that TSDE only requires the knowledge of S, A, c, and the prior distribution ?1 .
TSDE can operate without the knowledge of time horizon T , the bound H on span used in [7], and
any knowledge about the actual ?? such as the recurrent state needed in [15].
3.1
Main Result
Theorem 1. Under Assumption 1,
R(T, TSDE) ? (H + 1)
p
p
2SAT log(T ) + 49HS AT log(AT ).
The proof of Theorem 1 appears in Section 4.
Remark 2. Note that our regret bound has the same order in H, S, A and T as the optimistic
algorithm in [7] which is the best available bound for weakly communicating MDPs. Moreover, the
bound does not depend on the prior distribution or other problem-dependent parameters such as the
recurrent time of the optimal policy used in the regret bound of [15].
3.2
Approximation Error
At the beginning of each episode, TSDE computes the optimal stationary policy ?k for the parameter
?k . This step requires the solution to a fixed finite MDP. Policy iteration or value iteration can be used
to solve the sampled MDP, but the resulting stationary policy may be only approximately optimal in
practice. We call ? an ?approximate policy if
n
o
X
X
c(s, ?(s)) +
?(s0 |s, ?(s))v(s0 , ?) ? min c(s, a) +
?(s0 |s, a)v(s0 , ?) + .
s0 ?S
a?A
4
s0 ?S
When the algorithm returns an k ?approximate policy ?
?k instead of the optimal stationary policy ?k
at episode k, we have the following regret bound in the presence of such approximation error.
?k instead of the optimal stationary policy
Theorem 2. If TSDE computes an k ?approximate policy ?
?k at each episode k, the expected regret of TSDE satisfies
h X
i
?
?
AT ) + E
R(T, TSDE) ? O(HS
Tk k .
k:tk ?T
Furthermore, if k ?
1
k+1 ,
E
hP
i
k:tk ?T
Tk k ?
p
2SAT log(T ).
Theorem 2 shows that the approximation error in the computation of optimal
? stationary policy is only
?
additive to the regret under TSDE. The regret bound would remain O(HS AT ) if the approximation
1
error is such that k ? k+1
. The proof of Theorem 2 is in the appendix due to the lack of space.
4
4.1
Analysis
Number of Episodes
To analyze the performance of TSDE over T time steps, define KT = arg max {k : tk ? T } be the
number of episodes of TSDE until time T . Note that KT is a random variable because the number
of visits Nt (x, u) depends on the dynamical state trajectory. In the analysis for time T we use the
convention that t(KT +1) = T + 1. We provide an upper bound on KT as follows.
Lemma 1.
p
KT ? 2SAT log(T ).
Proof. Define macro episodes with start times tni , i = 1, 2, . . . where tn1 = t1 and
tni+1 = min{tk > tni :
Ntk (s, a) > 2Ntk?1 (s, a) for some (s, a)}.
The idea is that each macro episode starts when the second stopping criterion happens. Let M be the
number of macro episodes until time T and define n(M +1) = KT + 1.
Pni+1 ?1
Let T?i = k=n
Tk be the length of the ith macro episode. By the definition of macro episodes,
i
any episode except the last one in a macro episode must be triggered by the first stopping criterion.
Therefore, within the ith macro episode, Tk = Tk?1 + 1 for all k = ni , ni + 1, . . . , ni+1 ? 2. Hence,
ni+1 ?1
T?i =
X
ni+1 ?ni ?1
X
Tk =
(Tni ?1 + j) + Tni+1 ?1
j=1
k=ni
ni+1 ?ni ?1
?
X
(j + 1) + 1 = 0.5(ni+1 ? ni )(ni+1 ? ni + 1).
j=1
p
2T?i for all i = 1, . . . , M . From this property we obtain
M
M q
X
X
KT =nM +1 ? 1 =
(ni+1 ? ni ) ?
2T?i .
Consequently, ni+1 ? ni ?
i=1
Using (7) and the fact that
(7)
i=1
PM ?
i=1 Ti = T we get
v
u
M q
M
u X
X
?
KT ?
2T?i ?tM
2T?i = 2M T
i=1
(8)
i=1
where the second inequality is Cauchy-Schwarz.
From Lemma 6 in the appendix, the number of macro episodes M ? SA log(T ). Substituting this
bound into (8) we obtain the result of this lemma.
Remark 3. TSDE computes the optimal stationary policy of a finite MDP atpeach episode. Lemma 1
ensures that such computation only needs to be done at a sublinear rate of 2SAT log(T ).
5
4.2
Regret Bound
As discussed in [13, 20, 21], one key property of Thompson/Posterior Sampling algorithms is that for
any function f , E[f (?t )] = E[f (?? )] if ?t is sampled from the posterior distribution at time t. This
property leads to regret bounds for algorithms with fixed sampling episodes since the start time tk of
each episode is deterministic. However, our TSDE algorithm has dynamic episodes that requires us
to have the stopping-time version of the above property.
Lemma 2. Under TSDE, tk is a stopping time for any episode k. Then for any measurable function
f and any ?(htk )?measurable random variable X, we have
h
i
h
i
E f (?k , X) = E f (?? , X)
Proof. From the definition (6), the start time tk is a stopping-time, i.e. tk is ?(htk )?measurable.
Note that ?k is randomly sampled from the posterior distribution ?tk . Since tk is a stopping time,
tk and ?tk are both measurable with respect to ?(htk ). From the assumption, X is also measurable
with respect to ?(htk ). Then conditioned on htk , the only randomness in f (?k , X) is the random
sampling in the algorithm. This gives the following equation:
h
i
h
i Z
h
i
E f (?k , X)|htk = E f (?k , X)|htk , tk , ?tk = f (?, X)?tk (d?) = E f (?? , X)|htk
(9)
since ?tk is the posterior distribution of ?? given htk . Now the result follows by taking the expectation
of both sides.
For tk ? t < tk+1 in episode k, the Bellman equation (1) holds by Assumption 1 for s = st , ? = ?k
and action at = ?k (st ). Then we obtain
X
c(st , at ) = J(?k ) + v(st , ?k ) ?
?k (s0 |st , at )v(s0 , ?k ).
(10)
s0 ?S
Using (10), the expected regret of TSDE is equal to
E
KT tk+1
hX
X?1
i
h
i
c(st , at ) ? T E J(?? )
k=1 t=tk
=E
KT
hX
KT tk+1
i
h
i
hX
ii
X
X?1 h
Tk J(?k ) ? T E J(?? ) + E
v(st , ?k ) ?
?k (s0 |st , at )v(s0 , ?k )
s0 ?S
k=1 t=tk
k=1
=R0 + R1 + R2 ,
(11)
where R0 , R1 and R2 are given by
R0 = E
KT
hX
i
h
i
Tk J(?k ) ? T E J(?? ) ,
k=1
R1 = E
KT tk+1
hX
X?1 h
ii
v(st , ?k ) ? v(st+1 , ?k ) ,
k=1 t=tk
R2 = E
KT tk+1
hX
X?1 h
v(st+1 , ?k ) ?
X
ii
?k (s0 |st , at )v(s0 , ?k ) .
s0 ?S
k=1 t=tk
We proceed to derive bounds on R0 , R1 and R2 .
Based on the key property of Lemma 2, we derive an upper bound on R0 .
Lemma 3. The first term R0 is bounded as
R0 ? E[KT ].
6
Proof. From monotone convergence theorem we have
R0 = E
?
hX
?
i
h
i X
h
i
h
i
1{tk ?T } Tk J(?k ) ? T E J(?? ) =
E 1{tk ?T } Tk J(?k ) ? T E J(?? ) .
k=1
k=1
Note that the first stopping criterion of TSDE ensures that Tk ? Tk?1 + 1 for all k. Because
J(?k ) ? 0, each term in the first summation satisfies
h
i
h
i
E 1{tk ?T } Tk J(?k ) ? E 1{tk ?T } (Tk?1 + 1)J(?k ) .
Note that 1{tk ?T } (Tk?1 + 1) is measurable with respect to ?(htk ). Then, Lemma 2 gives
h
i
h
i
E 1{tk ?T } (Tk?1 + 1)J(?k ) = E 1{tk ?T } (Tk?1 + 1)J(?? ) .
Combining the above equations we get
R0 ?
?
X
h
i
h
i
E 1{tk ?T } (Tk?1 + 1)J(?? ) ? T E J(?? )
k=1
=E
KT
hX
i
h
i
(Tk?1 + 1)J(?? ) ? T E J(?? )
k=1
KT
h
i
h X
i
h i
= E KT J(?? ) + E
Tk?1 ? T J(?? ) ? E KT
k=1
where the last equality holds because J(?? ) ? 1 and
PKT
k=1
Tk?1 = T0 +
PKT ?1
k=1
Tk ? T .
Note that the first stopping criterion of TSDE plays a crucial role in the proof of Lemma 3. It allows
us to bound the length of an episode using the length of the previous episode which is measurable
with respect to the information at the beginning of the episode.
The other two terms R1 and R2 of the regret are bounded in the following lemmas. Their proofs
follow similar steps to those in [13, 16]. The proofs are in the appendix due to the lack of space.
Lemma 4. The second term R1 is bounded as
R1 ? E[HKT ].
Lemma 5. The third term R2 is bounded as
R2 ? 49HS
p
AT log(AT ).
We are now ready to prove Theorem 1.
Proof of Theorem 1. From (11), R(T, TSDE) = R0 + R1 + R2 ? E[KT ] + E[HKT ] + R2 where
the inequality comes from Lemma 3, Lemma 4. Then the claim of the theorem directly follows from
Lemma 1 and Lemma 5.
5
Simulations
In this section, we compare through simulations the performance of TSDE with three learning
algorithms with the same regret order: UCRL2 [8], TSMDP [15], and Lazy PSRL [16]. UCRL2 is
an optimistic algorithm with similar regret bounds. TSMDP and Lazy PSRL are TS algorithms for
infinite horizon MDPs. TSMDP has the same regret order in T given a recurrent state for resampling.
The original regret analysis for Lazy PSRL is incorrect, but the regret bounds are conjectured to
be correct [20]. We chose ? = 0.05 for the implementation of UCRL2 and assume an independent
Dirichlet prior with parameters [0.1, . . . , 0.1] over the transition probabilities for all TS algorithms.
We consider two environments: randomly generated MDPs and the RiverSwim example [22]. For
randomly generated MDPs, we use the independent Dirichlet prior over 6 states and 2 actions but
7
with a fixed cost. We select the resampling state s0 = 1 for TSMDP here since all states are recurrent
under the Dirichlet prior. The RiverSwim example models an agent swimming in a river who can
choose to swim either left or right. The MDP consists of six states arranged in a chain with the agent
starting in the leftmost state (s = 1). If the agent decides to move left i.e with the river current then
he is always successful but if he decides to move right he might fail with some probability. The cost
function is given by: c(s, a) = 0.8 if s = 1, a = left; c(s, a) = 0 if s = 6, a = right; and c(s, a) = 1
otherwise. The optimal policy is to swim right to reach the rightmost state which minimizes the cost.
For TSMDP in RiverSwim, we consider two versions with s0 = 1 and with s0 = 3 for the resampling
state. We simulate 500 Monte Carlo runs for both the examples and run for T = 105 .
600
6000
UCRL2
TSMDP
500
UCRL2
TSMDP with s 0 = 3
TSMDP with s 0 = 1
Lazy PSRL
TSDE
5000
Lazy PSRL
TSDE
4000
Regret
Regret
400
300
3000
200
2000
100
1000
0
0
2
4
6
T
8
0
10
0
2
4
6
T
4
x 10
(a) Expected Regret vs Time for random MDPs
8
10
4
x 10
(b) Expected Regret vs Time for RiverSwim
Figure 1: Simulation Results
From Figure 1(a) we can see that TSDE outperforms all the three algorithms in randomly generated
MDPs. In particular, there is a significant gap between the regret of TSDE and that of UCRL2
and TSMDP. The poor performance of UCRL2 assures the motivation to consider TS algorithms.
From the specification of TSMDP, its performance heavily hinges on the choice of an appropriate
resampling state which is not possible for a general unknown MDP. This is reflected in the randomly
generated MDPs experiment.
In the RiverSwim example, Figure 1(b) shows that TSDE significantly outperforms UCRL2, Lazy
PSRL, and TSMDP with s0 = 3. Although TSMDP with s0 = 1 performs slightly better than TSDE,
there is no way to pick this specific s0 if the MDP is unknown in practice. Since Lazy PSRL is also
equipped with the doubling trick criterion, the performance gap between TSDE and Lazy PSRL
highlights the importance of the first stopping criterion on the growth rate of episode length. We also
like to point out that in this example, the MDP is fixed and is not generated from the Dirichlet prior.
Therefore, we conjecture that TSDE also has the same regret bounds under a non-Bayesian setting.
6
Conclusion
We propose
? the Thompson Sampling with Dynamic Episodes (TSDE) learning algorithm and establish
?
O(HS
AT ) bounds on expected regret for the general subclass of weakly communicating MDPs.
Our result fills a gap in the theoretical analysis of Thompson Sampling for MDPs. Numerical results
validate that the TSDE algorithm outperforms other learning algorithms for infinite horizon MDPs.
The TSDE algorithm determines the end of an episode by two stopping criteria. The second criterion
comes from the doubling trick used in many reinforcement learning algorithms. But the first criterion
on the linear growth rate of episode length seems to be a new idea for episodic learning algorithms.
The stopping criterion is crucial in the proof of regret bound (Lemma 3). The simulation results of
TSDE versus Lazy PSRL further shows that this criterion is not only a technical constraint for proofs,
it indeed helps balance exploitation and exploration.
8
Acknowledgments
Yi Ouyang would like to thank Yang Liu from Harvard University for helpful discussions. Rahul Jain
and Ashutosh Nayyar were supported by NSF Grants 1611574 and 1446901.
References
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9
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10
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6,338 | 6,733 | Testing and Learning on Distributions with
Symmetric Noise Invariance
Ho Chung Leon Law
Department of Statistics
University Of Oxford
[email protected]
Christopher Yau
Centre for Computational Biology
University of Birmingham
[email protected]
Dino Sejdinovic
Department of Statistics
University Of Oxford
[email protected]
Abstract
Kernel embeddings of distributions and the Maximum Mean Discrepancy (MMD),
the resulting distance between distributions, are useful tools for fully nonparametric
two-sample testing and learning on distributions. However, it is rare that all possible
differences between samples are of interest ? discovered differences can be due to
different types of measurement noise, data collection artefacts or other irrelevant
sources of variability. We propose distances between distributions which encode
invariance to additive symmetric noise, aimed at testing whether the assumed
true underlying processes differ. Moreover, we construct invariant features of
distributions, leading to learning algorithms robust to the impairment of the input
distributions with symmetric additive noise.
1
Introduction
There are many sources of variability in data, and not all of them are pertinent to the questions that
a data analyst may be interested in. Consider, for example, a nonparametric two-sample testing
problem, which has recently been attracting significant research interest, especially in the context
N2
1
of kernel embeddings of distributions [2, 5, 7]. We observe samples {X1j }N
j=1 and {X2j }j=1 from
two data generating processes P1 and P2 , respectively, and would like to test the null hypothesis that
P1 = P2 without making any parametric assumptions on these distributions. With a large sample-size,
the minutiae of the two data generating processes are uncovered (e.g. slightly different calibration
of the data collecting equipment, different numerical precision), and we ultimately reject the null
hypothesis, even if the sources of variation across the two samples may be irrelevant for the analysis.
Similarly, we may be interested in learning on distributions [14, 23, 24], where the appropriate
level of granularity in the data is distributional. For example, each label yi in supervised learning
i
is associated to a whole bag of observations Bi = {Xij }N
j=1 ? assumed to come from a probability
distribution Pi , or we may be interested in clustering such bags of observations. Again, nonparametric
distances used in such contexts to facilitate a learning algorithm on distributions, such as Maximum
Mean Discrepancy (MMD) [5], can be sensitive to irrelevant sources of variation and may lead to
suboptimal or even misleading results, in which case building predictors which are invariant to noise
is of interest.
While it may be tempting to revert back to a parametric setup and work with simple, easy to interpret
models, we argue that a different approach is possible: we stay within a nonparametric framework,
exploit the irregular and complicated nature of real life distributions and encode invariances to sources
of variation assumed to be irrelevant. In this contribution, we focus on invariances to symmetric
additive noise on each of the data generating distributions. Namely, assume that the i-th sample
i
{Xij }N
j=1 we observe does not follow the distribution Pi of interest but instead its convolution Pi ? Ei
with some unknown noise distributions Ei assumed to be symmetric about 0 (we also require that it
has a positive characteristic function). We would like to assess the differences between Pi and Pi0
while allowing Ei and Ei0 to differ in an arbitrary way. We investigate two approaches to this problem:
(1) measuring the degree of asymmetry of the paired differences {Xij ? Xi0 j }, and (2) comparing
the phase functions of the corresponding samples. While the first approach is simpler and presents
a sensible solution for the two-sample testing problem, we demonstrate that phase functions give a
much better gauge on the relative comparisons between bags of observations, as required for learning
on distributions.
The paper is outlined as follows. In section 2, we provide an overview of the background. In section 3,
we provide details of the construction and implementation of phase features. In section 4, we discuss
the approach based on asymmetry in paired differences for two sample testing with invariances.
Section 5 provides experiments on synthetic and real data, before concluding in section 6.
2
Background and Setup
We will say that a random vector E on Rd is a symmetric
(SPD) component if its
positive definite
characteristic function is positive, i.e. ?E (?) = EX?E exp(i? > E) > 0, ?? ? Rd . This means
that E is (1) symmetric about zero, i.e. E and ?E have the same distribution and (2) if it has a
density, this density must be a positive definite function [20]. Note that many distributions used to
model additive noise, including the spherical zero-mean Gaussian distribution, as well as multivariate
Laplace, Cauchy or Student?s t (but not uniform), are all SPD components.
Following the terminology similar to that of [3], we will say that a random vector X on Rd is
decomposable if its characteristic function can be written as ?X = ?X0 ?E , with ?E > 0. Thus,
if X can be written in the form X = X0 + E, where X0 and E are independent and E is an
SPD noise component, then X is decomposable. We will say that X is indecomposable if it is
not decomposable. In this paper, we will assume that mostly the indecomposable components of
distributions are of interest and will construct tools to directly measure differences between these
indecomposable components, encoding invariance to other sources of variability. The class of Borel
Probability measures on Rd will be denoted M1+ (Rd ), while the class of indecomposable probability
measures will be denoted by I(Rd ) ? M1+ (Rd ).
2.1
Kernel Embeddings, Fourier Features and learning on distributions
For any positive definite function k : X ? X 7? R, there exists a unique reproducing kernel Hilbert
space (RKHS) Hk of real-valued functions on X . Function k(?, x) is an element of Hk and represents
evaluation at x, i.e. hf, k(?, x)iH = f (x), ?f ? Hk , ?x ? X . The kernel mean embedding
(cf.
R [15] for a recent review) of a probability measure P is defined by ?P = EX?P [k(?, X)] =
k(?, x)dP (x). The Maximum Mean Discrepancy (MMD) between probability measures P and Q
X
is then given by k?P ? ?Q kHk . For shift-invariant kernels on Rd , using Bochner?s characterisation of
positive definiteness [26, 6.2], the squared MMD can be written as a weighted L2 -distance between
characteristic functions [22, Corollary 4]
k?P ? ?Q k2Hk =
Z
2
|?P (?) ? ?Q (?)| d? (?) ,
(1)
Rd
where ? is the non-negative spectral measure (inverse Fourier transform) of kernel k as a function of
x ? y, while ?P (?) and ?Q (?) are the characteristic functions of probability measures P and Q.
Bochner?s theorem is also used to construct random Fourier features (RFF) [19] for fast approximations to kernel methods in order to approximate a pre-specified shift-invariant kernel by a finite
dimensional explicit feature map. If we can draw samples from its spectral measure ?, we can
2
approximate k by1
m
X
? y) = 1
k(x,
cos(?jT x) cos(?jT y) + sin(?jT x) sin(?jT y) = h?(x), ?(y)iR2m
m j=1
q
>
>
where ?1 , . . . , ?m ? ? and ?(x) := m1 cos ?1> x , sin ?1> x . . . , cos ?m
x , sin ?m
x . Thus, the
explicit computation of the kernel matrix is not needed and the computational complexity is
reduced. This also allows computation with the approximate, finite-dimensional embeddings
?
?P = ?(P ) = EX?P ?(X) ? R2m , which can be understood as the evaluations (real and complex
part stacked together) of the characteristic function ?P at frequencies ?1 , . . . , ?m . We will refer to
the approximate embeddings ?(P ) as Fourier features of distribution P .
Kernel embeddings can be used for supervised learning on distributions. Assume we have a training
i
set {Bi , yi }ni=1 , where input Bi = {xij }N
j=1 is a bag of samples taking values in X , and yi is
a response. Given a kernel k : X ? X ? R, we first map each Bi to the empirical embedding
PNi
?P?i = N1i j=1
k(?, xij ) ? Hk and then can apply any positive definite kernel on Hk as the kernel
? i , B 0 ) = h? ? , ? ? iH , in order to perform classification [14]
on bag inputs, e.g. linear kernel K(B
k
i
Pi
Pi0
or regression [24]. Approximate kernel embeddings have also been applied in this context [23].
3
Phase Discrepancy and Phase Features
While MMD and kernel embeddings are related to characteristic functions, and indeed the same
connection forms a basis for fast approximations to kernel methods using random Fourier features
[19], the relevant notion in our context is the phase function of a probability measure, recently used
for nonparametric deconvolution by [3]. In this section, we overview this formalism. Based on
the empirical phase functions, we will then derive and investigate hypothesis testing and learning
framework using phase features of distributions.
In nonparametric deconvolution [3], the goal is to estimate the density function f0 of a univariate r.v.
iid
X0 , but in general we only have noisy data samples X1 , . . . , Xn ? X = X0 + E, where E denotes
an independent noise term. Even though the distribution of E is unknown, making the assumption
that E is an SPD noise component, and that X0 is indecomposable, i.e. X0 itself does not contain
any SPD noise components, [3] show that it is possible to obtain consistent estimates of f0 .
They distinguish between the symmetric noise and the underlying indecomposable component by
matching phase functions, defined as
?X (?) =
?X (?)
|?X (?)|
where ?X (?) denotes the characteristic function of X. Observe that |?X (?)| = 1, and thus we
are effectively removing the amplitude information from the characteristic function. For a SPD
noise component E, the phase function is ?E (?) ? 1. But then since ?X = ?X0 ?E , we have that
?X0 = ?X = ?X /|?X |, i.e. the phase function is invariant to additive SPD noise components. This
motivates us to construct explicit feature maps of distributions with the same property and similarly
to the motivation of [3], we argue that real-world distributions of interest often exhibit certain amount
of irregularity and it is exactly this irregularity which is exploited in our methodology.
In analogy to the MMD, we first define the phase discrepancy (PhD) as a weighted L2 -distances
between the phase functions:
Z
2
PhD(X, Y ) =
|?X (?) ? ?Y (?)| d? (?)
(2)
Rd
for some non-negative measure ? (w.l.o.g. a probability measure). Now suppose we write X =
X0 + U , Y = Y0 + V , where U and V are SPD noise components. This then implies ?X = ?X0
and ?Y = ?Y0 ?-everywhere, so that PhD(X, Y ) = PhD(X0 , Y0 ). It is clear then that the PhD is
1
q
>
x can also be used, but we follow the cona complex feature map ?(x) = m1 exp i?1> x , . . . , exp i?m
vention of real-valued Fourier features, since kernels of interest are typically real-valued.
3
not affected by additive SPD noise components, so it captures desired invariance. However, the
PhD for ? supported everywhere is in fact not a proper metric on the indecomposable probability
measures I(Rd ), as one can find indecomposable random variables X and Y s.t. ?X = ?Y and thus
PhD(X, Y ) = 0. An example is given in Appendix A.
While such cases appear contrived, we hence restrict attention to a subset of indecomposable
probability measures P(Rd ) ? I(Rd ), which are uniquely determined by phase functions, i.e.
?P, Q ? P(Rd ) : ?P = ?Q ? P = Q.
We now have the two following propositions (proofs are given in Appendix B).
Proposition 1.
PhD(X, Y ) = 2 ? 2
R
E?? (X)
kE?? (X)k
>
E?? (Y )
kE?? (Y )k
d?(?)
>
where ?? (x) = cos ? > x , sin ? > x
and k ? k denotes the standard L2 norm.
Proposition 2.
K (PX , PY ) =
R
E?? (X)
kE?? (X)k
>
E?? (Y )
kE?? (Y )k
d?(?)
is a positive definite kernel on probability measures.
m
Now, we can construct an approximate explicit feature map
for kernel K. Taking a sample
{?i }i=1 ?
q
E?
(X)
(X)
E?
?1
1
?, we define ? : PX 7? R2m given by ?(PX ) = m
, . . . , kE???m (X)k . We will refer
m
kE??1 (X)k
to ?(?) as the phase features. Note that these are very similar to Fourier features, but the cos, sin-pair
corresponding to each frequency is normalised to have unit L2 norm. In other words, ?(?) can be
thought of as evaluations of the phase function at the selected frequencies. By construction, phase
features are invariant to additive SPD noise components. For an empirical measure, we simply have
the following:
q ?
? ? (X)
E??1 (X)
E?
1
m
?
?(PX ) = m E?
(3)
, . . . , E?
k ? ?1 (X)k
k ? ?m (X)k
where we have replaced the expectations by their empirical estimates. Because
?(P?X )
= 1, we
can construct
2
d P?X , P?Y ) =
PhD(
(4)
?(P?X ) ? ?(P?Y )
= 2 ? 2?(P?X )> ?(P?Y ),
which is a Monte Carlo estimator of PhD(P?X , P?Y ). In summary, ?(P? ) ? R2m is an explicit feature
vector of the empirical distribution which encodes invariance to additive SPD noise components
present in P 2 , as demonstrated in Figure F.1 in the Appendix. It can now be directly applied to (1)
two-sample testing up to SPD components, where the distance between the phase features, i.e. an
estimate (4) of the PhD, can be used as a test statistic, with details given in section 5.1 and (2) learning
on distributions, where we use phase features as the explicit feature map for a bag of samples.
Although we have assumed an indecomposable underlying distribution so far, this assumption is
not strict. For distribution regression, if the indecomposable assumption is invalid, given that the
underlying distribution is irregular, it may still be useful to encode invariance as long as the benefit
of removing the SPD components irrelevant for learning outweighs the signal in the SPD part of
the distribution, i.e. there is a trade off between SPD noise and SPD signal. In practice, the phase
features we propose can be used to encode such invariance where appropriate or in conjunction with
other features which do not encode invariance.
In order to construct the approximate mean embeddings for learning, we first compute an
?
explicit
q h feature map by taking
i averages of the Fourier features, as given by ?(PX ) =
1 ?
? ? (X) . For phase features, we need to compute an additional normalE?? (X), . . . , E?
m
1
m
isation term over each frequency as in (3). To obtain the set of frequencies {wi }m
i=1 , we can draw
2
Note that, unlike the population expression ?(P ), the empirical estimator ?(P? ) will in general have a
distribution affected by the noise components and is thus only approximately invariant, but we observe that it
captures invariance very well as long as the signal-to-noise regime remains relatively high (Section 5.1).
4
samples from a probability measure ? corresponding to an inverse Fourier transform of a shiftinvariant kernel, e.g. Gaussian Kernel. However, given a supervised signal, we can also optimise a set
of frequencies {wi }m
i=1 that will give us a useful representation and good discriminative performance.
In other words, we no longer focus on a specific shift-invariant kernel k, but are learning discriminative Fourier/phase features. To do this, we can construct a neural network (NN) with special
activation functions, pooling layers as shown in Algorithm D.1 and Figure D.1 in the Appendix.
4
Asymmetry in Paired Differences
We now consider a separate approach to nonparametric two-sample test, where we wish to test the
d
null hypothesis that H0 : P =Q vs. the general alternative, but we only have iid samples arising from
X ? P ? E1 and Y ? Q ? E2 . i.e.
X = X0 + U
Y = Y0 + V
where X0 ? P , Y0 ? Q lie in the space of P(Rd ) of indecomposable distributions uniquely
determined by phase functions and U and V are SPD noise components. With this setting (proof in
Appendix B):
d
Proposition 3. Under the null hypothesis H0 , X ? Y is SPD ?? X0 =Y0 .
This motivates us to simply perform a two-sample test on X ? Y and Y ? X since its rejection would
d
imply rejection of X0 =Y0 , as it tests for symmetry. However, note that this is a test for symmetry
only and that for consistency against all alternatives, positivity of characteristic function would need
to be checked separately. Now, given two i.i.d. samples {Xi }ni=1 and {Yi }ni=1 with n even, we split
the two samples into two halves and compute Wi = Xi ? Yi on one half and Zi = Yi ? Xi on the
other half, and perform a nonparametric two sample test on W and Z (which are, by construction,
independent of each other). The advantage of this regime is that we can use any two-sample test ?
in particular in this paper, we will focus on the linear time mean embedding (ME) test [7], which
was found to have performance similar to or better than the original MMD two-sample test [5], and
explicitly formulates a criterion which maximises the test power. We will refer to the resulting test on
paired differences as the Symmetric Mean Embedding (SME).
Although we have assumed here that X0 , Y0 lie in the space P(Rd ) of indecomposable distributions,
in practice, the SME test would not reject if the underlying distributions of interest differ only in the
symmetric components (or in the SPD components for the PhD test). We argue this to be unlikely due
to real life distributions being complex in nature with interesting differences often having a degree of
asymmetry. In practice, we recommend the use of the ME and SME or PhD test together to provide
an exploratory tool to understand the underlying differences, as demonstrated in the Higgs Data
experiment in section 5.1.
It is tempting to also consider learning on distributions with invariances using this formalism. However
note that the MMD on paired differences is not invariant to the additive SPD noise components under
the alternative, i.e. in general MMD(X ? Y, Y ? X) 6= MMD(X0 ? Y0 , Y0 ? X0 ). This means that
the paired differences approach to learning is sensitive to the actual type and scale of the additive
SPD noise components, hence not suitable for learning. The mathematical details and empirical
experiments to show this are presented in Appendix C and F.1.
5
5.1
Experimental Results
Two-Sample Tests with Invariances
In this section, we demonstrate the performance of the SME test and the PhD test on both artificial
d
and real-world data for testing the hypothesis H0 : X0 =Y0 based on samples {Xi }N
i=1 from X0 + U
and {Yi }N
i=1 from Y0 + V , where U and V are arbitrary SPD noise components (we assume the same
N/2
number of samples for simplicity). SME test follows the setup in [7] but applied to {Xi ? Yi }i=1 and
d ? ?
{Yi ? Xi }N
i=N/2+1 . For the PhD test, we use as the test statistic the estimate PhD(PX , PY ) of (2). It
is unclear what the exact form of the null distribution is, so we use a permutation test, by recomputing
this statistic on the samples which are first merged and then randomly split in the original proportions.
5
1.0
0.6
0.8
Power
Rejection Ratio
0.8
1.0
ME n11 = 0. 01, n12 = 0. 05
PhD n11 = 0. 01, n12 = 0. 05
SME n11 = 0. 01, n12 = 0. 05
ME n11 = 0. 25, n12 = 0. 5
PhD n11 = 0. 25, n12 = 0. 5
SME n11 = 0. 25, n12 = 0. 5
0.4
0.6
0.4
0.2
0.2
0.0 1000 2000 3000 4000 5000 6000 7000 8000
Sample Size
0.00
ME All levels
PhD n1 = 0. 0, n2 = 0. 0
SME n1 = 0. 0, n2 = 0. 0
PhD n1 = 0. 01, n2 = 0. 05
SME n1 = 0. 01, n2 = 0. 05
SME n1 = 0. 1, n2 = 0. 1
SME n1 = 0. 25, n2 = 0. 25
1000 2000 3000 4000 5000 6000 7000 8000
Sample Size
Figure 1: Type I error and Power under various additional symmetric noise in the synthetic ?2 dataset.
Dashed line is the 99% Wald interval here. Left: Type I error, n11 denotes the noise to signal ratio
for the first set of samples and n12 for the second set. Right: Power, n1 denotes the noise to signal
ratio for the X set of samples and n2 denotes the noise to signal ratio for the Y set of samples.
While we are combining samples with different distributions, the permutation test is still justified
d
since, under the null hypothesis X0 =Y0 , the resulting characteristic function ?null of the mixture
can be written as
1
1
1
1
?null = ?X0 ?U + ?X0 ?V = ?X0 ( ?U + ?V )
2
2
2
2
and since the mixture of the SPD noise terms is also SPD, we have that ?null = ?X0 = ?Y0 . For our
experiments, we denote by N the sample size, d the dimension of the samples, and we take ? = 0.05
to be the significance level. In the SME test, we take the number of test locations J to be 10, and
use 20% of the samples to optimise the test locations. All experimental results are averaged over
1000 runs, where each run repeats the simulation or randomly samples without replacement from the
dataset.
5.1.1
Synthetic example: Noisy ?2
We start by demonstrating our tests with invariances on a simulated dataset where X0 and Y0 are
random vectors with d = 5, each dimension is the same in distribution and follows ?2 (4)/4 and
?2 (8)/8 respectively, i.e. chi-squared random variables, with different degrees of freedom, rescaled to
have the same mean 1 (but have different variances, 1/2 and 1/4 respectively). An illustration of the
true and empirical phase and characteristic function with noise for these two distributions can be found
N
in Appendix F.2. We construct samples {Xn1 ,i }N
i=1 and {Yn2 ,i }i=1 such that Xn1 ? X0 + U , where
2
U ? N (0, ?1 I) and similarly Yn2 ? Y0 + V , where V ? N (0, ?22 I), ni denotes the noise-to-signal
ratio given by the ratio of variances in each dimension, i.e. n1 = 2?12 and n2 = 4?22 .
We first verify that Type I error is indeed controlled at our design level of ? = 0.05 up to various
d
additive SPD noise components. This is shown in Figure 1 (left), where X0 =Y0 , both constructed
2
using ? (4)/4, with the noiseless case found in Figure F.6 in the Appendix. It is noted here that the
ME test rejects the null hypothesis for even a small difference in noise levels, hence it is unable to
let us target the underlying distributions we are concerned with. This is unlike the SME test which
controls the Type I error even for large differences in noise levels. The PhD test, on the other hand,
while correctly controlling Type I at small noise levels, was found to have inflated Type I error rates
for large noise, with more results and explanation provided in Figure F.6 in the Appendix. Namely,
the test relies on the invariance to SPD of the population expression of PhD, but the estimator of the
null distribution of the corresponding test statistic will in general be affected by the differing noise
levels.
Next, we investigate the power, shown in Figure 1 (right). For a fair comparison, we have included
the PhD test power only for small noise levels, in which the Type I error is controlled at the design
level. In these cases, the PhD test has better power than the SME test. This is not surprising, as for the
SME we have to halve the sample size in order to construct a valid test. However, recall that the PhD
test has an inflated Type I error for large noises, which means that its results should be considered
with caution in practice. ME test rejects at all levels at all sample sizes as it picks up all possible
6
Figure 2: Rejection ratio vs. sample size for
extremely low level features for Higgs dataset.
Dashed line is the 99% Wald interval for 1000
repetitions for ? = 0.05. Note PhD is not used
here, due to its expensive computational cost.
Figure 3: RMSE on the Aerosol test set, corrupted by various levels of noise averaged
over 100 runs, with the 5th and the 95th percentile. The noiseless case is shown with one
run. RMSE from mean is 0.206.
differences. SME and PhD are by construction more conservative tests whose rejection provides a
much stronger statement: two samples differ even when all arbitrary additive SPD components have
been stripped off.
5.1.2
Higgs Dataset
The UCI Higgs dataset [1, 11] is a dataset with 11 million observations, where the problem is to
distinguish between the signal process where Higgs bosons are found, versus the background process
that do not produce Higgs bosons. In particular, we will consider a two-sample test with the ME
and SME test on the high level features derived by physicists, as well as a two-sample test on four
extremely low level features (azimuthal angular momentum ? measured by four particle jets in the
detector). The high level features here (in R7 ) have been shown to have good discriminative properties
in [1]. Thus, we expect them to have different distributions across two processes. Denoting by X the
high level features of the process without Higgs Boson, and Y as the corresponding distribution for
the processes where Higgs bosons are produced, we test the null hypothesis that the indecomposable
parts of X and Y agree. The results can be found in Table F.1 in the Appendix, which shows that the
high level features differ even up to additive SPD components, with a high power for the SME and
ME test even at small sample sizes (rejection rate of 0.94 at N = 500). Now we perform the same
experiment, but with the low level features ? R4 , commented in [1] to carry very little discriminating
information, using the setup from [2].
The results for the ME and SME test can be found in Figure 2. Here we observe that while ME
test clearly rejects and finds the difference between the two distributions, there is no evidence that
the indecomposable parts of the joint distributions of the angular momentum actually differ. In
fact, the test rejection rate remains around the chosen design level of ? = 0.05 for all sample sizes.
This highlights the significance in using the SME test, suggesting that the nature of the difference
between the two processes can potentially be explained by some additive symmetric noise components
which may be irrelevant for discrimination, providing an insight into the dataset. Furthermore, this
also highlights the argument that given two samples from complex data collection and generation
processes, a nonparametric two sample test like ME will likely reject given sufficient sample sizes,
even if the discovered difference may not be of interest. With the SME test however, we can ask a
much more subtle question about the differences between the assumed true underlying processes.
Figures showing that the Type I error is controlled at the design level of ? = 0.05 for both low and
high level features can be found in Figure F.7 in the Appendix.
5.2
5.2.1
Learning with Phase Features
Aerosol Dataset
To demonstrate the phase features invariance to SPD noise component, we use the Aerosol MISR1
dataset also studied by [24] and [25] and consider a situation with covariate shift [18] on distribution
inputs: the testing data is impaired by additive SPD components different to that in the training data.
7
Table 1: Mean Square Error (MSE) on dark
matter dataset for 500 runs with 5th and 95th
percentile.
Algorithm
MSE
Mean
PLRR
GLRR
0.16
0.021 (0.018, 0.024)
0.033 (0.030, 0.037)
LGRR
PGRR
GGRR
0.032 (0.028, 0.036)
0.021 (0.017, 0.024)
0.018 (0.015, 0.019)
Figure 4: MSE with various levels of noise
added on test set, with 5th and 95th percentile.
Here, we have an aerosol optical depth (AOD) multi-instance learning problem with 800 bags, where
each bag contains 100 randomly selected multispectral (potentially cloudy) pixels within 20km radius
around an AOD sensor. The label yi for each bag is given by the AOD sensor measurements and each
sample xi is 16-dimensional. This can be understood as a distribution regression problem where each
bag is treated as a set of samples from some distribution.
We use 640 bags for training and 160 bags for testing. Here in the bags for testing only, we add
varying levels of Gaussian noise ? N (0, Z) to each bag, where Z is a diagonal matrix with
diagonal components zi ? U [0, ?vi ] with vi being the empirical variance in dimension i across all
samples, accounting for different scales across dimensions. For comparisons, we consider linear
ridge regression on embeddings with respect to a Gaussian kernel, approximated with RFF (GLRR)
as described in section 2.1 (i.e. a linear kernel is applied on approximate embeddings), linear ridge
regression on phase features (PLRR) (i.e. normalisation step is applied to obtain (3)), and also the
phase and Fourier neural networks (NN), described in Appendix D, tuning all hyperparameters with
3-fold cross validation. With the same model, we now measure Root Mean Square Error (RMSE)
100 times with various noise-corrupted test sets and results are shown in figure 3. It is also noted that
? does not improve performance significantly on this problem [24].
a second level non-linear kernel K
We see that GLRR and PLRR are competitive (see Appendix Table F.2) in the noiseless case, and
these clearly outperform both the Fourier NN and Phase NN (likely due to the small size of the
dataset). For increasing noise, the performance of GLRR degrades significantly, and while the
performance of PLRR degrades also, the model is much more robust under additional SPD noise.
In comparison, the Phase NN implementation is almost insensitive to covariate shift in the test sets,
unlike the performance of PLRR, highlighting the importance of learning discriminative frequencies
w in a very low signal-to-noise setting.
It is noted that the Fourier NN performs similarly to that of the Phase NN on this example. Interestingly, discriminative frequencies learnt on the training data correspond to Fourier features that are
nearly normalised (i.e. they are close to unit norm - see Figure F.8 in the Appendix). This means
that the Fourier NN has learned to be approximately invariant based on training data, indicating that
the original Aerosol data potentially has irrelevant SPD noise components. This is reinforced by the
nature of the dataset (each bag contains 100 randomly selected potentially cloudy pixels, known to
be noisy [25]) and no loss of performance from going from GLRR to PLRR. The results highlights
that phase features are stable under additive SPD noise.
5.2.2
Dark Matter Dataset
We now study the use of phase features on the dark matter dataset, composing of a catalog of galaxy
clusters. In this setting, we would like to predict the total mass of galaxy clusters, using the dispersion
of velocities in the direction along our line of sight. In particular, we will use the ?ML1? dataset,
as obtained from the authors of [16, 17], who constructed a catalog of massive halos from the
MultiDark mdpl simulation [9]. The dataset contains 5028 bags, with each sample consisting of
its sub-object velocity and its mass label in R. By viewing each galaxy cluster at multiple lines of
sights, we obtain 15 000 bags, using the same experimental setup as in [10]. For experiments, we use
approximately 9000 bags for training, and 3000 bags each for validation and testing, keeping those
of multiple lines of sight in the same set. As before, we use GLRR and PLRR and we also include
8
in comparisons methods with a second level Gaussian kernel (with RFF) applied to phase features
(PGRR) and to approximate embeddings (GGRR). For a baseline, we also include a first level linear
kernel (equivalent to representing each bag with its mean), before applying a second level gaussian
kernel (LGRR). We use the same set of randomly sampled frequencies across the methods, tuning for
the scale of the frequencies and for regularisation parameters.
Table 1 shows the results of the methods across 10 different data splits, with 50 sets of randomised
frequencies for each data split. We see that PLRR is significantly better than GLRR. This suggests
that under this model structure, by removing SPD components from each bag, we can target the
underlying signal and obtain superior performance, highlighting the applicability of phase features.
Considering a second level gaussian kernel, we see that the GGRR has a slight advantage over PGRR,
with PGRR performing similar to PLRR. This suggests that the SPD components of the distribution
of sub-object velocity may be useful for predicting the mass of a galaxy cluster if an additional
nonlinearity is applied to embeddings ? whereas the benefits of removing them outweigh the signal
present in them without this additional nonlinearity. To show that indeed the phase features are robust
to SPD components, we perform the same covariate shift experiment as in the aerosol dataset, with
results given in Figure 4. Note that LGRR is robust to noise, as each bag is represented by its mean.
6
Conclusion
No dataset is immune from measurement noise and often this noise differs across different data
generation and collection processes. When measuring distances between distributions, can we
disentangle the differences in noise from the differences in the signal? We considered two different
ways to encode invariances to additive symmetric noise in those distances, each with different
strengths: a nonparametric measure of asymmetry in paired sample differences and a weighted
distance between the empirical phase functions. The former was used to construct a hypothesis test on
whether the difference between the two generating processes can be explained away by the difference
in postulated noise, whereas the latter allowed us to introduce a flexible framework for invariant
feature construction and learning algorithms on distribution inputs which are robust to measurement
noise and target underlying signal distributions.
Acknowledgements
We thank Dougal Sutherland for suggesting the use of of the dark matter dataset, Michelle Ntampaka
for providing the catalog, as well as Ricardo Silva, Hyunjik Kim and Kaspar Martens for useful
discussions. This work was supported by the EPSRC and MRC through the OxWaSP CDT
programme (EP/L016710/1). C.Y. and H.C.L.L. also acknowledge the support of the MRC Grant No.
MR/L001411/1.
The CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark database was developed in cooperation with the Spanish
MultiDark Consolider Project CSD2009-00064. The authors gratefully acknowledge the Gauss
Centre for Supercomputing e.V. (www.gauss-centre.eu) and the Partnership for Advanced
Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project
by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing
Centre (LRZ, www.lrz.de).
9
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11
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6,339 | 6,734 | A Dirichlet Mixture Model of Hawkes Processes for
Event Sequence Clustering
Hongteng Xu?
School of ECE
Georgia Institute of Technology
[email protected]
Hongyuan Zha
College of Computing
Georgia Institute of Technology
[email protected]
Abstract
How to cluster event sequences generated via different point processes is an interesting and important problem in statistical machine learning. To solve this problem,
we propose and discuss an effective model-based clustering method based on a
novel Dirichlet mixture model of a special but significant type of point processes ?
Hawkes process. The proposed model generates the event sequences with different
clusters from the Hawkes processes with different parameters, and uses a Dirichlet
distribution as the prior distribution of the clusters. We prove the identifiability
of our mixture model and propose an effective variational Bayesian inference
algorithm to learn our model. An adaptive inner iteration allocation strategy is
designed to accelerate the convergence of our algorithm. Moreover, we investigate
the sample complexity and the computational complexity of our learning algorithm
in depth. Experiments on both synthetic and real-world data show that the clustering method based on our model can learn structural triggering patterns hidden
in asynchronous event sequences robustly and achieve superior performance on
clustering purity and consistency compared to existing methods.
1
Introduction
In many practical situations, we need to deal with a huge amount of irregular and asynchronous
sequential data. Typical examples include the viewing records of users in an IPTV system, the
electronic health records of patients in hospitals, among many others. All of these data are so-called
event sequences, each of which contains a series of events with different types in the continuous time
domain, e.g., when and which TV program a user watched, when and which care unit a patient is
transferred to. Given a set of event sequences, an important task is learning their clustering structure
robustly. Event sequence clustering is meaningful for many practical applications. Take the previous
two examples: clustering IPTV users according to their viewing records is beneficial to the program
recommendation system and the ads serving system; clustering patients according to their health
records helps hospitals to optimize their medication resources.
Event sequence clustering is very challenging. Existing work mainly focuses on clustering synchronous (or aggregated) time series with discrete time-lagged observations [19, 23, 39]. Event
sequences, on the contrary, are in the continuous time domain, so it is difficult to find a universal and
tractable representation for them. A potential solution is constructing features of event sequences via
parametric [22] or nonparametric [18] methods. However, these feature-based methods have a high
risk of overfitting because of the large number of parameters. What is worse, these methods actually
decompose the clustering problem into two phases: extracting features and learning clusters. As a
result, their clustering results are very sensitive to the quality of learned (or predefined) features.
?
Corresponding author.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
To make concrete progress, we propose a Dirichlet Mixture model of Hawkes Processes (DMHP
for short) and study its performance on event sequence clustering in depth. In this model, the event
sequences belonging to different clusters are modeled via different Hawkes processes. The priors
of the Hawkes processes? parameters are designed based on their physically-meaningful constraints.
The prior of the clusters is generated via a Dirichlet distribution. We propose a variational Bayesian
inference algorithm to learn the DMHP model in a nested Expectation-Maximization (EM) framework.
In particular, we introduce a novel inner iteration allocation strategy into the algorithm with the help
of open-loop control theory, which improves the convergence of the algorithm. We prove the local
identifiability of our model and show that our learning algorithm has better sample complexity and
computational complexity than its competitors.
The contributions of our work include: 1) We propose a novel Dirichlet mixture model of Hawkes processes and demonstrate its local identifiability. To our knowledge, it is the first systematical research
on the identifiability problem in the task of event sequence clustering. 2) We apply an adaptive inner
iteration allocation strategy based on open-loop control theory to our learning algorithm and show
its superiority to other strategies. The proposed strategy achieves a trade-off between convergence
performance and computational complexity. 3) We propose a DMHP-based clustering method. It
requires few parameters and is robust to the problems of overfitting and model misspecification,
which achieves encouraging clustering results.
2
Related Work
A temporal point process [4] is a random process whose realization consists of an event sequence
{(ti , ci )}M
i=1 with time stamps ti ? [0, T ] and event types ci ? C = {1, ..., C}. It can be equivalently
represented as C counting processes {Nc (t)}C
c=1 , where Nc (t) is the number of type-c events
occurring at or before time t. A way to characterize point processes is via the intensity function
?c (t) = E[dNc (t)|HtC ]/dt, where HtC = {(ti , ci )|ti < t, ci ? C} collects historical events of all
types before time t. It is the expected instantaneous rate of happening type-c events given the history,
which captures the phenomena of interests, i.e., self-triggering [13] or self-correcting [44].
Hawkes Processes. A Hawkes process [13] is a kind of point processes modeling complicated event
sequences in which historical events have influences on current and future ones. It can also be viewed
as a cascade of non-homogeneous Poisson processes [8, 34]. We focus on the clustering problem of
the event sequences obeying Hawkes processes because Hawkes processes have been proven to be
useful for describing real-world data in many applications, e.g., financial analysis [1], social network
analysis [3, 51], system analysis [22], and e-health [30, 42]. Hawkes processes have a particular form
of intensity:
XC Z t
(1)
?c (t) = ?c +
?cc0 (s)dNc0 (t ? s),
0
c =1
0
PC R t
where ?c is the exogenous base intensity independent of the history while c0 =1 0 ?cc0 (s)dNc0 (t?s)
the endogenous intensity capturing the peer influence. The decay in the influence of historical type-c0
events on the subsequent type-c events is captured via the so-called impact function ?cc0 (t), which is
nonnegative. A lot of existing work uses predefined impact functions with known parameters, e.g.,
the exponential functions in [29, 50] and the power-law functions in [49]. To enhance the flexibility, a
nonparametric model of 1-D Hawkes process was first proposed in [16] based on ordinary differential
equation (ODE) and extended to multi-dimensional case in [22, 51]. Another nonparametric model
is the contrast function-based model in [30], which leads to a Least-Squares (LS) problem [7].
A Bayesian nonparametric model combining Hawkes processes with infinite relational model is
proposed in [3]. Recently, the basis representation of impact functions was used in [6, 15, 41] to avoid
discretization.
Sequential Data Clustering and Mixture Models. Traditional methods mainly focus on clustering
synchronous (or aggregated) time series with discrete time-lagged variables [19, 23, 39]. These
methods rely on probabilistic mixture models [46], extracting features from sequential data and
then learning clusters via a Gaussian mixture model (GMM) [25, 28]. Recently, a mixture model
of Markov chains is proposed in [21], which learns potential clusters from aggregate data. For
asynchronous event sequences, most of the existing clustering methods can be categorized into featurebased methods, clustering event sequences from learned or predefined features. Typical examples
2
include the Gaussian process-base multi-task learning method in [18] and the multi-task multidimensional Hawkes processes in [22]. Focusing on Hawkes processes, the feature-based mixture
models in [5, 17, 47] combine Hawkes processes with Dirichlet processes [2, 36]. However, these
methods aim at modeling clusters of events or topics hidden in event sequences (i.e., sub-sequence
clustering), which cannot learn clusters of event sequences. To our knowledge, the model-based
clustering method for event sequences has been rarely considered.
3
Proposed Model
3.1
Dirichlet Mixture Model of Hawkes Processes
Mn
Given a set of event sequences S = {sn }N
n=1 , where sn = {(ti , ci )}i=1 contains a series of events
ci ? C = {1, ..., C} and their time stamps ti ? [0, Tn ], we model them via a mixture model of
Hawkes processes. According to the definition of Hawkes process in (1), for the event sequence
belonging to the k-th cluster its intensity function of type-c event at time t is
X
X
XD
(2)
?kc (t) = ?kc +
?kcci (t ? ti ) = ?kc +
akcci d gd (t ? ti ),
ti <t
k
[?kc ]
ti <t
d=1
RC
+
where ? =
?
is the exogenous base intensity of the k-th Hawkes process. Following the
P
work in [41], we represent each impact function ?kcc0 (t) via basis functions as d akcc0 d gd (t ? ti ),
where gd (t) ? 0 is the d-th basis function and Ak = [akcc0 d ] ? RC?C?D
is the coefficient tensor.
0+
Here we use Gaussian basis function, and their number D can be decided automatically using the
basis selection method in [41].
In our mixture model, the probability of the appearance of an event sequence s is
X Z
X
Y
k
k
k
k
k
k
p(s; ?) =
? HP(s|? , A ), HP(s|? , A ) =
?ci (ti ) exp ?
k
i
k
k
c
T
?kc (s)ds . (3)
0
k
Here ? ?s are the probabilities of clusters and HP(s|? , A ) is the conditional probability of the
event sequence s given the k-th Hawkes process, which follows the intensity function-based definition
in [4]. According to the Bayesian graphical model, we regard the parameters of Hawkes processes,
{?k , Ak }, as random variables. For ?k ?s, we consider its positiveness and assume that they obey
C ? K independent Rayleigh distributions. For Ak ?s, we consider its nonnegativeness and sparsity
as the work in [22, 41, 50]) did, and assume that they obey C ? C ? D ? K independent exponential
distributions. The prior of cluster is a Dirichlet distribution. Therefore, we can describe the proposed
Dirichlet mixture model of Hawkes process in a generative way as
? ? Dir(?/K, ..., ?/K), k|? ? Category(?),
? ? Rayleigh(B), A ? Exp(?), s|k, ?, A ? HP(?k , Ak ),
Here ? = [?kc ] ? RC?K
and A = [akcc0 d ] ? RC?C?D?K
are parameters of Hawkes processes, and
+
0+
k
k
{B = [?c ], ? = [?cc0 d ]} are hyper-parameters. Denote the latent variables indicating the labels of
clusters as matrix Z ? {0, 1}N ?K . We can factorize the joint distribution of all variables as2
p(S, Z, ?, ?, A) = p(S|Z, ?, A)p(Z|?)p(?)p(?)p(A), where
Y
Y
p(S|Z, ?, A) =
HP(sn |?k , Ak )znk , p(Z|?) =
(? k )znk ,
n,k
n,k
Y
Y
p(?) = Dir(?|?), p(?) =
Rayleigh(?kc |?ck ), p(A) =
0
c,k
c,c ,d,k
(4)
k
Exp(akcc0 d |?cc
0 d ).
Our mixture model of Hawkes processes are different from the models in [5, 17, 47]. Those models
focus on the sub-sequence clustering problem within an event sequence. The intensity function is a
weighted sum of multiple intensity functions of different Hawkes processes. Our model, however,
aims at finding the clustering structure across different sequences. The intensity of each event is
generated via a single Hawkes process, while the likelihood of an event sequence is a mixture of
likelihood functions from different Hawkes processes.
2
2
Rayleigh(x|?) =
x
x ? 2? 2
e
?2
, Exp(x|?) =
x
1 ??
e
?
, x ? 0.
3
3.2
Local Identifiability
One of the most important questions about our mixture model is whether it is identifiable or not.
According to the definition of Hawkes process and the work in [26, 31], we can prove that our model
is locally identifiable. The proof of the following theorem is given in the supplementary file.
Theorem 3.1. When the time of observation goes to infinity, the mixture model ofh
the Hawkes proi
1
?K
cesses defined in (3) is locally identifiable, i.e., for each parameter point ? = vec ??1 ...
,
K
...
?
C?C?D
where ? k = {?k , Ak } ? RC
for k = 1, .., K, there exists an open neighborhood of ?
+ ? R0+
0
containing no other ? which makes p(s; ?) = p(s; ?0 ) holds for all possible s.
4
Proposed Learning Algorithm
4.1
Variational Bayesian Inference
Instead of using purely MCMC-based learning method like [29], we propose an effective variational
Bayesian inference algorithm to learn (4) in a nested EM framework. Specifically, we consider a
variational distribution having the following factorization:
Y
q(Z, ?, ?, A) = q(Z)q(?, ?, A) = q(Z)q(?)
q(?k )q(Ak ).
(5)
k
An EM algorithm can be used to optimize (5).
Update Responsibility (E-step). The logarithm of the optimized factor q ? (Z) is approximated as
log q ? (Z) = E? [log p(Z|?)] + E?,A [log p(S|Z, ?, A)] + C
X
=
znk E[log ? k ] + E[log HP(sn |?k , Ak )] + C
n,k
X
X
X Z Tn
znk E[log ? k ] + E[
log ?kci (ti ) ?
?kc (s)ds] + C
=
n,k
?
X
c
i
znk E[log ? k ] +
n,k
|
X
i
log E[?kci (ti )] ?
0
Z Tn
Var[?kci (ti )] X
?
E[
?kc (s)ds] +C.
2
k
c
2E [?ci (ti )]
0
{z
}
?nk
where C is a constant and Var[?] represents the variance of random variable. Each term E[log ?kc (t)]
is approximated via its second-order Taylor expansion log E[?kc (t)] ?
responsibility rnk is calculated as
X
rnk = E[znk ] = ?nk /(
?nj ).
Var[?k
c (t)]
2E2 [?k
c (t)]
[37]. Then, the
j
Denote Nk =
P
n rnk
(6)
for all k?s.
Update Parameters (M-step). The logarithm of optimal factor q ? (?, ?, A) is
log q ? (?, ?, A)
X
X
=
log(p(?k )p(Ak )) + EZ [log p(Z|?)] + log p(?) +
k
n,k
rnk log HP(sn |?k , Ak ) + C.
We can estimate the parameters of Hawkes processes via:
X
b = arg max?,A log(p(?)p(A)) +
? A
?,
rnk log HP(sn |?k , Ak ).
n,k
(7)
Following the work in [41, 47, 50], we need to apply an EM algorithm to solve (7) iteratively. After
b we update distributions as
? and A,
getting optimal ?
p
bk , B k = 2/? ?
(8)
?k.
?k = A
Update The Number of Clusters K. When the number of clusters K is unknown, we initialize K
randomly and update it in the learning phase. There are multiple methods to update the number of
4
3.44
2.37
5.1
0.2
0
7.4
7.3
7.2
7.1
Decreasing
#10 41 5 Clusters
0.8
0.6
0.4
0.2
0
8.6
8.5
8.4
8.3
Responsibility
0.4
7.5
Responsibility
2.38
Constant
#1014 4 Clusters
Negative Log-likelihood
3.46
2.39
Responsibility
2.4
Negative Log-likelihood
Negative Log-likelihood
Negative Log-likelihood
2.41
3.48
4 3 Clusters
Increasing
#10
1
Increasing
Increasing
5.2 0.8
Constant
Constant
Decreasing
Decreasing
0.6
OpenLoop
5.15
BayesOpt
Negative Log-likelihood
5
4 2 Clusters
#10
#10
3.5
2.42
0.8
0.6
0.4
0.2
0
200 400
800 1000
200 60
400 80
600100
800 1000
20 200
40 400
60 600
80 800
1001000
20 40
20 40
60 600
80 100
20 2040 4060 6080 80100100
Indices
of samples
of iterations
samples
Indices
samples
number
of inner
iterations The number
The The
number
of inner
iterations
of inner
iterations
The numberIndices
of inner
The number of
innerofiterations
Ground Truth
(a) Convergence curve
{rn} (15 Inner Iter.)
E(rn) (15 Inner Iter.)
(b) Responsibility and ground truth
Figure 1: The data contain 200 event sequences generated via two 5-dimensional Hawkes processes.
(a) Each curve is the average of 5 trials? results. In each trial, total 100 inner iterations are applied.
The increasing (decreasing) strategy changes the number of inner iterations from 2 to 8 (from 8 to 2).
The constant strategy fixes the number to 5. (b) The black line is the ground truth. The red dots are
responsibilities after 15 inner iterations, and the red line is their average.
clusters. Regrading our Dirichlet distribution as a finite approximation of a Dirichlet process, we set
a large initial K as the truncation level. A simple empirical method is discarding the empty cluster
(i.e., Nk = 0) and merging the cluster with Nk smaller than a threshold Nmin in the learning phase.
Besides this, we can apply the MCMC in [11, 48] to update K via merging or splitting clusters.
Repeating the three steps above, our algorithm maximizes the log-likelihood function (i.e., the
logarithm of (4)) and achieves optimal {?, B} accordingly. Both the details of our algorithm and its
computational complexity are given in the supplementary file.
4.2
Inner Iteration Allocation Strategy and Convergence Analysis
Our algorithm is in a nested EM framework, where the outer iteration corresponds to the loop of
E-step and M-step and the inner iteration corresponds to the inner EM in the M-step. The runtime of
our algorithm is linearly proportional to the total number of inner iterations. Given fixed runtime
(or the total number of inner iterations), both the final achievable log-likelihood and convergence
behavior of the algorithm highly depend on how we allocate the number of inner iterations across
the outer iterations. In this work, we test three inner iteration allocation strategies. The first
strategy is heuristic, which fixes, increases, or decreases the number of inner iterations as the outer
iteration progresses. Compared with constant inner iteration strategy, the increasing or decreasing
strategy might improve the convergence of algorithm [9]. The second strategy is based on open-loop
control [27]: in each outer iteration, we compute objective function via two methods respectively
? updating parameters directly (i.e., continuing current M-step and going to next inner iteration) or
first updating responsibilities and then updating parameters (i.e., going to a new loop of E-step and
M-step and starting a new outer iteration). The parameters corresponding to the smaller negative
log-likelihood are preserved. The third strategy is applying Bayesian optimization [33,35] to optimize
the number of inner iterations per outer iteration via maximizing the expected improvement.
We apply these strategies to a synthetic data set and visualize their impacts on the convergence of our
algorithm in Fig. 1(a). The open-loop control strategy and the Bayesian optimization strategy obtain
comparable performance on the convergence of algorithm. They outperform heuristic strategies (i.e.,
increasing, decreasing and fixing the number of inner iterations per outer iteration), which reduce the
negative log-likelihood more rapidly and reach lower value finally. Although adjusting the number of
inner iterations via different methodologies, both these two strategies tend to increase the number of
inner iterations w.r.t. the number of outer iterations. In the beginning of algorithm, the open-loop
control strategy updates responsibilities frequently, and similarly, the Bayesian optimization strategy
assigns small number of inner iterations. The heuristic strategy that increasing the number of inner
iterations follows the same tendency, and therefore, is just slightly worse than the open-loop control
and the Bayesian optimization. This phenomenon is because the estimated responsibility is not
reliable in the beginning. Too many inner iterations at that time might make learning results fall into
bad local optimums.
Fig. 1(b) further verifies our explanation. With the help of the increasing strategy, most of the
responsibilities converge to the ground truth with high confidence after just 15 inner iterations, because
5
40 Events per Sequence
0.4
0.2
1.2
1
0.8
0.6
0.4
0.2
0.2
0.3
0.4
0.1
Sample Percentage of Minor Cluster
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.3
0.2
0.3
0.4
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.5
0.4
0.3
0.4
0.2
0.1
0.2
0.3
0.4
Sample Percentage of Minor Cluster
20 Events per Sequence
80 Events per Sequence
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.1
Sample Percentage of Minor Cluster
0.6
0.6
0.4
0.2
0.1
0.7
1
0.8
40 Events per Sequence
Distance between Centers
Distance between Centers
0.2
Sample Percentage of Minor Cluster
80 Events per Sequence
1.8
0.8
1.2
0.2
Distance between Centers
0.1
0.9
1.4
0.2
0.3
0.4
Sample Percentage of Minor Cluster
1.8
1.6
1.4
40 Events per Sequence
1.2
1
0.8
0.6
0.4
0.2
0.1
0.2
0.3
0.4
0.2
0.3
1.4
1.2
1
0.8
0.6
0.4
0.4
1.8
0.2
0.3
0.8
1.4
0.7
1.2
0.6
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.1
0.2
0.3
0.4
Sample Percentage of Minor Cluster
(b) DMHP
40 Events per Sequence
1.8
0.9
1.6
0.4
Sample Percentage of Minor Cluster
80 Events per Sequence
Distance between Centers
20 Events per Sequence
1.8
0.2
0.1
Sample Percentage of Minor Cluster
(a) MMHP+DPGMM
1.6
0.2
0.1
Sample Percentage of Minor Cluster
1.8
Distance between Centers
0.6
1.4
1.6
20 Events per Sequence
Distance between Centers
1
0.8
20 Events per Sequence
1.8
Distance between Centers
1.2
1.6
Distance between Centers
1.4
1.8
Distance between Centers
1.6
Distance between Centers
Distance between Centers
Distance between Centers
80 Events per Sequence
1.8
1.8
Figure 2: Comparisons for various methods on F1 score of minor cluster.
1.6
1.4
1.2
1
0.8
0.6
0.4
1.6
1.4
1.2
1
0.8
0.6
0.4
1.6
1.4
1.2
1
0.8
0.6
0.4
the responsibilities has been updated over 5 times. On the contrary, the responsibilities corresponding
to the constant and the decreasing strategies have more uncertainty ? many responsibilities are
around 0.5 and far from the ground truth.
0.2
0.2
0.1
0.2
0.3
0.4
Sample Percentage of Minor Cluster
0.2
0.1
0.2
0.3
0.4
Sample Percentage of Minor Cluster
0.1
0.2
0.3
0.4
Sample Percentage of Minor Cluster
Based on the analysis above, the increasing allocation strategy indeed improves the convergence
of our algorithm, and the open-loop control and the Bayesian optimization are superior to other
competitors. Because the computational complexity of the open-loop control is much lower than
that of the Bayesian optimization, in the following experiments, we apply open-loop control strategy
to our learning algorithm. The scheme of our learning algorithm and more detailed convergence
analysis can be found in the supplementary file.
4.3
Empirical Analysis of Sample Complexity
Focusing on the task of clustering event sequences, we investigate the sample complexity of our
DMHP model and its learning algorithm. In particular, we want to show that the clustering method
based on our model requires fewer samples than existing methods to identify clusters successfully.
Among existing methods, the main competitor of our method is the clustering method based on the
multi-task multi-dimensional Hawkes process (MMHP) model in [22]. It learns a specific Hawkes
process for each sequence and clusters the sequences via applying the Dirichlet processes Gaussian
mixture model (DPGMM) [10, 28] to the parameters of the corresponding Hawkes processes.
Following the work in [14], we demonstrate the superiority of our DMHP-based clustering method
through the comparison on the identifiability of minor clusters given finite number of samples.
Specifically, we consider a binary clustering problem with 500 event sequences. For the k-th cluster,
k = 1, 2, Nk event sequences are generated via a 1-dimensional Hawkes processes with parameter
? k = {?k , Ak }. Taking the parameter as a representation of the clustering center, we can calculate
the distance between two clusters as d = k? 1 ? ? 2 k2 . Assume that N1 < N2 , we denote the first
1
cluster as ?minor? cluster, whose sample percentage is ? 1 = N1N+N
. Applying our DMHP model
2
and its learning algorithm to the data generated with different d?s and ? 1 ?s, we can calculate the F1
scores of the minor cluster w.r.t. {d, ?}. The high F1 score means that the minor cluster is identified
with high accuracy. Fig. 2 visualizes the maps of F1 scores generated via different methods w.r.t. the
number of events per sequence. We can find that the F1 score obtained via our DMHP-based method
is close to 1 in most situations. Its identifiable area (yellow part) is much larger than that of the
MMHP+DPGMM method consistently w.r.t. the number of events per sequence. The unidentifiable
cases happen only in the following two situations: the parameters of different clusters are nearly
equal (i.e., d ? 0); or the minor cluster is extremely small (i.e., ? 1 ? 0). The enlarged version of
Fig. 2 is given in the supplementary file.
5
Experiments
To demonstrate the feasibility and the efficiency of our DMHP-based sequence clustering method, we
compare it with the state-of-the-art methods, including the vector auto-regressive (VAR) method [12],
the Least-Squares (LS) method in [7], and the multi-task multi-dimensional Hawkes process (MMHP)
in [22]. All of the three competitors first learn features of sequences and then apply the DPGMM [10]
to cluster sequences. The VAR discretizes asynchronous event sequences to time series and learns
transition matrices as features. Both the LS and the MMHP learn a specific Hawkes process for each
event sequence. For each event sequence, we calculateRits infectivity matrix ? = [?cc0 ], where the
?
element ?cc0 is the integration of impact function (i.e., 0 ?cc0 (t)dt), and use it as the feature.
6
Table 1: Clustering Purity on Synthetic Data.
C
K
5
2
3
4
5
VAR+
DPGMM
0.5235
0.3860
0.2894
0.2543
Sine-like ?(t)
LS+
MMHP+
DPGMM DPGMM
0.5639
0.5917
0.5278
0.5565
0.4365
0.5112
0.3980
0.4656
VAR+
DPGMM
0.5222
0.3618
0.2901
0.2476
DMHP
0.9898
0.9683
0.9360
0.9055
Piecewise constant ?(t)
LS+
MMHP+
DPGMM DPGMM
0.5589
0.5913
0.4402
0.4517
0.3365
0.3876
0.2980
0.3245
DMHP
0.8085
0.7715
0.7056
0.6774
For the synthetic data with clustering labels, we use clustering purity [24] to evaluate various methods:
1 XK
maxj?{1,...,K 0 } |Wk ? Cj |,
Purity =
k=1
N
where Wk is the learned index set of sequences belonging to the k-th cluster, Cj is the real index set
of sequence belonging to the j-th class, and N is the total number of sequences. For the real-world
data, we visualize the infectivity matrix of each cluster and measure the clustering consistency via
a cross-validation method [38, 40]. The principle is simple: because random sampling does not
change the clustering structure of data, a clustering method with high consistency should preserve
the pairwise relationships of samples in different trials. Specifically, we test each clustering method
with J (= 100) trials. In the j-th trial, data is randomly divided into two folds. After learning the
corresponding model from the training fold, we apply the method to the testing fold. We enumerate
all pairs of sequences within a same cluster in the j-th trial and count the pairs preserved in all other
trials. The clustering consistency is the minimum proportion of preserved pairs over all trials:
0
Consistency = minj?{1,..,J}
X
X
j 0 6=j
(n,n0 )?Mj
0
1{knj = knj 0 }
,
(J ? 1)|Mj |
where Mj = {(n, n0 )|knj = knj 0 } is the set of sequence pairs within same cluster in the j-th trial, and
knj is the index of cluster of the n-th sequence in the j-th trial.
5.1
Synthetic Data
We generate two synthetic data sets with various clusters using sine-like impact functions and
piecewise constant impact functions respectively. In each data set, the number of clusters is set from 2
to 5. Each cluster contains 400 event sequences, and each event sequence contains 50 (= Mn ) events
and 5 (= C) event types. The elements of exogenous base intensity are sampled uniformly from [0, 1].
k
k
Each sine-like impact function in the k-th cluster is formulated as ?kcc0 = bkcc0 (1?cos(?cc
0 (t?scc0 ))),
? 2?
k
k
k
where {bcc0 , ?cc0 , scc0 } are sampled randomly from [ 5 , 5 ]. Each piecewise constant impact function
is the truncation of the corresponding sine-like impact function, i.e., 2bkcc0 ? round(?kcc0 /(2bkcc0 )).
Table 1 shows the clustering purity of various methods on the synthetic data. Compared with the
three competitors, our DMHP obtains much better clustering purity consistently. The VAR simply
treats asynchronous event sequences as time series, which loses the information like the order of
events and the time delay of adjacent events. Both the LS and the MMHP learn Hawkes process for
each individual sequence, which might suffer to over-fitting problem in the case having few events
per sequence. These competitors decompose sequence clustering into two phases: learning feature
and applying DPGMM, which is very sensitive to the quality of feature. The potential problems
above lead to unsatisfying clustering results. Our DMHP method, however, is model-based, which
learns clustering result directly and reduces the number of unknown variables greatly. As a result,
our method avoids the problems of these three competitors and obtains superior clustering results.
Additionally, the learning results of the synthetic data with piecewise constant impact functions prove
that our DMHP method is relatively robust to the problem of model misspecification ? although
our Gaussian basis cannot fit piecewise constant impact functions well, our method still outperforms
other methods greatly.
5.2
Real-world Data
We test our clustering method on two real-world data sets. The first is the ICU patient flow data used
in [43], which is extracted from the MIMIC II data set [32]. This data set contains the transition
7
Table 2: Clustering Consistency on Real-world Data.
Method
VAR+DPGMM LS+DPGMM MMHP+DPGMM DMHP
ICU Patient
0.0901
0.1390
0.3313
0.3778
IPTV User
0.0443
0.0389
0.1382
0.2004
Cluster 1
Cluster 1
2
2
4
4
4
6
DMHP
MMHP
60
2
4
6
6
8
6
8
10
8
Cluster 2
2
10
2
4
6
8 10
2
4
6
10
8
Cluster 2
Cluster 3
Cluster 3
2
2
4
4
6
4
6
8
10
2
4
6
8 10
2
10
2
4
6
8 10
4
4
6
4
6
8 10
2
4
6
8
2
4
6
2
4
6
2
4
4
8 10
2
4
6
6
8
8
10
10
8 10
10
8 10
Cluster 6
2
6
8
10
2
10
8 10
2
4
6
8 10
2
4
6
8 10
(b) DMHP
40
Cluster 1
20
Cluster 2
Cluster 9
Cluster 10
2
2
2
2
4
4
4
4
4
6
6
6
6
6
6
8
6
8 10
4
6
8 10
Cluster 1
6
8
10
10
2
4
6
8 10
2
4
6
4
6
2
4
4
2
6
6
8
8
4
10
8 10
6
6
8
8
10
10
4
4
6
8 10
2
8 10
2
4
4
4
6
2
4
6
8 10
4
6
6
6
8
8
8
6
8 10
4
10
2
4
6
8 10
2
4
6
(c) MMHP+DPGMM
4
6
8 10
2
Cluster 11
4
6
6
10
8 10
2
4
6
8 10
2
Cluster 19
2
4
4
2
6
6
8
8
4
10
4
6
8 10
Cluster 20
Cluster 6
Cluster 7
2
4
6
2
Cluster 9
2
2
2
4
4
4
4
6
6
6
4
6
8 10
2
4
6
10
8 10
2
4
6
8 10
6
8
8
8
8
8
8
8
8
10
10
10
10
10
10
10
10
8 10
2
Cluster 12
4
6
8 10
2
Cluster 13
4
6
8 10
2
Cluster 14
4
6
8 10
2
Cluster 15
4
6
8 10
2
Cluster 16
4
6
8 10
2
Cluster 17
4
6
8 10
2
Cluster 18
4
6
8 10
Cluster 19
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
4
4
4
4
6
6
6
6
6
6
6
6
6
8
8
8
8
8
8
8
8
8
8
10
10
10
10
10
10
10
10
10
10
processes of 30, 308 patients among different kinds of care units. The patients can be clustered
according to their transition processes. The second is the IPTV data set in [20, 22], which contains
7, 100 IPTV users? viewing records collected via Shanghai Telecomm Inc. The TV programs are
categorized into 16 classes and the viewing behaviors more than 20 minutes are recorded. Similarly,
the users can be clustered according to their viewing records. The event sequences in these two data
have strong but structural triggering patterns, which can be modeled via different Hawkes processes.
2
4
6
8 10
2
4
6
8 10
2
4
6
8 10
2
4
6
8 10
2
4
6
8 10
2
4
6
8 10
2
4
Table 2 shows the performance of various clustering methods on the clustering consistency. We can
find that our method outperforms other methods obviously, which means that the clustering result
obtained via our method is more stable and consistent than other methods? results. In Fig. 3 we
visualize the comparison for our method and its main competitor MMHP+DPGMM on the ICU
patient flow data. Fig. 3(a) shows the histograms of the number of clusters for the two methods. We
can find that MMHP+DPGMM method tends to over-segment data into too many clusters. Our DMHP
method, however, can find more compact clustering structure. The distribution of the number of
clusters concentrates to 6 and 19 for the two data sets, respectively. In our opinion, this phenomenon
reflects the drawback of the feature-based method ? the clustering performance is highly dependent
on the quality of feature while the clustering structure is not considered sufficiently in the phase of
extracting feature. Taking learned infectivity matrices as representations of clusters, we compare our
DMHP method with MMHP+DPGMM in Figs. 3(b) and 3(c). The infectivity matrices obtained by
our DMHP are sparse and with distinguishable structure, while those obtained by MMHP+DPGMM
are chaotic ? although MMHP also applies sparse regularizer to each event sequence? infectivity
matrix, it cannot guarantee the average of the infectivity matrices in a cluster is still sparse. Same
phenomena can also be observed in the experiments on the IPTV data. More experimental results are
given in the supplementary file.
6
Cluster 8
2
6
10
2
4
8
6
Figure 3: Comparisons on the ICU patient flow data.
2
4
2
8 10
6
8
10
2
Cluster 18
Cluster 5
4
10
4
8 10
2
2
2
6
Cluster 4
4
8 10
4
Cluster 17
2
8
10
2
Cluster 16
2
6
8
10
8 10
10
6
8
10
2
Cluster 15
Cluster 3
10
2
6
Cluster 14
2
8
10
8 10
2
4
8
10
2
Cluster 13
Cluster 2
2
8
8
10
2
Cluster 12
6
4
8
10
4
2
(a) Histogram of K
Cluster 8
2
4
6
4
25
Cluster 7
2
4
6
2
20
Cluster 6
2
4
6
2
15
Cluster 5
2
4
6
8
10
Cluster 4
2
4
10
5
Cluster 3
2
Cluster 11
0
4
6
10
8 10
Cluster 5
Cluster 6
2
2
4
8
86
Cluster 4
Cluster 5
2
6
6
8
10
8 10
Cluster 4
2
Conclusion and Future Work
In this paper, we propose and discuss a Dirichlet mixture model of Hawkes processes and achieve
a model-based solution to event sequence clustering. We prove the identifiability of our model
and analyze the convergence, sample complexity and computational complexity of our learning
algorithm. In the aspect of methodology, we plan to study other potential priors, e.g., the prior based
on determinantial point processes (DPP) in [45], to improve the estimation of the number of clusters,
and further accelerate our learning algorithm via optimizing inner iteration allocation strategy in near
future. Additionally, our model can be extended to Dirichlet process mixture model when K ? ?. In
that case, we plan to apply Bayesian nonparametrics to develop new learning algorithms. The source
code can be found at https://github.com/HongtengXu/Hawkes-Process-Toolkit.
8
6
8 10
2
4
6
8 10
6
2
4
6
8 10
Acknowledgment
This work is supported in part by NSF IIS-1639792, IIS-1717916, and CMMI-1745382.
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10
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6,340 | 6,735 | Deanonymization in the Bitcoin P2P Network
Giulia Fanti and Pramod Viswanath
Abstract
Recent attacks on Bitcoin?s peer-to-peer (P2P) network demonstrated that its
transaction-flooding protocols, which are used to ensure network consistency,
may enable user deanonymization?the linkage of a user?s IP address with her
pseudonym in the Bitcoin network. In 2015, the Bitcoin community responded
to these attacks by changing the network?s flooding mechanism to a different
protocol, known as diffusion. However, it is unclear if diffusion actually improves
the system?s anonymity. In this paper, we model the Bitcoin networking stack and
analyze its anonymity properties, both pre- and post-2015. The core problem is
one of epidemic source inference over graphs, where the observational model and
spreading mechanisms are informed by Bitcoin?s implementation; notably, these
models have not been studied in the epidemic source detection literature before.
We identify and analyze near-optimal source estimators. This analysis suggests
that Bitcoin?s networking protocols (both pre- and post-2015) offer poor anonymity
properties on networks with a regular-tree topology. We confirm this claim in
simulation on a 2015 snapshot of the real Bitcoin P2P network topology.
1
Introduction
The Bitcoin cryptocurrency has seen widespread adoption, due in part to its reputation as a privacypreserving financial system [17, 22]. In practice, though, Bitcoin exhibits serious privacy vulnerabilities [3, 19, 27, 28, 24]. Most of these vulnerabilities arise because of two key properties: (1)
Bitcoin associates each user with a pseudonym, and (2) pseudonyms can be linked to financial transactions through a public transaction ledger, called the blockchain [23]. If an attacker can associate a
pseudonym with a human identity, the attacker may learn the user?s transaction history.
In practice, there are several ways to link a user to her Bitcoin pseudonym. The most commonlystudied methods analyze transaction patterns in the public blockchain, and link those patterns using
side information [3, 19, 27, 28, 24]. In this paper, we are interested in a lower-layer vulnerability: the
networking stack. Like most cryptocurrencies, Bitcoin nodes communicate over a P2P network [23].
Whenever a user (Alice) generates a transaction (i.e., sends bitcoins to another user, Bob), she first
creates a ?transaction message? that contains her pseudonym, Bob?s pseudonym, and the transaction
amount. Alice subsequently floods this transaction message over the P2P network, which enables
other users to validate her transaction and incorporate it into the global blockchain.
The anonymity implications of transaction broadcasting were largely ignored until recently, when
researchers demonstrated practical deanonymization attacks on the P2P network [6, 15]. These
attacks use a ?supernode? to connect to all active Bitcoin nodes and listen to the transaction traffic
they relay [15, 6, 7]. By using simple estimators to infer the source IP of each transaction broadcast,
this eavesdropper adversary was able to link IP addresses to Bitcoin pseudonyms with an accuracy
of up to 30% [6]. We refer to such linkage as deanonymization.
Giulia Fanti ([email protected]) is in the ECE Department at Carnegie Mellon University. Pramod
Viswanath ([email protected]) is in the ECE Department at the University of Illinois at UrbanaChampaign. This work was funded by NSF grant CCF-1705007.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In 2015, the Bitcoin community responded to these attacks by changing its flooding protocols from
a gossip-style protocol known as trickle spreading to a diffusion spreading protocol that spreads
content with independent exponential delays [1]. We define these protocols precisely in Section 2.
However, no systematic motivation was provided for this shift. Indeed, it is unclear whether the
change actually defends against the deanonymization attacks in [6, 15].
Problem and contributions. The main point of our paper is to show that Bitcoin?s flooding protocols
have poor anonymity properties, and the community?s shift from trickle spreading (pre-2015) to
diffusion spreading (post-2015) did not help the situation. The problem of deanonymizing a user
in this context is mathematically equivalent to inferring the source of a random spreading process
over a graph, given partial observations of the spread. The optimal (maximum-likelihood) sourceidentification algorithms change between spreading protocols; identifying such algorithms and
quantifying their accuracy is the primary focus of this work. We find that despite having different
maximum-likelihood estimators, trickle and diffusion exhibit roughly the same, poor anonymity
properties. Our specific contributions are threefold:
(1) Modeling. We model the Bitcoin P2P network and an eavesdropper adversary, whose capabilities
reflect recent practical attacks in [6, 15]. Most Bitcoin network protocols are not explicitly documented, so modeling the system requires parsing a combination of documentation, papers, and code.
Several of the resulting models are new to the epidemic source detection literature.
(2) Analysis of Trickle (Pre-2015). We analyze the probability of deanonymization by an eavesdropper
adversary under trickle propagation. Our analysis is conducted over a regular tree-structured network.
Although the Bitcoin network topology is not a regular tree, we show in Section 2 that regular trees are
a reasonable first-order model. We consider graph-independent estimators (e.g., the first-timestamp
estimator), as well as maximum-likelihood estimators; both are defined precisely in Section 2. Our
analysis suggests that although the first-timestamp estimator performs poorly on high-degree trees,
maximum-likelihood estimators achieve high probabilities of detection for trees of any degree d.
(3) Analysis of Diffusion (Post-2015). We conduct a similar analysis of diffusion spreading, which was
adopted in 2015 as a fix for the anonymity weaknesses observed under trickle propagation [6, 15]. The
analysis of diffusion requires different theoretical tools, including nonlinear differential equations and
generalized P?lya urns. Although the analysis techniques and attack mechanisms are different, we find
that the anonymity properties of diffusion are similar to those of trickle. Namely, the first-timestamp
estimator?s probability of detection decays to 0 as degree d grows, but the maximum-likelihood
probability of detection remains high (in particular, non-vanishing) even as d ! 1.
2
Model and related work
Network model. We model the P2P network of Bitcoin nodes as a graph G(V, E), where V is
the set of all server nodes and E is the set of edges, or connections, between them. Each server is
represented by a (IP address, port) tuple; it can establish up to eight outgoing connections to other
Bitcoin nodes [6, 2]. The resulting sparse random graph between nodes can be modeled approximately
as a 16-regular graph; in practice, the average degree is closer to 8 due to nonhomogeneities across
nodes [20]. The graph is locally tree-like and (approximately) regular. For this reason, regular trees
are a natural class of graphs to study. In our theoretical analysis, we model G as a d-regular tree. We
validate this choice by running simulations on a snapshot of the true Bitcoin network [20] (Section 5).
Spreading protocols. Each transaction must be broadcast over the network; we analyze the spread
of a single message originating from source node v ? 2 V . Without loss of generality, we label v ? as
node ?0? when iterating over nodes. At time t = 0, the message starts spreading according to one of
two randomized protocols: trickle (pre-2015) or diffusion (post-2015).
Trickle spreading is a gossip-based flooding protocol. Each source or relay chooses a neighboring
peer (called the ?trickle? node) uniformly at random, every 200 ms. If the trickle node has not
yet received the message, the sender forwards the message [6].1 We model this by considering a
canonical, simpler spreading protocol of round-robin gossip. In round-robin gossip, each source or
relay randomly orders its neighbors who have not yet seen the message; we call these uninfected
neighbors. In each successive (discrete) timestep, the node transmits the message to the next neighbor
1
This description omits some details of trickle spreading, which we do not consider in our analysis. For
example, with probability 1/4, each relay forwards the message instantaneously to its neighbors without trickling.
2
in its ordering. Thus, if a node has d neighbors, all d neighbors will receive the message within d
timesteps. This differs from trickle spreading, where the time-to-infection is a coupon collector?s
problem, and therefore takes ?(d log d) timesteps in expectation [8]. We will henceforth abuse
terminology by referring to round-robin gossip as trickle spreading.
In diffusion, each source or relay node transmits the message to each of its uninfected neighbors with
an independent, exponential delay of rate . In practice, Bitcoin uses a higher rate on outgoing edges
than incoming ones [2]; we omit this distinction in our model. We assume a continuous-time system,
with each node starting the exponential clocks upon receipt (or creation) of a message.
For both protocols, we let Xv denote the timestamp at which node v 2 V receives a given message.
Note that server nodes cannot be infected more than once. We assume the message originates at time
t = 0, so Xv? = X0 = 0. Moreover, we let Gt (Vt , Et ) denote the infected subgraph of G at time
t, or the subgraph of nodes who have received the message (but not necessarily reported it to the
adversary) by time t.
Adversarial model. The adversary?s goal is to link a message with the source (IP address, port)?i.e.,
to identify the source node v ? 2 V . We consider an eavesdropper adversary, whose capabilities
are modeled on the practical deanonymization attacks in [6, 15]. These attacks use a supernode that
connects to most of the servers in the Bitcoin network. It can make multiple connections to each
honest server, with each connection coming from a different (IP address, port). Hence, the honest
server does not realize that the supernode?s connections are all from the same entity. We model this
by assuming that the eavesdropper adversary makes a fixed number ? of connections to each server,
where ? 1. We do not include these adversarial connections in the original server graph G, so G
remains a d-regular graph (see Figure 1). The supernode can learn the network structure between
servers [6], so we assume that G(V, E) is known to the eavesdropper.
The supernode in [6, 15] observes the timestamps at which
messages are relayed from each honest server, without
relaying or transmitting content. If the adversary maintains
Eavesdropper
multiple active connections to each server (? > 1), it
&=2
receives the message ? times from each server. We let
)=3
?v denote the time at which the adversary first observes
?
$
the message from node v 2 V . We let ? = (?v )v2V
denote the set of all observed first-timestamps. We assume
timestamps are relative to time t = 0, i.e., the adversary Figure 1: The eavesdropper adversary
knows when the message started spreading.
establishes ? links (in red) to each server.
Source estimation. The adversary?s goal is as follows: Honest servers are connected in a dgiven the observed timestamps ? (up to estimation time regular tree topology (edges in black).
t) and the graph G, find an estimator M(? , G) that outputs
the true source. Our metric of success for the adversary is probability of detection, P(M(? , G) = v ? ),
taken over the random spreading realization (captured by ? ) and any randomness in the estimator.
In [6, 15], the adversary uses a variant of the first-timestamp estimator MFT (? , G) = arg minv2Vt ?v ,
which outputs the first node (prior to estimation time t) to report the message to the adversary. The
first-timestamp estimator requires no knowledge of the graph, and it is computationally easy to
implement. We begin by analyzing this estimator for both trickle and diffusion propagation.
We also consider the maximum-likelihood (ML) estimator: MML (? , G) = arg maxv2V P(? |G, v ? =
v). The ML estimator depends on the time of estimation t to the extent that ? only contains timestamps
up to time t. Unlike the first-timestamp estimator, the ML estimator differs across spreading protocols,
depends on the graph, and may be computationally intractable in general.
Problem statement. Our goal is to understand whether the Bitcoin community?s move from trickle
spreading to diffusion actually improved the system?s anonymity guarantees. The problem at hand is
to characterize the maximum-likelihood (ML) probability of detection of the eavesdropper adversary
for both trickle and diffusion processes on d-regular trees, as a function of degree d, number of
corrupted connections ?, and detection time t. We meet this goal by computing lower bounds derived
from the analysis of suboptimal estimators (e.g., first-timestamp estimator and centrality-based
estimators), and upper bounds derived from fundamental limits on detection.
3
Related work. Although there has been much work on the anonymity properties of Bitcoin [19,
28, 24, 27], the ?epidemic source finding? interpretation of Bitcoin deanonymization is fairly new.
Prior work that (implicitly) adopts this interpretation has focused on Bitcoin?s protocol flaws more
than the inference aspect of the problem [6, 15]. As this is the focus of our paper, we include the
related source detection literature. Epidemic source detection has been widely studied under diffusion
spreading with a snapshot adversary, which observes the set of infected nodes at a single time t; in
our notation, the adversary would learn the set {v 2 V : Xv ? t} (no timestamps), along with graph
G. Shah and Zaman first characterized the ML probability of detection for diffusion observed by a
snapshot adversary when the underlying graph is a regular tree [29]. These results were later extended
to random, irregular trees [31], whereas other authors studied heuristic source detection methods on
general graphs [12, 26, 16] and related theoretical limits [32, 21, 14]. The eavesdropper adversary
differs in that it eventually observes a noisy timestamp ?v from every node, regardless of when the
node is infected. This changes both the analysis and the estimators that one can use. Another common
adversarial model is the spy-based adversary, which observes exact timestamps for a corrupted set
of nodes that does not include the source [25, 34]. In our notation, for a set of spies S ? V , the
spy-based adversary observes {(s, Xs ) : s 2 S}. Prior work on the spy-based adversary does not
characterize the ML probability of detection, but researchers have proposed efficient heuristics that
perform well in practice [25, 34, 35, 9]. Unlike the spy-based adversary, the eavesdropper only
observes delayed timestamps, and it does so for all nodes, including the source.
3
3.1
Analysis of trickle (pre-2015)
First-timestamp estimator
The analysis of trickle propagation is complicated by its combinatorial, time-dependent nature. As such, we lower-bound the first-timestamp estimator?s probability of detection. Let
?m , min(?1 , ?2 , . . .) denote the minimum observed timestamp among nodes that are not the
source. Then we compute P(?0 < ?m ), i.e., the probability that the true source reports the message
to the adversary strictly before any of the other nodes. This event (which causes the source to be
detected with probability 1) does not include cases where the true source is one of k nodes (k > 1)
that report the message to the adversary simultaneously, and before any other node in the system.
Nonetheless, for large node degree d, the ?simultaneous reporting? event is rare, so our lower bound
is close to the empirical probability of detection of the first-timestamp estimator.
Theorem 3.1 (Proof in Appendix C.1) Consider a message that propagates according to trickle
spreading over a d-regular tree of servers, where each node additionally has ? connections to an
eavesdropping adversary. The first-timestamp
estimator?s probability
of detection at time t = 1
?
?
?
1
d
satisfies P(MFT (? , G) = v ? )
Ei(2
log
?)
Ei
(log
?)
where
? = d d 1+?
, and Ei(x) ,
d log 2
R 1 e t dt
denotes the exponential integral.
t
x
Probability of Detection
We prove this bound by conditioning on the time at
which
the source reports to the adversary. The proof
0.65
Theoretical lower bound
then becomes a combinatorial counting problem. The
log(d) / (d log(2))
0.6
Simulation
expression in Theorem 3.1 can be simplified by exam0.55
ining its Taylor expansion (see Appendix A). In par0.5
ticular, for the special case of ? = 1 where the adver0.45
sary establishes only one connection per server,
? line
?
0.4
log d
(5) simplifies to P(MFT (? , G)) ? d?log 2 +o logd d .
0.35
This suggests that the first-timestamp estimator has
0.3
a probability of detection that decays to zero asymptotically as log(d)/d. Intuitively, the probability of
2
4
6
8
10
detection should decay to zero, because the higher
Tree degree, d
the degree of the tree, the higher the likelihood that
Figure 2: First-timestamp estimator accuracy a node other than the source reports to the adversary
before the source does. Nonetheless, this is only a
on d-regular trees when ? = 1.
lower bound on the first-timestamp?s probability of
detection, so we wish to understand how tight the bound is.
4
Simulation. To evaluate the lower bound in Theorem 3.1 and its approximation for ? = 1, we
simulate the first-timestamp estimator on regular trees. Figure 2 illustrates the simulation results
for ? = 1 compared to the approximation above. Each data point is averaged over 5,000 trials.
Empirically, the lower bound appears to be tight, especially as d grows. Figure 2 suggest a natural
solution to improve anonymity in the Bitcoin network: increase the degree of each node to reduce the
adversary?s probability of detection. However, we shall see in the next section that stronger estimators
(e.g., the ML estimator) may achieve high probabilities of detection, even for large d.
3.2
Maximum-likelihood estimator
At any time t, if one knew the ground truth timestamps (i.e., the Xv ?s), one could arrange the nodes
of the infected subgraph Gt in the order they received the message. We call such an arrangement an
ordering of nodes. Since propagation is in discrete time, multiple nodes may receive the message
simultaneously; such nodes are lumped together in the ordering. Of course, the true ordering is not
observed by the adversary, but the observed timestamps (i.e., ? ) restrict the set of possible orderings.
A feasible ordering is an ordering that respects the rules of trickle propagation over graph G, as well
as the observed timestamps ? . In this subsection only, we will abuse notation by using ? to refer
to all timestamps observed by the adversary, not just the first timestamp from each server. So if the
adversary has ? connections to each server, ? would include ? timestamps per honest server.
We propose an estimator called timestamp rumor centrality, which counts the number of feasible
orderings originating from each candidate source. The candidate with the most feasible orderings is
chosen as the estimator output. This estimator is similar to rumor centrality, an estimator devised for
snapshot adversaries in [29]. However, the presence of timestamps and the lack of knowledge of the
infected subgraph increases the estimator?s complexity. We first motivate timestamp rumor centrality.
Proposition 3.2 (Proof in Appendix C.2) Consider a trickle process over a d-regular graph, where
each node has ? connections to the eavesdropper adversary. Any feasible orderings o1 and o2 with
respect to observed timestamps ? and graph G have the same likelihood.
Proposition 3.2 implies that at any fixed time, the likelihood of observing ? given a candidate source
is proportional to the number of feasible orderings originating from that candidate source. Therefore,
an ML estimator (timestamp rumor centrality) counts the number of feasible orderings at estimation
time t. Timestamp rumor centrality is a message-passing algorithm that proceeds as follows: for
each candidate source, recursively determine the set of feasible times when each node could have
been infected, given the observed timestamps. This is achieved by passing a set of ?feasible times of
receipt" from the candidate source to the leaves of the largest feasible infected subtree rooted at the
candidate source. In each step, nodes prune receipt times that conflict with their observed timestamps.
Next, given each node?s set of feasible receipt times, they count the number of feasible orderings
that obey the rules of trickle propagation. This is achieved by passing sets of partial orderings from
the leaves to the candidate source, and pruning infeasible orderings. The timestamp rumor centrality
protocol is presented in Appendix A.2, along with minor modifications that reduce its complexity.
In [31], precise analysis of standard rumor centrality was possible because rumor centrality
. =4
can be reduced to a simple counting problem.
. =2
. =2
Such an analysis is more challenging for timestamp rumor centrality, because timestamps pre?
?3
?2
?1
1
2
$
3
vent us from using the same counting argument.
However, we identify a suboptimal, simplified
5 =3 5 =4
5 =3 5 =0
5 =4
5 =1
version of timestamp rumor centrality that apFigure 3: Example of ball centrality on a line with proaches optimal probabilities of detection as t
one link to the adversary per server (these links are grows. We call this estimator ball centrality.
not shown). The estimator is run at time t = 4.
Ball centrality checks whether a candidate
source v could have generated each of the observed timestamps, independently. For example, Figure 3 contains a sample spread on a line graph,
where the adversary has one connection per server (not shown). Therefore, d = 2 and ? = 1. The
ground truth infection time is written as Xv below each node, and the observed timestamps are written
above the node. In this figure, the estimator is run at time t = 4, so the adversary only sees three
timestamps. For each observed timestamp ?v , the estimator creates a ball of radius ?v 1, centered
210
/?0
10
270
210
/?0
10
70
80
5
at v. For example, in our figure, the green node (node 1) has ?1 = 2. Therefore, the adversary would
make a ball of radius 1 centered at node 1; this ball is depicted by the green bubble in our figure. The
ball represents the set of nodes that are close enough to node 1 to feasibly report to the adversary
from node 1 at time ?1 = 2. After constructing an analogous ball for every observed timestamp in ? ,
the protocol outputs a source selected uniformly from the intersection of these balls. In our example,
there are exactly two nodes in this intersection. We describe ball centrality precisely in Protocol 1
(Appendix A.2.1). Although ball centrality is not ML for a fixed time t, the following theorem lower
bounds the ML probability of detection by analyzing ball centrality and showing that its probability
of detection approaches a fundamental upper bound exponentially fast in detection time t.
Theorem 3.3 (Proof in Section C.3) Consider a trickle spreading process over a d-regular graph of
honest servers. In addition, each server has ? independent connections to an eavesdropper adversary.
The ML probability of detection at time t satisfies the following expression:
?
?t (a)
(b)
d
d
d
1
? P(MML (? , G) = v ? ) ? 1
(1)
2(? + d)
?+d
2(? + d)
Note that the right-hand side of equation (1) is always greater than 12 . As such, increasing the graph
degree would not significantly reduce the probability of detection; the adversary can still identify
the source with probability at least 12 given enough time. Second, the ML probability of detection
approaches its upper bound exponentially fast in time t. This suggests that the adversary can achieve
high probabilities of detection at small times t. These results highlight an important point: estimators
that exploit graph structure can reap significant, order-level gains in accuracy.
4
4.1
Analysis of diffusion (post-2015)
First-timestamp estimator
Although the first-timestamp estimator does not use knowledge of the underlying graph, its performance depends on the underlying graph structure. The following theorem exactly characterizes its
probability of detection on a regular tree as t ! 1.
Theorem 4.1 (Proof in Appendix C.4) Consider a diffusion process of rate = 1 over a d-regular
tree, d > 2. Suppose an adversary observes each node?s infection time with an independent,
exponential delay of rate 2 = ?, ? 1. Then the following expression describes the probability of
detection for the first-timestamp estimator at time t = 1: P(MFT (? , G) = v ? ) = d ? 2 log d+?? 2 .
The proof expresses the probability of detection as a nonlinear differential equation that can be
solved exactly. The expression highlights a few points: First, for a fixed degree d, the probability of
detection is strictly positive as t ! 1. This is straightforward to see, but under other adversarial
models (e.g., snapshot adversaries) it is not trivial to see that the probability of detection is positive as
t ! 1. Indeed, several papers are dedicated to making that point [30, 31]. Second, when ? = 1, i.e.,
the adversary has only one connection per node, the probability of detection approaches log(d)/d
asymptotically in d. This quantity tends to 0 as d ! 1, and it is order-equal to the probability of
detection of the first-timestamp adversary on the trickle protocol when ? = 1 (see Section 3.1).
Theorem 4.1 suggests that the Bitcoin community?s transition from trickle spreading to diffusion does
not provide order-level anonymity gains (asymptotically in the degree of the graph), at least for the
first-timestamp adversary. Next, we ask if the same is true for estimators that use the graph structure.
4.2
Centrality-based estimators
We compute a different lower bound on the ML probability of detection by analyzing a centralitybased estimator. Unlike the first-timestamp estimator, this reporting centrality estimator uses the
structure of the infected subgraph by selecting a candidate source that is close to the center (on the
graph) of the observed timestamps. However, it does not explicitly use the observed timestamps.
Also unlike the first-timestamp estimator, this centrality-based estimator improves as the degree d
of the underlying tree increases, with a strictly positive probability of detection as d ! 1. Thus
the eavesdropper adversary has an ML probability of detection that scales as ?(1) in d. Intuitively,
6
reporting centrality works as follows: for each candidate source v, the estimator counts the number
of nodes that have reported to the adversary from each of the node v?s adjacent subtrees. It picks a
candidate source for which the number of reporting nodes is approximately equal in each subtree.
To make this precise, suppose the infected subtree Gt is rooted at w; we use Tvw to denote the subtree
of Gt that contains v and all of v?s descendants, with respect to root node w. Consider a random
variable Yv (t),P
which is 1 if node v 2 V has reported to the adversary by time t, and 0 otherwise. We
let YTvw (t) = u2Tvw Yu (t) denote the number of nodes in Tvw that have reported to the adversary
P
by time t. We use Y (t) = v2Vt Yv (t) to denote the total number of reporting nodes in Gt at time t.
Similarly, we use NTvw (t) to denote the number of infected nodes in Tvw (so NTvw (t) YTvw (t)), and
we let N (t) denote the total number of infected nodes at time t (N (t) Y (t)). For each candidate
source v, we consider its d neighbors, which comprise the set N (v). We define a node v?s reporting
centrality at time t?denoted Rv (t)?as follows:
(
1 if maxu2N (v) YTuv (t) < Y 2(t)
Rv (t) =
(2)
0 otherwise.
That is, a node?s reporting centrality is 1 iff each of its adjacent subtrees has fewer than
Y (t)/2 reporting nodes. A node is a reporting center iff its reporting centrality is 1.
The estimator outputs v? chosen uniformly from all reporting centers. In Figure 4, v ? is the only reporting center.
Reporting centrality does not use the adversary?s observed
timestamps?it only counts the number of reporting nodes
in each of a node?s adjacent subtrees. This estimator is
inspired by rumor centrality [30], an ML estimator for the
source of a diffusion process under a snapshot adversary.
Recall that a snapshot adversary sees the infected subgraph
Gt at time t, but it does not learn timestamp information.
Y (t) = 5
N (t) = 7
Rv? (t) = 1
$?
#
Rw (t) = 0
The next theorem shows that for trees with high degree d,
reporting centrality has a strictly higher (in an order sense)
probability of detection than the first-timestamp estimator; Figure 4: Yellow nodes are infected; a
its probability of detection is strictly positive as d ! 1. red outline means the node has reported.
Rv? (t) = 1 since v ? ?s adjacent subtrees
Theorem 4.2 (Proof in Section C.5) Consider a diffu- have ? Y (t)/2 = 2.5 reporting nodes.
sion process of rate = 1 over a d-regular tree. Suppose this process is observed by an eavesdropper adversary, which sees each node?s timestamp
with an independent exponential delay of rate 2 = ?, ?
1. Then the reporting central?
ity estimator has a (time-dependent) probability of detection P(M
? RC (? , G)? = v ) that satisfies
??
lim inf t!1 P(MRC (? , G) = v ? )
Cd > 0. where Cd = 1
d 1
I1/2
1
d 2, 1
+
1
d 2
is a
constant that depends only on degree d, and I1/2 (a, b) is the regularized incomplete Beta function,
i.e., the probability a Beta random variable with parameters a and b takes a value in [0, 12 ).
To prove this, we relate two P?lya urn processes: one that represents the diffusion process over the
regular tree of honest nodes, and one that describes the full spreading process, which includes both
diffusion over the regular tree and random reporting to the adversary. The first urn can be posed as a
classic P?lya urn [10], which has been studied in the context of diffusion [31, 14]. The second urn
can be described by an unbalanced generalized P?lya urn (GPU) with negative coefficients [4, 13]?a
class of urns that does not typically appear in the study of diffusion (to the best of our knowledge).
As a side note, this approach can be used to analyze other epidemic source-finding problems that have
previously evaded analysis, as we show in Appendix B. Notice that the constant Cd in Theorem 4.2
does not depend on ??this is because the reporting centrality estimator makes no use of timestamp
information, so the delays in the timestamps ? do not affect the estimator?s asymptotic behavior.
Simulation results. To evaluate the lower bound in Theorem 4.2, we simulate reporting centrality
on diffusion over regular trees. Figure 5 illustrates the empirical performance of reporting centrality
averaged over 4,000 trials, compared to the theoretical lower bound on the liminf. The estimator is
run at time t = d + 2. Our simulations are run up to degree d = 5 due to computational constraints,
since the infected subgraph grows exponentially in the degree of the tree. By d = 5, reporting
centrality reaches the theoretical lower bound on the limiting detection probability.
7
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
3
4
5
6
7
8
9
0.9
0.8
0.7
0.6
0.5
Trickle, Theoretical (Lower bound)
Trickle, Simulated (Lower bound)
Trickle, Simulated (Exact)
Diffusion, Theoretical
Diffusion, Simulated
0.4
0
10
Prob. of Detection
1
First-timestamp, theoretical
First-timestamp, simulated
Reporting centrality, theoretical
Reporting centrality, simulated
Prob. of Detection
Prob. of Detection
1
0.9
5
10
15
20
Figure 5: First-timestamp vs. reporting centrality on diffusion
over regular trees, theoretically
and simulated. ? = 1, t = d+2.
0.8
0.7
0.6
0.5
Trickle, Theoretical lower bound
Trickle, Simulated
Trickle, Theoretical lower bound (d=2)
Diffusion, Theoretical
Diffusion, Simulation
0.4
0.3
0
5
10
15
20
Eavesdropper connections, ?
Eavesdropper connections, ?
Degree, d
0.9
Figure 6:
Comparison of Figure 7: Trickle vs. diffusion
trickle and diffusion under the under the first-timestamp estimafirst-timestamp estimator on tor, simulated on a snapshot of
the real Bitcoin network [20].
4-regular trees.
Table 1: Probability of detection on a d-regular tree. The adversary has ? connections per server.
Trickle
FirstTimestamp
MaximumLikelihood
All ?
?=1
All ?
?=1
? [Ei(2d log ?) Ei(log ?)]
d log 2?
?
log(d)
log d
d log(2) + o
d
1
1
d
2(?+d)
d
2(d+1)
Diffusion
(Thm 3.1)
?
d 2
(Sec. 3.1)
(Thm 3.3)
(Thm. 3.3)
1
d+? 2
(Thm. 4.1)
?
log(d 1)
(Thm. 4.1)
(d 2)
log
?
d 1
I1/2
?
1
d
2, 1 +
(Thm. 4.2)
1
d 2
??
For diffusion, neither lower bound on the first-timestamp or reporting centrality estimator strictly
outperforms the other. Figure 5 compares the two estimators as a function of degree d. We observe
that reporting centrality outstrips first-timestamp estimation for trees of degree 9 and higher; since
our theoretical result is only a lower bound on the performance of reporting centrality, the transition
may occur at even smaller d. Empirically, the true Bitcoin graph is approximately 8-regular [20], a
regime in which we expect reporting centrality to perform similarly to the first-timestamp estimator.
5
Discussion
Table 1 summarizes our theoretical results for trickle and diffusion. The probabilities of detection
for trickle and diffusion are similar, particularly when ? = 1. Although the maximum-likelihood
results are difficult to compare visually, they both approach a positive constant as d, t ! 1; for
trickle propagation, that constant is 12 , whereas for diffusion, it is approximately 0.307.
These results are asymptotic in degree d. In practice, the underlying Bitcoin graph is fixed; the
only variable quantity is the adversary?s resources, represented by ?. Figure 6 compares analytical
expressions and simulations for 4-regular trees under the first-timestamp estimator (as we lack an ML
estimator on general graphs), as a function of ?. It suggests nearly identical detection probabilities
for diffusion and trickle on regular trees; while our theoretical prediction for diffusion is exact, our
lower bound on trickle is loose since d is small.
To validate our decision to analyze regular trees, we simulate trickle and diffusion on a 2015 snapshot
of the Bitcoin network [20]. Figure 7 compares these results as a function of ?, for the first-timestamp
estimator. Unless specified otherwise, theoretical curves are calculated for a regular tree with d = 8,
the mean degree of our dataset. Diffusion performs close to the theoretical prediction; this is because
with high probability, the first-timestamp estimator uses only on a local neighborhood to estimate v ? ,
and the Bitcoin graph is locally tree-like. However, our trickle lower bound remains loose. This is
partially due to simultaneous reporting events, but the main contributing factor seems to be graph
irregularity. Understanding this effect more carefully is an interesting question for future work.
In summary, trickle and diffusion have similar probabilities of detection, both in an asymptotic-order
sense and numerically. We have analyzed the canonical class of d-regular trees and simulated these
protocols on a real Bitcoin graph topology. Our results omit certain details of the spreading protocols,
(Sec. 2); extending the analysis to include these details is practically relevant.
8
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6,341 | 6,736 | Accelerated consensus via Min-Sum Splitting
Patrick Rebeschini
Department of Statistics
University of Oxford
[email protected]
Sekhar Tatikonda
Department of Electrical Engineering
Yale University
[email protected]
Abstract
We apply the Min-Sum message-passing protocol to solve the consensus problem
in distributed optimization. We show that while the ordinary Min-Sum algorithm
does not converge, a modified version of it known as Splitting yields convergence to
the problem solution. We prove that a proper choice of the tuning parameters allows
Min-Sum Splitting to yield subdiffusive accelerated convergence rates, matching
the rates obtained by shift-register methods. The acceleration scheme embodied by
Min-Sum Splitting for the consensus problem bears similarities with lifted Markov
chains techniques and with multi-step first order methods in convex optimization.
1
Introduction
Min-Sum is a local message-passing algorithm designed to distributedly optimize an objective
function that can be written as a sum of component functions, each of which depends on a subset of
the decision variables. Due to its simplicity, Min-Sum has emerged as canonical protocol to address
large scale problems in a variety of domains, including signal processing, statistics, and machine
learning. For problems supported on tree graphs, the Min-Sum algorithm corresponds to dynamic
programming and is guaranteed to converge to the problem solution. For arbitrary graphs, the ordinary
Min-Sum algorithm may fail to converge, or it may converge to something different than the problem
solution [28]. In the case of strictly convex objective functions, there are known sufficient conditions
to guarantee the convergence and correctness of the algorithm. The most general condition requires
the Hessian of the objective function to be scaled diagonally dominant [28, 25]. While the Min-Sum
scheme can be applied to optimization problems with constraints, by incorporating the constraints
into the objective function as hard barriers, the known sufficient conditions do not apply in this case.
In [34], a generalization of the traditional Min-Sum scheme has been proposed, based on a
reparametrization of the original objective function. This algorithm is called Splitting, as it can be
derived by creating equivalent graph representations for the objective function by ?splitting? the
nodes of the original graph. In the case of unconstrained problems with quadratic objective functions,
where Min-Sum is also known as Gaussian Belief Propagation, the algorithm with splitting has been
shown to yield convergence in settings where the ordinary Min-Sum does not converge [35]. To date,
a theoretical investigation of the rates of convergence of Min-Sum Splitting has not been established.
In this paper we establish rates of convergence for the Min-Sum Splitting algorithm applied to solve
the consensus problem, which can be formulated as an equality-constrained problem in optimization.
The basic version of the consensus problem is the network averaging problem. In this setting, each
node in a graph is assigned a real number, and the goal is to design a distributed protocol that allows
the nodes to iteratively exchange information with their neighbors so to arrive at consensus on the
average across the network. Early work include [42, 41]. The design of distributed algorithms
to solve the averaging problem has received a lot of attention recently, as consensus represents a
widely-used primitive to compute aggregate statistics in a variety of fields. Applications include, for
instance, estimation problems in sensor networks, distributed tracking and localization, multi-agents
coordination, and distributed inference [20, 21, 9, 19]. Consensus is typically combined with some
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
form of local optimization over a peer-to-peer network, as in the case of iterative subgradient methods
[29, 40, 17, 10, 6, 16, 39]. In large-scale machine learning, consensus is used as a tool to distribute
the minimization of a loss function over a large dataset into a network of processors that can exchange
and aggregate information, and only have access to a subset of the data [31, 11, 26, 3].
Classical algorithms to solve the network averaging problem involve linear dynamical systems supported on the nodes of the graph. Even when the coefficients that control the dynamics are optimized,
these methods are known to suffer from a ?diffusive? rate of convergence, which corresponds to the
rate of convergence to stationarity exhibited by the ?diffusion? random walk naturally associated to
a graph [44, 2]. This rate is optimal for graphs with good expansion properties, such as complete
graphs or expanders. In this case the convergence time, i.e., the number of iterations required to
reach a prescribed level of error accuracy ? > 0 in the `2 norm relative to the initial condition, scales
independently of the dimension of the problem, as ?(log 1/?). For graphs with geometry this rate is
suboptimal [7], and it does not yield a convergence time that matches the lower bound ?(D log 1/?),
where D is the graph diameter [37, 36]. For example, in both cycle graphs and in grid-like topologies
2
the number
? of iterations scale like ?(D log 1/?) (if n is the number of nodes, D ? n in a cycle
and D ? n in a two-dimensional torus). ?(D2 log 1/?) is also the convergence time exhibited in
random geometric graphs, which represent the relevant topologies for many applications in sensor
networks [9]. In [7] it was established that for a class of graphs with geometry (polynomial growth or
finite doubling dimension), the mixing time of any reversible Markov chain scales at least like D2 ,
embodying the fact that symmetric walks on these graphs take D2 steps to travel distances of orderD.
Min-Sum schemes to solve the consensus problem have been previously investigated in [27]. The
authors show that the ordinary Min-Sum algorithm does not converge in graphs with cycles. They
investigate a modified version of it that uses a soft barrier function to incorporate the equality
constrains into the objective function. In the case of d-regular graphs, upon a proper choice of initial
conditions, the authors show that the algorithm they propose reduces to a linear process supported on
the directed edges of the graph, and they characterize the convergence time of the algorithm in terms
of the Ces?ro mixing time of a Markov chain defined on the set of directed edges of the original graph.
In the case of cycle graphs (i.e., d = 2), they prove that the mixing time scales like O(D), which
yields the convergence time O(D/? log 1/?). See Theorem 4 and Theorem 5 in [27]. In the case of
(d/2)-dimensional tori (D ? n2/d ), they conjecture that the mixing time is ?(D2(d?1)/d ), but do
not present bounds for the convergence time. See Conjecture 1 in [27]. For other graph topologies,
they leave the mixing time (and convergence time) achieved by their method as an open question.
In this paper we show that the Min-Sum scheme based on splitting yields convergence to the
consensus solution, and we analytically establish rates of convergence for any graph topology.
First, we show that a certain parametrization of the Min-Sum protocol for consensus yields a linear
message-passing update for any graph and for any choice of initial conditions. Second, we show that
the introduction of the splitting parameters is not only fundamental to guarantee the convergence
and correctness of the Min-Sum scheme in the consensus problem, but that proper tuning of these
parameters yields accelerated (i.e., ?subdiffusive?) asymptotic rates of convergence. We establish a
square-root improvement for the asymptotic convergence time over diffusive methods, which allows
Min-Sum Splitting to scale like O(D log(D/?)) for cycles and tori. Our results show that Min-Sum
schemes are competitive and get close to the optimal rate O(D log(1/?)) recently established for
some algorithms based on Nesterov?s acceleration [30, 36]. The main tool used for the analysis
involves the construction of an auxiliary linear process supported on the nodes of the original graph
to track the evolution of the Min-Sum Splitting algorithm, which is instead supported on the directed
edges. This construction allows us to relate the convergence time of the Min-Sum scheme to the
spectral gap of the matrix describing the dynamics of the auxiliary process, which is easier to analyze
than the matrix describing the dynamics on the edges as in [27].
In the literature, overcoming the suboptimal convergence rate of classical algorithms for network
averaging consensus has motivated the design of several accelerated methods. Two main lines of
research have been developed, and seem to have evolved independently of each others: one involves
lifted Markov chains techniques, see [37] for a review, the other involves accelerated first order
methods in convex optimization, see [13] for a review. Another contribution of this paper is to
show that Min-Sum Splitting bears similarities with both types of accelerated methods. On the one
hand, Min-Sum can be seen as a process on a lifted space, which is the space of directed edges in
the original graph. Here, splitting is seen to introduce a directionality in the message exchange of
the ordinary Min-Sum protocol that is analogous to the directionality introduced in non-reversible
2
random walks on lifted graphs to achieve faster convergence to stationarity. The advantage of the
Min-Sum algorithm over lifted Markov chain methods is that no lifted graph needs to be constructed.
On the other hand, the directionality induced on the edges by splitting translates into a memory term
for the auxiliary algorithm running on the nodes. This memory term, which allows nodes to remember
previous values and incorporate them into the next update, directly relates the Min-Sum Splitting
algorithm to accelerated multi-step first order methods in convex optimization. In particular, we show
that a proper choice of the splitting parameters recovers the same matrix that support the evolution
of shift-register methods used in numerical analysis for linear solvers, and, as a consequence, we
recover the same accelerated rate of convergence for consensus [45, 4, 24].
To summarize, the main contributions of this paper are:
1. First connection of Min-Sum schemes with lifted Markov chains techniques and multi-step
methods in convex optimization.
2. First proof of how the directionality embedded in Belief Propagation protocols can be tuned
and exploited to accelerate the convergence rate towards the problem solution.
3. First analysis of convergence rates for Min-Sum Splitting. New proof technique based on
the introduction of an auxiliary process to track the evolution of the algorithm on the nodes.
4. Design of a Min-Sum protocol for the consensus problem that achieves better convergence
rates than the ones established (and conjectured) for the Min-Sum method in [27].
Our results motivate further studies to generalize the acceleration due to splittings to other problems.
The paper is organized as follows. In Section 2 we introduce the Min-Sum Splitting algorithm in
its general form. In Section 3 we describe the consensus problem and review the classical diffusive
algorithms. In Section 4 we review the main accelerated methods that have been proposed in the
literature. In Section 5 we specialize the Min-Sum Splitting algorithm to the consensus problem, and
show that a proper parametrization yields a linear exchange of messages supported on the directed
edges of the graph. In Section 6 we derive the auxiliary message-passing algorithm that allows us to
track the evolution of the Min-Sum Splitting algorithm via a linear process with memory supported
on the nodes of the graph. In Section 7 we state Theorem 1, which shows that a proper choice of the
tuning parameters recovers the rates of shift-registers. Proofs are given in the supplementary material.
2
The Min-Sum Splitting algorithm
The Min-Sum algorithm is a distributed routine to optimize a cost function that is the sum of
components supported on a given graph structure. Given a simple graph G = (V, E) with n := |V |
vertices and m := |E| edges, let us assume that we are given a set of functions ?v : R ? R ? {?},
for each v ? V , and ?vw = ?wv : R ? R ? R ? {?}, for each {v, w} ? E, and that we want to
solve the following problem over the decision variables x = (xv )v?V ? RV :
X
X
minimize
?v (xv ) +
?vw (xv , xw ).
(1)
v?V
{v,w}?E
The Min-Sum algorithm describes an iterative exchange of messages?which are functions of the
decision variables?associated to each directed edge in G. Let E := {(v, w) ? V ? V : {v, w} ? E}
be the set of directed edges associated to the undirected edges in E (each edge in E corresponds
to two edges in E). In this work we consider the synchronous implementation of the Min-Sum
algorithm where at any given time step s, each directed edge (v, w) ? E supports two messages,
s
??vw
,?
?svw : R ? R ? {?}. Messages are computed iteratively. Given an initial choice of messages
?
?0 = (?
?0vw )(v,w)?E , the Min-Sum scheme that we investigate in this paper is given in Algorithm 1.
Henceforth, for each v ? V , let N (v) := {w ? V : {v, w} ? E} denote the neighbors of node v.
The formulation of the Min-Sum scheme given in Algorithm 1, which we refer to as Min-Sum
Splitting, was introduced in [34]. This formulation admits as tuning parameters the real number
? ? R and the symmetric matrix ? = (?vw )v,w?V ? RV ?V . Without loss of generality, we
assume that the sparsity of ? respects the structure of the graph G, in the sense that if {v, w} 6? E
then ?vw = 0 (note that Algorithm 1 only involves summations with respect to nearest neighbors
in the graph). The choice of ? = 1 and ? = A, where A is the adjacency matrix defined as
Avw := 1 if {v, w} ? E and Avw := 0 otherwise, yields the ordinary Min-Sum algorithm. For
3
Algorithm 1: Min-Sum Splitting
Input: Messages ?
?0 = (?
?0vw )(v,w)?E ; parameters ? ? R and ? ? RV ?V symmetric; time t ? 1.
for s ? {1, . . . , t} do
P
s
??wv
= ?v /? ? ?
?s?1
?s?1
wv +
zv , (w, v) ? E;
z?N (v) ?zv ?
s
s
s
?
?wv = minz?R {?vw ( ? , z)/?vw + (? ? 1)??wv
+ ? ??vw
(z)}, (w, v) ? E;
P
?tv = ?v + ? w?N (v) ?wv ?
?twv , v ? V ;
Output: xtv = arg minz?R ?tv (z), v ? V .
an arbitrary choice of strictly positive integer parameters, Algorithm 1 can be seen to correspond
to the ordinary Min-Sum algorithm applied to a new formulation of the original problem, where
an equivalent objective function is obtained from the original one in (1) by splitting each term
?
N \ {0}
term ?v into ? ? N \ {0} terms. Namely, minimize
Pvw into
P??vw ?
Pterms, and
P?each
vw
k
k
1
k
k
v?V
k=1 ?v (xv ) +
{v,w}?E
k=1 ?vw (xv , xw ), with ?v := ?v /? and ?vw := ?vw /?vw .
Hence the reason for the name ?splitting? algorithm. Despite this interpretation, Algorithm 1 is
defined for any real choice of parameters ? and ?.
In this paper we investigate the convergence behavior of the Min-Sum Splitting algorithm for some
choices of ? and ?, in the case of the consensus problem that we define in the next section.
3
The consensus problem and standard diffusive algorithms
Given a simple graph G = (V, E) with n := |V | nodes, for each v ? V let ?v : R ? R ? {?} be a
given function. The consensus problem is defined as follows:
X
minimize
?v (xv ) subject to xv = xw , {v, w} ? E.
(2)
v?V
We interpret G as a communication graph where each node represents an agent, and each edge
represent a communication channel between neighbor agents. Each agent v is given the function ?v ,
and agents collaborate by iteratively exchanging information with their neighbors in G with the goal
to eventually arrive to the solution of problem (2). The consensus problem amounts to designing
distributed algorithms to solve problem (2) that respect the communication constraints encoded by G.
A classical setting investigated in the literature is the least-square case yielding the network averaging
1 2
V
2
problem, where
P for a given b ? R we have ?v (z) := 2 z ? bv z and the solution of problem
(2) is ?b := n1 v?V bv . In this setup, each agent v ? V is given a number bv , and agents want
to exchange information with their neighbors according to a protocol that allows each of them to
eventually reach consensus on the average ?b across the entire network. Classical algorithms to
solve this problem involve a linear exchange of information of the form xt = W xt?1 with x0 = b,
for a given matrix W ? RV ?V that respects the topology of the graph G (i.e., Wvw 6= 0 only
if {v, w} ? E or v = w), so that W t ? 11T /n for t ? ?, where 1 is the all ones vector.
This linear iteration allows for a distributed exchange of information among agents, as at any
iteration each agent
Pv ? V only receives information from his/her neighbors N (v) via the update:
xtv = Wvv xt?1
+ w?N (v) Wvw xt?1
v
w . The original literature on this problem investigates the case
where the matrix W has non-negative coefficients and represents the transition matrix of a random
walk on the nodes of the graph G, so that Wvw is interpreted as the probability that a random walk at
node v visits node w in the next time step. A popular choice is given by the Metropolis-Hastings
MH
method [37], which involved the doubly-stochastic matrix W M H defined as Wvw
:= 1/(2dmax ) if
MH
MH
{v, w} ? E, Wvw := 1 ? dv /(2dmax ) if w = v, and Wvw := 0 otherwise, where dv := |N (v)|
is the degree of node v, and dmax := maxv?V dv is the maximum degree of the graph G.
1
As mentioned in [34], one can also consider a more general formulation of the splitting algorithm with
? ? (?v )v?V ? R (possibly also with time-varying parameters). The current choice of the algorithm is
motivated by the fact that in the present case the output of the algorithm can be tracked by analyzing a linear
system on the nodes of the graph, as we will show in Section
P 5.
2
In the literature, the classical choice is ?v (z) := 21 v?V (z ? bv )2 , which yields the same results as the
quadratic function that we define in the main text, as constant terms in the objective function do not alter the
optimal point of the problem but only the optimal value of the objective function.
4
In [44], necessary and sufficient conditions are given for a generic matrix W to satisfy W t ? 11T /n,
namely, 1T W = 1T , W 1 = 1, and ?(W ? 11T /n) < 1, where ?(M ) denotes the spectral radius
of a given matrix M . The authors show that the problem of choosing the optimal symmetric matrix
W that minimizes ?(W ? 11T /n) = kW ? 11T /nk ? where kM k denotes the spectral norm of a
matrix M that coincides with ?(M ) if M is symmetric ? is a convex problem and it can be cast as a
semi-definite program. Typically, the optimal matrix involves negative coefficients, hence departing
from the random walk interpretation. However, even the optimal choice of symmetric matrix is shown
to yield a diffusive rate of convergence, which is already attained by the matrix W M H [7]. This rate
corresponds to the speed of convergence to stationarity achieved by the diffusion random walk, defined
as the Markov chain with transition matrix diag(d)?1 A, where diag(d) ? RV ?V is the degree matrix,
i.e., diagonal with diag(d)vv := dv , and A ? RV ?V is the adjacency matrix, i.e., symmetric with
Avw := 1 if {v, w} ? E, and Avw := 0 otherwise. For instance, the condition kW ? 11T /nkt ? ?,
where k ? k is the `2 norm, yields a convergence time that scales like t ? ?(D2 log(1/?)) in cycle
graphs and tori [33], where D is the graph diameter. The authors in [7] established that for a class
of graphs with geometry (polynomial growth or finite doubling dimension) the mixing time of any
reversible Markov chain scales at least like D2 , and it is achieved by Metropolis-Hastings [37].
4
Accelerated algorithms
To overcome the diffusive behavior typical of classical consensus algorithms, two main types of
approaches have been investigated in the literature, which seem to have been developed independently.
b = (Vb , E)
b and of a linear system
The first approach involves the construction of a lifted graph G
cx
c ? RVb ?Vb is the transition matrix
supported on the nodes of it, of the form x
?t = W
?t?1 , where W
b This approach has its origins in the work of
of a non-reversible Markov chain on the nodes of G.
[8] and [5], where it was observed for the first time that certain non-reversible Markov chains on
properly-constructed lifted graphs yield better mixing times than reversible chains on the original
graphs. For some simple graph topologies, such as cycle graphs and two-dimensional grids, the
construction of the optimal lifted graphs is well-understood already from the works in [8, 5]. A general
theory of lifting in the context of Gossip algorithms has been investigated in [18, 37]. However, this
construction incurs additional overhead, which yield non-optimal computational complexity, even for
cycle graphs and two-dimensional grids. Typically, lifted random walks on arbitrary graph topologies
are constructed on a one-by-one case, exploiting the specifics of the graph at hand. This is the case,
for instance, for random geometric graphs [22, 23]. The key property that allows non-reversible lifted
Markov chains to achieve subdiffusive rates is the introduction of a directionality in the process to
break the diffusive nature of reversible chains. The strength of the directionality depends on global
properties of the original graph, such as the number of nodes [8, 5] or the diameter [37]. See Figure 1.
1/2
1/2
1?1/n
? 1?1/n
1
1/n
1/n
? ?1/n
1
1?1/n
(a)
(b)
(c)
(d)
Figure 1: (a) Symmetric Markov chain W on the nodes of the ring graph G. (b) Non-reversible
c on the nodes of the lifted graph G
b [8]. (c) Ordinary Min-Sum algorithm on the
Markov chain W
b ?), Algorithm 2, with ? = 1 and ? = A, where A is
directed edges E associated to G (i.e., K(?,
b ?), Algorithm 2, with ? = 1, ? = ?W ,
the adjacency
pmatrix of G). (d) Min-Sum Splitting K(?,
? = 2/(1 + 1 ? ?2W ) as in Theorem 1. Here, ?W is ?(1 ? 1/n2 ) and ? ? 2(1 ? 1/n) for n large.
b ?) has negative entries, departing from the Markov chain interpretation. This is also
The matrix K(?,
the case for the optimal tuning in classical consensus schemes [44] and for the ADMM lifting in [12].
The second approach involves designing linear updates that are supported on the original graph G and
keep track of a longer history of previous iterates. This approach relies on the fact that the original
consensus update xt = W xt?1 can be interpreted as a primal-dual gradient ascent method to solve
problem (2) with a quadratic objective function [32]. This allows the implementation of accelerated
5
gradient methods. To the best of our knowledge, this idea was first introduced in [14], and since then it
has been investigated in many other papers. We refer to [13, 24], and references in there, for a review
and comparison of multi-step accelerated methods for consensus. The simplest multi-step extension of
gradient methods is Polyak?s ?heavy ball,? which involves adding a ?momentum? term to the standard
update and yields a primal iterate of the form xt = W xt?1 + ?(xt?1 ? xt?2 ). Another popular
multi-step method involves Nesterov?s acceleration, and yields xt = (1 + ?)W xt?1 ? ?W xt?2 .
Aligned with the idea of adding a momentum term is the idea of adding a shift register term, which
yields xt = (1 + ?)W xt?1 ? ?xt?2 . For our purposes, we note that these methods can be written as
xt
xt?1
=K
xt?1
xt?2
,
(3)
for a certain matrix K ? R2n?2n . As in the case of lifted Markov chains techniques, also multi-step
methods are able to achieve accelerated rates by exploiting some form of global information: the
choice of the parameter ? that yields subdiffusive rates depends on the eigenvalues of W .
Remark 1. Beyond lifted Markov chains techniques and accelerated first order methods, many other
algorithms have been proposed to solve the consensus problem. The literature is vast. As we focus on
Min-Sum schemes, an exhaustive literature review on consensus is beyond the scope of our work. Of
particular interest for our results is the distributed ADMM approach [3, 43, 38]. Recently in [12],
for a class of unconstrained problems with quadratic objective functions, it has been shown that
message-passing ADMM schemes can be interpreted as lifting of gradient descent techniques. This
prompts for further investigation to connect Min-Sum, ADMM, and accelerated first order methods.
In the next two sections we show that Min-Sum Splitting bears similarities with both types of
accelerated methods described above. On the one hand, in Section 5 we show that the estimates xtv ?s
of Algorithm 1 applied to the network averaging problem can be interpreted as the result of a linear
process supported on a lifted space, i.e., the space E of directed edges associated to the undirected
edges of G. On the other hand, in Section 6 we show that the estimates xtv ?s can be seen as the result
of a linear multi-step process supported on the nodes of G, which can be written as in (3). Later on,
in Section 7 and Section 8, we will see that the similarities just described go beyond the structure
of the processes, and they extend to the acceleration mechanism itself. In particular, the choice of
splitting parameters that yields subdiffusive convergence rates, matching the asymptotic rates of shift
register methods, is also shown to depend on global information about G.
5
Min-Sum Splitting for consensus
We apply Min-Sum Splitting to solve network averaging. We show that in this case the messagepassing protocol is a linear exchange of parameters associated to the directed edges in E.
?
? wv :=
Given ? ? R and ? ? RV ?V symmetric, let h(?)
? RE be the vector defined as h(?)
E?E
b
bw + (1 ? 1/?)bv , and let K(?, ?) ? R
be matrix defined as
b ?)wv,zu
K(?,
?
?
???zw
?
?
?
??(?vw ? 1)
:= (? ? 1)?zv
?
?
?
(? ? 1)(?wv ? 1)
?
?
?0
if u = w, z ? N (w) \ {v},
if u = w, z = v,
if u = v, z ? N (v) \ {w},
if u = v, z = w,
otherwise.
(4)
0
0
? 0 = (R
? vw
Consider Algorithm 2 with initial conditions R
)(v,w)?E ? RE , r?0 = (?
rvw
)(v,w)?E ? RE .
Algorithm 2: Min-Sum Splitting, consensus problem, quadratic case
? 0 , r?0 ? RE ; ? ? R, ? ? RV ?V symmetric; K(?,
b ?) defined in (5); t ? 1.
Input: R
for s ? {1, . . . , t} do
?
? s = (2 ? 1/?)1 + K(?,
b ?)R
? s?1 ;
b ?)?
R
r?s = h(?)
+ K(?,
rs?1 ;
Output: xtv :=
P
t
bv +? w?N (v) ?wv r?wv
P
?t , v
1+? w?N (v) ?wv R
wv
?V.
6
Proposition 1. Let ? ? R and ? ? RV ?V symmetric be given. Consider Algorithm 1 applied to
0
? 0 z 2 ??
problem (2) with ?v (z) := 21 z 2 ?bv z and with quadratic initial messages: ?
?0vw (z) = 12 R
rvw
z,
vw
1 ?s
s
2
? 0 > 0 and r?0 ? R. Then, the messages will remain quadratic, i.e., ?
for some R
?
(z)
=
R
z
vw
vw
vw
2 vw ?
P
s
t
? wv
r?vw
z for any s ? 1, and the parameters evolve as in Algorithm 2. If 1 + ? w?N (v) ?wv R
>0
for any v ? V and t ? 1, then the output of Algorithm 2 coincides with the output of Algorithm 1.
6
Auxiliary message-passing scheme
We show that the output of Algorithm 2 can be tracked by a new message-passing scheme that
corresponds to a multi-step linear exchange of parameters associated to the nodes of G. This auxiliary
algorithm represents the main tool to establish convergence rates for the Min-Sum Splitting protocol,
i.e., Theorem 1 below. The intuition behind the auxiliary process is that while Algorithm 1 (hence,
Algorithm 2) involves an exchange of messages supported on the directed edges E, the computation
of the estimates xtv ?s only involve the belief functions ?tv ?s, which are supported on the nodes of G.
Due to the simple nature of the pairwise equality constraints in the consensus problem, in the present
case a reparametrization allows to track the output of Min-Sum via an algorithm that directly updates
the belief functions on the nodes of the graph, which yields Algorithm 3.
Given ? ? R and ? ? Rn?n symmetric, define the matrix K(?, ?) ? R2n?2n as
(1 ? ?)I ? (1 ? ?)diag(?1) + ??
?I
K(?, ?) :=
,
?I ? ?diag(?1) + (1 ? ?)?
(1 ? ?)I
(5)
V ?V
where I ?
is the identity matrix and diag(?1) ? RV ?V is diagonal with (diag(?1))vv =
PR
(?1)v = w?N (v) ?vw . Consider Algorithm 3 with initial conditions R0 , r0 , Q0 , q 0 ? RV .
Algorithm 3: Auxiliary message-passing
Input: R0 , r0 , Q0 , q 0 ? RV ; ? ? R, ? ? RV ?V symmetric; K(?, ?) defined in (5); t ? 1.
for s? {1,. . . , t} do
s
s?1
rs
rs?1
R
R
=
K(?,
?)
;
=
K(?,
?)
;
qs
Qs
q s?1
Qs?1
Output: xtv := rvt /Rvt , v ? V .
Proposition 2. Let ? ? R and ? ? RV ?V symmetric be given. The outputPof Algorithm 2 with initial
E
0
?0 0
?0
conditions
is the output ofP
Algorithm 3 with Rv0 := 1 + ? w?N
v :=
P R , r? ? R
P(v) ?wv Rwv , Q
0
0
0
0
0
?
1 ? ? w?N (v) ?wv Rwv , rv := bv + ? w?N (v) ?wv r?wv , and qv := bv ? ? w?N (v) ?vw r?vw .
Proposition 2 shows that upon proper initialization, the outputs of Algorithm 2 and Algorithm 3
are equivalent. Hence, Algorithm 3 represents a tool to investigate the convergence behavior of the
Min-Sum Splitting algorithm. Analytically, the advantage of the formulation given in Algorithm 3
over the one given in Algorithm 2 is that the former involves two coupled systems of n equations
whose convergence behavior can explicitly be linked to the spectral properties of the n ? n matrix ?,
as we will see in Theorem 1 below. On the contrary, the linear system of 2m equations in Algorithm
2 does not seem to exhibit an immediate link to the spectral properties of ?. In this respect, we note
that the previous paper that investigated Min-Sum schemes for consensus, i.e., [27], characterized the
convergence rate of the algorithm under consideration ? albeit only in the case of d-regular graphs,
and upon initializing the quadratic terms to the fix point ? in terms of the spectral gap of a matrix
that controls a linear system of 2m equations. However, the authors only list results on the behavior
of this spectral gap in the case of cycle graphs, i.e., d = 2, and present a conjecture for 2d-tori.
7
Accelerated convergence rates for Min-Sum Splitting
We investigate the convergence behavior of the Min-Sum Splitting algorithm to solve problem (2)
V
with quadratic objective functions. Henceforth, without loss of generality,
P let b ? R be given with
1 2
?
0 < bv < 1 for each v ? V , and let ?v (z) := 2 z ? bv z. Define b := v?V bv /n.
Recall from [27] that the ordinary Min-Sum algorithm (i.e., Algorithm 2 with ? = 1 and ? = A,
where A is the adjacency matrix of the graph G) does not converge if the graph G has a cycle.
7
We now show that a proper choice of the tuning parameters allows Min-Sum Splitting to converge
to the problem solution in a subdiffusive way. The proof of this result, which is contained in the
supplementary material, relies on the use of the auxiliary method defined in Algorithm 3 to track the
evolution of the Min-Sum Splitting scheme. Here, recall that kxk denotes the `2 norm of a given
vector x, kM k denotes the `2 matrix norm of the given matrix M , and ?(M ) its spectral radius.
T
Theorem 1. Let W ? RV ?V be a symmetric
p matrix with W 1t = 1 and ?W := ?(W ? 11 /n) < 1.
2
Let ? = 1 and ? = ?W , with ? = 2/(1 + 1 ? ?W ). Let x be the output at time t of Algorithm 2
? 0 = r?0 = 0. Define
with initial conditions R
1
?W
I
11T
11T
?
K :=
,
K :=
.
(6)
(1 ? ?)I 0
(1 ? ?)11T (1 ? ?)11T
(2 ? ?)n
?
2n
Then, for any v ? V we have limt?? xtv = ?b and kxt ? ?b1k ? 42??
k(K ? K ? )t k.
The asymptotic rate of convergence is given by
q
p
p
?
? t 1/t
?K := ?(K ? K ) = limt?? k(K ? K ) k = (1? 1??2W )/(1+ 1??2W ) < ?W < 1,
p
p
which satisfies 21 1/(1 ? ?W ) ? 1/(1 ? ?K ) ? 1/(1 ? ?W ).
Theorem 1 shows that the choice of splitting parameters ? = 1 and ? = ?W , where ? and W
are defined as in the statement of the theorem, allows the Min-Sum Splitting scheme to achieve
the asymptotic rate of convergence that is given by the second largest eigenvalue in magnitude of
the matrix K defined in (6), i.e., the quantity ?K . The matrix K is the same matrix that describes
shift-register methods for consensus [45, 4, 24]. In fact, the proof of Theorem 1 relies on the spectral
analysis previously established for shift-registers, which can be traced back to [15]. See also [13, 24].
?
Following [27], let us consider the absolute
measure of error given by kxt ? ?b1k/ n (recall that we
?
assume 0 <?bv < 1?so that kbk ? n). From Theorem 1 it follows that, asymptotically, we have
kxt ? ?b1k/ n . 4 2?tK /(2 ? ?). If we?define the asymptotic convergence time as the minimum
time t so that, asymptotically, kxt ? ?b1k/ n . ?, then the Min-Sum Splitting scheme investigated in
Theorem 1 has an asymptotic convergence time that is O(1/(1??K ) log{[1/(1??K )]/?}). Given the
last bound in Theorem 1, this result achieves (modulo logarithmic terms) a square-root improvement
over the convergence time of diffusive methods, which scale like ?(1/(1 ? ?W ) log 1/?). For cycle
graphs and, more generally, for higher-dimensional tori ? where 1/(1 ? ?W ) is ?(D2 ) so that
1/(1??K ) is ?(D) [33, 1] ? the convergence time is O(D log D/?), where D is the graph diameter.
As prescribed by Theorem 1, the choice of ? that makes the Min-Sum scheme achieve a subdiffusive
rate depends on global properties of the graph G. Namely, ? depends on the quantity ?W , the second
largest eigenvalue in magnitude of the matrix W . This fact connects the acceleration mechanism
induced by splitting in the Min-Sum scheme to the acceleration mechanism of lifted Markov chains
techniques (see Figure 1) and multi-step first order methods, as described in Section 4.
It remains to be investigated how choices of splitting parameters different than the ones investigated
in Theorem 1 affect the convergence behavior of the Min-Sum Splitting algorithm.
8
Conclusions
The Min-Sum Splitting algorithm has been previously observed to yield convergence in settings
where the ordinary Min-Sum protocol does not converge [35]. In this paper we proved that the
introduction of splitting parameters is not only fundamental to guarantee the convergence of the
Min-Sum scheme applied to the consensus problem, but that proper tuning of these parameters
yields accelerated convergence rates. As prescribed by Theorem 1, the choice of splitting parameters
that yields subdiffusive rates involves global type of information, via the spectral gap of a matrix
associated to the original graph (see the choice of ? in Theorem 1). The acceleration mechanism
exploited by Min-Sum Splitting is analogous to the acceleration mechanism exploited by lifted
Markov chain techniques ? where the transition matrix of the lifted random walks is typically chosen
to depend on the total number of nodes in the graph [8, 5] or on its diameter [37] (global pieces of
information) ? and to the acceleration mechanism exploited by multi-step gradient methods ? where
the momentum/shift-register term is chosen as a function of the eigenvalues of a matrix supported on
the original graph [13] (again, a global information). Prior to our results, this connection seems to
have not been established in the literature. Our findings motivate further studies to generalize the
acceleration due to splittings to other problem instances, beyond consensus.
8
Acknowledgements
This work was partially supported by the NSF under Grant EECS-1609484.
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6,342 | 6,737 | Generalized Linear Model Regression under
Distance-to-set Penalties
Jason Xu
University of California, Los Angeles
[email protected]
Eric C. Chi
North Carolina State University
[email protected]
Kenneth Lange
University of California, Los Angeles
[email protected]
Abstract
Estimation in generalized linear models (GLM) is complicated by the presence of
constraints. One can handle constraints by maximizing a penalized log-likelihood.
Penalties such as the lasso are effective in high dimensions, but often lead to
unwanted shrinkage. This paper explores instead penalizing the squared distance
to constraint sets. Distance penalties are more flexible than algebraic and regularization penalties, and avoid the drawback of shrinkage. To optimize distance
penalized objectives, we make use of the majorization-minimization principle. Resulting algorithms constructed within this framework are amenable to acceleration
and come with global convergence guarantees. Applications to shape constraints,
sparse regression, and rank-restricted matrix regression on synthetic and real data
showcase strong empirical performance, even under non-convex constraints.
1
Introduction and Background
In classical linear regression, the response variable y follows a Gaussian distribution whose mean
xt ? depends linearly on a parameter vector ? through a vector of predictors x. Generalized linear
models (GLMs) extend classical linear regression by allowing y to follow any exponential family
distribution, and the conditional mean of y to be a nonlinear function h(xt ?) of xt ? [24]. This
encompasses a broad class of important models in statistics and machine learning. For instance, count
data and binary classification come within the purview of generalized linear regression.
In many settings, it is desirable to impose constraints on the regression coefficients. Sparse regression
is a prominent example. In high-dimensional problems where the number of predictors n exceeds the
number of cases m, inference is possible provided the regression function lies in a low-dimensional
manifold [11]. In this case, the coefficient vector ? is sparse, and just a few predictors explain the
response y. The goals of sparse regression are to correctly identify the relevant predictors and to
estimate their effect sizes. One approach, best subset regression, is known to be NP hard. Penalizing
the likelihood by including an `0 penalty k?k0 (the number of nonzero coefficients) is a possibility,
but the resulting objective function is nonconvex and discontinuous. The convex relaxation of `0
regression replaces k?k0 by the `1 norm k?k1 . This LASSO proxy for k?k0 restores convexity
and continuity [31]. While LASSO regression has been a great success, it has the downside of
simultaneously inducing both sparsity and parameter shrinkage. Unfortunately, shrinkage often has
the undesirable side effect of including spurious predictors (false positives) with the true predictors.
Motivated by sparse regression, we now consider the alternative of penalizing the log-likelihood by
the squared distance from the parameter vector ? to the constraint set. If there are several constraints,
then we add a distance penalty for each constraint set. Our approach is closely related to the proximal
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
distance algorithm [19, 20] and proximity function approaches to convex feasibility problems [5].
Neither of these prior algorithm classes explicitly considers generalized linear models. Beyond
sparse regression, distance penalization applies to a wide class of statistically relevant constraint
sets, including isotonic constraints and matrix rank constraints. To maximize distance penalized loglikelihoods, we advocate the majorization-minimization (MM) principle [2, 18, 19]. MM algorithms
are increasingly popular in solving the large-scale optimization problems arising in statistics and
machine learning [22]. Although distance penalization preserves convexity when it already exists,
neither the objective function nor the constraints sets need be convex to carry out estimation. The
capacity to project onto each constraint set is necessary. Fortunately, many projection operators are
known. Even in the absence of convexity, we are able to prove that our algorithm converges to a
stationary point. In the presence of convexity, the stationary points are global minima.
In subsequent sections, we begin by briefly reviewing GLM regression and shrinkage penalties. We
then present our distance penalty method and a sample of statistically relevant problems that it can
address. Next we lay out in detail our distance penalized GLM algorithm, discuss how it can be
accelerated, summarize our convergence results, and compare its performance to that of competing
methods on real and simulated data. We close with a summary and a discussion of future directions.
GLMs and Exponential Families: In linear regression, the vector of responses y is normally
distributed with mean vector E(y) = X? and covariance matrix V(y) = ? 2 I. A GLM preserves
the independence of the responses yi but assumes that they are generated from a shared exponential
family distribution. The response yi is postulated to have mean ?i (?) = E[yi |?] = h(xti ?), where
xi is the ith row of a design matrix X, and the inverse link function h(s) is smooth and strictly
increasing [24]. The functional inverse h?1 (s) of h(s) is called the link function. The likelihood of
any exponential family can be written in the canonical form
y?i ? ?(?i )
p(yi |?i , ? ) = c1 (yi , ? ) exp
.
(1)
c2 (? )
Here ? is a fixed scale parameter, and the positive functions c1 and c2 are constant with respect to the
natural parameter ?i . The function ? is smooth and convex; a brief calculation shows that ?i = ? 0 (?i ).
The canonical link function h?1 (s) is defined by the condition h?1 (?i ) = xti ? = ?i . In this case,
h(?i ) = ? 0 (?i ), and the log-likelihood ln p(y|?, xj , ? ) is concave in ?. Because c1 and c2 are not
functions of ?, we may drop these terms and work with the log-likelihood up to proportionality. We
denote this by L(? | y, X) ? ln p(y|?, xj , ? ). The gradient and second differential of L(? | y, X)
amount to
m
m
X
X
?L =
[yi ? ? 0 (xti ?)]xi and d2 L = ?
? 00 (xti ?)xi xti .
(2)
i=1
i=1
As an example, when ?(?) = ?2 /2 and c2 (? ) = ? 2 , the density (1) is the Gaussian likelihood,
and GLM regression under the identity link coincides with standard linear regression. Choosing
?(?) = ln[1 + exp(?)] and c2 (? ) = 1 corresponds to logistic regression under the canonical link
s
es
h?1 (s) = ln 1?s
with inverse link h(s) = 1+e
s . GLMs unify a range of regression settings,
including Poisson, logistic, gamma, and multinomial regression.
Shrinkage penalties:
The least absolute shrinkage and selection operator (LASSO) [12, 31] solves
m
h
i
X
? = argmin ?k?k1 ? 1
?
L(? | yj , xj ) ,
?
m j=1
(3)
where ? > 0 is a tuning constant that controls the strength of the `1 penalty. The `1 relaxation is
a popular approach to promote a sparse solution, but there is no obvious map between ? and the
sparsity level k. In practice, a suitable value of ? is found by cross-validation. Relying on global
shrinkage towards zero, LASSO notoriously leads to biased estimates. This bias can be ameliorated
by re-estimating under the model containing only the selected variables, known as the relaxed LASSO
[25], but success of this two-stage procedure relies on correct support recovery in the first step.
In many cases, LASSO shrinkage is known to introduce false positives [30], resulting in spurious
covariates that cannot be corrected. To combat these shortcomings, one may replace the LASSO
penalty by a non-convex penalty with milder effects on large coefficients. The smoothly clipped
2
absolute deviation (SCAD) penalty [10] and minimax concave penalty (MCP) [34] are even functions
defined through their derivatives
(?? ? |?i |)+
|?i |
0
0
q? (?i , ?) = ? 1{|?i |??} +
1{|?i |>?}
and q? (?i , ?) = ? 1 ?
(? ? 1)?
?? +
for ?i > 0. Both penalties reduce bias, interpolate between hard thresholding and LASSO shrinkage,
and significantly outperform the LASSO in some settings, especially in problems with extreme
sparsity. SCAD, MCP, as well as the relaxed lasso come with the disadvantage of requiring an extra
tuning parameter ? > 0 to be selected.
2
Regression with distance-to-constraint set penalties
As an alternative to shrinkage, we consider penalizing the distance between the parameter vector ?
and constraints defined by sets Ci . Penalized estimation seeks the solution
?
?
m
X
X
1
? = argmin ? 1
L(? | yj , xj )? := argmin? f (?),
(4)
?
vi dist(?, Ci )2 ?
?
2 i
m j=1
where the vi are weights on the distance penalty to constraint set Ci . The Euclidean distance can
also be written as
dist(?, Ci ) = k? ? PCi (?)k2 ,
where PCi (?) denotes the projection of ? onto Ci . The projection operator is uniquely defined
when Ci is closed and convex. If Ci is merely closed, then PCi (?) may be multi-valued for a few
unusual external points ?. Notice the distance penalty dist(?, Ci )2 is 0 precisely when ? ? Ci .
The solution (4) represents a tradeoff between maximizing the log-likelihood and satisfying the
constraints. When each Ci is convex, the objective function is convex as a whole. Sending all of the
penalty constants vi to ? produces in the limit the constrained maximum likelihood estimate. This
is the philosophy behind the proximal distance algorithm [19, 20]. In practice, it often suffices to
find the solution (4) under fixed vi large. The reader may wonder why we employ squared distances
rather than distances. The advantage is that squaring renders the penalties differentiable. Indeed,
? 12 dist(x, Ci )2 = x ? PCi (x) whenever PCi (x) is single valued. This is almost always the case.
In contrast, dist(x, Ci ) is typically nondifferentiable at boundary points of Ci even when Ci is
convex. The following examples motivate distance penalization by considering constraint sets and
their projections for several important models.
Sparse regression: Sparsity can be imposed directly through the constraint set Ck =
{z ? Rn : kzk0 ? k} . Projecting a point ? onto C is trivially accomplished by setting all but
the k largest entries in magnitude of ? equal to 0, the same operation behind iterative hard thresholding algorithms. Instead of solving the `1 -relaxation (3), our algorithm approximately solves the
original `0 -constrained problem by repeatedly projecting onto the sparsity set Ck . Unlike LASSO
regression, this strategy enables one to directly incorporate prior knowledge of the sparsity level k in
an interpretable manner. When no such information is available, k can be selected by cross validation
just as the LASSO tuning constant ? is selected. Distance penalization escapes the NP hard dilemma
of best subset regression at the cost of possible convergence to a local minimum.
Shape and order constraints: As an example of shape and order restrictions, consider isotonic
regression [1]. For data y ? Rn , isotonic regression seeks to minimize 12 ky ? ?k22 subject to
the condition that the ?i are non-decreasing. In this case, the relevant constraint set is the isotone
convex cone C = {? : ?1 ? ?2 ? . . . ? ?n }. Projection onto C is straightforward and efficiently
accomplished using the pooled adjacent violators algorithm [1, 8]. More complicated order constraints
can be imposed analogously: for instance, ?i ? ?j might be required of all edges i ? j in a directed
graph model. Notably, isotonic linear regression applies to changepoint problems [32]; our approach
allows isotonic constraints in GLM estimation. One noteworthy application is Poisson regression
where the intensity parameter is assumed to be nondecreasing with time.
Rank restriction: Consider GLM regression where the predictors X i and regression coefficients
B are matrix-valued. To impose structure in high-dimensional settings, rank restriction serves as an
3
appropriate matrix counterpart to sparsity for vector parameters. Prior work suggests that imposing
matrix sparsity is much less effective than restricting the rank of B in achieving model parsimony
[37]. The matrix analog of the LASSO penalty is the nuclear norm penalty. The nuclear
? ? norm of a
P
matrix B is defined as the sum of its singular values kBk? = j ?j (B) = trace( B B). Notice
kBk? is a convex relaxation of rank(B). Including a nuclear norm penalty entails shrinkage and
induces low-rankness by proxy.
Distance penalization of rank involves projecting onto the set Cr = {Z ? Rn?n : rank(Z) ? r}
for a given rank r. Despite sacrificing convexity, distance penalization of rank is, in our view, both
more natural and more effective than nuclear norm penalization. Avoiding shrinkage works to the
advantage of distance penalization, which we will see empirically in Section 4. According to the
Eckart-Young theorem, the projection of a matrix B onto Cr is achieved by extracting the singular
value decomposition of B and truncating all but the top r singular values. Truncating the singular
value decomposition is a standard numerical task best computed by Krylov subspace methods [14].
Simple box constraints, hyperplanes, and balls: Many relevant set constraints reduce to closed
convex sets with trivial projections. For instance, enforcing non-negative parameter values is accomplished by projecting onto the non-negative orthant. This is an example of a box constraint.
Specifying linear equality and inequality constraints entails projecting onto a hyperplane or half-space,
respectively. A Tikhonov or ridge penalty constraint k?k2 ? r requires spherical projection.
Finally, we stress that it is straightforward to consider combinations of the aforementioned constraints.
Multiple norm penalties are already in common use. To encourage selection of correlated variables
[38], the elastic net includes both `1 and `2 regularization terms. Further examples include matrix
fitting subject to both sparse and low-rank matrix constraints [29] and LASSO regression subject
to linear equality and inequality constraints [13]. In our setting the relative importance of different
constraints can be controlled via the weights vi .
3
Majorization-minimization
Figure 1: Illustrative example of two MM iterates with surrogates g(x|xk ) majorizing f (x) = cos(x).
To solve the minimization problem (4), we exploit the principle of majorization-minimization. An
MM algorithm successively minimizes a sequence of surrogate functions g(? | ? k ) majorizing the
objective function f (?) around the current iterate ? k . See Figure 1. Forcing g(? | ? k ) downhill
automatically drives f (?) downhill as well [19, 22]. Every expectation-maximization (EM) algorithm
[9] for maximum likelihood estimation is an MM algorithm. Majorization requires two conditions:
tangency at the current iterate g(? k | ? k ) = f (? k ), and domination g(? | ? k ) ? f (?) for all
? ? Rm . The iterates of the MM algorithm are defined by
? k+1 := arg min g(? | ? k )
?
although all that is absolutely necessary is that g(? k+1 | ? k ) < g(? k | ? k ). Whenever this holds,
the descent property
f (? k+1 ) ? g(? k+1 | ? k ) ? g(? k | ? k ) = f (? k )
4
follows. This simple principle is widely applicable and converts many hard optimization problems
(non-convex or non-smooth) into a sequence of simpler problems.
2
To majorize the objective (4), it suffices to majorize each distance penalty dist (?, Ci ) . The ma2
jorization dist (?, Ci ) ? k? ? PCi (? k )k22 is an immediate consequence of the definitions of the set
2
distance dist (?, Ci ) and the projection operator PCi (?) [8]. The surrogate function
m
1X
1 X
g(? | ? k ) =
vi k? ? PCi (? k )k22 ?
L(? | yj , xj ).
2 i
m j=1
has gradient
m
?g(? | ? k )
=
X
vi [? ? PCi (? k )] ?
i
1 X
?L(? | yj , xj )
m j=1
and second differential
d2 g(? | ? k )
=
m
X
1 X 2
d L(? | yj , xj ) := H k .
vi I n ?
m j=1
i
(5)
The score ?L(? | yj , xj ) and information ?d2 L(? | yj , xj ) appear in equation (2). Note that for
GLMs under canonical link, the observed and expected information matrices coincide, and their
common value is thus positive semidefinite. Adding a multiple of the identity I n to the information
matrix is analogous to the Levenberg-Marquardt maneuver against ill-conditioning in ordinary
regression [26]. Our algorithm therefore naturally benefits from this safeguard.
Since solving the stationarity equation ?g(? | ? k ) = 0 is not analytically feasible in general, we
employ one step of Newton?s method in the form
? k+1 = ? k ? ?k d2 g(? k | ? k )?1 ?f (? k ),
where ?k ? (0, 1] is a stepsize multiplier chosen via backtracking. Note here our application of
the gradient identity ?f (? k ) = ?g(? k | ? k ), valid for all smooth surrogate functions. Because
the Newton increment is a descent direction, some value of ?k is bound to produce a decrease in
the surrogate and therefore in the objective. The following theorem, proved in the Supplement,
establishes global convergence of our algorithm under simple Armijo backtracking for choosing ?k :
Theorem 3.1 Consider the algorithm map
M(?) = ? ? ?? H(?)?1 ?f (?),
where the step size ?? has been selected by Armijo backtracking. Assume that f (?) is coercive in the
sense limk?k?? f (?) = +?. Then the limit points of the sequence ? k+1 = M(? k ) are stationary
points of f (?). Moreover, the set of limit points is compact and connected.
We remark that stationary points are necessarily global minimizers when f (?) is convex. Furthermore,
coercivity of f (?) is a very mild assumption, and is satisfied whenever either the distance penalty or
the negative log-likelihood is coercive. For instance, the negative log-likelihoods of the Poisson and
Gaussian distributions are coercive functions. While this is not the case for the Bernoulli distribution,
adding a small `2 penalty ?k?k22 restores coerciveness. Including such a penalty in logistic regression
is a common remedy to the well-known problem of numerical instability in parameter estimates
caused by a poorly conditioned design matrix X [27]. Since L(?) is concave in ?, the compactness
of one or more of the constraint sets Ci is another sufficient condition for coerciveness.
Generalization to Bregman divergences: Although we have focused on penalizing GLM likelihoods with Euclidean distance penalties, this approach holds more generally for objectives containing
non-Euclidean measures of distance. As reviewed in the Supplement, the Bregman divergence
D? (v, u) = ?(v) ? ?(u) ? d?(u)(v ? u) generated by a convex function ?(v) provides a general
notion of directed distance [4]. The Bregman divergence associated with the choice ?(v) = 12 kvk22 ,
for instance, is the squared Euclidean distance. One can rewrite the GLM penalized likelihood as a
sum of multiple Bregman divergences
m
h
i X
h
i
X
f (?) =
vi D? PC?i (?), ? +
wj D? y j , e
hj (?) .
(6)
i
j=1
5
Algorithm 1 MM algorithm to solve distance-penalized objective (4)
1: Initialize k = 0, starting point ? 0 , initial step size ? ? (0, 1), and halving parameter ? ? (0, 1):
2: repeat
Pm
P
1
3:
?fk ? i vi [? ? PCi (? k )] ? m
j=1 ?L(? | yj , ? j )
P
Pm 2
1
4:
Hk ?
i vi I n ? m
j=1 d L(? | yj , ? j )
5:
v ? ?H ?1
k ?fk
6:
??1
7:
while f (? k + ?v) > f (? k ) + ???fkt ? k do
8:
? ? ??
9:
end while
10:
? k+1 ? ? k + ?v
11:
k ?k+1
12: until convergence
The first sum in equation (6) represents the distance penalty to the constraint sets Ci . The projection
PC?i (?) denotes the closest point to ? in Ci measured under D? . The second sum generalizes the
GLM log-likelihood term where e
hj (?) = h?1 (xtj ?). Every exponential family likelihood uniquely
corresponds to a Bregman divergence D? generated
by the conjugate
of its cumulant
function ? = ? ?
Pm
1
?1
t
[28]. Hence, ?L(? | y, X) is proportional to m j=1 D? y j , h (xj ?) . The functional form (6)
immediately broadens the class of objectives to include quasi-likelihoods and distances to constraint
sets measured under a broad range of divergences. Objective functions of this form are closely related
to proximity function minimization in the convex feasibility literature [5, 6, 7, 33]. The MM principle
makes possible the extension of the projection algorithms of [7] to minimize this general objective.
Our MM algorithm for distance penalized GLM regression is summarized in Algorithm 1. Although
for the sake of clarity the algorithm is written for vector-valued arguments, it holds more generally for
matrix-variate regression. In this setting the regression coefficients B and predictors X i are matrix
valued, and response yj has mean h[trace(X ti B)] = h[vec(X i )t vec(B)]. Here the vec operator
stacks the columns of its matrix argument. Thus, the algorithm immediately applies if we replace B
by vec(B) and X 1 , . . . , X m by X = [vec(X 1 ), . . . , vec(X m )]t . Projections requiring the matrix
structure are performed by reshaping vec(B) into matrix form. In contrast to shrinkage approaches,
these maneuvers obviate the need for new algorithms in matrix regression [37].
Acceleration: Here we mention two modifications to the MM algorithm that translate to large
practical differences in computational cost. Inverting the n-by-n matrix d2 g(? k | ? k ) naively
requires O(n3 ) flops. When the number of cases m n, invoking the Woodbury formula requires
solving a substantially smaller m ? m linear system at each iteration. This computational savings is
crucial in the analysis of the EEG data of Section 4. The Woodbury formula says
?1
(vI n + U V )?1 = v ?1 I n ? v ?2 U I m + v ?1 V U
V
when U and V are n ? m and m ? n matrices, respectively. Inspection of equations (2) and (5) shows
that d2 g(? k | ? k ) takes the required form. Under Woodbury?s formula the dominant computation
is the matrix-matrix product V U , which requires only O(nm2 ) flops. The second modification to
the MM algorithm is quasi-Newton acceleration. This technique exploits secant approximations
derived from iterates of the algorithm map to approximate the differential of the map. As few as two
secant approximations can lead to orders of magnitude reduction in the number of iterations until
convergence. We refer the reader to [36] for a detailed description of quasi-Newton acceleration and
a summary of its performance on various high-dimensional problems.
4
Results and performance
We first compare the performance of our distance penalization method to leading shrinkage methods
in sparse regression. Our simulations involve a sparse length n = 2000 coefficient vector ? with 10
nonzero entries. Nonzero coefficients have uniformly random effect sizes. The entries of the design
matrix X are N (0, 0.1) Gaussian random deviates. We then recover ? from undersampled responses
6
0.09
0.07
Mean squared error
?0.2
0.0
0.03
?0.2
MM
MCP
SCAD
logistic
poisson
0.05
?0.6
?0.4
0.2
Relative Error, Logistic
Relative Error, Poisson
MM
MCP
SCAD
LASSO
600
Support indices of true coefficients
800
1000
1200
1400
1600
1800
Number of samples
Figure 2: The left figure displays relative errors among nonzero predictors in underdetermined
Poisson and logistic regression with m = 1000 cases. It is clear that LASSO suffers the most
shrinkage and bias, while MM appears to outperform MCP and SCAD. The right figure displays
MSE as a function of m, favoring MM most notably for logistic regression.
yj following Poisson and Bernoulli distributions with canonical links. Figure 2 compares solutions
obtained using our distance penalties (MM) to those obtained under MCP, SCAD, and LASSO
penalties. Relative errors (left) with m = 1000 cases clearly show that LASSO suffers the most
shrinkage and bias; MM seems to outperform MCP and SCAD. For a more detailed comparison, the
right side of the figure plots mean squared error (MSE) as a function of the number of cases averaged
over 50 trials. All methods significantly outperform LASSO, which is omitted for scale, with MM
achieving lower MSE than competitors, most noticeably in logistic regression. As suggested by an
anonymous reviewer, similar results from additional experiments for Gaussian (linear) regression
with comparison to relaxed lasso are included in the Supplement.
(a) Sparsity constraint
(b) Regularize kXk?
(c) Restrict rk(X) = 2 (d) Vary rk(X) = 1, . . . , 8
Figure 3: True B 0 in the top left of each set of 9 images has rank 2. The other 8 images in (a)?(c)
display solutions as varies over the set {0, 0.1, . . . , 0.7}. Figure (a) applies our MM algorithm
with sparsity rather than rank constraints to illustrate how failing to account for matrix structure
misses the true signal; Zhou and Li [37] report similar findings comparing spectral regularization to
`1 regularization. Figure (b) performs spectral shrinkage [37] and displays solutions under optimal ?
values via BIC, while (c) uses our MM algorithm restricting rank(B) = 2. Figure (d) fixes = 0.1
and uses MM with rank(B) ? {1, . . . , 8} to illustrate robustness to rank over-specification.
For underdetermined matrix regression, we compare to the spectral regularization method developed
by Zhou and Li [37]. We generate their cross-shaped 32 ? 32 true signal B0 and in all trials sample
m = 300 responses yi ? N [tr(X ti , B), ]. Here the design tensor X is generated with standard
normal entries. Figure 3 demonstrates that imposing sparsity alone fails to recover Y 0 and that
rank-set projections visibly outperform spectral norm shrinkage as varies. The rightmost panel also
shows that our method is robust to over-specification of the rank of the true signal to an extent.
We consider two real datasets. We apply our method to count data of global temperature anomalies
relative to the 1961-1990 average, collected by the Climate Research Unit [17]. We assume a non7
Global Temperature Anomalies
?0.6 ?0.2
0.2
0.6
50
100
150
200
1850
1900
1950
Year
2000
250
10
20
30
40
50
60
Figure 4: The leftmost plot shows our isotonic fit to temperature anomaly data [17]. The right figures
display the estimated coefficient matrix B on EEG alcoholism data using distance penalization,
nuclear norm shrinkage [37], and LASSO shrinkage, respectively.
decreasing solution, illustrating an instance of isotonic regression. The fitted solution displayed
in Figure 4 has mean squared error 0.009, clearly obeys the isotonic constraint, and is consistent
with that obtained on a previous version of the data [32]. We next focus on rank constrained matrix
regression for electroencephalography (EEG) data, collected by [35] to study the association between
alcoholism and voltage patterns over times and channels. The study consists of 77 individuals with
alcoholism and 45 controls, providing 122 binary responses yi indicating whether subject i has
alcoholism. The EEG measurements are contained in 256 ? 64 predictor matrices X i ; the dimension
m is thus greater than 16, 000. Further details about the data appear in the Supplement.
Previous studies apply dimension reduction [21] and propose algorithms to seek the optimal rank 1
solution [16]. These methods could not handle the size of the original data directly, and the spectral
shrinkage approach proposed in [37] is the first to consider the full EEG data. Figure 4 shows that
our regression solution is qualitatively similar to that obtained under nuclear norm penalization [37],
revealing similar time-varying patterns among channels 20-30 and 50-60. In contrast, ignoring matrix
structure and penalizing the `1 norm of B yields no useful information, consistent with findings in
[37]. However, our distance penalization approach achieves a lower misclassification error of 0.1475.
The lowest misclassification rate reported in previous analyses is 0.139 by [16]. As their approach is
strictly more restrictive than ours in seeking a rank 1 solution, we agree with [37] in concluding that
the lower misclassification error can be largely attributed to benefits from data preprocessing and
dimension reduction. While not visually distinguishable, we also note that shrinking the eigenvalues
via nuclear norm penalization [37] fails to produce a low-rank solution on this dataset.
We omit detailed timing comparisons throughout since the various methods were run across platforms
and depend heavily on implementation. We note that MCP regression relies on the MM principle,
and the LQA and LLA algorithms used to fit models with SCAD penalties are also instances of
MM algorithms [11]. Almost all MM algorithms share an overall linear rate of convergence. While
these require several seconds of compute time on a standard laptop machine, coordinate-descent
implementations of LASSO outstrip our algorithm in terms of computational speed. Our MM
algorithm required 31 seconds to converge on the EEG data, the largest example we considered.
5
Discussion
GLM regression is one of the most widely employed tools in statistics and machine learning. Imposing
constraints upon the solution is integral to parameter estimation in many settings. This paper considers
GLM regression under distance-to-set penalties when seeking a constrained solution. Such penalties
allow a flexible range of constraints, and are competitive with standard shrinkage methods for sparse
and low-rank regression in high dimensions. The MM principle yields a reliable solution method
with theoretical guarantees and strong empirical results over a number of practical examples. These
examples emphasize promising performance under non-convex constraints, and demonstrate how
distance penalization avoids the disadvantages of shrinkage approaches.
Several avenues for future work may be pursued. The primary computational bottleneck we face is
matrix inversion, which limits the algorithm when faced with extremely large and high-dimensional
datasets. Further improvements may be possible using modifications of the algorithm tailored to
8
specific problems, such as coordinate or block descent variants. Since the linear systems encountered
in our parameter updates are well conditioned, a conjugate gradient algorithm may be preferable to
direct methods of solution in such cases. The updates within our algorithm can be recast as weighted
least squares minimization, and a re-examination of this classical problem may suggest even better
iterative solvers. As the methods apply to a generalized objective comprised of multiple Bregman
divergences, it will be fruitful to study penalties under alternate measures of distance, and settings
beyond GLM regression such as quasi-likelihood estimation.
While our experiments primarily compare against shrinkage approaches, an anonymous referee points
us to recent work revisiting best subset selection using modern advances in mixed integer optimization
[3]. These exciting developments make best subset regression possible for much larger problems than
previously thought possible. As [3] focus on the linear case, it is of interest to consider how ideas in
this paper may offer extensions to GLMs, and to compare the performance of such generalizations.
Best subsets constitutes a gold standard for sparse estimation in the noiseless setting; whether it
outperforms shrinkage methods seems to depend on the noise level and is a topic of much recent
discussion [15, 23]. Finally, these studies as well as our present paper focus on estimation, and it
will be fruitful to examine variable selection properties in future work. Recent work evidences an
inevitable trade-off between false and true positives under LASSO shrinkage in the linear sparsity
regime [30]. The authors demonstrate that this need not be the case with `0 methods, remarking
that computationally efficient methods which also enjoy good model performance would be highly
desirable as `0 and `1 approaches possess one property but not the other [30]. Our results suggest
that distance penalties, together with the MM principle, seem to enjoy benefits from both worlds on a
number of statistical tasks.
Acknowledgements: We would like to thank Hua Zhou for helpful discussions about matrix
regression and the EEG data. JX was supported by NSF MSPRF #1606177.
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6,343 | 6,738 | Adaptive stimulus selection for optimizing neural
population responses
Benjamin R. Cowley1,2 , Ryan C. Williamson1,2,5 , Katerina Acar2,6 ,
Matthew A. Smith?,2,7 , Byron M. Yu?,2,3,4
1
Machine Learning Dept., 2 Center for Neural Basis of Cognition, 3 Dept. of Electrical
and Computer Engineering, 4 Dept. of Biomedical Engineering, Carnegie Mellon University
5
School of Medicine, 6 Dept. of Neuroscience, 7 Dept. of Ophthalmology, University of Pittsburgh
[email protected], {rcw30, kac216, smithma}@pitt.edu, [email protected]
?
denotes equal contribution.
Abstract
Adaptive stimulus selection methods in neuroscience have primarily focused on
maximizing the firing rate of a single recorded neuron. When recording from
a population of neurons, it is usually not possible to find a single stimulus that
maximizes the firing rates of all neurons. This motivates optimizing an objective
function that takes into account the responses of all recorded neurons together.
We propose ?Adept,? an adaptive stimulus selection method that can optimize
population objective functions. In simulations, we first confirmed that population
objective functions elicited more diverse stimulus responses than single-neuron
objective functions. Then, we tested Adept in a closed-loop electrophysiological
experiment in which population activity was recorded from macaque V4, a cortical
area known for mid-level visual processing. To predict neural responses, we used
the outputs of a deep convolutional neural network model as feature embeddings.
Images chosen by Adept elicited mean neural responses that were 20% larger
than those for randomly-chosen natural images, and also evoked a larger diversity
of neural responses. Such adaptive stimulus selection methods can facilitate
experiments that involve neurons far from the sensory periphery, for which it is
often unclear which stimuli to present.
1
Introduction
A key choice in a neurophysiological experiment is to determine which stimuli to present. Often, it
is unknown a priori which stimuli will drive a to-be-recorded neuron, especially in brain areas far
from the sensory periphery. Most studies either choose from a class of parameterized stimuli (e.g.,
sinusoidal gratings or pure tones) or present many randomized stimuli (e.g., white noise) to find the
stimulus that maximizes the response of a neuron (i.e., the preferred stimulus) [1, 2]. However, the
first approach limits the range of stimuli explored, and the second approach may not converge in a
finite amount of recording time [3]. To efficiently find a preferred stimulus, studies have employed
adaptive stimulus selection (also known as ?adaptive sampling? or ?optimal experimental design?)
to determine the next stimulus to show given the responses to previous stimuli in a closed-loop
experiment [4, 5]. Many adaptive methods have been developed to find the smallest number of
stimuli needed to fit parameters of a model that predicts the recorded neuron?s activity from the
stimulus [6, 7, 8, 9, 10, 11]. When no encoding model exists for a neuron (e.g., neurons in higher
visual cortical areas), adaptive methods rely on maximizing the neuron?s firing rate via genetic
algorithms [12, 13, 14] or gradient ascent [15, 16] to home in on the neuron?s preferred stimulus. To
our knowledge, all current adaptive stimulus selection methods focus solely on optimizing the firing
rate of a single neuron.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
A
B 100
V4 neuron 1
0
0 sorted image indices 1500
0
0
sorted image indices
V4 neuron 2
spikes/sec
100
1500
V4 neuron 2 (spikes/sec)
spikes/sec
100
0
0
V4 neuron 1 (spikes/sec)
100
Figure 1: Responses
of two macaque V4
neurons. A. Different neurons prefer different stimuli. Displayed images evoked
5 of top 25 largest
responses.
B. Images placed according to their responses.
Gray dots represent
responses to other images. Same neurons
as in A.
Developments in neural recording technologies now enable the simultaneous recordings of tens to
hundreds of neurons [17], each of which has its own preferred stimulus. For example, consider two
neurons recorded in V4, a mid-level visual cortical area (Fig. 1A). Whereas neuron 1 responds most
strongly to teddy bears, neuron 2 responds most strongly to arranged circular fruit. Both neurons
moderately respond to images of animals (Fig. 1B). Given that different neurons have different
preferred stimuli, how do we select which stimuli to present when simultaneously recording from
multiple neurons? This necessitates defining objective functions for adaptive stimulus selection that
are based on a population of neurons rather than any single neuron. Importantly, these objective
functions can go beyond simply maximizing the firing rates of neurons and instead can be optimized
for other attributes of the population response, such as maximizing the scatter of the responses in a
multi-neuronal response space (Fig. 1B).
We propose Adept, an adaptive stimulus selection method for a population of neurons that ?adeptly?
chooses the next stimulus to show based on a population objective function. Because the neural
responses to candidate stimuli are unknown, Adept utilizes feature embeddings of the stimuli to
predict to-be-recorded responses. In this work, we use the feature embeddings of a deep convolutional
neural network (CNN) for prediction. We first confirmed with simulations that Adept, using a
population objective, elicited larger mean responses and a larger diversity of responses than using
a single-neuron objective to optimize the response of each neuron in the population. Then, we ran
Adept on V4 population activity recorded during a closed-loop electrophysiological experiment.
Images chosen by Adept elicited higher mean firing rates and greater scatter of population responses
compared to randomly-chosen images. This demonstrates that Adept is effective at finding stimuli to
drive a population of neurons in brain areas far from the sensory periphery.
2
Population objective functions
Depending on the desired outcomes of an experiment, one may favor one objective function over
another. Here we discuss different objection functions for adaptive stimulus selection and the resulting
responses r ? Rp , where the ith element ri is the response of the ith neuron (i = 1, . . . , p) and p is the
number of neurons recorded simultaneously. To illustrate the effects of different objective functions,
we ran an adaptive stimulus selection method on the activity of two simulated neurons (see details
in Section 5.1). We first consider a single-neuron objective function employed by many adaptive
methods [12, 13, 14]. Using this objective function f (r) = ri , which maximizes the response for the
ith neuron of the population, the adaptive method for i = 1 chose stimuli that maximized neuron 1?s
response (Fig. 2A, red dots). However, images that produced large responses for neuron 2 were not
chosen (Fig. 2A, top left gray dots).
A natural population-level extension to this objective function is to maximize the responses of all
neurons by defining the objective function to be f (r) = krk2 . This objective function led to choosing
stimuli that maximized responses for neurons 1 and 2 individually, as well as large responses for
both neurons together (Fig. 2B). Another possible objective function is to maximize the scatter of the
responses. In particular, we would like to choose the next stimulus such that the response vector r is
far away from the previously-seen response vectors r1 , . . . , rM after M chosen stimuli. One way to
achieve this is to maximize the average Euclidean distance between r and r1 , . . . , rM , which leads
2
B 80
max r?
80
max r
C 80
2
max M1
M
j=1
unseen responses
responses to
chosen stimuli
neuron 2?s activity
A
0
0
neuron 1?s activity
80
0
0
neuron 1?s activity
80
0
0
r ? rj
neuron 1?s activity
D 80
2
80
0
max r
0
2
+
1
M
M
j=1
r ? rj
neuron 1?s activity
2
80
Figure 2: Different objective functions for adaptive stimulus selection yield different observed
population responses (red dots). Blue * denote responses to stimuli used to initialize the adaptive
method (the same for each panel).
PM
1
to the objective function f (r, r1 , . . . , rM ) = M
j=1 kr ? rj k2 . This objective function led to a
large scatter in responses for neurons 1 and 2 (Fig. 2C, red dots near and far from origin). This is
because choosing stimuli that yield small and large responses produces the largest distances between
responses.
Finally, we considered an objective function that favored large responses that are far away from
one another. To achieve this, we summed the objectives in Fig. 2B and 2C. The objective function
PM
1
f (r, r1 , . . . , rM ) = krk2 + M
j=1 kr ? rj k2 was able to uncover large responses for both neurons
(Fig. 2D, red dots far from origin). It also led to a larger scatter than maximizing the norm of r
alone (e.g., compare red dots in bottom right of Fig. 2B and Fig. 2D). For these reasons, we use this
objection function in the remainder of this work. However, the Adept framework is general and can
be used with many different objective functions, including all presented in this section.
3
Using feature embeddings to predict distances
We now formulate the optimization problem using the last objective function in Section 2. Consider
a pool of N candidate stimuli s1 , . . . , sN . After showing (t ? 1) stimuli, we are given previouslyrecorded response vectors rn1 , . . . , rnt?1 ? Rp , where n1 , . . . , nt?1 ? {1, . . . , N }. In other words,
rnj is the vector of responses to the stimulus snj . At the tth iteration of adaptive stimulus selection,
we choose the index nt of the next stimulus to show by the following:
t?1
1 X
nt =
arg max
krs k2 +
krs ? rnj k2
t ? 1 j=1
s?{1,...,N }\{n1 ,...,nt?1 }
(1)
where rs is the unseen population response vector to stimulus ss .
If the rs were known, we could directly optimize Eqn. 1. However, in an online setting, we do not
have access to the rs . Instead, we can directly predict the norm and average distance terms in Eqn. 1
by relating distances in neural response space to distances in a feature embedding space. The key
idea is that if two stimuli have similar feature embeddings, then the corresponding neural responses
will have similar norms and average distances. Concretely, consider feature embedding vectors
x1 , . . . , xN ? Rq corresponding to candidate stimuli s1 , . . . , sN . For example, we can use the
activity of q neurons from a CNN as a feature embedding vector for natural images [18]. To predict
the norm of unseen response vector rs ? Rq , we use kernel regression with the previously-recorded
response vectors rn1 , . . . , rnt?1 as training data [19]. To predict the distance between rs and a
previously-recorded response vector rnj , we extend kernel regression to account for the paired nature
of distances. Thus, the norm and average distance in Eqn. 1 for the unseen response vector rs to the
sth candidate stimulus are predicted by the following:
X K(xs , xn )
X K(xs , xn )
k
k
P
P
krs k2 =
krnk k2 ,
krs ? rnj k2 =
krnk ? rnj k2
K(xs , xn` )
K(xs , xn` )
`
`
k
k
(2)
where k, ` ? {1, . . . , t ? 1}. Here we use the radial basis function kernel K(xj , xk ) = exp(?kxj ?
xk k22 /h2 ) with kernel bandwidth h, although other kernels can be used.
V
V
We tested the performance of this approach versus three other possible prediction approaches. The
first two approaches use linear ridge regression and kernel regression, respectively, to predict rs . Their
3
prediction ?rs is then used to evaluate the objective in place of rs . The third approach is a linear ridge
regression version of Eqn. 2 to directly predict krs k2 and krs ? rnj k2 . To compare the performance
of these approaches, we developed a testbed in which we sampled two distinct populations of neurons
from the same CNN, and asked how well one population can predict the responses of the other
population using the different approaches described above. Formally, we let x1 , . . . , xN be feature
embedding vectors of q = 500 CNN neurons, and response vectors rn1 , . . . , rn800 be the responses of
p = 200 different CNN neurons to 800 natural images. CNN neurons were from the same GoogLeNet
CNN [18] (see CNN details in Results). To compute performance, we took the Pearson?s correlation ?
between the predicted and actual objective values on a held out set of responses not used for training.
We also tracked the computation time ? (computed on an Intel Xeon 2.3GHz CPU with 36GB RAM)
because these computations need to occur between stimulus presentations in an electrophysiological
experiment. The approach in Eqn. 2 performed the best (? = 0.64) and was the fastest (? = 0.2 s)
compared to the other prediction approaches (? = 0.39, 0.41, 0.23 and ? = 12.9 s, 1.5 s, 48.4 s,
for the three other approaches, respectively). The remarkably faster speed of Eqn. 2 over other
approaches comes from the evaluation of the objective function (fast matrix operations), the fact
that no training of linear regression weight vectors is needed, and the fact that distances are directly
predicted (unlike the approaches that first predict ?rs and then must re-compute distances between ?rs
and rn1 , . . . , rnt?1 for each candidate stimulus s). Due to its performance and fast computation time,
we use the prediction approach in Eqn. 2 for the remainder of this work.
4
Adept algorithm
We now combine the optimization problem in Eqn. 1 and prediction approach in Eqn. 2 to formulate
the Adept algorithm. We first discuss the adaptive stimulus selection paradigm (Fig. 3, left) and then
the Adept algorithm (Fig. 3, right).
For the adaptive stimulus selection paradigm (Fig. 3, left), the experimenter first selects a candidate
stimulus pool s1 , . . . , sN from which Adept chooses, where N is large. For a vision experiment,
the candidate stimulus pool could comprise natural images, textures, or sinusoidal gratings. For
an auditory experiment, the stimulus pool could comprise natural sounds or pure tones. Next,
feature embedding vectors x1 , . . . , xN ? Rq are computed for each candidate stimulus, and the
pre-computed N ? N kernel matrix K(xj , xk ) (i.e., similarity matrix) is input into Adept. For
visual neurons, the feature embeddings could come from a bank of Gabor-like filters with different
orientations and spatial frequencies [20], or from a more expressive model, such as CNN neurons in
a middle layer of a pre-trained CNN. Because Adept only takes as input the kernel matrix K(xj , xk )
and not the feature embeddings x1 , . . . , xN , one could alternatively use a similarity matrix computed
from psychophysical data to define the similarity between stimuli if no model exists. The previouslyrecorded response vectors rn1 , . . . , rnt?1 are also input into Adept, which then outputs the next
chosen stimulus snt to show. While the observer views snt , the response vector rnt is recorded and
appended to the previously-recorded response vectors. This procedure is iteratively repeated until the
end of the recording session. To show as many stimuli as possible, Adept does not choose the same
stimulus more than once.
For the Adept algorithm (Fig. 3, right), we initialize by randomly choosing a small number of stimuli
(e.g., Ninit = 5) from the large pool of N candidate stimuli and presenting them to the observer.
Using the responses to these stimuli R(:, 1:Ninit ), Adept then adaptively chooses a new stimulus
by finding the candidate stimulus that yields the largest objective (in this case, using the objective
defined by Eqns. 1 and 2). This search is carried out by evaluating the objective for every candidate
stimulus. There are three primary reasons why Adept is computationally fast enough to consider all
candidate stimuli. First, the kernel matrix KX is pre-computed, which is then easily indexed. Second,
the prediction of the norm and average distance is computed with fast matrix operations. Third, Adept
updates the distance matrix DR , which contains the pairwise distances between recorded response
vectors, instead of re-computing DR at each iteration.
5
Results
We tested Adept in two settings. First, we tested Adept on a surrogate for the brain?a pre-trained
CNN. This allowed us to perform comparisons between methods with a noiseless system. Second, in
a closed-loop electrophysiological experiment, we performed Adept on population activity recorded
in macaque V4. In both settings, we used the same candidate image pool of N ? 10,000 natural
4
candidate stimulus pool
s1 , . . . , s N
model
(e.g., CNN)
feature
embeddings
x1 , . . . , xN
compute similarity
K(xj , xk )
recorded
responses
Adept
rn1 , . . . , rnt?1
chosen
stimulus
s nt
response rnt
observer
(e.g., monkey)
Algorithm 1: Adept algorithm
Input: N candidate stimuli, feature embeddings X(q ? N ),
kernel bandwidth h (hyperparameter)
Initialization:
KX (j, k) = exp(?kX(:, j) ? X(:, k)k22 /h2 ) for all j, k
R(:, 1:Ninit ) ? responses to Ninit initial stimuli
DR (j, k) = kR(:, j) ? R(:, k)k2 for j, k = 1, . . . , Ninit
ind_obs ? indices of Ninit observed stimuli
Online algorithm:
for tth stimulus to show do
for sth candidate stimulus do
P
kX = KX (ind_obs, s)/ `?ind_obs KX (`, s)
% predict norm from recorded responses
?
norms(s) ? krs k2 = kX T diag( RT R)
% predict average distance from recorded responses
P
T
1
avgdists(s) ? t?1
` krs ? rn` k2 = mean(kX DR )
end
ind_obs(Ninit + t) ? argmax(norms + avgdists)
R(:, Ninit + t) ? recorded responses to chosen stimulus
update DR with kR(:, Ninit + t) ? R(:, `)k2 for all `
end
V
V
Figure 3: Flowchart of the adaptive sampling paradigm (left) and the Adept algorithm (right).
images from the McGill natural image dataset [21] and Google image search [22]. For the predictive
feature embeddings in both settings, we used responses from a pre-trained CNN different from the
CNN used as a surrogate for the brain in the first setting. The motivation to use CNNs was inspired
by the recent successes of CNNs to predict neural activity in V4 [23].
5.1
Testing Adept on CNN neurons
The testbed for Adept involved two different CNNs. One CNN is the surrogate for the brain. For
this CNN, we took responses of p = 200 neurons in a middle layer of the pre-trained ResNet CNN
[24] (layer 25 of 50, named ?res3dx?). A second CNN is used for feature embeddings to predict
responses of the first CNN. For this CNN, we took responses of q = 750 neurons in a middle layer of
the pre-trained GoogLeNet CNN [18] (layer 5 of 10, named ?icp4_out?). Both CNNs were trained for
image classification but had substantially different architectures. Pre-trained CNNs were downloaded
from MatConvNet [25], with the PVT version of GoogLeNet [26]. We ran Adept for 2,000 out of
the 10,000 candidate images (with Ninit = 5 and kernel bandwidth h = 200?similar results were
obtained for different h), and compared the CNN responses to those of 2,000 randomly-chosen
images. We asked two questions pertaining to the two terms in the objective function in Eqn. 1. First,
are responses larger for Adept than for randomly-chosen images? Second, to what extent does Adept
produce larger scatter of responses than if we had chosen images at random? A larger scatter implies
a greater diversity in evoked population responses (Fig. 1B).
To address the first question, we computed the mean response across all 2,000 images for each CNN
neuron. The mean responses using Adept were on average 15.5% larger than the mean responses to
randomly chosen images (Fig. 4A, difference in means was significantly greater than zero, p < 10?4 ).
For the second question, we assessed the amount of response scatter by computing the amount of
variance captured by each dimension. We applied PCA separately to the responses to images chosen
by Adept and those to images selected randomly. For each dimension, we computed the ratio between
the Adept eigenvalue divided by the randomly-chosen-image eigenvalue. In this way, we compared
the dimensions of greatest variance, followed by the dimensions of the second-most variance, and so
on. Ratios above 1 indicate that Adept explored a dimension more than the corresponding ordered
dimension of random selection. We found that Adept produced larger response scatter compared to
randomly-chosen images for many dimensions (Fig. 4B). Ratios for dimensions of lesser variance
(e.g., dimensions 10 to 75) are nearly as meaningful as those of the dimensions of greatest variance
5
?Adept ?random
1.0
0.8
1.0
equal to random selection
0
20
40
60
dimension index
75
D
1.4
equal to Adept
1.0
equal to Adept
corr predicted vs. actual
75
single multineuron neuron
2
2
?Adept
/? method
2
2
?Adept
/?random
mean response
0.8
dim
1.2
0.6
0.4
better prediction
0.4
0%
0
1.4
C
ist
vgd
pt-a
Ade t-norm
p
Ade
-50
etic 0
gen
pt-5
Adept-1
Ade
om
rand
0
%?
1.8
Adept
random
ist
vgd
pt-a
Ade t-norm
p
Ade
-50
etic 0
gen
pt-5
Adept-1
Ade
om
rand
0
-0.4
5%
2
?Adept / ?method
B
*
0.2
Adept better
fraction of CNN neurons
A
0.2
0.0
1 2 3 4 5 6 7 8 9 10
CNN layer index
Figure 4: CNN testbed for Adept. A. Mean responses (arbitrary units) to images chosen by Adept
were greater than to randomly-chosen images. B. Adept produced higher response variance for each
PC dimension than when randomly choosing images. Inset: Percent variance explained. C. Relative
to the full objective function in Eqn. 1, population objective functions (green) yielded higher response
mean and variance than those of single-neuron objective functions (blue). D. Feature embeddings for
all CNN layers were predictive. Error bars are ? s.d. across 10 runs.
(i.e., dimensions 1 to 10), as the top 10 dimensions explained only 16.8% of the total variance
(Fig. 4B, inset).
Next, we asked to what extent does optimizing a population objective function perform better than
optimizing a single-neuron objective function. For the single-neuron case, we implemented three
different methods. First, we ran Adept to optimize the response of a single CNN neuron with the
largest mean response (?Adept-1?). Second, we applied Adept in a sequential manner to optimize
the response of 50 randomly-chosen CNN neurons individually. After optimizing a CNN neuron
for 40 images, optimization switched to the next CNN neuron (?Adept-50?). Third, we sequentially
optimized 50 randomly-chosen CNN neurons individually using a genetic algorithm (?genetic-50?),
similar to the ones proposed in previous studies [12, 13, 14]. We found that Adept produced higher
mean responses than the three single-neuron methods (Fig. 4C, blue points in left panel), likely
because Adept chose images that evoked large responses across neurons together. All methods
produced higher mean responses than randomly choosing images (Fig. 4C, black point above blue
points in left panel). Adept also produced higher mean eigenvalue ratios across the top 75 PCA
dimensions than the three single-neuron methods (Fig. 4C, blue points in right panel). This indicates
that Adept, using a population objective, is better able to optimize population responses than using a
single-neuron objective to optimize the response of each neuron in the population.
We then modified the Adept objective function to include only the norm term (?Adept-norm?, Fig. 2B)
and only the average distance term (?Adept-avgdist?, Fig. 2C). Both of these population methods
performed better than single-neuron methods (Fig. 4C, green points below blue points). While their
performance was comparable to Adept using the full objective function, upon closer inspection,
we observed differences in performance that matched our intuition about the objective functions.
The mean response ratio for Adept using the full objection function and Adept-norm was close to
1 (Fig. 4C, left panel, Adept-norm on red-dashed line), but the eigenvalue ratio was greater than 1
(Fig. 4C, right panel, Adept-norm above red-dashed line, p < 0.005). Thus, Adept-norm maximizes
mean responses at the expense of less scatter. On the other hand, Adept-avgdist produced a lower
mean response than that of Adept using the full objective function (Fig. 4C, left panel, Adept-avgdist
above red-dashed line, p < 10?4 ), but an eigenvalue ratio of 1 (Fig. 4C, right panel, Adept-avgdist
on red-dashed line). Thus, Adept-avgdist increases the response scatter at the expense of a lower
mean response.
The results in this section were based on middle layer neurons in the GoogleNet CNN predicting
middle layer neurons in the ResNet CNN. However, it is possible that CNN neurons in other layers
may be better predictors than those in a middle layer. To test for this, we asked which layers of the
GoogLeNet CNN were most predictive of the objective values of the middle layer of the ResNet CNN.
For each layer of increasing depth, we computed the correlation between the predicted objective
(using 750 CNN neurons from that layer) and the actual objective of the ResNet responses (200
CNN neurons) (Fig. 4D). We found that all layers were predictive (? ? 0.6), although there was
variation across layers. Middle layers were slightly more predictive than deeper layers, likely because
6
deeper layers of GoogLeNet have a different embedding of natural images than the middle layer of
the ResNet CNN.
5.2
Testing Adept on V4 population recordings
Next, we tested Adept in a closed-loop neurophysiological experiment. We implanted a 96-electrode
array in macaque V4, whose neurons respond differently to a wide range of image features, including
orientation, spatial frequency, color, shape, texture, and curvature, among others [27]. Currently, no
existing parametric encoding model fully captures the stimulus-response relationship of V4 neurons.
The current state-of-the-art model for predicting the activity of V4 neurons uses the output of middle
layer neurons in a CNN previously trained without any information about the responses of V4 neurons
[23]. Thus, we used a pre-trained CNN (GoogLeNet) to obtain the predictive feature embeddings.
The experimental task flow proceeded as follows. On each trial, a monkey fixated on a central
dot while an image flashed four times in the aggregate receptive fields of the recorded V4 neurons.
After the fourth flash, the monkey made a saccade to a target dot (whose location was unrelated to
the shown image), for which he received a juice reward. During this task, we recorded threshold
crossings on each electrode (referred to as ?spikes?), where the threshold was defined as a multiple
of the RMS voltage set independently for each channel. This yielded 87 to 96 neural units in each
session. The spike counts for each neural unit were averaged across the four 100 ms flashes to
obtain mean responses. The mean response vector for the p neural units was then appended to the
previously-recorded responses and input into Adept. Adept then output an image to show on the
next trial. For the predictive feature embeddings, we used q = 500 CNN neurons in the fifth layer
of GoogLeNet CNN (kernel bandwidth h = 200). In each recording session, the monkey typically
performed 2,000 trials (i.e., 2,000 of the N =10,000 natural images would be sampled). Each Adept
run started with Ninit = 5 randomly-chosen images.
We first recorded a session in which we used Adept during one block of trials and randomly chose
images in another block of trials. To qualitatively compare Adept and randomly selecting images,
we first applied PCA to the response vectors of both blocks, and plotted the top two PCs (Fig. 5A,
left panel). Adept uncovers more responses that are far away from the origin (Fig. 5A, left panel,
red dots farther from black * than black dots). For visual clarity, we also computed kernel density
estimates for the Adept responses (pAdept ) and responses to randomly-chosen images (prandom ), and
plotted the difference pAdept ? prandom (Fig. 5A, right panel). Responses for Adept were denser than
for randomly-chosen images further from the origin, whereas the opposite was true closer to the
origin (Fig. 5A, right panel, red region further from origin than black region). These plots suggest
that Adept uncovers large responses that are far from one another. Quantitatively, we verified that
Adept chose images with larger objective values in Eqn. 1 than randomly-chosen images (Fig. 5B).
This result is not trivial because it relies on the ability of the CNN to predict V4 population responses.
If the CNN predicted V4 responses poorly, the objective evaluated on the V4 responses to images
chosen by Adept could be lower than that evaluated on random images.
We then compared Adept and random stimulus selection across 7 recording sessions, including the
above session (450 trials per block, with three sessions with the Adept block before the random
selection block, three sessions with the opposite ordering, and one session with interleaved trials).
We found that the images chosen by Adept produced on average 19.5% higher mean responses than
randomly-chosen images (Fig. 5C, difference in mean responses were significantly greater than zero,
p < 10?4 ). We also found that images chosen by Adept produced greater response scatter than for
randomly-chosen images, as the mean ratios of eigenvalues were greater than 1 (Fig. 5D, dimensions
1 to 5). Yet, there were dimensions for which the mean ratios of eigenvalues were less than 1 (Fig. 5D,
dimensions 9 and 10). These dimensions explained little overall variance (< 5% of the total response
variance).
Finally, we asked to what extent do the different CNN layers predict the objective of V4 responses,
as in Fig. 4D. We found that, using 500 CNN neurons for each layer, all layers had some predictive
ability (Fig. 5E, ? > 0). Deeper layers (5 to 10) tended to have better prediction than superficial
layers (1 to 4). To establish a noise level for the V4 responses, we also predicted the norm and
average distance for one session (day 1) with the V4 responses of another session (day 2), where
the same images were shown each day. In other words, we used the V4 responses of day 2 as
feature embeddings to predict V4 responses of day 1. The correlation of prediction was much higher
7
0
-200
-200
0
PC1 (spikes/sec)
*
D
2
2
?Adept
/?random
fraction of neural units
0.3
600
0
-40
-20
0
20
mean response (spikes/sec)
?Adept ?random
40
0
600
PC1 (spikes/sec)
40%
1.6
%?
1.4
2
0%
1
dim
1.2
15
1.0
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5
10
dimension index
pAdept
<
prandom
E
Adept
random
15
800
0.6
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1600
Adept
random
avgdist + norm
p Adept
=
prandom
0
C
pAdept B
>
prandom
400
Adept
random
corr predicted vs. actual
PC2 (spikes/sec)
A 400
0
trial number
1200
predict day 1 responses
with day 2 responses
predict day 1 responses
with CNN responses
0.2
0.0
1
2
3
4
5 6
7
8
9 10 responses
from
day 2
CNN layer index
Figure 5: Closed-loop experiments in V4. A. Top 2 PCs of V4 responses to stimuli chosen by Adept
and random selection (500 trials each). Left: scatter plot, where each dot represents the population
response to one stimulus. Right: difference of kernel densities, pAdept ? prandom . Black * denotes a
zero response for all neural units. B. Objective function evaluated across trials (one stimulus per trial)
using V4 responses. Same data as in A. C. Difference in mean responses across neural units from
7 sessions. D. Ratio of eigenvalues for different PC dimensions. Error bars: ? s.e.m. E. Ability of
different CNN layers to predict V4 responses. For comparison, we also used V4 responses from a
different day to predict the same V4 responses. Error bars: ? s.d. across 100 runs.
(? ? 0.5) than that of any CNN layer (? < 0.25). This discrepancy indicates that finding feature
embeddings that are more predictive of V4 responses is a way to improve Adept?s performance.
5.3
Testing Adept for robustness to neural noise and overfitting
A potential concern for an adaptive method is that stimulus responses are susceptible to neural noise.
Specifically, spike counts are subject to Poisson-like variability, which might not be entirely averaged
away based on a finite number of stimulus repeats. Moreover, adaptation to stimuli and changes
in attention or motivation may cause a gain factor to scale responses dynamically across a session
[9]. To examine how Adept performs in the presence of noise, we first recorded a ?ground-truth?,
spike-sorted dataset in which 2,000 natural images were presented (100 ms flashes, 5 to 30 repeats
per image randomly presented throughout the session). We then re-ran Adept on simulated responses
under three different noise models (whose parameters were fit to the ground truth data): a Poisson
model (?Poisson noise?), a model that scales each response by a gain factor that varies independently
from trial to trial [28] (?trial-to-trial gain?), and the same gain model but where the gain varies
smoothly across trials (?slowly-drifting gain?). Because the drift in gain was randomly generated and
may not match the actual drift in the recorded dataset, we also considered responses in which the drift
was estimated across the recording session and added to the mean responses as their corresponding
images were chosen (?recorded drift?). For reference, we also ran Adept on responses with no noise
(?no noise?). To compare performance across the different settings, we computed the mean response
and variance ratios between responses based on Adept and random selection (Fig. 6A). All settings
showed better performance using Adept than random selection (Fig. 6A, all points above red-dashed
line), and Adept performed best with no noise (Fig. 6, ?no noise? bar at or above others). For a fair
comparison, ratios were computed with the ground truth responses, where only the chosen images
could differ across settings. These results indicate that, although Adept would benefit from removing
neural noise, Adept continues to outperform random selection in the presence of noise.
Another concern for an adaptive method is overfitting. For example, when no relationship exists
between the CNN feature embeddings and neural responses, Adept may overfit to a spurious stimulus8
1.0
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random selection
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Figure 6: A. Adept
is robust to neural
noise. B. Adept
shows no overfitting when responses are shuffled across images.
Error bars: ? s.d.
across 10 runs.
response mapping and perform worse than random selection. To address this concern, we performed
two analyses using the same ground truth dataset as in Fig. 6A. For the first analysis, we ran Adept
on the ground truth responses (choosing 500 of the 2,000 candidate images) to yield on average a
6% larger mean response and a 21% larger response scatter (average over top 5 PCs) than random
selection (Fig 6B, unshuffled embeds.). Next, to break any stimulus-response relationship, we shuffled
all of the ground truth responses across images, and re-ran Adept. Adept performed no worse than
random selection (Fig 6B, shuffled embeds., blue points on red-dashed line). For the second analysis,
we asked if Adept focuses on the most predictable neurons to the detriment of other neurons. We
shuffled all of the ground truth responses across images for half of the neurons, and ran Adept on
the full population. Adept performed better than random selection for the subset of neurons with
unshuffled responses (Fig 6B, unshuffled subset), but no worse than random selection for the subset
with shuffled responses (Fig 6B, shuffled subset, green points on red-dashed line). Adept showed no
overfitting in either scenario, likely because Adept cannot choose exceedingly similar images (i.e.,
differing by a few pixels) from its discrete candidate pool.
6
Discussion
Here we proposed Adept, an adaptive method for selecting stimuli to optimize neural population
responses. To our knowledge, this is the first adaptive method to consider a population of neurons
together. We found that Adept, using a population objective, is better able to optimize population
responses than using a single-neuron objective to optimize the response of each neuron in the
population (Fig. 4C). While Adept can flexibly incorporate different feature embeddings, we take
advantage of the recent breakthroughs in deep learning and apply them to adaptive stimulus selection.
Adept does not try to predict the response of each V4 neuron, but rather uses the similarity of CNN
feature embeddings to different images to predict the similarity of the V4 population responses to
those images.
Widely studied neural phenomena such as changes in responses due to attention [29] and trial-to-trial
variability [30, 31] likely depend on mean response levels [32]. When recording from a single neuron,
one can optimize to produce large mean responses in a straightforward manner. For example, one can
optimize the orientation and spatial frequency of a sinusoidal grating to maximize a neuron?s firing
rate [9]. However, when recording from a population of neurons, identifying stimuli that optimize
the firing rate of each neuron can be infeasible due to limited recording time. Moreover, neurons far
from the sensory periphery tend to be more responsive to natural stimuli [33], and the search space
for natural stimuli is vast. Adept represents a principled way to efficiently search through a space of
natural stimuli to optimize the responses of a population of neurons. Experimenters can run Adept for
a recording session, and then present the Adept-chosen stimuli in subsequent sessions when probing
neural phenomena.
A future challenge for adaptive stimulus selection is to generate natural images rather than selecting
from a pre-existing pool of candidate images. For Adept, one could use a parametric model to
generate natural images, such as a generative adversarial network [34], and optimize Eqn. 1 with
gradient-based or Bayesian optimization.
9
Acknowledgments
B.R.C. was supported by a BrainHub Richard K. Mellon Fellowship. R.C.W. was supported by NIH
T32 GM008208, T90 DA022762, and the Richard K. Mellon Foundation. K.A. was supported by
NSF GRFP 1747452. M.A.S. and B.M.Y. were supported by NSF-NCS BCS-1734901/1734916.
M.A.S. was supported by NIH R01 EY022928 and NIH P30 EY008098. B.M.Y. was supported
by NSF-NCS BCS-1533672, NIH R01 HD071686, NIH R01 NS105318, and Simons Foundation
364994.
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6,344 | 6,739 | Nonbacktracking Bounds on the Influence in
Independent Cascade Models
1
Emmanuel Abbe1 2 Sanjeev Kulkarni2 Eun Jee Lee1
Program in Applied and Computational Mathematics 2 The Department of Electrical Engineering
Princeton University
{eabbe, kulkarni, ejlee}@princeton.edu
Abstract
This paper develops upper and lower bounds on the influence measure in a network,
more precisely, the expected number of nodes that a seed set can influence in the
independent cascade model. In particular, our bounds exploit nonbacktracking
walks, Fortuin?Kasteleyn?Ginibre (FKG) type inequalities, and are computed by
message passing algorithms. Nonbacktracking walks have recently allowed for
headways in community detection, and this paper shows that their use can also
impact the influence computation. Further, we provide parameterized versions
of the bounds that control the trade-off between the efficiency and the accuracy.
Finally, the tightness of the bounds is illustrated with simulations on various
network models.
1
Introduction
Influence propagation is concerned with the diffusion of information from initially influenced nodes,
called seeds, in a network. Understanding how information propagates in networks has become
a central problem in a broad range of fields, such as viral marketing [17], sociology [8, 19, 23],
communication [12], epidemiology [20], and social network analysis [24].
One of the most fundamental questions on influence propagation is to estimate the influence, i.e. the
expected number of influenced nodes at the end of the propagation given a set of seeds. Estimating
the influence is central to diverse research problems related to influence propagation, such as the
widely-known influence maximization problem ? finding a set of k nodes that maximizes the
influence.
Recent studies on influence propagation have proposed various algorithms [11, 18, 3, 7, 22, 21] for
the influence maximization problem while using Monte Carlo (MC) simulations to approximate the
influence. The submodularity argument and the probabilistic error bound on MC give a probabilistic
lower bound on the influence that is obtainable by the algorithms in terms of the true maximum
influence. Despite its benefits on the influence maximization problem, approximating the influence
via MC simulations is far from ideal for large networks; in particular, MC may require a large amount
of computations in order to stabilize the approximation.
To overcome the limitations of Monte Carlo simulations, many researchers have been taking theoretical approaches to approximate the influence of given seeds in a network. Draief et al., [5] introduced
an upper bound for the influence by using the spectral radius of the adjacency matrix. Tighter upper
bounds were later suggested in [16] which relate the ratio of influenced nodes in a network to the
spectral radius of the so-called Hazard matrix. Further, improved upper bounds which account for
sensitive edges were introduced in [15].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In contrast, there has been little work on finding a tight lower bound for the influence. An exception
is a work by Khim et al. [13], where the lower bound is obtained by only considering the influence
through the maximal-weighted paths.
In this paper, we propose both upper and lower bounds on the influence using nonbacktracking walks
and Fortuin?Kasteleyn?Ginibre (FKG) type inequalities. The bounds can be efficiently obtained by
message passing implementation. This shows that nonbacktracking walks can also impact influence
propagation, making another case for the use of nonbacktracking walks in graphical model problems
as in [14, 9, 2, 1], discussed later in the paper. Further, we provide a parametrized version of the
bounds that can adjust the trade-off between the efficiency and the accuracy of the bounds.
2
Background
We introduce here the independent cascade model and provide background for the main results.
Definition 1 (Independent Cascade Model). Consider a directed graph G = (V, E) with |V | = n, a
transmission probability matrix P ? [0, 1]n?n , and a seed set S0 ? V . For all u ? V , let N + (u)
be the set of out-neighbors of node u. The independent cascade model IC(G, P, S0 ) sequentially
generates the influenced set St ? V for each discrete time t ? 1 as follows. At time t, St is initialized
to be an empty set. Then, each node u ? St?1 attempts to influence v ? N + (u) \ ?t?1
i=0 Si with
probability Puv , i.e. node u influences its uninfluenced out-neighbor v with probability Puv . If v is
influenced at time t, add v to St . The process stops at T if ST = ? at the end of the step t = T . The
?1
set of the influenced nodes at the end of propagation is defined as S = ?Ti=0
St .
We often refer an edge (u, v) being open if node u influences node v. The IC model is equivalent
to the live-arc graph model, where the influence happens at once, rather than sequentially. The
live-arc graph model first decides the state of every edge with a Bernoulli trial, i.e. edge (u, v) is
open independently with probability Puv and closed, otherwise. Then, the set of influenced nodes is
defined as the nodes that are reachable from at least one of the seeds by the open edges.
Definition 2 (Influence). The expected number of nodes that are influenced at the end of the propagation process is called the influence (rather than the expected influence, with a slight abuse of
terminology) of IC(G, P, S0 ), and is defined as
X
?(S0 ) =
P(v is influenced).
(1)
v?V
It is shown in [4] that computing the influence ?(S0 ) in the independent cascade model IC(G, P, S0 )
is #P-hard, even with a single seed, i.e. |S0 | = 1.
Next, we define nonbacktracking (NB) walks on a directed graph. Nonbacktracking walks have
already been used for studying the characteristics of networks. To the best of our knowledge, the use
of NB walks in the context of epidemics was first introduced in the paper of Karrer et al. [10] and
later applied to percolation in [9]. In particular, Karrer et al. reformulate the spread of influence as a
message passing process and demonstrate how the resulting equations can be used to calculate an
upper bound on the number of nodes that are susceptible at a given time. As we shall see, we take a
different approach to the use of the NB walks, which focuses on the effective contribution of a node
in influencing another node and accumulates such contributions to obtain upper and lower bounds.
More recently, nonbacktracking walks are used for community detection [14, 2, 1].
Definition 3 (Nonbacktracking Walk). Let G = (V, E) be a directed graph. A nonbacktracking walk
of length k is defined as w(k) = (v0 , v1 , . . . , vk ), where vi ? V and (vi?1 , vi ) ? E for all i ? [k],
and vi?1 6= vi+1 for i ? [k ? 1].
We next recall a key inequality introduced by Fortuin et. al [6].
Theorem 1 (FKG Inequality). Let (?, ?) be a distributive lattice, where ? is a finite partially ordered
set, ordered by ?, and let ? be a positive measure on ? satisfying the following condition: for all
x, y ? ?,
?(x ? y)?(x ? y) ? ?(x)?(y),
where x ? y = max{z ? ? : z x, z y} and x ? y = min{z ? ? : y z, y z}. Let f and g
be both increasing (or both decreasing) functions on ?. Then,
X
X
X
X
(
?(x))(
f (x)g(x)?(x)) ? (
f (x)?(x))(
g(x)?(x)).
(2)
x??
x??
x??
2
x??
FKG inequality is instrumental in studying influence propagation since the probability that a node is
influenced is nondecreasing with respect to the partial order of random variables describing the states,
open or closed, of the edges.
3
Nonbacktracking bounds on the influence
In this section, we present upper and lower bounds on the influence in the independent cascade model
and explain the motivations and intuitions of the bounds. The bounds utilize nonbacktracking walks
and FKG inequalities and are computed efficiently by message passing algorithms. In particular,
the upper bound on a network based on a graph G(V, E) runs in O(|V |2 + |V ||E|) and the lower
bound runs in O(|V | + |E|), whereas Monte Carlo simulation would require O(|V |3 + |V |2 |E|)
computations without knowing the variance of the influence, which is harder to estimate than the
influence. The reason for the large computational complexity of MC is that in order to ensure
that the standard error of the estimation does not grow with respect to |V |, MC requires O(|V |2 )
computations. Hence, for large networks, where MC may not be feasible, our algorithms can still
provide bounds on the influence.
Furthermore, from the proposed upper ? + and lower bounds ? ? , we can compute an upper bound on
the variance given by (? + ? ? ? )2 /4. This could be used to estimate the number of computations
needed by MC. Computing the upper bound on the variance with the proposed bounds can be done in
O(|V |2 + |V ||E|), whereas computing the variance with MC simulation requires O(|V |5 + |V |4 |E|).
3.1
Nonbacktracking upper bounds (NB-UB)
We start by defining the following terms for the independent cascade model IC(G, P, S0 ), where
G = (V, E) and |V | = n.
Definition 4. For any v ? V , we define the set of in-neighbors N ? (v) = {u ? V : (u, v) ? E} and
the set of out-neighbors N + (v) = {u ? V : (v, u) ? E}.
Definition 5. For any v ? V and l ? [n ? 1], the set Pl (S0 ? v) is defined as the set of all paths
with length l from any seed s ? S0 to v. We call a path P is open iff every edge in P is open. For
l = 0, we define P0 (S0 ? v) as the set (of size one) of the zero-length path containing node v and
assume the path P ? P0 (S0 ? v) is open iff v ? S0 .
Definition 6. For any v ? V and l ? {0, . . . , n ? 1}, we define
p(v)
pl (v)
pl (u ? v)
= P(v is influenced)
= P(?P ?Pl (S0?v) {P is open})
(3)
(4)
= P(?P ?Pl (S0?u),P 6?v {P is open and edge (u, v) is open})
(5)
In other words, pl (v) is the probability that node v is influenced by open paths of length l, i.e. there
exists an open path of length l from a seed to v, and pl (u ? v) is the probability that v is influenced
by node u with open paths of length l + 1, i.e. there exists an open path of length l + 1 from a seed to
v that ends with edge (u, v).
Lemma 1. For any v ? V ,
n?1
Y
p(v) ? 1 ?
(1 ? pl (v)).
(6)
l=0
For any v ? V and l ? [n ? 1],
pl (v) ? 1 ?
Y
(1 ? pl?1 (u ? v)).
(7)
u?N ? (v)
Lemma 1, which can be proved by FKG inequalities, suggests that given pl?1 (u ? v), we may
compute an upper bound on the influence. Ideally, pl?1 (u ? v) can be computed by considering all
paths that end with (u, v) having length l. However, this results in exponential complexity O(nl ),
as l goes up to n ? 1. Thus, we present an efficient way to compute an upper bound UBl?1 (u ? v)
on pl?1 (u ? v), which in turns gives an upper bound UBl (v) on pl (v), with the following recursion
formula.
3
Definition 7. For all l ? {0, . . . , n?1} and u, v ? V such that (u, v) ? E, UBl (u) ? [0, 1] and
UBl (u ? v) ? [0, 1] are defined recursively as follows.
Initial condition: For every s ? S0 , s+? N + (s), u ? V \S0 , and v ? N + (u),
UB0 (s) = 1, UB0 (s ? s+ ) = Pss+
UB0 (u) = 0, UB0 (u ? v) = 0.
(8)
(9)
Recursion: For every l ? [n?1], s ? S0 , s+? N + (s), s?? N ? (s), u ? V \S0 , and v ? N + (u)\S0 ,
UBl (s) = 0, UBl (s ? s+ ) = 0, UBl (s? ? s) = 0
Y
UBl (u) = 1 ?
(1 ? UBl?1 (w ? u))
(10)
(11)
w?N ? (u)
(
UBl (u ? v) =
1?UBl (u)
), if v ? N ? (u)
Puv (1 ? 1?UB
l?1 (v?u)
Puv UBl (u),
otherwise.
(12)
Equation (10) follows from that for any seed node s ? S0 and for all l > 0, the probabilities
pl (s) = 0, pl (s ? s+ ) = 0, and pl (s? ? s) = 0. A naive way to compute UBl (u ? v) is
UBl (u ? v) = Puv UBl?1 (u), but this results in an extremely loose bound due to the backtracking.
For a tighter bound, we use nonbacktracking in Equation (12), i.e. when computing UBl (u ? v), we
ignore the contribution of UBl?1 (v ? u).
Theorem 2. For any independent cascade model IC(G, P, S0 ),
?(S0 ) ?
X
v?V
(1 ?
n?1
Y
(1 ? UBl (v))) =: ? + (S0 ),
(13)
l=0
where UBl (v) is obtained recursively as in Definition 7.
Next, we present Nonbacktracking Upper Bound (NB-UB) algorithm which computes UBl (v) and
UBl (u ? v) by message passing. At l-th iteration, the variables in NB-UB represent as follows.
? Sl : set of nodes that are processed at l-th iteration.
? Mcurr (v) = {(u, UBl?1 (u ? v)) : u is an in-neighbor of v, and u ? Sl?1 },
the set of pairs (previously processed in-neighbor u of v, incoming message from u to v).
? MSrc(v) = {u : u is a in-neighbor of v, and u ? Sl?1 },
the set of in-neighbor nodes of v that were processed at the previous step.
? Mcurr (v)[u] = UBl?1 (u ? v), the incoming message from u to v.
? Mnext (v) = {(u, UBl (u ? v)) : u is an in-neighbor of v, and u ? Sl }, the set of pairs (currently
processed in-neighbor u, next iteration?s incoming message from u to v).
Algorithm 1 Nonbacktracking Upper Bound (NB-UB)
Initialize: UBl (v) = 0 for all 0 ? l ? n ? 1 and v ? V
Initialize: Insert (s, 1) to Mnext (s) for all s ? S0
for l = 0 to n ? 1 do
for u ? Sl do
Mcurr (u) = Mnext (u)
Clear Mnext (u)
UBl (u) = ProcessIncomingMsgUB (Mcurr (u))
for v ? N + (u) \ S0 do
Sl+1 .insert(v)
if v ? MSrc(u) then
UBl (u ? v) = GenerateOutgoingMsgUB (Mcurr (u)[v], UBl (u), Puv )
Mnext (v).insert((u, UBl (u ? v))).
else
UBl (u ? v) = GenerateOutgoingMsgUB (0, UBl (u), Puv )
Mnext (v).insert((u, UBl (u ? v))).
Output: UBl (u) for all l, u
4
At the beginning, every seed node s ? S0 is initialized such that Mcurr (s) = {(s, 1)} in order to
satisfy the initial condition, UB0 (s) = 1. For each l-th iteration, every node u in Sl is processed as
follows. First, ProcessIncomingMsgUB (Mcurr (u)) computes UBl (u) as in Equation (11). Second, u
passes a message to its neighbor v ? N + (u) \ S0 along the edge (u, v), and v stores (inserts) the
message in Mnext (v) for the next iteration. The message contains 1) the source of the message, u, and
2) UBl (u ? v), which is computed as in Equation (12), by the function GenerateOutgoingMsgUB .
Finally, the algorithm outputs UBl (u) for all u ? V and l ? {0, . . . , n?1}, and the upper bound
? + (S0 ) is computed by Equation (13). The description of how the algorithm runs on a small network
can be found in the supplementary material.
Computational complexity: Notice that for each iteration l ? {0, . . . , n ? 1}, the algorithm
accesses at most n nodes, and for each node v, the functions ProcessIncomingMsgUB and
GenerateOutgoingMsgUB are computed in O(deg(v)) and O(1), respectively. Therefore, the worst
case computational complexity is O(|V |2 + |V ||E|).
3.2
Nonbacktracking lower bounds (NB-LB)
A naive way to compute a lower bound on the influence in a network IC(G, P, S0 ) is to reduce the
network to a (spanning) tree network, by removing edges. Then, since there is a unique path from
a node to another, we can compute the influence of the tree network, which is a lower bound on
the influence in the original network, in O(|V |). We take this approach of generating a subnetwork
from the original network, yet we avoid the significant gap between the bound and the influence by
considering the following directed acyclic subnetwork, in which there is no backtracking walk.
Definition 8 (Min-distance Directed Acyclic Subnetwork). Consider an independent cascade model
IC(G, P, S0 ) with G = (V, E) and |V | = n. Let d(S0 , v) := mins?S0 d(s, v), i.e. the minimum distance from a seed in S0 to v. A minimum-distance directed acyclic subnetwork (MDAS),
IC(G0 , P 0 , S0 ), where G0 = (V 0 , E 0 ), is obtained as follows.
? V 0 = {v1 , ..., vn } is an ordered set of nodes such that d(S0 , vi ) ? d(S0 , vj ), for every i < j.
? E 0 = {(vi , vj ) ? E : i < j}, i.e. remove edges from E whose source node comes later in the
order than its destination node to obtain E 0 .
? Pv0 i vj = Pvi vj , if (vi , vj ) ? E 0 , and Pv0 i vj = 0, otherwise.
If there are multiple ordered sets of vertices satisfying the condition, we may choose one arbitrarily.
For any k ? [n], let p(vk ) be the probability that vk ? V 0 is influenced in the MDAS, IC(G0 , P 0 , S0 ).
Since p(vk ) is equivalent to the probability of the union of the events that an in-neighbor ui ? N ? (vk )
influences vk , p(vk ) can be computed by the principle of inclusion and exclusion. Thus, we may
compute a lower bound on p(vk ), using Bonferroni inequalities, if we know the probabilities that
in-neighbors u and v both influences vk , for every pair u, v ? N ? (vk ). However, computing such
probabilities can take O(k k ). Hence, we present LB(vk ) which efficiently computes a lower bound
on p(vk ) by the following recursion.
Definition 9. For all vk ? V 0 , LB(vk ) ? [0, 1] is defined by the recursion on k as follows.
Initial condition: For every vs ? S0 ,
LB(vs ) = 1.
(14)
0
Recursion: For every vk ? V \ S0 ,
?
?
m?
i?1
X
X
?Pu0 v LB(ui )(1 ?
LB(vk ) =
Pu0 j vk )? ,
(15)
i k
i=1
j=1
?
where N (vk ) = {u1 , . . . , um } is the ordered set of in-neighbors of vk in IC(G0 , P 0 , S0 ) and
Pm0 ?1
m? = max{m0 ? m : j=1 Pu0 j vk ? 1}.
Pi?2
Remark. Since the i-th summand in Equation (15) can utilize j=1 Pu0 j vk , which is already
Pi?1 0
computed in (i?1)-th summand, to compute j=1 Puj vk , the summation takes at most O(deg(vk )).
Theorem 3. For any independent cascade model IC(G, P, S0 ) and its MDAS IC(G0 , P 0 , S0 ),
X
?(S0 ) ?
LB(vk ) =: ? ? (S0 ),
(16)
vk ?V 0
where LB(vk ) is obtained recursively as in Definition 9.
5
Next, we present Nonbacktracking Lower Bound (NB-LB) algorithm which efficiently computes
LB(vk ). At k-th iteration, the variables in NB-LB represent as follow.
? M(vk ) = {(LB(vj ), Pv0 j vk ) : vj is an in-neighbor of vk }, set of pairs (incoming message from
an in-neighbor vj to vk , the transmission probability of edge (vj , vk )).
Algorithm 2 Nonbacktracking Lower Bound (NB-LB)
Input: directed acyclic network IC(G0 , P 0 , S0 )
Initialize: ? ? = 0
Initialize: Insert (1, 1) to M(vi ) for all vi ? S0
for k = 1 to n do
LB(vk ) = ProcessIncomingMsgLB (M(vk ))
? ? += LB(vk )
for vl ? N + (vk ) \ S0 do
M(vl ).insert((LB(vk ), Pv0 k vl ))
Output: ? ?
At the beginning, every seed node s ? S0 is initialized such that M(s) = {(1, 1)} in order to satisfy
the initial condition, LB(s) = 1. For each k-th iteration, node vk is processed as follows. First,
LB(vk ) is computed as in the Equation (15), by the function ProcessIncomingMsgLB , and added
to ? ? . Second, vk passes the message (LB(vk ), Pv0 k vl ) to its out-neighbor vl ? N + (vk )\S0 , and vl
stores (inserts) it in M(vl ). Finally, the algorithm outputs ? ? , the lower bound on the influence. The
description of how the algorithm runs on a small network can be found in the supplementary material.
Computational complexity: Obtaining an arbitrary directed acyclic subnetwork from the original
network takes O(|V | + |E|). Next, the algorithm iterates through the nodes V 0 = {v1 , . . . , vn }. For
each node vk , ProcessIncomingMsgLB takes O(deg(vk )) and vk sends messages to its out-neighbors
in O(deg(vk )). Hence, the worst case computational complexity is O(|V | + |E|).
3.3
Tunable bounds
In this section, we briefly introduce the parametrized version of NB-UB and NB-LB which provide
control to adjust the trade-off between the efficiency and the accuracy of the bounds.
Upper bounds (tNB-UB): Given a non-negative integer t ? n ? 1, for every node u ? V , we
compute the probability p?t (u) that node u is influenced by open paths whose length is less than or
equal to t, and for each v ? N + (u), we compute the probability pt (u ? v). Then, we start NB-UB
from l = t + 1 with the new initial conditions that UBt (u ? v) = pt (u ? v) and UBt (u) = p?t (u),
P
Qn?1
and compute the upper bound as v?V (1 ? l=t (1 ? UBl (v))).
For higher values of t, the algorithm results in tighter upper bounds, while the computational
complexity may increase exponentially for dense networks. Thus, this method is most applicable in
sparse networks, where the degree of each node is bounded.
Lower bounds (tNB-LB): We first order the set of nodes {v1 , . . . , vn } such that d(S0 , vi ) ?
d(S0 , vj ) for every i < j. Given a non-negative integer t ? n, we obtain a subnetwork
IC(G[Vt ], P[Vt ], S0 ? Vt ) of size t, where G[Vt ] is the subgraph induced by the set of nodes
Vt = {v1 , . . . , vt }, and P[Vt ] is the corresponding transmission probability matrix. For each
vi ? Vt , we compute the exact probability pt (vi ) that node vi is influenced in the subnetwork
IC(G[Vt ], P[Vt ], S0 ? Vt ). Then, we start NB-LB from i = t + 1 with the new initial conditions that
LB(vk ) = pt (vk ), for all k ? t.
For larger t, the algorithm results in tighter lower bounds. However, the computational complexity
may increase exponentially with respect to t, the size of the subnetwork. This algorithm can
adopt Monte Carlo simulations on the subnetwork to avoid the large computational complexity.
However, this modification results in probabilistic lower bounds, rather than theoretically guaranteed
lower bounds. Nonetheless, this can still give a significant improvement, because the Monte Carlo
simulations on a smaller size of network require less computation to stabilize the estimation.
6
4
Experimental Results
In this section, we evaluate the NB-UB and NB-LB in independent cascade models on a variety of
classical synthetic networks.
Network Generation. We consider 4 classical random graph models with the parameters shown
as follows: Erdos Renyi random graphs with ER(n = 1000, p = 0.003), scale-free networks
SF (n = 1000, ? = 2.5), random regular graphs Reg(n = 1000, d = 3), and random tree graphs
with power-law degree distributions T (n = 1000, ? = 3). For each graph model, we generate 100
networks IC(G, pA, {s}) as follows. The graph G is the largest connected component of a graph
drawn from the graph model, the seed node s is a randomly selected vertex, and A is the adjacency
matrix of G. The corresponding IC model has the same transmission probability p for every edge.
Evaluation of Bounds. For each network generated, we compute the following quantities for
each p ? {0.1, 0.2, . . . , 0.9}.
? ?mc : the estimation of the influence with 106 Monte Carlo simulations.
? ? + : the upper bound obtained by NB-UB.
+
? ?spec
: the spectral upper bound by [16].
? ? ? : the lower bound obtained by NB-LB.
?
? ?prob
: the probabilistic lower bound obtained by 10 Monte Carlo simulations.
Figure 1: This figure compares the average relative gap of the bounds: NB-UB, the spectral upper bound in [16],
NB-LB, and the probabilistic lower bound computed by MC simulations, for various types of networks.
The probabilistic lower bound is chosen for the experiments since there has not been any tight lower
bound. The sample size of 10 is determined to overly match the computational complexity of NB-LB
algorithm. In Figure 1, we compare the average relative gap of the bounds for every network model
and for each transmission probability, where the true value is assumed to be ?mc . For example,
the average relative gap of NB-UB for 100 Erdos Renyi networks {Ni }100
i=1 with the transmission
P
? + [Ni ]??mc [Ni ]
1
+
probability p is computed by 100
,
where
?
[N
i ] and ?mc [Ni ] denote the
i?[100]
?mc [Ni ]
NB-UB and the MC estimation, respectively, for the network Ni .
Results. Figure 1 shows that NB-UB outperforms the upper bound in [16] for the Erdos-Renyi and
random 3-regular networks, and performs comparably for the scale-free networks. Also, NB-LB gives
tighter bounds than the MC bounds on the Erdos-Renyi, scale-free, and random regular networks
when the transmission probability is small, p < 0.4. The NB-UB and NB-LB compute the exact
influence for the tree networks since both algorithms avoid backtracking walks.
Next, we show the bounds on exemplary networks.
4.1
Upper Bounds
Selection of Networks. In order to illustrate a typical behavior of the bounds, we have chosen
the network in Figure 2a as follows. First, we generate 100 random 3-regular graphs G with 1000
nodes and assign a random seed s. Then, the corresponding IC model is defined as IC(G, P =
7
Upper bounds of the influence
100
100
180.92
90
90
900
80
80
2000
800
70
70
700
60
60
1500
600
50
50
Influence
Influence
Lower bounds of the influence
2500
1000
605.8
500
40
40
400
30
30
1000
300
20
20
500
200
10
10
100
0
00
0.05
0
0.1
0.1
0.2
0.15
0.2
0.3
0.4
0.25
0.5
0
0
0.3
0.5
0.35
0.6
0.4
0.7
0.45
0.8
0
0.5
0.9
0
1
0.1
0.2
MC (5)
MC (10)
MC (30)
MC (300)
MC (3000)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Transmission Probability
Transmission Probability
NB-UB
Spectral
MC (5)
(a)
MC (12)
MC (30)
MC (300)
MC (3000)
NB-LB
(b)
Figure 2: (a) The figure compares various upper bounds on the influence in the 3-regular network in section 4.1.
The MC upper bounds are computed with various simulation sizes and shown with the data points indicated
with MC(N ), where N is the number of simulations. The spectral upper bound in [16] is shown in red line, and
NB-UB is shown in green line.
(b) The figure shows lower bounds on the influence of a scale-free network in section 4.2. The probabilistic
lower bounds shown with points are obtained from Monte Carlo simulations with various simulation sizes, and
the data points indicated with MC(N ) are obtained by N number of simulations. NB-LB is shown in green line.
pA, S0 = {s}). For each network, we compute NB-UB and MC estimation. Then, we compute the
score for each network, where the score is defined as the sum of the square differences between the
upper bounds and MC estimations over the transmission probability p ? {0.1, 0.2,. . ., 0.9}. Finally,
a graph whose score is the median from all 100 scores is chosen for Figure 2a.
Results. In figure 2a, we compare 1) the upper bounds introduced [16] and 2) the probabilistic upper
bounds obtained by Monte Carlo simulations with 99% confidence level, to NB-UB. The MC upper
bounds are computed with the various sample sizes N ? {5, 10, 30, 300, 3000}. It is evident from
the figure that a larger sample size provides a tighter probabilistic upper bound. NB-UB outperforms
the bound by [16] and the probabilistic MC bound when the transmission probability is relatively
small. Further, it shows a similar trend as the MC simulations with a large sample size.
4.2
Lower Bounds
Selection of Networks. We adopt a similar selection process as in the selection for the upper
bounds, but with the scale free networks, with 3000 nodes and ? = 2.5.
Results. We compare probabilistic lower bounds obtained by MC with 99% confidence level
to NB-LB. The lower bounds from Monte Carlo simulations are computed with various sample
sizes N ? {5, 12, 30, 300, 3000}, which accounts for a constant, log(|V |), 0.01|V |, 0.1|V |, and
|V |. NB-LB outperforms the probabilistic bounds by MC with small sample sizes. Recall that
the computational complexity of the lower bound in algorithm 2 is O(|V | + |E|), which is the
computational complexity of a constant number of Monte Carlo simulations. In figure 2b, it shows
that NB-LB is tighter than the probabilistic lower bounds with the same computational complexity,
and it also agrees with the behavior of the MC simulations.
5
Conclusion
In this paper, we propose both upper and lower bounds on the influence in the independent cascade
models and provide algorithms to efficiently compute the bounds. We extend the results by proposing
tunable bounds which can adjust the trade-off between the efficiency and the accuracy. Finally,
the tightness and the performance of bounds are shown with experimental results. One can further
improve the bounds considering r-nonbacktracking walks, i.e. avoiding cycles of length r rather than
just backtracks, and we leave this for future study.
Acknowledgement. The authors thank Colin Sandon for helpful discussions. This research was partly supported by the NSF CAREER Award CCF-1552131 and the ARO grant
W911NF-16-1-0051
8
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6,345 | 674 | Directional-Unit Boltzmann Machines
Richard S. Zemel
Computer Science Dept.
University of Toronto
Toronto, ONT M5S lA4
Christopher K. I. Williams
Computer Science Dept.
University of Toronto
Toronto, ONT M5S lA4
Michael C. Mozer
Computer Science Dept.
University of Colorado
Boulder, CO 80309-0430
Abstract
We present a general formulation for a network of stochastic directional units. This formulation is an extension of the Boltzmann
machine in which the units are not binary, but take on values in a
cyclic range, between 0 and 271' radians. The state of each unit in
a Directional-Unit Boltzmann Machine (DUBM) is described by a
complex variable, where the phase component specifies a direction;
the weights are also complex variables. We associate a quadratic
energy function, and corresponding probability, with each DUBM
configuration. The conditional distribution of a unit's stochastic
state is a circular version of the Gaussian probability distribution,
known as the von Mises distribution. In a mean-field approximation to a stochastic DUBM, the phase component of a unit's state
represents its mean direction, and the magnitude component specifies the degree of certainty associated with this direction. This
combination of a value and a certainty provides additional representational power in a unit. We describe a learning algorithm and
simulations that demonstrate a mean-field DUBM'S ability to learn
interesting mappings.
Many kinds of information can naturally be represented in terms of angular, or
directional, variables. A circular range forms a suitable representation for explicitly
directional information, such as wind direction, as well as for information where
the underlying range is periodic, such as days of the week or months of the year.
In computer vision, tangent fields and optic flow fields are represented as fields of
oriented line segments, each of which can be described by a magnitude and direction.
Directions can also be used to represent a set of symbolic labels, e.g., object label
A at 0, and object label B at 71'/2 radians. We discuss below some advantages of
representing symbolic labels with directional units.
172
Directional-Unit Boltzmann Machines
These and many other phenomena can be usefully encoded using a directional
representation-a polar coordinate representation of complex values in which the
phase parameter indicates a direction between 0 and 27r radians. We have devised a
general formulation of networks of stochastic directional units. This paper describes
a directional-unit Boltzmann machine (DUBM), which is a novel generalization of a
Boltzmann machine (Ackley, Hinton and Sejnowski, 1985) in which the units are
not binary, but instead take on directional values between 0 and 27r.
1
STOCHASTIC DUBM
A stochastic directional unit takes on values on the unit circle. We associate with
unit j a random variable Zj; a particular state of j is described by a complex
number with magnitude one and direction, or phase Tj: Zj = eiTj ?
The weights of a DUBM also take on complex values. The weight from unit k to
unit j is: Wj k = hj ke ifJ'k . We constrain the weight matrix W to be Hermitian :
W T = W*, where the diagonal elements of the matrix are zero, and the asterisk
indicates the complex conjugate operation. Note that if the components are real,
then W T = W, which is a real symmetric matrix. Thus, the Hermitian form is a
natural generalization of weight symmetry to the complex domain.
This definition of W leads to a Hermitian quadratic form that generalizes the real
quadratic form of the Hopfield energy function:
E(z)
=
-1/2 z*TWz = -1/2 LZjZZWjk
j,k
(1)
where z is the vector of the units' complex states in a particular global configuration.
Noest (1988) independently proposed this energy function. It is similar to that
used in Fradkin, Huberman, and Shenker's (1978) generalization of the XY model
of statistical mechanics to allow arbitary weight phases OJ k, and coupled oscillator
models, e.g., Baldi and Meir (1990).
We can define a probability distribution over the possible states of a stochastic
network using the Boltzmann factor. In a DUBM, we can describe the energy as a
function of the state of a particular unit j:
E(Zj
= Zj) =
-1/2 [L ZjZZWjk
+L
k
ZkZIWkj]
k
We define
Xj = LZkwlk
k
to be the net input to unit j, where
x j, respectively.
aj
and
O:j
denote the magnitude and phase of
Applying the Boltzmann factor, we find that the probability that unit j
particular state is proportional to:
p(Zj
= Zj)
ex
e-/3E(Zj=zj)
= e/3 a j
where f3 is the reciprocal of the system temperature.
COS(Tj-crj)
IS 10
a
(2)
173
174
Zemel, Williams, and Mozer
0?
--->
Figure 1: A circular normal density function laid over a unit circle. The dots
along the circle represent samples of the circular normal random variable Zj. The
expected direction of Zj, Tj, is 7r /4; rj is its resultant length.
This probability distribution for a unit's state corresponds to a distribution known
as the von Mises, or circular normal, distribution (Mardia, 1972). Two parameters
completely characterize this distribution: a mean direction r = (0,27r] and a concentration parameter m > 0 that behaves like the reciprocal of the variance of a
Gaussian distribution on a linear random variable. The probability density function
of a circular normal random variable Z is l :
p (T;
r, m ) =
1
() em cos( T-T)
27r1o m
(3)
From Equations 2 and 3, we see that if a unit adopts states according to its contribution to the system energy, it will be a circular normal variable with mean direction
Cl:j and concentration parameter mj = f3aj. These parameters are directly determined by the net input to the unit.
Figure 1 shows a circular normal density function for Zj, the state of unit j. This
figure also shows the expected value of its stochastic state, which we define as:
Yj
= < Zj > =
(4)
rjei'yi
where Ij, the phase of Yj, is the mean direction and rj, the magnitude of Yj, is the
resultant length. For a circular normal random variable, Ij = Tj, and rj = ~~~:!j.2
When samples of Zj are concentrated on a small arc about the mean (see Figure 1),
rj will approach length one. This corresponds to a large concentration parameter
(mj
f3aj). Conversely, for small mj, the distribution approaches the uniform
distribution on the circle, and the resultant length falls toward zero. For a uniform
distribution, rj = O. Note that the concentration parameter for a unit's circular
=
IThe normalization factor Io(m) is the modified Bessel function of the first kind and
order zero. An integral representation of this function is Io(m) = ~
e?mcos()d6. It can
be computed by numerical routines.
2 An integral representation of the modified Bessel function of the first kind and order
k is h(m) = ~ o7r emcos() cos(k6)d6. Note that II(m) = dlo(m)/dm.
J:
J
Directional-Unit Boltzmann Machines
normal density function is proportional to /3, the reciprocal of the system temperature. Higher temperatures will thus have the effect of making this distribution more
uniform, just as they do in a binary-unit Boltzmann machine.
2
EMERGENT PROPERTIES OF A DUBM
A network of directional units as defined above contains two important emergent
properties. The first property is that the magnitude of the net input to unit j
describes the extent to which its various inputs "agree". Intuitively, one can think
of each component Zk wj k of the sum that comprises x j as predicting a phase for
unit j. When the phases of these components are equal, the magnitude of Xj, aj, is
maximized. If these phase predictions are far apart, then they will act to cancel each
other out, and produce a small aj. Given Xj, we can compute the expected value
of the output of unit j. The expected direction of the unit roughly represents the
weighted average of the phase predictions, while the resultant length is a monotonic
function of aj and hence describes the agreement between the various predictions.
The key idea here is that the resultant length directly describes the degree of certainty in the expected direction of unit j. Thus, a DUBM naturally incorporates a
representation of the system's confidence in a value. This ability to combine several
sources of evidence, and not only represent a value but also describe the certainty
of that value is an important property that may be useful in a variety of domains.
The second emergent property is that the DUBM energy is globally rotationinvariant-E is unaffected when the same rotation is applied to all units' states
in the network. For each DUBM configuration, there is an equivalence class of configurations which have the same energy. In a similar way, we find that the magnitude
of Xj is rotation-invariant. That is, when we translate the phases of all units but
one by some phase, the magnitude of that unit is unaffected. This property underlies one of the key advantages of the representation: both the magnitude of a unit's
state as well as system energy depend on the relative rather than absolute phases
of the units.
3
DETERMINISTIC DUBM
Just as in deterministic binary-unit Boltzmann machines (Peterson and Anderson,
1987; Hinton, 1989), we can greatly reduce the computational time required to
run a large stochastic system if we invoke the mean-field approximation, which
states that once the system has reached equilibrium, the stochastic variables can be
approximated by their mean values. In this approximation, the variables are treated
as independent, and the system probability distribution is simply the product of
the probability distributions for the individual units.
Gislen, Peterson, and Soderberg (1992) originally proposed a mean-field theory for
networks of directional (or "rotor") units, but only considered the case of realvalued weights. They derived the mean-field consistency equations by using the
saddle-point method. Our approach provides an alternative, perhaps more intuitive
derivation, due to the use of the circular normal distribution.
175
176
Zemel, Williams, and Mozer
We can directly describe these mean values based on the circular normal interpretation. We still denote the net input to a unit j as Xj:
xj
=
~
*
~ Yk W j k
=
iao
aj e
(5)
1
k
Once equilibrium has been reached, the state of unit j is
Zj given the mean-field approximation:
Yj,
the expected value of
(6)
In the stochastic as well as the deterministic system, units evolve to minimize the
free energy, F = < E > - T H. The calculation of H, the entropy of the system,
follows directly from the circular normal distribution and the mean-field approximation. We can derive mean-field consistency equations for Xj and Yj by minimizing
the mean-field free energy, FM F, with respect to each variable independently. The
resulting equations match the mean-field equations (Equations 5 and 6) derived
directly from the circular normal probability density function. They also match the
special case derived by Gislen et al. for real-valued weights.
We have implemented a DUBM using the mean-field approximation. We solve for a
consistent set of x and y values by performing synchronous updates of the discretetime approximation of the set of differential equations based on the net input to
each unit j. We update the x j variables using the following differential equation:
dXj
--;It
*
= -Xj + ~
~ YkWjk
(7)
k
which has Equation 5 as its steady-state solution. In the simulations, we use simulated annealing to help find good minima of FM F.
Just as for the Hopfield binary-state network, it can be shown that the free energy
always decreases during the dynamical evolution described in Equation 7 (Zemel,
Williams and Mozer, 1992). The equilibrium solutions are free energy minima.
4
DUBM LEARNING
The units in a DUBM can be arranged in a variety of architectures. The appropriate
method for determining weight values for the network depends on the particular
class of network architecture. In an autoassociative network containing a single set
of interconnected units, the weights can be set directly from the training patterns.
If hidden units are required to perform a task, then an algorithm for learning the
weights is required. We use an algorithm that generalizes the Boltzmann machine
training algorithm (Ackley, Hinton and Sejnowski, 1985; Peterson and Anderson,
1987) to these networks.
As in the standard Boltzmann machine learning algorithm, the partial derivative of
the objective function with respect to a weight depends on the difference between
the partials of two mean-field free energies: one when both input and output units
are clamped, and the other when only the input units are clamped. On a given
Directional-Unit Boltzmann Machines
training case, for each of these stages we let the network settle to equilibrium and
then calculate the following derivatives:
OFMF/objk
OFM F / O(}j k
+ (}jk)
'Yk + (}jk)
-rjTk COS(-yj - 'Yk
rjrkbjk
sin('Yj -
The learning algorithm uses these gradients to find weight values that will minimize
the objective over a training set.
5
EXPERIMENTAL RESULTS
We present below some illustrative examples to show that an adaptive network of
directional units can be used in a range of paradigms, including associative memory,
input/output mappings, and pattern completion.
5.1
SIMPLE AUTOASSOCIATIVE DUBM
The first set of experiments considers a simple autoassociative DUBM, which contains
no hidden units, and the units are fully connected. As in a standard Hopfield
network, the weights are set directly from the training patterns; they equal the
superposition of the outer product of the patterns.
We have run several experiments with simple autoassociative DUBMs. The empirical
results parallel those for binary-unit autoassociative networks. We find, for example,
that a network containing 30 fully interconnected units is capable of reliably settling
from a corrupted version of one of 4 stored patterns to a state near the pattern.
These patterns thus form stable attractors, as the network can perform pattern
completion and clean-up from noisy inputs. The rotation-invariance property of the
energy function allows any rotated version of a training pattern to also act as an
attractor. The network's performance rapidly degrades for more than 4 orthogonal
patterns; the patterns themselves no longer act as fixed-points, and many random
initial states end in states far from any stored pattern. In addition, more orthogonal
patterns can be stored than random patterns. See Noest (1988) for an analysis of
the capacity of an autoassociative DUBM with sparse and asymmetric connections.
5.2
LEARNING INPUT/OUTPUT MAPPINGS
We have also used the mean-field DUBM learning algorithm to learn the weights in
networks containing hidden units. We have experimented with a task that is wellsuited to a directional representation. There is a single-jointed robot arm, anchored
at a point, as shown in Figure 2. The input consists of two angles: the angle
between the first arm segment and the positive x-axis (A), and the angle between
the two arm segments (p). The two segments each have a fixed length, A and B;
these are not explicitly given to the network. The output is the angle between the
line connecting the two ends of the arm and the x-axis (J.t). This target angle is
related in a complex, non-linear way to the input angles-the network must learn
to approximate the following trigonometric relationship:
J.1.
A sin A - B sin( A + p) )
= arctan (
A cos A - B cos( A + p)
177
178
Zemel, Williams, and Mozer
/
I
I
,\ ~ p,
----------- -j~----------~
Figure 2: A sample training case for the robot arm problem. The arm consists of
two fixed-length segments, A and B, and is anchored on the x-axis. The two angles,
,\ and p, are given as input for each case, and the target output is the angle p,.
With 500 training cases, a DUBM with 2 input units and 8 hidden units is able to
learn the task so that it can accurately estimate p, for novel patterns. The learning
requires 200 iterations of a conjugate gradient training algorithm. On each of 100
testing patterns, the resultant length of the output unit exceeds .85, and the mean
error on the angle is less than .05 radians. The network can also learn the task
with as few as 5 hidden units, with a concomitant decrease in learning speed. The
compact nature of this network shows that the directional units form a natural,
efficient representation for this problem.
5.3
COMPLEX PATTERN COMPLETION
Our earlier work described a large-scale DUBM that attacks a difficult problem in
computer vision: image segmentation. In MAGIC (Mozer et al., 1992), directional
values are used to represent alternative labels that can be assigned to image features.
The goal of MAGIC is to learn to assign appropriate object labels to a set of image
features (e.g., edge segments) based on a set of examples. The idea is that the
features of a given object should have consistent phases, with each object taking on
its own phase. The units in the network are arranged into two layers-feature and
hidden-and the computation proceeds by randomly initializing the phases of the
units in the feature layer, and settling on a labeling through a relaxation procedure.
The units in the hidden layer learn to detect spatially local configurations of the
image features that are labeled in a consistent manner across the training examples.
successfully learns to segment novel scenes consisting of overlapping geometric objects. The emergent DUBM properties described above are essential to MAGIC'S
ability to perform this task. The complex weights are necessary in MAGIC, as the
weights encode statistical regularities in the relationships between image features,
e.g., that two features typically belong to the same object (i.e., have similar phase
values) or to different objects (i.e., are out of phase). The fact that a unit's resultant length reflects the certainty in a phase label allows the system to decide
which phase labels to use when updating labels of neighboring features: the initially random phases are ignored, while confident labels are propagated. Finally,
the rotation-invariance property allows the system to assign labels to features in a
manner consistent with the relationships described in the weights, where it is the
relative rather than absolute phases of the units that are important.
MAGIC
Directional-Unit Boltzmann Machines
6
CURRENT DIRECTIONS
We are currently extending this work in a number of directions. We are extending
the definition of a DUBM to combine binary and directional units (Radford Neal,
personal communication). This expanded representation may be useful in domains
with directional data that is not present everywhere. For example, it can be directly
applied to the object labeling problem explored in MAGIC. The binary aspect of
the unit can describe whether a particular image feature is present or absent. This
may enable the system to handle various complications, particularly labeling across
gaps along the contour of an object. Finally, we are applying a DUBM network to
the interesting and challenging problem of time-series prediction of wind directions.
Acknowledgements
The authors thank Geoffrey Hinton for his generous support and guidance. We
thank Radford Neal, Peter Dayan, Conrad Galland, Sue Becker, Steve Nowlan, and
other members of the Connectionist Research Group at the University of Toronto
for helpful comments regarding this work. This research was supported by a grant
from the Information Technology Research Centre of Ontario to Geoffrey Hinton,
and NSF Presidential Young Investigator award IRI-9058450 and grant 90-21 from
the James S. McDonnell Foundation to MM.
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6,346 | 6,740 | Learning with Feature Evolvable Streams
Bo-Jian Hou Lijun Zhang Zhi-Hua Zhou
National Key Laboratory for Novel Software Technology,
Nanjing University, Nanjing, 210023, China
{houbj,zhanglj,zhouzh}@lamda.nju.edu.cn
Abstract
Learning with streaming data has attracted much attention during the past few years.
Though most studies consider data stream with fixed features, in real practice the
features may be evolvable. For example, features of data gathered by limitedlifespan sensors will change when these sensors are substituted by new ones. In
this paper, we propose a novel learning paradigm: Feature Evolvable Streaming
Learning where old features would vanish and new features would occur. Rather
than relying on only the current features, we attempt to recover the vanished
features and exploit it to improve performance. Specifically, we learn two models
from the recovered features and the current features, respectively. To benefit from
the recovered features, we develop two ensemble methods. In the first method,
we combine the predictions from two models and theoretically show that with the
assistance of old features, the performance on new features can be improved. In the
second approach, we dynamically select the best single prediction and establish a
better performance guarantee when the best model switches. Experiments on both
synthetic and real data validate the effectiveness of our proposal.
1
Introduction
In many real tasks, data are accumulated over time, and thus, learning with streaming data has attracted
much attention during the past few years. Many effective approaches have been developed, such
as hoeffding tree [7], Bayes tree [27], evolving granular neural network (eGNN) [17], Core Vector
Machine (CVM) [29], etc. Though these approaches are effective for certain scenarios, they have a
common assumption, i.e., the data stream comes with a fixed stable feature space. In other words,
the data samples are always described by the same set of features. Unfortunately, this assumption
does not hold in many streaming tasks. For example, for ecosystem protection one can deploy many
sensors in a reserve to collect data, where each sensor corresponds to an attribute/feature. Due to its
limited-lifespan, after some periods many sensors will wear out, whereas some new sensors can be
spread. Thus, features corresponding to the old sensors vanish while features corresponding to the
new sensors appear, and the learning algorithm needs to work well under such evolving environment.
Note that the ability of adapting to environmental change is one of the fundamental requirements for
learnware [37], where an important aspect is the ability of handling evolvable features.
A straightforward approach is to rely on the new features and learn a new model to use. However,
this solution suffers from some deficiencies. First, when new features just emerge, there are few data
samples described by these features, and thus, the training samples might be insufficient to train a
strong model. Second, the old model of vanished features is ignored, which is a big waste of our data
collection effort. To address these limitations, in this paper we propose a novel learning paradigm:
Feature Evolvable Streaming Learning (FESL). We formulate the problem based on a key observation:
in general features do not change in an arbitrary way; instead, there are some overlapping periods in
which both old and new features are available. Back to the ecosystem protection example, since the
lifespan of sensors is known to us, e.g., how long their battery will run out is a prior knowledge, we
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
usually spread a set of new sensors before the old ones wear out. Thus, the data stream arrives in a
way as shown in Figure 1, where in period T1 , the original set of features are valid and at the end of
T1 , period B1 appears, where the original set of features are still accessible, but some new features
are included; then in T2 , the original set of features vanish, only the new features are valid but at the
end of T2 , period B2 appears where newer features come. This process will repeat again and again.
Note that the T1 and T2 periods are usually long, whereas the B1 and B2 periods are short because,
as in the ecosystem protection example, the B1 and B2 periods are just used to switch the sensors
and we do not want to waste a lot of lifetime of sensors for such overlapping periods.
Data Streaming
Feature Evolution
In this paper, we propose to solve the FESL
Feature Set
problem by utilizing the overlapping period to
?1
?2
?3
?
discover the relationship between the old and
data with
new features, and exploiting the old model even
feature set ?1
?1
when only the new features are available. Specif?1
ically, we try to learn a mapping from new feadata with feature set ?1 and ?2
tures to old features through the samples in the
data with
feature set ?2
overlapping period. In this way, we are able to
?2
data with feature set ?2 and ?3
?2
reconstruct old features from new ones and thus
the old model can still be applied. To benefit
from additional features, we develop two ensemFigure 1: Illustration that how data stream comes.
ble methods, one is in a combination manner
and the other in a dynamic selection manner. In
the first method, we combine the predictions from two models and theoretically show that with the
assistance of old features, the performance on new features can be improved. In the second approach,
we dynamically select the best single prediction and establish a better performance guarantee when
the best model switches at an arbitrary time. Experiments on synthetic and real datasets validate the
effectiveness of our proposal.
The rest of this paper is organized as follows. Section 2 introduces related work. Section 3 presents
the formulation of FESL. Our proposed approaches with corresponding analyses are presented in
section 4. Section 5 reports experimental results. Finally, Section 6 concludes.
2
Related Work
Data stream mining contains several tasks, including classification, clustering, frequency counting,
and time series analysis. Our work is most related to the classification task and we can also solve
the regression problem. Existing techniques for data stream classification can be divided into two
categories, one only considers a single classifier and the other considers ensemble classifiers. For the
former, several methods origin from approaches such as decision tree [7], Bayesian classification [27],
neural networks [17], support vector machines [29], and k-nearest neighbour [1]. For the latter,
various ensemble methods have been proposed including Online Bagging & Boosting [22], Weighted
Ensemble Classifiers [30, 20], Adapted One-vs-All Decision Trees (OVA) [12] and Meta-knowledge
Ensemble [33]. For more details, please refer to [9, 10, 2, 6, 21]. These traditional streaming data
algorithms often assume that the data samples are described by the same set of features, while in
many real streaming tasks feature often changes. We want to emphasize that though concept-drift
happens in streaming data where the underlying data distribution changes over time [2, 10, 4], the
number of features in concept-drift never changes which is different from our problem. Most studies
correlated to features changing are focusing on feature selection and extraction [26, 35] and to the
best of our knowledge, none of them consider the evolving of feature set during the learning process.
Data stream mining is a hot research direction in the area of data mining while online learning [38, 14]
is a related topic from the area of machine learning. Yet online learning can also tackle the streaming
data problem since it assumes that the data come in a streaming way. Online learning has been
extensively studied under different settings, such as learning with experts [5] and online convex
optimization [13, 28]. There are strong theoretical guarantees for online learning, and it usually uses
regret or the number of mistakes to measure the performance of the learning procedure. However,
most of existing online learning algorithms are limited to the case that the feature set is fixed.
Other related topics involving multiple feature sets include multi-view learning [18, 19, 32], transfer
learning [23, 24] and incremental attribute learning [11]. Although both our approaches and multiview learning exploit the relation between different sets of features, there exists a fundamental
2
difference: multi-view learning assumes that every sample is described by multiple feature sets
simultaneously, whereas in FESL only few samples in the feature switching period have two sets
of features, and no matter how many periods there are, the switching part involves only two sets of
features. Transfer learning usually assumes that data are in batch mode, few of them consider the
streaming cases where data arrives sequentially and cannot be stored completely. One exception is
online transfer learning [34] in which data from both sets of features arrive sequentially. However,
they assume that all the feature spaces must appear simultaneously during the whole learning process
while such an assumption is not available in FESL. When it comes to incremental attribute learning,
old sets of features do not vanish or do not vanish entirely while in FESL, old ones will vanish
thoroughly when new sets of features come.
The most related work is [15], which also handles evolving features in streaming data. Different to
our setting where there are overlapping periods, [15] handles situations where there is no overlapping
period but there are overlapping features. Thus, the technical challenges and solutions are different.
3
Preliminaries
Feature Evolution
We focus on both classification and regression
Feature Space ?1
Feature Space ?2
tasks. On each round of the learning process, the
?
?1
algorithm observes an instance and gives its prediction. After the prediction has been made, the
?
true label is revealed and the algorithm suffers
?
?1
? ? ??
a loss which reflects the discrepancy between
?
?
?
? ? ??+1
? ??+1
the prediction and the groundtruth. We define
?
?
?
?feature space" in our paper by a set of features.
?
?
??
??
That the feature space changes means both the
?
? ? +1
underlying distribution of the feature set and the
number of features change. Consider the pro?
?2
cess with three periods where in the first period
large amount of data stream come from the old
?
? ? +?
feature space; then in the second period named
as overlapping period, few of data come from
Figure 2: Specific illustration with one cycle.
both the old and the new feature space; soon
afterwards in the third period, data stream only
come from the new feature space. We call this whole process a cycle. As can be seen from Figure 1,
each cycle merely includes two feature spaces. Thus, we only need to focus on one cycle and it is
easy to extend to the case with multiple cycles. Besides, we assume that the old features in one cycle
will vanish simultaneously by considering the example that in ecosystem protection, all the sensors
share the same expected lifespan and thus they will wear out at the same time. We will study the case
where old features do not vanish simultaneously in the future work.
1
Data Streaming
1
1
1
1
2
1
1
1
2
1
2
1
2
1
2
Based on the above discussion, we only consider two feature spaces denoted by S1 and S2 , respectively. Suppose that in the overlapping period, there are B rounds of instances both from S1 and S2 .
As can be seen from Figure 2, the process can be concluded as follows.
? For t = 1, . . . , T1 ? B, in each round, the learner observes a vector xSt 1 ? Rd1 sampled
from S1 where d1 is the number of features of S1 , T1 is the number of total rounds in S1 .
? For t = T1 ? B + 1, . . . , T1 , in each round, the learner observes two vectors xSt 1 ? Rd1 and
xSt 2 ? Rd2 from S1 and S2 , respectively where d2 is the number of features of S2 .
? For t = T1 + 1, . . . , T1 + T2 , in each round, the learner observes a vector xSt 2 ? Rd2
sampled from S2 where T2 is the number of rounds in S2 . Note that B is small, so we can
omit the streaming data from S2 on rounds T1 ? B + 1, . . . , T1 since they have minor effect
on training the model in S2 .
We use kxk to denote the `2 -norm of a vector x ? Rdi , i = 1, 2. The inner product is denoted by
h?, ?i. Let ?1 ? Rd1 and ?2 ? Rd2 be two sets of linear models that we are interested in. We define
the projection ??i (b) = argmina??i ka ? bk, i = 1, 2. We restrict our prediction function in i-th
feature space and t-th round to be linear which takes the form hwi,t , xSt i i where wi,t ? Rdi , i = 1, 2.
The loss function `(w> x, y) is convex in its first argument whereas in implementing algorithms, we
3
Algorithm 1 Initialize
1: Initialize w1,1 ? ?1 randomly, M1 = 0, and M2 = 0;
2: for t = 1, 2, . . . , T1 do
d1
1
Receive xS
and predict ft = w>
xS1 ? R; Receive the target yt ? R, and suffer loss `(ft , yt );
3:
t ? R
?1,t t
4:
Update w1,t using (1) where ?t = 1/ t;
>
>
2 S1
2 S2
;
and M2 = M2 + xS
5:
if t > T1 ? B then M1 = M1 + xS
t xt
t xt
?1
6: M? = M1 M2 .
use logistic loss for classification task, namely `(w> x, y) = (1/ ln 2) ln(1 + exp(?y(w> x))) and
square loss for regression task, namely `(w> x, y) = (y ? w> x)2 .
The most straightforward or baseline algorithm is to apply online gradient descent [38] on rounds
1, . . . , T1 with streaming data xSt 1 , and invoke it again on rounds T1 + 1, . . . , T1 + T2 with streaming
data xSt 2 . The models are updated according to (1), where ?t is a varied step size:
Si
wi,t+1 = ??i wi,t ? ?t ?`(w>
(1)
i,t xt , yt ) , i = 1, 2.
4
Our Proposed Approach
In this section, we first introduce the basic idea of the solution to FESL, then two different kinds of
approaches with the corresponding analyses are proposed.
The major limitation of the baseline algorithm mentioned above is that the model learned on rounds
1, . . . , T1 is ignored on rounds T1 + 1, . . . , T1 + T2 . The reason is that from rounds t > T1 , we
cannot observe data from feature space S1 , and thus the model w1,T1 , which operates in S1 , cannot
be used directly. To address this challenge, we assume there is a certain relationship ? : Rd2 ? Rd1
between the two feature spaces, and we try to discover it in the overlapping period. There are several
methods to learn a relationship between two sets of features including multivariate regression [16],
streaming multi-label learning [25], etc. In our setting, since the overlapping period is very short, it is
unrealistic to learn a complex relationship between the two spaces. Instead, we use a linear mapping
to approximate ?. Assume the coefficient matrix of the linear mapping is M , then during rounds
T1 ? B + 1, . . . , T1 , the estimation of M can be based on least squares
min
XT1
M ?Rd2 ?d1
t=T1 ?B+1
kxSt 1 ? M > xSt 2 k22 .
The optimal solution M? to the above problem is given by
M? =
T1
X
>
xSt 2 xSt 2
!?1
T1
X
>
xSt 2 xSt 1
!
.
t=T1 ?B+1
t=T1 ?B+1
Then if we only observe an instance xSt 2 ? Rd2 from S2 , we can recover an instance in S1 by
?(xS2 ) ? Rd1 , to which w1,T1 can be applied. Based on this idea, we will make two changes to the
baseline algorithm:
? During rounds T1 ?B +1, . . . , T1 , we will learn a relationship ? from (xST11 ?B+1 , xST12 ?B+1 ),
. . . , (xST11 , xST12 ).
? From rounds t > T1 , we will keep on updating w1,t using the recovered data ?(xSt 2 ) and
predict the target by utilizing the predictions of w1,t and w2,t .
In round t > T1 , the learner can calculate two base predictions based on models w1,t and w2,t :
S2
> S2
f1,t = w>
1,t (?(xt )) and f2,t = w2,t xt . By utilizing the two base predictions in each round, we
propose two methods, both of which are able to follow the better base prediction empirically and
theoretically. The process to obtain the relationship mapping ? and w1,T1 during rounds 1, . . . , T1
are concluded in Algorithm 1.
4
Algorithm 2 FESL-c(ombination)
1:
2:
3:
4:
5:
6:
7:
8:
4.1
Initialize ? and w1,T1 during 1, . . . , T1 using Algorithm 1;
?1,T1 = ?2,T1 = 12 ;
Initialize w2,T1 +1 randomly and w1,T1 +1 by w1,T1 ;
for t = T1 + 1, T1 + 2, . . . , T1 + T2 do
S2
S2
> S2
2
Receive xS
and predict f1,t = w>
1,t (?(xt )) and f2,t = w2,t xt ;
t ? R
Predict pbt ? R using (2), then receivep
the target yt ? R, and suffer loss `(b
pt , yt );
Update weights using (3) where ? = 8(ln 2)/T2 ;
?
Update w1,t and w2,t using (4) and (1) respectively where ?t = 1/ t ? T1 ;
Weighted Combination
We first propose an ensemble method by combining predictions with weights based on exponential of
the cumulative loss [5]. The prediction at time t is the weighted average of all the base predictions:
pbt = ?1,t f1,t + ?2,t f2,t
(2)
where ?i,t is the weight of the i-th base prediction. With the previous loss of each base model, we
can update the weights of the two base models as follows:
?i,t e??`(fi,t ,yt )
?i,t+1 = P2
, i = 1, 2,
??`(fj,t ,yt )
j=1 ?j,t e
(3)
where ? is a tuned parameter. The updating rule of the weights shows that if the loss of one of
the models on previous round is large, then its weight will decrease in an exponential rate in next
round, which is reasonable and can derive a good theoretical result shown in Theorem 1. Algorithm 2
summarizes our first approach for FESL named as FESL-c(ombination). We first learn a model w1,T1
using online gradient descent on rounds 1, . . . , T1 , during which, we also learn a relationship ? for
t = T1 ? B + 1, . . . , T1 . For t = T1 + 1, . . . , T1 + T2 , we learn a model w2,t on each round and
keep updating w1,t on the recovered data ?(xSt 2 ) showed in (4) where ?t is a varied step size:
S2
w1,t+1 = ??i w1,t ? ?t ?`(w>
(?(x
)),
y
)
.
(4)
t
t
1,t
Then we combine the predictions of the two models by weights calculated in (3).
Analysis In this paragraph, we borrow the regret from online learning to measure the performance
of FESL-c. Specifically, we give a loss bound as follows which shows that the performance will be
improved with assistance of the old feature space. For the sake of soundness, we put the proof of our
theorems in the supplementary file. We define that LS1 and LS2 are two cumulative losses suffered
by base models on rounds T1 + 1, . . . , T1 + T2 ,
LS1 =
TX
1 +T2
`(f1,t , yt ), LS2 =
t=T1 +1
TX
1 +T2
`(f2,t , yt ),
(5)
t=T1 +1
and LS12 is the cumulative loss suffered by our methods: LS12 =
PT1 +T2
t=T1 +1
`(b
pt , yt ). Then we have:
Theorem 1. Assume that the loss function ` is convex in its first argument and that it takes value
S12
in [0,1].
with parameter
p For all T2 > 1 and for all yt ? Y with t = T1 + 1, . . . , T1 + T2 , L
?t = 8(ln 2)/T2 satisfies
p
LS12 ? min(LS1 , LS2 ) + (T2 /2) ln 2
(6)
This theorem implies that the cumulative loss LS12 of Algorithm 2 over rounds T1 +
p1, . . . , T1 + T2
is comparable to the minimum of LS1 and LS2 . Furthermore, we define C = (T2 /2) ln 2. If
LS2 ? LS1 > C, it is easy to verify that LS12 is smaller than LS2 . In summary, on rounds T1 +
1, . . . , T1 + T2 , when w1,t is better than w2,t to certain degree, the model with assistance from S1 is
better than that without assistance.
5
Algorithm 3 FESL-s(election)
1:
2:
3:
4:
5:
6:
7:
8:
4.2
Initialize ? and w1,T1 during 1, . . . , T1 using Algorithm 1;
?1,T1 = ?2,T1 = 12 ;
Initialize w2,T1 +1 randomly and w1,T1 +1 by w1,T1 ;
for t = T1 + 1, T1 + 2, . . . , T1 + T2 do
S2
S2
> S2
2
Receive xS
and predict f1,t = w>
1,t (?(xt )) and f2,t = w2,t xt ;
t ? R
Draw a model wi,t according to the distribution (7) and predict pbt = fi,t according to the model;
Receive the target yt ? R, and suffer loss `(b
pt , yt ); Update the weights
? using (8);
Update w1,t and w2,t using (4) and (1) respectively, where ?t = 1/ t ? T1 .
Dynamic Selection
The combination approach mentioned in the above subsection combines several base models to
improve the overall performance. Generally, combination of several classifiers performs better than
selecting only one single classifier [36]. However, it requires that the performance of base models
should not be too bad, for example, in Adaboost the accuracy of the base classifiers should be no less
than 0.5 [8]. Nevertheless, in our FESL problem, on rounds T1 + 1, . . . , T1 + T2 , w2,t cannot satisfy
the requirement in the beginning due to insufficient training data and w1,t may become worse when
more and more data come causing a cumulation of recovered error. Thus, it may not be appropriate to
combine the two models all the time, whereas dynamically selecting the best single may be a better
choice. Hence we propose a method based on a new strategy, i.e., dynamic selection, similar to the
Dynamic Classifier Selection [36] which only uses the best single model rather than combining both
of them in each round. Note that, though we only select one of the models, we retain and utilize both
of them to update their weights. So it is still an ensemble method. The basic idea of dynamic selection
is to select the model of larger weight with higher probability. Algorithm 3 summarizes our second
approach for FESL named as FESL-s(election). Specifically, the steps in Algorithm 3 on rounds
1, . . . , T1 is the same as that in Algorithm 2. For t = T1 + 1, . . . , T1 + T2 , we still update weights
of each model. However, when doing prediction, we do not combine all the models? prediction, we
adopt the result of the ?best" model?s according to the distribution of their weights
?i,t?1
pi,t = P2
i = 1, 2.
(7)
j=1 ?j,t?1
To track the best model, we have a different way of updating weights which is given as follows [5].
Wt
+ (1 ? ?)vi,t , i = 1, 2,
(8)
vi,t = ?i,t?1 e??`(fi,t ,yt ) , i = 1, 2, ?i,t = ?
2
p
where we define Wt = v1,t + v2,t , ? = 1/(T2 ? 1), ? = 8/T2 (2 ln 2 + (T2 ? 1)H(1/(T2 ? 1)))
and H(x) = ?x ln x ? (1 ? x) ln(1 ? x) is the binary entropy function defined for x ? (0, 1).
Analysis From rounds t > T1 , the first model w1,t would become worse due to the cumulative
recovered error while the second model will become better by the large amount of coming data. Since
w1,t is initialized by w1,T 1 which is learnt from the old feature space and w2,t is initialized randomly,
it is reasonable to assume that w1,t is better than w2,t in the beginning, but inferior to w2,t after
sufficient large number of rounds. Let s be the round after which w1,t is worse than w2,t . We define
Ps
PT2
Ls = t=T1 +1 `(f1,t , yt ) + t=s+1
`(f2,t , yt ), we can verify that
min
T1 +1?s?T1 +T2
Ls ? min LSi .
i=1,2
(9)
Then a more ambitious goal is to compare the proposed algorithm against w1,t from rounds T1 + 1
to s, and against the w2,t from rounds s to T1 + T2 , which motivates us to study the following
performance measure LS12 ? Ls . Because the exact value of s is generally unknown, we need to
bound the worst-case LS12 ? minT1 +1?s?T1 +T2 Ls . An upper bound of LS12 is given as follows.
Theorem
2. For all T2 > 1, if the model is run with parameter ? = 1/(T2 ? 1) and ? =
p
8/T2 (2 ln 2 + (T2 ? 1)H(1/T2 ? 1)), then
s
T2
H(?)
S12
s
L
?
min
L +
2 ln 2 +
(10)
T1 +1?s?T1 +T2
2
?
where H(x) = ?x ln x ? (1 ? x) ln(1 ? x) is the binary entropy function.
6
Table 1: Detail description of datasets: let n be the number of examples, and d1 and d2 denote the dimensionality
of the first and second feature space, respectively. The first 9 datasets in the left column are synthetic datasets,
?r.EN-GR" means the dataset EN-GR comes from Reuter and ?RFID" is the real dataset.
Dataset
Dataset
Dataset
n
d1
d2
n
d1 d2
n
d1
d2
Australian
690
42 29 r.EN-FR 18,758 21,531 24,892 r.GR-IT 29,953 34,279 15,505
Credit-a
653
15 10 r.EN-GR 18,758 21,531 34,215 r.GR-SP 29,953 34,279 11,547
1,000 20 14 r.EN-IT 18,758 21,531 15,506 r.IT-EN 24,039 15,506 21,517
Credit-g
Diabetes
768
8
5 r.EN-SP 18,758 21,531 11,547 r.IT-FR 24,039 15,506 24,892
DNA
940 180 125 r.FR-EN 26,648 24,893 21,531 r.IT-GR 24,039 15,506 34,278
1,000 59 41 r.FR-GR 26,648 24,893 34,287 r.IT-SP 24,039 15,506 11,547
German
Kr-vs-kp
3,196 36 25 r.FR-IT 26,648 24,893 15,503 r.SP-EN 12,342 11,547 21,530
3,175 60 42 r.FR-SP 26,648 24,893 11,547 r.SP-FR 12,342 11,547 24,892
Splice
Svmguide3 1,284 22 15 r.GR-EN 29,953 34,279 21,531 r.SP-GR 12,342 11,547 34,262
940
78 72 r.GR-FR 29,953 34,279 24,892 r.SP-IT 12,342 11,547 15,500
RFID
According to Theorem 2 we know that LS12 is comparable to minT1 +1?s?T1 +T2 Ls . Due to (9), we
can conclude that the upper bound of LS12 in Algorithm 3 is tighter than that of Algorithm 2.
5
Experiments
In this section, we first introduce the datasets we use. We want to emphasize that we collected one
real dataset by ourselves since our setting of feature evolving is relatively novel so that the required
datasets are not widely available yet. Then we introduce the compared methods and settings. Finally
experiment results are given.
5.1
Datasets
We conduct our experiments on 30 datasets consisting of 9 synthetic datasets, 20 Reuter datasets
and 1 real dataset. To generate synthetic data, we randomly choose some datasets from different
domains including economy and biology, etc1 whose scales vary from 690 to 3,196. They only have
one feature space at first. We artificially map the original datasets into another feature space by
random Gaussian matrices, then we have data both from feature space S1 and S2 . Since the original
data are in batch mode, we manually make them come sequentially. In this way, synthetic data are
completely generated. We also conduct our experiments on 20 datasets from Reuter [3]. They are
multi-view datasets which have large scale varying from 12,342 to 29,963. Each dataset has two
views which represent two different kinds of languages, respectively. We regard the two views as the
two feature spaces. Now they do have two feature spaces but the original data is in batch mode, so
we will artificially make them come in streaming way.
We use the RFID technique to collect the real data which has 450 instances from S1 and S2 respectively. RFID technique is widely used to do moving goods detection [31]. In our case, we want to
utilize the RFID technique to predict the location?s coordinate of the moving goods attached by RFID
tag. Concretely, we arranged several RFID aerials around the indoor area. In each round, each RFID
aerial received the tag signals, then the goods with tag moved, at the same time, we recorded the
goods? coordinate. Before the aerials expired, we arranged new aerials beside the old ones to avoid
the situation without aerials. So in this overlapping period, we have data from both old and new
feature spaces. After the old aerials expired, we continue to use the new ones to receive signals. Then
we only have data from feature space S2 . So the RFID data we collect totally satisfy our assumptions.
The details of all the datasets we use are presented in Table 1.
5.2
Compared Approaches and Settings
We compare our FESL-c and FESL-s with three approaches. One is mentioned in Section 3, where
once the feature space changed, the online gradient descent algorithm will be invoked from scratch,
named as NOGD (Naive Online Gradient Descent). The other two approaches utilize the model
learned from feature space S1 by online gradient descent to do predictions on the recovered data. The
1
Datasets can be found in http://archive.ics.uci.edu/ml/.
7
0.9
1.2
1.0
0.8
2
0.8
0.6
0.08
0.06
1.0
3
Loss
0.10
4
Loss
1.0
Loss
1.2
0.12
Loss
Loss
0.14
1
70
139
208
0.4
277
Time
66
131
196
261
301
401
77
Time
Time
(a) australian
201
153
229
305
Time
(c) credit-g
(b) credit-a
0.6
0.4
0.7
101
326
0.8
381
0.2
376
751 1126 1501
Time
(d) diabetes
(e) r.EN-SP
1.0
0.4
0.2
0.6
0.2
Time
(f) r.FR-SP
0.6
0.4
0.4
533 1065 1597 2129 2661
Loss
0.8
Loss
Loss
Loss
0.6
3.0
0.8
1.0
0.8
600 1199 1798 2397
Time
(g) r.GR-EN
0.2
2.5
2.0
1.5
481
961 1441 1921 2401
Time
(h) r.IT-FR
1.0
91
181
271
361
Time
(i) RFID
legend
Figure 3: The trend of loss with three baseline methods and the proposed methods on synthetic data. The smaller
the cumulative loss, the better. All the average cumulative loss at any time of our methods is comparable to the
best of baseline methods and 8 of 9 are smaller.
difference between them is that one keeps updating with the recovered data while the other does not.
The one which keeps updating is called Updating Recovered Online Gradient Descent (ROGD-u)
and the other which keeps fixed is called Fixed Recovered Online Gradient Descent (ROGD-f). We
evaluate the empirical performances of the proposed approaches on classification and regression tasks
on rounds T1 + 1, . . . , T1 + T2 . To verify that our analysis is reasonable, we present the trend of
average cumulative loss. Concretely, at each time t0 , the loss `?t0 of every method is the average of
Pt0
the cumulative loss over 1, . . . , t0 , namely `?t0 = (1/t0 ) t=1 `t . We also present the classification
performance over all instances on rounds T1 + 1, . . . , T1 + T2 on synthetic and Reuter data. The
performances of all approaches are obtained by average results over 10 independent runs on synthetic
data. Due to the large scale of Reuter data, we only conduct 3 independent runs on Reuter data and
report the average results.
The parameters we need to set are the number of instances in overlapping period, i.e., B, the number
of instances in S1 and S2 , i.e., T1 and T2 and the step size, i.e., ?t where t is time. For all baseline
methods and our methods, the parameters are the same. In our experiments, we set B 5 or 10 for
synthetic data, 50 for Reuter data and 40
?for RFID data. We set almost T1 and T2 to be half of the
number of instances, and ?t to be 1/(c t) where c is searched in the range {1, 10, 50, 100, 150}.
The detailed setting of c in ?t for each dataset is presented in supplementary file.
5.3
Results
Here we only present part of the loss trend results, and other results are presented in the supplementary
file. Figure 3 gives the trend of average cumulative loss. (a-d) are the results on synthetic data, (e-h)
are the results on Reuter data, (i) is the result of the real data. The smaller the average cumulative loss,
the better. From the experimental results, we have the following observations. First, all the curves
with circle marks representing NOGD decrease rapidly which conforms to the fact that NOGD on
rounds T1 + 1, . . . , T1 + T2 becomes better and better with more and more data coming. Besides,
the curves with star marks representing ROGD-u also decline but not very apparent since on rounds
1, . . . , T1 , ROGD-u already learned well and tend to converge, so updating with more recovered data
could not bring too much benefits. Moreover, the curves with plus marks representing ROGD-f does
not drop down but even go up instead, which is also reasonable because it is fixed and if there are
some recovering error, it will perform worse. Lastly, our methods are based on NOGD and ROGD-u,
so their average cumulative loss also decrease. As can be seen from Figure 3, the average cumulative
loss of our methods is comparable to the best of baseline methods on all datasets and are smaller than
them on 8 datasets. And FESL-s exhibits slightly smaller average cumulative loss than FESL-c. You
may notice that NOGD is always worse than ROGD-u on synthetic data and real data while on Reuter
data NOGD becomes better than ROGD-u after a few rounds. This is because on synthetic data and
real data, we do not have enough rounds to let all methods converge while on Reuter data, large
amounts of instances ensure the convergence of every method. So when all the methods converge, we
can see that NOGD is better than other baseline methods since it always receives the real instances
while ROGD-u and ROGD-f receive the recovered instances which may contain recovered error. As
can be seen from (e-h), in the first few rounds, our methods are comparable to ROGD-u. When
NOGD is better than ROGD-u, our methods are comparable to NOGD which shows that our methods
8
Table 2: Accuracy with its variance on synthetic datasets and Reuter datasets. The larger the better. The best
ones among all the methods are bold.
Dataset
NOGD
ROGD-u
ROGD-f
FESL-c
FESL-s
australian
.767?.009 .849?.009 .809?.025 .849?.009 .849?.009
credit-a
.811?.006 .826?.018 .785?.051 .827?.014 .831?.009
credit-g
.659?.010 .733?.006 .716?.011 .733?.006 .733?.006
diabetes
.650?.002 .652?.009 .651?.006 .652?.007 .652?.009
dna
.610?.013 .691?.023 .608?.064 .691?.023 .692?.021
german
.684?.006 .700?.002 .700?.002 .700?.001 .703?.004
kr-vs-kp
.612?.005 .621?.036 .538?.024 .626?.028 .630?.016
splice
.568?.005 .612?.022 .567?.057 .612?.022 .612?.022
svmguide3 .680?.010 .779?.010 .748?.012 .779?.010 .778?.010
r.EN-FR
.902?.004 .849?.003 .769?.069 .903?.003 .902?.005
r.EN-GR
.867?.005 .836?.007 .802?.036 .870?.002 .870?.003
r.EN-IT
.858?.014 .847?.014 .831?.018 .861?.010 .863?.013
r.EN-SP
.900?.002 .848?.002 .825?.001 .901?.001 .899?.002
r.FR-EN
.858?.007 .776?.009 .754?.012 .858?.007 .858?.007
r.FR-GR
.869?.004 .774?.019 .753?.021 .870?.004 .868?.003
r.FR-IT
.874?.005 .780?.022 .744?.040 .874?.005 .873?.005
r.FR-SP
.872?.001 .778?.022 .735?.013 .872?.001 .871?.002
.907?.000 .850?.007 .801?.035 .907?.001 .906?.000
r.GR-EN
r.GR-FR
.898?.001 .827?.009 .802?.023 .898?.001 .898?.000
r.GR-IT
.847?.011 .851?.017 .816?.006 .850?.018 .851?.017
r.GR-SP
.902?.001 .845?.003 .797?.012 .902?.001 .902?.001
.854?.003 .760?.006 .730?.024 .856?.002 .854?.003
r.IT-EN
r.IT-FR
.863?.002 .753?.012 .730?.020 .864?.002 .862?.003
r.IT-GR
.849?.004 .736?.022 .702?.012 .849?.004 .846?.004
r.IT-SP
.839?.006 .753?.014 .726?.005 .839?.007 .839?.006
r.SP-EN
.926?.002 .860?.005 .814?.021 .926?.002 .924?.001
.876?.005 .873?.017 .833?.042 .876?.014 .878?.012
r.SP-FR
r.SP-GR
.871?.013 .827?.025 .810?.026 .873?.013 .873?.013
r.SP-IT
.928?.002 .861?.005 .826?.005 .928?.003 .927?.002
are comparable to the best one all the time. Moreover, FESL-s performs worse than FESL-c in the
beginning while afterwards, it becomes slightly better than FESL-c.
Table 2 shows the accuracy results on synthetic datasets and Reuter datasets. We can see that for
synthetic datasets, FESL-s outperforms other methods on 8 datasets, FESL-c gets the best on 5
datasets and ROGD-u also gets 5. NOGD performs worst since it starts from scratch. ROGD-u is
better than NOGD and ROGD-f because ROGD-u exploits the old better trained model from old
feature space and keep updating with recovered instances. Our two methods are based on NOGD
and ROGD-u. We can see that our methods can follow the best baseline method or even outperform
it. For Reuter datasets, we can see that FESL-c outperforms other methods on 17 datasets, FESL-s
gets the best on 9 datasets and NOGD gets 8 while ROGD-u gets 1. In Reuter datasets, the period
on new feature space is longer than that in synthetic datasets so that NOGD can update itself to a
good model. Whereas ROGD-u updates itself with recovered data, so the model will become worse
when recovered error accumulates. ROGD-f does not update itself, thus it performs worst. Our two
methods can take the advantage of NOGD and ROGD-f and perform better than them.
6
Conclusion
In this paper, we focus on a new setting: feature evolvable streaming learning. Our key observation is
that in learning with streaming data, old features could vanish and new ones could occur. To make
the problem tractable, we assume there is an overlapping period that contains samples from both
feature spaces. Then, we learn a mapping from new features to old features, and in this way both
the new and old models can be used for prediction. In our first approach FESL-c, we ensemble two
predictions by learning weights adaptively. Theoretical results show that the assistance of the old
feature space can improve the performance of learning with streaming data. Furthermore, we propose
FESL-s to dynamically select the best model with better performance guarantee.
9
Acknowledgement This research was supported by NSFC (61333014, 61603177), JiangsuSF
(BK20160658), Huawei Fund (YBN2017030027) and Collaborative Innovation Center of Novel
Software Technology and Industrialization.
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11
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6,347 | 6,741 | Online Convex Optimization with Stochastic
Constraints
Hao Yu, Michael J. Neely, Xiaohan Wei
Department of Electrical Engineering, University of Southern California?
{yuhao,mjneely,xiaohanw}@usc.edu
Abstract
This paper considers online convex optimization (OCO) with stochastic constraints,
which generalizes Zinkevich?s OCO over a known simple fixed set by introducing
multiple stochastic functional constraints that are i.i.d. generated at each round
and are disclosed to the decision maker only after the decision is made. This
formulation arises naturally when decisions are restricted by stochastic environments or deterministic environments with noisy observations. It also includes
many important problems as special case, such as OCO with long term constraints,
stochastic constrained convex optimization, and deterministic constrained convex optimization.
p To solve this problem, this paper proposes a newpalgorithm
that achieves O( T ) expected regret and constraint violations and O( T log(T ))
high probability regret and constraint violations. Experiments on a real-world data
center scheduling problem further verify the performance of the new algorithm.
1
Introduction
Online convex optimization (OCO) is a multi-round learning process with arbitrarily-varying convex
loss functions where the decision maker has to choose decision x(t) 2 X before observing the
corresponding loss function f t (?). For a fixed time horizon T , define the regret of a learning algorithm
with respect to the best fixed decision in hindsight (with full knowledge of all loss functions) as
regret(T ) =
T
X
t=1
f t (x(t))
min
x2X
T
X
f t (x).
t=1
The goal of OCO is to develop dynamic learning algorithms such that regret grows sub-linearly with
respect to T . The setting of OCO is introduced in a series of work [3, 14, 9, 29] and is formalized in
[29]. OCO has gained considerable amount of research interest recently with various applications
such as online regression, prediction with expert advice, online ranking, online shortest paths, and
portfolio selection. See [23, 11] for more applications and backgrounds.
In [29], Zinkevich shows that using an online gradient descent (OGD) update given by
?
?
x(t + 1) = PX x(t)
rf t (x(t))
(1)
p
where rf t (?) is a subgradient of f t (?) and PX [?] is the projection onto set X can achieve O( T )
regret. Hazan et al. in [12] show that
p better regret is possible under the assumption that each loss
function is strongly convex but O( T ) is the best possible if no additional assumption is imposed.
It is obvious that Zinkevich?s OGD in (1) requires the full knowledge of set X and low complexity
of the projection PX [?]. However, in practice, the constraint set X , which is often described by
many functional inequality constraints, can be time varying and may not be fully disclosed to the
?
This work is supported in part by grant NSF CCF-1718477
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
decision maker. In [18], Mannor et al. extend OCO by considering time-varying constraint functions
g t (x) which can arbitrarily vary and are only disclosed to us after each x(t) is chosen. In this
setting, Mannor et al. in [18] explore the possibility of designing learning algorithms such that
PT
regret grows sub-linearly and lim supT !1 T1 t=1 g t (x(t)) ? 0, i.e., the (cumulative) constraint
PT
violation t=1 g t (x(t)) also grows sub-linearly. Unfortunately, Mannor et al. in [18] prove that this
is impossible even when both f t (?) and g t (?) are simple linear functions.
Given the impossibility results shown by Mannor et al. in [18], this paper considers OCO where
constraint functions g t (x) are not arbitrarily varying but independently and identically distributed
(i.i.d.) generated from an unknown probability model (and functions f t (x) are still arbitrarily varying
and possibly non-i.i.d.). More specifically, this paper considers online convex optimization (OCO)
with stochastic constraint X = {x 2 X0 : E! [gk (x; !)] ? 0, k 2 {1, 2, . . . , m}} where X0 is a
known fixed set; the expressions of stochastic constraints E! [gk (x; !)] (involving expectations with
respect to ! from an unknown distribution) are unknown; and subscripts k 2 {1, 2, . . . , m} indicate
the possibility of multiple functional constraints. In OCO with stochastic constraints, the decision
maker receives loss function f t (x) and i.i.d. constraint function realizations gkt (x) = gk (x; !(t))
at each round t. However, the expressions of gkt (?) and f t (?) are disclosed to the decision maker
only after decision x(t) 2 X0 is chosen. This setting arises naturally when decisions are restricted
by stochastic environments or deterministic environments with noisy observations. For example,
if we consider online routing (with link capacity constraints) in wireless networks [18], each link
capacity is not a fixed constant (as in wireline networks) but an i.i.d. random variable since wireless
channels are stochastically time-varying by nature [25]. OCO with stochastic constraints also covers
important special cases such as OCO with long term constraints [16, 5, 13], stochastic constrained
convex optimization [17] and deterministic constrained convex optimization [21].
PT
Let x? = argmin{x2X0 :E[gk (x;!)]?0,8k2{1,2,...,m}} t=1 f t (x) be the best fixed decision in hindsight (knowing all loss functions f t (x) and the distribution of stochastic constraint functions
gk (x; !)). Thus, x? minimizes the T -round cumulative loss and satisfies all stochastic constraints in
PT
expectation, which also implies lim supT !1 T1 t=1 gkt (x? ) ? 0 almost surely by the strong law
of large numbers. Our goal is to develop dynamic learning algorithms that guarantee both regret
PT
PT
PT
t
t ?
t
t=1 f (x(t))
t=1 f (x ) and constraint violations
t=1 gk (x(t)) grow sub-linearly.
Note that Zinkevich?s algorithm in (1) is not applicable to OCO with stochastic constraints since X
is unknown and it can happen that X (t) = {x 2 X0 : gk (x; !(t)) ? 0, 8k 2 {1, 2, . . . , m}} = ;
for certain realizations !(t), such that projections PX [?] or PX (t) [?] required in (1) are not even
well-defined.
Our Contributions: This paper solves online convex optimization with stochastic
constraints. In
p
particular, we propose a new learning
algorithm
that
is
proven
to
achieve
O(
T
)
expected
regret
p
and constraint violations and O( T log(T )) high probability regret and constraint violations. The
proposed new algorithm also improves upon state-of-the-art results in the following special cases:
? OCO with long term constraints: This is a special case where each gkt (x) ? gk (x) is known
and does not depend on time. Note that X = {x 2 X0 : gk (x) ? 0, 8k 2 {1, 2, . . . , m}} can
be complicated while X0 might be a simple hypercube. To avoid high complexity involved in
the projection onto X as in Zinkevich?s algorithm, work in [16, 5, 13] develops low complexity
algorithms that use projections onto a simpler set X0 by allowing gk (x(t)) > 0 for certain
PT
rounds but ensuring lim supT !1 T1 t=1 gk (x(t)) ? 0. The best existing performance is
1
/2
O(T max{ ,1 } ) regret and O(T
) constraint violations where 2 (0, 1) is an algorithm
p
p
parameter [13]. This gives O( T ) regret with worse O(T 3/4 ) constraint violations or O( T )
constraint violations with worse O(T ) regret. In contrast,pour algorithm, which
p only uses
projections onto X0 as shown in Lemma 1, can achieve O( T ) regret and O( T ) constraint
violations simultaneously. Note that by adapting the methodology presented in this paper, our
other work [27] developed a different algorithm thatpcan only solve the special case problem
?OCO with long term constraints? but can achieve O( T ) regret and O(1) constraint violations.
? Stochastic constrained convex optimization: This is a special case where each f t (x) is i.i.d.
generated from an unknown distribution. This problem has many applications in operations
research and machine learning such as Neyman-Pearson classification and risk-mean portfolio.
The work [17] develops a (batch) offline algorithm that produces a solution with high probability
2
performance guarantees only after sampling the problems for sufficiently many times. That is,
during the process of sampling, there is no performance guarantees. The work [15] proposes
a stochastic approximation based (batch) offline algorithm for stochastic convex optimization
with one single stochastic functional inequality constraint. In contrast, our algorithm is an
online algorithm with online performance guarantees and can deal with an arbitrary number of
stochastic constraints.
? Deterministic constrained convex optimization: This is a special case where each f t (x) ? f (x)
and gkt (x) ? gk (x) are known and do not depend on time. In this case, the goal is to develop
a fast algorithm that converges to a good
p solution (with a small error) with a few number of
iterations; and our algorithm with O( Tp) regret and constraint violations is equivalent to an
iterative numerical algorithm with O(1/ T ) convergence rate. Our algorithm is subgradient
based and does not require the smoothness or differentiability of the
p convex program. The
primal-dual subgradient method considered in [19] has the same O(1/ T ) convergence rate but
requires an upper bound of optimal Lagrange multipliers, which is usually unknown in practice.
2
Formulation and New Algorithm
Let X0 be a known fixed compact convex set. Let f t (x) be a sequence of arbitrarily-varying convex
functions. Let gk (x; !(t)), k 2 {1, 2, . . . , m} be sequences of functions that are i.i.d. realizations of
stochastic constraint functions g?k (x) = E! [gk (x; !)] with random variable ! 2 ? from an unknown
distribution. That is, !(t) are i.i.d. samples of !. Assume that each f t (?) is independent of all !(? )
with ? t + 1 so that we are unable to predict future constraint functions based on the knowledge of
the current loss function. For each ! 2 ?, we assume gk (x; !) are convex with respect to x 2 X0 . At
the beginning of each round t, neither the loss function f t (x) nor the constraint function realizations
gk (x; !(t)) are known to the decision maker. However, the decision maker still needs to make a
decision x(t) 2 X0 for round t; and after that f t (x) and gk (x, !(t)) are disclosed to the decision
maker at the end of round t.
For convenience, we often suppress the dependence of each gk (x; !(t)) on !(t) and write
gkt (x) = gk (x; !(t)). Recall g?k (x) = E! [gk (x; !)] where the expectation is with respect to !.
Define X = {x 2 X0 : g?k (x) = E[gk (x; !)] ? 0, 8k 2 {1, 2, . . . , m}}. We further define the
t
t
stacked vector of multiple functions g1t (x), . . . , gm
(x) as gt (x) = [g1t (x), . . . , gm
(x)]T and define
T
? (x) = [E! [g1 (x; !)], . . . , E! [gm (x; !)]] . We use k ? k to denote the Euclidean norm for a vector.
g
Throughout this paper, we have the following assumptions:
Assumption 1 (Basic Assumptions).
? Loss functions f t (x) and constraint functions gk (x; !) have bounded subgradients on X0 .
That is, there exists D1 > 0 and D2 > 0 such that krf t (x)k ? D1 for all x 2 X0 and all
t 2 {0, 1, . . .} and krgk (x; !)k ? D2 for all x 2 X0 , all ! 2 ? and all k 2 {1, 2, . . . , m}.2
? There exists constant G > 0 such that kg(x; !)k ? G for all x 2 X0 and all ! 2 ?.
? There exists constant R > 0 such that kx
yk ? R for all x, y 2 X0 .
? 2 X0 such that g?k (?
Assumption 2 (The Slater Condition). There exists ? > 0 and x
x) =
E! [gk (?
x; !)] ? ? for all k 2 {1, 2, . . . , m}.
2.1
New Algorithm
Now consider the following algorithm described in Algorithm 1. This algorithm chooses x(t + 1) as
the decision for round t + 1 based on f t (?) and gt (?) without requiring f t+1 (?) or gt+1 (?).
For each stochastic constraint function gk (x; !), we introduce Qk (t) and call it a virtual queue since
its dynamic is similar to a queue dynamic. The next lemma summarizes that x(t + 1) update in (2)
can be implemented via a simple projection onto X0 .
?
?
1
Lemma 1. The x(t + 1) update in (2) is given by x(t + 1) = PX0 x(t) 2?
d(t) , where d(t) =
Pm
V rf t (x(t)) + k=1 Qk (t)rgkt (x(t)) and PX0 [?] is the projection onto convex set X0 .
2
The notation rh(x) is used to denote a subgradient of a convex function h at the point x.; it is the same as
the gradient whenever the gradient exists.
3
Algorithm 1
Let V > 0 and ? > 0 be constant algorithm parameters. Choose x(1) 2 X0 arbitrarily and let
Qk (1) = 0, 8k 2 {1, 2, . . . , m}. At the end of each round t 2 {1, 2, . . .}, observe f t (?) and gt (?)
and do the following:
? Choose x(t + 1) that solves
m
X
min V [rf t (x(t))]T [x x(t)] +
Qk (t)[rgkt (x(t))]T [x x(t)] + ?kx x(t)k2
(2)
x2X0
k=1
as the decision for the next round t + 1, where rf t (x(t)) is a subgradient of f t (x) at point
x = x(t) and rgkt (x(t)) is a subgradient of gkt (x) at point x = x(t).
? Update each virtual queue Qk (t + 1), 8k 2 {1, 2, . . . , m} via
Qk (t + 1) = max Qk (t) + gkt (x(t)) + [rgkt (x(t))]T [x(t + 1)
x(t)], 0 ,
(3)
where max{?, ?} takes the larger one between two elements.
Proof. The projection by definition is minx2X0 kx
2.2
[x(t)
1
2
2? d(t)]k
and is equivalent to (2).
Intuitions of Algorithm 1
Note that if there are no stochastic constraints gkt (x), i.e., X = X0 , then Algorithm 1 has Qk (t) ?
V
0, 8t and becomes Zinkevich?s algorithm with = 2?
in (1) since
(a)
x(t + 1) = argmin V [rf t (x(t))]T [x
|
x2X0
x(t)] + ?kx
{z
penalty
x(t)k2
}
?
= PX0 x(t)
(b)
?
V
rf t (x(t))
2?
(4)
where (a) follows from (2); and (b) follows from Lemma 1 by noting that d(t) = V rf t (x(t)). Call
the term marked by an underbrace in (4) the penalty. Thus, Zinkevich?s algorithm is to minimize the
penalty term and is a special case of Algorithm 1 used to solve OCO over X0 .
?
?T
Let Q(t) = Q1 (t), . . . , Qm (t) be the vector of virtual queue backlogs. Let L(t) = 12 kQ(t)k2 be
a Lyapunov function and define Lyapunov drift
(t) = L(t + 1)
1
[kQ(t + 1)k2
2
L(t) =
kQ(t)k2 ].
(5)
The intuition behind Algorithm 1 is to choose x(t + 1) to minimize an upper bound of the expression
(t) + V [rf t (x(t))]T [x
|{z} |
drift
x(t)] + ?kx
{z
penalty
x(t)k2
}
(6)
The intention to minimize penalty is natural since Zinkevich?s algorithm (for OCO without stochastic
constraints) minimizes penalty, while the intention to minimize drift is motivated by observing that
gkt (x(t)) is accumulated into queue Qk (t + 1) introduced in (3) such that we intend to have small
queue backlogs. The drift (t) can be complicated and is in general non-convex. The next lemma
(proven in Supplement 7.1) provides a simple upper bound on (t) and follows directly from (3).
Lemma 2. At each round t 2 {1, 2, . . .}, Algorithm 1 guarantees
m
X
?
? 1
p
(t) ?
Qk (t) gkt (x(t)) + [rgkt (x(t))]T [x(t + 1) x(t)] + [G + mD2 R]2 ,
(7)
2
k=1
where m is the number of constraint functions; and D2 , G and R are defined in Assumption 1.
Pm
p
At the end of round t, k=1 Qk (t)gkt (x(t)) + 12 [G + mD2 R]2 is a given constant that is not
affected by decision x(t + 1). The algorithm decision in (2) is now transparent: x(t + 1) is chosen to
minimize the drift-plus-penalty expression (6), where (t) is approximated by the bound in (7).
2.3
Preliminary Analysis and More Intuitions of Algorithm 1
The next lemma (proven in Supplement 7.2) relates constraint violations and virtual queue values and
follows directly from (3).
4
PT
PT
Lemma 3. For any T 1, Algorithm 1 guarantees t=1 gkt (x(t)) ? kQ(T +1)k+D2 t=1 kx(t+
1) x(t)k, 8k 2 {1, 2, . . . , m}, where D2 is defined in Assumption 1.
Recall that function h : X0 ! R is said to be c-strongly convex if h(x) 2c kxk2 is convex over
x 2 X0 . It is easy to see that if q : X0 ! R is a convex function, then for any constant c > 0
and any vector b, the function q(x) + 2c kx bk2 is c-strongly convex. Further, it is known that if
h : X ! R is a c-strongly convex function that is minimized at a point xmin 2 X0 , then (see, for
example, Corollary 1 in [28]):
c
h(xmin ) ? h(x)
kx xmin k2 8x 2 X0
(8)
2
Note that the expression involved in minimization (2) in Algorithm 1 is strongly convex with modulus
2? and x(t + 1) is chosen to minimize it. Thus, the next lemma follows.
Lemma 4. Let z 2 X0 be arbitrary. For all t 1, Algorithm 1 guarantees
V [rf t (x(t))]T [x(t + 1)
x(t)] +
m
X
Qk (t)[rgkt (x(t))]T [x(t + 1)
x(t)] + ?kx(t + 1)
x(t)k2
k=1
?V [rf t (x(t))]T [z
x(t)] +
m
X
Qk (t)[rgkt (x(t))]T [z
x(t)] + ?kz
x(t)k2
?kz
x(t + 1)k2 .
k=1
The next corollary follows by taking z = x(t) in Lemma 4 and is proven in Supplement 7.3.
Corollary 1. For all t
1, Algorithm 1 guarantees kx(t + 1)
x(t)k ?
V D1
2?
+
p
mD2
2? kQ(t)k.
The next corollary follows directly from Lemma 3 and Corollary 1 and shows that constraint violations
are ultimately bounded by sequence kQ(t)k, t 2 {1, 2, . . . , T + 1}.
PT
D1 D2
Corollary 2. For any T 1, Algorithm 1 guarantees t=1 gkt (x(t)) ? kQ(T + 1)k + V T 2?
+
p
mD22 PT
t=1 kQ(t)k, 8k 2 {1, 2, . . . , m} where D1 and D2 are defined in Assumption 1.
2?
This corollary further justifies why Algorithm 1 intends to minimize drift (t). As illustrated in
the next section, controlled drift can often lead to boundedness of a stochastic process. Thus, the
intuition of minimizing drift (t) is to yield small kQ(t)k bounds.
3
Expected Performance Analysis of Algorithm 1
p
This section shows that if we choose V =p T and ? = T in Algorithm 1, then both expected regret
and expected constraint violations are O( T ).
3.1
A Drift Lemma for Stochastic Processes
Let {Z(t), t 0} be a discrete time stochastic process adapted3 to a filtration {F(t), t 0}. For
example, Z(t) can be a random walk, a Markov chain or a martingale. The drift analysis is the
method of deducing properties, e.g., recurrence, ergodicity, or boundedness, about Z(t) from its drift
E[Z(t + 1) Z(t)|F(t)]. See [6, 10] for more discussions or applications on drift analysis. This
paper proposes a new drift analysis lemma for stochastic processes as follows:
Lemma 5. Let {Z(t), t 0} be a discrete time stochastic process adapted to a filtration {F(t), t
0} with Z(0) = 0 and F(0) = {;, ?}. Suppose there exists an integer t0 > 0, real constants ? > 0,
max > 0 and 0 < ? ? max such that
|Z(t + 1)
E[Z(t + t0 )
Z(t)| ?
Z(t)|F(t)] ?
(9)
max ,
?
t0
max ,
t0 ?,
if Z(t) < ?
.
if Z(t) ?
(10)
hold for all t 2 {1, 2, . . .}. Then, the following holds
1. E[Z(t)] ? ? + t0
3
max
+ t0
4
2
max
?
log
?8
2
max
?2
?
, 8t 2 {1, 2, . . .}.
Random variable Y is said to be adapted to -algebra F if Y is F -measurable. In this case, we often write
Y 2 F. Similarly, random process {Z(t)} is adapted to filtration {F(t)} if Z(t) 2 F(t), 8t. See e.g. [7].
5
2. For any constant 0 < ? < 1, we have Pr(Z(t)
?8 2 ?
4 2
4 2
? + t0 max + t0 max
log ?max
+ t0 max
log( ?1 ).
2
?
?
z) ? ?, 8t 2 {1, 2, . . .} where z =
The above lemma is proven in Supplement 7.4 and provides both expected and high probability
bounds for stochastic processes based on a drift condition. It will be used to establish upper bounds of
virtual queues kQ(t)k, which further leads to expected and high probability constraint performance
bounds of our algorithm. For a given stochastic process Z(t), it is possible to show the drift condition
(10) holds for multiple t0 with different ? and ?. In fact, we will show in Lemma 7 that kQ(t)k
yielded by Algorithm 1 satisfies (10) for any integer t0 > 0 by selecting ? and ? according to t0 .
One-step drift conditions, corresponding to the special case t0 = 1 of Lemma 5, have been previously
considered in [10, 20]. However, Lemma 5 (with general t0 > 0) allows us to choose the best t0 in
performance analysis such that sublinear regret and constraint violation bounds are possible.
3.2
Expected Constraint Violation Analysis
Define filtration {W(t), t
0} with W(0) = {;, ?} and W(t) = (!(1), . . . , !(t)) being the
-algebra generated by random samples {!(1), . . . , !(t)} up to round t. From the update rule
in Algorithm 1, we observe that x(t + 1) is a deterministic function of f t (?), g(?; !(t)) and Q(t)
where Q(t) is further a deterministic function of Q(t 1), g(?; !(t 1)), x(t) and x(t 1). By
inductions, it is easy to show that (x(t)) ? W(t 1) and (Q(t)) ? W(t 1) for all t 1 where
(Y ) denotes the -algebra generated by random variable Y . For fixed t 1, since Q(t) is fully
determined by !(? ), ? 2 {1, 2, . . . , t 1} and !(t) are i.i.d., we know gt (x) is independent of Q(t).
This is formally summarized in the next lemma.
? (x? ) = E! [g(x? ; !)] ? 0, then Algorithm 1 guarantees:
Lemma 6. If x? 2 X0 satisfies g
E[Qk (t)gkt (x? )] ? 0, 8k 2 {1, 2, . . . , m}, 8t
(11)
1.
Proof. Fix k 2 {1, 2, . . . , m} and t
1. Since gkt (x? ) = gk (x? ; !(t)) is independent
of Qk (t), which is determined by {!(1), . . . , !(t
1)}, it follows that E[Qk (t)gkt (x? )] =
(a)
E[Qk (t)]E[gkt (x? )] ? 0, where (a) follows from the fact that E[gkt (x? )] ? 0 and Qk (t)
0.
To establish a bound on constraint violations, by Corollary 2, it suffices to derive upper bounds for
kQ(t)k. In this subsection, we derive upper bounds for kQ(t)k by applying the new drift lemma
(Lemma 5) developed at the beginning of this section. The next lemma shows that random process
Z(t) = kQ(t)k satisfies the conditions in Lemma 5.
Lemma 7. Let t0 > 0 be an arbitrary integer. At each round t 2 {1, 2, . . . , } in Algorithm 1, the
following holds
p
kQ(t + 1)k kQ(t)k ?G + mD2 R, and
?
p
t0 (G + mD2 R), if kQ(t)k < ?
E[kQ(t + t0 )k kQ(t)k W(t 1)] ?
,
t0 2? ,
if kQ(t)k ?
p
p
2
2V D1 R+[G+ mD2 R]2
where ? = 2? t0 + (G + mD2 R)t0 + 2?R
, m is the number of constraint
t0 ? +
?
functions; D1 , D2 , G and R are defined in Assumption 1; and ? is defined in Assumption 2. (Note
that ? < G by the definition of G.)
Lemma 7 (proven in Supplement
5 to random
process Z(t) = kQ(t)k
p 7.5) allows us to apply Lemma
p
p
p
and obtain E[kQ(t)k] = O( T ), 8t by taking
t
=
d
T
e,
V
=
T
and
? = T , where d T e
0
p
represents the smallest integer no less than T . By
pCorollary 2, this further implies the expected
PT
constraint violation bound E[ t=1 gk (x(t))] ? O( T ) as summarized in the next theorem.
p
Theorem 1 (Expected Constraint Violation Bound). If V = T and ? = T in Algorithm 1, then for
all T 1, we have
E[
T
X
t=1
p
gkt (x(t))] ? O( T ), 8k 2 {1, 2, . . . , m}.
where the expectation is taken with respect to all !(t).
6
(12)
Proof. Define random process Z(t) with Z(0) = 0 and Z(t) = kQ(t)k, t
1 and filtration
F(t) with F(0) = {;, ?} and F(t) = W(t 1), t
1. Note that Z(t) ispadapted to F(t). By
Lemma 7, Z(t) satisfies the conditions in Lemma 5 with max = G + mD2 R, ? = 2? and
p
p
2
2V D1 R+[G+ mD2 R]2
? = 2? t0 + (G + mD2 R)t0 + 2?R
. Thus, by part (1) of Lemma 5, for all
t0 ? +
?
p
p
2
2V D1 R+[G+ mD2 R]2
?
t 2 {1, 2, . . .}, we have E[kQ(t)k] ? 2 t0 + 2(G + mD2 R)t0 + 2?R
+
+
t
?
?
0
p
p
p
p
8[G+ mD2 R]2
32[G+ mD2 R]2
t0
log[
]. Taking t0 = d T e, V = T and ? = T , we have
?
?2
p
E[kQ(t)k] ? O( T ) for all t 2 {1, 2, . . .}.
p
PT
Fix T
1. By Corollary 2 (with V = T and ? = T ) , we have t=1 gkt (x(t)) ? kQ(T +
p
p
2 P
mD
T
1)k + T D21 D2 + 2T 2 t=1 kQ(t)k, 8k 2 {1, 2, . . . , m}. Taking expectations on both sides and
p
p
PT
substituting E[kQ(t)k] = O( T ), 8t into it yields E[ t=1 gkt (x(t))] ? O( T ).
3.3
Expected Regret Analysis
The next lemma (proven in Supplement 7.6) refines Lemma 4 and is useful to analyze the regret.
Lemma 8. Let z 2 X0 be arbitrary. For all T 1, Algorithm 1 guarantees
T
T
X
X
T
m
?
p
?
V D12
1
1 X? X
2 T
t
f t (x(t)) ?
f t (z) + R2 +
T + [G + mD2 R]
+
Qk (t)gk (z)
(13)
t=1
V
|
t=1
4?
2
{z
(I)
V
}
V
|
t=1
k=1
{z
(II)
}
where m is the number of constraint functions; and D1 , D2 , G and R are defined in Assumption 1.
p
p
Note that if we take V = T and ? = T , then term (I) in (13) is O( T ). Recall that the expectation
of term (II) in (13) with z = x? is non-positive by Lemma 6. The expected regret bound of Algorithm
1 follows by taking expectations on both sides of (13) and is summarized in the next theorem.
? (x? ) ? 0,
Theorem 2 (Expected Regret Bound). Let xp? 2 X0 be any fixed solution that satisfies g
PT
?
t
e.g., x = argminx2X t=1 f (x). If V = T and ? = T in Algorithm 1, then for all T 1,
E[
T
X
t=1
f t (x(t))] ? E[
T
X
p
f t (x? )] + O( T ).
t=1
where the expectation is taken with respect to all !(t).
PT
PT
1. Taking z = x? in Lemma 8 yields t=1 f t (x(t)) ? t=1 f t (x? ) + V? R2 +
?
?
PT
Pm
p
V
1
1
2T
t
?
t=1
k=1 Qk (t)gk (x ) . Taking expectations on both sides
4? T + 2 [G + mD2 R] V + V
PT
P
p
D2
T
and using (11) yields t=1 E[f t (x(t))] ? t=1 E[f t (x? )] + R2 V? + 41 V? T + 12 [G + mD2 R]2 VT .
p
p
PT
P
T
Taking V = T and ? = T yields t=1 E[f t (x(t))] ? t=1 E[f t (x? )] + O( T ).
Proof. Fix T
D12
3.4
Special Case Performance Guarantees
Theorems 1 and 2 provide expected performance guarantees of Algorithm 1 for OCO with stochastic
constraints. The results further imply the performance guarantees in the following special cases:
? OCO with long term constraints: In this case, gk (x; !(t)) ? gk (x) and there is no randomness. Thus,p
the expectations in Theorems 1 and
p 2 disappear. For this problem, Algorithm 1 can
achieve O( T ) (deterministic) regret and O( T ) (deterministic) constraint violations.
? Stochastic constrained convex optimization: Note that i.i.d. time-varying f (x; !(t)) is a
special case of arbitrarily-varying f t (x) as considered in our OCO setting. Thus, Theorems 1
and 2 still hold when Algorithm 1 is applied to stochastic
constrained convex optimization.
That
p
p
PT
PT
PT
is, t=1 E[f t (x(t))] ? t=1 E[f t (x? )] + O( T ) and t=1 E[gkt (x(t))] ? O( T ), 8k 2
{1, 2, . . . , n}. This online performance
guarantee also implies Algorithm 1 can be used as a
p
(batch) offline algorithm with O(1/ T ) convergence for stochastic constrained convex optimizaPT
tion. That is, after running Algorithm 1 for T slots, if we use x(T ) = T1 t=1 x(t) as a fixed
solution, then E[f (x(T ); !)] = E[f t (x(T ))] ? E[f t (x? )] + O( p1T ) and E[gk (x(T ); !)] =
7
E[gkt (x(T ))] ? O( p1T ), 8k 2 {1, 2, . . . , m} with t
T + 1 by the i.i.d. property of each
t
t
f and g and Jensen?s inequality. If we use Algorithm 1 as a (batch) offline algorithm, its
performance ties with the algorithm developed in [15], which is by design a (batch) offline
algorithm and can only solve stochastic optimization with a single constraint function.
? Deterministic constrained convex optimization: Similarly to OCO with long term constraints, the expectations in Theorems 1 and 2 disappear in this case since f t (x) ? f (x)
PT
and gk (x; !(t)) ? gk (x). If we use x(T ) = T1 t=1 x(t) as the solution, then f (x(T )) ?
f (x? ) + O( p1T ) and gk (x(T )) ? O( p1T ), which follows by dividing inequalities in Theorems 1 and 2 by T on both sides and applying Jensen?s inequality. Thus, Algorithm 1 solves
deterministic constrained convex optimization with O( p1T ) convergence.
4
High Probability Performance Analysis
p
This section shows that if we choose V = Tpand ? = T in Algorithm 1, then for any 0 < < 1,
with probability at least 1
, regret is O( T log(T ) log1.5 ( 1 )) and constraint violations are
p
O T log(T ) log( 1 ) .
4.1
High Probability Constraint Violation Analysis
Similarly to the expected constraint violation analysis, we can use part (2) of the new drift lemma
(Lemma 5) to obtain a high probability bound of kQ(t)k, which together with Corollary 2 leads to a
high probability constraint violation bound summarized in Theorem 3 (proven in Supplement 7.7).
p
Theorem 3 (High Probability Constraint Violation Bound). Let 0 < < 1 be arbitrary. If V = T
and ? = T in Algorithm 1, then for all T 1 and all k 2 {1, 2, . . . , m}, we have
T
?X
p
1 ?
Pr
gk (x(t)) ? O T log(T ) log( )
1
.
t=1
4.2
High Probability Regret Analysis
To obtain a high probability regret bound from Lemma 8, it remains to derive a high probability
bound of term (II) in (13) with z = x? . The main challenge is that term (II) is a supermartingale with
unbounded differences (due to the possibly unbounded virtual queues Qk (t)). Most concentration
inequalities, e.g., the Hoeffding-Azuma inequality, used in high probability performance analysis of
online algorithms are restricted to martingales/supermartingales with bounded differences. See for
example [4, 2, 16]. The following lemma considers supermartingales with unbounded differences.
Its proof (provided in Supplement 7.8) uses the truncation method to construct an auxiliary wellbehaved supermargingale. Similar proof techniques are previously used in [26, 24] to prove different
concentration inequalities for supermartingales/martingales with unbounded differences.
Lemma 9. Let {Z(t), t
0} be a supermartingale adapted to a filtration {F(t), t
0} with
Z(0) = 0 and F(0) = {;, ?}, i.e., E[Z(t + 1)|F(t)] ? Z(t), 8t 0. Suppose there exits a constant
c > 0 such that {|Z(t + 1) Z(t)| > c} ? {Y (t) > 0}, 8t 0, where Y (t) is process with Y (t)
adapted to F(t) for all t 0. Then, for all z > 0, we have
t 1
X
2
2
Pr(Z(t) z) ? e z /(2tc ) +
Pr(Y (? ) > 0), 8t 1.
? =0
Note that if Pr(Y (t) > 0) = 0, 8t 0, then Pr({|Z(t + 1) Z(t)| > c}) = 0, 8t 0 and Z(t) is a
supermartingale with differences bounded by c. In this case, Lemma 9 reduces to the conventional
Hoeffding-Azuma inequality.
The next theorem (proven in Supplement 7.9) summarizes the high probability regret performance of
Algorithm 1 and follows from Lemmas 5-9 .
Theorem 4 (High Probability Regret Bound). Let x? 2 X0 be any fixed solution that p
satisfies
PT
?
?
t
?
g(x ) ? 0, e.g., x = argminx2X t=1 f (x). Let 0 < < 1 be arbitrary. If V = T and
8
? = T in Algorithm 1, then for all T 1, we have
T
T
?X
X
p
1 ?
Pr
f t (x(t)) ?
f t (x? ) + O( T log(T ) log1.5 ( ))
t=1
5
1
.
t=1
Experiment: Online Job Scheduling in Distributed Data Centers
Consider a geo-distributed data center infrastructure consisting of one front-end job router and 100
geographically distributed servers, which are located at 10 different zones to form 10 clusters (10
servers in each cluster). See Fig. 1(a) for an illustration. The front-end job router receives job
tasks and schedules them to different servers to fulfill the service. To serve the assigned jobs, each
server purchases power (within its capacity) from its zone market. Electricity market prices can vary
significantly across time and zones. For example, see Fig. 1(b) for a 5-minute average electricity
price trace (between 05/01/2017 and 05/10/2017) at New York zone CENTRL [1]. This problem
is to schedule jobs and control power levels at each server in real time such that all incoming jobs
are served and electricity cost is minimized. In our experiment, each server power is adjusted every
5 minutes, which is called a slot. (In practice, server power can not be adjusted too frequently due
to hardware restrictions and configuration delay.) Let x(t) = [x1 (t), . . . , x100 (t)] be the power
vector at slot t, where each xi (t) must be chosen from an interval [xmin
, xmax
] restricted by the
i
i
hardware, and the service rate at each server i satisfies ?i (t) = hi (xi (t)), where hi (?) is an increasing
concave function. At each slot t, the job router schedules ?i (t) amount of jobs to server i. The
P100
electricity cost at slot t is f t (x(t)) = i=1 ci (t)xi (t) where ci (t) is the electricity price at server
i?s zone. We use ci (t) from real-world 5-minute average electricity price data at 10 different zones
in New York city between 05/01/2017 and 05/10/2017 obtained from NYISO [1]. At each slot
t, the incoming job is given by !(t) and satisfies a Poisson distribution. Note that the amount of
incoming jobs and electricity price ci (t) are unknown to us at the beginning of each slot t but can
be observed at the end of each slot. This is an example of OCO with stochastic constraints, where
we aim to minimize the electricity cost subject to the constraint that incoming jobs must be served
in time. In particular, at each round t, we receive loss function f t (x(t)) and constraint function
P100
g t (x(t)) = !(t)
i=1 hi (xi (t)).
We compare our proposed algorithm with 3 baselines: (1) best fixed decision in hindsight; (2) react
[8] and (3) low-power [22]. Both ?react" and ?low-power" are popular power control strategies
used in distributed data centers. See Supplement 7.10 for more details of these 2 baselines and our
experiment. Fig. 1(c)(d) plot the performance of 4 algorithms, where the running average is the
time average up to the current slot. Fig. 1(c) compares electricity cost while Fig. 1(d) compares
unserved jobs. (Unserved jobs accumulate if the service rate provided by an algorithm is less than
the job arrival rate, i.e., the stochastic constraint is violated.) Fig. 1(c)(d) show that our proposed
algorithm performs closely to the best fixed decision in hindsight over time, both in electricity cost
and constraint violations. ?React" performs well in serving job arrivals but yields larger electricity
cost, while ?low-power" has low electricity cost but fails to serve job arrivals.
Electricity market price
Running average electricity cost
450
Running average unserved jobs
15000
1200
400
300
10000
Cost (dollar)
Price (dollar/MWh)
350
250
200
5000
150
Our algorithm
Best fixed strategy in hindsight
React (Gandhi et al. 2012)
Low-power (Qureshi et al. 2009)
100
50
0
0
0
500
1000
1500
2000
2500
(a)
(b)
800
Our algorithm
Best fixed decision in hindsight
React (Gandhi et al. 2012)
Low-power (Qureshi et al. 2009)
600
400
200
0
-200
0
500
1000
1500
2000
Number of slots (each 5 min)
Number of slots (each 5 min)
Unserved jobs (per slot)
1000
(c)
2500
0
500
1000
1500
2000
2500
Number of slots (each 5 min)
(d)
Figure 1: (a) Geo-distributed data center infrastructure; (b) Electricity market prices at zone CENTRAL New York; (c) Running average electricity cost; (d) Running average unserved jobs.
6
Conclusion
This paper studies OCO with stochastic constraints, where the objective function varies arbitrarily but
thep
constraint functions are i.i.d. over time. A novel learning
p algorithm is developed that guarantees
O( T ) expected regret and constraint violations and O( T log(T )) high probability regret and
constraint violations.
9
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6,348 | 6,742 | Max-Margin Invariant Features from Transformed
Unlabeled Data
Dipan K. Pal, Ashwin A. Kannan?, Gautam Arakalgud?, Marios Savvides
Department of Electrical and Computer Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
{dipanp,aalapakk,garakalgud,marioss}@cmu.edu
Abstract
The study of representations invariant to common transformations of the data
is important to learning. Most techniques have focused on local approximate
invariance implemented within expensive optimization frameworks lacking explicit
theoretical guarantees. In this paper, we study kernels that are invariant to a unitary
group while having theoretical guarantees in addressing the important practical
issue of unavailability of transformed versions of labelled data. A problem we
call the Unlabeled Transformation Problem which is a special form of semisupervised learning and one-shot learning. We present a theoretically motivated
alternate approach to the invariant kernel SVM based on which we propose MaxMargin Invariant Features (MMIF) to solve this problem. As an illustration, we
design an framework for face recognition and demonstrate the efficacy of our
approach on a large scale semi-synthetic dataset with 153,000 images and a new
challenging protocol on Labelled Faces in the Wild (LFW) while out-performing
strong baselines.
1
Introduction
It is becoming increasingly important to learn well generalizing representations that are invariant
to many common nuisance transformations of the data. Indeed, being invariant to intra-class
transformations while being discriminative to between-class transformations can be said to be one of
the fundamental problems in pattern recognition. The nuisance transformations can give rise to many
?degrees of freedom? even in a constrained task such as face recognition (e.g. pose, age-variation,
illumination etc.). Explicitly factoring them out leads to improvements in recognition performance as
found in [10, 7, 6]. It has also been shown that that features that are explicitly invariant to intra-class
transformations allow the sample complexity of the recognition problem to be reduced [2]. To this
end, the study of invariant representations and machinery built on the concept of explicit invariance is
important.
Invariance through Data Augmentation. Many approaches in the past have enforced invariance by
generating transformed labelled training samples in some form such as [13, 17, 19, 9, 15, 4]. Perhaps,
one of the most popular method for incorporating invariances in SVMs is the virtual support method
(VSV) in [18], which used sequential runs of SVMs in order to find and augment the support vectors
with transformed versions of themselves.
Indecipherable transformations in data leads to shortage of transformed labelled samples. The
above approaches however, assume that one has explicit knowledge about the transformation. This
is a strong assumption. Indeed, in most general machine learning applications, the transformation
?
Authors contributed equally
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
present in the data is not clear and cannot be modelled easily, e.g. transformations between different
views of a general 3D object and between different sentences articulated by the same person. Methods
which work on generating invariance by explicitly transforming or augmenting labelled training data
cannot be applied to these scenarios. Further, in cases where we do know the transformations that
exist and we actually can model them, it is difficult to generate transformed versions of very large
labelled datasets. Hence there arises an important problem: how do we train models to be invariant to
transformations in test data, when we do not have access to transformed labelled training samples ?
Transformed unlabeled data
Non-transformed labeled data
Train
Train
Availability of unlabeled transformed data.
Although it is difficult to obtain or generate
transformed labelled data (due to the reasons
mentioned above), unlabeled transformed data
is more readily available. For instance, if different views of specific objects of interest are not
available, one can simply collect views of general objects. Also, if different sentences spoken
by a specific group of people are not available,
one can simply collect those spoken by members of the general population. In both these
scenarios, no explicit knowledge or model of
the transformation is needed, thereby bypassing
the problem of indecipherable transformations.
This situation is common in vision e.g. only
unlabeled transformed images are observed, but
has so far mostly been addressed by the community by intense efforts in large scale data collection. Note that the transformed data that is
collected is not required to be labelled. We now
are in a position to state the central problem that
this paper addresses.
Test image
invariant to
not invariant to
Figure 1: Max-Margin Invariant Features (MMIF) can
solve an important problem we call the Unlabeled Transformation Problem. In the figure, a traditional classifier
F (x) "learns" invariance to nuisance transformations
directly from the labeled dataset X . On the other hand,
our approach (MMIF) can incorporate additional invariance learned from any unlabeled data that undergoes the
nuisance transformation of interest.
The Unlabeled Transformation (UT) Problem:
Having access to transformed versions of the training unlabeled data but not of labelled data, how
do we learn a discriminative model of the labelled data, while being invariant to transformations
present in the unlabeled data ?
Overall approach. The approach presented in this paper however (see Fig. 1), can solve this problem
and learn invariance to transformations observed only through unlabeled samples and does not need
labelled training data augmentation. We explicitly and simultaneously address both problems of
generating invariance to intra-class transformation (through invariant kernels) and being discriminative
to inter or between class transformations (through max-margin classifiers). Given a new test sample,
the final extracted feature is invariant to the transformations observed in the unlabeled set, and thereby
generalizes using just a single example. This is an example of one-shot learning.
Prior Art: Invariant Kernels. Kernel methods in machine learning have long been studied to
considerable depth. Nonetheless, the study of invariant kernels and techniques to extract invariant
features has received much less attention. An invariant kernel allows the kernel product to remain
invariant under transformations of the inputs. Most instances of incorporating invariances focused
on local invariances through regularization and optimization such as [18, 19, 3, 21]. Some other
techniques were jittering kernels [17, 3] and tangent-distance kernels [5], both of which sacrificed
the positive semi-definite property of its kernels and were computationally expensive. Though these
methods have had some success, most of them still lack explicit theoretical guarantees towards
invariance. The proposed invariant kernel SVM formulation on the other hand, develops a valid PSD
kernel that is guaranteed to be invariant. [4] used group integration to arrive at invariant kernels
but did not address the Unlabeled Transformation problem which our proposed kernels do address.
Further, our proposed kernels allow for the formulation of the invariant SVM and application to large
scale problems. Recently, [14] presented some work with invariant kernels. However, unlike our
non-parametric formulation, they do not learn the group transformations from the data itself and
assume known parametric transformations (i.e. they assume that transformation is computable).
Key ideas. The key ideas in this paper are twofold.
2
1. The first is to model transformations using unitary groups (or sub-groups) leading to unitarygroup invariant kernels. Unitary transforms allow the dot product to be preserved and
allow for interesting generalization properties leading to low sample complexity and also
allow learning transformation invariance from unlabeled examples (thereby solving the
Unlabeled Transformation Problem). Classes of learning problems, such as vision, often
have transformations belonging to a unitary-group, that one would like to be invariant
towards (such as translation and rotation). In practice however, [8] found that invariance to
much more general transformations not captured by this model can been achieved.
2. Secondly, we combine max-margin classifiers with invariant kernels leading to non-linear
max-margin unitary-group invariant classifiers. These theoretically motivated invariant
non-linear SVMs form the foundation upon which Max-Margin Invariant Features (MMIF)
are based. MMIF features can effectively solve the important Unlabeled Transformation
Problem. To the best of our knowledge, this is the first theoretically proven formulation of
this nature.
Contributions. In contrast to many previous studies on invariant kernels, we study non-linear
positive semi-definite unitary-group invariant kernels guaranteeing invariance that can address the UT
Problem. One of our central theoretical results to applies group integration in the RKHS. It builds on
the observation that, under unitary restrictions on the kernel map, group action in the input space is
reciprocated in the RKHS. Using the proposed invariant kernel, we present a theoretically motivated
approach towards a non-linear invariant SVM that can solve the UT Problem with explicit invariance
guarantees. As our main theoretical contribution, we showcase a result on the generalization of
max-margin classifiers in group-invariant subspaces. We propose Max-Margin Invariant Features
(MMIF) to learn highly discriminative non-linear features that also solve the UT problem. On the
practical side, we propose an approach to face recognition to combine MMIFs with a pre-trained
deep learning feature extractor (in our case VGG-Face [12]). MMIF features can be used with deep
learning whenever there is a need to focus on a particular transformation in data (in our application
pose in face recognition) and can further improve performance.
2
Unitary-Group Invariant Kernels
Premise: Consider a dataset of normalized samples along with labels X = {xi }, Y = {yi } ?i ?
1...N with x ? Rd and y ? {+1, ?1}. We now introduce into the dataset a number of unitary transformations g part of a locally compact unitary-group G. We note again that the set of transformations
under consideration need not be the entire unitary group. They could very well be a subgroup. Our
augmented normalized dataset becomes {gxi , yi } ?g ? G ?i. For clarity, we denote by gx the action
of group element g ? G on x, i.e. gx = g(x). We also define an orbit of x under G as the set
XG = {gx} ?g ? G. Clearly, X ? XG . An invariant function is defined as follows.
Definition 2.1 (G-Invariant Function). For any group G, we define a function f : X ? Rn to be
G-invariant if f (x) = f (gx) ?x ? X ?g ? G.
One method of generating an invariant towards a group is through group integration. Group integration
has stemmed from classical invariant theory and can be shown to be a projection onto a G-invariant
subspace for vector spaces. In such a space x = gx ?g ? G and thus the representation x is invariant
under the transformation of any element from the group G. This is ideal for recognition problems
where one would want to be discriminative to between-class transformations (for e.g. between distinct
subjects in face recognition) but be invariant to within-class transformations (for e.g. different images
of the same subject). The set of transformations we model as G are the within-class transformations
that we would like to be invariant towards. An invariant to any group G can be generated through the
following basic (previously) known property (Lemma 2.1) based on group integration.
d
Lemma 2.1. (Invariance Property) Given a vector ? ?
R R , and any
R affine group G, for any fixed
0
0
g ? G and a normalized Haar measure dg, we have g G g? dg = G g? dg
The Haar measure (dg) exists for every locally compact group and is unique up to a positive
multiplicative constant (hence
normalized). A similar property holds for discrete groups. Lemma 2.1
R
results in the quantity G g? dg enjoy global invariance (encompassing all elements) to group G.
This property allows one to generate a G-invariant subspace in the inherent space Rd through group
integration. In practice, the integral corresponds to a summation over transformed samples. The
3
following two lemmas (novel results, and part
R of our contribution) (Lemma 2.2 and 2.3) showcase
elementary properties of the operator ? = G g dg for a unitary-group G 2 . These properties would
prove useful in the analysis of unitary-group invariant kernels and features.
R
Lemma 2.2. If ? = G g dg for unitary G, then ?T = ?
R
Lemma 2.3. (Unitary Projection) If ? = G g dg for any affine G, then ?? = ?, i.e. it is a
projection operator. Further, if G is unitary, then h?, ?? 0 i = h??, ? 0 i ??, ? 0 ? Rd
Sample Complexity and Generalization. On applying the operator ? to the dataset X , all points
in the set {gx | g ? G} for any x ? X map to the same point ?x in the G-invariant subspace
thereby reducing the number of distinct points by a factor of |G| (the cardinality of G, if G is finite).
Theoretically, this would drastically reduce sample complexity while preserving linear feasibility
(separability). It is trivial to observe that a perfect linear separator learned in X? = {?x | x ?
X } would also be a perfect separator for XG , thus in theory achieving perfect generalization.
Generalization here refers to the ability to perform correct classification even in the presence of the
set of transformations G. We prove a similar result for Reproducing Kernel Hilbert Spaces (RKHS)
in Section 2.2. This property is theoretically powerful since cardinality of G can be large. A classifier
can avoid having to observe transformed versions {gx} of any x and yet generalize perfectly.
is
The case of Face Recognition. As an illustration, if the group G of transformations considered
R
pose (it is hypothesized that small changes in pose can be modeled as unitary [10]), then ? = G g dg
represents a pose invariant subspace. In theory, all poses of a subject will converge to the same point
in that subspace leading to near perfect pose invariant recognition.
We have not yet leveraged the power of the unitary structure of the groups which is also critical in
generalization to test cases as we would see later. We now present our central result showcasing that
unitary kernels allow the unitary group action to reciprocate in a Reproducing Kernel Hilbert Space.
This is critical to set the foundation for our core method called Max-Margin Invariant Features.
2.1
Group Actions Reciprocate in a Reproducing Kernel Hilbert Space
Group integration provides exact invariance as seen in the previous section. However, it requires
the group structure to be preserved, i.e. if the group structure is destroyed, group integration does
not provide an invariant function. In the context of kernels, it is imperative that the group relation
between the samples in XG be preserved in the kernel Hilbert space H corresponding to some kernel
k with a mapping ?. If the kernel k is unitary in the following sense, then this is possible.
Definition 2.2 (Unitary Kernel). A kernel k(x, y) = h?(x), ?(y)i is a unitary kernel if, for a unitary
group G, the mapping ?(x) : X ? H satisfies h?(gx), ?(gy)i = h?(x), ?(y)i ?g ? G, ?x, y ? X .
The unitary condition is fairly general, a common class of unitary kernels is the RBF kernel. We now
define a transformation within the RKHS itself as gH : ?(x) ? ?(gx) ??(x) ? H for any g ? G
where G is a unitary group. We then have the following result of significance.
Theorem 2.4. (Covariance in the RKHS) If k(x, y) = h?(x), ?(y)i is a unitary kernel in the sense of
Definition 2.2, then gH is a unitary transformation, and the set GH = {gH | gH : ?(x) ? ?(gx) ?g ?
G} is a unitary-group in H.
Theorem 2.4 shows that the unitary-group structure is preserved in the RKHS. This paves the way for
new theoretically motivated approaches to achieve invariance to transformations in the RKHS. There
have been a few studies on group invariant kernels [4, 10]. However, [4] does not examine whether
the unitary group structure is actually preserved in the RKHS, which is critical. Also, DIKF was
recently proposed as a method utilizing group structure under the unitary kernel [10]. Our result is a
generalization of the theorems they present. Theorem 2.4 shows that since the unitary group structure
is preserved in the RKHS, any method involving group integration would be invariant in the original
space. The preservation of the group structure allows more direct group invariance results to be
applied in the RKHS. It also directly allows one to formulate a non-linear SVM while guaranteeing
invariance theoretically leading to Max-Margin Invariant Features.
2
All proofs are presented in the supplementary material
4
2.2
Invariant Non-linear SVM: An Alternate Approach Through Group Integration
We now apply the group integration approach to the kernel SVM. The decision function of SVMs
can be written in the general form as f? (x) = ? T ?(x) + b for some bias b ? R (we agglomerate all
parameters of f in ?) where ? is the kernel feature map, i.e. ? : X ? H. Reviewing the SVM, a
maximum margin separator is found by minimizing loss functions such as the hinge loss along with a
regularizer. In order to invoke invariance, we can now utilize group integration in the the kernel space
H using Theorem 2.4. All points in the set {gx ? XG } get mapped to ?(gx) = gH ?(x) for a given
g ? G in
R the input space X . Group integration then results in a G-invariant subspace within H through
?H = GH gH dgH using Lemma 2.1. Introducing Lagrange multipliers ? = (?1 , ?2 ...?N ) ? RN ,
the dual formulation (utilizing Lemma 2.2 and Lemma 2.3) then becomes
min ?
?
X
i
?i +
1X
yi yj ?i ?j h?H ?(xi ), ?H ?(xj )i
2 i,j
(1)
P
under the constraints
0 ? ?i ? N1 ?i. The SVM separator is then given by
i ?i yi = 0,
P
?
?
?H = ?H ? = i yi ?i ?H ?(xi ) thereby existing in the GH -invariant (or equivalently G-invariant)
subspace ?H within H (since g ? gH is a bijection). Effectively, the SVM observes samples from
?
X?H = {x | ?(x) = ?H ?(u), ?u ? XG } and therefore ?H
enjoys exact global invariance to G.
?
Further, ?H ? is a maximum-margin separator of {?(XG )} (i.e. the set of all transformed samples).
This can be shown by the following result.
For a unitary group G and unitary kernel k(x, y) = h?(x), ?(y)i,
Theorem 2.5. (Generalization)
R
?
if ?H
= ?H ? ? = ( GH gH dgH ) ? ? is a perfect separator for {?H ?(X )} = {?H ?(x) | ?x ? X },
then ?H ? ? is also a perfect separator for {?(XG )} = {?(x) | x ? XG } with the same margin.
Further, a max-margin separator of {?H ?(X )} is also a max-margin separator of {?(XG )}.
The invariant non-linear SVM in objective 1, observes samples in the form of ?H ?(x) and obtains a
max-margin separator ?H ? ? . This allows for the generalization properties of max-margin classifiers
to be combined with those of group invariant classifiers. While being invariant to nuisance transformations, max-margin classifiers can lead to highly discriminative features (more robust than DIKF
[10] as we find in our experiments) that are invariant to within-class transformations.
Theorem 2.5 shows that the margins of ?(XG ) and {?H ?(XG )} are deeply related and implies that
?H ?(x) is a max-margin separator for both datasets. Theoretically, the invariant non-linear SVM is
able to generalize to XG on just observing X and utilizing prior information in the form of G for all
unitary kernels k. This is true in practice for linear kernels. For non-linear kernels in practice, the
invariant SVM still needs to observe and integrate over transformed training inputs.
Leveraging unitary group properties. During test time to achieve invariance, the SVM would
require to observe and integrate over all possible transformations of the test sample. This is a huge
computational and design bottleneck. We would ideally want to achieve invariance and generalize
by observing just a single test sample, in effect perform one shot learning. This would not only
be computationally much cheaper but make the classifier powerful owing to generalization to full
transformed orbits of test samples by observing just that single sample. This is where unitarity of g
helps and we leverage it in the form of the following
Lemma.
R
Lemma 2.6. (Invariant Projection) If ? = G g dg for any unitary group G, then for any fixed g 0 ? G
(including the identity element) we have h?x0 , ?? 0 i = hg 0 x0 , ?? 0 i ??, ? 0 ? Rd
Assuming ?? 0 is the learned SVM classifier, Lemma 2.6 shows that for any test x0 , the invariant dot
product h?x0 , ?? 0 i which involves observing all transformations of x0 is equivalent to the quantity
hg 0 x0 , ?? 0 i which involves observing only one transformation of x0 . Hence one can model the entire
orbit of x0 under G by a single sample g 0 x0 where g 0 ? G can be any particular transformation
including identity. This drastically reduces sample complexity and vastly increases generalization
capabilities of the classifier since one only need to observe one test sample to achieve invariance
Lemma 2.6 also helps us in saving computation, allowing us to apply the computationally expensive
? (group integration) operation only once on he classifier and not the test sample. Thus, the kernel in
the Invariant SVM formulation can be replaced by the form k? (x, y) = h?(x), ?H ?(y)i.
For kernels in general, the GH -invariant subspace cannot be explicitly
R computed since it lies in the
RKHS. It is only implicitly projected upon through ?H ?(xi ) = G ?(gxi )dgH . It is important to
5
Integration over the group
(pooling)
Class 1
Test Image
Class 2
Class 3
Kernel Invariant Feature
Test Image
Class 4
(a) Invariant kernel feature extraction
(b) SVM feature extraction leading to MMIF features
Figure 2: MMIF Feature Extraction. (a) l(x) denotes the invariant kernel feature of any x which is invariant
to the transformation G. Invariance is generated by group integration (or pooling). The invariant kernel feature
learns invariance form the unlabeled transformed template set TG . Also, the faces depicted are actual samples
from the large-scale mugshots data (? 153, 000 images). (b) Once the invariant features have been extracted
for the labelled non-transformed dataset X , then the SVMs learned act as feature extractors. Each binary class
SVM (different color) was trained on the invariant kernel feature of a random subset of l(X ) with random class
assignments. The final MMIF feature for x is the concatenation of all SVM inner-products with l(x).
note that during testing however, the SVM formulation will be invariant to transformations of the test
sample regardless of a linear or non-linear kernel.
Positive
R Semi-Definiteness. The G-invariant kernel map is now of the form k? (x, y) =
h?(x), G ?(gy)dgH i. This preserves the positive semi-definite property of the kernel k while
guaranteeing global invariance to unitary transformations., unlike jittering kernels [17, 3] and
tangent-distance kernels [5]. If we wish to include invariance to scaling however (in the sense of
scaling an image), then we would lose positive-semi-definiteness (it is also not a unitary transform).
Nonetheless, [20] show that conditionally positive definite kernels still exist for transformations
including scaling, although we focus of unitary transformations in this paper.
3
Max-Margin Invariant Features
The previous section utilized a group integration approach to arrive a theoretically invariant non-linear
SVM. It however does not
Transformation problem i.e. the kernel k? (x, y) =
R address the Unlabeled
R
h?H ?(x), ?H ?(y)i = h G ?(gx)dgH , G ?(gy)dgH i still requires observing transformed versions
of the labelled input sample namely {gx | gx ? XG } (or atleast one of the labelled samples if we
utilize Lemma 2.6). We now present our core approach called Max-Margin Invariant Features (MMIF)
that does not require the observation of any transformed labelled training sample whatsoever.
Assume that we have access to an unlabeled set of M templates T = {ti }i={1,...M } . We assume that
we can observe all transformations under a unitary-group G, i.e. we have access to TG = {gti | ?g ?
G}i={1,...M } . Also, assume we have access to a set X = {xj }i={1,...D} of labelled data with N
classes which are not transformed. We can extract an M -dimensional invariant kernel feature for each
xj ? X as follows. Let the invariant kernel feature be l(x) ? RM to explicitly show the dependence
on x. Then the ith dimension of l for any particular x is computed as
Z
l(x)i = h?(x), ?H ?(ti )i = h?(x),
Z
gH ?(ti )dgH i = h?(x),
G
?(gti )dgH i
(2)
G
The first equality utilizes Lemma 2.6 and the third equality uses Theorem 2.4. This is equivalent
to observing all transformations of x since h?(x), ?H ?(ti )i = h?H ?(x), ?(ti )i using Lemma 2.3.
Thereby we have constructed a feature l(x) which is invariant to G without ever needing to observe
transformed versions of the labelled vector x. We now briefly the training of the MMIF feature
extractor. The matching metrics we use for this study is normalized cosine distance.
6
Training MMIF SVMs. To learn a K-dimensional MMIF feature (potentially independent of
N ), we learn K independent binary-class linear SVMs. Each SVM trains on the labelled dataset
l(X ) = {l(xj ) | j = {1, ...D}} with each sample being label +1 for some subset of the N classes
(potentially
P just one class) and the rest being labelled ?1. This leads us to a classifier in the form
of ?k = j yj ?j l(xj ). Here, yj is the label of xj for the k th SVM. It is important to note that the
unlabeled data was only used to extract l(xj ). Having multiple classes randomly labelled as positive
allows the SVM to extract some feature that is common between them. This increases generalization
by forcing the extracted feature to be more general (shared between multiple classes) rather than
being highly tuned to a single class. Any K-dimensional MMIF feature can be trained through this
technique leading to a higher dimensional feature vector useful in case where one has limited labelled
samples and classes (N is small). During feature extraction, the K inner products (scores) of the
test sample x0 with the K distinct binary-class SVMs provides the K-dimensional MMIF feature
vector. This feature vector is highly discriminative due to the max-margin nature of SVMs while
being invariant to G due to the invariant kernels.
0
MMIF. Given TG and X , the MMIF feature is defined as MMIF(x
) ? RK for any test x0 with
P
0
each dimension k being computed as hl(x ), ?k i for ?k =
j yj ?j l(xj ) ?xj ? X . Further,
l(x0 ) ? RM ?x with each dimension i being l(x0 )i = h?(x0 ), ?H ?(ti )i. The process is illustrated in
Fig. 2.
Inheriting transformation invariance from transformed unlabeled data: A special case of semisupervised learning. MMIF features can learn to be invariant to transformations (G) by observing
them only through TG . It can then transfer the invariance knowledge to new unseen samples from
X thereby becoming invariant to XG despite never having observed any samples from XG . This
is a special case of semi-supervised learning where we leverage on the specific transformations
present in the unlabeled data. This is a very useful property of MMIFs allowing one to learn
transformation invariance from one source and sample points from another source while having
powerful discrimination and generalization properties. The property is can be formally stated as the
following Theorem.
Theorem 3.1. (MMIF is invariant to learnt transformations) MMIF(x0 ) = MMIF(gx0 ) ?x0 ?g ? G
where G is observed only through TG = {gti | ?g ? G}i={1,...M } .
Thus we find that MMIF can solve the Unlabeled Transformation Problem. MMIFs have an invariant
and a discriminative component. The invariant component of MMIF allows it to generalize to
new transformations of the test sample whereas the discriminative component allows for robust
classification due to max-margin classifiers. These two properties allow MMIFs to be very useful as
we find in our experiments on face recognition.
Max and Mean Pooling in MMIF. Group integration in practice directly results in mean pooling.
Recent work however, showed that group integration can be treated as a subset of I-theory where one
tries to measure moments (or a subset of) of the distribution hx, g?i g ? G since the distribution itself
is also an invariant [1]. Group integration can be seen as measuring the mean or the first moment of
the distribution. One can also characterize using the infinite moment or the max of the distribution.
We find in our experiments that max pooling outperforms mean pooling in general. All results in this
paper however, still hold under the I-theory framework.
MMIF on external feature extractors (deep networks). MMIF does not make any assumptions
regarding its input and hence one can apply it to features extracted from any feature extractor in
general. The goal of any feature extractor is to (ideally) be invariant to within-class transformation
while maximizing between-class discrimination. However, most feature extractors are not trained
to explicitly factor out specific transformations. If we have access to even a small dataset with
the transformation we would like to be invariant to, we can transfer the invariance using MMIFs
(e.g. it is unlikely to observe all poses of a person in datasets, but pose is an important nuisance
transformation).
Modelling general non-unitary transformations. General non-linear transformations such as
out-of-plane rotation or pose variation are challenging to model. Nonetheless, a small variation
in these transformations can be approximated by some unitary G assuming piece wise linearity
through transformation-dependent sub-manifold unfolding [11]. Further, it was found that in practice,
integrating over general transformations produced approximate invariance [8].
7
1
0.9
0.9
0.8
Verification Rate
Verification Rate
1
0.95
0.85
? -DIKF (0.74)
1 -DIKF (0.61)
NDP-? (0.41)
NDP-1 (0.32)
MMIF (Ours) (0.78)
VGG Features (0.55)
MMIF-VGG (Ours) (0.61)
0.8
0.75
0.7
0.65
0.6
0.55
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -8
0.5
0.1
0.2
0.3
0.4
0.5
0.6
MMIF VGG (Ours)(0.71)
VGG (0.56)
0.7
False Accept Rate
(a) Invariant kernel feature extraction
10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
False Accept Rate
(b) SVM feature extraction leading to MMIF features
Figure 3: (a) Pose-invariant face recognition results on the semi-synthetic large-scale mugshot database (testing
on 114,750 images). Operating on pixels: MMIF (Pixels) outperforms invariance based methods DIKF [10]
and invariant NDP [8]. Operating on deep features: MMIF trained on VGG-Face features [12] (MMIF-VGG)
produces a significant improvement in performance. The numbers in the brackets represent VR at 0.1% FAR.
(b) Face recognition results on LFW with raw VGG-Face features and MMIF trained on VGG-Face features.
The values in the bracket show VR at 0.1% FAR.
4
Experiments on Face Recognition
As illustration, we apply MMIFs using two modalities overall 1) on raw pixels and 2) on deep features
from the pre-trained VGG-Face network [12]. We provide more implementation details and results
discussion in the supplementary.
A. MMIF on a large-scale semi-synthetic mugshot database (Raw-pixels and deep features).
We utilize a large-scale semi-synthetic face dataset to generate the sets TG and X for MMIF. In this
dataset, only two major transformations exist, that of pose variation and subject variation. All other
transformations such as illumination, translation, rotation etc are strictly and synthetically controlled.
This provides a very good benchmark for face recognition. where we want to be invariant to pose
variation and be discriminative for subject variation. The experiment follows the exact protocol
and data as described in [10] 3 We test on 750 subjects identities with 153 pose varied real-textured
gray-scale image each (a total of 114,750 images) against each other resulting in about 13 billion
pair-wise comparisons (compared to 6,000 for the standard LFW protocol). Results are reported as
ROC curves along with VR at 0.1% FAR. Fig. 3(a) shows the ROC curves for this experiment. We
find that MMIF features out-performs all baselines including VGG-Face features (pre-trained), DIKF
and NDP approaches thereby demonstrating superior discriminability while being able to effectively
capture pose-invariance from the transformed template set TG . MMIF is able to solve the Unlabeled
Transformation problem by extracting transformation information from unlabeled TG .
B. MMIF on LFW (deep features): Unseen subject protocol. In order to be able to effectively
train under the scenario of general transformations and to challenge our algorithms, we define a new
much harder protocol on LFW. We choose the top 500 subjects with a total of 6,300 images for
training MMIF on VGG-Face features and test on the remaining subjects with 7,000 images. We
perform all versus all matching, totalling upto 49 million matches (4 orders more than the official
protocol). The evaluation metric is defined to be the standard ROC curve with verification rate
reported at 0.1% false accept rate. We split the 500 subjects into two sets of 250 and use as TG and
X . We do not use any alignment for this experiment, and the faces were cropped according to [16].
Fig. 3(b) shows the results of this experiment. We see that MMIF on VGG features significantly
outperforms raw VGG on this protocol, boosting the VR at 0.1% FAR from 0.56 to 0.71. This
demonstrates that MMIF is able to generate invariance for highly non-linear transformations that are
not well-defined rendering it useful in real-world scenarios where transformations are unknown but
observable.
3
We provide more details in the supplementary. Also note that we do not need utilize identity information, all
that is required is the fact that a set of pose varied images belong to the same subject. Such data can be obtained
through temporal sampling.
8
References
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9
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6,349 | 6,743 | Regularized Modal Regression with Applications in
Cognitive Impairment Prediction
Xiaoqian Wang1 , Hong Chen1 , Weidong Cai2 , Dinggang Shen3 , Heng Huang1?
Department of Electrical and Computer Engineering, University of Pittsburgh, USA
2
School of Information Technologies, University of Sydney, Australia
3
Department of Radiology and BRIC, University of North Carolina at Chapel Hill, USA
[email protected],[email protected]
[email protected],[email protected],[email protected]
1
Abstract
Linear regression models have been successfully used to function estimation and
model selection in high-dimensional data analysis. However, most existing methods
are built on least squares with the mean square error (MSE) criterion, which
are sensitive to outliers and their performance may be degraded for heavy-tailed
noise. In this paper, we go beyond this criterion by investigating the regularized
modal regression from a statistical learning viewpoint. A new regularized modal
regression model is proposed for estimation and variable selection, which is robust
to outliers, heavy-tailed noise, and skewed noise. On the theoretical side, we
establish the approximation estimate for learning the conditional mode function,
the sparsity analysis for variable selection, and the robustness characterization. On
the application side, we applied our model to successfully improve the cognitive
impairment prediction using the Alzheimer?s Disease Neuroimaging Initiative
(ADNI) cohort data.
1
Introduction
Modal regression [21, 5] has gained increasing attention recently due to its effectiveness on function
estimation and robustness to outliers and heavy-tailed noise. Unlike the traditional least-square
estimator pursuing the conditional mean, modal regression aims to estimate the conditional mode of
output Y given the input X = x. It is well known that the conditional modes can reveal the structure
of outputs and the trends of observation, which is missed by the conditional mean [29, 4]. Thus,
modal regression often achieves better performance than the traditional least square regression in
practical applications.
There are some studies for modal regression with (semi-)parametric or nonparametric methods, such
as [29, 28, 4, 6]. For parametric approaches, a parametric form is required for the global conditional
mode function. Recent works in [29, 28] belong to this category, where the method in [28] is based
on linear mode function assumption and the algorithm in [29] is associated with the local polynomial
regression. For non-parametric approaches, the conditional mode is usually derived by maximizing a
conditional density or a joint density. Typical work for this setting is established in [4], where a local
modal regression is proposed based on kernel density estimation and theoretical analysis is provided
to characterize asymptotic error bounds.
Most of the above mentioned works consider the asymptotic theory on the conditional mode function
estimation. Recently, several studies on variable selection under modal regression were also conducted in [30, 27]. These approaches addressed the problem from statistical theory viewpoint (e.g.,
?
X. Wang and H. Chen made equal contributions to this paper. H. Huang is the corresponding author.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
asymptotic normality) and were implemented by modified EM algorithm. Although these studies
provide us good understanding for modal regression, the following problems still remain unclear
in theory and applications. Can we design new modal regression following the line of structural
risk minimization? Can we provide its statistical guarantees and computing algorithm for designed
model? This paper focuses on answering the above questions.
To illustrate the effectiveness of our model, we looked into a practical problem, i.e., cognitive
impairment prediction via neuroimaging data. As the most common cause of dementia, Alzheimer?s
Disease (AD) imposes extensive and complex impact on human thinking and behavior. Accurate and
automatic study of the relationship between brain structural changes and cognitive impairment plays
a crucial role in early diagnosis of AD. In order to increase the diagnostic capabilities, neuroimaging
provides an effective approach for clinical detection and treatment response monitoring of AD [13].
Several cognitive tests were presented to assess the individual?s cognitive level, such as Mini-Mental
State Examination (MMSE) [8] and Trail Making Test (TMT) [1]. With the development of these
techniques, a wide range of work employed regression models to study the correlations between
neuroimaging data and cognitive measures [23, 16, 26, 25, 24].
However, existing methods use mean regression models based on the least-square estimator to predict
the relationship between neuroimaging features and cognitive assessment, which may fail when
the noise in the data is heavy-tailed or skewed. According to the complex data collection process
[13], the assumption of symmetric noise may not be guaranteed in biomedical data. Under such a
circumstance, modal regression model proves to be more appropriate due to its robustness to outliers,
heavy-tailed noise, and skewed noise. We applied our method to the ADNI cohort for the association
study between neuroimaging features and cognitive assessment. Experimental results illustrated the
effectiveness of our model. Moreover, with sparse constraints, our model found several imaging
features that have been reported to be crucial to the onset and progression of AD. The replication of
these results further support the validity of our model.
Our main works can be summarized as below:
1) Following the Tikhonov regularization and kernel density estimation, we develop a new Regularized
Modal Regression (RMR) for estimating the conditional mode function and selecting informative
variables, which can be considered as a natural extension of Lasso [22] and can be implemented
efficiently by half-quadratic minimization methods.
2) Learning theory analysis is established for RMR from three aspects: approximation ability, sparsity,
and robustness, which provide the theoretical foundations of the proposed approach.
3) By applying our RMR model to the ADNI cohort, we reveal interesting findings in cognitive
impairment prediction of Alzheimer?s disease.
2
2.1
Regularized Modal Regression
Modal regression
We consider learning problem with input space X ? Rp and output space Y ? R. Let pY |X=x be the
conditional density of Y ? Y for given X = x ? X . In the prediction of cognitive assessment, we
denote the neuroimaging data for the i-th sample as xi and the cognitive measure for the i-th sample
as yi . Suppose that training samples z = {(xi , yi )}ni=1 ? X ? Y are generated independently by:
Y = f ? (X) + ?,
(1)
where mode(?|X = x) = arg max p?|X (t|X = x) = 0 for any x ? X . Here, p?|X , as the
t
conditional density of ? conditioned on X, is well defined. Then, the target function of modal
regression can be written as:
f ? (x) = mode(Y |X = x) = arg max pY |X (t|X = x), ?x ? X .
(2)
t
To assure f ? is well defined on X , we require that the existence and uniqueness of pY |X (t|X = x)
for any given x ? X . The relationship (2) means f ? is the maximum of the conditional density
pY |X , and also equals to maximize the joint density pX,Y [4, 29, 28]. Here, we formulate the modal
regression following the dimension-insensitive statistical learning framework [7].
2
For feasibility, we denote ? on X ? Y as the intrinsic distribution for data generated by (1), and
denote ?X as the corresponding marginal distribution on X . It has been proved in Theorem 3 [6] that
f ? is the maximizer of
Z
R(f ) =
pY |X (f (x)|X = x)d?X (x)
(3)
X
over all measurable function. Hence, we can adopt R(f ) as the evaluation measure of modal
regression estimator f : X ? R. However, we can not get the estimator directly by maximizing
this criterion since pY |X and ?X are unknown. Recently, Theorem 5.1 in [6] shows R(f ) = p?f (0),
where p?f is the density function of random variable ?f = Y ? f (X). Then, the problem of
maximizing R(f ) over some hypothesis spaces can be transformed to maximize the density of ?f at
0. This density p?f can be estimated by nonparametric kernel density estimation.
0
u?u
0
For a kernel K? : R ? R ? R+ , we denote its representing function
R ?( ? ) = K? (u, u ), which
usually satisfies ?(u) = ?(?u), ?(u) ? ?(0) for any u ? R and R ?(u)du = 1. Typical examples
of kernel include Gaussian kernel, Epanechnikov kernel, quadratic kernel, triwight kernel, and
sigmoid function. The empirical estimation of R(f ) (also pf (0)) can be obtained by kernel density
estimation, which is defined as:
n
n
1 X
1 X yi ? f (xi )
R?z (f ) =
).
K? (yi ? f (xi ), 0) =
?(
n? i=1
n? i=1
?
Hence, the approximation of f ? can be found by learning algorithms associated with R?z (f ). In
theory, for any f : X ? R, the expectation version of R?z (f ) is:
Z
y ? f (x)
1
?
?(
)d?(x, y).
R (f ) =
? X ?Y
?
In particular, there holds R(f ) ? R? (f ) ? 0 as ? ? 0 [6].
2.2
Modal regression with coefficient-based regularization
In this paper, we assume that f ? (x) = mode(Y |X = x) = w?T x for some w? ? Rp . Following the
ideas of ridge regression and Lasso [22], we consider the robust linear estimator for learning the
conditional mode function.
Let F be a linear hypothesis space defined by:
F = {f (x) = wT x : w = (w1 , ..., wp ) ? Rp , x ? X }.
For any given positive tuning parameters {?j }pj=1 , we denote:
p
nX
o
?(f ) = inf
?j |wj |q : f (x) = wT x, q ? [1, 2] .
j=1
Given training set z, the regularized modal regression (RMR) can be formulated as below:
n
o
fz = arg max R?z (f ) ? ??(f ) ,
(4)
f ?F
where regularization parameter ? > 0 is used to balance the modal regression measure and hypothesis
space complexity. It is easy to deduce that fz (x) = wzT x with
p
n
o
n 1 X
X
yi ? wT xi
?(
)??
?j |wj |q .
(5)
wz = arg max
n? i=1
?
w?Rp
j=1
When ?j ? 1 for 1 ? j ? p and q = 1, (5) can be considered as an natural extension of Lasso in
[22] from learning the conditional mean function to estimating the conditional mode function. When
?j ? 1 for 1 ? j ? p and q = 2, (5) also can be regarded as the corresponding version of ridge
regression by replacing the MSE criterion with modal regression criterion. In particular, when K? is
Gaussian kernel and ?j ? 1 for 1 ? j ? p, (5) can be rewritten as:
n
n 1 X
n (y ? wT x )2 o
o
i
i
q
wz = arg max
exp
?
?kwk
,
q
n? i=1
?2
w?Rp
which is equivalent to correntropy regression under maximum correntropy criterion [19, 9, 7].
3
2.3
Optimization algorithm
We employ the half-quadratic (HQ) theory [18] in the optimization. For a convex problem min u(s),
s
it is equivalent to solve the following half-quadratic reformulation:
min Q(s, t) + v(t),
s,t
where Q(s, t) is quadratic for any t ? R and v : R ? R satisfies:
u(s) = min Q(s, t) + v(t), ?s ? R.
t
Such a dual potential function v can be determined via convex conjugacy as shown below.
According to the convex optimization theory [20], for a closed convex function f (a), there exists a
convex function g(b), such that:
f (a) = max(ab ? g(b)),
b
where g is the conjugate of f , i.e., g = f ? . Symmetrically, it is easy to prove f = g ? .
Theorem 1 For a closed convex function f (a) = max(ab ? g(b)), we have arg max(ab ? g(b)) =
b
f 0 (a) for any a ? R.
b
When K? is Gaussian kernel, the optimization steps can be found in [9]. Here we take Epanechnikov
kernel (a.k.a., parabolic kernel) as an example to show the optimization of Problem (5) via HQ theory.
The kernel-induced representing function of Epanechnikov kernel is ?(e) = 34 (1 ? e2 )1[|e|?1] .
Define a closed convex function f as:
f (a) =
3
4 (1
0,
? a), 0 ? a ? 1
a ? 1.
There exists a convex function g such that f (a) = max(ab ? g(b)) and ?(e) = f (e2 ) = max(e2 b ?
b
b
g(b)). Thus, when ?j ? 1 for 1 ? j ? p, the optimization problem (5) can be rewritten as:
p
n
n 1 X
o
X
yi ? wT xi 2
q
max
b
(
)
?
g(b
)
?
?
?
|w
|
.
i
i
j
j
w?Rp ,b?Rn n?
?
i=1
j=1
(6)
Problem (6) can be easily optimized via alternating optimization algorithm. Note that according
T
xi 2
to Theorem 1, when w is fixed, b can be updated as bi = f 0 (( yi ?w
for
) ) = ? 43 1 yi ?wT xi
?
[|
?
|?1]
i = 1, 2, . . . , n. For the space limitation, we provide the proof of Theorem 1 and the optimization
steps of RMR in the supplementary material.
3
Learning Theory Analysis
This section presents the theoretical foundations of RMR from approximation ability, variable sparsity,
and algorithmic robustness. Detail proofs of these results can be found in the supplementary material.
3.1
Approximation ability analysis
Besides the linear requirement for the conditional mode function, we also need some basic conditions
on the kernel-induced representing function ? [6, 28].
Assumption 1 The representing function ? satisfies the following
conditions: 1)
R
R ?u ? R, ?(u) ?
?(0) < ?, 2) ? is Lipschitz continuous with constant L? , 3) R ?(u)du = 1 and R u2 ?(u)du < ?.
It is easy to verify that most of kernels used for density estimation satisfy the above conditions,
e.g., Gaussian kernel, Epanechnikov kernel, quadratic kernel, etc. Since RMR is associated with
R?z (f ), we need to establish quantitative relationship between R? (f ) and R(f ). Recently, the modal
regression calibration has been illustrated in Theorem 10 [6] under the following restrictions on the
conditional density p?|X .
4
Assumption 2 The conditional density p?|X is second-order continuously differentiable and uniform
bounded.
Now, we present the approximation bound on R(f ? ) ? R(fz ).
q
Theorem 2 Let kxk q?1
? a for q ? (1, 2] for any x ? X and f ? ? F. Under Assumptions 1-2, for
q
q ? (1, 2], by taking ? = ? 2 = O(n? 4q+3 ), we have:
q
R(f ? ) ? R(fz ) ? C log(4/?)n? 4q+3
1
with confidence at least 1 ? ?. In particular, for q = 1 and kxk? ? a, choosing ? = ? 2 = ( lnnp ) 7 ,
we have:
ln p 17
R(f ? ) ? R(fz ) ? C log(4/?)
n
with confidence at least 1 ? ?, Here C1 , C2 is a constant independent of n, ?.
Theorem 2 shows that the excess risk of R(f ? ) ? R(fz ) ? 0 with the polynomial decay and the
estimation consistency is guaranteed as n ? ?. Moreover, under Assumption 3 in [6], we can derive
q
1
that fz tends to f ? with approximation order O(n? 4q+3 ) for q ? (1, 2] and O( lnnp ) 7 ) for q = 1.
Although approximation analysis has been provided for modal regression in [6, 28], both of them are
limited to the empirical risk minimization. This is different from our result for regularized modal
regression under structural risk minimization.
3.2
Sparsity analysis
To characterize the variable selection ability of RMR, we first present the properties for nonzero
component of wz .
Theorem 3 Assume that ? is differentiable for any t ? R. For j ? {1, 2, ..., p} satisfying wzj 6= 0,
there holds:
n
1 X
p?? |w |p?1
yi ? fz (xi )
j
zj
?0 (
)xij =
.
2
n? i=1
?
2
Observe that the condition on ? holds true for Gaussian kernel, sigmoid function, and logistic function.
Theorem 3 demonstrates the necessary condition for the non-zero wzj . Without loss of generality, we
set S0 = {1, 2, ..., p0 } as the index set of truly informative variables and denote Sz = {j : wzj 6= 0}
as the set of identified informative variables by RMR in (4).
Theorem 4 Assume that kxk? ? a for any x ? X and ??j ? k?0 k? ? for any j > p0 . Then, for
RMR (4) with q = 1, there holds Sz ? S0 for all z ? (X ? Y)n .
Theorem 4 assures that RMR has the capacity to identify the truly informative variable in theory.
Combining Theorem 4 and Theorem 2, we provide the asymptotic theory of RMR on estimation and
model selection.
3.3
Robustness analysis
To quantify the robustness of RMR, we calculate its finite sample breakdown point, which reflects the
largest amount of contamination points that an estimator can tolerate before returning arbitrary values
[11, 12]. Recently, this index has been used to investigate the robustness of modal linear regression
[28] and kernel-based modal regression [6].
Recall that the derived weight wz defined in (5) is dependent on any given sampling set z =
{(xi , yi )}ni=1 . By adding m arbitrary points z0 = {(xn+j , yn+j )}m
j=1 ? X ? Y, we obtain the
corrupted sample set z ? z0 . For given ?, ?, {?j }pj=1 , we denote wz?z0 be the maximizer of (5). Then,
the finite sample breakdown point of wz is defined as:
n m
o
(wz ) = min
: sup kwz?z0 k2 = ? .
1?m?n n + m
z0
5
Theorem 5 Assume that ?(u) = ?(?u) and ?(t) ? 0 as t ? ?. For given ?, ?, {?j }pj=1 , we
denote:
n
M=
1 X y?i ? fz (xi ))
?(
) ? ??(?(0))?1 ?(fz ).
?(0) i=1
?
Then the finite sample breakdown point of wz in (5) is (wz ) =
is the smallest integer not less than M .
m?
n+m? ,
where m? ? dM e and dM e
From Theorem 5, we know that the finite breakdown point of RMR depends on ?, ?, and the sample
configuration, which is similar with re-descending M-estimator and recent analysis for modal linear
regression in [28]. As illustrated in [11, 12], the finite sample breakdown point is high when the
bandwidth ? only depends on the training samples. Hence, RMR can achieve satisfactory robustness
when ?, ?j are chosen properly and ? is determined by data-driven techniques.
4
Experimental Analysis
In this section, we conduct experiments on both toy data, benchmark data as well as the ADNI
cohort data to evaluate our RMR model. We compare several regression methods in the experiments,
including: LSR (traditional mean regression based on the least square estimator), LSR-L2 (LSR with
squared `2 -norm regularization, i.e., ridge regression) LSR-L1 (LSR with `1 -norm regularization),
MedianR (median regression), HuberR (regression with huber loss), RMR-L2 (RMR with squared
`2 -norm regularization), and RMR-L1 (RMR with `1 -norm regularization).
For evaluation, we calculate root mean square error (RMSE) between the predicted value and
ground truth in out-of-sample prediction. The RMSE value is normalized via Frobenius norm of
the ground truth matrix. We employ 2-fold cross validation and report the average performance for
each method. For each method, we set the hyper-parameter of the regularization term in the range of
{10?4 , 10?3.5 , . . . , 104 }. We tune the hyper-parameters via 2-fold cross validation on the training
data and report the best parameter w.r.t. RMSE of each method. For RMR methods, we adopt the
Epanechnikov kernel and set the bandwidth as ? = max(|y ? wT x|).
4.1
Performance comparison on toy data
Following the design in [28], we generate the toy data by sampling i.i.d. from the model: Y =
?2 + 3X + ? (X), where X ? U(0, 1), ?(X) = 1 + 2X and ? 0.5N (?2, 32 ) + 0.5N (2, 12 ).
We can derive that E() = 0, Mode() = 1.94 and Median() = 1, hence the conditional mean
regression function of the toy data is E(Y |X) = ?2 + 3X, the conditional median function is
Median(Y |X) = 1 + 5X, while the conditional mode is Mode(Y |X) = ?0.06 + 6.88X.
We consider three different number of samples: 100,200,500, and repeat the experiments 100 times
for each setting. We present the RMSE in Table 1, which shows that RMR models get lower RMSE
values than all comparing methods. It indicates that RMR models make better estimation of the
output when the noise in data is skewed and relatively heavy-tailed. Moreover, we compare the
coverage probabilities for prediction intervals centered around the predicted value from each method.
We set the length of coverage intervals to be {0.1?, 0.2?, 0.3?} respectively with ? = 3 being the
approximate standard error of . From Table 2 we can find that RMR models provide larger coverage
probabilities than the counterparts.
4.2
Performance comparison on benchmark data
Here we present the comparison results on six benchmark datasets from UCI repository [15] and
StatLib2 , which include: slumptest, forestfire, bolts, cloud, kidney, and lupus. We summarize the
results in Table 3. From the comparison we notice that RMR models tend to perform better on all
datasets. Also, RMR-L1 obtains lower RMSE value since the RMR-L1 model is more robust with
the `1 -norm regularization term.
2
http://lib.stat.cmu.edu/datasets/
6
Table 1: Average RMSE and standard deviation with different number (n) of toy samples.
n=100
0.9687?0.0699
0.9671?0.0685
0.9672?0.0685
0.9944?0.0806
0.9725?0.0681
0.9663?0.0683
0.9662?0.0679
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
n=200
0.9477?0.0294
0.9469?0.0284
0.9473?0.0288
0.9568?0.0350
0.9485?0.0296
0.9466?0.0282
0.9465?0.0281
n=500
0.9495?0.0114
0.9495?0.0114
0.9495?0.0114
0.9542?0.0120
0.9502?0.0116
0.9493?0.0114
0.9492?0.0114
Table 2: Average coverage possibilities and standard deviation on toy data.
0.1?
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
n=100
0.0730?0.0247
0.0753?0.0247
0.0747?0.0246
0.0563?0.0255
0.0710?0.0258
0.0760?0.0254
0.0760?0.0255
n=200
0.0702?0.0166
0.0731?0.0155
0.0719?0.0161
0.0626?0.0124
0.0698?0.0160
0.0740?0.0161
0.0742?0.0156
n=500
0.0702?0.0106
0.0709?0.0108
0.0706?0.0106
0.0654?0.0097
0.0694?0.0101
0.0719?0.0111
0.0720?0.0111
0.2?
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
0.1313?0.0338
0.1337?0.0334
0.1337?0.0337
0.1087?0.0351
0.1237?0.0347
0.1340?0.0336
0.1343?0.0340
0.1450?0.0255
0.1461?0.0251
0.1458?0.0258
0.1331?0.0239
0.1442?0.0257
0.1477?0.0256
0.1481?0.0247
0.1430?0.0193
0.1429?0.0196
0.1430?0.0193
0.1377?0.0182
0.1421?0.0188
0.1441?0.0199
0.1441?0.0198
0.3?
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
0.1923?0.0402
0.1940?0.0415
0.1940?0.0415
0.1750?0.0414
0.1873?0.0389
0.1943?0.0420
0.1950?0.0406
0.2142?0.0342
0.2165?0.0331
0.2153?0.0334
0.2031?0.0299
0.2132?0.0333
0.2179?0.0327
0.2177?0.0323
0.2150?0.0229
0.2156?0.0222
0.2153?0.0226
0.2095?0.0233
0.2144?0.0224
0.2168?0.0220
0.2167?0.0219
Table 3: Average RMSE and standard deviation on benchmark data.
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
slumptest
0.2689?0.0295
0.2616?0.0266
0.2571?0.0277
0.2810?0.0024
0.2669?0.0268
0.2538?0.0185
0.2517?0.0240
forestfire
0.9986?0.0874
0.9822?0.0064
0.9822?0.0079
0.9964?0.0050
0.9874?0.0299
0.9817?0.0093
0.9802?0.0198
bolts
0.4865?0.0607
0.4687?0.0137
0.4713?0.0172
0.4436?0.0232
0.4841?0.0661
0.4782?0.0107
0.3298?0.1313
7
cloud
0.6178?0.0190
0.5782?0.0029
0.5802?0.0043
0.6457?0.0301
0.6178?0.0190
0.5702?0.0131
0.5663?0.0305
kidney
0.5077?0.0264
0.5106?0.0219
0.5196?0.0089
0.5432?0.0160
0.5447?0.0270
0.4871?0.0578
0.4989?0.0398
lupus
0.8646?0.3703
0.8338?0.3282
0.8408?0.3366
1.2274?0.6979
0.9198?0.4226
0.8071?0.3053
0.7885?0.2910
Table 4: Average RMSE and standard deviation on the ADNI data.
LSR
LSR-L2
LSR-L1
MedianR
HuberR
RMR-L2
RMR-L1
4.3
Fluency
0.3856?0.0034
0.3269?0.0069
0.3295?0.0035
0.4164?0.0291
0.3856?0.0034
0.3256?0.0049
0.3269?0.0057
ADAS
0.4397?0.0112
0.4116?0.0208
0.4121?0.0100
0.4700?0.0151
0.4383?0.0133
0.4105?0.0216
0.4029?0.0234
TRAILS
0.6798?0.0538
0.5443?0.0127
0.5476?0.0115
0.6702?0.1184
0.6621?0.0789
0.5342?0.0186
0.5423?0.0123
Performance comparison on the ADNI cohort data
Now we look into a practical problem in Alzheimer?s disease, i.e., prediction of cognitive scores
via neuroimaging features. Data used in this article were obtained from the ADNI database (adni.
loni.usc.edu). We extract 93 regions of interest (ROIs) as neuroimaging features and use
cognitive scores from three tests: Fluency Test, Alzheimer?s Disease Assessment Scale (ADAS) and
Trail making test (TRAILS). 795 sample subjects were involved in our study, including 180 AD
samples, 390 MCI samples and 225 normal control (NC) samples. Detailed data description can be
found in the supplementary material.
Our goal is to construct an appropriate model to predict cognitive performance given neuroimaging
data. Meanwhile, we expect the model to illustrate the importance of different features in the
prediction, which is fundamental to understanding the role of each imaging marker in the study of AD.
From Table 4, we find that RMR models always perform equal or better than the comparing methods,
which verifies that RMR is more appropriate to learn the association between neuroimaging markers
and cognitive performance. We can notice that RMR-L2 always performs better than LSR-L2, and
RMR-L1 outperforms LSR-L1. This is because the symmetric noise assumption in least square
models may not be guaranteed on the ADNI cohort. Compared with HuberR, our RMR model is
shown to be less sensitive to outliers. Moreover, from the comparison between MedianR and RMR
models, we can infer that conditional mode is more suitable than conditional median for the prediction
of cognitive scores.
RMR-L1 imposes sparse constraints on the learnt weight matrix, which naturally achieves the goal
of feature selection in the association study. Here we take TRAILS cognitive assessment as an
example and look into the important neuroimaging features in the prediction. From the heat map
and brain map in Fig. 1 and 2, we obtain several interesting findings. In the prediction, temporal
lobe white matter has been picked out as a predominant feature. [10, 2] reported decreased fractional
anisotropy (FA) and increased radial diffusivity (DR) in the white matter of the temporal lobe among
AD and Mild Cognitive Impairment (MCI) subjects. [10] also revealed the correlation between
temporal lobe FA and episodic memory, which may account for the influence of temporal lobe to
TMT results. Besides, there is evidence in [17] supporting the association between left temporal lobe
and the working memory component involving letters and numbers in TMT. Moreover, angular gyrus
indicates high correlation with TRAILS scores in our analysis. Previous research has revealed that
angular gyrus share many clinical features with AD. [14] presented structural MRI findings showing
more left anular gyrus in MCI converters than non-converters, which pointed out the role of atrophy
of structures like angular gyrus in the progression of dementia. [3] showed evidence for the role
of angular gyrus in orienting spatial attention, which serves as a key factor in TMT results. The
replication of these results supports the effectiveness of our model.
5
Conclusion
This paper proposes a new regularized modal regression and establishes its theoretical foundations
on approximation ability, sparsity, and robustness. These characterizations fill in the theoretical
gaps for modal regression under Tikhonov regularization. Empirical results verify the competitive
performance of the proposed approach on simulated data, benchmark data and real biomedical data.
8
Figure 1: Heatmap showing the weights of each neuroimaging feature via RMR-L1 model for the
prediction of TRAILS cognitive measures. We draw two matrices, where the upper figure is for the
left hemisphere and the lower figure for the right hemisphere. Imaging markers (columns) with larger
weights indicate higher correlation with corresponding cognitive measure in the prediction.
Figure 2: Cortical maps of ROIs identified in RMR-L1 model for the prediction of TRAILS cognitive
measures. The brain maps show one slice of multi-view. The three maps correspond to three different
cognitive measures in TRAILS cognitive test, respectively.
With the sparsity property of our model, we identified several biological meaningful neuroimaging
markers, showing the potential to enhance the understanding of onset and progression of AD.
Acknowledgments
This work was partially supported by U.S. NSF-IIS 1302675, NSF-IIS 1344152, NSF-DBI 1356628,
NSF-IIS 1619308, NSF-IIS 1633753, NIH AG049371. Hong Chen was partially supported by
National Natural Science Foundation of China (NSFC) 11671161. We are grateful to the anonymous
NIPS reviewers for the insightful comments.
9
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11
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6,350 | 6,744 | Translation Synchronization via Truncated Least
Squares
Xiangru Huang?
The University of Texas at Austin
2317 Speedway, Austin, 78712
[email protected]
Zhenxiao Liang?
Tsinghua University
Beijing, China, 100084
[email protected]
Chandrajit Bajaj
The University of Texas at Austin
2317 Speedway, Austin, 78712
[email protected]
Qixing Huang
The University of Texas at Austin
2317 Speedway, Austin, 78712
[email protected]
Abstract
In this paper, we introduce a robust algorithm, TranSync, for the 1D translation
synchronization problem, in which the aim is to recover the global coordinates of a
set of nodes from noisy measurements of relative coordinates along an observation
graph. The basic idea of TranSync is to apply truncated least squares, where the
solution at each step is used to gradually prune out noisy measurements. We analyze
TranSync under both deterministic and randomized noisy models, demonstrating
its robustness and stability. Experimental results on synthetic and real datasets
show that TranSync is superior to state-of-the-art convex formulations in terms of
both efficiency and accuracy.
1
Introduction
In this paper, we are interested in solving the 1D translation synchronization problem, where the
input is encoded as an observation graph G = (V, E) with n nodes (i.e. V = {1, ? ? ? , n}). Each node
is associated with a latent coordinate x?i ? R, 1 ? i ? n, and each edge (i, j) ? E is associated with
a noisy measurement tij = x?i ? x?j + N (ij ) of the coordinate difference xi ? xj under some noise
model N (ij ). The goal of translation synchronization is to recover the latent coordinates (up to a
global shift) from these noisy pairwise measurements. Translation synchronization is a fundamental
problem that arises in many application domains, including joint alignment of point clouds [7] and
ranking from relative comparisons [8, 16].
A standard approach to translation synchronization is to solve the following linear program:
minimize
X
(i,j)?E
|tij ? (xi ? xj )|,
subject to
n
X
xi = 0,
(1)
i=1
Where the constraint ensures that the solution is unique. The major drawback of the linear programming formulation is that it can only tolerate up to 50% of measurements coming from biased
noise models (e.g., uniform samples with non-zero mean). Moreover, it is challenging to solve (1)
efficiently at scale. Solving (1) using interior point method becomes impractical for large-scale
datasets, while more scalable methods such as coordinate descent usually exhibit slow convergence.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this paper, we introduce a robust and scalable algorithm, TranSync, for translation synchronization.
The algorithm is rather simple, we solve a truncated least squares at each iteration k:
(k)
{xi } = argmin
{xi }
X
wij |tij ? (xi ? xj )|2 ,
subject to
n p
X
di xi = 0,
i=1
(i,j)?E
di :=
X
wij .
j?N (i)
(2)
(k?1)
(k?1)
where the weights wij = Id(|tij ? (xi
? xj
)| < ?k ) are obtained from the solution at
the previous iteration using a geometrically decaying truncation parameter ?k . Although TranSync
requires solving a linear system at each step, these linear systems are fairly similar to each other,
meaning that the solution at the previous iteration provides an excellent warm-start for solving the
linear system at the current iteration. As a result, the computational efficiency of TranSync is superior
to state-of-the-art methods for solving the linear programming formulation. We analyze TranSync
under both deterministic and randomized noise models, demonstrating its robustness and stability. In
particular, we show that TranSync is able to handle biased noisy measurements.
We have evaluated TranSync on both synthetic datasets and real datasets used in the applications of
joint alignment of point clouds and ranking from pair-wise measurements. Experimental results show
that TranSync is superior to state-of-the-art solvers for the linear programming formulation in terms
of both computational efficiency and accuracy.
1.1
Related Work
Translation synchronization falls into the general problem of map synchronization, which takes maps
computed between pairs of objects as input, and outputs consistent maps across all the objects. Map
synchronization appears as a crucial step in many scientific problems, including fusing partially
overlapping range scans [15], assembling fractured surfaces [14], solving jigsaw puzzles [5, 11],
multi-view structure from motion [25], data-driven shape analysis and processing [17], and structure
from motion [27].
Early methods for map synchronization focused on applying greedy algorithms [14, 15, 18] or
combinatorial optimization [20, 23, 27]. Although these methods exhibit certain empirical success,
they lack theoretical understanding (e.g. we do not know under what conditions can the underlying
ground-truth maps be exactly recovered).
Recent methods for map synchronization apply modern optimization techniques such as convex
optimization and non-convex optimization. In [13], Huang and Guibas introduce a semidefinite
programming formulation for permutation synchronization and its variants. Chen et al. [4] generalize
the method to partial maps. In [26], Wang and Singer introduce a method for rotation synchronization.
Although these methods provide tight, exact recovery conditions, the computational cost of the convex
optimizations provide an obstruction for applying these methods to large-scale data sets.
In contrast to convex optimization, very recent map synchronization methods leverage non-convex
optimization approaches such as spectral techniques and gradient-based optimization. In [21, 22],
Pachauri et al. study map synchronization from the perspective of spectral decomposition. Recently,
Shen et al. [24] provide an analysis of spectral techniques for permutation synchronization. Beyond
spectral techniques, Zhou et al. [28] apply alternating minimization for permutation synchronization.
Finally, Chen and Candes [3] introduce a method for the generalized permutation synchronization
problem using the projected power method. To the best of our knowledge, we are the first to
develop and analyze continuous map synchronizations (e.g., translations or rotations) beyond convex
optimization.
Our approach can be considered as a special case of reweighted least squares (or RLS) [9, 12], which
is a powerful method for solving convex and non-convex optimizations. The general RLS framework
has been applied for map synchronization (e.g. see [1, 2]). Despite the empirical success of these
approaches, the theoretical understanding of RLS remains rather limited. The analysis in this paper
provides a first step towards the understanding of RLS for map synchronization.
1.2
Notation
Before proceeding to the technical part of this paper, we introduce some notation that will be used
later. The unnormalized graph Laplacian of a graph G is denoted as LG . If it is obvious from the
2
Algorithm 1 TranSync(c, kmax )
1. x(?1) ? 0. ??1 ? ?.
for k = 0, 1, 2, kmax do
2. Obtain the truncated graph G (k) using x(k?1) and ?k?1 .
3. Break if G (k) is disconnected
if G is non-bipartite then
4. Solve (2) using (3) to obtain x(k) .
else
5. Solve (2) using conjugate gradient to obtain x(k) .
end if.
(0)
(0)
6. ?k = min max |tij ? (xi ? xj )|, c?k?1 .
(i,j)?E
end for
Output: x(k) .
context, we will always shorten LG as L to make the notation uncluttered. Similarly, we will use
D = diag(d1 , ? ? ? , dn ) to collect the vertex degrees and denote the vertex adjacency and vertex-edge
adjacency matrices as A and B respectively. The peusdo-inverse of a matrix X is given by X + . In
addition, we always sort the eigenvalues of a symmetric matrix X ? Rn?n in increasing order (i.e.
?1 (X) ? ?2 (X) ? ? ? ? ? ?n (X)). Moreover, we will consider several matrix norms k ? k, k ? k1,?
and k ? kF , which are defined as follows:
kXk = ?max (X),
kXk1,? = max
1?i?n
n
X
|xij |,
kXkF =
X
j=1
x2ij
12
.
i,j
Note that kXk1,? is consistent with the L? -norm of vectors.
2
Algorithm
In this section, we provide the algorithmic details of TranSync. The iterative scheme (1) requires an
initial solution x(0) , an initial truncation parameter ?0 , and a stopping condition. The initial solution
can be determined by solving for x(0) from (2) w.r.t. wij = 1. We set the initial truncation parameter
(0)
(0)
?0 = max |tij ? (xi ? xj )|, so that the edge with the biggest residual is removed. We stop
(i,j)?E
TranSync either after the maximum number of iterations is reached or the truncated graph becomes
disconnected. Algorithm 1 provides the pseudo code of TranSync.
Clearly, the performance of TranSync is driven by the efficiency of solving (2) at each iteration.
TranSync takes an iterative approach, in which we utilize a warm-start x(k?1) provided by the
solution obtained at the previous iteration. When the truncated graph is non-bipartite, we find
a simple weighted average scheme delivers satisfactory computational efficiency. Specifically, it
generates a series of vectors xk,0 = x(k?1) , xk,1 , ? ? ? , xk,nmax via the following recursion:
xik,l+1 =
X
j?N (i)
wij (xk,l
j + tij )/
X
xk,l+1
= xk,l+1
? Pn
i
i
wij ,
1
?
i0 =1
j?N (i)
n p
X
di
di0 xk,l+1
,
i0
i0 =1
(3)
which may be written in the following matrix form:
1 ? 1 T 1 ?1
D 2 11 D 2 )D
Axk,l + Bt(k) ,
(4)
n
P?
Remark 2.1 The corresponding normalization constraint in (3), i.e.,
di xi = 0, only changes
xk,l+1 = (In ?
i
the solution to (2) by a constant factor. We utilize this modification for the purpose of obtaining a
concise convergence property of the iterative scheme detailed below.
The following proposition states that (3) admits a geometric convergence rate:
3
Proposition 2.1 xk,l geometrically converges to x(k+1) . Specifically, ?l ? 0,
kD
1
2
k,l
x
?
(k)
xshift k
l
? (1 ? ?) kD
1
2
k,0
x
?
(k)
xshift k,
(k)
xshift
(k)
=x
P ? (k)
i di xi
?
1.
? P
i di
where ? < 1 is the spectral gap of the normalized Graph Laplacian of the truncated graph.
Proof. The proof is straightforward from (4) and is omitted for brevity.
Since the intermediate solutions are mainly used to prune outlier observations, it is clear that
O(log(n)) iterations of (4), which induce a O(1/n) error for solving (2), are sufficient. The complexity of checking
if the graph is non-bapriatite is O(|E|). The total running time for solving (2) is
thus O |E| log(n) . This means the total running time of TranSync is O(|E| log(n)kmax ), making it
scalable to large-scale datasets.
3
Analysis of TranSync
In this section, we provide exact recovery conditions of TranSync. We begin with describing an exact
recovery condition under a deterministic noise model in Section 3.1. We then study an exact recovery
condition to demonstrate that TranSync can handle biased noisy samples in Section 3.2.
3.1
Deterministic Exact Recovery Condition
We consider the following deterministic noisy model: We are given the ground-truth location xgt .
gt
Then, for each correct measurement tij , (i, j) ? G, |tij ? (xgt
i ? xj )| ? ? for a threshold ?. In
contrast, each incorrect measurement tij , (i, j) ? G could take any real number. The following
theorem provides an exact recovery condition under this noisy model.
Theorem 3.1 Let dbad be the maximum number of incorrect measurements per node. Define
n
max
|L?G,ki ? L?G,kj |,
? = max L?G,kk + max L?G,ij +
i,j,k
k
i6=j
2
pairwisely different
and
dbad ?
(n ? dbad )?
, q=
.
1 ? 2h
1 ? 2h
Suppose h < 61 (or p < 14 ), then starting from any initial solution x(0) , and for any large enough
initial truncation threshold ? 2kx(0) k? + ? and iterative step size c satisfying 4p < c < 1, we
have
kx(k) ? xgt k? ? q? + 2pck?1 ,
where
(c ? 4p)
k ? ? log
/ log c + 1.
(1 + 2q) ?
h = ?dbad ,
p=
Moreover, we can eventually reach an x(k) such that
kx(k) k? ?
2p + cq
?
c ? 4p
which is independent of the initial solution x(0) , initial truncation threshold , and values of all wrong
measurements tG\Ggood .
Proof: See Appendix A.
Theorem 3.1 essentially says that TransSync can tolerate a constant fraction of arbitrary noise. To
understand how strong this condition is, we consider the case where G = Kn is given by a clique.
Moreover, we assume the nodes are divided into two clusters of equal size, where all the measurements
within each cluster are correct. For measurements between different clusters, half of them are correct
and the other half are wrong. In this case, 25% of all measurements are wrong. However, we cannot
recover the original xgt in this case. In fact, we can set the wrong measurements in a consistent
4
gt
manner, i.e tij = xgt
i ? xj + b for a constant b 6= 0, leading to two competing clusters (one correct
and the other one incorrect) with equal strength. Hence, in the worst case, any algorithm can only
tolerate at most 25% of measurements being wrong.
We now try to use Theorem 3.1 to analyze the case where the observation graph is a clique. In
this case, it is clear that ? = n1 , and p = dbad
n , i.e the fraction of wrong measurements out of all
measurements. Hence, in the clique case, we have shown that TranSync converges to a neighborhood
of the ground truth from any initial solution if the fraction of wrong measurements is less that 16 (i.e.,
2/3 of the upper bound).
3.2
Biased Random Noisy Model
We proceed to provide an exact recovery condition of TranSync under a biased random noisy model.
To simplify the discussion, we assume the observation graph G = Kn is a clique. However, our
analysis framework can be extended to handle arbitrary graphs.
Assume ? << a + b. We consider the following noise model, where the noisy measurements are
independent, and they follow
gt
xi ? xgt
with probability p
j + U [??, ?]
tij =
(5)
gt
gt
xi ? xj + U [?a, b] with probability 1 ? p
It is easy to check that the linear programming formulation is unable to recover the ground-truth
b
(1 ? p) > 21 . The following theorem shows that TranSync achieves a sub-constant
solution if a+b
recovery rate instead.
p
Theorem 3.2 There exists a constant c so that if p > c/ log(n), then w.h.p,
kx(k) ? xgt k? ? (1 ? p/2)k (b ? a),
? k = 0, ? ? ? , [? log(
b+a
)/log(1 ? p/2)].
2?
The major difficulty of proving Theorem 3.2 is that x(k) is dependent on tk , making it hard to
control x(k) using existing concentration bounds. We address this issue by showing that the solutions
x(k) , k = 0, ? ? ? , stay close to the segment between xgt and xgt + (1 ? p) a+b
2 1. Specifically, for
points on this segment, we can leverage the independence of tij to derive the following concentration
bound for one step of TranSync:
Lemma 3.1 Consider a fixed observation graph G. Let r =
(a+b)p
(a+b)p+2(1?p)?
and dmin be the
minimum degree of G. Suppose dmin = ?(log (n)), and p + r(1 ? p) = ?(log2 (n)/dmin ) .
Consider an initial point x(0) (independent from tij ) and a threshold parameter ? such that ?a + ? ?
(0)
(0)
mini xi ? maxi xi ? b ? ?. Then w.h.p., one step of TranSync outputs x(1) which satisfies
2
kx(1) ? (1 ? r)x(0) + rxgt )k?
s
! r
p
log(n)
=O
) ? max(kx(0) k2d,? , r2 ) + O
?2 ,
r
(p + r(1 ? p))dmin ?2 (LG )
(0)
where kx(0) kd,? = max |xi
1?i,j?n
(0)
? xj |, and LG is the normalized graph Laplacian of G.
Proof: See Appendix B.1.
Remark 3.1 Note
qthat when G is a clique orqa graph sampled from the standard Erd?os-R?nyi model
? log(n)
log(n)
) = O( (p+r(1?p))n
).
G(n, q), then O( (p+r(1?p))?
2 (LG )
3
To prove Theorem 3.2, we show that when k = O(log 4 (n)), the L? distance between x(k) to the
line segment between xgt and xgt + (1 ? p) a+b
2 1 only grows geometrically, and this distance is in
the order of o(p). On the other hand, (1 ? p/2)k = o(p). So when k ? k, that distance decays with
a geometrical rate that is small than c. The details are deferred to Appendix B.2.
5
Improving recovery rate via sample splitting. Note that Lemma 3.1 enables us to apply standard
sampling tricks to improve the recovery rate. To simplify the discussion, we will assume ? is
sufficiently small. First of all, it is clear that if re-sampling is allowed at each iteration, then TranSync
log(n)
admits a recovery rate of O( ?
). When re-sampling is not allowed, we can improve the recovery
d
min
?
) independent sets, and apply one set of observations
rate by dividing the observations into O( log(n)
n
2
at each iteration. In this case, the recovery rate is O( log?n(n) ). These recovery rates suggest that the
recovery rate in Theorem 3.2 could potentially be improved. Nevertheless, Theorem 3.2 still shows
that TranSync can tolerate a sub-constant recovery rate, which is superior to the linear programming
formulation.
4
Experimental Results
In this section, we provide a detailed experimental evaluation of the proposed translation synchronization (TranSync) method. We begin with describing the experimental setup in Section 4.1. We
then perform evaluations on synthetic and real datasets in Section 4.2 and Section 4.3 respectively.
4.1
Experimental Setup
Datasets. We employ both synthetic datasets and real datasets for evaluation. The synthetic data is
generated following the noisy model described in (5). In the following, we encode the noisy model
as M(G, p, ?), where G is the observation graph, p is the fraction of correct measurements, and ?
describes the interval of correct measurements. Besides the synthetic data, we also consider two real
datasets coming from the applications of joint alignment of point clouds and global ranking from
relative rankings.
Baseline comparison. We choose coordinate descent for solving (1) as the baseline algorithm.
(k)
(k)
Specifically, denote the solution of xi , 1 ? i ? n at iteration k as xi . Then {xi } are given by the
following recursion:
X
(k)
(k?1)
xi = arg min
|xi ? (xj
? tij )|
xi
j?N (i)
(k?1)
= median{xj
j?N (i)
? tij },
1 ? i ? n,
k = 1, 2, ? ? ? ,
(6)
We use the same initial starting point as TranSync. We also tested interior point methods, and all the
datasets used in our experiments are beyond their reach.
Evaluation protocol. We report the min, median, and max of the coordinate-wise difference between
the solution of each algorithm and the underlying ground-truth. We also report the total running time
of each algorithm on each dataset (See Table 1).
4.2
Experimental Evaluation on Synthetic Datasets
We generate the synthetic datasets by sampling from four kinds of observation graphs and two values
of ?, i.e. ? ? {0.01, 0.04}. The graphs are generated according to two modes: 1) dense graphs
versus sparse graphs, and 2) regular graphs versus irregular graphs. To illustrate the strength of
TranSync, we choose p ? {0.4, 0.8} for dense graphs and p ? {0.8, 1.0} for sparse graphs. Below is
a detailed descriptions for all kinds of observation graphs generated.
? Gdr (dense, regular): The first graph contains n = 2000 nodes. Independently, we connect
an edge between a pair of vertices vi , vj with a fixed probability p = 0.1. The expected
degree of each vertex is 200.
? Gdi (dense, irregular): The second graph contains n = 2000 nodes. Independently, we
connect an edge between a pair of vertices vi , vj with probability p = 0.4si sj , where
i?1
si = 0.2 + 0.6 n?1
, 1 ? i ? n are scalar values associated the vertices. The expected degree
of each vertex is about 200.
6
G
Gdr
Gdr
Gdr
Gdr
Gdi
Gdi
Gdi
Gdi
Gsr
Gsr
Gsr
Gsr
Gsi
Gsi
Gsi
Gsi
p
0.4
0.4
0.8
0.8
0.4
0.4
0.8
0.8
0.8
0.8
1.0
1.0
0.8
0.8
1.0
1.0
?
0.01
0.04
0.01
0.04
0.01
0.04
0.01
0.04
0.01
0.04
0.01
0.04
0.01
0.04
0.01
0.04
min
0.95e-2
3.87e-2
0.30e-2
1.19e-2
2.17e-2
5.46e-2
0.34e-2
1.39e-2
0.58e-2
2.35e-2
0.45e-2
1.84e-2
0.72e-2
2.88e-2
0.53e-2
2.24e-2
Coordinate Descent
median
max
1.28e-2
11.40e-2
4.73e-2
18.59e-2
0.34e-2
0.41e-2
1.35e-2
1.78e-2
17.59e-2 50.51e-2
19.40e-2 53.88e-2
0.42e-2
0.58e-2
1.66e-2
2.30e-2
0.65e-2
0.79e-2
2.62e-2
3.54e-2
0.50e-2
0.58e-2
1.99e-2
2.36e-2
0.85e-2
75.85e-2
3.38e-2
11.48e-2
0.62e-2
0.77e-2
2.52e-2
3.12e-2
time
0.939s
1.325s
0.781s
1.006s
0.865s
1.043s
0.766s
0.972s
10.062s
12.375s
9.798s
11.626s
10.236s
12.350s
9.388s
12.200s
min
0.30e-2
1.04e-2
0.16e-2
0.57e-2
0.39e-2
1.25e-2
0.17e-2
0.68e-2
0.38e-2
1.35e-2
0.28e-2
1.14e-2
0.52e-2
1.79e-2
0.37e-2
1.44e-2
TranSync
median
max
0.37e-2 0.60e-2
1.22e-2 1.59e-2
0.18e-2 0.28e-2
0.70e-2 0.87e-2
0.52e-2 0.93e-2
1.55e-2 2.42e-2
0.24e-2 0.33e-2
0.86e-2 1.16e-2
0.45e-2 0.61e-2
1.55e-2 2.05e-2
0.32e-2 0.39e-2
1.29e-2 1.60e-2
0.64e-2 1.10e-2
2.16e-2 3.59e-2
0.43e-2 0.57e-2
1.72e-2 2.47e-2
time
0.178s
0.155s
0.149s
0.133s
0.179s
0.169s
0.159s
0.141s
1.852s
1.577s
0.188s
0.179s
1.835s
1.610s
0.180s
0.187s
Table 1: Experimental results comparing TranSync and Coordinate Descent (CD) under different
settings. All statistics (min, median, max) and mean running time are computed among 100 independent experiments with the same setting. As observed, TranSync outperforms Coordinate Descent in
all experiments.
? Gsr (sparse, regular): The third graph is generated in a similar fashion as the first graph,
except that the number of nodes n = 20K, and the connecting probability is set to p = 0.003.
The expected degree of each vertex is 60.
? Gsi (sparse, irregular): The fourth graph is generated in a similar fashion as the second
graph, except that the number of nodes n = 20K, and the connecting probability between a
i?1
, 1 ? i ? n are scalar values
pair of vertices is p = 0.1si sj , where si = 0.07 + 0.21 n?1
associated the vertices. The expected degree of each vertex is about 60.
For this experiment, instead of using kmax as stopping condition as in Algorithm 1, we stop when we
observe ?k < ?min . Here ?min does not need to be close to ?. In fact, we choose ?min = 0.05, 0.1 for
? = 0.01, 0.04, respectively. We also claim that if a small validation set (with size significantly less
than n) of correct observations is available, our performance could be further improved.
As illustrated in Table 1, TranSync dominates coordinate descent in terms of both accuracy and
prediction. In particular, TranSync is significantly better than coordinate descent on dense graphs in
terms of accuracy. In particular, on dense but irregular graphs, coordinate descent did not converge
at all when p = 0.8. The main advantage of TranSync on sparse graphs is the computational cost,
although the accuracy is still considerably better than coordinate descent.
4.3
Experimental Evaluation on Real Datasets
Translation synchronization for joint alignment of point clouds. In the first application, we
consider the problem of joint alignment of point clouds from pair-wise alignment [10]. To this end,
we utilize the Patriot Circle Lidar dataset1 . We uniformly subsampled the dataset to 6K scans. We
applied Super4PCS [19] to match each scan to 300 randomly selected scans, where each match
returns a pair-wise rigid transformation and a score. We then pick the top-30 matches for each scan,
this results in a graph with 140K edges. To create the input data for translation synchronization, we
run the state-of-the-art rotation synchronization algorithm described in [2] to estimate a global pose
Ri for each scan. The pair-wise measurement tij from node i to node j is then given by RiT tlocal
ij ,
where tlocal
is
the
translation
vector
obtained
in
pair-wise
matching.
The
average
outlier
ratio
of
the
ij
pair-wise matches per node is 35%, which is relatively high since the observation graph is fairly
sparse. Since tij is a 3D vector, we run TranSync three times, one for each coordinate. As illustrated
in Figure 1, TranSync is able to recover the the global shape of the underlying scanning trajectory. In
contrast, coordinate descent completely fails on this dataset.
1
http://masc.cs.gmu.edu/wiki/MapGMU
7
Figure 1: The application of TranSync in joint alignment of 6K Lidar scans around a city block. (a)
Snapshot of the underlying scanning trajectory. (b) Reconstruction using TranSync (c) Reconstruction
using Coordinate Descent.
Movie
Shakespeare in Love
Witness
October Sky
The Waterboy
Interview with the Vampire
Dune
MRQE
1(85)
2(77)
3(76)
4(66)
5(65)
6(44)
Hodge-Diff.
1(0.247)
2(0.217)
3(0.213)
6(-0.464)
4(-0.031)
5(-0.183)
Global ranking (score)
Hodge-Ratio Hodge-Binary
2(0.078)
1 (0.138)
1(0.088)
3(0.107)
3(0.078)
2(0.111)
6(-0.162)
6(-0.252)
4(-0.012)
4(-0.120)
5(-0.069)
5(-0.092)
TS-Init
1(0.135)
3(0.076)
2(0.092)
5(-0.134)
4 (-0.098)
6(-0.216)
TS-Final
1(0.219)
2(0.095)
3(0.0714)
4(-0.112)
5(-0.140)
6(-0.281)
Table 2: Global ranking of selected six movies via different methods: MRQE, HodgeRank[16] with
1) arithmetic mean score difference, 2) geometric mean score ratio and 3) and binary comparisons,
and the initial and final predictions of TranSync. TranSync results in the most consistent result with
MRQE.
Ranking from relative comparisons. In the second application, we apply TranSync to predict
global rankings of Netflix movies from their relative comparisons provided by users. The Netflix
dataset contains 17070 movies that were rated between October, 1998 and December, 2005. We adapt
the procedure described in [16] to generate the input data. Specifically, for each pair of movies, we
average the relative ratings from the same users within the same month. We only consider a relative
measurement if we collect more than 10 such relative ratings. We then apply TranSync to predict
the global rankings of all the movies. We report the initial prediction obtained by the first step of
TranSync (i.e., all the relative comparisons are used) and the final prediction suggested by TranSync
(i.e., after removing inconsistent relative comparisons).
Table 2 compares TranSync with HodgeRank [16] on six representative movies that are studied
in [16]. The experimental results show that both predictions appear to be more consistent with
MRQE2 (the largest online directory of movie reviews on the internet) than HodgeRank [16] and its
variants, which were only applied on these six movies in isolation. Moreover, the final prediction is
superior to the initial prediction. These observations indicate two key advantages of TranSync, i.e.,
scalability on large-scale datasets and robustness to noisy relative comparisons.
5
Conclusions and Future Work
In this paper, we have introduced an iterative algorithm for solving the translation synchronization
problem, which estimates the global locations of objects from noisy measurements of relative
locations. We have justified the performance of our approach both experimentally and theoretically
under both deterministic and randomized conditions. Our approach is more scalable and accurate
than the standard linear programming formulation. In particular, when the pair-wise measurement
2
http://www.mrqe.com
8
is biased, our approach can still achieve sub-constant recovery rate, while the linear programming
approach can tolerate no more than 50% of the measurements being biased.
In the future, we plan to extend this iterative scheme to other synchronization problems, such as
synchronizing rotations and point-based maps. Moreover, it would also be interesting to study
variants of the iterative scheme such as re-weighted least squares. We would also like to close the
gap between the current recovery rate and the lower bound, which exhibits a poly-log factor. This
requires developing new tools for analyzing the iterative algorithm.
Acknowledgement. Qixing Huang would like to acknowledge support this research from NSF DMS1700234. Chandrajit Bajaj would like to acknowledge support for this research from the National
Institute of Health grants #R41 GM116300 and #R01 GM117594.
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10
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6,351 | 6,745 | From which world is your graph?
Cheng Li
College of William & Mary
Felix M. F. Wong
Independent Researcher?
Zhenming Liu
College of William & Mary
Varun Kanade
University of Oxford
Abstract
Discovering statistical structure from links is a fundamental problem in the analysis of social networks. Choosing a misspecified model, or equivalently, an incorrect inference algorithm will result in an invalid analysis or even falsely uncover
patterns that are in fact artifacts of the model. This work focuses on unifying two
of the most widely used link-formation models: the stochastic blockmodel (SBM)
and the small world (or latent space) model (SWM). Integrating techniques from
kernel learning, spectral graph theory, and nonlinear dimensionality reduction, we
develop the first statistically sound polynomial-time algorithm to discover latent
patterns in sparse graphs for both models. When the network comes from an SBM,
the algorithm outputs a block structure. When it is from an SWM, the algorithm
outputs estimates of each node?s latent position.
1
Introduction
Discovering statistical structures from links is a fundamental problem in the analysis of social
networks. Connections between entities are typically formed based on underlying feature-based
similarities; however these features themselves are partially or entirely hidden. A question of great
interest is to what extent can these latent features be inferred from the observable links in the network. This work focuses on the so-called assortative setting, the principle that similar individuals
are more likely to interact with each other. Most stochastic models of social networks rely on this assumption, including the two most famous ones ? the stochastic blockmodel [1] and the small-world
model [2, 3], described below.
Stochastic Blockmodel (SBM). In a stochastic blockmodel [4, 5, 6, 7, 8, 9, 10, 11, 12, 13], nodes
are grouped into disjoint ?communities? and links are added randomly between nodes, with a higher
probability if nodes are in the same community. In its simplest incarnation, an edge is added between
nodes within the same community with probability p, and between nodes in different communities
with probability q, for p > q. Despite arguably na??ve modelling choices, such as the independence
of edges, algorithms designed with SBM work well in practice [14, 15].
Small-World Model (SWM). In a small-world model, each node is associated with a latent variable
xi , e.g., the geographic location of an individual. The probability that there is a link between two
nodes is proportional to an inverse polynomial of some notion of distance, dist(xi , xj ), between
them. The presence of a small number of ?long-range? connections is essential to some of the most
intriguing properties of these networks, such as small diameter and fast decentralized routing algorithms [3]. In general, the latent position may reflect geographic location as well as more abstract
concepts, e.g., position on a political ideology spectrum.
The Inference Problem. Without observing the latent positions, or knowing which model generates
the underlying graph, the adjacency matrix of a social graph typically looks like the one shown in
?
Currently at Google.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Fig. 5(a) (App. A.1). However, if the model generating the graph is known, it is then possible to
run a suitable ?clustering algorithm? [14, 16] that reveals the hidden structure. When the vertices
are ordered suitably, the SBM?s adjacency matrix looks like the one shown in Fig. 5(b) (App. A.1)
and that of the SWM looks like the one shown in Fig. 5(c) (App. A.1). Existing algorithms typically
depend on knowing the ?true? model and are tailored to graphs generated according to one of these
models, e.g., [14, 16, 17, 18].
Our Contributions. We consider a latent space model that is general enough to include both these
models as special cases. In our model, an edge is added between two nodes with a probability
that is a decreasing function of the distance between their latent positions. This model is a fairly
natural one, and it is quite likely that a variant has already been studied; however, to the best of
our knowledge there is no known statistically sound and computationally efficient algorithm for
latent-position inference on a model as general as the one we consider.
1. A unified model. We propose a model that is a natural generalization of both the stochastic
blockmodel and the small-world model that captures some of the key properties of real-world social
networks, such as small out-degrees for ordinary users and large in-degrees for celebrities. We focus
on a simplified model where we have a modest degree graph only on ?celebrities?; the full paper
material contains an analysis of the more realistic model using somewhat technical machinery [19].
2. A provable algorithm. We present statistically sound and polynomial-time algorithms for inferring
latent positions in our model(s). Our algorithm approximately infers the latent positions of almost all
?celebrities? (1 ? o(1)-fraction), and approximately infers a constant fraction of the latent positions
of ordinary users. We show that it is statistically impossible to err on at most o(1) fraction of
ordinary users by using standard lower bound arguments.
3. Proof-of-concept experiments. We report several experiments on synthetic and real-world data
collected on Twitter from Oct 1 and Nov 30, 2016. Our experiments demonstrate that our model and
inference algorithms perform well on real-world data and reveal interesting structures in networks.
Additional Related Work. We briefly review the relevant published literature. 1. Graphon &
Latent-space techniques. Studies using graphons and latent-space models have focused on the statistical properties of the estimators [20, 21, 22, 23, 24, 25, 26, 27, 28], with limited attention paid
to computational efficiency. The ?USVT? technique developed recently [29] estimates the kernel
well when the graph is dense. Xu et al. [30] consider a polynomial time algorithm for a sparse
model similar to ours, but focus on edge classification rather than latent position estimation. 2.
Correspondence analysis in political science. Estimating the ideology scores of politicians is an important research topic in political science [31, 32, 33, 34, 35, 36, 17, 18]. High accuracy heuristics
developed to analyze dense graphs include [17, 18].
Organization. Section 2 describes background, our model and results. Section 3 describes our
algorithm and an gives an overview of its analysis. Section 4 contains the experiments.
2
Preliminaries and Summary of Results
Basic Notation. We use c0 , c1 , etc. to denote constants which may be different in each case. We
use whp to denote with high probability, by which we mean with probability larger 1 ? n1c for any
c. All notation is summarized in Appendix B for quick reference.
Stochastic Blockmodel. Let n be the number of nodes in the graph with each node assigned a label
from the set {1, . . . , k} uniformly at random. An edge is added between two nodes with the same
label with probability p and between the nodes with different labels with probability q, with p > q
(assortative case). In this work, we focus on the k = 2 case, where p, q = ? ((log n)c /n) and the
community sizes are exactly the same. (Many studies of the regimes where recovery is possible have
been published [37, 9, 5, 8].)
P
Q
Let A be the adjacency matrix of the realized graph and let M = E[A] = Q
, where
P
n
n
P and Q ? R 2 ? 2 with every entry equal to p and q, respectively. We next explain the inference
algorithm, which uses two key observations. 1. Spectral Properties of M . M has rank 2 and the
non-trivial eigenvectors are (1, . . . , 1)T and (1, . . . , 1, ?1, . . . , ?1) corresponding to eigenvalues
n(p + q)/2 and n(p ? q)/2, respectively. If one has access to M , the hidden structure in the graph
is revealed merely by reading off the second eigenvector. 2. Low Discrepancy between A and
2
M . Provided the average degree n(p + q)/2 and the gap p ? q are large enough, the spectrum
and eigenspaces of the matrices A and M can be shown to be close using matrix concentration
inequalities and the Davis-Kahan theorem [38, 39]. Thus, it is sufficient to look at the projection of
the columns of A onto the top two eigenvectors of A to identify the hidden latent structure.
Small-World Model (SWM). In a 1-dim. SWM, each node vi is associated with an independent
latent variable xi ? [0, 1] that is drawn from the uniform distribution on [0, 1]. The probability of a
link between two nodes is Pr[{vi , vj } ? E] ? |xi ?xj1|? +c0 , where ? > 1 is a hyper-parameter.
The inference algorithm for small-world models uses different ideas. Each edge in the graph
is considered as either ?short-range? or ?long-range.? Short-range edges are those between nodes
that are nearby in latent space, while long-range edges have end-points that are far away in latent
space. After removing the long-range edges, the shortest path distance between two nodes scales
proportionally to the corresponding latent space distance (see Fig. 6 in App. A.2). After obtaining
estimates for pairwise distances, standard buidling blocks are used to find the latent positions xi [40].
The key observation used to remove the long-range edges is: an edge {vi , vj } is a short-range edge
if and only if vi and vj will share many neighbors.
A Unified Model. Both SBM and SWM are special cases of our unified latent space model. We
begin by describing the full-fledged bipartite (heterogeneous) model that is a better approximation
of real-world networks, but requires sophisticated algorithmic techniques (see [19] for a detailed
analysis). Next, we present a simplified (homogeneous) model to explain the key ideas.
Bipartite Model. We use latent-space model to characterize the stochastic interactions between
users. Each individual is associated with a latent variable in [0, 1]. The bipartite graph model
consists of two types of users: the left side of the graph Y = {y1 , . . . , ym } are the followers
(ordinary users) and the right side X = {x1 , . . . , xn } are the influencers (celebrities). Both yi and
xi are i.i.d. random variables from a distribution D. This assumption follows the convention of
existing heterogeneous models [41, 42]. The probability that two individuals yi and xj interact is
?(yi , xj )/n, where ? : [0, 1] ? [0, 1] ? (0, 1] is a kernel function. Throughout this paper we assume
that ? is a small-world kernel, i.e., ?(x, y) = c0 /(kx ? yk? + c1 ) for some ? > 1 and suitable
constants c0 , c1 , and that m = ?(n ? polylog(n)). Let B ? Rm?n be a binary matrix that Bi,j = 1
if and only if there is an edge between yi and xj . Our goal is to estimate {xi }i?[n] based on B for
suitably large n.
Simplified Model. The graph only has the node set is X = {x1 , ..., xn } of celebrity users. Each
xi is again an i.i.d. random variable from D. The probability that two users vi and vj interact
is ?(xi , xj )/C(n). The denominator is a normalization term that controls the edge density of the
graph. We assume C(n) = n/polylog(n), i.e., the average degree is polylog(n). Unlike the SWM
where the xi are drawn uniformly from [0, 1], in the unified model D can be flexible. When D is the
uniform distribution, the model is the standard SWM. When D has discrete support (e.g., xi = 0
with prob. 1/2 and xi = 1 otherwise), then the unified model reduces to the SBM. Our distributionagnostic algorithm can automatically select the most suitable model from SBM and SWM, and infer
the latent positions of (almost) all the nodes.
Bipartite vs. Simplified Model. The simplified model suffers from the following problem: If the
average degree is O(1), then we err on estimating every individual?s latent position with a constant
probability (e.g., whp the graph is disconnected), but in practice we usually want a high prediction
accuracy on the subset of nodes corresponding to high-profile users. Assuming that the average
degree is ?(1) mismatches empirical social network data. Therefore, we use a bipartite model that
introduces heterogeneity among nodes: By splitting the nodes into two classes, we achieve high
estimation accuracy on the influencers and the degree distribution more closely matches real-world
data. For example, in most online social networks, nodes have O(1) average degree, and a small
fraction of users (influencers) account for the production of almost all ?trendy? content while most
users (followers) simply consume the content.
Additional Remarks on the Bipartite Model. 1. Algorithmic contribution. Our algorithm computes B T B and then regularizes the product by shrinking the diagonal entries before carrying out
spectral analysis. Previous studies of the bipartite graph in similar settings [43, 44, 45] attempt to
construct a regularized product using different heuristics. Our work presents the first theoretically
sound regularization technique for spectral algorithms. In addition, some studies have suggested running SVD on B directly (e.g., [28]). We show that the (right) singular vectors of B do not converge
3
to the eigenvectors of K (the matrix with entries ?(xi , xj )). Thus, it is necessary to take the product
and use regularization. 2. Comparison to degree-corrected models (DCM). In DCM, each node vi is
associated with a degree parameter D(vi ). Then we have Pr[{vi , vj } ? E] ? D(vi )?(xi , xj )D(vj ).
The DCM model implies the subgraph induced by the highest degree nodes is dense, which is inconsistent with real-world networks. There is a need for better tools to analyze the asymptotic behavior
of such models and we leave this for future work (see, e.g., [41, 42]).
Theoretical Results. Let F be the cdf of D. We say F and ? are well-conditioned if:
(1) F has finitely many points of discontinuity, i.e., the closure of the support of F can be expressed
as the union of non-overlapping closed intervals I1 , I2 , ..., Ik for a finite number k.
(2)
R F is near-uniform, i.e., for any interval I that has non-empty overlap with F ?s support,
dF (x) ? c0 |I|, for some constant c0 .
I
(3) Decay Condition: The eigenvalues
of the integral operator based on ? and F decay sufficiently
R
fast. We define the Kf (x) = ?(x, x0 )f (x0 )dF (x0 ) and let (?i )i?1 denote the eigenvalues of K.
Then, it holds that ?i = O(i?2.5 ).
If we use the small-word kernel ?(x, y) = c0 /(|x ? y|? + c1 ) and choose F that gives rise
to SBM or SWM, in each case the pair F and ? are well-conditioned, as described below. As the
decay condition is slightly more invoved, we comment upon it. The condition is a mild one. When
F is uniformly distributed on [0, 1], it is equivalent to requiring K to be twice differentiable, which
is true for the small world kernel. When F has a finite discrete support, there are only finitely many
non-zero eigenvalues, i.e., this condition also holds. The decay condition holds in more general
settings, e.g., when F is piecewise linear [46] (see [19]). Without the decay condition, we would
require much stronger assumptions: Either the graph is very dense or ? 2. Neither of these
assumptions is realistic, so effectively our algorithm fails to work. In practice, whether the decay
condition is satisfied can be checked by making a log-log plot and it has been observed that for
several real-world networks, the eigenvalues follow a power-law distribution [47].
Next, we define the notion of latent position recovery for our algorithms.
Definition 2.1 ((?, ?, ?)-Aproximation Algorithm). Let Ii , F , and K be defined as above, and let
Ri = {xj : xj ? Ii }. An algorithm is called an (?, ?, ?)-approximation algorithm if
1. It outputs a collection of disjoint points C1 , C2 , . . . , Ck such that Ci ? Ri , which correspond to
subsets of reconstructed latent variables.
2. For each Ci , it produces a distance matrix D(i) . Let Gi ? Ci be such that for any ij , ik ? Gi
(i)
(i)
Dij ,ik ? |xij ? xik | ? (1 + ?)Dij ,ik + ?.
(1)
S
3. | i Gi | ? (1 ? ?)n.
In bipartite graphs, Eq.(1) is required only for influencers.
We do not attempt to optimize constants in this paper. We set ? = o(1), ? a small constant,
and ? = o(1). Definition 2.1 allows two types of errors: Ci s are not required to form a partition
i.e., some nodes can be left out, and a small fraction of estimation errors is allowed in each Ci ,
e.g., if xj = 0.9 but x
bj = 0.2, then the j-th ?row? in D(i) is incorrect. To interpret the definition,
consider the blockmodel with 2 communities. Condition 1 means that our algorithm will output two
disjoint groups of points. Each group corresponds to one block. Condition 2 means that there are
pairwise distance estimates within each group. Since the true distances for nodes within the same
block are zero, our estimates must also be zero to satisfy Eq.1. Condition 3 says that the proportion
of misclassified nodes is ? = o(1). We can also interpret the definition when we consider a smallworld graph, in which case k = 1. The algorithm outputs pairwise distances for a subset C1 . We
know that there is a sufficiently large G1 ? C1 such that the pairwise distances are all correct in C1 .
Our algorithm does not attempt to estimate the distance between Ci and Cj for i 6= j. When
the support contains multiple disjoint intervals, e.g., in the SBM case, it first pulls apart the nodes in
different communities. Estimating the distance between intervals, given the output of our algorithm
is straightforward. Our main result is the following.
Theorem 2.2. Using the notation above, assume F and ? are well-conditioned, and C(n) and
m/n are ?(logc n) for some suitably large c. The algorithm for the simplified model shown in
Figure 1 and that for the bipartite model (appears in [19]) give us an (1/ log2 n, , O(1/ log n))approximation algorithm w.h.p. for any constant . Furthermore, the distance estimates D(i) for
each Ci are constructed using the shortest path distance of an unweighted graph.
4
L ATENT-I NFERENCE(A)
1 // Step 1. Estimate ? .
b = SM-E ST(A).
2 ?
3 // Step 2. Execute isomap algo.
b
4 D = I SOMAP -A LGO(?)
5 // Step 3. Find latent variables.
6 Run a line embedding algorithm [48, 49].
b `)
I SOMAP -A LGO(?,
b (See Section 3.2)
1 Execute S ? D ENOISE(?)
2 // S is a subset of [n].
3 Build G = {S, E} s.t. {i, j} ? E iff
? d )i ? (?
? d )j | ? `/ log n (` a constant).
4
|(?
5 Compute D such D(i, j) is the shortest
6
path distance between i and j when i, j ? S.
7 return D
SM-E ST(A, t)
?A , S?A , V?A ] = svd(A).
1 [U
2 Let also ?i be i-th singular value of A.
3 // let t be a suitable parameter.
4 d = D ECIDE T HRESHOLD(t, ?(n)).
5 SA : diagonal matrix comprised of {?i }i?d
6 UA , VA : the singular vectors
7
corresponding
to SA .
p
b = C(n)UA S 1/2 .
8 Let ?
A
b
9 return ?
D ECIDE T HRESHOLD(t, ?(n))
1 // This procedure decides d the number
2
of Eigenvectors to keep.
3 // t is a tunable parameter. See Proposition 3.1.
A
A
4 d = arg maxd {?d ( ?(n)
) ? ?d+1 ( ?(n)
) ? ?}.
24/59
5 where ? = 10(t/?(n))
Figure 1: Subroutines of our Latent Inference Algorithm.
Pairwise Estimation to Line-embedding and High-dimensional Generalization. Our algorithm builds estimates on pairwise latent distance and uses well-studied metric-embedding methods [48, 49] as blackboxes to infer latent positions. Our inference algorithm can be generalized to
d-dimensional space with d being a constant. But the metric-embedding on `dp becomes increasingly
?
difficult, e.g., when d = 2, the approximation ratio for embedding a graph is ?( n) [50].
3
Our algorithms
As previously noted, SBM and SWM are special cases of our unified model and both require
different algorithmic techniques. Given that it is not surprising that our algorithm blends ingredients
from both sets of techniques. Before proceeding, we review basics of kernel learning.
Notation. Let A be the adjacency matrix of the observed graph (simplified model) and let ?(n) ,
?K S?K V? T (U
?A S?A V? T ) be the SVD of K
n/C(n). Let K be the matrix with entries ?(xi , xj ). Let U
K
A
(A). Let d be a parameter to be chosen later. Let SK (SA ) be a d ? d diagonal matrix comprising the
d-largest eigenvalues of K (A). Let UK (UA ) and VK (VA ) be the corresponding singular vectors of
? = UK SK V T (A? = UA SA V T ) be the low-rank approximation of K (A). Note
K (A). Finally, let K
A
K
that when a matrix is positive definite and symmetric SVD coincides with eigen-decomposition; as
a consequence UK = VK and UA = VA .
R
Kernel Learning. Define an integral operator K as Kf (x) =
?(x, x0 )f (x0 )dF (x0 ). Let
?1 , ?2 , . . . be the eigenfunctions of K and ?1 , ?2 , . . . be the corresponding eigenvalues such that
?1 ? ?2 ? ? ? ? and ?i ? 0 for each i. Also let NH be the number of eigenfunctions/eigenvalues
of K, which is either finite or countably infinite. p
We recall some important properties of K [51, 25].
For x ? [0, 1], define the feature map ?(x) = ( ?j ?j (x) : j = 1, 2, ...), so that h?(x), ?(x0 )i =
p
?(x, x0 ). We also consider a truncated feature ?d (x) = ( ?j ?j (x) : j = 1, 2, ..., d). Intuitively,
if ?j is too small for sufficiently large j, then the first d coordinates (i.e., ?d ) already
p approximate
the feature map well. Finally, let ?d (X) ? Rn?d such that its (i, j)-th entry is ?j ?j (xi ). Let?s
further write (?d (X)):,i be the i-th column of ?d (X). Let ?(X) = limd?? ?d (X). When the
context is clear, shorten ?d (X) and ?(X) to ?d and ?, respectively.
There are two main steps in our algorithm which we explain in the following two subsections.
3.1
Estimation of ? through K and A
The mapping ? : [0, 1] ? RNH is bijective so a (reasonably) accurate estimate of ?(xi ) can
be used to recover xi . Our main result is the design of a data-driven procedure to choose a suitable
number of eigenvectors and eigenvalues of A to approximate ? (see SM-E ST(A) in Fig. 1).
5
Proposition 3.1. Let t be a tunable parameter such that t = o(?(n)) and t2 /?(n) = ?(log n).
b ? RNH be such that its first d-coordinates are
Let d be chosen by D ECIDE T HRESHOLD(?). Let ?
p
1/2
equal to C(n)UA SA , and its remaining entries are 0. If ?(n) = ?(log n) and K (F and ?) is
well-conditioned, then with high probability:
?
2
b ? ?kF = O
(2)
k?
n (t/(?(n))) 29
b ? ?kF = O ?n??2/87 (n) . We remark that
Specifically, by letting t = ?2/3 (n), we have k?
our result is stronger than an analogous result for sparse graphs in [25] as our estimate is close to ?
rather than the truncated ?d .
Remark on the Eigengap. In our analysis, there are three groups of eigenvalues: the eigenvalues
of K, those of K, and those of A. They are in different scales: ?i (K) ? 1 (resulting from the
fact that ?(x, y) ? 1 for all x and y), and ?i (A/?(n)) ? ?i (K/n) ? ?i (K) if n and ?(n) are
sufficiently large. Thus, ?d (K) are independent of n for a fixed d and should be treated as ?(1).
Also ?d , ?d (K) ? ?d+1 (K) ? 0 as d ? ?. Since the procedure of choosing d depends on C(n)
(and thus also on n), ?d depends on n and can be bounded by a function in n. This is the reason why
Proposition 3.1 does not explicitly depend on the eigengap. We also note that we cannot directly
find ?d based on the input matrix A. But standard interlacing results can give ?d = ?(?d (A/?(n))?
?d+1 (A/?(n))) (cf. [19]).
Intuition of the algorithm.
Using Mercer?s theorem, we have h?(xi ), ?(xj )i =
limd?? h?d (xi ), ?d (xj )i = ?(xi , xj ). Thus, limd?? ?d ?T
d = K. On the other hand, we have
?K S?1/2 )(U
?K S?1/2 )T = K. Thus, ?d (X) and U
?K S?1/2 are approximately the same, up to a uni(U
K
K
K
tary transformation. We need to identify different sources of errors to understand the approximation
quality.
Error source 1. Finite samples to learn the kernel. We want to infer about ?continuous objects? ?
and D (specifically the eigenfunctions of K) but K only contains the kernel values of a finite set of
pairs. From standard results in Kernel PCA [52, 25], we have with probability ? 1 ? ,
p
p
?
?
log ?1
log ?1
1/2
kUK SK W ? ?d (X)kF ? 2 2
=2 2
.
?d (K) ? ?d+1 (K)
?d
Error source 2. Only observe A. We observe only the realized graph A and not K, though it holds
? ?1/2
that EA =
K/C(n). Thus, we can only
use singular
? vectors
of C(n)A to approximate UK SK .
p
1/2
1/2
dn
. When A is dense (i.e., C(n) = O(1)),
We have:
C(n)UA SA W ? UK SK
= O ?t2 ?(n)
d
F
the problem is analyzed in [25]. We generalize the results in [25] for the sparse graph case. See [19]
for a complete analysis.
?A ):,i ?outweighs? the
Error source 3. Truncation error. When i is large, the noise in ?i (A)(U
signal. Thus, we need to choose a d such that only the first d eigenvectors/eigenvalues
of A are
?
used to approximate ?d . Here, we need to address the truncation error: the tail { ?i ?i (xj )}i>d is
thrown away.
Next we analyze the magitude of the tail. We abuse notation so that ?d (x) refers to both a
d-dimensional vector and a NH -dimensional
after the d-th one
R
? vector in which
P
P all entries
P are 0.
We have Ek?(x) ? ?d (x)k2 = i>d E[( ?i ?i (x))2 ] = i>d ?i |?i (x)|2 dF (x) = i>d ?i .
? pP
(A Chernoff bound is used to obtain that k? ? ?d kF = O( n/(
i>d ?i ))). Using the decay
condition, we show that a d can be identified so that the tail can be bounded by a polynomial in ?d .
The details are technical and are provided in [19].
3.2
b i ) through Isomap
Estimating Pairwise Distances from ?(x
b d , we
See I SOMAP -A LGO(?) in Fig. 1 for the pseudocode. After we construct our estimate ?
T
?
b
b
b
estimate K by letting K = ?d ?d . Recalling Ki,j = c0 /(|xi ? xj | + c1 ), a plausible approach is
b i,j ? c1 )1/? . However, ?(xi , xj ) is a convex function in |xi ? xj |.
to estimate |xi ? xj | = (c0 /K
6
(a) True features
(b) Estimated features (c) Isomap w/o denoising (d) Isomap + denoising
Figure 2: Using the Isomap Algorithm to recover pairwise distances. (a) The true curve C = {?(x)}x?[0,1]
b (c) Shows that an undesirable short-cut may exist when we run the Isomap algorithm and (d)
(b) Estimate ?
Shows the result of running the Isomap algorithm after removal of the corrupted nodes.
Thus, when Ki,j is small, a small estimation error here will result in an amplified estimation error
in |xi ? xj | (see also Fig. 7 in App. A.3). But when |xi ? xj | is small, Ki,j is reliable (see the
?reliable? region in Fig. 7 in App. A.3).
Thus, our algorithm only uses large values of Ki,j to construct estimates. The isomap technique
introduced in topological learning [53, 54] is designed to handle this setting. Specifically, the set
b i )}i?[n] will be a noisy
C = {?(x)}x?[0,1] forms a curve in RNH (Fig. 2(a)). Our estimate {?(x
approximation of the curve (Fig. 2(b)). Thus, we build up a graph on {?(xi )}i?n so that xi and xj
b i ) and ?(x
b j ) are close (Fig. 2(c-d)). Then the shortest path distance
are connected if and only if ?(x
on G approximates the geodesic distance on C. By using the fact that ? is a radial basis kernel, the
geodesic distance will also be proportional to the latent distance.
Corrupted nodes. Excessively corrupted nodes may help build up ?undesirable bridges? and interfere with the shortest-path based estimation (cf.Fig. 2(c)). Here, the shortest path between two green
nodes ?jumps through? the excessively corrupted nodes (labeled in red) so the shortest path distance
is very different from the geodesic distance.
Below, we describe a procedure to remove excessively corrupted nodes and then explain how
to analyze the isomap technique?s performance after their removal. Note that d in this section mostly
refers to the shortest path distance.
Step 1. Eliminate corrupted nodes. Recall that x1 , x2 , ..., xn are the latent variables. Let zi =
b i ). For any z ? RNH and r > 0, we let Ball(z, r) = {z 0 : kz 0 ?zk ? r}. Define
?(xi ) and zbi = ?(x
projection Proj(z) = arg minz0 ?C kz 0 ? zk, where C is the curve formed by {?(x)}x?[0,1] . Finally,
for any point z ? C, define ??1 (z) such that ?(??1 (z)) = z (i.e., z?s original latent position). For
the points that fall outside of C, define ??1 (z) = ??1 (Proj(z)). Let us re-parametrize the error
b ? ?kF ? ?n/f (n), where f (n) = ?2/87 (n) =
term in Propostion 3.1. Let f (n) be such that k?
2
b i ) ? ?(xi )k2 ?
?(log
we have Pri [k?(x
p
p n) for sufficiently large ?(n). By Markov?s inequality,
2
b
1/ f (n)] ? 1/f (n). Intuitively, when k?(xi ) ? ?(xi )k ? 1/ f (n), i becomes a candidate that
can serve to build up undesirable shortcuts. Thus, we want to eliminate these nodes.
p
Looking at a ball of radius O(1/ f (n)) centered at a point zbi , consider two cases.
Case 1. If zbi is close to Proj(b
zi ), i.e., corresponding to the blue nodes in Figure 2(c). For the purpose
of exposition, let us assume
z
b
| = O(f ?1/? (n)), then we
p i = zi . Now for any point zj , if |xi ? xjp
have kb
zi ? zbj k = O(1/ f (n)), which means zj is in Ball(zi , O(1/ f (n))). The total number
of such nodes will be in the order of ?(n/f 1/? (n)), by using the near-uniform density assumption.
Case 2. If zbi is far awaypfrom any point in C, i.e., corresponding to the red ball in Figure 2(c), any
points in Ball(b
zi , O(1/ f (n))) will also be far from C. Then the total number of such nodes will
be O(n/f (n)).
As n/f 1/? (n) = ?(n/f (n)) for ? > 1, there is a phase-transition phenomenon: When zbi
is far from C, then a neighborhood of zbi contains O(n/f (n)) nodes. When zbi is close to C, then a
neighborhood of zbi contains ?(n/f (n)) nodes. We can leverage this intuition to design a countingbased algorithm to eliminate nodes that are far from C:
D ENOISE(b
zi ) : If |Ball(b
zi , 3/
p
f (n))| < n/f (n), remove zbi .
7
(3)
Algo.
Ours
Mod. [55]
CA [18]
Maj [56]
RW [54]
MDS [49]
?
0.53
0.16
0.20
0.13
0.01
0.05
Slope of ?
9.54
1.14
0.11
0.09
1.92
30.91
S.E.
0.28
0.02
7e-4
0.02
0.65
120.9
p-value
< 0.001
< 0.001
< 0.001
< 0.001
< 0.001
0.09
Figure 3: Latent Estimates vs. Ground-truth.
(a) Inferred kernel
(b) SWM
(c) SBM
Figure 4: Visualization of real and synthetic networks. (a) Our inferred kernel matrix, which is ?in-between?
(b) the small-world model and (c) the stochastic blockmodel.
Theoretical result. We classify a point i into three groups:
p
1. Good: Satisfying kb
zi ? Proj(b
zi )k ? 1/ f (n). We
p further partition the set of good points into
two parts. Good-I are points such that kb
zi ? zi k ? 1/ f (n), while Good-II are points that are good
but not in Good-I.
p
2. Bad: when kzi ? Proj(zi )k > 4/ f (n).
3. Unclear: otherwise.
Lemma 3.2. (cf. [19] ) After running D ENOISE that uses the counting-based decision rule, all good
points are kept, all bad points are eliminated, and all unclear points have no performance guarantee.
The total number of eliminated nodes is ? n/f (n).
Step 2. An isomap-based algorithm. Wlog assume there is only one closed interval for
support(F ). Wepbuild a graph G on [n] so that two nodes zbi and zbj are connected if and only
if kb
zi ? zbj k ? `/ f (n), where ` is a sufficiently large constant (say 10). Consider the shortest path
distance between arbitrary pairs of nodes i and j (that are not eliminated.) Because the corrupted
nodes are removed, the whole path is around C. Also, by the uniform density assumption, walking
on the shortest path in G is equivalent to walking on C with ?uniform speed?, i.e., each edge on the
path will map to an approximately fixed distance on C. Thus, the shortest path distance scales with
2/?
2/?
1/?
1/?
?`?3
?`+8
the latent distance, i.e., (d ? 1) 2c
? |xi ? xj | ? d 2c
, which
f (n)
f (n)
implies Theorem 2.2 (cf. [19] for details).
Discussion: ?Gluing together? two algorithms? The unified model is much more flexible than
SBM and SWM. We were intrigued that the generalized algorithm needs only to ?glue together?
important techniques used in both models: Step 1 uses the spectral technique inspired by SBM
inference methods, while Step 2 resembles techniques used in SWM: the isomap G only connects
between two nodes that are close, which is akin to throwing away the long-range edges.
4
Experiments
We apply our algorithm to a social interaction graph from Twitter to construct users? ideology
scores. We assembled a dataset by tracking keywords related to the 2016 US presidential election for
10 million users. First, we note that as of 2016 the Twitter interaction graph behaves ?in-between?
the small-world and stochastic blockmodels (see Figure 4), i.e., the latent distributions are bi-modal
but not as extreme as the SBM.
Ground-truth data. Ideology scores of the US Congress (estimated by third parties [57]) are usually considered as a ?ground-truth? dataset, e.g., [18]. We apply our algorithm and other baselines
on Twitter data to estimate the ideology score of politicians (members of the 114th Congress), and
8
observe that our algorithm has the highest correlation with ground-truth. See Fig. 3. Beyond correlation, we also need to estimate the statistical significance of our estimates. We set up a linear model
y ? ?1 x
b + ?0 , in which x
b?s are our estimates and y?s are ground-truth. We use bootstrapping to
compute the standard error of our estimator, and use the standard error to estimate the p-value of our
estimator. The details of this experiment and additional empirical evaluation are available in [19].
Acknowlegments
The authors thank Amazon for partly providing AWS Cloud Credits for this research.
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6,352 | 6,746 | A New Alternating Direction Method for Linear
Programming
Sinong Wang
Department of ECE
The Ohio State University
[email protected]
Ness Shroff
Department of ECE and CSE
The Ohio State University
[email protected]
Abstract
It is well known that, for a linear program (LP) with constraint matrix A ? Rm?n ,
the Alternating Direction Method of Multiplier converges globally and linearly at a
rate O((kAk2F + mn) log(1/)). However, such a rate is related to the problem
dimension and the algorithm exhibits a slow and fluctuating ?tail convergence? in
practice. In this paper, we propose a new variable splitting method of LP and prove
that our method has a convergence rate of O(kAk2 log(1/)). The proof is based
on simultaneously estimating the distance from a pair of primal dual iterates to the
optimal primal and dual solution set by certain residuals. In practice, we result
in a new first-order LP solver that can exploit both the sparsity and the specific
structure of matrix A and a significant speedup for important problems such as
basis pursuit, inverse covariance matrix estimation, L1 SVM and nonnegative
matrix factorization problem compared with the current fastest LP solvers.
1
Introduction
We are interested in applying the Alternating Direction Method of Multiplier (ADMM) to solve a
linear program (LP) of the form
min
x?Rn
cT x
s.t. Ax = b, xi ? 0, i ? [nb ].
(1)
where c ? Rn , A ? Rm?n is the constraint matrix, b ? Rm and [nb ] = {1, . . . , nb }. This problem
plays a major role in numerical optimization, and has been used in a large variety of application areas.
For example, several important machine learning problems including the nonnegative matrix factorization (NMF) [1], l1 -regularized SVM [2], sparse inverse covariance matrix estimation (SICE) [3]
and the basis pursuit (BP) [4], and the MAP inference [5] problem can be cast into an LP setting.
The complexity of the traditional LP solver is still at least quadratic in the problem dimension, i.e., the
Interior Point method (IPM) with a weighted path finding strategy. However, many recent problems
in machine learning have extremely large-scale targeting data but exhibit a sparse structure, i.e.,
nnz(A) mn, where nnz(A) is the number of non-zero elements in the constraint matrix A. This
characteristic severely limits the ability of the IPM or Simplex technique to solve these problems. On
the other hand, first-order methods have received extensive attention recently due to their ability to
deal with large data sets. These methods require a matrix vector multiplication Ax in each iteration
with complexity linear in nnz(A). However, the key challenge in designing a first-order algorithm is
that LPs are usually non-smooth and non-strongly convex optimization problems (may not have a
unique solution). Utilizing the standard primal and dual stochastic sub-gradient descent method will
result in an extremely slow convergence rate, i.e., O(1/2 ) [6].
The ADMM was first developed in 1975 [7], and since then there have been several LP solvers
based on this technique. Compared with the traditional Augmented Lagrangian Method (ALM), this
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
method splits the variable into several blocks, and optimizes the augmented Lagrangian (AL) function
in a Gauss-Seidel fashion, which often results in relatively easier subproblems to solve. However,
this method suffers from a slow convergence when the number of blocks increases. Moreover, the
challenge of applying the ADMM to the LP is that the LP problem does not exhibit an explicit
separable structure among variables, which are difficult to split in the traditional sense. The notable
work [8] first applies the ADMM to solve the LP by augmenting the original n-dimensional variables
into nm?dimensions, and the resultant Augmented Lagrangian function is separable among n blocks
of variables. They prove that this method converges globally and linearly. However, the rate of this
method is dependent on the problem dimension m, n, and converges quite slowly when m, n are
large. Thus, they leave an open question on whether other efficient splitting methods exist, resulting
in convergence analysis in the space with lower dimension m or n.
In this paper, we propose a new splitting method for LP, which splits the equality and inequality
constraints into two blocks. The resultant subproblems in each iteration are a linear system with a
positive definite matrix, and n one-dimensional truncation operations. We prove our new method
converges globally and linearly at a faster rate compared with the method in [8]. Specifically, the main
contributions of this paper can be summarized as follows: (i) We show that the existing ADMM in [8]
exhibits a slow and fluctuating ?tail convergence?, and provide a theoretical understanding of why
this phenomenon occurs. (ii) We propose a new ADMM method for LP and provide a new analysis
of the linear convergence rate of this new method, which only involves O(m + n)?dimensional
iterates. This result answers the open question proposed in [8]. (iii) We show that when the matrix A
possesses some specific structure, the resultant subproblem can be solved in closed form. For the
general constraint matrix A, we design an efficiently implemented Accelerated Coordinate Descent
Method (ACDM) to solve the subproblem in O(log(1/)nnz(A)) time. (iv) Practically, we show that
our proposed algorithm significantly speeds up solving the basis pursuit, l1 -regularized SVM, sparse
inverse covariance matrix estimation, and the nonnegative matrix factorization problem compared
with existing splitting method [8] and the current fastest first-order LP solver in [9].
2
Preliminaries
In this section, we first review several definitions that will be used in the sequel. Then we illustrate
some observations from the existing method. We also include several LP-based machine problems
that can be cast into the LP setting in the Appendix.
2.1
Notation
A twice differentiable function f : Rn ? R has strong convexity parameter ? if and only if its
Hessian satisfies ?2 f (x) ?I, ?x. We use k?k to denote standard l2 norm for vector or spectral norm
for matrix, k ? k1 to denote the l1 norm and k ? kF to denote the Frobenius norm. A twice differentiable
function f : Rn ? R has a component-wise Lipschitz continuous gradient with constant Li if
and only if k?i f (x) ? ?i f (y)k ? Li kx ? yk, ?x, y. For example, for the quadratic function
F (x) = 12 kAx ? bk2 , the gradient ?F (x) = AT (Ax ? b) and the Hessian ?2 F (x) = AT A.
Hence the parameter ? and Li satisfy (choose y = x + tei , where t ? R, ei ? Rn is the unit
vector), xAT Ax ? ?kxk2 and tATi Aei ? Li |t|, ?x, t. Thus, the ? is the smallest eigenvalue of
AT A and Li = kAi k2 , where Ai is the ith column of the matrix A. The projection operator of
point x into convex set S is defined as [x]S = arg minu?S kx ? uk. If S is the non-negative cone,
let [x]+ , [x]S . Let Vi = [0, ?) for i ? [nb ] and Vi = R for i ? [nf ].
2.2
Tail Convergence of the Existing ADMM Method
The existing ADMM in [8] solves the LP (1) by following procedure: in each iteration k, go through
the following two steps:
h
T
i
k
Ai (b?Axk )
ci ?AT
1
k
i z
1. Primal update: xk+1
=
x
+
?
, i = 1, . . . , n.
i
i
kAi k2
q
?
Vi
2. Dual update: z
k+1
k
=z ?
k
?
q (Ax
? b).
We plot the solving accuracy versus the number of iterations for solving three kinds of problems (see
Fig.1 in Appendix). We can observe that it converges fast in the initial phase, but exhibits a slow and
2
fluctuating convergence when the iterates approach the optimal set. This method originates from
a specific splitting method in the standard 2?block ADMM [10]. To provide some understanding
of this phenomenon, we show that this method can be actually recovered by an inexact Uzawa
method [11]. The Augmented Lagrangian function of the problem (1) is denoted by L(x, z) =
cT x + ?2 kAx ? b ? z/?k2 . In each iteration k, the inexact Uzawa method first minimizes a local
second-order approximation of the quadratic term in L(x, zk ) with respect to primal variables x,
specifically,
1
xk+1 = arg min cT x + h?AT (Axk ? b ? zk /?), x ? xk i + kx ? xk kD ,
xi ?Vi
2
(2)
then update the dual variables by zk+1 = zk ? ?(Axk+1 ? b). Let the proximity parameter ? = ?/q
and matrix D equal to the diagonal matrix diag{. . . , 1/qkAi k2 , . . .}, then we can recover the above
algorithm by the first-order optimality condition of (2). This equivalence allows us to illustrate the
main reason for the slow and fluctuating ?tail convergence? comes from the inefficiency of such a
local approximation of the Augmented Lagrangian function when the iterates approach the optimal
set.
One straightforward idea to resolve this issue is to minimize the Augmented Lagrangian function
exactly instead of its local approximation, which leads to the classic ALM. There exists a line of
works focusing on analyzing the convergence of applying ALM to LP [9, 12, 13]. This method will
produce a sequence of constrained quadratic programs (QP) that are difficult to solve. The work [9]
proves that the proximal Coordinate Descent method can solve each QPs at a linear rate even when
matrix A is not full column rank. However, there exists several drawbacks in this approach: (i) the
practical solving time of each subproblem is quite long when A is rank-deficient; (ii) the theoretical
performance and complexity of using recent accelerated techniques in proximal optimization [14]
with the ALM is unknown; (iii) it cannot exploit the specific structure of matrix A when solving
each constrained QP. Therefore, it motivates us to investigate the new and efficient variable splitting
method for such a problem.
3
New Splitting Method in ADMM
We first separate the equality and inequality constraints of the above LP (1) by adding another group
of variables y ? Rn .
min cT x
s.t. Ax = b, x = y,
yi ? 0, i ? [nb ].
(3)
The dual of problem (3) takes the following form.
min
bT zx
(4)
T
s.t. ?A zx ? zy = c,
zy,i ? 0, i ? [nb ], zy,i = 0, i ? [n]\[nb ].
Let zx , zy be the Lagrange multipliers for constraints Ax = b, x = y,respectively. Define the
indicator function g(y) of the non-negative cone: g(y) = 0 if yi ? 0, ?i ? [nb ]; otherwise
g(y) = +?. Then the augmented Lagrangian function of the primal problem (3) is defined as
?
L(x, y, z) = cT x + g(y) + zT (A1 x + A2 y ? b) + kA1 x + A2 y ? bk2 ,
(5)
2
where z = [zx ; zy ]. The matrix A1 , A2 and vector b are denoted by
A
0
b
, and b =
A1 =
, A2 =
.
I
?I
0
In each iteration k, the standard ADMM go through following three steps:
1. Primal update: xk+1 = arg minn L(x, yk , zk ).
x?R
2. Primal update: y
k+1
= arg minn L(xk+1 , y, zk ).
y?R
3
(6)
Algorithm 1 Alternating Direction Method of Multiplier with Inexact Subproblem Solver
Initialize z0 ? Rm+n , choose parameter ? > 0.
repeat
n Fk (x) ? k .
1. Primal update: find xk+1 such that Fk(xk+1 ) ? minx?R
k
2. Primal update: for each i, let yik+1 = xk+1
+
z
/?
.
y,i
i
Vi
k
k+1
k+1
k
k+1
3. Dual update: zk+1
=
z
+
?(Ax
?
b),
z
=
z
? yk+1 ).
x
x
y
y + ?(x
k+1
until kAx
? bk? ? and kxk+1 ? yk+1 k? ?
3. Dual update: zk+1 = zk + ?(A1 xk+1 + A2 yk+1 ? b).
The first step is an unconstrained quadratic program, which can be simplified as
?
xk+1 = arg min Fk (x) , cT x + (zk )T A1 x + kA1 x + A2 yk ? bk2 .
x
2
The gradient of the function Fk (x) can be expressed as
?Fk (x) = ?(AT A + I)x + AT1 [zk + ?(A2 yk ? b)] + c,
(7)
(8)
and the Hessian of function Fk (x) is
?2 Fk (x) = ?(AT A + I).
(9)
Further, based on the first-order optimality condition, the first step is equivalent to solving a linear
system, which requires inverting the Hessian matrix (9). In practice, the complexity is quite high to
be exactly solved unless the Hessian exhibits some specific structures. Thus, we relax the first step
into the inexact minimization: find xk+1 such that
Fk (xk+1 ) ? minn Fk (x) ? k ,
x?R
(10)
where k is the given accuracy. Transforming the indicator function g(y) back to the constraints, the
second step can be separated into n one?dimensional optimization problems: for each i,
?
k
k
yik+1 = arg min ?zy,i
yi + (yi ? xk+1
)2 = xk+1
+ zy,i
/? V .
i
i
i
yi ?Vi
2
The resultant algorithm is sketched in Algorithm 1. In some applications such as l1 -regularized
SVMs and basis pursuit problem, the objective function contains the l1 norm of the variables.
Transforming to the canonical form (1) will introduce additional n variables and 2n constraints. One
important feature in our method is that we can split the objective function by adding variable y. The
corresponding subproblems are similar with Algorithm 1 and the only difference is that the second
step will be n one?dimensional shrinkage operations. (Details can be seen in Appendix.)
4
Convergence Analysis of New ADMM
In this section, we prove that the Algorithm 1 converges at a global and linear rate, and provide
a roadmap of the main technical development. We can first write the primal problem (3) as the
following standard 2?block form.
min f (x) + g(y) s.t. A1 x + A2 y = b,
x,y
(11)
where f (x) = cT x and g(y) is the indicator function as defined before. Most works in the literature
prove that the 2-block ADMM converges globally and linearly via assuming that one of the functions
f and g is strongly convex [15, 16, 17]. Unfortunately, both the linear function f and the indicator
function g in the LP do not satisfy this property, which poses a significant challenge on the current
analytical framework. There exists several recent works trying to address this problem in some sense.
In work [18], they have demonstrated that when the dual step size ? is sufficiently small (impractical),
the ADMM converges globally linearly, while no implicit rate is given. The work [13] shows that
the ADMM is locally linearly converged when applying to LP. They utilize a unique combination of
iterates and conduct a spectral analysis. However, they still leave an open question whether ADMM
converges globally and linearly when applying to the LP in the above form.
4
In the sequel, we will answer this question positively and provide an accurate analysis of such a
splitting method. The main technical development is based on a geometric argument: we first prove
that the set formed by optimal primal and dual solutions of LP (3) is a (3n + m)?dimensional
polyhedron S ? ; then we utilize certain global error bound to simultaneously estimate the distance
from iterates xk+1 , yk , zk to S ? . All detailed proofs are given in the Appendix.
Lemma 1. (Convergence of 2-block ADMM [10]) Let pk = zk ? ?A2 yk , we have
kpk+1 ? [pk+1 ]G? k2 ? kpk ? [pk ]G? k2 ? kpk+1 ? pk k2 ,
where G? , {p? ? Rm+n |T (p? ) = p? }, and the definition of operator T is given in (54) in
Appendix. Moreover, if the LP (3) has a pair of optimal primal and dual solution, the iterates xk ,yk
and zk converges to an optimal solution; Otherwise, at least one of the iterates is unbounded.
Lemma 1 is tailored from applying the classic Douglas-Rachford splitting method to the LP. This result
guarantees that the sequence pk produced by ADMM globally converges under a mild assumption.
However, to establish the linear convergence rate, the key lies in estimating the other side inequality,
kpk ? [pk ]G? k ? ?kpk+1 ? pk k, ? > 0.
(12)
k
Then one can combine these two results together to prove that sequence p converges globally and
linearly with kpk+1 ? [pk+1 ]G? k2 ? (1 ? 1/? 2 ) ? kpk ? [pk ]G? k2 , which further can be used to show
the R?linear convergence of iterates xk , yk and zk . To estimate the constant ?, we first describe the
geometry formed by the optimal primal solutions x? , y? and dual solutions z? of the LP (3).
Lemma 2. (Geometry of the optimal solution set of LP) The variables (x? , y? ) are the optimal
primal solutions and z? are optimal dual solutions of LP (3) if and only if (i) Ax? = b, x? = y? ; (ii)
?
?
?AT z?x ? z?y = c; (iii) yi? ? 0, zy,i
? 0, i ? [nb ]; zy,i
= 0, i ? [n]\[nb ]; (iv) cT x? + bT z?x = 0.
In Lemma 2, one interesting element is to utilize the strong duality condition (iv) to eliminate the
complementary slackness in the standard KKT condition. Then, the set of optimal primal and dual
solutions is described only by affine constraints, which further implies that the optimal solution set is
an (m + 3n)?dimensional polyhedron. We use S ? to denote such a polyhedron.
Lemma 3. (Hoffman bound [19, 20]) Consider a polyhedron set S = {x ? Rd |Ex = t, Cx ? d}.
For any point x ? Rd , we have
Ex ? t
,
kx ? [x]S k ? ?S
(13)
[Cx ? d]+
where ?S is the Hoffman constant that depends on the structure of polyhedron S.
According to the result in Lemma 2, it seems that we can use the Hoffman bound to estimate the
distance between the current iterates (xk , yk , zk ) and the solution set S ? via the their primal and
dual residual. However, to obtain the form of inequality (12), we need to bound such a residual in
terms of kpk ? pk+1 k. Indeed, we have these results.
Lemma 4. (Estimation of residual) The sequence (xk+1 , yk , zk ) produced by Algorithm 1 satisfies
?
A1 xk+1 + A2 yk ? b = (pk+1 ? pk )/?,
?
?
?
?
? c + AT zk = AT (pk ? pk+1 ),
1
1
T k+1
T k
k+1
?
c
x
+
b
z
? zk /?)T (pk ? pk+1 ),
?
x = (A1 x
?
?
? k
k
k
yi ? 0, zy,i
? 0, i ? [nb ]; zy,i
= 0, i ? [n]\[nb ].
One observation from Lemma 4 is that Algorithm 1 automatically preserves the boundness and the
complementary slackness of both primal and dual iterates. Instead, in the previous algorithm in [8],
the complementary slackness is not preserved during the iteration. Combining the results in Lemma 2,
Lemma 3 and Lemma 4, we are readily to estimate the constant ?.
Lemma 5. (Estimation of linear rate) The sequence pk = zk ? ?A2 yk produced by Algorithm 1
satisfies kpk ? [pk ]G? k ? ?kpk+1 ? pk k, where the rate ? is given by
Rz + 1
T
? = (1 + ?)
+ Rx kA1 k + kA1 k ?S ? .
(14)
?
Rx = supk kxk k < +?, Rz = supk kzk k < +? are the maximum radius of iterates xk and zk .
5
Then we can establish the global and linear convergence of Algorithm 1.
Theorem 1. (Linear convergence of Algorithm 1) Denote zk as the primal iterates produced by
Algorithm 1. To guarantee that there exists an optimal dual solution z? such that kzk ? z? k ? , it
suffices to run Algorithm 1 for number of iterations K = 2? 2 log(2D0 /) with the solving accuracy
k satisfying k ? 2 /8K 2 , where D0 = kp0 ? [p0 ]G? k.
The proof of Theorem 1 consists of two steps: first, we establish the global and linear convergence
rate of Algorithm 1 when k = 0, ?k (exact subproblem solver); then we relax this condition and
prove that when k is less than a specified threshold, the algorithm still shares a convergence rate of
the same order. The results of primal iterates xk and yk are similar.
5
Efficient Subproblem Solver
In this section, we will show that, due to our specific splitting method, each subproblem in line 1 of
Algorithm 1 can be either solved in closed-form expression or efficiently solved by the Accelerated
Coordinate Descent Method.
5.1
Well-structured Constraint Matrix
Let the gradient (8) vanish, then the primal iterates xk+1 can be exactly determined by
xk+1 = ??1 (I + AT A)?1 dk , with dk = ?AT1 [zk + ?(A2 yk ? b)] ? c,
(15)
which requires inverting an n ? n positive definite matrix I + AT A, or equivalently, inverting an
m ? m positive definite matrix I + AAT via the following Sherman?Morrison?Woodbury identity,
(I + AT A)?1 = I ? AT (I + AAT )?1 A.
(16)
One basic fact is that we only need to invert such a matrix once and then use this cached factorization
in subsequent iterations. Therefore, there are several cases for which the above factorization can
be efficiently calculated: (i) Factorization has a closed-form expression. For example, in the LPbased MAP inference [5], the matrix I + AT A is block diagonal, and each block has been shown
to possess a closed-form factorization. Another important application is that, in the basis pursuit
problem, the encoding matrices such as DFT (discrete Fourier transform) and DWHT (discrete
Walsh-Hadamard transform) matrices have orthonormal rows and satisfy AAT = I. Based on
(15), each xk+1 = ??1 (I ? 12 AT A)dk and can be calculated in O(n log(n)) time by certain
fast transforms. (ii) Factorization has a low-complexity: the dimension m (or n) is small, i.e.,
m = 104 . Such a factorization can be calculated in O(m3 ) and the complexity of each iteration is
only O(nnz(A) + m2 ). Detailed applications can be viewed in Appendix.
Remark 1. In the traditional Augmented Lagrangian method, the resultant subproblem is a constrained and non-strongly convex QP (Hessian is not invertible), which does not allow the above
close-form expression. Besides, in the ALCD [9], the coordinate descent (CD) step only picks one
column in each iteration and cannot exploit the nice structure of matrix A. One idea is to modify the
CD step in [9] to the proximal gradient descent. However, it will greatly increase the computation
time due to the large number of inner gradient descent steps.
5.2
General Constraint Matrix
However, in other applications, the constraint matrix A only exhibits the sparsity, which is difficult
to invert. To resolve this issue, we resort to the current fastest
accelerated coordinate descent
?
method [21]. This method has an order improvement up to O( n) of iteration complexity compared
with previous accelerated coordinate descent methods [22]. However, the naive evaluation of partial
derivative of function Fk (x) in ACDM takes O(nnz(A)) time; second, the time cost of full vector
operation in each iteration of ACDM is O(n). We will show that these difficulties can be tackled by a
carefully designed implementation technique1 and the main procedure is listed in Algorithm 2. Here
the iterates st and matrix M in Algorithm 2 is defined as
"
#
?
??
+ 1??
?i Fk (ut )eTi
1 ? ?v
?v
?v
i
1+??
p
(1+??)
L
i
i
M=
with
= ??
and st =
,
?
T
?u
1 ? ?u
?u
1+??
pi (1+??) ?i Fk (ut )ei
(17)
1
This technique is motivated by [22].
6
Algorithm 2 Efficiently Subproblem Solver
Initialize p
u0 , v0 , u0 = Au0 , v0 = Av0 , matrix M, parameter ? , ?, S by (17) and distribution
p = [. . . , 1 + kAi k2 /S, . . . ] and let dk = AT1 [zk + ?(A2 yk ? b)] + c.
repeat
[ut , vt ]T = Mt?1 ? [u, v]T and [ut , vt ]T = Mt?1 ? [u, v]T .
Sample i from [n] based on probability distribution p.
k
i
?i Fk (ut ) = ?(Ai )T ut + ?u
(17).
t,i
+ di , and
calculate st byT
T
T
u
u
u
uT
?1 i
i T
Mt = M ? Mt?1 . Update T = T ? Mt st , T = T ? M?1
t st A ,
v
v
v
v
until Converge
Output xk+1 = (uT ? ? vT )/(1 ? ? ).
where ? =
1
? S2 , ?
=
1+
?
2
,
4S 2 /?+1
S=
Pn
i=1
p
kAi k2 + 1. See more details in Appendix.
Lemma 6. (Inner complexity) In each iteration of Algorithm 2, if the current picked coordinate
is i, the update can be finished in O(nnz(Ai )) time, moreover, to guarantee that Fk (xk+1 ) ?
minx Fk (x) ? k with probability 1 ? p, it suffices to run Algorithm 2 for number of iterations
k
n
X
D0
Tk ? O(1) ?
kAi k log
, D0k = kF k (u0 ) ? min F k (x)k.
(18)
x
p
k
i=1
The above iteration complexity is obtained by choosing parameter ? = 0 in [21] and utilizing the
Theorem 1 in [23] to transform the convergence in expectation to the form of probability.
Theorem 2. (Overall complexity) Denote zk as the dual iterates produced by Algorithm 1. To
guarantee that there exists an optimal solution z? such that kzk ? z? k ? with probability 1 ? p, it
suffices to run Algorithm 1 for k ? 2? 2 log(2D0 /) outer iterations and solve each sub-problem (7)
for the number of inner iterations
!
1
n
X
?(D0k ) 3 ? 2
2D0
T ? O(1) ?
kAi k log
log
.
(19)
2
1
3 p3
i=1
The results for the primal iterates xk and yk are similar. In the existing ADMM [8], each primal and
dual update only requires O(nnz(A)) time to solve. The complexity of this method is
?
O(am ?2 (am Rx + dm Rz )2 ( mn + kAkF )2 nnz(A) log(1/)),
where am = maxi kAi k, dm is the largest number of non-zero elements of each row of matrix A,
and ? is the Hoffman constant depends on the optimal solution set of LP. Based on Theorem 2, an
estimation of the worst-case complexity of Algorithm 1 is
O(am ?S2 ? (Rx kAk + Rz )2 nnz(A) log2 (1/)).
Remark that our method has a weak dependence on the problem dimension compared with the
existing ADMM. Since the Frobenius norm of a matrix satisfies kAk2 ? kAkF , our method is faster
than the one in [8].
6
Numerical Results
In this section, we examine the performance of our algorithm and compare it with the state-of-art
of algorithms developed for solving the LP. The first is the existing ADMM in [8]. The second is
the ALCD method in [9], which is reported to be the current fastest first-order LP solver. They have
shown that this algorithm can significantly speed up solving several important machine learning
problems compared with the Simplex and IPM. We name our Algorithm 1 as LPADMM. In the
experiments, we require that the accuracy of subproblem solver k = 10?3 and the stopping criteria
is that both primal residual kA1 xk + A2 yk ? bk? and dual residual kAT1 zk + ck? is less than
10?3 . All the LP instances are generated from the basis pursuit, L1 SVM, SICE and NMF problems.
The data source and statistics are included in the supplementary material.
7
1
10
ADMM
LPADMM
ALCD
100
10
-1
Duality gap
Duality gap
10
10-2
10-3
5000
10000
15000
100
10-1
10-3
ADMM
LPADMM
ALCD
100
10-1
0
Number of iterations
50
100
150
200
10-3
10-1
10-2
0
1000 2000 3000 4000 5000 6000 7000
Number of iterations
Number of iterations
ADMM
LPADMM
ALCD
100
10-2
10-2
0
101
101
ADMM
LPADMM
ALCD
Duality gap
102
1
Duality gap
102
10-3
1000
2000
3000
4000
5000
Number of iterations
Figure 1: The duality gap versus the number of iterations. From left to right figures are the BP, NMF,
the L1 SVM and and the SICE problem.
Table 1: Timing Results for BP, SICE, NMF and L1 SVM Problem (in sec. long means > 60 hours)
Data
bp1
bp2
bp3
arcene
real-sim
sonar
colon
w2a
news20
m
17408
34816
69632
50095
176986
80912
217580
12048256
2785205
n
16384
32768
65536
30097
135072
68224
161040
12146960
2498375
nnz(A)
8421376
33619968
134348800
1151775
7609186
2756832
8439626
167299110
53625267
LPADMM
Time
Iterations
22
3155
79
4657
217
6287
801
15198
955
4274
258
5446
395
216
19630
2525
7765
2205
Time
864
2846
12862
1978
1906
659
455
45388
9173
ALCD
Iterations
14534
19036
24760
176060
18262
13789
1288
8492
6174
ADMM
Time
Iterations
long
long
long
long
long
long
21329
2035415
19697
249363
3828
151972
7423
83680
long
long
long
long
We first compare the convergence rate of different algorithms in solving the above problems. We
use the bp1 for BP problem, data set colon cancer for NMF problem, news20 for L1 SVM problem
and real-sim for SICE problem. We set proximity parameter ? = 1. We adopt the relative duality
gap as the comparison metric, which is defined as kcT xk + bT zkx k/kcT x? k, where x? is obtained
approximately by running our method with a strict stopping condition. In our simulation, one iteration
represents n coordinate descent steps for ALCD and LPADMM, and one dual updating step for
ADMM. As can be seen in the Fig. 1, our new method exhibits a global and linear convergence rate
and matches our theoretical performance bound. Besides, it converges faster than both the ALCD and
existing ADMM method, especially in solving the BP and NMF problem. The sensitivity analysis of
? is listed in Appendix.
We next examine the performance of our algorithm from the perspective of time efficiency (both
clocking time and number of iterations). We adopt the dynamic step size rule for ALCD to optimize
its performance. Note that, exchanging the role of the primal and dual problem in (3), we can obtain
the dual version of both ADMM and ACLD, which can be used to tackle the primal or dual sparse
problem. We run both methods and adopt the minimum time. The stopping criterion requires that the
primal and dual residual and the relative duality gap is less than 10?3 . The data set bp1,bp2,bp3 is
used for basis pursuit problem, news20 is used for L1 SVM problem; arcene, real-sim are used for
SICE problem; sonar, colon and w2a are used for NMF problem. Among all experiments, we can
observe that our proposed algorithm requires approximately 10% ? 40% iterations and 10% ? 85%
time of the ALCD method, and become particularly advantageous for basis pursuit problem (50?
speed up) or ill posed problems such as SICE and NMF problem. In particular, for the basis pursuit
problem, the primal iterates xk is updated by closed-form expression (15), which can be calculated in
O(n log(n)) time by Fast Walsh?Hadamard transform.
7
Conclusions
In this paper, we proposed a new variable splitting method to solve the linear programming problem.
The theoretical contribution of this work is that we prove that 2?block ADMM converges globally and
linearly when applying to the linear program. The obtained convergence rate has a weak dependence
of the problem dimension and is less than the best known result. Compared with the existing LP
solvers, our algorithms not only provides a flexibility to exploit the specific structure of constraint
matrix A, but also can be naturally combined with the existing acceleration techniques to significantly
speed up solving the large-scale machine learning problems. The future work focuses on generalizing
our theoretical framework and exhibiting the global linear convergence rate when applying ADMM
to solve a convex quadratic program.
Acknowledgments: This work is supported by ONR N00014-17-1-2417, N00014-15-1-2166, NSF
CNS-1719371 and ARO W911NF-1-0277.
8
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9
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6,353 | 6,747 | Regret Analysis for Continuous Dueling Bandit
Wataru Kumagai
Center for Advanced Intelligence Project
RIKEN
1-4-1, Nihonbashi, Chuo, Tokyo 103-0027, Japan
[email protected]
Abstract
The dueling bandit is a learning framework wherein the feedback information in
the learning process is restricted to a noisy comparison between a pair of actions.
In this research, we address a dueling bandit problem based on a cost function over
a continuous space. We propose a?stochastic mirror descent algorithm and show
that the algorithm achieves an O( T log T )-regret bound under strong convexity
and smoothness assumptions for the cost function. Subsequently, we clarify the
equivalence between regret minimization in dueling bandit and convex optimization for the cost function. Moreover, when considering a lower bound in convex
optimization, our algorithm is shown to achieve the optimal convergence rate in
convex optimization and the optimal regret in dueling bandit except for a logarithmic factor.
1
Introduction
Information systems and computer algorithms often have many parameters which should be tuned.
When cost or utility are explicitly given as numerical values or concrete functions, the system parameters can be appropriately determined depending on the the values or the functions. However,
in a human-computer interaction system, it is difficult or impossible for users of the system to provide user preference as numerical values or concrete functions. Dueling bandit is introduced to
model such situations in Yue and Joachims (2009) and enables us to appropriately tune the parameters based only on comparison results on two parameters by the users. In the learning process of
a dueling bandit algorithm, the algorithm chooses a pair of parameters called actions (or arms) and
receives only the corresponding comparison result. Since dueling bandit algorithms do not require
an individual evaluation value for each action, they can be applied for wider areas that cannot be
formulated using the conventional bandit approach.
When action cost (or user utility) implicitly exists, the comparison between two actions is modeled
via a cost (or utility) function, which represents the degree of the cost (or utility), and a link function,
which determines the noise in the comparison results. We refer to such a modeling method as costbased (or utility-based) approach and employ it in this research. Yue and Joachims (2009) first
introduced the utility-based approach as a model for a dueling bandit problem.
The cost-based dueling bandit relates to function optimization with noisy comparisons (Jamieson
et al., 2012; Matsui et al., 2016) because in both frameworks an oracle compare two actions and
the feedback from the oracle is represented by binary information. In particular, the same algorithm
can be applied to both frameworks. However, as different performance measures are applied to
the algorithms in function optimization and dueling bandit, it has not been demonstrated that an
algorithm that works efficiently in one framework will also perform well in the other framework.
This study clarifies relation between function optimization and dueling bandit thorough their regret
analysis.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1.1
Problem Setup
In the learning process of the dueling bandit problem, a learner presents two points, called actions in
a space A, to an oracle and the oracle returns one-bit feedback to the learner based on which action
wins (i.e., which action is more preferable for the oracle). Here, we denote by a ? a? the event that
a wins a? and by P (a ? a? ) the probability that a ? a? happens. In other words, we assume that the
feedback from the oracle follows the following two-valued random variable:
{
1 w.p. P (a ? a? )
?
F (a, a ) :=
(1)
0 w.p. 1 ? P (a ? a? ),
where the probability P (a ? a? ) is determined by the oracle. We refer to this type of feedback as
noisy comparison feedback. Unlike conventional bandit problems, the leaner has to make a decision
that is based only on the noisy comparison feedback and cannot access the individual values of
the cost (or utility) function. We further assume that each comparison between a pair of actions is
independent of other comparisons.
The learner makes a sequence of decisions based on the noisy comparisons provided by the oracle.
After receiving F (at , a?t ) at time t, the learner chooses the next two action (at+1 , a?t+1 ). As a
performance measure for an action a, we introduce the minimum win probability:
P ? (a) = inf
P (a ? a? ).
?
a ?A
We next quantify the performance of the algorithm using the expected regret as follows:1)
[ T
]
?
DB
?
?
?
? ?
RegT = sup E
{(P (a) ? P (at )) + (P (a) ? P (at ))} .
a?A
1.2
(2)
t=1
Modeling Assumption
In this section, we clarify some of the notations and assumptions. Let an action space A ? Rd be
compact convex set with non-empty interior. We denote the Euclidean norm by ? ? ?.
Assumption 1. There exist functions f : A ? R and ? : R ? [0, 1] such that the probability in
noisy comparison feedback can be represented as follows:
P (a ? a? ) = ?(f (a? ) ? f (a)).
(3)
In the following, we call f in Assumption 1 a cost function and and ? a link function. Here, the
cost function and the link function are fixed for each query to the oracle. In this sense, our setting is
different from online optimization where the objective function changes.
Definition 1. (Strong Convexity) A function f : Rd ? R is ?-strongly convex over the set A ? Rd
if for all x, y ? A it holds that
?
f (y) ? f (x) + ?f (x)? (y ? x) + ?y ? x?2 .
2
Definition 2. (Smoothness) A function f : Rd ? R is ?-smooth over the set A ? Rd if for all
x, y ? A it holds that
?
?y ? x?2 .
2
Assumption 2. The cost function f : A ? R is twice continuously differentiable, L-Lipschitz,
?-strongly convex and ?-smooth with respect to the Euclidean norm.
f (y) ? f (x) + ?f (x)? (y ? x) +
From Assumption 2, there exists a unique minimizer a? of the cost function f since f is strictly
convex. We set B := supa,a? ?A f (a? ) ? f (a).
Assumption 3. The link function ? : R ? [0, 1] is three times differentiable and rotation-symmetric
(i.e., ?(?x) = 1 ? ?(x)). Its first derivative is positive and monotonically non-increasing on [0, B].
1)
Although the regret in (2) appears superficially different from that in Yue and Joachims (2009), two regrets
can be shown to coincide with each other under Assumptions 1-3 in Subsection 1.2.
2
For examples, the standard logistic distribution function, the cumulative standard Gaussian distribution function and the linear function ?(x) = (1 + x)/2 can be taken to be link functions that
satisfy Assumption 3. We note that link functions often behave like cumulative probability distribution functions. This is because the sign of the difference between two noisy function values can
be regarded as the feedback (1) which satisfies Assumption 1, and then, the link function ? coincides with the cumulative probability distribution function of the noise (see Section 2 of Jamieson
et al. (2012) for more details). We will discuss the relation of noisy comparison feedback to noisy
function values in Section 5.
1.3
Related Work and Our Contributions
Dueling bandit on the continuous action space relates with various optimization methods. We summarize related studies in the following.
Dueling bandit problem: Yue and Joachims (2009) formulated information retrieval systems as a
dueling bandit problem. They reduced this to a problem of optimizing an ?almost"-concave function
and presented a stochastic gradient ascent algorithm based on one-point bandit feedback. Subsequently, they showed that their algorithm achieves an O(T 3/4 )-regret bound under the differentiability and the strict concavity for a utility function. Ailon et al. (2014) presented reduction methods
from dueling bandit to the conventional bandit under the?strong restriction that the link function
is linear and showed that their algorithm achieves an O( T log3 T )-regret bound. We note that
dueling bandit has a number of other formulations (Yue and Joachims, 2011; Yue et al., 2012; BusaFekete et al., 2013, 2014; Urvoy et al., 2013; Zoghi et al., 2014; Jamieson et al., 2015).
Optimization with one-point bandit feedback: In conventional bandit settings, various convex optimization methods have been studied. Flaxman et al. (2005) showed that the gradient of smoothed
version of a convex function can be estimated from a one-point bandit feedback and proposed a
stochastic gradient descent algorithm which achieves an O(T 3/4 ) regret bound under the Lipschitzness condition. Moreover, assuming the strong convexity and the smoothness for the convex function
such as?(2), Hazan and Levy (2014) proposed a stochastic mirror descent algorithm which achieves
an O( T log T ) regret bound and showed
that the algorithm is near optimal because the upper
?
bound matched the lower bound of ?( T ) derived by Shamir (2013) up to a logarithmic factor in
bandit convex optimization.
Optimization with two-point bandit feedback: Dueling bandit algorithms require two actions at
each round in common with two-point bandit optimization. In the context of online optimization,
Agarwal et al. (2010) first considered convex optimization with two-point feedback. They proposed
a gradient descent-based algorithm and showed that the algorithm achieves the regret bounds of under the Lipschitzness condition and O(log T ) under the strong convexity condition. In stochastic
convex?optimization, Duchi et al. (2015) showed that a stochastic mirror descent algorithm achieves
an O( T ) regret bound under the Lipschitzness (or the smoothness)
condition and proved the up?
per bound to be optimal deriving a matching lower bound ?( T ). Moreover, in both of online
and stochastic convex
optimization, Shamir (2017) showed that a gradient descent-based algorithm
?
achieves an O( T ) regret bound with optimal dependence on the dimension under the Lipschitzness condition. However, those two-point bandit algorithms strongly depend on the availability of
the difference of function values and cannot be directly applied to the case of dueling bandit where
the difference of function values is compressed to one bit in noisy comparison feedback.
Optimization with noisy comparison feedback: The cost-based dueling bandit relates to function
optimization with noisy comparisons (Jamieson et al., 2012; Matsui et al., 2016) because in both
frameworks, the feedback is represented by preference information. Jamieson et al. (2012) proposed
a coordinate descent algorithm and proved that the convergence rate of the algorithm achieved an
optimal order.2) Matsui et al. (2016) proposed a Newton method-based algorithm and proved that its
convergence rate was almost equivalent to that of Jamieson et al. (2012). They further showed that
their algorithm could easily be parallelized and performed better numerically than the dueling bandit
algorithm in Yue and Joachims (2009). However, since they considered only the unconstrained case
in which A = Rd , it is not possible to apply their algorithm to the setting considered here, in which
the action space is compact.
The optimal order changes depending on the model parameter ? ? 1 of the pairwise comparison oracle in
Jamieson et al. (2012).
2)
3
Optimization with one-bit feedback: The optimization method of the dueling bandit algorithm is
based on one-bit feedback. In related work, Zhang et al. (2016) considered stochastic optimization
under one-bit feedback. However, since their approach was restricted to the problem of linear optimization with feedback generated by the logit model, it cannot be applied to the problem addressed
in the current study.
Our contributions: In this paper, we consider the cost-based dueling bandit under Assumptions
1-3. While the formulation is similar to that of Yue and Joachims (2009), Assumptions 2 and 3 are
stronger than those used in that study. On the other hand, we impose the weaker assumption on the
link function than that of Ailon et al. (2014). Yue and Joachims (2009) showed that a stochastic
gradient descent algorithm can be applied to dueling bandit. Thus, it is naturally expected that a
stochastic mirror descent algorithm, which achieves the (near) optimal order in convex optimization with one/two-point bandit feedback, can be applied to dueling bandit setting and achieves good
performance. Following this intuition, we propose a mirror descent-based algorithm. Our key contributions can be summarized as follows:
? We propose a stochastic mirror descent algorithm with noisy comparison feedback.
?
? We provide an O( T log T )-regret bound for our algorithm in dueling bandit.
? We clarify the relation between the cost-based dueling bandit and convex optimization in terms
of their regrets and show that our algorithm can be applied to convex optimization.
?
? We show that the convergence rate of our algorithm is O( log T /T ) in convex optimization.
? We derive a lower bound in convex optimization with noisy comparison feedback and show
our algorithm to be near optimal in both dueling bandit and convex optimization.
2
Algorithm and Main Result
2.1
Stochastic Mirror Descent Algorithm
We first prepare the notion of a self-concordant function on which our algorithm is constructed (see
e.g., Nesterov et al. (1994), Appendix F in Griva et al. (2009)).
Definition 3. A function R : int(A) ? R is considered self-concordant if the following two conditions hold:
1. R is three times continuously differentiable and convex, and approaches infinity along any
sequence of points approaching the boundary of int(A).
2. For every h ? Rd and x ? int(A), |?3 R(x)[h, h, h]| ? 2(h? ?2 R(x)h) 2 holds, where
3
(x
+
t
h
+
t
h
+
t
h)
.
?3 R(x)[h, h, h] := ?t1??tR
1
2
3
?t
t1 =t2 =t3 =0
2
3
3
In addition to these two conditions, if R satisfies the following condition for a positive real number
?, it is called a ?-self-concordant function:
3. For every h ? Rd and x ? int(A), |?R(x)? h| ? ? 2 (h? ?2 R(x)h) 2 .
1
1
In this paper, we assume the Hessian ?2 R(a) of a ?-self-concordant function to be full-rank over
A and ?R(int(A)) = Rd . Bubeck and Eldan (2014) showed that such a ?-self-concordant function
satisfying ? = (1 + o(1))d will always exist as long as the dimension d is sufficiently large. We next
propose Algorithm 1, which we call NC-SMD. This can be regarded as stochastic mirror descent
with noisy comparison feedback.
We make three remarks on Algorithm 1. First, the function Rt is self-concordant though not ?self-concordant.
The second remark is as follows. Let us denote the local norms by ?a?w =
?
a? ?2 R(w)a. Then, if R is a self-concordant function for A, the Dikin Ellipsoid {a? ? A| ?a? ?
a?a ? 1} centered at a is entirely contained in int(A) for any a ? int(A). Thus, a?t := at +
1
?2 Rt (at )? 2 ut in Algorithm 1 is contained in int(A) for any at ? int(A) and a unit vector ut .
This shows a comparison between actions at and a?t to be feasible. Our third remark is as follows.
Since the self-concordant function Rt at round t depends on the past actions {ai }ti=1 , it may be
thought that those past actions are stored during the learning process. However, note that only ?Rt
4
Algorithm 1 Noisy Comparison-based Stochastic Mirror Descent (NC-SMD)
Input: Learning rate ?, ?-self-concordant function R, time horizon T , tuning parameters ?, ?
Initialize: a1 = argmina?A R(a).
for t = 1 to T do
?t
2
2
Update Rt (a) = R(a) + ??
i=1 ?a ? ai ? + ??a?
2
Pick a unit vector ut uniformly at random
1
Compare at and a?t := at + ?2 Rt (at )? 2 ut and receive F (a?t , at )
1
Set g?t = F (a?t , at )d?2 Rt (at ) 2 ut
Set at+1 = ?R?1
gt )
t (?Rt (at ) ? ??
end for
Output: aT +1
?t
and ?2 Rt are used in the algorithm; ?Rt depends only on i=1 at and ?2 Rt does not depend on
the past actions. Thus, only the sum of past actions must be stored, rather than all past actions.
2.2
Main Result: Regret Bound
From Assumption 2 and the compactness of A, the diameter R and B := supa,a? ?A f (a? ) ? f (a)
are finite. From Assumption 3, there are exist positive constants l0 , L0 , B2 and L2 such that the first
derivative ? ? of the link function is bounded as l0 ? ? ? ? L0 on [?B, B] and the second derivative
? ?? is bounded above by B2 and L2 -Lipschitz on [?B, B]. We use the constants below.
The following
theorem shows that with appropriate parameters, NC-SMD (Algorithms 1) achieves
?
an O( T log T )-regret bound.
2
0?
Theorem 4. We set C := ? + B2 L +(L+1)L
. When the tuning parameters satisfy ? ? l0 ?/2,
2?
( 3
)2
? ? L0 L2 /? and the total number T of rounds satisfies T ? C log T . Algorithm 1 with a ??
C log T
1
self-concordant function and the learning parameter ? = 2d
achieves the following regret
T
bound under Assumptions 1-3:
?
RegTDB ? 4d CT log T + 2LL0 R.
(4)
3
Regret Analysis
We prove Theorem 4 in this section. The proofs of lemmas in this section are provided in supplementary material.
3.1
Reduction to Locally-Convex Optimization
We first reduce the dueling bandit problem to a locally-convex optimization problem. We define
Pb (a) := ?(f (a) ? f (b)) for a, b ? A and Pt (a) := Pat (a). For a cost function f and a selfconcordant function R, we set a? := argmina?A f (a), a1 := argmina?A R(a) and a?T := T1 a1 +
(1 ? T1 )a? . The regret of dueling bandit is bounded as follows.
Lemma 5. The regret of Algorithm 1 is bounded as follows:
[ T
]
?
LL0 ?
DB
?
RegT
? 2E
(Pt (at ) ? Pt (aT )) +
(5)
log T + 2LL0 R.
??
t=1
The following lemma shows that Pb inherits the smoothness of f globally.
Lemma 6. The function Pb is (L0 ? + B2 L2 )-smooth for an arbitrary b ? A.
Let B be the unit Euclidean ball, B(a, ?) the ball centered at a with radius ? and L(a, b) the line
segment between a and b. In addition, for a, b ? A, let A? (a, b) := ?a? ?L(a,b) B(a? , ?) ? A. The
following lemma guarantees the local strong convexity of Pb .
Lemma 7. The function Pb is 12 l0 ?-strongly convex on A? (a? , b) when ? ?
5
l0 ?
.
2L30 L2
3.2
Gradient Estimation
We note that at + ?2 Rt (at )? 2 x for x ? B is included in A due to the properties of the Dikin
ellipsoid. We introduce the smoothed version of Pt over int(A):
1
P?t (a) := Ex?B [Pt (a + ?2 Rt (at )? 2 x)]
(6)
1
Ex?B [?(f (a + ?2 Rt (at )? 2 x) ? f (at ))].
Next, we adopt the following estimator for the gradient of P?t :
1
=
(7)
g?t := F (at + ?2 Rt (at )? 2 ut , at )d?2 Rt (at ) 2 ut ,
where ut is drawn from the unit surface S uniformly. We then derive the unbiasedness of g?t as
follows.
Lemma 8.
E[?
gt |at ] = ?P?t (at ).
1
3.3
1
Regret Bound with Bregman Divergence
From Lemma 5, the regret analysis in dueling bandit is reduced to the minimization problem of the
regret-like value of Pt . Since Pt is globally smooth and locally strongly convex from Lemmas 6
and 7, we can employ convex-optimization methods. Moreover, since g?t is an unbiased estimator
for the gradient of the smoothed version of Pt from Lemma 8, it is expected that stochastic mirror
descent (Algorithm 1) with g?t is effective to the minimization problem. In the following, making use
of the property of stochastic mirror descent, we bound the regret-like value of Pt by the Bregman
divergence, and subsequently prove Theorem 4.
Definition 9. Let R be a continuously differentiable strictly convex function on int(A). Then, the
Bregman divergence associated with R is defined by
DR (a, b) = R(a) ? R(b) ? ?R(b)? (a ? b).
(
)2
Lemma 10. When ? ? l0 ?/2 and ? ? L30 L2 /? , the regret of Algorithm 1 is bounded for any
a ? int(A) as follows:
[ T
]
?
E
(Pt (at ) ? Pt (a))
t=1
1
?
?
(
[
R(a) ? R(a1 ) + E
T
?
])
gt , ?R(at ))
DR?t (?R(at ) ? ??
t=1
+
L0 ? + B2 L2
log T, (8)
??
where R?t (a) := supx?Rd ?x, a? ? Rt (a) is the Fenchel dual of Rt .
The Bregman divergence in Lemma 10 is bounded as follows.
1
Lemma 11. When ? ? 2d
, the sequence at output by Algorithm 1 satisfies
DR?t (?Rt (at ) ? ??
gt , ?Rt (at )) ? 4d2 ? 2 .
(9)
[Proof of Theorem 4] From Lemma 4 of Hazan and Levy (2014), the ?-self-concordant function R
satisfies
1
R(a?T ) ? R(a1 ) ? ? log
,
1 ? ?a1 (a?T )
where ?a (a? ) := inf{r ? 0|a + r?1 (a? ? a)} is the Minkowsky function. Since ?a1 (a?T ) ? 1 ? T ?1
from the definition of a?T , we obtain
R(a?T ) ? R(a1 ) ? ? log T.
1
Note that the condition ? ? 2d in Lemma 11 is satisfied due to T ? C log T . Combining Lemmas
5, 10 and 11, we have
) L0 ? + D??? L2
2 (
LL0 ?
RegTDB ?
? log T + 4d2 ? 2 T +
log T +
+ 2LL0 R
?
??
l0 ??
(
)
B2 L2 + (L + 1)L0 ? log T
?
2? +
+ 8d2 T ? + 2LL0 R.
?
?
Thus, when ? is defined in Theorem 4, the regret bound (4) is obtained.
6
4
Convergence Rate in Convex Optimization
In the previous sections, we considered the minimization problem for the regret of dueling bandit.
In this section, as an application of the approach, we show that the averaged action of NC-SMD
(Algorithm 1) minimize the cost function f in (3).
To derive the convergence rate of our algorithm, we introduce the regret of function optimization and
establish a connection between the regrets of dueling bandit and function optimization. In convex
optimization with noisy comparison feedback, the learner chooses a pair (at , a?t ) of actions in the
learning process and suffers a loss f (at ) + f (a?t ). Then, the regret of the algorithms in function
optimization is defined as follows:
[ T
]
?
FO
?
?
?
RegT
:= sup E
(f (at ) ? f (a )) + (f (at ) ? f (a )) ,
(10)
a?A
t=1
?
where a = argmina?A f .
Recalling that the positive constants l0 and L0 satisfy l0 ? ? ? ? L0 on [?B, B], where B :=
supa,a? ?A f (a? ) ? f (a), the regrets of function optimization (10) and dueling bandit (2) are related
as follows:
Lemma 12.
RegTDB
RegTDB
? RegTF O ?
.
L0
l0
(11)
?
Theorem 4 and Lemma 12 give an O( T log T )-upper bound of the regret of function optimization
in Algorithm 1 under the same conditions as Theorem 4. Given the convexity of f , the average of
?T
1
?
the chosen actions of any dueling bandit algorithm a
?T := 2T
t=1 (at + at ) satisfies
RegTF O
.
(12)
2T
Thus, if an optimization algorithm has a sub-linear regret bound, the above online-to-batch conversion guarantees convergence to the optimal point.
Theorem 13. Under Assumptions 1-3, the averaged action a
?T satisfies the following when T ?
C log T :
( ?
)
? log T + C
1
LL0 R
?
E[f (?
aT ) ? f (a )] ?
2d
+
,
l0
T
T
E[f (?
aT ) ? f (a? )] ?
where C is the constant defined in Theorem 4.
?
Theorem 13 shows the convergence rate of NC-SMD (Algorithm 1) to be O(d log T /T ).
5
Lower Bound
We next derive a lower bound in convex optimization with noisy comparison feedback. To do so, we
employ a lower bound of convex optimization with noisy function feedback. In a setting where the
function feedback is noisy, we query a point a ? A and obtain a noisy function value f (a)+?, where
? is a zero-mean random variable with a finite second moment and independent for each query. 3)
Theorem 14. Assume that the action space A is the d-dimensional unit Euclidean ball Bd and
that the link function ?G is the cumulative distribution function of the zero-mean Gaussian random
variable with variance 2. Let the number of rounds T be fixed. Then, for any algorithm with noisy
comparison feedback, there exists a function f over Bd which is twice continuously differentiable,
0.5-strongly convex and 3.5-smooth such that the output aT of the algorithm satisfies
}
{
d
.
E[f (aT ) ? f (a? )] ? 0.004 min 1, ?
(13)
2T
3)
In general, the noise ? can depend on the action a. See e.g. Shamir (2013) for more details.
7
[Proof] The probability distribution of noisy comparison feedback F (a, a? ) with the link function
?G can be realized by noisy function feedback with thestandard Gaussian noise as follows. Two
noisy function values f (a) + ? and f (a? ) + ? ? can be obtained using the noisy function feedback
twice, where ? and ? ? are independent standard Gaussian random variables. Then, the probability
distribution of the following random variable coincide with that of F (a, a? ) for arbitrary a, a? ? A:
sign(f (a) + ? ? (f (a? ) + ? ? )) = sign(f (a) ? f (a? ) + (? ? ? ? )).
(14)
Here, note that ? ? ? ? is the zero-mean Gaussian random variable with variance 2. Thus, single
noisy comparison feedback with the link function ?G for any actions can be obtained by using
noisy function feedback with standard Gaussian noise twice. This means that if any algorithm with
2T -times noisy function feedback is unable to achieve a certain performance, any algorithm with
T -times noisy comarison feedback is similarly unable to achieve that performance. Thus, to derive
Theorem 14, it is sufficient to show a lower bound of convergence rate with noisy function feedback.
The following lower bound is derived by Theorem 7 of Shamir (2013).
Theorem 15. (Shamir, 2013) Let the number of rounds T be fixed. Suppose that the noise ? at
each round is a standard Gaussian random variable. Then, for any algorithm with noisy function
feedback, there exists a function f over Bd which is twice continuously differentiable, 0.5-strongly
convex and 3.5-smooth such that the output aT satisfies
{
}
d
?
E[f (aT ) ? f (a )] ? 0.004 min 1, ?
.
T
By the above discussion and from Theorem 15, we obtain Theorem 14.
Combining Theorem 13 and Theorem 14, the convergence rate of NC-SMD (Algorithm 1) is near
optimal with respect to the number of rounds T . In addition, when the parameter ? of the selfconcordant function is of constant order with respect to the dimension d of the space A, the convergence rate of NC-SMD is optimal with respect to d. However, it should be noted that the parameter
? of a self-concordant function is often of the order of ?(d) for compact convex sets including the
simplex and the hypercube.
As a consequece of Lemma 12, (12), and Theorems
4 and 14, the optimal regrets of dueling ban?
dit and function optimization are of the order T except for the logarithmic factor and NC-SMD
achieves the order. To the best of our knowledge, this is the first algorithm with the optimal order in
the continuous dueling bandit setting with the non-linear link function.
Finally, we provide an interesting observation on convex optimization. When noisy
? function feedback is available, the optimal regret of function optimization is of the order ?( T ) under strong
convexity and smoothness conditions (Shamir, 2013). However, even when noisy function feedback
is "compressed" into one-bit information
as in (14), our results show that NC-MSD (Algorithm 1)
?
achieves almost the same order O( T log T ) about the regret of function optimization as long as
the cumulative probability distribution of the noise satisfies Assumption 3.4)
6
Conclusion
We considered a dueling bandit problem over a continuous action space and proposed a stochastic ?
mirror descent. By introducing Assumptions 1-3, we proved that our algorithm achieves an
O( T log T )-regret bound. We further considered convex optimization under noisy comparison
feedback and showed that the regrets of dueling bandit and function optimization are essentially
equivalent. Using the connection
between the two regrets, it was shown that our algorithm achieves
?
a convergence rate of O( log T /T ) in the framework of function optimization with noisy comparison feedback. Moreover, we derived a lower bound of the convergence rate in convex optimization
and showed that our algorithm achieves near optimal performance in dueling bandit and convex optimization. Some open questions still remain. While we have only dealt with bounds which hold
in expectation, the derivation of the high-probability bound is a problem that has not been solved.
While the analysis of our algorithm relies on strong convexity and smoothness, a regret bound without these conditions is also important.
4)
Jamieson et al. (2012) provided a similar observation. However, their upper bound of the regret was
derived only when the action space is the whole of Euclidean space (i.e., A = Rd ) and the assumption for
noisy comparison feedback is different from ours (Assumption 1).
8
Acknowledgment
We would like to thank Professor Takafumi Kanamori and Professor Kota Matsui for helpful comments. This work was supported by JSPS KAKENHI Grant Number 17K12653.
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10
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6,354 | 6,748 | Best Response Regression
Omer Ben-Porat
Technion - Israel Institute of Technology
Haifa 32000 Israel
[email protected]
Moshe Tennenholtz
Technion - Israel Institute of Technology
Haifa 32000 Israel
[email protected]
Abstract
In a regression task, a predictor is given a set of instances, along with a real value
for each point. Subsequently, she has to identify the value of a new instance as
accurately as possible. In this work, we initiate the study of strategic predictions in
machine learning. We consider a regression task tackled by two players, where the
payoff of each player is the proportion of the points she predicts more accurately
than the other player. We first revise the probably approximately correct learning
framework to deal with the case of a duel between two predictors. We then devise
an algorithm which finds a linear regression predictor that is a best response to
any (not necessarily linear) regression algorithm. We show that it has linearithmic
sample complexity, and polynomial time complexity when the dimension of the
instances domain is fixed. We also test our approach in a high-dimensional setting,
and show it significantly defeats classical regression algorithms in the prediction
duel. Together, our work introduces a novel machine learning task that lends itself
well to current competitive online settings, provides its theoretical foundations, and
illustrates its applicability.
1
Introduction
Prediction is fundamental to machine learning and statistics. In a prediction task, an algorithm is
given a sequence of examples composed of labeled instances, and its goal is to learn a general rule
that maps instances to labels. When the labels take continuous values, the task is typically referred
to as regression. The quality of a regression algorithm is measured by its success in predicting the
value of an unlabeled instance. Literature on regression is mostly concerned with minimizing the
discrepancy of the prediction, i.e. the difference between the true value and the predicted one.
Despite the tremendous amount of work on prediction and regression, online commerce presents new
challenges. In this context, prediction is not carried out in isolation. New entrants can utilize knowledge of previous expert predictions and the corresponding true values, to maximize their probability
of predicting better than that expert, treated as the new entrant?s opponent. This fundamental task is
the main challenge we tackle in this work.
We initiate the study of strategic predictions in machine learning. We present a regression learning
setting that stems from a game-theoretic point of view, where the goal of the learner is to maximize
the probability of being the most accurate among a set of predictors. Note that this approach may be
in conflict with the traditional prediction goal.
Consider an online real estate expert, who frequently predicts the sale value of apartments. This
expert, having been in the market for a while, has historical data on the values and characteristics
of similar apartments. For simplicity, assume the expert uses simple linear regression to predict the
value of an apartment as a function of its size. When a new apartment comes on the market, the
expert uses her gathered historical data to predict the new apartment?s value. When the apartment is
sold, the true value (and the accuracy of the prediction) is revealed.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Value (dollars)
Expert
Agent
Size (square feet)
Figure 1: A case where minimizing the square error can be easily beaten. Each point is an instancevalue pair, where the circles are historical points (i.e. their value has been revealed) and the triangles
are new points, unseen by either the expert and the agent. The red (solid) line represents the linear
least squares estimators, employed by the expert. After collecting a sufficient amount of historical
data (circles) on apartments along with their true value and the value predicted by the expert, the
agent comes up with the response represented by the green (dashed) line. For each of the unseen
apartment sizes, both the expert and the agent declare their predictions of the apartment?s value.
Notice that the agent outperforms the expert in the majority of the historical points. In addition, the
agent produces a more accurate prediction in the majority of the new (unseen) points.
At first glance this seems extremely effective, however it is also extremely fragile. An agent who
enters the real estate business may come up with a linear predictor for which the probability (over
all apartments and their values) of being more accurate is high, making it the preferable predictor.
Figure 1 illustrates our approach. The expert uses linear least square estimators (LSE) to minimize
the mean square error (MSE). The agent, after having collected "enough" historical data (circles) and
having observed the predictions of the expert, produces a strategy (regression line). Both the expert
and the agent predict the value of new apartments coming on the market (triangles). As illustrated,
the prediction of the agent is the most accurate in the majority of new instances.
One criticism of this novel approach is that while maximizing the probability of being the most
accurate, the agent may produce "embarrassing" predictions for some instances. Current prediction
algorithms are designed to minimize some measure of overall loss, such as the MSE. Notice that in
many, and perhaps even most, practical scenarios, being a better predictor on more instances is more
important than avoiding such sporadic "embarrassing predictions". In particular, our approach fits
any commerce and advertising setting where the agent offers predictions to users on the value of
different goods or services, aiming at maximizing the number of users that will find her predictions
more accurate than the one provided by the expert. For example, an agent, serving users searching
for small apartments, would be happy to fail completely in predicting the value of very large sized
apartments if this allowed predicting the value of smaller apartments better than an opponent.
Our novel perspective suggests several new fundamental problems:
1. Given a prediction algorithm ALG (e.g. LSE), what would be the best response to ALG, if
we aim at maximizing the probability that the new algorithm would be more accurate than
ALG?
2. In case ALG is unknown, but the agent has access to a labeled set of instances along with
the prediction made by ALG for each instance, how many i.i.d. samples are needed in order
to learn a best response to ALG over the whole population?
3. How poorly do classical regression algorithms preform against such a best response algorithm?
In this work, we focus on a two player scenario and analyze the best response of the agent against an
opponent. We examine the agent?s perspective, and introduce a rigorous treatment of Problems 1-3
above. We model the task of finding a best response as a supervised learning task, and show that it
2
fits the probably approximately correct (PAC) learning framework. Specifically, we show that when
the strategy space of the agent is restricted, a best response over a large enough sample set is likely to
be an approximate best response over the unknown distribution.
Our main result deals with an agent employing linear regression in Rn for any constant n. We present
a polynomial time algorithm which computes a linear best response (i.e. from the set of all linear
predictors) to any regression algorithm employed by the opponent. We also show a linearithmic
bound in the number of training samples needed in order to successfully learn a best response. In
addition, we show that in some cases our algorithm can be adapted to have an MSE score arbitrarily
close to that of the given regression algorithm ALG. The theoretical analysis is complemented by
an experimental study, which illustrates the effectiveness of our approach. In order to find a best
linear response in high dimensional space, we provide a mixed integer linear programming (MILP)
algorithm. The MILP algorithm is tested on the Boston housing dataset [5]. Indeed, we show that we
can outperform classical regression algorithms in up to 70% of the points. Moreover, we outperform
classical regression algorithms even in the case where they have full access to both training and test
data, while we restrict our responder algorithm to the use of the training data only.
Our contribution. Our contributions are 3-fold. The main conceptual contribution of this paper is
the explicit suggestion that a prediction task may have strategic aspects. We introduce the setting
of best response regression, applicable to a huge variety of scenarios, and revise the PAC-learning
framework to deal with such a duel framework. Then, we show an efficient algorithm dealing with
finding a best-response linear regression in Rn for any constant n, against any regression algorithm.
This best response algorithm maximizes the probability of beating the latter on new instances. Finally,
we present an experimental study showing the applicability of our approach. Together, this work
offers a new machine learning challenge, addresses some of its theoretical properties and algorithmic
challenges, while also showing its applicability.
1.1
Related work
The intersection of learning theory with multi-agent systems is expanding with the rise of data science.
In the field of mechanism design [8], [3, 7] considered prediction tasks with strategic aspects. In their
model, the instances domain is to be labeled by one agent, and the dataset is constructed of points
controlled by selfish users, who have their own view on how to label the instances domain. Hence,
the users can misreport the points in order to sway decisions in their favor. A different line of work
that is related to our model is the analysis of sample complexity in revenue maximizing auctions. In
a recent work [2] the authors reconsider an auction setting where the auctioneer can sample from
the valuation functions of the bidders, thereby relaxing the ubiquitous assumption of knowing the
underlying distribution over bidders? valuations.
While the above papers consider mechanism design problems inspired by machine learning, our work
considers a novel machine learning problem inspired by game theory.
In work on dueling algorithms [6], an optimization problem is analyzed from the perspective of
competition, rather than from the point of view of a single optimizer. That work examines the dueling
form of several optimization problems, e.g. the shortest path from the source vertex to the target
vertex in a graph with random weights. While minimizing the expected length is a probable solution
concept for a single optimizer, this is no longer the case in the defined duel. While [6] assumes a
commonly-known distribution over a finite set of instances, we have no such assumption. Instead,
we consider a sample set drawn from the underlying distribution with the aim of predicting a new
instance better than the opponent.
Our formulation is also related to the Learning Using Privileged Information paradigm (see, e.g.,
[9, 14, 15]), in which the learner (agent) is supplied with additional information along with the label
of each instance. In this paper, we assume the agent has access to predictions made by another
algorithm (the opponent?s), which can be treated as additional information.
2
Problem formulation
The environment is composed of instances and labels. In the motivating example given above, the
instances are the characteristics of the apartments, and the labels are the values of these apartments.
3
A set of N players offer predictive services, where a strategy of a player is a labeling function.
For each instance-label pair (x, y), the players see x, and subsequently each player i, predicts
the value of the y. We call this label estimate y?i . The player who wins a point (x, y) is the one
with the smallest discrepancy, i.e. mini |y?i ? y|. Under the strategy profile (h1 , . . . hN ), where
each entry is the labeling function chosen by the corresponding player, the payoff of Player i is
Pr ({(x, y) : Player i wins (x, y)}).
A strategy of a player is called a best response if it maximizes the payoff of that player, when the
strategies of all the other players are fixed. In this work, we analyze the best response of a player, and
w.l.o.g. we assume she has only one opponent. The model is as follows:
1. We assume a distribution over the examples domain, which is the cross product of the
instances domain X ? Rn and the labels domain Y ? R.
2. The agent and the opponent both predict the label of each instance. The opponent uses a
? which is a conditional distribution over R given x ? X .
strategy h,
?
3. The agent is unaware of the distribution over X ? Y or the strategy of the opponent h.
Hence, we explicitly address the joint distribution D over Z = X ? Y ? R, where a triplet
(x, y, p) represents an instance x, its label y, and the discrepancy of the opponent?s predicted
?
value p, i.e. p = |h(x)
? y|. We stress that D is unknown to the agent.
4. The payoff of the agent under the strategy h : X ? Y is given by
?D (h) =
E
1|h(x)?y)|<p .
(x,y,p)?D
5. The agent has access to a sequence of examples S, with which she wishes to maximize her
payoff.
Note that a strategy which outputs yi for every instance xi in S may look promising, but will probably
lead to overfitting, and low payoff for the agent. Since the agent wishes to generalize from S to D,
restricting the strategy set to H ? Y X seems justified. We define the goal of the agent:
6. The agent is willing to restrict herself to a strategy from H ? Y X . Her goal: to find an
algorithm which, given , ? ? (0, 1) and a sequence of m = m(, ?) examples S sampled
i.i.d. from D, outputs a strategy h? such that with probability at least 1 ? ? (over the choices
of S) it holds that
?D (h? ) ? sup ?D (h) ? .
h?H
Indeed, the access to a sequence of examples seems realistic, and the size of S depends on the amount
of resources at the agent?s disposal. The size of S also affects the selection of H: if the agent can
gather "many" examples, she might be able to learn a "good" strategy from a more complex strategy
space.
We say that h ? H is an approximate best response with factor if for all h0 ? H it holds that
?D (h0 ) ? ?D (h) ? . Note that the goal of the agent can be interpreted as finding an approximate
best response with high probability. The empirical payoff of the agent is defined by
1
?S (h) =
? {i : 1|h(xi )?yi )|<pi } ,
m
and a strategy h ? arg maxh0 ?H ?S (h0 ) is called an empirical best response (w.r.t S). Next, we
adopt the PAC framework [12] to define under which strategy spaces an empirical best response is
likely to be an approximate best response.
2.1
Approximate best response with PAC learnability
The field of statistical learning addresses the problem of finding a predictive function based on data.
We briefly define some key concepts in learning theory, that will be used later. For a more gentle
introduction the reader is referred to [11].
Let G be a class of functions from Z to {0, 1} and let S = {z1 , . . . , zm } ? Z. The restriction of
G to S, denoted G(S), is defined by G(S) = {(g(z1 ), g(z2 ), . . . , g(zm )) : g ? G}. Namely, G(S)
contains all the binary vectors induced by the functions in G on the items of S. We say that G shatters
S if G(S) contains all binary vectors of size m, i.e. |G(S)| = 2m .
4
Definition 1 (VC dimension,[13]). The VC dimension of a class G, denoted VCdim(G), is the
maximal size of a set S ? Z that can be shattered by G.
Definition 2 (PAC learnability,[12]). A hypothesis class H is PAC-learnable with respect to a domain
set Z and a loss function l : H ? Z ? R+ , if there exists a function ?H : (0, 1)2 ? N and a
learning algorithm ALG such that for every , ? ? (0, 1) and for every distribution D over Z, when
running ALG on m ? ?H (, ?) i.i.d. examples generated by D, it returns a hypothesis h ? H such
that with probability of at least 1 ? ? it holds that
LD (h) ? inf
LD (h0 ) + ,
0
(1)
h ?H
where LD (h) = Ez?Z l(h, z).
Let H be a class of functions from X to Y, and let Z = X ? Y ? R, as defined earlier in this section.
Typically in a regression task, the hypothesis class is restricted in order to decrease the distance
between the predicted labels and the true label. In the aforementioned model, however, the agent may
want to deliberately harm her accuracy on some subset of the instances domain. She will do this as
long as it increases the number of instances having a better prediction, thereby improving her payoff.
Since h ? H can either win a point (x, y, p) or lose it, the model resembles a binary classification
task, where the "label" of (x, y, p) is the identity of the winner. That is, a triplet (x, y, p) would be
labeled 1 if the agent produced a better prediction than the opponent, and zero otherwise. However,
notice that the agent?s strategy is involved in the labeling. This is, of course, not the case of binary
classification. Our approach is to introduce a corresponding binary classification problem, and by
leveraging former results obtained on binary classification, deduce sufficient learnability conditions
for our model. The complete reduction is described in detail in the appendix.
Adjusting to the loss function framework, define:
?z = (x, y, p) ? Z : l(h, z) =
1
0
|h(x) ? y| ? p
.
|h(x) ? y| < p
Observe that l(h, z) = 0 whenever the agent wins a point and l(h, z) = 1 otherwise. If we set
LD (h) = Ez?D l(h, z), Equation (1) can be reformulated as ?D (h) ? suph0 ?H ?D (h0 ) ? . Our
goal is to find sufficient conditions for H to be PAC-learnable w.r.t Z and l.
Given H, let GH = {gh : h ? H} such that
?h ? H, ?z ? Z : gh (z) = 1 ? l(h, z) =
1
0
|h(x) ? y| < p
.
|h(x) ? y| ? p
Note that GH is a class of functions from Z to {0, 1}. Sufficient learnability conditions can now be
stated.
Lemma 1. Let H be a class of functions from X to Y with VCdim(GH ) = d < ?. Then there is a
constant C, such that for every , ? ? (0, 1) and every distribution D over Z = X ? Y ? R, if we
d+log 1
sample a sequence of examples S of size m ? C ? 2 ? i.i.d. from D and pick an empirical best
response h ? H w.r.t. S, then with probability of at least 1 ? ? it holds that
?D (h) ? sup ?D (h0 ) ? .
h0 ?H
3
Best linear response
We assume throughout this section that the agent uses a linear response. In what follows, we first
show that H is PAC-learnable with respect to Z and the payoff function. Afterwards, we devise an
empirical best response algorithm with respect to a sequence of examples. Hence, according to the
previous section, this empirical payoff maximization algorithm outputs, with high probability, an
approximate best response with respect to D. The proofs of all theorems and the supporting lemmas
are in the appendix.
For ease of presentation, we re-denote the dimension of the instances domain to be n ? 1, i.e.
X ? Rn?1 . Every h ? Rn defines a linear predictor of a point x ? Rn?1 via dot product, namely
5
h ? (xi , 1). Thus, Rn is referred to as the strategy space H, where axis i represents the i?th entry in h,
1 ? i ? n + 1. We study the case where n is fixed, although the complementary case is discussed in
the end of the section.
m
Recall that the empirical
Pmpayoff of the agent w.r.t to a sequence of examples S = (xi , yi , pi )i=1 is de1
fined as ?S (h) = m i=1 1|h?(xi ,1)?yi |<pi , and the best response w.r.t. to S is arg maxh?H ?S (h).
m
Observe that there is a mapping MH
from any h ? H to a vector v ? {0, 1}m
S : H ? {0, 1}
such that entry i in v equals one if h gains the i?th point, and zero otherwise. Put differently,
MH
S (h) = v = (v1 , . . . vm ) such that:
?i ? [m] : vi = 1 ? |h ? (xi , 1) ? yi | < pi .
Hence, the target set of MH
S is GH (S), which is the restriction of GH to S. The size of GH (S) is
essentially the effective size of H, since any two strategies which are mapped to the same vector will
gain the same points, and thus are equivalent. The following theorem puts a bound on the size of
GH (S).
Theorem 1. Let H be the hypothesis class of all linear functions in Rn?1
P. nFor anysequence of
examples S of size m, GH (S) is polynomial in m. Specifically, |GH (S)| ? i=0 2i mi .
The VC-dimension of GH can be bounded using the Sauer - Shelah lemma [10]:
Lemma 2. It holds that VCdim(GH ) ? max{b2n ? log(n)c, 20}.
We now devise an empirical payoff maximizing algorithm. Our approach is to first explicitly
characterize the vectors in GH (S), and afterwards to pick a strategy from
h :
MH
(h)
=
max
kvk
S
1 .
1
v?GH (S)
For each vector v, one can formulate a linear program which outputs a strategy in {h : MH
S (h) = v}
in case this set is not empty, or outputs none in case it is. Naively, 2m such feasibility problems can
be solved, although this is very inefficient. Instead, we will recursively construct the set of feasible
vectors. The Partial Vector Feasibility problem aids in recursively partitioning the hypothesis space.
Note that it is solvable in time poly(n, m) using Linear Programming.
Problem: PARTIAL V ECTOR F EASIBILITY (PVF)
m
Input: a sequence of examples S = (xi , yi , pi )m
i=1 , and a vector v ? {1, 0, a, b}
n
Output: a point h ? R satisfying
1. If vi = 1 then |h ? (xi , 1) ? yi | < pi .
2. If vi = a then h ? (xi , 1) ? yi > pi .
3. If vi = b then h ? (xi , 1) ? yi < ?pi .
if such exists, and ? otherwise.
The following algorithm partitions Rn according to GH (S), where in each iteration it "discovers" one
more point in the sequence S.
Algorithm: E MPIRICAL PAYOFF M AXIMIZATION (EPM)
Input: S = (xi , yi , pi )m
i=1
Output: Empirical payoff maximizer w.r.t. S
m
1 v ? {0}
// v = (v1 , v2 , . . . , vm )
2 R0 ? {v}
3 for i = 1 to m do
4
Ri ? ?
5
for v ? Ri?1 do
6
for ? ? {1, a, b} do
7
if PVF (S, (v ?i , ?)) 6= ? then
8
add (v ?i , ?) to Ri
// (v ?i , ?) = (v1 , . . . vi?1 , ?, vi+1 , . . . , vm )
?
9 return v ? arg maxv?Rm kvk1
Theorem 2. When running EPM on a sequence of examples S, it finds an empirical best response in
poly(|S|) time.
6
1
R1
2
b
y
y=a
? ? x + ?b
y = a? ? x + b?
(?
a, ?b)
3
R2
(a? , b? )
x
R3
a
Figure 2: An example of simple linear regression with linear strategies. On the left we have
? = (?
a, ?b) of the opponent (the solid line)
a sample sequence of size 3, along with the strategy h
and a best response strategy of the agent (the dashed line). On the right the hypothesis space is
presented,
where each pair (a, b) represents a possible strategy, and each bounded set Ri is defined by
Ri = (a, b) ? R2 : |a ? xi + b ? yi | < pi , i.e. the set of hypotheses which give xi better prediction
? Notice that (?
than h.
a, ?b) relies on the boundaries of all Ri , 1 ? i ? 3. In addition, since (a? , b? ) is
inside R1 ? R2 ? R3 , the strategy h? = (a? , b? ), i.e. the line y = a? ? x + b? , predicts all the points
? the agent not only
better than the opponent. Observe that by taking any convex combination of h? , h,
perserves her empirical payoff but also improves her MSE score.
When we combine Theorem 2 with Lemmas 2 and 1, we get:
Corollary 1. Given , ? ? (0, 1), if we run EPM on m ? C2 ? max{b2n ? log(n)c, 20} + log 1?
examples sampled i.i.d. from D (for a constant C), then it outputs h? such that with probability at
least 1 ? ? satisfies
?D (h? ) ? sup ?D (h0 ) ? .
h0 ?H
A desirable achievement would be if the best response prediction algorithm would also keep the loss
small in the original (e.g. MSE) measure. We now show that in some cases the agent can, by slightly
modifying the output of EPM, find a strategy that is not only an approximate best response, but is
also robust with respect to additive functions of discrepancies. See Figure 2 for illustration.
? and denote by h? the strategy output by
Lemma 3. Assume the opponent uses a linear predictor h,
?
EPM. Then, h can be efficiently modified to a strategy which is not only an empirical best response,
? w.r.t. to any additive function of the discrepancies.
but also performs arbitrarily close to h
Finaly, we discuss the case where the dimension of the instances domain is a part of the input. It is
known that learning the best halfspace is NP-hard in binary classification (w.r.t. to a given sequence
of points), when the dimension of the data is not fixed (see e.g. [1]). We show that the empirical best
(linear) response problem is of the same flavor.
Lemma 4. In case H is the set of linear functions in Rn?1 and n is not fixed, the empirical best
response problem is NP-hard.
4
Experimental results
We note that when n is large, the proposed method for finding an empirical best response may not be
suitable. Nevertheless, if the agent is interested in finding a "good" response to her opponents, she
should come up with something. With slight modifications, the linear best response problem can be
formulated as a mixed integer linear program (MILP).1 Hence, the agent can exploit sophisticated
solvers and use clever heuristics. Further, one implication of Lemma 1 is that the true payoffs
1
See the appendix for the mixed integer linear programming formulation.
7
Table 1: Experiments on Boston Housing dataset
The opponent?s strategy
Least square errors (LSE)
Least absolute errors (LAE)
Scenario
TRAIN
ALL
TRAIN
ALL
Train payoff
0.699
0.711
0.621
0.625
Test payoff
0.641
0.645
0.570
0.528
Results obtained on the Boston Housing dataset. Each cell in the table represents the average payoff
of the agent over 1000 simulations (splits into 80% train and 20% test). The "train payoff" is the
proportion of points in the training set on which the agent is more accurate, and the "test payoff"
payoff is the equivalent proportion with respect to the test (unseen) data.
uniformly converge, and hence any empirical payoff obtained by the MILP is close to its real payoff
with high probability.
In this section, we show the extent to which classical linear regression algorithms can be beaten
using the Boston housing dataset [5], a built-in dataset in the leading data science packages (e.g.
scikit-learn in Python and MASS in R). The Boston housing dataset contains 506 instances, where
each instance has 13 continuous attributes and one binary attribute. The label is the median value of
owner-occupied homes, and among the attributes are the per capita crime rate, the average number of
rooms per dwelling, the pupil-teacher ratio by town and more. The R-squared measure for minimizing
the square error in the Boston housing dataset is 0.74, indicating that the use of linear regression is
reasonable.
As possible strategies of the opponent, we analyzed the linear least squares estimators (LSE) and
linear least absolute estimators (LAE). The dataset was split into training (80%) and test (20%) sets,
and two scenarios were considered:
Scenario TRAIN - the opponent?s model is learned from the training set only.
Scenario ALL - the opponent?s model is learned from both the training and the test sets.
In both scenarios the agent had access to the training set only, along with the opponent?s discrepancy
for each point in the training set. Obviously, achieving payoff of more than 0.5 (that is, more than
50% of the points) in the ALL scenario is a real challenge, since the opponent has seen the test set in
her learning process. We ran 1000 simulations, where each simulation is a random split of the dataset.
We employed the MILP formulation, and used Gurobi software [4] in order to find a response, where
the running time of the solver was limited to one minute.2
Our findings are reported in Table 1. Notice that against both opponent strategies, and even in
case where the opponent had seen the test set, the agent still gets more than 50% of the points. In
both scenarios, LAE guarantees the opponent more than LSE. This is because absolute error is less
sensitive to large deviations. We also noticed that when the opponent learns from the whole dataset,
the empirical payoff of the agent is greater. Indeed, the latter is reasonable as in the ALL scenario the
agent?s strategy fits the training set while the opponent strategy does not.
Beyond the main analysis, we examined the success (or lack thereof) of the agent with respect to
the additive loss function optimized by the opponent (corresponding to the MSE for LSE, and the
MAE (mean absolute error) for LAE), hereby referred to as the "classical loss". Recall that Lemma 3
guarantees that the agent?s classical loss can be arbitrarily close to that of the opponent when she
plays a best response; however, the response we consider in this section (using the MILP) does not
necessarily converge to a best response. Therefore, we find it interesting to consider the classical loss
as well, thereby presenting the complementary view.
We report in Table 2 the average ratio between the agent?s classical loss and that of the opponent
under the TRAIN scenario with respect to the training and test sets. Notice that the agent suffers from
less than a 0.7% increase with respect to the classical loss optimized by the opponent. In particular,
2
Code for reproducing the
Best-Response-Regression
experiments
is
8
available
at
https://github.com/omerbp/
Table 2: Ratio of the classical loss
The opponent?s strategy
Training set
Test set
LSE
LAE
1.007
0.999
1.005
1.002
Ratio of the agent?s loss and the opponent?s loss, where the loss function corresponds to the original
optimization function of the opponent, under scenario TRAIN. For example, the upper leftmost cell
represents the agent?s MSE divided by the opponents MSE on the training set, where the opponent
uses LSE. Similarly, the lower rightmost cell represents the agent?s MAE (mean absolute error)
divided by the opponents MAE on the test data, when the opponent uses LAE.
the MSE of the agent (when she responds to LSE) on the test set is less than that of the opponent.
The same phenomenon, albeit on a smaller scale, occurs against LAE: the training set ratio is greater
than the test set ratio.
To conclude, the agent is not only able to obtain the majority of the points (and in some cases, up to
70%), but also to keep the classical loss optimized by her opponent within less than 0.2% from the
optimum on the test set.
5
Discussion
This work introduces a game theoretic view of a machine learning task. After finding sufficient
conditions for learning to occur, we analyzed the induced learning problem, when the agent is
restricted to a linear response. We showed that a best response with respect to a sequence of examples
can be computed in polynomial time in the number of examples, as long as the instance domain has a
constant dimension. Further, we showed an algorithm that for any , ? computes an -best response
with a probability of at least 1 ? ?, when it is given a sequence of poly 12 n log n + log 1?
examples drawn i.i.d.
As the reader may notice, our analysis holds as long as the hypothesis is linear in its parameters,
and therefore is much more general than linear regression. Interestingly, this is a novel type of
optimization problem and so rich hypothesis, which are somewhat unnatural in the traditional task of
regression, might be successfully employed in the proposed setting.
From an empirical standpoint, the gap between the empirical payoff and the true payoff calls for
applying regularization methods for the best response problem and encourages further algorithmic
research. Exploring whether or not a response in the form of hyperplanes can be effective against a
more complex strategy employed by the opponent will be intriguing. For instance, showing that a
deep learner is beatable in this setting will be remarkable.
The main direction to follow is the analysis of the competitive environment introduced in the beginning
of Section 2 as a simultaneous game: is there an equilibrium strategy? Namely, is there a linear
predictor which, when used by both the agent and the opponent, is a best response to one another?
Acknowledgments
We thank Gili Baumer and Argyris Deligkas for helpful discussions, and anonymous reviewers for
their useful suggestions. This project has received funding from the European Research Council (ERC)
under the European Union?s Horizon 2020 research and innovation programme (grant agreement n?
740435).
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6,355 | 6,749 | TernGrad: Ternary Gradients to Reduce
Communication in Distributed Deep Learning
Wei Wen1 , Cong Xu2 , Feng Yan3 , Chunpeng Wu1 , Yandan Wang4 , Yiran Chen1 , Hai Li1
1
Duke University, 2 Hewlett Packard Labs, 3 University of Nevada ? Reno, 4 University of Pittsburgh
1
{wei.wen, chunpeng.wu, yiran.chen, hai.li}@duke.edu
2
[email protected], 3 [email protected], 4 [email protected]
Abstract
High network communication cost for synchronizing gradients and parameters
is the well-known bottleneck of distributed training. In this work, we propose
TernGrad that uses ternary gradients to accelerate distributed deep learning in data
parallelism. Our approach requires only three numerical levels {?1, 0, 1}, which
can aggressively reduce the communication time. We mathematically prove the
convergence of TernGrad under the assumption of a bound on gradients. Guided
by the bound, we propose layer-wise ternarizing and gradient clipping to improve its convergence. Our experiments show that applying TernGrad on AlexNet
doesn?t incur any accuracy loss and can even improve accuracy. The accuracy
loss of GoogLeNet induced by TernGrad is less than 2% on average. Finally, a
performance model is proposed to study the scalability of TernGrad. Experiments
show significant speed gains for various deep neural networks. Our source code is
available 1 .
1
Introduction
The remarkable advances in deep learning is driven by data explosion and increase of model size.
The training of large-scale models with huge amounts of data are often carried on distributed systems [1][2][3][4][5][6][7][8][9], where data parallelism is adopted to exploit the compute capability
empowered by multiple workers [10]. Stochastic Gradient Descent (SGD) is usually selected as the
optimization method because of its high computation efficiency. In realizing the data parallelism
of SGD, model copies in computing workers are trained in parallel by applying different subsets of
data. A centralized parameter server performs gradient synchronization by collecting all gradients
and averaging them to update parameters. The updated parameters will be sent back to workers, that
is, parameter synchronization. Increasing the number of workers helps to reduce the computation
time dramatically. However, as the scale of distributed systems grows up, the extensive gradient
and parameter synchronizations prolong the communication time and even amortize the savings
of computation time [4][11][12]. A common approach to overcome such a network bottleneck is
asynchronous SGD [1][4][7][12][13][14], which continues computation by using stale values without
waiting for the completeness of synchronization. The inconsistency of parameters across computing
workers, however, can degrade training accuracy and incur occasional divergence [15][16]. Moreover,
its workload dynamics make the training nondeterministic and hard to debug.
From the perspective of inference acceleration, sparse and quantized Deep Neural Networks (DNNs)
have been widely studied, such as [17][18][19][20][21][22]. However, these methods generally
aggravate the training effort. Researches such as sparse logistic regression and Lasso optimization
problems [4][12][23] took advantage of the sparsity inherent in models and achieved remarkable
1
https://github.com/wenwei202/terngrad
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
speedup for distributed training. A more generic and important topic is how to accelerate the
distributed training of dense models by utilizing sparsity and quantization techniques. For instance,
Aji and Heafield [24] proposed to heuristically sparsify dense gradients by dropping off small values in
order to reduce gradient communication. For the same purpose, quantizing gradients to low-precision
values with smaller bit width has also been extensively studied [22][25][26][27].
Our work belongs to the category of gradient quantization, which is an orthogonal approach to
sparsity methods. We propose TernGrad that quantizes gradients to ternary levels {?1, 0, 1} to
reduce the overhead of gradient synchronization. Furthermore, we propose scaler sharing and
parameter localization, which can replace parameter synchronization with a low-precision gradient
pulling. Comparing with previous works, our major contributions include: (1) we use ternary values
for gradients to reduce communication; (2) we mathematically prove the convergence of TernGrad
in general by proposing a statistical bound on gradients; (3) we propose layer-wise ternarizing
and gradient clipping to move this bound closer toward the bound of standard SGD. These simple
techniques successfully improve the convergence; (4) we build a performance model to evaluate the
speed of training methods with compressed gradients, like TernGrad.
2
Related work
Gradient sparsification. Aji and Heafield [24] proposed a heuristic gradient sparsification method
that truncated the smallest gradients and transmitted only the remaining large ones. The method
greatly reduced the gradient communication and achieved 22% speed gain on 4 GPUs for a neural
machine translation, without impacting the translation quality. An earlier study by Garg et al. [28]
adopted the similar approach, but targeted at sparsity recovery instead of training acceleration. Our
proposed TernGrad is orthogonal to these sparsity-based methods.
Gradient quantization. DoReFa-Net [22] derived from AlexNet reduced the bit widths of weights,
activations and gradients to 1, 2 and 6, respectively. However, DoReFa-Net showed 9.8% accuracy
loss as it targeted at acceleration on single worker. S. Gupta et al. [27] successfully trained neural
networks on MNIST and CIFAR-10 datasets using 16-bit numerical precision for an energy-efficient
hardware accelerator. Our work, instead, tends to speedup the distributed training by decreasing the
communicated gradients to three numerical levels {?1, 0, 1}. F. Seide et al. [25] applied 1-bit SGD
to accelerate distributed training and empirically verified its effectiveness in speech applications. As
the gradient quantization is conducted by columns, a floating-point scaler per column is required. So
it cannot yield speed benefit on convolutional neural networks [26]. Moreover, ?cold start? of the
method [25] requires floating-point gradients to converge to a good initial point for the following 1-bit
SGD. More importantly, it is unknown what conditions can guarantee its convergence. Comparably,
our TernGrad can start the DNN training from scratch and we prove the conditions that promise the
convergence of TernGrad.
Very recently, a preprint by D. Alistarh et al. [26] presented QSGD that explores the trade-off between
accuracy and gradient precision. The effectiveness of gradient quantization was justified and the
convergence of QSGD was provably guaranteed. Compared to QSGD developed simultaneously,
our TernGrad shares the same concept but advances in the following three aspects: (1) we prove
the convergence from the perspective of statistic bound on gradients. The bound also explains why
multiple quantization buckets are necessary in QSGD; (2) the bound is used to guide practices and
inspires techniques of layer-wise ternarizing and gradient clipping; (3) TernGrad using only 3-level
gradients achieves 0.92% top-1 accuracy improvement for AlexNet, while 1.73% top-1 accuracy loss
is observed in QSGD with 4 levels. The accuracy loss in QSGD can be eliminated by paying the cost
of increasing the precision to 4 bits (16 levels) and beyond.
3
Problem Formulation and Our Approach
3.1 Problem Formulation and TernGrad
Figure 1 formulates the distributed training problem of synchronous SGD using data parallelism. At
iteration t, a mini-batch of training samples are split and fed into multiple workers (i ? {1, ..., N }).
(i)
(i)
Worker i computes the gradients gt of parameters w.r.t. its input samples zt . All gradients are
first synchronized and averaged at parameter server, and then sent back to update workers. Note
that parameter server in most implementations [1][12] are used to preserve shared parameters, while
here we utilize it in a slightly different way of maintaining shared gradients. In Figure 1, each
2
worker keeps a copy of parameters locally. We name this technique as parameter localization. The
parameter consistency among workers can be maintained by random initialization with an identical
seed. Parameter localization changes the communication of parameters in floating-point form to the
transfer of quantized gradients that require much lighter traffic. Note that our proposed TernGrad can
be integrated with many settings like Asynchronous SGD [1][4], even though the scope of this paper
only focuses on the distributed SGD in Figure 1.
Algorithm 1 formulates the t-th iteration of TernGrad algorithm according to Figure 1. Most steps of
TernGrad remain the same as traditional distributed training, except that gradients shall be quantized
into ternary precision before sending to parameter server. More specific, ternarize(?) aims to reduce
the communication volume of gradients. It randomly quantizes gradient gt 2 to a ternary vector with
values ? {?1, 0, +1}. Formally, with a random binary vector bt , gt is ternarized as
g?t = ternarize(gt ) = st ? sign (gt ) ? bt ,
(1)
st , max (abs (gt ))
(2)
where
is a scaler that can shrink ?1 to a much smaller amplitude. ? is the Hadamard product. sign(?) and
abs(?) respectively returns the sign and absolute value of each element. Giving a gt , each element of
bt independently follows the Bernoulli distribution
P (btk = 1 | gt ) = |gtk |/st
,
(3)
P (btk = 0 | gt ) = 1 ? |gtk |/st
where btk and gtk is the k-th element of bt and gt , respectively. This stochastic rounding, instead
of deterministic one, is chosen by both our study and QSGD [26], as stochastic rounding has an
unbiased expectation and has been successfully studied for low-precision processing [20][27].
Theoretically, ternary gradients can at least reduce the worker-to-server traffic by a factor of
32/log2 (3) = 20.18?. Even using 2 bits to encode a ternary gradient, the reduction factor is
still 16?. In this work, we compare TernGrad with 32-bit gradients, considering 32-bit is the default
precision in modern deep learning frameworks. Although a lower-precision (e.g. 16-bit) may be
enough in some scenarios, it will not undervalue TernGrad. As aforementioned, parameter localization reduces server-to-worker traffic by pulling quantized gradients from servers. However, summing
P (i)
up ternary values in i g?t will produce more possible levels and thereby the final averaged gradient
gt is no longer ternary as shown in Figure 2(d). It emerges as a critical issue when workers use
(i)
different scalers st . To minimize the number of levels, we propose a shared scaler
(i)
st = max({st } : i = 1...N )
(4)
across all the workers. We name this technique as scaler sharing. The sharing process has a small
overhead of transferring 2N floating scalars. By integrating parameter localization and scaler
sharing, the maximum number of levels in gt decreases to 2N + 1. As a result, the server-to-worker
communication reduces by a factor of 32/log2 (1 + 2N ), unless N ? 230 .
?"
?"
($)
?"
Worker 1
?"#$ ? ?" ? ?"
Algorithm 1 TernGrad: distributed SGD training
using ternary gradients.
Parameter
server
?"
??
(*)
?"
Worker 2
?"#$ ? ?" ? ?"
1
(+)
?"
2
3
Worker N
?"#$ ? ?" ? ?"
4
5
6
Figure 1: Distributed SGD with data parallelism.
2
7
Worker : i = 1, ..., N
(i)
Input zt , a part of a mini-batch of training samples
zt
(i)
(i)
Compute gradients gt under zt
(i)
(i)
Ternarize gradients to g?t = ternarize(gt )
(i)
Push ternary g?t to the server
Pull averaged gradients gt from the server
Update parameters wt+1 ? wt ? ? ? gt
Server :
P (i)
Average ternary gradients gt = i g?t /N
Here, the superscript of gt is omitted for simplicity.
3
3.2 Convergence Analysis and Gradient Bound
We analyze the convergence of TernGrad in the framework of online learning systems. An online
learning system adapts its parameter w to a sequence of observations to maximize performance. Each
observation z is drawn from an unknown distribution, and a loss function Q(z, w) is used to measure
the performance of current system with parameter w and input z. The minimization target then is the
loss expectation
C(w) , E {Q(z, w)} .
(5)
In General Online Gradient Algorithm (GOGA) [29], parameter is updated at learning rate ?t as
wt+1 = wt ? ?t gt = wt ? ?t ? ?w Q(zt , wt ),
(6)
g , ?w Q(z, w)
(7)
where
and the subscript t denotes observing step t. In GOGA, E {g} is the gradient of the minimization
target in Eq. (5).
According to Eq. (1), the parameter in TernGrad is updated, such as
wt+1 = wt ? ?t (st ? sign (gt ) ? bt ) ,
(8)
where st , max (abs (gt )) is a random variable depending on zt and wt . As gt is known for given
zt and wt , Eq. (3) is equivalent to
P (btk = 1 | zt , wt ) = |gtk |/st
.
(9)
P (btk = 0 | zt , wt ) = 1 ? |gtk |/st
At any given wt , the expectation of ternary gradient satisfies
E {st ? sign (gt ) ? bt } = E {st ? sign (gt ) ? E {bt |zt }} = E {gt } = ?w C(wt ),
(10)
which is an unbiased gradient of minimization target in Eq. (5).
The convergence analysis of TernGrad is adapted from the convergence proof of GOGA presented
in [29]. We adopt two assumptions, which were used in analysis of the convergence of standard
GOGA in [29]. Without explicit mention, vectors indicate column vectors here.
Assumption 1. C(w) has a single minimum w? and gradient ??w C(w) always points to w? , i.e.,
? > 0,
inf
||w?w? ||2 >
T
(w ? w? ) ?w C(w) > 0.
(11)
Convexity is a subset of Assumption 1, and we can easily find non-convex functions satisfying it.
Assumption 2. Learning rate ?t is positive and constrained as
P+? 2
?t < +?
Pt=0
,
(12)
+?
t=0 ?t = +?
which ensures ?t decreases neither very fast nor very slow respectively.
We define the square of distance between current parameter wt and the minimum w? as
2
ht , ||wt ? w? || ,
(13)
where || ? || is `2 norm. We also define the set of all random variables before step t as
Xt , (z1...t?1 , b1...t?1 ) .
(14)
Under Assumption 1 and Assumption 2, using Lyapunov process and Quasi-Martingales convergence
theorem, L. Bottou [29] proved
Lemma 1. If ?A, B > 0 s.t.
E ht+1 ? 1 + ?t2 B ht |Xt ? ?2?t (wt ? w? )T ?w C(wt ) + ?t2 A,
(15)
then C(z, w) converges almost surely toward minimum w? , i.e., P (limt?+? wt = w? ) = 1.
4
We further make an assumption on the gradient as
Assumption 3 (Gradient Bound). The gradient g is bounded as
2
E {max(abs(g)) ? ||g||1 } ? A + B ||w ? w? || ,
(16)
where A, B > 0 and || ? ||1 is `1 norm.
With Assumption 3 and Lemma 1, we prove Theorem 1 ( in Supplementary Material):
Theorem 1. When online learning systems update as
wt+1 = wt ? ?t (st ? sign (gt ) ? bt )
(17)
?
using stochastic ternary gradients, they converge almost surely toward minimum w , i.e.,
P (limt?+? wt = w? ) = 1.
Comparing with the gradient bound of standard GOGA [29]
2
E ||g||2 ? A + B ||w ? w? || ,
(18)
the bound in Assumption 3 is stronger because
max(abs(g)) ? ||g||1 ? ||g||2 .
(19)
We propose layer-wise ternarizing and gradient clipping to make two bounds closer, which shall be
explained in Section 3.3. A side benefit of our work is that, by following the similar proof procedure,
we can prove the convergence of GOGA when Gaussian noise N (0, ? 2 ) is added to gradients [30],
under the gradient bound of
2
E ||g||2 ? A + B ||w ? w? || ? ? 2 .
(20)
Although the bound is also stronger, Gaussian noise encourages active exploration of parameter
space and improves accuracy as was empirically studied in [30]. Similarly, the randomness of ternary
gradients also encourages space exploration and improves accuracy for some models, as shall be
presented in Section 4.
3.3
Feasibility Considerations
The gradient bound of TernGrad in Assumption 3 is stronger than the bound in standard GOGA. Pushing the two bounds closer can improve the convergence of TernGrad. In Assumption 3, max (abs (g))
is the maximum absolute value of all the gradients in the DNN. So, in a large DNN, max (abs (g))
could be relatively much larger than most gradients, implying that the bound in TernGrad becomes
much stronger. Considering the situation, we propose layer-wise ternarizing and gradient clipping to
reduce max (abs (g)) and therefore shrink the gap between these two bounds.
Layer-wise ternarizing is proposed based on the observation that the range of gradients in each
layer changes as gradients are back propagated. Instead of adopting a large global maximum scaler,
Iteration #
conv
Iteration #
fc
(a) original
(b) clipped
(c) ternary
(d) final
Figure 2: Histograms of (a) original floating gradients, (b) clipped gradients, (c) ternary gradients
and (d) final averaged gradients. Visualization by TensorBoard. The DNN is AlexNet distributed
on two workers, and vertical axis is the training iteration. As examples, top row visualizes the third
convolutional layer and bottom one visualizes the first fully-connected layer.
5
we independently ternarize gradients in each layer using the layer-wise scalers. More specific, we
separately ternarize the gradients of biases and weights by using Eq. (1), where gt could be the
gradients of biases or weights in each layer. To approach the standard bound more closely, we can
split gradients to more buckets and ternarize each bucket independently as D. Alistarh et al. [26] does.
However, this will introduce more floating scalers and increase communication. When the size of
bucket is one, it degenerates to floating gradients.
Layer-wise ternarizing can shrink the bound gap resulted from the dynamic ranges of the gradients
across layers. However, the dynamic range within a layer still remains as a problem. We propose
gradient clipping, which limits the magnitude of each gradient gi in g as
gi
|gi | ? c?
f (gi ) =
,
(21)
sign(gi ) ? c? |gi | > c?
where ? is the standard derivation of gradients in g. In distributed training, gradient clipping is
applied to every worker before ternarizing. c is a hyper-parameter to select, but we cross validate
it only once and use the constant in all our experiments. Specifically, we used a CNN [31] trained
on CIFAR-10 by momentum SGD with staircase learning rate and obtained the optimal c = 2.5.
Suppose the distribution of gradients is close to Gaussian distribution as shown in Figure 2(a), very
few gradients can drop out of [?2.5?, 2.5?]. Clipping these gradients in Figure 2(b) can significantly
reduce the scaler but slightly changes the length and direction of original g. Numerical analysis
shows that gradient clipping with c = 2.5 only changes the length of g by 1.0% ? 1.5% and its
direction by 2? ? 3? . In our experiments, c = 2.5 remains valid across multiple databases (MNIST,
CIFAR-10 and ImageNet), various network structures (LeNet, CifarNet, AlexNet, GoogLeNet, etc)
and training schemes (momentum, vanilla SGD, adam, etc).
The effectiveness of layer-wise ternarizing and gradient clipping can also be explained as follows.
When the scalar st in Eq. (1) and Eq. (3) is very large, most gradients have a high possibility to be
ternarized to zeros, leaving only a few gradients to large-magnitude values. The scenario raises a
severe parameter update pattern: most parameters keep unchanged while others likely overshoot.
This will introduce large training variance. Our experiments on AlexNet show that by applying both
layer-wise ternarizing and gradient clipping techniques, TernGrad can converge to the same accuracy
as standard SGD. Removing any of the two techniques can result in accuracy degradation, e.g., 3%
top-1 accuracy loss without applying gradient clipping as we shall show in Table 2.
4
Experiments
We first investigate the convergence of TernGrad under various training schemes on relatively small
databases and show the results in Section 4.1. Then the scalability of TernGrad to large-scale
distributed deep learning is explored and discussed in Section 4.2. The experiments are performed
by TensorFlow[2]. We maintain the exponential moving average of parameters by employing an
exponential decay of 0.9999 [15]. The accuracy is evaluated by the final averaged parameters. This
gives slightly better accuracy in our experiments. For fair comparison, in each pair of comparative
experiments using either floating or ternary gradients, all the other training hyper-parameters are the
same unless differences are explicitly pointed out. In experiments, when SGD with momentum is
adopted, momentum value of 0.9 is used. When polynomial decay is applied to decay the learning
rate (LR), the power of 0.5 is used to decay LR from the base LR to zero.
4.1 Integrating with Various Training Schemes
We study the convergence of TernGrad using LeNet on MNIST and a ConvNet [31] (named as
CifarNet) on CIFAR-10. LeNet is trained without data augmentation. While training CifarNet, images
Accuracy
100.00%
(a) momentum SGD
99.50%
(b) vanilla SGD
baseline
TernGrad
99.00%
98.50%
N workers
98.00%
2
4
8
16
32
2
64
4
8
16
32
64
Figure 3: Accuracy vs. worker number for baseline and TernGrad, trained with (a) momentum SGD
or (b) vanilla SGD. In all experiments, total mini-batch size is 64 and maximum iteration is 10K.
6
Table 1: Results of TernGrad on CifarNet.
SGD
base LR
total mini-batch size
iterations
gradients
workers
2
2
86.56%
85.64% (-0.92%)
16
16
83.19%
82.80% (-0.39%)
Adam
0.0002
128
300K
floating
TernGrad
Adam
0.0002
2048
18.75K
floating
TernGrad
accuracy
are randomly cropped to 24 ? 24 images and mirrored. Brightness and contrast are also randomly
adjusted. During the testing of CifarNet, only center crop is used. Our experiments cover the scope
of SGD optimizers over vanilla SGD, SGD with momentum [32] and Adam [33].
Figure 3 shows the results of LeNet. All are trained using polynomial LR decay with weight decay of
0.0005. The base learning rates of momentum SGD and vanilla SGD are 0.01 and 0.1, respectively.
Given the total mini-batch size M and the worker number N , the mini-batch size per worker is
M/N . Without explicit mention, mini-batch size refers to the total mini-batch size in this work.
Figure 3 shows that TernGrad can converge to the similar accuracy within the same iterations, using
momentum SGD or vanilla SGD. The maximum accuracy gain is 0.15% and the maximum accuracy
loss is 0.22%. Very importantly, the communication time per iteration can be reduced. The figure
also shows that TernGrad generalizes well to distributed training with large N . No degradation is
observed even for N = 64, which indicates one training sample per iteration per worker.
Table 1 summarizes the results of CifarNet, where all trainings terminate after the same epochs.
Adam SGD is used for training. Instead of keeping total mini-batch size unchanged, we maintain the
mini-batch size per worker. Therefore, the total mini-batch size linearly increases as the number of
workers grows. Though the base learning rate of 0.0002 seems small, it can achieve better accuracy
than larger ones like 0.001 for baseline. In each pair of experiments, TernGrad can converge to the
accuracy level with less than 1% degradation. The accuracy degrades under a large mini-batch size in
both baseline and TernGrad. This is because parameters are updated less frequently and large-batch
training tends to converge to poorer sharp minima [34]. However, the noise inherent in TernGrad can
help converge to better flat minimizers [34], which could explain the smaller accuracy gap between
the baseline and TernGrad when the mini-batch size is 2048. In our experiments of AlexNet in
Section 4.2, TernGrad even improves the accuracy in the large-batch scenario. This attribute is
beneficial for distributed training as a large mini-batch size is usually required.
4.2 Scaling to Large-scale Deep Learning
We also evaluate TernGrad by AlexNet and GoogLeNet trained on ImageNet. It is more challenging to
apply TernGrad to large-scale DNNs. It may result in some accuracy loss when simply replacing the
floating gradients with ternary gradients while keeping other hyper-parameters unchanged. However,
we are able to train large-scale DNNs by TernGrad successfully after making some or all of the
following changes: (1) decreasing dropout ratio to keep more neurons; (2) using smaller weight
decay; and (3) disabling ternarizing in the last classification layer. Dropout can regularize DNNs by
adding randomness, while TernGrad also introduces randomness. Thus, dropping fewer neurons helps
avoid over-randomness. Similarly, as the randomness of TernGrad introduces regularization, smaller
weight decay may be adopted. We suggest not to apply ternarizing to the last layer, considering
that the one-hot encoding of labels generates a skew distribution of gradients and the symmetric
ternary encoding {?1, 0, 1} is not optimal for such a skew distribution. Though asymmetric ternary
levels could be an option, we decide to stick to floating gradients in the last layer for simplicity. The
overhead of communicating these floating gradients is small, as the last layer occupies only a small
percentage of total parameters, like 6.7% in AlexNet and 3.99% in ResNet-152 [35].
All DNNs are trained by momentum SGD with Batch Normalization [36] on convolutional layers.
AlexNet is trained by the hyper-parameters and data augmentation depicted in Caffe. GoogLeNet is
trained by polynomial LR decay and data augmentation in [37]. Our implementation of GoogLeNet
does not utilize any auxiliary classifiers, that is, the loss from the last softmax layer is the total loss.
More training hyper-parameters are reported in corresponding tables and published source code.
Validation accuracy is evaluated using only the central crops of images.
The results of AlexNet are shown in Table 2. Mini-batch size per worker is fixed to 128. For fast
development, all DNNs are trained through the same epochs of images. In this setting, when there are
7
Table 2: Accuracy comparison for AlexNet.
base LR
mini-batch size
workers
iterations
gradients
weight decay
DR?
top-1
top-5
0.01
256
2
370K
floating
TernGrad
TernGrad-noclip ?
0.0005
0.0005
0.0005
0.5
0.2
0.2
57.33%
57.61%
54.63%
80.56%
80.47%
78.16%
0.02
512
4
185K
floating
TernGrad
0.0005
0.0005
0.5
0.2
57.32%
57.28%
80.73%
80.23%
0.5
0.2
56.62%
57.54%
80.28%
80.25%
floating
0.0005
TernGrad
0.0005
DR: dropout ratio, the ratio of dropped neurons. ? TernGrad without gradient clipping.
0.04
?
1024
8
92.5K
Table 3: Accuracy comparison for GoogLeNet.
base LR
mini-batch size
workers
iterations
gradients
weight decay
DR
top-5
0.04
128
2
600K
floating
TernGrad
4e-5
1e-5
0.2
0.08
88.30%
86.77%
0.08
256
4
300K
floating
TernGrad
4e-5
1e-5
0.2
0.08
87.82%
85.96%
0.10
512
8
300K
floating
TernGrad
4e-5
2e-5
0.2
0.08
89.00%
86.47%
more workers, the number of iterations becomes smaller and parameters are less frequently updated.
To overcome this problem, we increase the learning rate for large-batch scenario [10]. Using this
scheme, SGD with floating gradients successfully trains AlexNet to similar accuracy, for mini-batch
size of 256 and 512. However, when mini-batch size is 1024, the top-1 accuracy drops 0.71% for the
same reason as we point out in Section 4.1.
TernGrad converges to approximate accuracy levels regardless of mini-batch size. Notably, it
improves the top-1 accuracy by 0.92% when mini-batch size is 1024, because its inherent randomness
encourages to escape from poorer sharp minima [30][34]. Figure 4 plots training details vs. iteration
when mini-batch size is 512. Figure 4(a) shows that the convergence curve of TernGrad matches
well with the baseline?s, demonstrating the effectiveness of TernGrad. The training efficiency can be
further improved by reducing communication time as shall be discussed in Section 5. The training
data loss in Figure 4(b) shows that TernGrad converges to a slightly lower level, which further proves
the capability of TernGrad to minimize the target function even with ternary gradients. A smaller
dropout ratio in TernGrad can be another reason of the lower loss. Figure 4(c) simply illustrate that
on average 71.32% gradients of a fully-connected layer (fc6) are ternarized to zeros.
Finally, we summarize the results of GoogLeNet in Table 3. On average, the accuracy loss is less
than 2%. In TernGrad, we adopted all that hyper-parameters (except dropout ratio and weight decay)
that are well tuned for the baseline [38]. Tuning these hyper-parameters specifically for TernGrad
could further optimize TernGrad and obtain higher accuracy.
5
Performance Model and Discussion
Our proposed TernGrad requires only three numerical levels {?1, 0, 1}, which can aggressively
reduce the communication time. Moreover, our experiments in Section 4 demonstrate that within the
(a) top-1 accuracy vs iteration
70%
60%
50%
40%
30%
20%
10%
0%
baseline
terngrad
(b) training loss vs iteration
8
6
baseline
terngrad
4
2
50000
100000
150000
60%
40%
20%
0
0
(c) gradient sparsity of terngrad in fc6
80%
0%
0
50000
100000
150000
0
50000
100000
150000
Figure 4: AlexNet trained on 4 workers with mini-batch size 512: (a) top-1 validation accuracy, (b)
training data loss and (c) sparsity of gradients in first fully-connected layer (fc6) vs. iteration.
8
Training throughput on GPU cluster with
Ethernet and PCI switch
100000
AlexNet FP32
GoogLeNet FP32
VggNet-A FP32
80000
Training throughput on GPU cluster with
InfiniBand and NVLink
240000
AlexNet TernGrad
GoogLeNet TernGrad
VggNet-A TernGrad
200000
Images/sec
Images/sec
AlexNet TernGrad
GoogLeNet TernGrad
VggNet-A TernGrad
6000
4000
60000 3000
2000
40000
1000
160000
4000
120000
80000
2000
0
20000
40000
0
AlexNet FP32
GoogLeNet FP32
VggNet-A FP32
1
2
4
8
32
16
# of GPUs
64
128
256 512
(a)
0
0
1
2
4
8
16
32
# of GPUs
64
128 256 512
(b)
Figure 5: Training throughput on two different GPUs clusters: (a) 128-node GPU cluster with
1Gbps Ethernet, each node has 4 NVIDIA GTX 1080 GPUs and one PCI switch; (b) 128-node GPU
cluster with 100 Gbps InfiniBand network connections, each node has 4 NVIDIA Tesla P100 GPUs
connected via NVLink. Mini-batch size per GPU of AlexNet, GoogLeNet and VggNet-A is 128, 64
and 32, respectively
same iterations, TernGrad can converge to approximately the same accuracy as its corresponding
baseline. Consequently, a dramatical throughput improvement on the distributed DNN training is
expected. Due to the resource and time constraint, unfortunately, we aren?t able to perform the
training of more DNN models like VggNet-A [39] and distributed training beyond 8 workers. We plan
to continue the experiments in our future work. We opt for using a performance model to conduct
the scalability analysis of DNN models when utilizing up to 512 GPUs, with and without applying
TernGrad. Three neural network models?AlexNet, GoogLeNet and VggNet-A?are investigated.
In discussions of performance model, performance refers to training speed. Here, we extend the
performance model that was initially developed for CPU-based deep learning systems [40] to estimate
the performance of distributed GPUs/machines. The key idea is combining the lightweight profiling
on single machine with analytical modeling for accurate performance estimation. In the interest of
space, please refer to Supplementary Material for details of the performance model.
Figure 5 presents the training throughput on two different GPUs clusters. Our results show that
TernGrad effectively increases the training throughput for the three DNNs. The speedup depends on
the communication-to-computation ratio of the DNN, the number of GPUs, and the communication
bandwidth. DNNs with larger communication-to-computation ratios (e.g. AlexNet and VggNet-A)
can benefit more from TernGrad than those with smaller ratios (e.g., GoogLeNet). Even on a very
high-end HPC system with InfiniBand and NVLink, TernGrad is still able to double the training
speed of VggNet-A on 128 nodes as shown in Figure 5(b). Moreover, the TernGrad becomes more
efficient when the bandwidth becomes smaller, such as 1Gbps Ethernet and PCI switch in Figure 5(a)
where TernGrad can have 3.04? training speedup for AlexNet on 8 GPUs.
Acknowledgments
This work was supported in part by NSF CCF-1744082 and DOE SC0017030. Any opinions,
findings, conclusions or recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of NSF, DOE, or their contractors.
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6,356 | 675 | How Oscillatory Neuronal Responses Reflect
Bistability and Switching of the Hidden
Assembly Dynamics
K. Pawelzik, H.-V. Bauert, J. Deppisch, and T. Geisel
Institut fur Theoretische Physik and SFB 185 Nichtlineare Dynamik
Universitat Frankfurt, Robert-Mayer-Str. 8-10, D-6000 Frankfurt/M. 11, FRG
ttemporary adress:CNS-Program, Caltech 216-76, Pasadena
email: [email protected]
Abstract
A switching between apparently coherent (oscillatory) and stochastic
episodes of activity has been observed in responses from cat and monkey
visual cortex. We describe the dynamics of these phenomena in two parallel approaches, a phenomenological and a rather microscopic one. On the
one hand we analyze neuronal responses in terms of a hidden state model
(HSM). The parameters of this model are extracted directly from experimental spike trains. They characterize the underlying dynamics as well
as the coupling of individual neurons to the network. This phenomenological model thus provides a new framework for the experimental analysis
of network dynamics. The application of this method to multi unit activities from the visual cortex of the cat substantiates the existence of
oscillatory and stochastic states and quantifies the switching behaviour
in the assembly dynamics. On the other hand we start from the single
spiking neuron and derive a master equation for the time evolution of the
assembly state which we represent by a phase density. This phase density
dynamics (PDD) exhibits costability of two attractors, a limit cycle, and
a fixed point when synaptic interaction is nonlinear. External fluctuations
can switch the bistable system from one state to the other. Finally we
show, that the two approaches are mutually consistent and therefore both
explain the detailed time structure in the data.
977
978
Pawelzik, Bauer, Deppisch, and Geisel
1
INTRODUCTION
A few years ago, oscillatory and synchronous neuronal activity was discovered in
cat visual cortex [1-3]. These experiments backed earlier considerations about synchrony in neuronal activity as a mechanism to bind features, e.g., of an object in a
visual scene [4]. They triggered broad experimental and theoretical investigations
of detailed neuronal dynamics as a means for information processing and, in particular, for feature binding. Many theoretical contributions tried to reproduce and
explain aspects of the experimentally observed phenomena [5]. Motivated by the
experiments, the models where particularly designed to exhibit spatial synchronization of permanent oscillatory responses upon stimulation by a common, connected
stimulus like a bar. Most models consist of elements which exhibit a limit cycle
after a simple Hopf bifurcation.
The experimental data, however, contain many details which the present models do
not yet completely incorporate. One of these details is the coexistence of regular
and irregular episodes in the data, which interchange in an apparently stochastic
manner. This interchange can be observed in the signals from a single electrode [6]
as well as in the time-resolved correlation of the signals from two electrodes [7]. In
this contribution we show, that the observed time structure reflects a switching in
the dynamics of the underlying neuronal system. This will be demonstrated by two
complementary approaches:
On the one hand we present a new method for a quantitative analysis of the dynamical system underlying the measured spike trains. Our approach gives a quantitative
description of the dynamical phenomena and furthermore explains the relation between the collective excitation in the network which is not accessible experimentally
(i.e. hidden) and the contributions of the single observed neurons in terms of transition probability functions. These probabilities are the parameters of our Ansatz and
can be estimated directly from multi unit activities (MUA) using the Baum-Welchalgorithm. Especially for the data from cat visual cortex we find that indeed there
are two states dominating the dynamics of collective excitation, namely a state of
repeated excitation and a state in which the observed neurons fire independently
and stochastically.
On the other hand using simple statistical considerations we derive a description
for a local neuronal subpopulation which exhibits bistability. The dynamics of
the subpopulation can either rest on a fixed point - corresponding to the irregular
firing pat terns - or can follow a limit cycle - corresponding to the oscillatory firing
patterns. The subpopulation can alternate between both states under the influence
of noise in the external excitation. It turns out that the dynamics of this formal
model reproduces the observed local cortical signals in much detail.
2
Excitability of Neurons and Neuronal Assemblies
An abstract model of a neuron under external excitation e is given by its threshold
dynamics. The state of the neuron is represented by its phase 4>8, which is the
time passed by since the last action potential (4)8 = 0). The threshold 6 is high
directly after a spike and falls off in time and the neuron can fire again when e
exceeds 6. In case of noise or internal stochasticity, an excitability description of
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics
the dynamics of the neuron is more adequate. It gives the probability PI to fire
again in dependence of the state l/J6 with PI (l/J6) = 0'( e - B( l/JS)) and 0' some sigmoid
function. A monotonously falling threshold B then corresponds to a monotonously
increasing excitability PI' Such a description neglects any memory in the neuron
going beyond the last spike. In particular this means for an isolated neuron, that
PI can be easily calculated from the inter-spike interval histogram (ISIH) Ph using
the relation Ph(t)
PI(t) . (1- Ph (t')dt'). In that case also the autocorrelation
Ph(T)G(T - t)dt.
function can be calculated from Ph(t) via G(T) Ph(T) +
=
f:
f;
=
The excitability formulation sketched above is not valid for a neuron which is embedded in a neuronal assembly. However, we may use this Ansatz of a renewal
process to describe the activation dynamics of the whole assembly (see section 5).
The phase l/Jb
0 here corresponds to the state of synchronous activity of many
neurons in the assembly, which we call burst for convenience. Since the dynamics
of the network can differ from the dynamics of the elements we expect the function pj(l/Jb) which now describes the burst excitability of the whole assembly to be
different from?the spike excitability PI(l/Js) of the single neuron.
=
A simple example for this is a system of integrate and fire neurons in which oscillatory and irregular phases emerge under fixed stimulus conditions( [8, 9] and
section 5). Contrary to the excitability of the single refractory element the burst
excitability pj of the system has a maximum at l/Jb
T which expresses the increased probability to burst again after the typical oscillation period T, i.e. the
maximum represents a state 0 of oscillation. The assembly, however, can miss to
burst around l/Jb = T with a probability PO-+6 and switch into a second state s
in which the probability Ps-+o to burst again is reduced to a constant level. The
switching probabilities PO-+6 and Ps-+o can be easily calculated from Pj. In this
way the shape of pj distinguishes a system with an oscillatory state from a system
which is purely refractory but which nevertheless can still have strong modulations
in the autocorrelogram [13].
=
3
Hidden states and stochastic observables
The single neuron in an assembly, however, need not be strictly coupled to the state
of the assembly, i.e. a neuron may also spike for l/Jb > 0 and it may not take part
in a burst. This stochastic coupling to an underlying process suffices to destroy
the equivalence of Ph and the autocorrelogram C( T) =< s(t)s(t + T) >t of the
spike train set) E {O, I} (Fig 1). We therefore include the probability PObs(l/Jb) to
observe a spike when the assembly is in the state l/Jb into our description (Fig. 2).
The unlikely case where the spike represents the burst corresponds to the choice
Pobs = 84>b,O'
4
Application to Experimental Data
While our approach is quite general, we here concentrate on the measurements
of Gray et al. [2] in cat visual cortex. Because our hidden state model has the
structure of a hidden Markov model we can obtain all the parameters POb6(l/J) and
979
980
Pawelzik, Bauer, Deppisch, and Geisel
0.10
0.08
C(T)
0.0(3
0.04
0.02
20
40
60
80
T[mS]
Figure 1: Correlogram of multi unit activities from cat visual cortex (line). Correlaograms predicted from the ISIH (6) and from the hidden state model (+) .
Measurement Level
Network Level
4>:
,. p (n)
f
Figure 2: The hidden state model. While Pj(q}) governs the dynamics of assembly
states q;b, Pobs (q;b) represents the probability to observe a spike of a single neuron.
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics
0.25[
0.20
Pj( 4>b)
0.15
010
005
O.Oob~~~:::t:!~~_ __
o
----:,::--_ _ _~.
020
0.10
o.oo L---_ _ _---=-~:=:!:?.""--~:!T~~~~.
o
10
20
pJ
Figure 3: Network excitability
and single neuron contribution Pobs estimated
from experimental spike trains (A17, cat).
pJ (?;)
directly from the multi unit activities using the well known Baum-Welch
algorithm[lO]. The results can be seen in Fig. 3. The excitability shows a typical
peak around the main period at T = 19ms, which indicates a state of oscillation.
For larger phases we see a reduced excitability which reveals a state of stochastic
activity (Pobs (?;b > T) > 0). The spike observation probability Pobs (?;) is peaked
near the burst and is about constant elsewhere. This means that we can characterize
the data by a stochastic switching between two dynamical states in the underlying
system. Because of the stochastic coupling of the single neuron to the assembly state
this can only hardly be observed directly. The switching probabilities between either
states calculated from
coincide with results from other methods [11].
pJ
From the excitability pJ and the spike probabilities Pobs we now obtain the autocor-
relation function C(r) = f4>f4>' Pobs(?;')M(?;',?;fPobs(?;)p(?;)d?;'d?;, with M being
the transition matrix of the Markov model (see also below). The result is compared
to the true autocorrelation C( r) in Fig. 1. The excellent agreement confirms our
simple Ansatz of a renewal process for the hidden burst dynamics of the assembly.
5
Bistability and Switching in Networks of Spiking Neurons
The above results indicate that the dynamics of a cortical assembly includes bistability rather than a simple Hopf bifurcation. In order to understand how this bistability emerges in a network we go one step back and derive a model for a neuronal
subpopulation on the basis of spiking neurons. We assume again that the internal
state of the neuron is given by the threshold function e depending on the time since
the last spike event and that the excitability of the neuron can be described by a
981
982
Pawelzik, Bauer, Deppisch, and Geisel
c
o
':=J
Q)
c:
4 ~----'l.r---....Il ,..----"
3
2
t
ti me
Figure 4: Illustration of assembly state representation by a phase density.
firing probability Pf. In a network, however, the input to the neuron has external
contributions i ext as well as from within iint i.e.
= sigm(wextiext(t) + Wintiint(t) - 8(<p)).
For a more formal treatment of the dynamics of such a network we characterize the
assembly state by a phase density p( <P, t) which gives the relative amount of neurons
in the assembly which are in phase <p at time t (Fig 4).
Pf(<P,t)
Discretizing the internal phases <p, we transform p(<p, t) to p(j), a vector whose
components i give the probability to be in phase <Pi E [(i -1)ilt,iilt] at time
tj = jilt. The number T of components is chosen large enough to ensure that
Pf (T, j) = Pf (T ilt, jilt) does not change any more. This vector evolves in time
according to
p(j + 1) = M(j)p(j)
(1)
with
0
Pf(I,})
Pf(2,j)
Pf(T-l , j)
Pf(T,j)
1
0
0
0
0
I-Pf (1, j)
0
M(j) =
I-Pi (2,j)
0
0
0
1-Pf(T-1,j)
I-Pf(T,j)
beeing a matrix that incorporates the effects offiring (reset) via the firing probability
Pf(i,j).
It remains to define the lateral interaction in the sUbpopulation. Clearly only the
fraction of neurons that fire can interact, therefore we have
l.int
=
g(po).
Oscillatory Neuronal Responses Reflect Bistability & Switching of Assembly Dynamics
~
0
...
.c
1.0
0.8 -
-
0.6 -
-
0.4 l0.2 t-0.0 ",,"IA~
o
ltv
200
~~
If'"'"
400
~
600
IA
.AlIJ
BOO
1000
Figure 5: Switching in the assembly dynamics under external fluctuations. Note
that Po denotes the time dependent firing rate.
Numerically iterating the dynamics (1) we find, that the distribution can evolve in
two distinct ways, depending on the lateral interaction function 9 and the initialization. First Ii can relax to a fixed point, corresponding to a constant fraction of
the neurons firing at a particular time. On the level of an individual neuron, this
corresponds to a stochastic firing characteristics, and in the measurements this state
corresponds to the irregular periods. Secondly the distribution can evolve according
to a limit cycle, with a rather large fraction of the neurons in a narrow band of phases
i.e. the neurons are synchronous. We find parameter combinations Wext, Wint where
both states coexist, i.e. we find bistability, if the interaction function is nonlinear,
like g( x) ex: x 2 or g( x) ex: x4. Nonlinearities of this kind can be brought about by
a different type of preferred synapses with a nonlinear transmission characteristic
(like an NMDA-receptor synapse) for the corticocortical projections, as compared
to the thalamocortical projections. At this point we only want to make the point
that our simple model system can exhibit bistability under reasonable assumptions.
In the bistable regime some initializations of Ii lead to the oscillatory state, some
to the stochastic state. Adding some noise to the external input the system can
also switch between the two states(Fig. 5). In this way our simple local model can
capture the switching phenomenon which is inherent in the experimental data [12].
6
Summary and Synthesis
We presented two complementary approaches to the dynamics of neuronal subpopulations, a phenomenological one which captures the time structure in measured spike
trains and a neuronal one which provides a formal description of the dynamics of
assemblies of spiking neurons. In the phenomenological approach we introduced
the hidden state model which revealed that the system underlying the multi unit
activities from the cat switches between states of oscillatory and stochastic activity.
The analysis of the phase density dynamics showed, that bistability and switching
emerges in networks of spiking neurons when the neuronal interaction is nonlinear.
It remains to show that these approaches are also quantitatively consistent, i.e. that
they are two sides of the same medal. Instead of a formal proof we only remark here
that the parameters of the HSM can be extracted directly from the dynamics of the
PDD under external noise. For this purpose one only needs to evaluate the interburst interval distribution from which pj can be calculated. Pobs is easily estimated
as the average shape of Po(t) between successive bursts. We find, that already this
procedure gives HSMs which accurately reproduce the correlation function of the
983
984
Pawelzik, Bauer, Deppisch, and Geisel
full firing dynamics poet). This means that the HSM captures relevant aspects of
the assembly dynamics including the relation between the network dynamics and
the contributions of single neurons.
References
[1] Gray C.M., Singer W., Stimulus-Specific Neuronal Oscillations in Cat Visual
Cortex: a Cortical Functional Unit, Soc.Neurosci.Abstr. 13.404.3 (1989).
[2] Gray C.M., Konig P., Engel A.K., and Singer W., Oscillatory Responses in Cat
Visual Cortex Exhibit Inter-Columnar Synchronization which Reflects Global
Stimulus Properties Nature 338, pp. 334-337 (1989).
[3] Eckhorn R., Bauer R., Jordan W., Brosch M., Kruse W., Munk M., and Reitboeck H.J., Coherent Oscillations: a Mechanism of Feature Linking in the
Visual Cortex'l, BioI. Cyb. 60, pp121-130 (1988).
[4] C. v.d.Malsburg The Correlation Theory of Brain Function Internal Report 812, Max-Planck-Institute for Biophysical Chemistry, Gottingen, F.R.G. (1981).
[5] Schuster H.G. (Ed.), Nonlinear Dynamics and Neuronal Networks, VCH Weinheim, Heidelberg (1991)
[6] Pawelzik K., Bauer H.-U., Geisel T. Switching between predictable and unpredictable states in data from cat visual cortex, talk at CNS San Francisco 1992,
to appear in the CNS Proceedings.
[7] Gray C.M., Engel A.K., Konig P., Singer W., Temporal Properties of Synchronous Oscillatory Interactions in Cat Striate Cortex, in: Nonlinear Dynamics and Neuronal Networks, Ed. H.G. Schuster, VCH Weinheim, pp. 27-55
(1991)
[8] Deppisch J., Bauer H.-U., Schillen T., Konig P., Pawelzik K., Geisel T.,
Stochastic and Oscillatory Burst Activities, accepted for ICANN'92, Brighton,
UK. (1992).
[9] Deppisch J., Bauer H.-U., Schillen T., Konig P., Pawelzik K., Geisel T., Alternating Oscillatory and Stochastic States in a Network of Spiking Neurons,
submitted to Biol.Cyb. (1992).
[10] Rabiner, L.R., A Tutorial on Hidden-Markov Models and Selected Applications
in Speech Recognition Proc. IEEE 11, 2 pp. 257-286 (1989).
[11] Bauer H.TU., Deppisch J., Geisel T., Pawelzik K., in preparation.
[12] Bauer H.U., Pawelzik K., Alternating Oscillatory and Stochastic Dynamics in
a Model for a Neuronal Assembly, Physica D, submitted.
[13] Schuster H.G., Koch C., Burst Synchronization Without Frequency-Locking in
a Completely Solvable Network Model, in Moody J.E., Hanson S.J., Lippmann
R.P. (Eds.), Neural Information Processing Systems 4, p. 117, Morgan Kauffmann (1992).
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6,357 | 6,750 | Learning Affinity via Spatial Propagation Networks
Sifei Liu
UC Merced, NVIDIA
Guangyu Zhong
Dalian University of Technology
Shalini De Mello
NVIDIA
Ming-Hsuan Yang
UC Merced, NVIDIA
Jinwei Gu
NVIDIA
Jan Kautz
NVIDIA
Abstract
In this paper, we propose spatial propagation networks for learning the affinity matrix for vision tasks. We show that by constructing a row/column linear propagation
model, the spatially varying transformation matrix exactly constitutes an affinity
matrix that models dense, global pairwise relationships of an image. Specifically,
we develop a three-way connection for the linear propagation model, which (a)
formulates a sparse transformation matrix, where all elements can be outputs from
a deep CNN, but (b) results in a dense affinity matrix that effectively models
any task-specific pairwise similarity matrix. Instead of designing the similarity
kernels according to image features of two points, we can directly output all the
similarities in a purely data-driven manner. The spatial propagation network is a
generic framework that can be applied to many affinity-related tasks, such as image
matting, segmentation and colorization, to name a few. Essentially, the model
can learn semantically-aware affinity values for high-level vision tasks due to the
powerful learning capability of deep CNNs. We validate the framework on the
task of refinement of image segmentation boundaries. Experiments on the HELEN
face parsing and PASCAL VOC-2012 semantic segmentation tasks show that the
spatial propagation network provides a general, effective and efficient solution for
generating high-quality segmentation results.
1
Introduction
An affinity matrix is a generic matrix that determines how close, or similar, two points are in a space. In
computer vision tasks, it is a weighted graph that regards each pixel as a node, and connects each pair
of pixels by an edge [25, 16, 15, 10, 29]. The weight on that edge should reflect the pairwise similarity
with respect to different tasks. For example, for low-level vision tasks such as image filtering, the
affinity values should reveal the low-level coherence of color and texture [29, 28, 10, 9]; for mid to
high-level vision tasks such as image matting and segmentation [16, 22], the affinity measure should
reveal the semantic-level pairwise similarities. Most techniques explicitly or implicitly assume a
measurement or a similarity structure over the space of configurations. The success of such algorithms
depends heavily on the assumptions made to construct these affinity matrices, which are generally
not treated as part of the learning problem.
In this paper, we show that the problem of learning the affinity matrix can be equivalently expressed
as learning a group of small row/column-wise, spatially varying linear transformation matrices.
Since a linear transformation can be easily implemented as a differentiable module in a deep neural
network, the transformation matrix can be learned in a purely data-driven manner as opposed to
being constructed by hand. Specifically, we adopt an independent deep CNN with the original RGB
images as inputs to output all entities of the matrix, such that the affinity is learned by a deep model
conditioned on the specific inputs. We show that using a three-way connection, instead of the full
connection between adjoining rows/columns, is sufficient for learning a dense affinity matrix and
requires much fewer output channels of a deep CNN. Therefore, instead of using designed features
and kernel tricks, our network outputs all entities of the affinity matrix in a data-driven manner.
The advantages of learning an affinity matrix in a data-driven manner are multifold. First, a handdesigned similarity matrix based on a distance metric in a certain space (e.g., RGB or Euclidean [10,
25, 5, 36, 14]) may not adequately describe the pairwise relationships in the mid-to-high-level feature
spaces. To apply such designed pairwise kernels to tasks such as semantic segmentation, multiple
iterations are required [14, 5, 36] for satisfactory performance. In contrast, the proposed method
learns and outputs all entities of an affinity matrix under direct supervision of ultimate objectives,
where no iteration, specific design or assumption about the kernel function is needed. Second, we can
learn the high-level semantic affinity measures by initializing with hierarchical deep features from
pre-trained VGG [26] and ResNet [11] networks where conventional metrics and kernels may not be
applied. Due to the above properties, the framework is far more efficient than the related graphical
models, such as Dense CRF.
Our proposed architecture, namely spatial propagation network (SPN), contains a deep CNN that
learns the entities of the affinity matrix and a spatial linear propagation module, which propagates
information in an image using the learned affinity values. Images or general 2D matrices are input
into the module, and propagated under the guidance of the learned affinity values. All modules are
differentiable and jointly trained using the stochastic gradient descent (SGD) method. The spatial
linear propagation module is computationally efficient for inference due to the linear time complexity
of its recurrent architecture.
2
Related Work
Numerous methods explicitly design affinity matrices for image filtering [29, 10], colorization [15],
matting [16] and image segmentation [14] based on the characterstics of the problem. Other methods,
such as total variation (TV) [23] and learning to diffuse [18] improve the modeling of pairwise
relationships by utilizing different objectives, or incorporating more priors into diffusion partial
differential equations (PDEs). However, due to the lack of an effective learning strategy, it is still
challenging to produce learning-based affinity for complex visual analysis problems. Recently, Maire
et al. [22] trained a deep CNN to directly predict the entities of an affinity matrix, which demonstrated
good performance on image segmentation. However, since the affinity is followed by a solver of
spectral embedding as an independent part, it is not directly supervised for the classification/prediction
task. Bertasius et al. [2] introduced a random walk network that optimizes the objectives of pixel-wise
affinity for semantic segmentation. Differently, their affinity matrix is additionally supervised by
ground-truth sparse pixel similarities, which limits the potential connections between pixels.
On the other hand, many graphical model-based methods have successfully improved the performance
of image segmentation. In the deep learning framework, conditional random fields (CRFs) with
efficient mean field inference are frequently used [14, 36, 17, 5, 24, 1] to model the pairwise relations
in the semantic labeling space. Some methods use CFR as a post-processing module [5], while others
integrate it as a jointly-trained part [36, 17, 24, 1]. While both methods describe the densely connected
pairwise relationships, dense CRFs rely on designed kernels, while our method directly learns all
pairwise links. Since in this paper, SPN is trained as a universal segmentation refinement module, we
specifically compare it with one of the methods [5] that relies on dense CRF [14] as a post-processing
strategy. Our architecture is also related to the multi-dimensional RNN or LSTM [30, 3, 8]. However,
both the standard RNN and LSTM contain multiple non-linear units and thus do not fit into our
proposed affinity framework.
3
Proposed Approach
In this work, we construct a spatial propagation network that can transform a two-dimensional
(2D) map (e.g., coarse image segmentation) into a new one with desired properties (e.g., refined
segmentation). With spatially varying parameters that supports the propagation process, we show
theoretically in Section 3.1 that this module is equivalent to the standard anisotropic diffusion
process [32, 18]. We prove that the transformation of maps is controlled by a Laplacian matrix that is
constituted by the parameters of the spatial propagation module. Since the propagation module is
differentiable, its parameters can be learned by any type of neural network (e.g., a typical deep CNN)
that is connected to this module, through joint training. We introduce the spatial propagation network
in Section 3.2, and specifically analyze the properties of different types of connections within its
framework for learning the affinity matrix.
2
3.1
Linear Propagation as Spatial Diffusion
We apply a linear transformation by means of the spatial propagation network, where a matrix is
scanned row/column-wise in four fixed directions: left-to-right, top-to-bottom, and verse-vise. This
strategy is used widely in [8, 30, 19, 4]. We take the left-to-right direction as an example for the
following discussion. Other directions are processed independently in the same manner.
We denote X and H as two 2D maps of size n ? n, with exactly the same dimensions as the matrix
before and after spatial propagation, where xt and ht , respectively, represent their tth columns with
n ? 1 elements each. We linearly propagate information from left-to-right between adjacent columns
using an n ? n linear transformation matrix wt as:
ht = (I ? dt ) xt + wt ht?1 ,
t ? [2, n]
(1)
where I is the n ? n identity matrix, the initial condition h1 = x1 , and dt (i, i) is a diagonal matrix,
whose ith element is the sum of all the elements of the ith row of wt except wt (i, j) as:
dt (i, i) =
n
X
wt (i, j).
(2)
j=1,j6=i
To propagate across the entire image, the matrix H, where {ht ? H, t ? [1, n]}, is updated in a
column-wise manner recursively. For each column, ht is a linear, weighted combination of the
previous column ht?1 , and the corresponding column xt in X. When the recursive scanning is
finished, the updated 2D matrix H can be expressed with an expanded formulation of Eq. (1):
?
I
? w2
?
?w3 w2
Hv = ?
? ..
? .
?
..
.
0
?2
w3 ?2
..
.
..
.
???
0
?3
..
.
???
???
0
..
.
???
???
?
0
? ? ??
?
? ? ??
.. ?
? Xv = GXv ,
. ?
?
?n
(3)
where G is a lower triangular, N ? N (N = n2 ) transformation matrix, which relates X and
H. Hv and Xv are vectorized versions of X and H, respectively, with the dimension of N ? 1.
Specifically, they are created by concatenating ht and xt along the same, single dimension, i.e.,
T
T
Hv = hT1 , ..., hTn
and Xv = xT1 , ..., xTn . All the parameters {?t , wt , dt , I} , t ? [2, n] are
n ? n sub-matrices, where ?t = I ? dt .
In the following section, we validate that Eq. (3) can be expressed as a spatial anisotropic diffusion
process, with the corresponding propagation affinity matrix constituted by all wt for t ? [2, n].
Theorem 1. The summation of elements in each row of G equals to one.
Since G contains n ? n sub-matrices, each representing the transformation between the corresponding
columns of H and X, we denote all the weights used to compute ht as the tth block-row Gt . On
setting ?1 = I, the k th constituent n ? n sub-matrix of Gt is:
Gtk =
?
t
Y
?
?
?
w? ?k ,
? =k+1
?
?
?
?k ,
k ? [1, t ? 1]
(4)
k=t
To prove that the summation of any row in G equals to one, we instead prove that for ?t ? [1, n],
each row of Gt has the summation of one.
T
T
Proof. Denoting E = [1, 1, ..., 1] as an n ? 1 vector, we need to prove that Gt [1, ..., 1]N ?1 = E.
Pt
Equivalently k=1 Gtk E = E, because G is a lower triangular matrix. In the following part, we first
Pm
Qt
prove that when m ? [1, t ? 1], we have k=1 Gtk E = ? =m+1 wt E by mathematical induction .
Initial step. When m = 1,
Pm
k=1
Gtk E = Gt1 E =
3
Qt
? =2
w? E, which satisfies the assertion.
Figure 1: Different propagation ranges for (a) one-way connections; and (b) three-way connections. Each pixel
(node) receives information from a single line with one-way connection, and from a 2 dimensional plane with
three-way connection. Integration of four directions w.r.t. (a) results in global, but sparsely connected pairwise
relations, while (b) formulates global and densely connected pairwise relations.
Inductive step. Assume there is an n ? [1, t ? 1], such that
prove the formula is true for n + 1 ? [1, t ? 1].
n+1
X
k=1
Gtk E =
n
X
Gtk E + Gt(n+1) E =
t
Y
w? E +
? =n+1
k=1
t
Y
? =n+2
Pn
k=1
w? =
Gtk E =
t
Y
Qt
? =n+1
wt E, we must
w? [(wn+1 + I ? dn+1 ) E] .
? =n+2
(5)
According to the formulation of the diagonal matrix in Eq. (2) we have
Qt
? =n+2 w? E. Therefore, the assertion is satisfied. When m = t, we have:
t
X
k=1
Gtk E =
t?1
X
Gtk E + Gtt E =
t
Y
Pn+1
k=1 Gtk E =
w? E + ?t E = w? E + (I ? dt ) E = E,
(6)
? =t
k=1
which yields the equivalence of Theorem 1.
Theorem 2. We define the evolution of a 2D matrix as a time sequence {U }T , where U (T = 1) = U1
is the initial state. When the transformation between any two adjacent states follows Eq. (3), the
sequence is a diffusion process expressed with a partial differential equation (PDE):
?T U = ?LU
(7)
where L = D ? A is the Laplacian matrix, D is the degree matrix composed of dt in Eq. (2), and A
is the affinity matrix composed by the off-diagonal elements of G.
Proof. We substitute the X and H as two consecutive matrices UT +1 and UT in (3). According to
Theorem 1, we ensure that the sum of each row I ? G is 0 that can formulate a standard Laplacian
matrix. Since G has the diagonal sub-matrix I ? dt , we can rewrite (3) as:
UT +1 = (I ? D + A) UT = (I ? L) UT
(8)
where G = (I ? D + A), D is an N ? N diagonal matrix containing all the dt and A is the offdiagonal part of G. It then yields UT +1 ? UT = ?LUT , a discrete formulation of (7) with the time
discretization interval as one.
Theorem 2 shows the essential property of the row/column-wise linear propagation in Eq. (1): it
is a standard diffusion process where L defines the spatial propagation and A, the affinity matrix,
describes the similarities between any two points. Therefore, learning the image affinity matrix A in
Eq. (8) is equivalent to learning a group of transformation matrices wt in Eq. (1).
In the following section, we show how to build the spatial propagation (1) as a differentiable module
that can be inserted into a standard feed-forward neural network, so that the affinity matrix A can be
learned in a data-driven manner.
3.2 Learning Data-Driven Affinity
Since the spatial propagation in Eq.(1) is differentiable, the transformation matrix can be easily
configured as a row/column-wise fully-connected layer. However, we note that since the affinity
matrix indicates the pairwise similarities of a specific input, it should also be conditioned on the
4
content of this input (i.e., different input images should have different affinity matrices). Instead of
setting the wt matrices as fixed parameters of the module, we design them as the outputs of a deep
CNN, which can be directly conditioned on an input image.
One simple way is to set the output of the deep CNN to use the same size as the input matrix. When
the input has c channels (e.g., an RGB image has c = 3), the output needs n ? c ? 4 channels
(there are n connections from the previous row/column per pixel per channel, and with four different
directions). Obviously, this is too many (e.g., an 128 ? 128 ? 16 feature map needs an output of
128 ? 128 ? 8192) to be implemented in a real-world system. Instead of using full connections
between the adjacent rows/columns, we show that certain local connections, corresponding to a
sparse row/column-wise transform matrix, can also formulate densely connected affinity. Specifically,
we introduce the (a) one-way connection and the (b) three-way connection as two different ways to
implement Eq. (1).
One-way connection. The one-way connection enables every pixel to connect to only one pixel
from the previous row/column (see Figure 1(a)). It is equivalent to one-dimensional (1D) linear
recurrent propagation that scans each row/column independently as a 1D sequence. Following Eq. (1),
we denote xk,t and hk,t as the k th pixels in the tth column, where the left-to-right propagation for
one-way connection is:
hk,t = (1 ? pk,t ) ? xk,t + pk,t ? hk,t?1 ,
(9)
where p is a scaler weight indicating the propagation strength between the pixels at {k, t ? 1}
and {k, t}. Equivalently, wt in Eq. (1) is a diagonal matrix, with the elements constituted by
pk,t , k ? [1, n]. The one-way connection is a direct extension of sequential recurrent propagation [8,
31, 13]. The exact formulation of Eq. (9) has been used previously for semantic segmentation [4]
and for learning low-level vision filters [19]. In [4], Chen et al.explain it as domain transform,
where for semantic segmentation, p corresponds to the object edges. Liu et al. [19] explain it by
arbitrary-order recursive filters, where p corresponds to more general image properties (e.g., low-level
image/color edges, missing pixels, etc.). Both of these formulations can be explained as the same
linear propagation framework of Eq. (1) with one-way connections.
Three-way connection. We propose a novel three-way connection in this paper. It enables each
pixel to connect to three pixels from the previous row/column, i.e., the left-top, middle and bottom
pixels from the previous column for the left-to-right propagation direction (see Figure. 2(b)). With the
same notations, we denote N as the set of these three pixels. Then the propagation for the three-way
connection is:
!
hk,t =
1?
X
pk,t
xk,t +
k?N
X
pk,t hk,t?1
(10)
k?N
Equivalently, wt forms a tridiagonal matrix, with p:,k , k ? N constitute the three non-zero elements
of each row/column.
Relations to the affinity matrix. As introduced in Theorem 2, the affinity matrix A with linear
propagation is composed of the off-diagonal elements of G in Eq. (3). The one-way connection
formulates a spares affinity matrix, since each sub-matrix of A has nonzero elements only along its
diagonal, and the multiplication of several individual diagonal matrics will also results in a diagonal
matrix. On the other hand, the three-way connection, also with a sparse wt , can form a relatively
dense A with the multiplication of several different tridiagonal matrices. It means pixels can be
densely and globally associated, by simply increasing the number of connections of each pixel during
spatial propagation from one to three. As shown in Figures 2(a) and 2(b), the propagation of one-way
connections is restricted to a single row, while the three-way connections can expand the region to a
triangular 2D plane with respect to each direction. The summarization of the four directions result in
dense connections of all pixels to each other (see Figure. 2(b)).
Stability of linear propagation. Model stability is of critical importance for designing linear systems.
In the context of spatial propagation (Eq. 1), it refers to restricting the responses or errors that flow
in the module from going to infinity, and preventing the network from encountering the vanishing
of gradients in the backpropagation process [37]. Specifically, the norm of the temporal Jacobian
?ht \ ?ht?1 should be equal to or less than one. In our case, it is equivalent to regularizing each
transformation matrix wt with its norm satisfying
k?ht \ ?ht?1 k = kwt k ? ?max ,
5
(11)
Figure 2: We illustrate the general architecture of the SPN using a three-way connection for segmentation
refinement. The network, divided by the black dash line, contains a propagation module (upper) and a guidance
network (lower). The guidance network outputs all entities that can constitute four affinity matrices, where each
sub-matrix wt is a tridiagonal matrix. The propagation module, being guided by the affinity matrices, deforms
the input mask to a desired shape. All modules are differentiable and jointly learned via SGD.
where ?max denotes the largest singularity value of wt . This condition, ?max ? 1 provides a
sufficient condition for stability.
n
o
P
be the weight in wt , the model can be stabilized if k?N pK
Theorem 3. Let pK
t,k
t,k ? 1. See
k?N
the supplementary material for proof.
Theorem 3 shows that the stability of a linear propagation model can be maintained by regularizing
the all weights of each pixel in the hidden layer H, with the summation of their absolute values less
than one. For the one-way connection, Chen et al. [4] limited each scalar output p to be within (0, 1).
Liu et al. [19] extended the range to (?1, 1), where the negative weights showed preferable effects
for learning image enhancers. It indicates that the affinity matrix is not necessarily restricted to be
positive/semi-positive definite (e.g., the setting is also applied in [16].) For the three-way connection,
we simply regularize the three weights (the output of a deep CNN) according to Theorem 3 without
restriction to be any positive/semi-positive definite.
4
Implementation
We specify two separate branches: (a) a deep CNN, namely the guidance network that outputs
all elements of the transformation matrix, and (b) a linear propagation module that outputs the
transformation matrix entities (see Figure 2). The propagation module receives an input map and
output a refined or transformed result. It also takes the weights learned by the deep CNN guidance
network as the second input. The structure of a guidance network can be any regular CNN, which
is designed for the task at hand. Examples of this network are described in Section 5. It takes, as
input, any 2D matrix that can help with learning the affinity matrix (e.g., typically an RGB image),
and outputs all the weights that constitute the transformation matrix wt .
Suppose that we have a map of size n ? n ? c that is input into the propagation module, the guidance
network needs to output a weight map with the dimensions of n ? n ? c ? (3 ? 4), i.e., each pixel in
the input map is paired with 3 scalar weights per direction, and 4 directions in total. The propagation
module contains 4 independent hidden layers for the different directions, where each layer combines
the input map with its respective weight map using Eq. (10). All submodules are differentiable
and jointly trained using stochastic gradient descent (SGD). We use node-wise max-pooling [19] to
integrate the hidden layers and to obtain the final propagation result.
We implement the network with a modified version of CAFFE [12]. We employ a parallel version
of the SPN implemented in CUDA for propagating each row/column to the next one. We use the
SGD optimizer, and set the base learning rate to 0.0001. In general, we train the networks for the
HELEN and VOC segmentation tasks for about 40 and 100 epochs, respectively. The inference time
(we do not use cuDNN) of SPN on HELEN and Pascal VOC is about 7ms and 84ms for an image of
size 512 ? 512 pixels, respectively. In comparison, the dense CRF (CPU only) takes about 1s [14],
3.2s [5] and 4.4s [36] with different publicly available implementations. We note that the majority
of the time for the SPN is spend in the guidance network, which can be accelerated by utilizing
various existing network compressing strategies, applying smaller models, or sharing weights with
the segmentation model if they are trained jointly. During inference, a single 64 ? 64 ? 32 SPN
hidden layer takes 1.3ms with the same computational settings.
6
original
CNN-base
CNN-Highres
one-way SPN
three-way SPN
ground truth
Figure 3: Results of face parsing on the HELEN dataset with detailed regions cropped from the high resolution
images. (The images are all in high resolution and can be viewed by zooming in.)
5
Experimental Results
The SPN can be trained jointly with any segmentation CNN model by being inserted on top of the
last layer that outputs probability maps, or trained separately as a segmentation refinement model.
In this paper we choose the second option. Given a coarse image segmentation mask as the input
to the spatial propagation module, we show that the SPN can produce higher-quality masks with
significantly refined details at object boundaries. Many models [21, 5] generate low-resolution
segmentation masks with coarse boundary shapes to seek a balance between computational efficiency
and semantic accuracy. The majority of work [21, 5, 36] choose to first produce an output probability
map with 8? smaller resolution, and then refine the result using either post-processing [5] or jointly
trained modules [36]. Hence, producing high-quality segmentation results with low computational
complexity is a non-trivial task. In this work, we train only one SPN model for a specific task, and
treat it as a universal refinement tool for the different publicly available CNN models for each of
these tasks.
We carry out the refinement of segmentation masks on two tasks: (a) generating high-resolution
segmentations on the HELEN face parsing dataset [27]; and (b) refining generic object segmentation
maps generated by pretrained models (e.g., VGG based model [21, 5]. For the HELEN dataset, we
directly use low-resolution RGB face images to train a baseline parser, which successfully encapsulates the global semantic information. The SPN is then trained on top of the coarse segmentations to
generate high-resolution outputs. For the Pascal VOC dataset, we train the SPN on top of the coarse
segmentation results generated by the FCN-8s [21], and directly generalize it to any other pretrained
model.
General network settings. For both tasks, we train the SPN as a patch refinement model on top of
the coarse map with basic semantic information. It is trained with smaller patches cropped from
the original high-resolution images, their corresponding coarse segmentation maps produced by a
baseline segmentor, and with the corresponding high-resolution ground-truth segmentation masks
for supervision. All coarse segmentation maps are obtained by applying a baseline (for HELEN) or
pre-trained (for Pascal VOC) image segmentation CNN to their standard training splits [6, 5]. Since
the baseline HELEN parser produces low-resolution segmentation results, we upsample them using a
bi-linear filter to be of the same size as the desired higher output resolution. We fix the size of our
input patches to 128 ? 128, use the softmax loss, and use the SGD solver for all the experiments.
During training, the patches are sampled from image regions that contain more than one ground-truth
segmentation label (e.g., a patch with all pixels labeled as ?background? will not be sampled). During
testing, for the VOC dataset, we restrict the classes in the refined results to be contained within the
corresponding coarse input. More specific settings are specified in the supplementary material.
HELEN Dataset. The HELEN dataset provides high-resolution photography-style face images
(2330 in total), with high-quality manually labeled facial components including eyes, eyebrows, nose,
lips, and jawline, which makes the high-resolution segmentation tasks applicable. All previous work
utilize low-resolution parsing output as their final results for evaluation. Although many [27, 33, 20]
achieve preferable performance, their results cannot be directly adopted by high-quality facial image
editing applications. We use the same settings as the state-of-the work [20]. We use similarity
transformation according to the results of 5-keypoint detection [35] to align all face images to the
center. Keeping the original resolution, we then crop or pad them to the size of 1024 ? 1024.
7
Table 1: Quantitative evaluation results on the HELEN dataset. We denote the upper and lower lips as ?U-lip?
and ?L-lip?, and overall mouth part as ?mouth?, respectively. The label definitions follow [20].
Method
skin
brows
eyes
nose
mouth U-lip
L-lip in-mouth
Liu et al. [20]
90.87 69.89 74.74 90.23 82.07 59.22 66.30
81.70
baseline-CNN
90.53 70.09 74.86 89.16 83.83 55.61 64.88
71.72
Highres-CNN
91.78 71.84 74.46 89.42 81.83 68.15 72.00
71.95
SPN (one-way)
92.26 75.05 85.44 91.51 88.13 77.61 70.81
79.95
83.13
SPN (three-way) 93.10 78.53 87.71 92.62 91.08 80.17 71.63
overall
83.68
82.89
83.21
87.09
89.30
We first train a baseline CNN with a symmetric U-net structure, where both the input image and the
output map are 8? smaller than the original image. The detailed settings are in the supplementary
meterial. We apply the multi-objective loss as [20] to improve the accuracy along the boundaries. We
note that the symmetric structure is powerful, since the results we obtained for the baseline CNN
are comparable (see Table. 1) to that of [20], who apply a much larger model (38 MB vs. 12 MB).
We then train a SPN on top of the baseline CNN results on the training set, with patches sampled
from the high-resolution input image and the coarse segmentations masks. For the guidance network,
we use the same structure as that of the baseline segmentation network, except that its upsampling
part ends at a resolution of 64 ? 64, and its output layer has 32 ? 12 = 384 channels. In addition,
we train another face parsing CNN with 1024 ? 1024 sized inputs and outputs (CNN-Highres) for
better comparison. It has three more sub-modules at each end of the baseline network, where all are
configured with 16 channels to process higher resolution images.
We show quantitative and qualitative results in Table. 1 and 3 respectively. We compared the one/three
way connection SPNs with the baseline, the CNN-Highres and the most relevant state-of-the-art
technique for face parsing [20]. Note that the results of baseline and [20]1 are bi-linearly upsampled to
1024 ? 1024 before evaluation. Overall, both SPNs outperform the other techniques with a significant
margin of over 6 intersection-over-union (IoU) points. Especially for the smaller facial components
(e.g., eyes and lips) where with smaller resolution images, the segmentation network performs poorly.
We note that the one-way connection-based SPN is quite successful on relatively simple tasks such
as the HELEN dataset, but fails for more complex tasks, as revealed by the results of Pascal VOC
dataset in the following section.
Pascal VOC Dataset. The PASCAL VOC 2012 segmentation benchmark [6] involves 20 foreground
object classes and one background class. The original dataset contains 1464 training, 1499 validation
and 1456 testing images, with pixel-level annotations. The performance is mainly measured in terms
of pixel IoU averaged across the 21 classes. We train our SPNs on the train split with the coarse
segmentation results produced by the FCN-8s model [21]. The model is fine-tuned on a pre-trained
VGG-16 network, where different levels of features are upsampled and concatenated to obtain the
final, low-resolution segmentation results (8? smaller than the original image size). The guidance
network of the SPN also fine-tunes the VGG-16 structure from the beginning till the pool5 layer
as the downsampling part. Similar to the settings for the HELEN dataset, the upsampling part
has a symmetric structure with skipped links until the feature dimensions of 64 ? 64. The spatial
propagation module has the same configuration as that of the SPN that we employed for the HELEN
dataset. The model is applied on the coarse segmentation maps of the validation and test splits
generated by any image segmentation algorithm without fine-tuning. We test the refinement SPN
on three base models: (a) FCN-8s [21], (b) the atrous spatial pyramid pooling (ASPP-L) network
fine-tuned with VGG-16, denoted as Deeplab VGG, and (c) the ASPP-L: a multi-scale network
fine-tuned with ResNet-101 [11] (pre-trained on the COCO dataset), denoted as Deeplab ResNet-101.
Among them, (b) and (c) are the two basic models
from [5], which are then refined with dense CRF [14]
Table 2:
Quantitative comparison (mean
conditioned on the original image.
IoU) with dense CRF-based refinement [5] on
Table 3 shows that through the three-way SPN, the Deeplab pre-trained models.
accuarcy of segmentation is significantly improved over
CNN +dense CRF
mIoU
VGG
68.97
71.57
the coarse segmentation results for all the three baseline
ResNet 76.40
77.69
models. It has strong capability of generalization and
can successfully refine any coarse maps from different
pre-trained models by a large margin. Different with
1
The original output (also for evaluation) size it 250 ? 250.
8
+SPN
73.12
79.76
Figure 4: Visualization of Pascal VOC segmentation results (left) and object probability (by 1 ? Pb , Pb is the
probability of background). The ?pretrained? column denotes the base Deeplab ResNet-101 model, while the
rest 4 columns show the base model combined with the dense CRF [5] and the proposed SPN, respectively.
Table 3: Quantitative evaluation results on the Pascal VOC dataset. We compare the two connections of SPN
with the corresponding pre-trained models, including: (a) FCN-8s (F), (b) Deeplab VGG (V) and (c) Deeplab
ResNet-101 (R). AC denotes accuracy, ?+? denote added on top of the base model.
F
+1 way +3 way
V
+1 way +3 way
R
+1 way +3 way
Model
overall AC 91.22
90.64
92.90
92.61
92.16
93.83
94.63
94.12
95.49
mean AC
77.61
70.64
79.49
80.97
73.53
83.15
84.16
77.46
86.09
mean IoU
65.51
60.95
69.86
68.97
64.42
73.12
76.46
72.02
79.76
the Helen dataset, the one-way SPN fails to refine the segmentation, which is probably due to
its limited capability of learning preferable affinity with a sparse form, especially when the data
distribution gets more complex. Table 2 shows that by replacing the dense CRF module with the
same refinement model, the performance is boosted by a large margin, without fine-tuning. One the
test split, the DeepNet ResNet-101 based SPN achieves the mean IoU of 80.22, while the dense
CRF gets 79.7. The three-way SPN produces fine visual results, as shown in the red bounding box
of Figure 4. By comparing the probability maps (column 3 versus 7), SPN exhibits fundamental
improvement in object details, boundaries, and semantic integrity.
In addition, we show in table 4 that the same refinement model can also be generalize to dilated
convolution based networks [34]. It significantly improves the quantitative performance on top of the
?Front end? base model, as well as adding a multi-scale refinement module, denoted as ?+Context?.
Specifically, the SPN improves the base model with much larger margin compared to the context
aggregation module (see ?+3 way? vs ?+Context? in table 4).
6
Conclusion
We propose spatial propagation networks for learning pairwise affinities for vision tasks. It is a generic
framework that can be applied to numerous tasks, and in this work we demonstrate its effectiveness
for semantic object segmentation. Experiments on the HELEN face parsing and PASCAL VOC
object semantic segmentation tasks show that the spatial propagation network is general, effective
and efficient for generating high-quality segmentation results.
Table 4: Quantitative evaluation results on the Pascal VOC dataset. We refine the base models proposed with
dilated convolutions [34]. ?+? denotes additions on top of the ?Front end? model.
Model
Front end +3 way
+Context +Context+3 way
overall AC
93.03
93.89
93.44
94.35
mean AC
80.31
83.47
80.97
83.98
mean IoU
69.75
73.14
71.86
75.28
9
Acknowledgement. This work is supported in part by the NSF CAREER Grant #1149783, gifts
from Adobe and NVIDIA.
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6,358 | 6,751 | Linear regression without correspondence
Daniel Hsu
Columbia University
New York, NY
[email protected]
Kevin Shi
Columbia University
New York, NY
[email protected]
Xiaorui Sun
Microsoft Research
Redmond, WA
[email protected]
Abstract
This article considers algorithmic and statistical aspects of linear regression when
the correspondence between the covariates and the responses is unknown. First, a
fully polynomial-time approximation scheme is given for the natural least squares
optimization problem in any constant dimension. Next, in an average-case and
noise-free setting where the responses exactly correspond to a linear function of
i.i.d. draws from a standard multivariate normal distribution, an efficient algorithm
based on lattice basis reduction is shown to exactly recover the unknown linear
function in arbitrary dimension. Finally, lower bounds on the signal-to-noise ratio
are established for approximate recovery of the unknown linear function by any
estimator.
1
Introduction
? 2 Rd from noisy linear measurements
Consider the problem of recovering an unknown vector w
when the correspondence between the measurement vectors and the measurements themselves is
unknown. The measurement vectors (i.e., covariates) from Rd are denoted by x1 , x2 , . . . , xn ; for
each i 2 [n] := {1, 2, . . . , n}, the i-th measurement (i.e., response) yi is obtained using x?? (i) :
? > x?? (i) + "i ,
yi = w
i 2 [n] .
(1)
Above, ?
? is an unknown permutation on [n], and the "1 , "2 , . . . , "n are unknown measurement errors.
This problem, which has been called unlabeled sensing [22], linear regression with an unknown
permutation [18], and linear regression with shuffled labels [1], arises in many settings; see the
aforementioned references for more details. In short, sensing limitations may create ambiguity in
or even completely lose the ordering of measurements. The problem is also interesting because the
missing correspondence makes an otherwise well-understood problem into one with very different
computational and statistical properties.
Prior works. Unnikrishnan et al. [22] study conditions on the measurement vectors that permit
? under noiseless measurements. They show that when the entries of
recovery of any target vector w
the xi are drawn i.i.d. from a continuous distribution, and n 2d, then almost surely, every vector
? 2 Rd is uniquely determined by noiseless correspondence-free measurements as in (1). (Under
w
? can be recovered when an appropriate signal-to-noise ratio
noisy measurements, it is shown that w
tends to infinity.) It is also shown that n 2d is necessary for such a guarantee that holds for all
? 2 Rd .
vectors w
Pananjady et al. [18] study statistical and computational limits on recovering the unknown permutation
?
? . On the statistical front, they consider necessary and sufficient conditions on the signal-to-noise ratio
? 22 / 2 when the measurement errors ("i )ni=1 are i.i.d. draws from the normal distribution
SNR := kwk
N(0, 2 ) and the measurement vectors (xi )ni=1 are i.i.d. draws from the standard multivariate normal
distribution N(0, I d ). Roughly speaking, exact recovery of ?
? is possible via maximum likelihood
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
when SNR nc for some absolute constant c > 0, and approximate recovery is impossible for any
0
method when SNR ? nc for some other absolute constant c0 > 0. On the computational front, they
show that the least squares problem (which is equivalent to maximum likelihood problem)
n ?
?2
X
min
w> x?(i) yi
(2)
w,?
i=1
given arbitrary x1 , x2 , . . . , xn 2 R and y1 , y2 , . . . , yn 2 R is NP-hard when d = ?(n)1 , but admits
a polynomial-time algorithm (in fact, an O(n log n)-time algorithm based on sorting) when d = 1.
d
? in
Abid et al. [1] observe that the maximum likelihood estimator can be inconsistent for estimating w
certain settings (including the normal setting of Pananjady et al. [18], with SNR fixed but n ! 1).
One of the alternative estimators they suggest is consistent under additional assumptions in dimension
d = 1. Elhami et al. [8] give a O(dnd+1 )-time algorithm that, in dimension d = 2, is guaranteed to
? when the measurement vectors are chosen in a very particular way from the
approximately recover w
unit circle and the measurement errors are uniformly bounded.
Contributions. We make progress on both computational and statistical aspects of the problem.
1. We give an approximation algorithm for the least squares problem from (2) that, any given
(xi )ni=1 , (yi )ni=1 , and ? 2 (0, 1), returns a solution with objective value at most 1 + ? times that
of the minimum in time (n/?)O(d) . This a fully polynomial-time approximation scheme for any
constant dimension.
? in the measurement model from (1), under the
2. We give an algorithm that exactly recovers w
assumption that there are no measurement errors and the covariates (xi )ni=1 are i.i.d. draws
from N(0, I d ). The algorithm, which is based on a reduction to a lattice problem and employs
the lattice basis reduction algorithm of Lenstra et al. [16], runs in poly(n, d) time when the
? are appropriately quantized. This result may
covariate vectors (xi )ni=1 and target vector w
? which
also be regarded as for each-type guarantee for exactly recovering a fixed vector w,
complements the for all-type results of Unnikrishnan et al. [22] concerning the number of
measurement vectors needed for recovering all possible vectors.
3. We show that in the measurement model from (1) where the measurement errors are i.i.d. draws
from N(0, 2 ) and the covariate vectors are i.i.d. draws from N(0, I d ), then no algorithm can
? unless SNR C min {1, d/ log log(n)} for some absolute constant
approximately recover w
C > 0. We also show that when the covariate vectors are i.i.d. draws from the uniform
distribution on [ 1/2, 1/2]d , then approximate recovery is impossible unless SNR C 0 for
some other absolute constant C 0 > 0.
Our algorithms are not meant for practical deployment, but instead are intended to shed light on the
computational difficulty of the least squares problem and the average-case recovery problem. Indeed,
note that a na?ve brute-force search over permutations requires time ?(n!) = n?(n) , and the only
other previous algorithms (already discussed above) were restricted to d = 1 [18] or only had some
form of approximation guarantee when d = 2 [8]. We are not aware of previous algorithms for the
average-case problem in general dimension d.2
Our lower bounds on SNR stand in contrast to what is achievable in the classical linear regression
model (where the covariate/response correspondence is known): in that model, the SNR requirement
? scales as d/n, and hence the problem becomes easier with n. The
for approximately recovering w
lack of correspondence thus drastically changes the difficulty of the problem.
2
Approximation algorithm for the least squares problem
In this section, we consider the least squares problem from Equation (2). The inputs are an arbitrary
matrix X = [x1 |x2 | ? ? ? |xn ]> 2 Rn?d and an arbitrary vector y = (y1 , y2 , . . . , yn )> 2 Rn , and the
1
Pananjady et al. [18] prove that PARTITION reduces to the problem of deciding if the optimal value of (2) is
zero or non-zero. Note that PARTITION is weakly, but not strongly, NP-hard: it admits a pseudo-polynomial-time
algorithm [10, Section 4.2]. In Appendix A, we prove that the least squares problem is strongly NP-hard by
reduction from 3-PARTITION (which is strongly NP-complete [10, Section 4.2.2]).
2
A recent algorithm of Pananjady et al. [19] exploits a similar average-case setting but only for a somewhat
easier variant of the problem where more information about the unknown correspondence is provided.
2
Algorithm 1 Approximation algorithm for least squares problem
input Covariate matrix X = [x1 |x2 | ? ? ? |xn ]> 2 Rn?k ; response vector y = (y1 , y2 , . . . , yn )> 2
Rn ; approximation parameter ? 2 (0, 1).
assume X > X = I k .
? 2 Pn .
? 2 Rk and permutation matrix ?
output Weight vector w
1: Run ?Row Sampling? algorithm with input matrix X to obtain a matrix S 2 Rr?n with r = 4k.
2: Let B be the set of vectors b = (b1 , b2 , . . . , bn )> 2 Rn satisfying the following: for each i 2 [n],
? if the i-th column of S is all zeros, then bi = 0;
? otherwise, bi 2 {y1 , y2 , . . . , yn }.
p
3: Let c := 1 + 4(1 + n/(4k))2 .
4: for each b 2 B do
2
2
?p
? b ?> yk2 .
5:
Compute w
b)k2 , and let rb := min?2Pn kX w
b 2 arg minw2Rk kS(Xw
p
? b , so that for each
6:
Construct a ?rb /c-net Nb for the Euclidean ball of radius crb around w
p
p
? b k2 ? crb , there exists v 0 2 Nb such that kv v 0 k2 ? ?rb /c.
v 2 Rk with kv w
7: end for
2
2
? 2 arg min kX w
? 2 arg
? ?> yk2 .
8: return w
min min kXw ?> yk2 and ?
S
w2
b2B
Nb ?2Pn
?2Pn
goal is to find a vector w 2 Rd and permutation matrix ? 2 Pn (where Pn denotes the space of n?n
2
permutation matrices3 ) to minimize kXw ?> yk2 . This problem is NP-hard in the case where
d = ?(n) [18] (see also Appendix A). We give an approximation scheme that, for any ? 2 (0, 1),
returns a (1 + ?)-approximation in time (n/?)O(k) + poly(n, d), where k := rank(X) ? min{n, d}.
We assume without loss of generality that X 2 Rn?k and X > X = I k . This is because we can
always replace X with its matrix of left singular vectors U 2 Rn?k , obtained via singular value
decomposition X = U ?V > , where U > U = V > V = I k and ?
0 is diagonal. A solution
(w, ?) for (U , y) has the same cost as the solution (V ? 1 w, ?) for (X, y), and a solution (w, ?)
for (X, y) has the same cost as the solution (?V > w, ?) for (U , y).
2.1
Algorithm
Our approximation algorithm, shown as Algorithm 1, uses a careful enumeration to beat the na?ve
brute-force running time of ?(|Pn |) = ?(n!). It uses as a subroutine a ?Row Sampling? algorithm
of Boutsidis et al. [5] (described in Appendix B), which has the following property.
Theorem 1 (Specialization of Theorem 12 in [5]). There is an algorithm (?Row Sampling?) that,
given any matrix A 2 Rn?k with n k, returns in poly(n, k) time a matrix S 2 Rr?n with r = 4k
such that the following hold.
1. Every row of S has at most one non-zero entry.
2
2. For every b 2 Rn , every w0 2 arg minp
b)k2 satisfies kAw0
w2Rk kS(Aw
2
minw2Rk kAw bk2 for c = 1 + 4(1 + n/(4k))2 = O(n/k).
2
bk2 ? c ?
The matrix S returned by Row Sampling determines a (weighted) subset of O(k) rows of A such
that solving a (ordinary) least squares problem (with any right-hand side b) on this subset of rows and
corresponding right-hand side entries yields a O(n/k)-approximation to the least squares problem
over all rows and right-hand side entries. Row Sampling does not directly apply to our problem
because (1) it does not minimize over permutations of the right-hand side, and (2) the approximation
factor is too large. However, we are able to use it to narrow the search space in our problem.
An alternative to Row Sampling is to simply enumerate all subsets of k rows of X. This is justified
by a recent result of Derezi?nski and Warmuth [7], which shows that for any right-hand side b 2 Rn ,
k?k
using ?volume sampling? [3] to choose a matrix S 2 {0, 1}
(where each row has one non-zero
entry) gives a similar guarantee as that of Row Sampling, except with the O(n/k) factor replaced by
k + 1 in expectation.
3
Each permutation matrix ? 2 Pn corresponds to a permutation ? on [n]; the (i, j)-th entry of ? is one if
?(i) = j and is zero otherwise.
3
2.2
Analysis
The approximation guarantee of Algorithm 1 is given in the following theorem.
? 2 Pn satisfying
? 2 Rk and ?
Theorem 2. Algorithm 1 returns w
? >y
?
?
Xw
2
2
? (1 + ?)
min
w2Rk ,?2Pn
?> y
Xw
2
2
.
2
Proof. Let opt := minw,? kXw ?> yk2 be the optimal cost, and let (w? , ?? ) denote a solution
achieving this cost. The optimality implies that w? satisfies the normal equations X > Xw? =
>
X > ?>
? y. Observe that there exists a vector b? 2 B satisfying Sb? = S?? y. By Theorem 1 and
? b? and cost value rb? satisfy
the normal equations, the vector w
? b?
opt ? rb? ? X w
?>
?y
2
2
? b?
= X(w
? b?
Moreover, since X > X = I k , we have that kw
tion of Nb? , there exists w 2 Nb? satisfying kw
this w, the normal equations imply
min kXw
?2Pn
2
?> yk2 ? kXw
2
w? )
p
w? k2 ? (c
2
w? k2 = kX(w
?>
? yk2 = kX(w
2
2
+ opt ? c ? opt .
p
1) opt ? crb? . By construc2
w? )k2 ? ?rb? /c ? ? opt. For
2
w? )k2 + opt ? (1 + ?) opt .
Therefore, the solution returned by Algorithm 1 has cost no more than (1 + ?) opt.
By the results of Pananjady et al. [18] for maximum likelihood estimation, our algorithm enjoys recov? and ?
ery guarantees for w
? when the data come from the Gaussian measurement model (1). However,
the approximation guarantee also holds for worst-case inputs without generative assumptions.
Running time. We now consider the running time of Algorithm 1. There is the initial cost for
singular value decomposition (as discussed at the beginning of the section), and also for ?Row
Sampling?; both of these take poly(n, d) time. For the rest of the algorithm, we need to consider the
size of B and the size of the net Nb for each b 2 B. First, we have |B| ? nr = nO(k) , since S has
only 4k rows and each
entry. Next, for each b 2 B, we construct the
p row has at most a single non-zero
p
p
-net Nb (for := ?rb /c) by constructing a / k-net for the `1 -ball of radius crb centered at
? b (using an appropriate axis-aligned grid). This has size |Nb | ? (4c2 k/?)k/2 = (n/?)O(k) . Finally,
w
each arg minw2Rk computation takes O(nk 2 ) time, and each (arg) min?2Pn takes O(nk + n log n)
time [18] (also see Appendix B). So, the overall running time is (n/?)O(k) + poly(n, d).
3
Exact recovery algorithm in noiseless Gaussian setting
To counter the intractability of the least squares problem in (2) confronted in Section 2, it is natural
to explore distributional assumptions that may lead to faster algorithms. In this section, we consider
the noiseless measurement model where the (xi )ni=1 are i.i.d. draws from N(0, I d ) (as in [18]). We
? with high probability when n d + 1. The algorithm runs
give an algorithm that exactly recovers w
? are appropriately quantized.
in poly(n, d)-time when (xi )ni=1 and w
It will be notationally simpler to consider n + 1 covariate vectors and responses
? > x?? (i) ,
yi = w
i = 0, 1, . . . , n .
(3)
Here, (xi )ni=0 are n + 1 i.i.d. draws from N(0, I d ), the unknown permutation ?
? is over {0, 1, . . . , n},
and the requirement of at least d + 1 measurements is expressed as n d.
In fact, we shall consider a variant of the problem in which we are given one of the values of the
unknown permutation ?
? . Without loss of generality, assume we are given that ?
? (0) = 0. Solving this
variant of the problem suffices because there are only n + 1 possible values of ?
? (0): we can try them
all, incurring just a factor n + 1 in the computation time. So henceforth, we just consider ?
? as an
unknown permutation on [n].
4
Algorithm 2 Find permutation
input Covariate vectors x0 , x1 , x2 , . . . , xn in Rd ; response values y0 , y1 , y2 , . . . , yn in R; confidence parameter 2 (0, 1); lattice parameter > 0.
? 2 Rd and permutation ?
? > x?? (i) for each i 2 [n], and
assume there exists w
? on [n] such that yi = w
>
? x0 .
that y0 = w
output Permutation ?
? on [n] or failure.
1: Let X = [x1 |x2 | ? ? ? |xn ]> 2 Rn?d , and its pseudoinverse be X ? = [?
x1 |?
x2 | ? ? ? |?
xn ].
?>
2: Create Subset Sum instance with n2 source numbers ci,j := yi x
j x0 for (i, j) 2 [n] ? [n] and
target sum y0 .
3: Run Algorithm 3 with Subset Sum instance and lattice parameter .
4: if Algorithm 3 returns a solution S ? [n] ? [n] then
5:
return any permutation ?
? on [n] such that ?
? (i) = j implies (i, j) 2 S.
6: else
7:
return failure.
8: end if
Algorithm 3 Lagarias and Odlyzko [12] subset sum algorithm
input Source numbers {ci }i2I ? R; target sum t 2 R; lattice parameter > 0.
output Subset S? ? I or failure.
1: Construct lattice basis B 2 R(|I|+2)?(|I|+1) where
"
#
I
|I|+1
B :=
2 R(|I|+2)?(|I|+1) .
t
ci : i 2 I
2: Run basis reduction [e.g., 16] to find non-zero lattice vector v of length at most 2|I|/2 ? 1 (B).
I
3: if v = z(1, >?, 0)> , with z 2 Z and S? 2 {0, 1} is characteristic vector for some S? ? I then
S
?
4:
return S.
5: else
6:
return failure.
7: end if
3.1
Algorithm
Our algorithm, shown as Algorithm 2, is based on a reduction to the Subset Sum problem. An
instance of Subset Sum is specified by an unordered collection ofPsource numbers {ci }i2I ? R, and
a target sum t 2 R. The goal is to find a subset S ? I such that i2S ci = t. Although Subset Sum
is NP-hard in the worst case, it is tractable for certain structured instances [12, 9]. We prove that
Algorithm 2 constructs such an instance with high probability. A similar algorithm based on such a
reduction was recently used by Andoni et al. [2] for a different but related problem.
Algorithm 2 proceeds by (i) solving a Subset Sum instance based on the covariate vectors and
response values (using Algorithm 3), and (ii) constructing a permutation ?
? on [n] based on the
solution to the Subset Sum instance. With the permutation ?
? in hand, we (try to) find a solution
w 2 Rd to the system of linear equations yi = w> x?? (i) for i 2 [n]. If ?
?=?
? , then there is a unique
such solution almost surely.
3.2
Analysis
The following theorem is the main recovery guarantee for Algorithm 2.
Theorem 3. Pick any 2 (0, 1). Suppose (xi )ni=0 are i.i.d. draws from N(0, I d ), and (y0 )ni=1
? 2 Rd and permutation ?
follow the noiseless measurement model from (3) for some w
? on [n] (and
?
? (0) = 0), and that n d. Furthermore, suppose Algorithm 2 is run with inputs (xi )ni=0 , (yi )ni=0 , ,
2
and , and also that
2n /" where " is defined in Equation (8). With probability at least 1
,
Algorithm 2 returns ?
?=?
?.
? 2 , and Algorithm 2
Remark 1. The value of " from Equation (8) is directly proportional to kwk
requires a lower bound on " (in the setting of the lattice parameter ). Hence, it suffices to determine
5
? 2 . Such a bound can be obtained from the measurement
a lower bound on kwk
pPn values: a standard
2
tail bound (Lemma 6 in Appendix C) shows that with high probability,
i=1 yi /(2n) is a lower
? 2 , and is within a constant factor of it as well.
bound on kwk
Remark 2. Algorithm 2 strongly exploits the assumption of noiseless measurements, which is
expected given the SNR lower bounds of Pananjady et al. [18] for recovering ?
? . The algorithm,
however, is also very brittle and very likely fails in the presence of noise.
Remark 3. The recovery result does not contradict the results of Unnikrishnan et al. [22], which
? even in the
show that a collection of 2d measurement vectors are necessary for recovering all w,
? 2 Rd , with high
noiseless measurement model of (3). Indeed, our result shows that for a fixed w
? but this
probability d + 1 measurements in the model of (3) suffice to permit exactly recovery of w,
? 0.
same set of measurement vectors (when d + 1 < 2d) will fail for some other w
The proof of Theorem 3 is based on the following theorem?essentially due to Lagarias and Odlyzko
[12] and Frieze [9]?concerning certain structured instances of Subset Sum that can be solved using
the lattice basis reduction algorithm of Lenstra et al. [16]. Given a basis B = [b1 |b2 | ? ? ? |bk ] 2 Rm?k
for a lattice
8
9
k
<X
=
L(B) :=
zi bi : z1 , z2 , . . . , zk 2 Z ? Rm ,
:
;
i=1
this algorithm can be used to find a non-zero vector v 2 L(B) \ {0} whose length is at most 2(k
times that of the shortest non-zero vector in the lattice
min kvk2 .
1 (B) :=
1)/2
v2L(B)\{0}
Theorem 4 ([12, 9]). Suppose the Subset Sum instance specified by source numbers {ci }i2I ? R
and target sum t 2 R satisfy the following properties.
P
1. There is a subset S ? ? I such that i2S ? ci = t.
p
P
2. Define R := 2|I|/2 |S ? | + 1 and ZRP:= {(z0 , z) 2 Z ? ZI : 0 < z02 + i2I zi2 ? R2 }.
There exists " > 0 such that |z0 ? t
" for each (z0 , z) 2 ZR that is not
i2I zi ? ci |
I
?
?
an integer multiple of (1, ), where
2 {0, 1} is the characteristic vector for S ? .
Let B be the lattice basis B constructed by Algorithm 3, and assume
2|I|/2 /". Then every
|I|/2
non-zero vector in the lattice ?(B) with length at most 2
times the length of the shortest non-zero
vector in ?(B) is an integer multiple of the vector (1, S ? , 0), and the basis reduction algorithm
of Lenstra et al. [16] returns such a non-zero vector.
The Subset Sum instance constructed in Algorithm 2 has n2 source numbers {ci,j : (i, j) 2 [n] ? [n]}
and target sum y0 . We need to show that it satisfies the two conditions of Theorem 4.
? = (?
? i,j )(i,j)2[n]?[n] 2 Pn be the permutation
Let S?? := {(i, j) : ?
? (i) = j} ? [n] ? [n], and let ?
? is the ?characteristic vector?
? i,j := 1{?
matrix with ?
? (i) = j} for all (i, j) 2 [n] ? [n]. Note that ?
p
2
n
/2
for S?? . Define R := 2
n + 1 and
(
)
X
n?n
2
2
2
: 0 < z0 +
ZR := (z0 , Z) 2 Z ? Z
Zi,j ? R
.
1?i,j?n
O(n4 )
A crude bound shows that|ZR | ? 2
.
The following lemma establishes the first required property in Theorem 4.
Lemma
1. The random matrix X has rank d almost surely, and the subset S?? satisfies y0 =
P
ci,j .
(i,j)2S?
?
Proof. That X has rank d almost surely follows fromPthe fact that the probability density of X is
n
? j x>
supported on all of Rn?d . This implies that X ? X = j=1 x
j = I d , and
y0 =
n
X
j=1
? j x>
? =
x>
0x
jw
X
1?i,j?n
? j ? yi ? 1{?
x>
? (i) = j} =
0x
6
X
1?i,j?n
ci,j ? 1{?
? (i) = j} .
The next lemma establishes the second required property in Theorem 4. Here, we use the fact that
? Z
the Frobenius norm z0 ?
is at least one whenever (z0 , Z) 2 Z ? Zn?n is not an integer
F
?
multiple of (1, ?).
Lemma 2. Pick any ?, ? 0 > 0 such that 3|ZR | ? + ? 0 < 1. With probability at least 1 3|ZR | ? ? 0 ,
every (z0 , Z) 2 ZR with Z = (Zi,j )(i,j)2[n]?[n] satisfies
p
1
X
(?/4) ? (d 1)/n ? ? 2+ d 1
? Z
? 2.
z0 ? y 0
Zi,j ? ci,j
?kwk
?p
? 2 ? z0 ?
p
p
F
i,j
n + d + 2 ln(1/? 0 )
? satisfies y0 = P ?
?
Proof. By Lemma 1, the matrix ?
i,j i,j ? ci,j . Fix any (z0 , Z) 2 ZR with
Z = (Zi,j )(i,j)2[n]?[n] . Then
X
X
? i,j Zi,j ) ? x>
?j ? w
? > x?? (i) .
z0 ? y 0
Zi,j ? ci,j =
(z0 ? ?
0x
i,j
i,j
?
?
Using matrix and vector notations, this can be written compactly as the inner product x>
0 (X (z0 ?
> ?
? Since x0 ? N(0, I d ) and is independent of X, the distribution of the inner product is
Z) ?X w).
? Z)> ?X
? wk
? 2 . By Lemma 7 (in
normal with mean zero and standard deviation equal to kX ? (z0 ?
Appendix C), with probability at least 1 ?,
r
?
?
?
> ?
> ?
?
?
?
?
x>
X
(z
?
Z)
?X
w
kX
(z
?
Z)
?X
wk
?
??.
(4)
0
0
0
2
2
Observe that X ? = (X > X)
?
kX ? (z0 ?
1
X > since X has rank d by Lemma 1, so
?
kX > (z0 ?
? wk
? 2
Z)> ?X
? wk
? 2
Z)> ?X
2
kXk2
(5)
.
By Lemma 4 (in Appendix C), with probability at least 1 ? 0 ,
?p
?2
p
p
2
kXk2 ?
n + d + 2 ln(1/? 0 ) .
And by Lemma 9 (in Appendix C), with probability at least 1
?
kX > (z0 ?
? wk
? 2
Z)> ?X
?
(z0 ?
?
Z)> ?
F
2?,
r
? 2?
?kwk
(d
(6)
1)? 1+1/(d
??
8n
1)
.
(7)
? is orthogonal, we have that k(z0 ?
? Z)> ?k
?
? Zk . Combining this with (4),
Since ?
F = kz0 ?
F
(5), (6), and (7), and union bounds over all (z0 , Z) 2 ZR proves the claim.
Proof of Theorem 3. Lemma 1 and Lemma 2 (with ? 0 := /2 and ? := /(6|ZR |)) together imply
that with probability at least 1
, the source numbers {ci,j : (i, j) 2 [n] ? [n]} and target sum y0
satisfy the conditions of Theorem 4 with
S ? := {(i, j) 2 [n] ? [n] : ?
? (i) = j} ,
p
1
(?/4) ? (d 1)/n ? ( /(6|ZR |))2+ d 1
:=
? 2
"
?kwk
?p
?2
p
p
n + d + 2 ln(2/ )
? 2 . (8)
2 poly(n, log(1/ )) ?kwk
2
Thus, in this event, Algorithm 3 (with satisfying
2n /2 /") returns S? = S ? , which uniquely
determines the permutation ?
?=?
? returned by Algorithm 2.
Running time. The basis reduction algorithm of Lenstra et al. [16] is iterative, with each iteration
primarily consisting of Gram-Schmidt orthogonalization and another efficient linear algebraic process
called ?size reduction?. The total number of iterations required is
0
!1
p maxi2[k] kbi k2
k(k
+
1)
A.
O@
log
k?
2
1 (B)
7
p
In our case, k = n2 and 1 (B) = n + 1; and by Lemma 10 (in Appendix C), each of the basis
? 22 . Using the tight
vectors constructed has squared length at most 1 + 2 ? poly(d, log(n), 1/ ) ?kwk
setting of required in Theorem 3, this gives a poly(n, d, log(1/ )) bound on the total number of
iterations as well as on the total running time.
However, the basis reduction algorithm requires both arithmetic and rounding operations, which are
typically only available for finite precision rational inputs. Therefore, a formal running time analysis
? to be
would require the idealized real-valued covariate vectors (xi )ni=0 and unknown target vector w
quantized to finite precision values. This is doable, and is similar to using a discretized Gaussian
? is a vector of finite precision
distribution for the distribution of the covariate vectors (and assuming w
values), but leads to a messier analysis incomparable to the setup of previous works. Nevertheless, it
would be desirable to find a different algorithm that avoids lattice basis reduction that still works with
just d + 1 measurements.
4
Lower bounds on signal-to-noise for approximate recovery
In this section, we consider the measurement model from (1) where (xi )ni=1 are i.i.d. draws from either
N(0, I d ) or the uniform distribution on [ 1/2, 1/2]d , and ("i )ni=1 are i.i.d. draws from N(0, 2 ). We
establish lower bounds on the signal-to-noise ratio (SNR),
2
SNR =
? 2
kwk
2
,
n
n
? = w((x
?
? to approximately recover w
? in expectation.
required by any estimator w
i )i=1 , (yi )i=1 ) for w
? 2 and 2 .
The estimators may have a priori knowledge of the values ofkwk
Theorem 5. Assume ("i )ni=1 are i.i.d. draws from N(0,
2
).
1. There is an absolute constant C > 0 such that the following holds. If n 3, d 22, (xi )ni=1
are i.i.d. draws from N(0, I d ), (yi )ni=1 follow the measurement model from (1), and
?
d
SNR ? C ? min
,1 ,
log log(n)
? there exists some w
? 2 Rd such that
then for any estimator w,
?
?
E kw
? 2
wk
?
1
? 2.
kwk
24
2. If (xi )ni=1 are i.i.d. draws from the uniform distribution on [ 1/2, 1/2]d , and (yi )ni=1 follow the
measurement model from (1), and
SNR ? 2 ,
? there exists some w
? 2 Rd such that
then for any estimator w,
?
?
?
?
1
1
? wk
? 2
? 2.
E kw
1 p kwk
2
2
? > xi + "i for i 2 [n], the maximum
Note that in the classical linear regression model where p
yi = w
? mle satisfies Ekw
? mle wk
? 2?C
likelihood estimator w
d/n, where C > 0 is an absolute constant.
? up to (say) Euclidean distancekwk
? 2 /24
Therefore, the SNR requirement to approximately recover w
is SNR 242 Cd/n. Compared to this setting, Theorem 5 implies that with the measurement model
of (1), the SNR requirement (as a function of n) is at substantially higher (d/ log log(n) in the normal
covariate case, or a constant not even decreasing with n in the uniform covariate case).
p
For the normal covariate case, Pananjady et al. [18] show that if n > d, ? < n, and
SNR
nc? n
n
d +?
,
? mle , ?
then the maximum likelihood estimator (w
?mle ) (i.e., any minimizer of (2)) satisfies ?
?mle = ?
?
with probability at least 1 c0 n 2? . (Here, c > 0 and c0 > 0 are absolute
constants.) It is
p
? mle wk
? 2?C
straightforward to see that, on the same event, we havekw
d/n for some absolute
8
constant C > 0. Therefore, the necessary and sufficient conditions on SNR for approximate recovery
00
? lie between C 0 d/ log log(n) and nC (for absolute constants C 0 , C 00 > 0). Narrowing this
of w
range remains an interesting open problem.
A sketch of the proof in the normal covariate case is as follows. Without
p loss of generality, we restrict
? is a unit vector. We construct a 1/ 2-packing of the unit sphere in
attention to the case where w
? will be chosen from from this set. Observe that for any distinct u, u0 2 U , each
Rd ; the target w
>
n
0 n
of (xi u)i=1 and (x>
i u )i=1 is an i.i.d. sample from N(0, 1) of size n; we prove that they therefore
determine empirical distributions that are close to each other in Wasserstein-2 distance with high
probability. We then prove that conditional on this event, the resulting distributions of (yi )ni=1 under
? = u and x
? = u0 (for any pair u, u0 2 U ) are close in Kullback-Leibler divergence. Hence, by (a
x
generalization of) Fano?s inequality [see, e.g., 11], no estimator can determine the correct u 2 U
with high probability.
The proof for the uniform case is similar, using U = {e1 , e1 } where e1 = (1, 0, . . . , 0)> . The full
proof of Theorem 5 is given in Appendix D.
Acknowledgments
We are grateful to Ashwin Pananjady, Micha? Derezi?nski, and Manfred Warmuth for helpful discussions. DH was supported in part by NSF awards DMR-1534910 and IIS-1563785, a Bloomberg Data
Science Research Grant, and a Sloan Research Fellowship. XS was supported in part by a grant from
the Simons Foundation (#320173 to Xiaorui Sun). This work was done in part while DH and KS were
research visitors and XS was a research fellow at the Simons Institute for the Theory of Computing.
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6,359 | 6,752 | NeuralFDR: Learning Discovery Thresholds
from Hypothesis Features
Fei Xia? ,
Martin J. Zhang?, James Zou? , David Tse?
Stanford University
{feixia,jinye,jamesz,dntse}@stanford.edu
Abstract
As datasets grow richer, an important challenge is to leverage the full features
in the data to maximize the number of useful discoveries while controlling for
false positives. We address this problem in the context of multiple hypotheses
testing, where for each hypothesis, we observe a p-value along with a set of
features specific to that hypothesis. For example, in genetic association studies,
each hypothesis tests the correlation between a variant and the trait. We have a
rich set of features for each variant (e.g. its location, conservation, epigenetics
etc.) which could inform how likely the variant is to have a true association.
However popular testing approaches, such as Benjamini-Hochberg?s procedure
(BH) and independent hypothesis weighting (IHW), either ignore these features or
assume that the features are categorical. We propose a new algorithm, NeuralFDR,
which automatically learns a discovery threshold as a function of all the hypothesis
features. We parametrize the discovery threshold as a neural network, which
enables flexible handling of multi-dimensional discrete and continuous features
as well as efficient end-to-end optimization. We prove that NeuralFDR has strong
false discovery rate (FDR) guarantees, and show that it makes substantially more
discoveries in synthetic and real datasets. Moreover, we demonstrate that the
learned discovery threshold is directly interpretable.
1
Introduction
In modern data science, the analyst is often swarmed with a large number of hypotheses ? e.g. is a
mutation associated with a certain trait or is this ad effective for that section of the users. Deciding
which hypothesis to statistically accept or reject is a ubiquitous task. In standard multiple hypothesis
testing, each hypothesis is boiled down to one number, a p-value computed against some null
distribution, with a smaller value indicating less likely to be null. We have powerful procedures to
systematically reject hypotheses while controlling the false discovery rate (FDR) Note that here the
convention is that a ?discovery? corresponds to a ?rejected? null hypothesis.
These FDR procedures are widely used but they ignore additional information that is often available
in modern applications. Each hypothesis, in addition to the p-value, could also contain a set of
features pertinent to the objects being tested in the hypothesis. In the genetic association setting
above, each hypothesis tests whether a mutation is correlated with the trait and we have a p-value
for this. Moreover, we also have other features about both the mutation (e.g. its location, epigenetic
status, conservation etc.) and the trait (e.g. if the trait is gene expression then we have features on the
gene). Together these form a feature representation of the hypothesis. This feature vector is ignored
by the standard multiple hypotheses testing procedures.
In this paper, we present a flexible method using neural networks to learn a nonlinear mapping
from hypothesis features to a discovery threshold. Popular procedures for multiple hypotheses
?
?
These authors contributed equally to this work and are listed in alphabetical order.
These authors contributed equally.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Discovery
Threshold
Input
H1
H2
H3
H4
p1
p2
p3
p4
x1
x2
x3
x4
discovery
H1
H2
H3
H4
End-to-end learning of the
neural network t(x; ?)
Covariate X
true
alternative
yes
yes
no
yes
true
true
false
false
FDP = 1/3
Figure 1: NeuralFDR: an end-to-end learning procedure.
testing correspond to having one constant threshold for all the hypotheses (BH [2]), or a constant
for each group of hypotheses (group BH [12], IHW [13]). Our algorithm takes account of all the
features to automatically learn different thresholds for different hypotheses. Our deep learning
architecture enables efficient optimization and gracefully handles both continuous and discrete multidimensional hypothesis features. Our theoretical analysis shows that we can control false discovery
proportion (FDP) with high probability. We provide extensive simulation on synthetic and real
datasets to demonstrate that our algorithm makes more discoveries while controlling FDR compared
to state-of-the-art methods.
Contribution. As shown in Fig. 1, we provide NeuralFDR, a practical end-to-end algorithm
to the multiple hypotheses testing problem where the hypothesis features can be continuous and
multi-dimensional. In contrast, the currently widely-used algorithms either ignore the hypothesis
features (BH [2], Storey?s BH [17]) or are designed for simple discrete features (group BH [12],
IHW [13]). Our algorithm has several innovative features. We learn a multi-layer perceptron as
the discovery threshold and use a mirroring technique to robustly estimate false discoveries. We
show that NeuralFDR controls false discovery with high probability for independent hypotheses
and asymptotically under weak dependence [17, 12], and we demonstrate on both synthetic and real
datasets that it controls FDR while making substantially more discoveries. Another advantage of
our end-to-end approach is that the learned discovery threshold are directly interpretable. We will
illustrate in Sec. 4 how the threshold conveys biological insights.
Related works. Holm [11] investigated the use of p-value weights, where a larger weight suggests
that the hypothesis is more likely to be an alternative. Benjamini and Hochberg [3] considered
assigning different losses to different hypotheses according to their importance. Some more recent
works are [9, 8, 12]. In these works, the features are assumed to have some specific forms, either
prespecified weights for each hypothesis or the grouping information. The more general formulation
considered in this paper was purposed quite recently [13, 15, 14]. It assumes that for each hypothesis,
we observe not only a p-value Pi but also a feature Xi lying in some generic space X . The feature
is meant to capture some side information that might bear on the likelihood of a hypothesis to be
significant, or on the power of Pi under the alternative, but the nature of this relationship is not fully
known ahead of time and must be learned from the data.
The recent work most relevant to ours is IHW [13]. In IHW, the data is grouped into G groups based
on the features and the decision threshold is a constant for each group. IHW is similar to NeuralFDR
in that both methods optimize the parameters of the decision rule to increase the number of discoveries
while using cross validation for asymptotic FDR control. IHW has several limitations: first, binning
the data into G groups can be difficult if the feature space X is multi-dimensional; second, the
decision rule, restricted to be a constant for each group, is artificial for continuous features; and third,
the asymptotic FDR control guarantee requires the number of groups going to infinity, which can
be unrealistic. In contrast, NeuralFDR uses a neural network to parametrize the decision rule which
is much more general and fits the continuous features. As demonstrated in the empirical results, it
works well with multi-dimensional features. In addition to asymptotic FDR control, NeuralFDR also
has high-probability false discovery proportion control guarantee with a finite number of hypotheses.
SABHA [15] and AdaPT [14] are two recent FDR control frameworks that allow flexible methods to
explore the data and compute the feature dependent decision rules. To compute the decision rule,
SABHA estimates the null proportion using non-parametric methods while AdaPT estimates the
joint density of the p-value and the features with spline regression. Both methods have a similar
2
limitation to IHW in that the estimation becomes very hard for multi-dimensional features. This issue
is addressed in [4], where the null proportion is modeled as a linear combination of some hand-crafted
transformation of the features. NeuralFDR models this relation in a more flexible way.
2
Preliminaries
We have n hypotheses and each hypothesis i is characterized by a tuple (Pi , Xi , Hi ), where Pi 2
(0, 1) is the p-value, Xi 2 X is the hypothesis feature, and Hi 2 {0, 1} indicates if this hypothesis
is null ( Hi = 0) or alternative ( Hi = 1). The p-value Pi represents the probability of observing
an equally or more extreme value compared to the testing statistic when the hypothesis is null, and
is calculated based on some data different from Xi . The alternate hypotheses (Hi = 1) are the
true signals that we would like to discover. A smaller p-value presents stronger evidence for a
hypothesis to be alternative. In practice, we observe Pi and Xi but do not know Hi . We define
the null proportion ?0 (x) to be the probability that the hypothesis is null conditional on the feature
Xi = x. The standard assumption is that under the null (Hi = 0), the p-value is uniformly distributed
in (0, 1). Under the alternative (Hi = 1), we denote the p-value distribution by f1 (p|x). In most
applications, the p-values under the alternative are systematically smaller than those under the null. A
detailed discussion of the assumptions can be found in Sec. 5.
The general goal of multiple hypotheses testing is to claim a maximum number of discoveries based
on the observations {(Pi , Xi )}ni=1 while controlling the false positives. The most popular quantities
that conceptualize the false positives are the family-wise error rate (FWER) [7] and the false discovery
rate (FDR) [2]. We specifically consider FDR in this paper. FDR is the expected proportion of false
discoveries, and one closely related quantity, the false discovery proportion (FDP), is the actual
proportion of false discoveries. We note that FDP is the actual realization of FDR. Formally,
Definition 1. (FDP and FDR) For any decision rule t, let D(t) and F D(t) be the number of
discoveries and the number of false discoveries. The false discovery proportion F DP (t) and the
false discovery rate F DR(t) are defined as F DP (t) , F D(t)/D(t) and F DR(t) , E[F DP (t)].
In this paper, we aim to maximize D(t) while controlling F DP (t) ? ? with high probability. This
is a stronger statement than those in FDR control literature of controlling FDR under the level ?.
Motivating example. Consider a genetic association study where the genotype and phenotype (e.g.
height) are measured in a population. Hypothesis i corresponds to testing the correlation between the
variant i and the individual?s height. The null hypothesis is that there is no correlation, and Pi is the
probability of observing equally or more extreme values than the empirically observed correlation
conditional on the hypothesis is null Hi = 0. Small Pi indicates that the null is unlikely. Here Hi = 1
(or 0) corresponds to the variant truly is (or is not) associated with height. The features Xi could
include the location, conservation, etc. of the variant. Note that Xi is not used to compute Pi , but it
could contain information about how likely the hypotheses is to be an alternative. Careful readers
may notice that the distribution of Pi given Xi is uniform between 0 and 1 under the null and f1 (p|x)
under the alternative, which depends on x. This implies that Pi and Xi are independent under
the null and dependent under the alternative.
To illustrate why modeling the features could improve discovery power, suppose hypothetically that
all the variants truly associated with height reside on a single chromosome j ? and the feature is
the chromosome index of each SNP (see Fig. 2 (a)). Standard multiple testing methods ignore this
feature and assign the same discovery threshold to all the chromosomes. As there are many purely
noisy chromosomes, the p-value threshold must be very small in order to control FDR. In contrast, a
method that learns the threshold t(x) could learn to assign a higher threshold to chromosome j ? and
0 to other chromosomes. As a higher threshold leads to more discoveries and vice versa, this would
effectively ignore much of the noise and make more discoveries under the same FDR.
3
Algorithm Description
Since a smaller p-value presents stronger evidence against the null hypothesis, we consider the
threshold decision rule without loss of generality. As the null proportion ?0 (x) and the alternative
distribution f1 (p|x) vary with x, the threshold should also depend on x. Therefore, we can write
the rule as t(x) in general, which claims hypothesis i to be significant if Pi < t(Xi ). Let I be the
indicator function. For t(x), the number of discoveries D(t) and the number of false discoveries
3
Train
Train
d
F
D(t)
p
t*(x; ?) Optimize (3)
CV
?*t*(x; ?)
Rescale
D(t)
Test
(a)
t*(x; ?) Optimize (3)
CV
?*t*(x; ?)
Covariate X
(b)
Mirroring estimator
Rescale
Test
(c)
Mirroring estimato
Figure 2: (a) Hypothetical example where small p-values are enriched at chromosome j ? . (b) The
mirroring estimator. (c) The training and cross validation procedure.
Pn
Pn
F D(t) can be expressed as D(t) = i=1 I{Pi <t(Xi )} and F D(t) = i=1 I{Pi <t(Xi ),Hi =0} . Note
that computing F D(t) requires the knowledge of Hi , which is not available from the observations.
Ideally we want to solve t for the following problem:
maximizet D(t), s.t. F DP (t) ? ?.
(1)
Directly solving (1) is not possible. First, without a parametric representation, t can not be optimized.
Second, while D(t) can be calculated from the data, F D(t) can not, which is needed for evaluating
F DP (t). Third, while each decision rule candidate tj controls FDP, optimizing over them may yield
a rule that overfits the data and loses FDP control. We next address these three difficulties in order.
First, the representation of the decision rule t(x) should be flexible enough to address different
structures of the data. Intuitively, to have maximal discoveries, the landscape of t(x) should be similar
to that of the alternative proportion ?1 (x): t(x) is large in places where the alternative hypotheses
abound. As discussed in detail in Sec. 4, two structures of ?1 (x) are typical in practice. The first is
bumps at a few locations, and the second is slopes that vary with x. Hence the representation should
at least be able to address these two structures. In addition, the number of parameters needed for the
representation should not grow exponentially with the dimensionality of x. Hence non-parametric
models, such as the spline-based methods or the kernel based methods, are infeasible. Take kernel
density estimation in 5D as example. If we let the kernel width be 0.1, each kernel contains on
average 0.001% of the data. Then we need at least a million alternative hypothesis data to have a
reasonable estimate of the landscape of ?1 (x). In this work, we investigate the idea of modeling
t(x) using a multilayer perceptron (MLP), which has a high expressive power and has a number of
parameters that does not grow exponentially with the dimensionality of the features. As demonstrated
in Sec. 4, it can efficiently recover the two common structures, bumps and slopes, and yield promising
results in all real data experiments.
Second, although F D(t) can not be calculated from the data, if it can be overestimated by some
d
d
\
F
D(t), then the corresponding estimate of FDP, namely F
DP (t) = F
D(t)/D(t), is also an
\
overestimate. Then if F
DP (t) ? ?, then F DP (t) ? ?, yielding the desired FDP control. Moreover,
d
if F
D(t) is close to F D(t), the FDP control is tight. Conditional on X = x, the rejection region of
p, namely (0, t(x)), contains a mixture of nulls and alternatives. As the null distribution Unif(0, 1)
is symmetrical w.r.t. p = 0.5 while the alternative distribution f1 (p|x) is highly asymmetrical, the
mirrored region (1 t(x), 1) will contain roughly the same number of nulls but very few alternatives.
Then the number of hypothesis in (t(x), 1) can be a proxy of the number of nulls in (0, t(x)). This
idea is illustrated in Fig. 2 (b) and we refer to this estimator as the mirroring estimator.
Definition 2. (The mirroring estimator) For any decision rule t, let C(t) = {(p, x) : p < t(x)} be the
rejection region of t over (Pi , Xi ) and let its mirrored region be C M (t) = {(p, x) : p > 1 t(x)}.The
P
d
mirroring estimator of F D(t) is defined as F
D(t) = i I{(Pi ,Xi )2C M (t)} .
The mirroring estimator overestimates the number of false discoveries in expectation:
Lemma 1. (Positive bias of the mirroring estimator)
n
X
?
?
d
E[F D(t)] E[F D(t)] =
P (Pi , Xi ) 2 C M (t), Hi = 1
0.
i=1
4
(2)
Remark 1. In practice, t(x) is always very small and f1 (p|x) approaches 0 very fast as p ! 1.
Then for any hypothesis with (Pi , Xi ) 2 C M (t), Pi is very close to 1 and hence P(Hi = 1) is very
small. In other words, the bias in (2) is much smaller than E[F D(t)]. Thus the estimator is accurate.
d
In addition, F
D(t) and F D(t) are both sums of n terms. Under mild conditions, they concentrate
d
well around their means. Thus we should expect that F
D(t) approximates F D(t) well most of the
times. We make this precise in Sec. 5 in the form of the high probability FDP control statement.
Third, we use cross validation to address the overfitting problem introduced by optimization. To
be more specific, we divide the data into M folds. For fold j, the decision rule tj (x; ?), before
applied on fold j, is trained and cross validated on the rest of the data. The cross validation is done by
\
rescaling the learned threshold tj (x) by a factor j so that the corresponding mirror estimate F
DP
on the CV set is ?. This will not introduce much of additional overfitting since we are only searching
over a scalar . The discoveries in all M folds are merged as the final result. We note here distinct
folds correspond to subsets of hypotheses rather than samples used to compute the corresponding
p-values. This procedure is shown in Fig. 2 (c). The details of the procedure as well as the FDP
control property are also presented in Sec. 5.
Algorithm 1 NeuralFDR
1: Randomly divide the data {(Pi , Xi )}n
i=1 into M folds.
2: for fold j = 1, ? ? ? , M do
3:
Let the testing data be fold j, the CV data be fold j 0 6= j, and the training data be the rest.
4:
Train tj (x; ?) based on the training data by optimizing
\
maximize? D(t(?)) s.t. F
DP (t?j (?)) ? ?.
\
Rescale t?j (x; ?) by j? so that the estimated FDP on the CV data F
DP (
? ?
Apply j tj (?) on the data in fold j (the testing data).
7: Report the discoveries in all M folds.
5:
6:
(3)
? ?
j tj (?))
= ?.
The proposed method NeuralFDR is summarized as Alg. 1. There are two techniques that enabled
robust training of the neural network. First, to have non-vanishing gradients, the indicator functions
in (3) are substituted by sigmoid functions with the intensity parameters automatically chosen based
on the dataset. Second, the training process of the neural network may be unstable if we use random
initialization. Hence, we use an initialization method called the k-cluster initialization: 1) use
k-means clustering to divide the data into k clusters based on the features; 2) compute the optimal
threshold for each cluster based on the optimal group threshold condition ((7) in Sec. 5); 3) initialize
the neural network by training it to fit a smoothed version of the computed thresholds. See Supp. Sec.
2 for more implementation details.
4
Empirical Results
We evaluate our method using both simulated data and two real-world datasets3 . The implementation
details are in Supp. Sec. 2. We compare NeuralFDR with three other methods: BH procedure
(BH) [2], Storey?s BH procedure (SBH) with threshold = 0.4 [17], and Independent Hypothesis
Weighting (IHW) with number of bins and folds set as default [13]. BH and SBH are two most
popular methods without using the hypothesis features and IHW is the state-of-the-art method that
utilizes hypothesis features. For IHW, in the multi-dimensional feature case, k-means is used to
group the hypotheses. In all experiments, k is set to 20 and the group index is provided to IHW as the
hypothesis feature. Other than the FDR control experiment, we set the nominal FDR level ? = 0.1.
Simulated data. We first consider DataIHW, the simulated data in the IHW paper ( Supp. 7.2.2
[13]). Then, we use our own data that are generated to have two feature structures commonly seen
in practice, the bumps and the slopes. For the bumps, the alternative proportion ?1 (x) is generated
from a Gaussian mixture (GM) to have a few peaks with abundant alternative hypotheses. For the
slopes, ?1 (x) is generated linearly dependent with the features. After generating ?1 (x), the p-values
are generated following a beta mixture under the alternative and uniform (0, 1) under the null. We
3
We released the software at https://github.com/fxia22/NeuralFDR
5
5/19/17, 12)45 AM
(a)
5/19/17, 12)44 AM
(b)
Figure 3: FDP for (a) DataIHW and (b) 1DGM. Dashed line indicate 45 degrees, which is optimal.
Table 1: Simulated data: # of discoveries and gain over BH at FDR = 0.1.
DataIHW
DataIHW(WD) 1D GM
BH
2259
6674
8266
SBH
2651(+17.3%)
7844(+17.5%)
9227(+11.62%)
IHW
5074(+124.6%) 10382(+55.6%) 11172(+35.2%)
NeuralFDR 6222(+175.4%) 12153(+82.1%) 14899(+80.2%)
1D slope
2D GM
2D slope
5D GM
BH
11794
9917
8473
9917
SBH
13593(+15.3%) 11334(+14.2%) 9539(+12.58%) 11334(+14.28%)
IHW
12658(+7.3%)
12175(+22.7%) 8758(+3.36%)
11408(+15.0%)
NeuralFDR 15781(+33.8%) 18844(+90.0%) 10318(+21.7%) 18364(+85.1%)
http://localhost:8894/files/sideinfo/FDR2.svg
http://localhost:8894/files/sideinfo/FDR1.svg
Page 1 of 1
generated the data for both 1D and 2D cases, namely 1DGM, 2DGM, 1Dslope, 2Dslope. For example,
Fig. 4 (a) shows the alternative proportion of 2Dslope. In addition, for the high dimensional feature
scenario, we generated a 5D data, 5DGM, which contains the same alternative proportion as 2DGM
with 3 addition non-informative directions.
We first examine the FDR control property using DataIHW and 1DGM. Knowing the ground truth,
we plot the FDP (actual FDR) over different values of the nominal FDR ? in Fig. 3. For a perfect
FDR control, the curve should be along the 45-degree dashed line. As we can see, all the methods
control FDR. NeuralFDR controls FDR accurately while IHW tends to make overly conservative
decisions. Second, we visualize the learned threshold by both NeuralFDR and IWH. As mentioned in
Sec. 3, to make more discoveries, the learned threshold should roughly have the same shape as ?1 (x).
The learned thresholds of NeuralFDR and IHW for 2Dslope are shown in Fig. 3 (b,c). As we can see,
NeuralFDR well recovers the slope structure while IHW fails to assign the highest threshold to the
bottom right block. IHW is forced to be piecewise constant while NeuralFDR can learn a smooth
threshold, better recovering the structure of ?1 (x). In general, methods that partition the hypotheses
into discrete groups would not scale for higher-dimensional features. In Appendix 1, we show that
NeuralFDR is also able to recover the correct threshold for the Gaussian signal. Finally, we report
the total numbers of discoveries in Tab. 1.
In addition, we ran an experiment with dependent p-values with the same dependency structure as
Sec. 3.2 in [13]. We call this dataset DataIHW(WD). The number of discoveries are shown in Tab.
1. NeuralFDR has the actual FDP 9.7% while making more discoveries than SBH and IHW. This
empirically shows that NeuralFDR also works for weakly dependent data.
All numbers are averaged over 10 runs of the same simulation setting. We can see that NeuralFDR
outperforms IHW in all simulated datasets. Moreover, it outperforms IHW by a large margin
multi-dimensional feature settings.
Airway RNA-Seq data. Airway data [10] is a RNA-Seq dataset that contains n = 33469 genes
and aims to identify glucocorticoid responsive (GC) genes that modulate cytokine function in airway
smooth muscle cells. The p-values are obtained by a standard two-group differential analysis using
DESeq2 [16]. We consider the log count for each gene as the hypothesis feature. As shown in the
first column in Tab. 2, NeuralFDR makes 800 more discoveries than IHW. The learned threshold
6
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(a) Actual alternative proportion
for 2Dslope.
http://localhost:8894/files/sideinfo/2dslope1.png
(b) NeuralFDR?s learned threshold.
(c) IHW?s learned threshold
http://localhost:8894/files/sideinfo/2dslope3.png
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Page 1 of 1
http://localhost:8894/files/sideinfo/2dslope2.png
Page 1 of 1
(d) NeuralFDR?s learned thresh- (e) NeuralFDR?s learned thresh- (f) NeuralFDR?s learned threshold for Airway log count.
old for GTEx log distance.
old for GTEx expression level.
Figure 4: (a-c) Results for 2Dslope: (a) the alternative proportion for 2Dslope; (b) NeuralFDR?s
learned threshold; (c) IHW?s learned threshold. (d-f): Each dot corresponds to one hypothesis. The
red curves shows the learned threshold by NeuralFDR: (d) for log count for airway data; (e) for log
distance for GTEx data; (f) for expression level for GTEx data.
Table 2: Real data: # of discoveries at FDR = 0.1.
http://deepfei:8894/files/airway.png
BH
SBH
IHW
NeuralFDR
Airway
4079
4038(-1.0%)
4873(+19.5%)
6031(+47.9%)
GTEx-PhastCons
29348
29758(+1.4%)
30241(+3.0%)
30525(+4.0%)
http://deep.fxia.me:8894/files/gtex-distance.pngPage 1 of 1
BH
SBH
IHW
NeuralFDR
GTEx-dist
29348
29758(+1.4%)
35771(+21.9%)
36127(+23.1%)
GTEx-2D
29348
29758(+1.4%)
35705(+21.7%)
37095(+26.4%)
GTEx-exp
29348
29758(+1.4%)
32195(+9.7%)
32214(+9.8%)
GTEx-3D
29348
29758(+1.4%)
35598(+21.3%)
37195(+26.7%)
http://deep.fxia.me:8894/files/gtex-expression.png
Page 1 of 1
by NeuralFDR is shown in Fig. 4 (d). It increases monotonically with the log count, capturing the
positive dependency relation. Such learned structure is interpretable: low count genes tend to have
higher variances, usually dominating the systematic difference between the two conditions; on the
contrary, it is easier for high counts genes to show a strong signal for differential expression [16, 13].
GTEx data. A major component of the GTEx [5] study is to quantify expression quantitative
trait loci (eQTLs) in human tissues. In such an eQTL analysis, each pair of single nucleotide
polymorphism (SNP) and nearby gene forms one hypothesis. Its p-value is computed under the null
hypothesis that the SNP?s genotype is not correlated with the gene expression.We obtained all the
GTEx p-values from chromosome 1 in a brain tissue (interior caudate), corresponding to 10, 623, 893
SNP-gene combinations. In the original GTEx eQTL study, no features were considered in the FDR
analysis, corresponding to running the standard BH or SBH on the p-values. However, we know many
biological features affect whether a SNP is likely to be a true eQTL; i.e. these features could vary
the alternative proportion ?1 (x) and accounting for them could increase the power to discover true
eQTL?s while guaranteeing that the FDR remains the same. For each hypothesis, we generated three
features: 1) the distance (GTEx-dist) between the SNP and the gene (measured in log base-pairs) ; 2)
the average expression (GTEx-exp) of the gene across individuals (measured in log rpkm); 3) the
evolutionary conservation measured by the standard PhastCons scores (GTEx-PhastCons).
7
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The numbers of discoveries are shown in Tab. 2. For GTEx-2D, GTEx-dist and GTEx-exp are used.
For NeuralFDR, the number of discoveries increases as we put in more and more features, indicating
that it can work well with multi-dimensional features. For IHW, however, the number of discoveries
decreases as more features are incorporated. This is because when the feature dimension becomes
higher, each bin in IHW will cover a larger space, decreasing the resolution of the piecewise constant
function, preventing it from capturing the informative part of the feature.
The learned discovery thresholds of NeuralFDR are directly interpretable and match prior biological
knowledge. Fig. 4 (e) shows that the threshold is higher when SNP is closer to the gene. This allows
more discoveries to be made among nearby SNPs, which is desirable since we know there most
of the eQTLs tend to be in cis (i.e. nearby) rather than trans (far away) from the target gene [5].
Fig. 4 (f) shows that the NeuralFDR threshold for gene expression decreases as the gene expression
becomes large. This also confirms known biology: the highly expressed genes tend to be more
housekeeping genes which are less variable across individuals and hence have fewer eQTLs [5].
Therefore it is desirable that NeuralFDR learns to place less emphasis on these genes. We also show
that NeuralFDR learns to give higher threshold to more conserved variants in Supp. Sec. 1, which
also matches biology.
5
Theoretical Guarantees
We assume the tuples {(Pi , Xi , Hi )}ni=1 are i.i.d. samples from an empirical Bayes model:
?
[P |H = 0, X = x] ? Unif(0, 1)
i.i.d.
Xi ? ?(X), [Hi |Xi = x] ? Bern(1 ?0 (x)), [Pi |Hi = 1, X = x] ? f (p|x)
i
i
1
(4)
The features Xi are drawn i.i.d. from some unknown distribution ?(x). Conditional on the feature
Xi = x, hypothesis i is null with probability ?0 (x) and is alternative otherwise. The conditional
distributions of p-values are Unif(0, 1) under the null and f1 (p|x) under the alternative.
FDR control via cross validation. The cross validation procedure is described as follows. The data
is divided randomly into M folds of equal size m = n/M . For fold j, let the testing set Dte (j) be
itself, the cross validation set Dcv (j) be any other fold, and the training set Dtr (j) be the remaining.
The size of the three are m, m, (M 2)m respectively. For fold j, suppose at most L decision rules
are calculated based on the training set, namely tj1 , ? ? ? , tjL . Evaluated on the cross validation set,
let l? -th rule be the rule with most discoveries among rules that satisfies 1) its mirroring estimate
\
F
DP (tjl ) ? ?; 2) D(tjl )/m > c0 , for some small constant c0 > 0. Then, tjl? is selected to apply
on the testing set (fold j). Finally, discoveries from all folds are combined.
The FDP control follows a standard argument of cross validation. Intuitively, the FDP of the rules
{tjl }L
l=1 are estimated based on Dcv (j), a dataset independent of the training set. Hence there is no
overfitting and the overestimation property of the mirroring estimator, as in Lemma 1, is statistical
valid, leading to a conservative decision that controls FDP. This is formally stated as below.
Theorem 1. (FDP control) Let M be the number of folds and let L be the maximum number of
decision rule candidates evaluated by the cross validation
,
?q set. Then with
? probability at least 1
M
ML
the overall FDP is less than (1 + )?, where = O
log
.
?n
Remark 2. There are two subtle points. First, L can not be too large. Otherwise Dcv (j) may
eventually be overfitted by being used too many times for FDP estimation. Second, the FDP estimates
may be unstable if the probability of discovery E[D(tjl )/m] approaches 0. Indeed, the mirroring
d
F
D(tjl )
d
\
method estimates FDP by F
DP (tjl ) =
, where both F
D(tjl ) and D(tjl ) are i.i.d. sums of n
D(tjl )
Bernoulli random variables with mean roughly ?E[D(tjl )/m] and E[D(tjl )/m]. When their means
are small, the concentration property will fail. So we need E[D(tjl )/m] to be bounded away from
zero. Nevertheless this is required in theory but may not be used in practice.
Remark 3. (Asymptotic FDR control under weak dependence) Besides the i.i.d. case, NeuralFDR can
also be extended to control FDR asymptotically under weak dependence [17, 12]. Generalizing the
concept in [12] from discrete groups to continuous features X, the data are under weak dependence
if the CDF of (Pi , Xi ) for both the null and the alternative proportion converge almost surely to
their true values respectively. The linkage disequilibrium (LD) in GWAS and the correlated genes
in RNA-Seq can be addressed by such dependence structure. In this case, if learned threshold
is c-Lipschitz continuous for some constant c, NeuralFDR will control FDR asymptotically. The
8
Lipschitz continuity can be achieved, for example, by weight clipping [1], i.e. clamping the weights to
a bounded set after each gradient update when training the neural network. See Supp. 3 for details.
Optimal decision rule with infinite hypotheses. When n = 1, we can recover the joint density fP X (p, x). Based on that, the explicit form of the optimal decision rule can be obtained
if we are willing to further assumer f1 (p|x) is monotonically non-increasing w.r.t. p. This
rule is used for the k-cluster initialization for NeuralFDR as mentioned in Sec. 3. Now suppose we know fP X (p, x). Then ?(x) and fP |X (p|x) can also be determined. Furthermore, as
f1 (p|x) = 1 ?10 (x) (fP |X (p|x) ?0 (x)), once we specify ?0 (x), the entire model is specified.
Let S(fP X ) be the set of null proportions ?0 (x) that produces the model consistent with fP X .
Because f1 (p|x)
0, we have 8p, x, ?0 (x) ? fP |X (p|x). This can be further simplified as
?0 (x) ? fP |X (1|x) by recalling that fP |X (p|x) is monotonically decreasing w.r.t. p. Then we know
S(fP X ) = {?0 (x) : 8x, ?0 (x) ? fP |X (1|x)}.
(5)
Given fP X (p, x), the model is not fully identifiable. Hence we should look for a rule t that
maximizes the power while controlling FDP for all elements in S(fP X ). For (P1 , X1 , H1 ) ?
(fP X , ?0 , f1 ) following (4), the probability of discovery and the probability of false discovery are
PD (t, fP X ) = P(P1 ? t(X1 )), PF D (t, fP X , ?0 ) = P(P1 ? t(X1 ), H1 = 0). Then the FDP
P X ,?0 )
is F DP (t, fP X , ?0 ) = PFPDD(t,f
(t,fP X ) . In this limiting case, all quantities are deterministic and
FDP coincides with FDR. Given that the FDP is controlled, maximizing the power is equivalent to
maximizing the probability of discovery. Then we have the following minimax problem:
max
t
min
?0 2S(fP X )
PD (t, fP X )
s.t.
max
?0 2S(fP X )
F DP (t, fP X , ?0 ) ? ?,
(6)
where S(fP X ) is the set of possible null proportions consistent with fP X , as defined in (5).
Theorem 2. Fixing fP X and let ?0? (x) = fP |X (1|x). If f1 (p|x) is monotonically non-increasing
w.r.t. p, the solution to problem (6), t? (x), satisfies
1.
fP X (1, x)
= const, almost surely w.r.t. ?(x)
fP X (t? (x), x)
2.
F DR(t? , fP X , ?0? ) = ?. (7)
Remark 4. To compute the optimal rule t? by the conditions (7), consider any t that satisfies (7.1).
According to (7.1), once we specify the value of t(x) at any location x, say t(0), the entire function is
determined. Also, F DP (t, fP X , ?0? ) is monotonically non-decreasing w.r.t. t(0). These suggests the
following strategy: starting with t(0) = 0, keep increasing t(0) until the corresponding FDP equals
?, which gives us the optimal threshold t? . Similar conditions are also mentioned in [14, 13].
6
Discussion
We proposed NeuralFDR, an end-to-end algorithm to the learn discovery threshold from hypothesis
features. We showed that the algorithm controls FDR and makes more discoveries on synthetic and
real datasets with multi-dimensional features. While the results are promising, there are also a few
challenges. First, we notice that NeuralFDR performs better when both the number of hypotheses
and the alternative proportion are large. Indeed, in order to have large gradients for the optimization,
we need a lot of elements at the decision boundary t(x) and the mirroring boundary 1 t(x). It
is important to improve the performance of NeuralFDR on small datasets with small alternative
proportion. Second, we found that a 10-layer MLP performed well to model the decision threshold
and that shallower networks performed more poorly. A better understanding of which network
architectures optimally capture signal in the data is also an important question.
References
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6,360 | 6,753 | Cost efficient gradient boosting
Sven Peter
Ferran Diego
Heidelberg Collaboratory for Image Processing
Interdisciplinary Center for Scientific Computing
University of Heidelberg
69115 Heidelberg, Germany
Robert Bosch GmbH
Robert-Bosch-Stra?e 200
31139 Hildesheim, Germany
[email protected]
[email protected]
Fred A. Hamprecht
Boaz Nadler
Heidelberg Collaboratory for Image Processing
Interdisciplinary Center for Scientific Computing
University of Heidelberg
69115 Heidelberg, Germany
Department of Computer Science
Weizmann Institute of Science
Rehovot 76100, Israel
[email protected]
[email protected]
Abstract
Many applications require learning classifiers or regressors that are both accurate
and cheap to evaluate. Prediction cost can be drastically reduced if the learned
predictor is constructed such that on the majority of the inputs, it uses cheap features
and fast evaluations. The main challenge is to do so with little loss in accuracy. In
this work we propose a budget-aware strategy based on deep boosted regression
trees. In contrast to previous approaches to learning with cost penalties, our method
can grow very deep trees that on average are nonetheless cheap to compute. We
evaluate our method on a number of datasets and find that it outperforms the
current state of the art by a large margin. Our algorithm is easy to implement and
its learning time is comparable to that of the original gradient boosting. Source
code is made available at http://github.com/svenpeter42/LightGBM-CEGB.
1
Introduction
Many applications need classifiers or regressors that are not only accurate, but also cheap to evaluate
[32, 29]. Prediction cost usually consists of two different components: The acquisition or computation
of the features used to predict the output, and the evaluation of the predictor itself. A common
approach to construct an accurate predictor with low evaluation cost is to modify the classical
empirical risk minimization objective, such that it includes a prediction cost penalty, and optimize
this modified functional [32, 29, 22, 23].
In this work we also follow this general approach, and develop a budget-aware strategy based on
deep boosted regression trees. Despite the recent re-emergence and popularity of neural networks,
our choice of boosted regression trees is motivated by three observations:
(i) Given ample training data and computational resources, deep neural networks often give the most
accurate results. However, standard feed-forward architectures route a single input component (for
example, a single coefficient in the case of vectorial input) through most network units. While the
computational cost can be mitigated by network compression or quantization [14], in the extreme
case to binary activations only [16], the computational graph is fundamentally dense. In a standard
decision tree, on the other hand, each sample is routed along a single path from the root to a leaf, thus
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
visiting typically only a small subset of all split nodes, the "units" of a decision tree. In the extreme
case of a balanced binary tree, each sample visits only log(N ) out of a total of N nodes.
(ii) Individual decision trees and their ensembles, such as Random Forest [4] and Gradient Boosting
[12], are still among the most useful and highly competitive methods in machine learning, particularly
in the regime of limited training data, little training time and little expertise for parameter tuning [11].
(iii) When features and/or decisions come at a premium, it is convenient but wasteful to assume that
all instances in a data set are created equal (even when assumed i.i.d.). Some instances may be easy
to classify based on reading a single measurement / feature, while others may require a full battery of
tests before a decision can be reached with confidence [34]. Decision trees naturally lend themselves
to such a "sequential experimental design" setup: after first using cheap features to split all instances
into subsets, the subsequent decisions can be based on more expensive features which are, however,
only elicited if truly needed. Importantly, the set of more expensive features is requested conditionally
on the values of features used earlier in the tree.
In this work we address the challenge of constructing an ensemble of trees that is both accurate and
yet cheap to evaluate. We first describe the problem setup in Section 2, and discuss related work in
Section 3. Our key contribution appears in Section 4, where we propose an extension of gradient
boosting [12] which takes prediction time penalties into account. In contrast to previous approaches
to learning with cost penalties, our method can grow very deep trees that on average are nonetheless
cheap to compute. Our algorithm is easy to implement and its learning time is comparable to that
of the original gradient boosting. As illustrated in Section 5, on a number of datasets our method
outperforms the current state of the art by a large margin.
2
Problem setup
Consider a regression problem where the response Y ? R and each instance X is represented by M
features, X ? RM . Let L : R ? R ? R be a loss function, and T be a set of admissible functions.
In supervised learning, given a training set of N pairs (xi , yi ) sampled i.i.d. from (X, Y ), a classical
approach to learn a predictor T ? T is to minimize the empirical loss L on the training set,
min
T ?T
N
X
L(yi , T (xi )).
(1)
i=1
In this paper we restrict ourselves to the set T that consists of an ensemble of trees, namely predictors
PK
of the form T (x) = k=1 tk (x). Each single decision tree tk can be represented as a collection
of Lk leaf nodes with corresponding responses ? k = (?k,1 , . . . , ?1,Lk ) ? RLk and a function
qk : RM ? {1, . . . , Lk } that encodes the tree structure and maps an input to its corresponding
terminal leaf index. The output of the tree is tk (x) = ? k,qk (x) .
Learning even a single tree that exactly minimizes the functional in Eq. (1) is NP-hard under several
aspects of optimality [15, 18, 24, 35]. Yet, single trees and ensemble of trees are some of the most
successful predictors in machine learning and there are multiple greedy based methods to construct
tree ensembles that approximately solve Eq. (1) [4, 12, 11].
In many practical applications, however, it is important that the predictor T is not only accurate but
also fast to compute. Given a prediction cost function ? : T ? RM ? R+ a standard approach is to
add a penalty to the empirical risk minimization above [32, 29, 34, 22, 23]:
X
min
L(yi , T (xi )) + ??(T, xi ).
(2)
T ?T
i
The parameter ? controls the tradeoff between accuracy and prediction cost.
Typically, the prediction cost function ? consists of two components. The first is the cost of acquiring
or computing relevant input features. For example, think of a patient at the emergency room where
taking his temperature and blood oxygen levels are cheap, but a CT-scan is expensive. The second
component is the cost of evaluating the function T , which in our case is the sum of the cost of
evaluating the K individual trees tk .
In more detail, the first component of feature computation cost may also depend on the specific
prediction problem. In some scenarios, test instances are independent of each other and the features
2
can be computed for each input instance on demand. But there are also others. In image processing,
for example, where the input is an image which consists of many pixels and the task is to predict some
function at all pixels. In such cases, even though specific features can be computed for each pixel
independently, it may be cheaper or more efficient to compute the same feature, such as a separable
convolution filter, at all pixels at once [1, 13]. The cost function ? may be dominated in these cases
by the second component - the time it takes to evaluate the trees.
After discussing related work in Section 3, in Section 4 we present a general adaptation of gradient
boosting [12] to minimize Eq. (2), that takes into account both prediction cost components.
3
Related work
The problem of learning with prediction cost penalties has been extensively studied. One particular
case is that of class imbalance, where one class is extremely rare and yet it is important to accurately
annotate it. For example, the famous Viola-Jones cascades [30] use cheap features to discard examples
belonging to the negative class. Later stages requiring expensive features are only used for the rare
suspected positive class. While such an approach is very successful, due to its early exit strategy it
cannot use expensive features for different inputs [19, 29, 9].
To overcome the limitations imposed by early exit strategies, various methods [33, 34, 17, 31]
proposed single tree constructions but with more complicated decisions at the individual split nodes.
The tree is first learned without taking prediction cost into account followed by an optimization step
that includes this cost. Unfortunately, in practice these single-tree methods are inferior to current
state-of-the-art algorithms that construct tree ensembles [22, 23].
B UDGET RF [22] is based on Random Forests and modifies the impurity function that decides which
split to make, to take feature costs into account. B UDGET RF has several limitations: First, it assumes
that tree evaluation cost is negligible compared to feature acquisition, and hence is not suitable for
problems where features are cheap to compute and the prediction cost is dominated by predictor
evaluation or were both components contribute equally. Second, during its training phase, each usage
of a feature incurs its acquisition cost so repeated feature usage is not modeled, and the probability
for reaching a node is not taken into account. At test time, in contrast, they do allow "free" reuse of
expensive features and do compute the precise cost of reaching various tree branches. B UDGET RF
thus typically does not yield deep but expensive branches which are only seldomly reached.
B UDGET P RUNE [23] is a pruning scheme for ensembles of decision trees. It aims to mitigate
limitations of B UDGET RF by pruning expensive branches from the individual trees. An Integer
Linear Program is formulated and efficiently solved to take repeated feature usage and probabilities
for reaching different branches into account. This method results in a better tradeoff but still cannot
create deep and expensive branches which are only seldomly reached if these were not present in the
original ensemble. This method is considered to be state of the art when prediction cost is dominated
by the feature acquisition cost [23]. We show in Section 5 that constructing deeper trees with our
methods results in a significantly better performance.
G REEDY M ISER [32], which is most similar to our work, is a stage-wise gradient-boosting type
algorithm that also aims to minimize Eq. (2) using an ensemble of regression trees. When both
prediction cost components are assumed equally significant, G REEDY M ISER is considered state
of the art. Yet, G REEDY M ISER also has few limitations: First, all trees are assumed to have the
same prediction cost for all inputs. Second, by design it constructs shallow trees all having the
same depth. We instead consider individual costs for each leaf and thus allow construction of deeper
trees. Our experiments in Section 5 suggest that constructing deeper trees with our proposed method
significantly outperforms G REEDY M ISER.
4
Gradient boosting with cost penalties
We build on the gradient boosting framework [12] and adapt it to allow optimization with cost
penalties. First we briefly review the original algorithm. We then present our cost penalty in Section
4.1, the step wise optimization in 4.2 and finally our tree growing algorithm that builds trees with
deep branches but low expected depth and feature cost in Section 4.3 (such a tree is shown in Figure
1b and compared to a shallow tree that is more expensive and less accurate in Figure 1a).
3
0
0
4
1
2
A
1
B
2
5
E
6
K
F
C
G
H
D
A
4
B
C
5
6
G
7
10
11
12
3
3
H
L
M N
9
O
8
13
E
F
I
J
(b) CEGB
D
(a) Other methods
Figure 1: Illustration of trees generated by the different methods: Split nodes are numbered
in the order they have been created, leaves are represented with letters. The vertical position of
nodes corresponds to the feature cost required for each sample and the edge?s thickness represents
the number of samples moving along this edge. A tree constructed by GreedyMiser is shown in
(a): The majority of samples travel along a path requiring a very expensive feature. BudgetPrune
could only prune away leaves E,F,G and H which does not correspond to a large reduction in costs.
CEGB however only uses two very cheap splits for almost all samples (leaves A and B) and builds a
complex subtree for the minority that is hard to classify. The constructed tree shown in (b) is deep
but nevertheless cheap to evaluate on average.
Gradient boosting tries to minimize the empirical risk of Eq. (1), by constructing a linear combination
of K weak predictors tk : RM ? R from a set F of admissible functions (not necessarily decision
trees). Starting with T0 (x) = 0 each iteration k > 0 constructs a new weak function tk aiming to
reduce the current loss. These boosting updates can be interpreted as approximations of the gradient
descent direction in function space. We follow the notation of [8] who use gradient boosting with
weak predictors tk from the set of regression trees T to minimize regularized empirical risk
min
t1 ,...,tK ?T
N
X
i=1
"
L(yi ,
K
X
k=1
#
tk (xi )) +
K
X
?(tk ).
(3)
k=1
The regularization term ?(tk ) penalizes the complexity of the regression tree functions. They assume
that ?(tk ) only depends on the number of leaves Lk and leaf responses wk and derive a simple
algorithm to directly learn these. We instead use a more complicated prediction cost penalty ? and
use a different tree construction algorithm that allows optimization with cost penalties.
4.1
Prediction cost penalty
Recall that for each individual tree the prediction cost penalty ? consists of two components: (i) the
feature acquisition cost ?f and (ii) the tree evaluation cost ?ev . However, this prediction cost for
the k-th tree, which is fitted to the residual of all previous iterations, depends on the earlier trees.
Specifically, for any input x, features used in the trees of the previous iterations do not contribute to
the cost penalty again. We thus use the indicator function C : N0?K ? N?N ? N?M ? {0, 1} with
C(k, i, m) = 1 if and only if feature m was used to predict xi by any tree constructed prior to and
including iteration k. Furthermore ?m ? 0 is the cost for computing or acquiring feature m for a
single input x. Then the feature cost contribution ?f : N0?K ? N?N ? R+ of xi for the first k trees
4
is calculated as
M
X
?f (k, i) =
?m C(k, i, m)
(4)
m=1
Features computed for all inputs at once (e.g. separable convolution filters) contribute to the penalty
independent of the instance x being evaluated. For those we use ?m as their total computation cost
and define the indicator function D : N0?K ? N?M ? {0, 1} with D(k, m) = 1 if and only if feature
m was used for any input x in any tree constructed prior to and including iteration k. Then
M
X
?c (k) =
?m D(k, m)
(5)
m=1
The evaluation cost ?ev,k : N?Lk ? R+ for a single input x passing through a tree is the number of
split nodes between the root node and the input?s terminal leaf qk (x), multiplied by a suitable constant
? ? 0 which captures the cost to evaluate a single split. The total cost ?ev : N0?K ? N?N ? R+ for
the first k trees is the sum of the costs of each tree
?ev (k, i) =
k
X
?ev,k? (qk? (xi )).
(6)
?
k=1
4.2
Tree Boosting with Prediction Costs
We have now defined all components of Eq. (2). Simultaneous optimization of all trees tk is
intractable. Instead, as in gradient boosting , we minimize the objective by starting with T0 (x) = 0
and iteratively adding a new tree at each iteration.
At iteration k we construct the k-th regression tree tk by minimizing the following objective
Ok =
N
X
[L(yi , Tk?1 (xi ) + tk (xi )) + ??(k, xi )] + ??c (k)
(7)
i=1
with ?(k, xi ) = ?ev (k, i) + ?f (k, i). Note that the penalty for features, which are computed for all
inputs at once, ?c (k) does not depend on x but only on the structure of the current and previous trees.
Directly optimizing the objective Ok w.r.t. the tree tk is difficult since the argument tk appears inside
the loss function. Following [8] we use a second order Taylor expansion of the loss around Tk?1 (xi ).
Removing constant terms from earlier iterations the objective function can be approximated by
N
X
1 2
?
Ok ? Ok =
gi tk (xi ) + hi tk (xi ) + ???(xi ) + ???c
(8)
2
i=1
where
gi = ?y?i L(yi , y?i )
,
(9a)
y?i =Tk?1 (xi )
??(xi ) = ?(k, xi ) ? ?(k ? 1, xi ),
(9c)
hi = ?y2?i L(yi , y?i )
,
y?i =Tk?1 (xi )
??c = ?c (k) ? ?c (k ? 1).
As in [8] we rewrite Eq. (8) for a decision tree tk (x) = ? k,qk (x) with a fixed structure qk ,
!
!#
!
"
Lk
X
X
X
X
1
hi ? 2k,l + ?
??(xi )
+ ???c
O?k =
gi ? k,l +
2
l
i?Il
i?Il
(9b)
(9d)
(10)
i?Il
with the set Il = {i|qk (xi ) = l} containing inputs in leaf l. For this fixed structure the optimal
weights and the corresponding best objective reduction can be calculated explicitly:
? ?k,l
" P
P
L
X
( i?Il gi )2
1
i?Il gi
?
?k = ?
P
+?
= ?P
, (11a) O
2
i?Il hi
i?Il hi
l
5
!#
X
i?Il
??(xi )
+???c (11b)
As we shall see in the next section, our cost-aware impurity function depends on the difference of
Eq. (10) which results by replacing a terminal leaf with a split node [8]. Let p be any leaf of the tree
that can be converted to a split node and two new children r and l then the difference of Eq. (10)
evaluated for the original and the modified tree is
" P
#
P
P
( i?Ip gi )2
( i?Il gi )2
( i?Ir gi )2
1
split
?
P
?O
=
+ P
? P
? ? ??split
(12)
k
k
2
h
h
h
i
i
i
i?Ir
i?Il
i?Ip
Let m be the feature used by the node s that we are considering to split. Then
is feature m
used to split x for the first time?
is feature m
used for the first time?
??split
= |Ip |? + ?m
k
| {z }
?split
|
ev,k
4.3
i
z
}|
{ X
z
}|
{
(1 ? D(k, m)) +
?m (1 ? C(k, i, m))
(13)
i?Ip
{z
?split
f,k
}
Learning a weak regressor with cost penalties
With these preparations we can now construct the regression trees. As mentioned above, this is a
NP-hard problem. We use a greedy algorithm to grow a tree that approximately minimizes Eq. (10).
Standard algorithms that grow trees start from a single leaf containing all inputs. The tree is then
iteratively expanded by replacing a single leaf with a split node and two new child leaves [4]. Typically
this expansion happens in a predefined leaf order (breadth- or depth-first). Splits are only evaluated
locally at a single leaf to select the best feature. The expansion is stopped once leaves are pure or
once a maximum depth has been reached. Here, in contrast, we adopt the approach of [28] and grow
the tree in a best-first order. Splits are evaluated for all current leaves and the one with the best
objection reduction according to Eq. (12) is chosen. The tree can thus grow at any location. This
allows to compare splits across different leaves and features at the same time (figure 1b shows an
example for a best-first tree while figure 1a shows a tree constructed in breadth-first order). Instead
of limiting the depth we limit the number of leaves in each tree to prevent over fitting.
This procedure has an important advantage when optimizing with cost penalties: Growing in a
predefined order usually leads to balanced trees - all branches are grown independent of the cost.
Deep and expensive branches using only a tiny subset of inputs are not easily possible. In contrast,
growing at the leaf that promises the best tradeoff as given by Eq. (12) encourages growth on branches
that contain few instances or growth using cheap features. Growth on branches that contain many
instances or growth that requires expensive features is penalized. This strategy results in deep trees
that are nevertheless cheap to compute on average. Figure 1 compares an individual tree constructed
by others methods to the deeper tree constructed by CEGB.
We briefly compare our proposed strategy to G REEDY M ISER: When we limit Eq. (8) to first
order terms only, use breadth-first instead of best-first growth, assume only features that have to be
computed for all instances at once and limit the tree depth to four we minimize Eq. (18) from [32].
GreedyMiser can therefore be represented as a special case of our proposed algorithm.
5
Experiments
The Yahoo! Learning to Rank (Yahoo! LTR) challenge dataset [7] consists of 473134 training,
71083 validation and 165660 test document-query pairs with labels {0, 1, 2, 3, 4} where 0 means the
document is irrelevant and 4 that it is highly relevant to the query. Computation cost for the 519
features used in the dataset are provided [32] and take the values {1, 5, 10, 20, 50, 100, 150, 200}.
Prediction performance is evaluated using the Average Precision@5 metric which only considers the
five most relevant documents returned for a query by the regressor [32, 22, 23]. We use the dataset
provided by [7] and used in [22, 23].
We consider two different settings for our experiments, (i) feature acquisition and classifier evaluation
time both contribute to prediction cost and (ii) classifier evaluation time is negligible w.r.t feature
acquisition cost.
The first setting is used by G REEDY M ISER. Regression trees with depth four are constructed and
assumed to approximately cost as much as features with feature cost ?m = 1. We therefore set the
6
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2: Comparison against state of the art algorithms: The Yahoo! LTR dataset has been used
for (2a) and (2b) in different settings. In (2a) both tree evaluation and feature acquisition cost is
considered. In (2b) only feature acquisition cost is shown. (2c) shows results on the MiniBooNE
dataset with uniform feature costs. G REEDY M ISER and B UDGET P RUNE results for (2b), (2c) and
(2d) from [23]. B UDGET P RUNE did not finish training on the HEPMASSS datasets to due their size
and the associated CPU time and RAM requirements. CEGB is our proposed method.
split cost ? = 14 to allow a fair comparison with our trees which will contain deeper branches. We
also use our algorithm to construct trees similar to G REEDY M ISER by limiting the trees to 16 leaves
with a maximum branch depth of four. Figure 2a shows that even the shallow trees are already always
strictly better than G REEDY M ISER. This happens because our algorithm correctly accounts for the
different probabilities of reaching different leaves (see also figure 1). When we allow deep branches
the proposed method gives significantly better results than G REEDY M ISER and learns a predictor
with better accuracy at a much lower cost.
The second setting is considered by B UDGET P RUNE. It assumes that feature computation is much
more expensive than classifier evaluation. We set ? = 0 to adapt our algorithm to this setting.
7
(a)
(b)
Figure 3: In (3a) we study the influence of the feature penalty on the learned classifier. (3b) shows
how best-first training results in better precision given the same cost budget.
The dataset is additionally binarized by setting all targets y > 0 to y = 1. G REEDY M ISER has a
disadvantage in this setting since it works on the assumption that the cost of each tree is independent
of the input x. We still include it in our comparison as a baseline. Figure 3b shows that our proposed
method again performs significantly better than others. This confirms that we learn a classifier with
very expected cheap prediction cost in terms of both feature acquisition and classifier evaluation time.
The MiniBooNE dataset [26, 20] consists of 45523 training, 19510 validation and 65031 test instances
with labels {0, 1} and 50 features. The Forest Covertype dataset [20, 3] consists of 36603 training,
15688 validation and 58101 test instances with 54 features restricted to two classes as done in [23].
Feature costs are not available for either dataset and assumed to be uniform, i.e. ?m = 1. Since no
relation between classifier evaluation and feature cost is known we only compute the latter to allow
a fair comparison, as in [23]. Figure 2c and 2d show that our proposed method again results in a
significantly better predictor than both G REEDY M ISER and B UDGET P RUNE.
We additionally use the HEPMASS-1000 and HEPMASS-not1000 datasets [2, 20]. Similar to
MiniBooNE no feature costs are known and we again uniformly set them to one for all features, i.e.
?m = 1. Both datasets contain over ten million instances which we split into 3.5 million training, 1.4
million validation and 5.6 million test instances. These datasets are much larger than the others and
we did not manage to successfully run B UDGET P RUNE due to its RAM and CPU time requirements.
We only report results on G REEDY M ISER and our algorithm in Figure 2e and 2f. CEGB again results
in a classifier with a better tradeoff than G REEDY M ISER.
5.1
Influence of feature cost and tradeoff parameters
We use the Yahoo! LTR dataset to study the influence of the features costs ? and the tradeoff
parameter ? on the learned regressor. Figure 3a shows that regressors learned with a large ? reach
similar accuracy as those with smaller ? at a much cheaper cost. Only ? = 0.001 converges to a
lower accuracy while others approximately reach the same final accuracy. The tradeoff is shifted
towards using cheap features too strongly. Such a regressor is nevertheless useful when the problems
requires very cheap results and the final improvement in accuracy does not matter.
Next, we set all ? m = 1 during training time only and use the original cost during test time. The
learned regressor behaves similar to one learned with ? = 0. This shows that the regressors save
most of the cost by limiting usage of expensive features to a small subset of inputs.
Finally we compare breadth-first to best-first training in Figure 3b. We use the same number of leaves
and trees and try to build a classifier that is as cheap as possible. Best-first training always reaches
a higher accuracy for a given prediction cost budget. This supports our observation that deep trees
which are cheap to evaluate on average are important for constructing cheap and accurate predictors.
5.2
Multi-scale classification / tree structure optimization
In images processing, classification using multiple scales has been extensively studied and used
to build fast or more more accurate classifiers [6, 30, 10, 25]. The basic idea of these schemes is
8
(a)
(b)
(c)
Figure 4: Multi-scale classification: (4a) shows a single frame from the dataset we used. (4b) shows
how our proposed algorithm CEGB is able to build significantly cheaper trees than normal gradient
boosting. (4c) zooms into the region showing the differences between the various patch sizes.
that a large image is downsampled to increasingly coarse resolutions. A multi-scale classifier first
analyzes the coarsest resolution and decides whether a pixel on the coarse level represents a block of
homogeneous pixels on the original resolution, or if analysis on a less coarse resolution is required.
Efficiency results from the ability to label many pixels on the original resolution at once by labeling a
single pixel on a coarser image.
We use this setting as an example to show how our algorithm is also capable of optimizing problems
where feature cost is negligible compare to predictor evaluation cost. Inspired by average pooling
layers in neural networks [27] and image pyramids [5] we first compute the average pixel values
across non-overlapping 2x2, 4x4 and 8x8 blocks of the original image. We compute several commonly
used and very fast convolutional filters on each of those resolutions. We then replicated these features
values on the original resolution, e.g. the feature response of a single pixel on the 8x8-averaged image
is used for all 64 pixels We modify Eq. (12) and set ??split
= |Ip |?m where m is the number
k
of pixels that share this feature value, e.g. m = 64 when feature m was computed on the coarse
8x8-averaged image.
We use forty frames with a resolution of 1024x1024 pixels taken from a video studying fly ethology.
Our goal here is to detect flies as quickly as possible, as preprocessing for subsequent tracking. A
single frame is shown in Figure 4a. We use twenty of those for training and twenty for evaluation.
Accuracy is evaluated using the SEGMeasure score as defined in [21]. Comparison is done against
regular gradient boosting by setting ? = 0.
Figure 4b shows that our algorithm constructs an ensemble that is able to reach similar accuracy with
a significantly smaller evaluation cost. Figure 4c shows more clearly how the different available
resolutions influence the learned ensemble. Coarser resolutions allow a very efficient prediction at
the cost of accuracy. Overall these experiments show that our algorithm is also capable of learning
predictors that are cheap while maintaining accuracy even when the evaluation cost of these dominates
w.r.t the feature acquisition cost.
6
Conclusion
We presented an adaptation of gradient boosting that includes prediction cost penalties, and devised
fast methods to learn an ensemble of deep regression trees. A key feature of our approach is its ability
to construct deep trees that are nevertheless cheap to evaluate on average. In the experimental part we
demonstrated that this approach is capable of handing various different settings of prediction cost
penalties consisting of feature cost and tree evaluation cost. Specifically, our method significantly
outperformed state of the art algorithms G REEDY M ISER and B UDGET P RUNE when feature cost
either dominates or contributes equally to the total cost.
We additionally showed an example where we are able to optimize the decision structure of the trees
itself when evaluation of these is the limiting factor.
Our algorithm can be easily implemented using any gradient boosting library and does not slow down
training significantly. For these reasons we believe it will be highly valuable for many applications.
Source code is available at http://github.com/svenpeter42/LightGBM-CEGB.
9
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11
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6,361 | 6,754 | Probabilistic Rule Realization and Selection
Haizi Yu? ?
Department of Computer Science
University of Illinois at Urbana-Champaign
Urbana, IL 61801
[email protected]
Tianxi Li?
Department of Statistics
University of Michigan
Ann Arbor, MI 48109
[email protected]
Lav R. Varshney?
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801
[email protected]
Abstract
Abstraction and realization are bilateral processes that are key in deriving intelligence and creativity. In many domains, the two processes are approached through
rules: high-level principles that reveal invariances within similar yet diverse examples. Under a probabilistic setting for discrete input spaces, we focus on the
rule realization problem which generates input sample distributions that follow
the given rules. More ambitiously, we go beyond a mechanical realization that
takes whatever is given, but instead ask for proactively selecting reasonable rules to
realize. This goal is demanding in practice, since the initial rule set may not always
be consistent and thus intelligent compromises are needed. We formulate both rule
realization and selection as two strongly connected components within a single and
symmetric bi-convex problem, and derive an efficient algorithm that works at large
scale. Taking music compositional rules as the main example throughout the paper,
we demonstrate our model?s efficiency in not only music realization (composition)
but also music interpretation and understanding (analysis).
1
Introduction
Abstraction is a conceptual process by which high-level principles are derived from specific examples;
realization, the reverse process, applies the principles to generalize [1, 2]. The two, once combined,
form the art and science in developing knowledge and intelligence [3, 4]. Neural networks have
recently become popular in modeling the two processes, with the belief that the neurons, as distributed
data representations, are best organized hierarchically in a layered architecture [5, 6]. Probably the
most relevant such examples are auto-encoders, where the cascaded encoder and decoder respectively
model abstraction and realization. From a different angle that aims for interpretability, this paper first
defines a high-level data representation as a partition of the raw input space, and then formalizes
abstraction and realization as bi-directional probability inferences between the raw inputs and its
high-level representations.
While abstraction and realization is ubiquitous among knowledge domains, this paper embodies the
two as theory and composition in music, and refers to music high-level representations as compositional rules. Historically, theorists [7, 8] devised rules and guidelines to describe compositional
?
Equal contribution.
Supported in part by the IBM-Illinois Center for Cognitive Computing Systems Research (C3SR), a research
collaboration as part of the IBM Cognitive Horizons Network.
?
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
regularities, resulting in music theory that serves as the formal language to speak of music style and
composers? decisions. Automatic music theorists [9?11] have also been recently developed to extract
probabilistic rules in an interpretable way. Both human theorists and auto-theorists enable teaching
of music composition via rules such as avoiding parallel octaves and resolving tendency tones. So,
writing music, to a certain extent (e.g. realizing a part-writing exercise), becomes the process of
generating ?legitimate? music realizations that satisfy the given rules.
This paper focuses on the realization process in music, assuming rules are given by a preceding
abstraction step. There are two main challenges. First, rule realization: problem occurs when one
asks for efficient and diverse music generation satisfying the given rules. Depending on the rule
representation (hard or probabilistic), there are search-based systems that realize hard-coded rules to
produce music pieces [12, 13], as well as statistical models that realize probabilistic rules to produce
distributions of music pieces [9, 14]. Both types of realizations typically suffer from the enormity of
the sample space, a curse of input dimensionality. Second, rule selection (which is subtler): not all
rules are equally important nor are they always consistent. In some cases, a perfect and all-inclusive
realization is not possible, which requires relaxation/sacrifice of some rules. In other cases, composers
intentionally break certain rules to establish unique styles. So the freedom and creativity in selecting
the ?right? rules for realization poses the challenge.
The main contribution of the paper is to propose and implement a unified framework that makes
reasonable rule selections and realizes them in an efficient way, tackling the two challenges in one
shot. As one part of the framework, we introduce a two-step dimensionality reduction technique?a
group de-overlap step followed by a screening step?to efficiently solve music rule realization. As the
other part, we introduce a group-level generalization of the elastic net penalty [15] to weight the rules
for a reasonable selection. The unified framework is formulated as a single bi-convex optimization
problem (w.r.t. a probability variable and a weight variable) that coherently couples the two parts
in a symmetric way. The symmetry is beneficial in both computation and interpretation. We run
experiments on artificial rule sets to illustrate the operational characteristics of our model, and further
test it on a real rule set that is exported from an automatic music theorist [11], demonstrating the
model?s selectivity in music rule realization at large scale.
Although music is the main case study in the paper, we formulate the problem in generality so the
proposed framework is domain-agnostic and applicable anywhere there are rules (i.e. abstractions) to
be understood. Detailed discussion at the end of the paper demonstrates that the framework applies
directly to general real-world problems beyond music. In the discussion, we also emphasize how our
algorithm is non-trivial, not just a simple combinatorial massaging of standard models. Therefore,
the techniques introduced in this paper offer broader algorithmic takeaways and are worth further
studying in the future.
2
The Formalism: Abstraction, Realization, and Rule
Abstraction and Realization We restrict our attention to raw input spaces that are discrete and
finite: X = {x1 , . . . , xn }, and assume the raw data is drawn from a probability distribution pX ,
where the subscript refers to the sample space (not a random variable). We denote a high-level
representation space (of X ) by a partition A (of X ) and its probability distribution by pA . Partitioning
the raw input space gives one way of abstracting low-level details by grouping raw data into clusters
and ignoring within-cluster variations. Following this line of thought, we define an abstraction as the
process: (X , pX ) ? (A, pA ) for some high-level representation A, where pA is inferred from pX by
summing up the probability masses within each partition cluster. Conversely, we define a realization
as the process: (A, pA ) ? (X , pX ), where pX is any probability distribution that infers pA .
Probabilistic Compositional Rule To put the formalism in the context of music, we first follow
the convention [9] to approach a music piece as a sequence of sonorities (a generic term for chord)
and view each moment in a composition as determining a sonority that fits the existing music context.
If we let ? be a finite collection of pitches specifying the discrete range of an instrument, e.g. the
collection of the 88 keys on a piano, then a k-part sonority?k simultaneously sounding pitches?is a
point in ?k . So X = ?k is the raw input space containing all possible sonorities. Although discrete
and finite, the raw input size is typically large, e.g. |X | = 884 considering piano range and 4-part
chorales. Therefore, theorists have invented various music parameters such as quality and inversion,
to abstract specific sonorities. In this paper, we inherit the approach in [11] to formalize a high2
level representation of X by a feature-induced partition A, and call the output of the corresponding
abstraction (A, pA ) a probabilistic compositional rule.
Probabilistic Rule System The interrelation between abstraction and realization (X , pX ) ?
(A, pA ) can be formalized by a linear equation: Ap = b, where A ? {0, 1}m?n represents a partition
(Aij = 1 if and only if xj is assigned to the ith cluster in the partition), and p = pX , b = pA are
probability distributions of the raw input space and the high-level representation space, respectively.
In the sequel, we represent a rule by the pair (A, b), so realizing this rule becomes solving the
linear equation Ap = b. More interestingly, given a set of rules: (A(1) , b(1) ), . . . , (A(K) , b(K) ), the
realization of all of them involves finding a p such that A(r) p = b(r) , for all r = 1, . . . , K. In this
case, we form a probabilistic rule system by stacking all rules into one single linear system:
? (1) ?
? (1) ?
A
b
?
?
?
?
A = ? ... ? ? {0, 1}m?n , b = ? ... ? ? [0, 1]m .
(1)
A(K)
(r)
(r)
We call Ai,: p = bi
3
b(K)
a rule component, and mr = dim(b(r) ) the size (# of components) of a rule.
Unified Framework for Rule Realization and Selection
In this section, we detail a unified framework for simultaneous rule realization and selection. Recall
rules themselves can be inconsistent, e.g. rules learned from different music contexts can conflict.
So given an inconsistent rule system, we can only achieve Ap ? b. To best realizePthe possibly
inconsistent rule system, we solve for p ? ?n by minimizing the error kAp ? bk22 = r kA(r) p ?
b(r) k22 , the sum of the Brier scores from every individual rule. This objective does not differentiate
rules (or their components) in the rule system, which typically yields a solution that satisfies all
rules approximately and achieves a small error on average. This performance, though optimal in the
averaged sense, is somewhat disappointing since most often no rule is satisfied exactly (error-free).
Contrarily, a human composer would typically make a clear separation: follow some rules exactly
and disregard others even at the cost of a larger realization error. The decision made on rule selection
usually manifests the style of a musician and is a higher level intelligence that we aim for. In this
pursuit, we introduce a fine-grained set of weights w ? ?m to distinguish not only individual rules
but also their components. The weights are estimates of relative importance, and are further leveraged
for rule selection. This yields a weighted error, which is used herein to measure realization quality:
E(p, w; A, b) = (Ap ? b)> diag(w)(Ap ? b).
(2)
minimize E(p, w; A, b) + ?p Pp (p) + ?w Pw (w)
subject to p ? ?n , w ? ?m .
(3)
2
0 k ,
Pp (p) = kpg10 k21 + ? ? ? + kpgK
0 1
?
?
Pw0 (w) = m1 kwg1 k12 + ? ? ? + mK kwgK k12 .
(4)
If we revisit the two challenges mentioned in Sec. 1, we see that under the current setting, the first
challenge concerns the curse of dimensionality for p, while the second concerns the selectivity for w.
We introduce two penalty terms, one each for p and w, to tackle the two challenges, and propose the
following bi-convex optimization problem as the unified framework:
Despite contrasting purposes, both penalty terms, Pp (p) and Pw (w), adopt the same high-level
strategy of exploiting group structures in p and w. Regarding the curse of dimensionality, we exploit
the group structure of p by grouping pj and pj 0 together if the jth and j 0 th columns of A are identical,
0
0
partitioning p?s coordinates into K 0 groups: g10 , . . . , gK
0 where K is the number of distinct columns
of A. This grouping strategy uses the fact that in a simplex-constrained linear system, we cannot
determine the individual pj s within each group but only their sum. We later show (Sec. 4.1) the
resulting group structure of p is essential in dimensionality reduction (when K 0 n ) and has a
deeper interpretation regarding abstraction levels. Regarding the rule-level selectivity, we exploit
the group structure of w by grouping weights together if they are associated with the same rule,
partitioning w?s coordinates into K groups: g1 , . . . , gK where K is the number of given rules. Based
on the group structures of p and w, we introduce their corresponding group penalties as follows:
3
(5)
One can see the symmetry here: group penalty (4) on p is a squared, unweighted L2,1 -norm, which is
designed to secure a unique solution that favors more randomness in p for the sake of diversity in
sonority generation [9]; group penalty (5) on w is a weighted L1,2 -norm (group lasso), which enables
rule selection. However, there is a pitfall of the group lasso penalty when deployed in Problem (3):
the problem has multiple global optima that are indefinite about the number of rules to pick (e.g.
selecting one rule and ten consistent rules are both optimal). To give more control over the number of
selections, we finalize the penalty on w as the group elastic net that blends between a group lasso
penalty and a ridge penalty:
Pw (w) = ?Pw0 (w) + (1 ? ?)kwk22 ,
0 ? ? ? 1,
(6)
where ? balances the trade-off between rule elimination (less rules) and selection (more rules).
Model Interpretation Problem (3) is a bi-convex problem: fixing p it is convex in w; fixing w it is
convex in p. The symmetry between the two optimization variables further gives us the reciprocal
interpretations of the rule realization and selection problem: given p, the music realization, we can
analyze its style by computing w; given w, the music style, we can realize it by computing p and
further sample from it to obtain music that matches the style. The roles of the hyperparameters ?p
and (?w , ?) are quite different. In setting ?p sufficiently small, we secure a unique solution for the
rule realization part. However, for the rule selection part, what is more interesting is that adjusting
?w and ? allows us to guide the overall composition towards different directions, e.g. conservative
(less strictly obeyed rules) versus liberal (more loosely obeyed rules).
Model Properties We state two properties of the bi-convex problem (3) as the following theorems
whose proofs can be found in the supplementary material. Both theorems involve the notion of group
selective weight. We say w ? ?m is group selective if for every rule in the rule set, w either drops
it or selects it entirely, i.e. either wgr = 0 or wgr > 0 element-wisely, for any r = 1, . . . , K. For a
group selective w, we further define suppg (w) to be the selected rules, i.e. suppg (w) = {r | wgr >
0 element-wisely} ? {1, . . . , K}.
Theorem 1. Fix any ?p > 0, ? ? [0, 1]. Let (p? (?w ), w? (?w )) be a solution path to problem (3).
(1) w? (?w ) is group selective, if ?w > 1/?.
?
(2) kwg?r (?w )k2 ? mr /m as ?w ? ?, for r = 1, . . . , K.
Theorem 2. For ?p = 0 and any ?w > 0, ? ? [0, 1], let (p? , w? ) be a solution to problem (3). We
define C ? 2{1,...,K} such
(error-free) subset of the given rule set. If
P
P that any C ? C is a consistent
suppg (w? ) ? C, then r?suppg (w? ) mr = max
r?C mr | C ? C .
Thm. 1 implies a useful range of the ?w -solution path: if ?w is too large, w? will converge to a
known value that always selects all the rules; if ?w is too small, w? can lose the guarantee to be
group selective. This further suggests the termination criteria used later in the experiments. Thm. 2
considers rule selection in the consistent case, where the solution selects the largest number of rule
components among all other consistent rule selections. Despite the condition ?p = 0, in practice, this
theorem suggests one way of using model for a small ?p : if the primary interest is to select consistent
rules, the model is guaranteed to pick as many rule components as possible (Sec. 5.1). Yet, a more
interesting application is to slightly compromise consistency to achieve better selection (Sec. 5.2).
4
Alternating Solvers for Probability and Weight
It is natural to solve the bi-convex problem (3) by iteratively alternating the update of one optimization
variable while fixing the other, yielding two alternating solvers.
4.1
The p-Solver: for Rule Realization
If we fix w, the optimization problem (3) boils down to:
minimize E(p, w; A, b) + ?p Pp (p)
subject to p ? ?n .
4
(7)
8
>
>(1, 0, 0),
>
>
>
(0, 1, 0),
De-Overlap
>
>
>
>
>
G06 0 G05
<(0, 0, 1),
G7
g(x) = (0, 1, 1),
>
G02 G0 G03
>
>
4
(1, 0, 1),
G2
G3
>
>
>
>
>
(1, 1, 0),
>
>
:
DeO(G) = {G01 , G02 , G03 , G04 , G05 , G06 , G07 }
G = {G1 , G2 , G3 }
(1, 1, 1),
G1
G01
x 2 G01
x 2 G02
x 2 G03
x 2 G04
x 2 G05
x 2 G06
x 2 G07
Figure 1: An example of group de-overlap.
Making a change of variable: qk = 1> pgk0 = kpgk0 k1 for k = 1, . . . , K 0 and letting q = (q1 , . . . , qK 0 ),
problem (7) is transformed to its reduced form:
minimize E(p, w; A0 , b) + ?p kqk22
(8)
0
subject to q ? ?K ,
where A0 is obtained from A by removing its column duplicates. Problem (8) is a convex problem
with a strictly convex objective, so it has a unique solution q ? . However, the solution to the original
problem (7) may not be unique: any p? satisfying qk? = 1> p?g0 is a solution to (7). To favor a
k
more random p (as discussed in Sec. 3), we can uniquely determine p? by uniformly distributing the
probability mass qk within the group gk0 : p?g0 = (qk / dim(pgk0 ))1, k = 1, . . . , K 0 .
k
Dimensionality Reduction: Group De-Overlap Problem (7) is of dimension n, while its reduced
form (8) is of dimension K 0 (? n) from which we can attain dimensionality reduction. In cases where
K 0 n, we have a huge speed-up for the p-solver; in other cases, there is still no harm to always run
the p-solve from the reduced problem (8). Recall that we have achieved this type of dimensionality
reduction by exploiting the group structure of p purely from a computational perspective (Sec. 3).
However, the resulting group structure has a deeper interpretation regarding abstraction levels, which
is closely related to the concept of de-overlapping a family of groups, group de-overlap in short.
(Group De-Overlap) Let G = {G1 , . . . , Gm } be a family of groups (a group is a non-empty set), and
m
G = ?m
i=1 Gi . We introduce a group assignment function g : G 7? {0, 1} , such that for any x ? G,
g(x)i = 1{x ? Gi }, and further introduce an equivalence relation ? on G: x ? x0 if g(x) = g(x0 ).
We then define the de-overlap of G, another family of groups, by the quotient space
DeO(G) = {G01 , . . . , G0m0 } := G/ ? .
(9)
The idea of group de-overlap is simple (Fig. 1), and DeO(G) indeed comprises non-overlapping
groups, since it is a partition of G that equals the set of equivalence classes under ?.
Now given a set of rules (A(1) , b(1) ), . . . , (A(K) , b(K) ), we denote their corresponding high-level
representation spaces by A(1) , . . . , A(K) , each of which is a partition of the raw input space X
(k)
(Sec. 2). Let G = ?K
, then DeO(G) is a new partition?hence a new high-level representation
k=1 A
space?of G = X , and is finest (may be tied) among all partitions A(1) , . . . , A(K) . Therefore,
DeO(G), as a summary of the rule system, delimits a lower bound on the level of abstraction produced
by the given set of rules/abstractions. What coincides with DeO(G), is the group structure of p (recall:
pj and pj 0 are grouped together if the jth and j 0 th columns of A are identical), since for any xj ? X ,
the jth column of A is precisely the group assignment vector g(xj ). Therefore, the decomposed
solve step from q ? to p? reflects the following realization chain:
n
o
(A(1) , pA(1) ), . . . , (A(K) , pA(K) ) ? (DeO(G), q ? ) ? (X , pX ),
(10)
where the intermediate step not only computationally achieves dimensionality reduction, but also
conceptually summarizes the given set of abstractions and is further realized in the raw input space.
Note that the ?-algebra of the probability space associated with (8) is precisely generated by DeO(G).
When rules are inserted into a rule system sequentially (e.g. the growing rule set from an automatic
music theorist), the successive solve of (8) is conducted along a ?-algebra path that forms a filtration:
nested ?-algebras that lead to finer and finer delineations of the raw input space. In a pedagogical
setting, the filtration reflects the iterative refinements of music composition from high-level principles
that are taught step by step.
5
Dimensionality Reduction: Screening We propose an additional technique for further dimensionality reduction when solving the reduced problem (8). The idea is to perform screening, which quickly
identifies the zero components in q ? and removes them from the optimization problem. Leveraging
DPC screening for non-negative lasso [16], we introduce a screening strategy for solving a general
simplex-constrained linear least-squares problem (one can check problem (8) is indeed of this form):
minimize kX? ? yk22 ,
subject to ? 0, k?k1 = 1.
(11)
We start with the following non-negative lasso problem, which is closely related to problem (11):
minimize ?? (?) := kX? ? yk22 + ?k?k1 ,
subject to ? 0,
(12)
and denote its solution by ? (?). One can show that if k? (? )k1 = 1, then ? (? ) is a solution
to problem (11). Our screening strategy for problem (11) runs the DPC screening algorithm on the
non-negative lasso problem (12), which applies a repeated screening rule (called EDPP) to solve a
solution path specified by a ?-sequence: ?max = ?0 > ?1 > ? ? ? . The `1 -norms along the solution
path are non-decreasing: 0 = k? ? (?0 )k1 ? k? ? (?1 )k1 ? ? ? ? . We terminate the solution path at ?t
if k? ? (?t )k1 ? 1 and k? ? (?t?1 )k1 < 1. Our goal is to use ? ? (?t ) to predict the zero components in
? ? (?? ), a solution to problem (11). More specifically, we assume that the zero components in ? ? (?t )
are also zero in ? ? (?? ), hence we can remove those components from ? (also the corresponding
columns of X) in problem (11) and reduce its dimensionality.
?
?
?
?
?
While in practice this assumption is usually true provided that we have a delicate solution path, the
monotonicity of ? ? (?)?s support along the solution path does not hold in general [17]. Nevertheless,
the assumption does hold when k? ? (?t )k1 ? 1, since the solution path is continuous and piecewise
linear [18]. Therefore, we carefully design a solution path in the hope of a ? ? (?t ) whose `1 -norm is
close to 1 (e.g. let ?i = ??i?1 with a large ? ? (0, 1), while more sophisticated design is possible
such as a bi-section search). To remedy the (rare) situations where ? ? (?t ) predicts some incorrect
zero components in ? ? (?? ), one can always leverage the KKT conditions of problem (11) as a final
check to correct those mis-predicted components [19]. Finally, note that the screening strategy may
fail when the `1 -norms along the solution path converge to a value less than 1. In these cases we can
never find a desired ?t with k? ? (?t )k1 ? 1. In theory, such failure can be avoided by a modified
lasso problem which in practice does not improve efficiency much (see the supplementary material).
4.2
The w-Solver: for Rule Selection
If we fix p, the optimization problem (3) boils down to:
minimize E(p, w; A, b) + ?w Pw (w)
subject to w ? ?m .
(13)
w(k+1) = arg min e> w + ?w Pw (w) + ?2 kw ? z (k) + u(k) k22 ,
(14)
z (k+1) = arg min I?m (z) + ?2 kw(k+1) ? z + u(k) k22 ,
(15)
u(k+1) = u
(16)
We solve problem (13) via ADMM [20]:
w
z
(k)
+ w(k+1) ? z (k+1) .
In the w-update (14), we introduce the error vector e = (Ap ? b) (element-wise square), and obtain
a closed-form solution by a soft-thresholding procedure [21]: for r = 1, . . . , K,
!
?
?w ? mr
?(z (k) ? u(k) ) ? e
(k+1)
(k)
(k)
w gr
= 1?
e
?
,
where
e
?
=
. (17)
g
r
(k)
? + 2?w (1 ? ?)
(? + 2?w (1 ? ?)) ? k?
egr k2
2
+
In the z-update (15), we introduce the indicator function I?m (z) = 0 if z ? ?m and ? otherwise,
and recognize it as a (Euclidean) projection onto the probability simplex:
z (k+1) = ??m (w(k+1) + u(k) ),
(18)
which can be solved efficiently by a non-iterative method [22]. Given that ADMM enjoys a linear
convergence rate in general [23] and the problem?s dimension m n, one execution of the wsolver is cheaper than that of the p-solver. Indeed, the result from the w-solver can speed up the
subsequent execution of the p-solver, since we can leverage the zero components in w? to remove the
corresponding rows in A, yielding additional savings in the group de-overlap of the p-solver.
6
1.2
group norm
2.0
0
wt. err.
rule 1
rule 2
rule 3
rule 4
rule 5
4.0
1.0
0
?8
?6
?4
?2
0
2
log2 (?w)
(a) Case A1: ? = 0.8.
? 10?1
rule 1
rule 2
rule 3
rule 4
rule 5
0.6
0
? 10?4
wt. err.
group norm
? 10?2
? 10?4
4.0
2.0
0
?8
?6
?4
?2
0
2 log2 (?w)
(b) Case A2: ? = 0.8.
Figure 2: The ?w -solution paths obtained from the two artificial rule sets. Each path is depicted by
the trajectories of the group norms (top) and the trajectory of the weighted errors (bottom).
5
5.1
Experiments
Artificial Rule Set
We generate two artificial rule sets: Case A1 and A2, both of which are derived from the same raw
input space X = {x1 , . . . , xn } for n = 600, and comprise K = 5 rules. The rules in Case A1 are of
size 80, 50, 60, 60, 60, respectively; the rules in Case A2 are of size 70, 50, 65, 65, 65, respectively.
For both cases, rule 1&2 and rule 3&4 are the only two consistent sub rule sets of size ? 2. The main
difference between the two cases is: in Case A1, rule 1&2 has a combined size of 130 which is larger
than rule 3&4 and in Case A2 it is opposite. Under different settings of the hyperparameters ?w and
?, our model selects different rule combinations exhibiting unique ?personal? styles.
Tuning the blending factor ? ? [0, 1] is relatively easy, since it is bounded and has a nice interpretation.
Intuitively, if ? ? 0, the effect of the group lasso vanishes, yielding a solution w? that is not selective;
if ? ? 1, the group elastic net penalty reduces to the group lasso, exposing the pitfall mentioned
in Sec. 3. Experiments show that if we fix a small ?, the model picks either all five rules or none;
if we fix a large ?, the group norms associated with each rule are highly unstable as ?w varies.
Fortunately in practice, ? has a wide middle range (typically between 0.4 and 0.9), within which all
corresponding ?w -solution paths look similar and perform stable rule selection. Therefore, for all
experiments herein, we fix ? = 0.8 and study the behavior of the corresponding ?w -solution path.
We show the ?w -solution paths in Fig. 2. Along the path, we plot the group norms (top, one curve per
rule) and the weighted errors (bottom). The former, formulated as kwg?r (?w )k2 , describes the options
for rule selection; the latter, formulated as E(p? (?w ), w? (?w ); A, b), describes the quality of rule
realization. To produce the trajectories, we start with a moderate ?w (e.g. ?w = 1), and gradually
increase and decrease its value to bi-directionally grow the curves. We terminate the descending
direction when w? (?w ) is not group selective and terminate the ascending direction when the group
norms converge. Both terminations are indicated by Thm. 1, and work well in practice. As ?w grows,
the model transitions its compositional behavior from a conservative style (sacrifice a number of rules
for accuracy) towards a more liberal one (sacrifice accuracy for more rules). If we further focus on
the ?w s that give us zero weighted error, Fig. 2a reveals rule 1&2, and Fig. 2b reveals rule 3&4, i.e.
the largest consistent subset of the given rule set in both cases (Thm. 2). Finally, we mention the
efficiency of our algorithm. Averaged over several runs on multiple artificial rule sets of the same size,
the run-time of our solver is 27.2 ? 5.5 seconds, while that of a generic solver (CVX) is 41.4 ? 3.8
seconds. We attribute the savings to the dimensionality reduction techniques introduced in Sec. 4.1,
which will be more significant at large scale.
5.2
Real Compositional Rule Set
As a real-world application, we test our unified framework on rule sets from an automatic music
theorist [11]. The auto-theorist teaches people to write 4-part chorales by providing personalized
7
385
A.1
The KKT Condition of Simplex Constrained Linear Least-Squares
386
A.2
An Equivalent Formulation of Simplex Constrained Linear Least-Squares
387
A.3
The Convergence of Group Norms
388
A.4
The Global Minimum of Problem (3) Under Consistency
389
A.5
Miscellaneous
group norm
? 10?2
2.0
Table 1: Compositional rule selections
rule 3
rule 6
rule 9
rule 10
others
1.0
wt. err.
0
? 10?4
1.0
0
?8
?6
w)
selected rule set
# of rules
# of rule
components
[ 12, 6]
[ 5, 2]
[ 1, 0]
1
2
3
{10}
{3, 6, 10}
{3, 6, 9, 10}
{3, 6, 8, 9, 10, 11, 13}
{1, 3, 7, 9, 10, 11, 13}
all
1
3
4
7
7
16
1540
1699
2154
2166
2312
2417
log2 (
?4
?2
0
2
log2 (?w)
Figure 3: The ?w -solution path obtained from a real compositional rule set.
rules at every stage of composition. In this experiment, we exported a set of 16 compositional rules
which aims to guide a student in writing the next sonority that follows well with the existing music
content. Each voice in a chorale is drawn from ? = {R, G1 , . . . , C6 } that includes the rest (R) and
54 pitches (G1 to C6 ) from human vocal range. The resulting raw input space X = ?4 consists of
n = 554 ? 107 sonorities, whose distribution lives in a very high dimensional simplex. This curse of
dimensionality typically fails most of the generic solvers in obtaining an acceptable solution within a
reasonable amount of time.
We show the ?w -solution path associated with this rule set in Fig. 3. Again, the general trend shows
the same pattern here: the model turns into a more liberal style (more rules but less accurate) as
?w increases. Along the solution path, we also observe that the consistent range (i.e. the error-free
zone) is wider than that in the artificial cases. This is intuitive, since a real rule set should be largely
consistent with minor contradictions, otherwise it will confuse the student and lose its pedagogical
purpose. A more interesting phenomenon occurs when the model is about to leave the error-free
zone. When log2 (?w ) goes from 1 to 2, the combined size of the selected rules increases from 2166
to 2312 but the realization error increases only a little. Will sacrificing this tiny error be a smarter
decision to make? The difference between the selected rules at these two moments shows that rule
1 and 7 were added into the selection at log2 (?w ) = 2 replacing rule 6 and 8. Rule 1 is about the
bass line, while rule 6 is about tenor voice. It is known in music theory that outer voices (soprano
and bass) are more characteristic and also more identifiable than inner voices (alto and tenor) which
typically stay more or less stationary as background voices. So it is understandable that although
larger variety in the bass increases the opportunity for inconsistency (in this case not too much), it is
a more important rule to keep. Rule 7 is about the interval between soprano and tenor, while rule 8
describes a small feature between the upper two voices but does not have a meaning yet in music
theory. So unlike rule 7 that brings up the important concept of voicing (i.e. classifying a sonority
11
into open/closed/neutral position), rule 8 could simply be a miscellaneous
artifact. To conclude, in
this particular example, we would argue that the rule selection happens at log2 (?w ) = 2 is a better
decision, in which case the model makes a good compromise on exact consistency.
To compare a selective rule realization with its non-selective counterpart [11], we plot the errors
kA(r) p ? b(r) k2 for each rule r = 1, . . . , 16 as histograms in Fig. 4. The non-selective realization
takes all rules into consideration with equal importance, which turns out to be a degenerate case along
our model?s solution path for log2 (?w ) ? ?. This realization yields a ?well-balanced? solution but
no rules are satisfied exactly. In constrast, a selective realization (e.g. log2 (?w ) = 1) gives near-zero
errors on selected rules, producing more human-like compositional decisions.
1 error
1 error
0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 rule
(a) selective
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 rule
(b) non-selective
Figure 4: Comparison between a selective rule realization (log2 (?w ) = 1) and its non-selective
counterpart. The boldfaced x-tick labels designate the indices of the selected rules.
8
6
Discussion
Generality of the Framework The formalism of abstraction and realization in Sec. 2, as well
as the unified framework for simultaneous rule realization and selection in Sec. 3, is general and
domain-agnostic, not specific to music. The problem formulation as a bi-convex problem (3) admits
numerous real-world applications that can be cast as (quasi-)linear systems, possibly equipped with
some group structure. For instance, many problems in physical science involve estimating unknowns
x from their observations y via a linear (or linearized) equation y = Ax [24], where a grouping of
yi ?s (say, from a single sensor or sensor type) itself summarizes x as a rule/abstraction. In general, the
observations are noisy and inconsistent due to errors from the measuring devices or even the failure of
a sensor. It is then necessary to assign a different reliability weight to every individual sensor reading,
and ask for a ?selective? algorithm to ?realize? the readings respecting the group structure. So in
cases where some devices fail and give inconsistent readings, we can run the proposed algorithm to
filter them out.
Linearity versus Expressiveness The linearity with respect to p in the rule system Ap = b results
directly from adopting the probability-space representation. However, this does not imply that the
underlying domain (e.g. music) is as simple as linear. In fact, the abstraction process can be highly
nonlinear which involves hierarchical partitioning of the input space [11]. So, instead of running the
risk of losing expressiveness, the linear equation Ap = b hides the model complexity in the A matrix.
On the other hand, the linearity with respect to w in the bi-convex objective (3) is a design choice. We
start with a simple linear model to represent relative importance for the sake of interpretability, which
may sacrifice the model?s expressiveness like other classic linear models. To push the boundary of
this trade off in the future, we will pursue more expressive models without compromising (practically
important) interpretability.
Differences from (Group) Lasso Component-wise, both subproblems (7) and (13) of the unified
framework look similar to regular feature selection settings such as lasso [25] and group lasso [26].
However, not only does the strong coupling between the two subproblems exhibit new properties
(Thm. 1 and 2), but also the differences in the formulation present unique algorithmic challenges.
First, the weighted error term (2) in the objective is in stark contrast with the regular regression
formulation where (group) lasso is paired with least-squares or other similar loss functions. Whereas
dropping features in a regression model typically increases training loss (under-fitting), dropping
rules, on the contrary, helps drive the error to zero since a smaller rule set is more likely to achieve
consensus. Hence, the tendency to drop rules in a regular (group) lasso is against the pursuit of a
largest consistent rule set as desired. This stresses the necessity of a more carefully designed penalty
like our proposed group elastic net. Second, the additional simplex constraint weakens the grouping
property of group lasso: failures in group selection (i.e. there exists a rule that is not entirely selected)
are observed for small ?w s. The simplex constraint, effectively an `1 constraint, also incurs an ?`1
cancellation?, which nullifies a simple lasso (also an `1 ) on a simple parameterization of the rules
(one weight per rule). These differences pose new model behaviors and deserve further study.
Local Convergence We solve the bi-convex problem (3) via alternating minimizations in which
the algorithm decreases the non-negative objective in every iteration thus assures its convergence.
Nevertheless, neither a global optimum nor a convergence in solution can be guaranteed. The former
leaves the local convergence susceptible to different initializations, demanding further improvements
through techniques such as random start and noisy updates. The latter leaves the possibility for the
optimization variables to enter a limit cycle. However, we consider this as an advantage, especially in
music where one prefers multiple realizations and interpretations that are equally optimal.
More Microscopic Views The weighting scheme in this paper presents the rule selection problem
in a most general setting, where a different weight is assigned to every rule component. Hence, we
can study the relative importance not only between rules by the group norms kwgr k2 , but also within
every single rule. The former compares compositional rules in a macroscopic level, e.g. restricting to
a diatonic scale is more important than avoiding parallel octaves; while the latter in a microscopic
level, e.g. changing the probability mass within a diatonic scale creates variety in modes: think about
C major versus A minor. We can further study the rule system microscopically by sharing weights of
the same component but from different rules, yielding an overlapping group elastic net.
9
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2010.
11
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6,362 | 6,755 | Nearest-Neighbor Sample Compression:
Efficiency, Consistency, Infinite Dimensions
Aryeh Kontorovich
Department of Computer Science
Ben-Gurion University of the Negev
[email protected]
Sivan Sabato
Department of Computer Science
Ben-Gurion University of the Negev
[email protected]
Roi Weiss
Department of Computer Science and Applied Mathematics
Weizmann Institute of Science
[email protected]
Abstract
We examine the Bayes-consistency of a recently proposed 1-nearest-neighbor-based
multiclass learning algorithm. This algorithm is derived from sample compression
bounds and enjoys the statistical advantages of tight, fully empirical generalization
bounds, as well as the algorithmic advantages of a faster runtime and memory
savings. We prove that this algorithm is strongly Bayes-consistent in metric
spaces with finite doubling dimension ? the first consistency result for an efficient
nearest-neighbor sample compression scheme. Rather surprisingly, we discover
that this algorithm continues to be Bayes-consistent even in a certain infinitedimensional setting, in which the basic measure-theoretic conditions on which
classic consistency proofs hinge are violated. This is all the more surprising, since
it is known that k-NN is not Bayes-consistent in this setting. We pose several
challenging open problems for future research.
1
Introduction
This paper deals with Nearest-Neighbor (NN) learning algorithms in metric spaces. Initiated by
Fix and Hodges in 1951 [16], this seemingly naive learning paradigm remains competitive against
more sophisticated methods [8, 46] and, in its celebrated k-NN version, has been placed on a solid
theoretical foundation [11, 44, 13, 47].
Although the classic 1-NN is well known to be inconsistent in general, in recent years a series of
papers has presented variations on the theme of a regularized 1-NN classifier, as an alternative to the
Bayes-consistent k-NN. Gottlieb et al. [18] showed that approximate nearest neighbor search can
act as a regularizer, actually improving generalization performance rather than just injecting noise.
In a follow-up work, [27] showed that applying Structural Risk Minimization to (essentially) the
margin-regularized data-dependent bound in [18] yields a strongly Bayes-consistent 1-NN classifier.
A further development has seen margin-based regularization analyzed through the lens of sample
compression: a near-optimal nearest neighbor condensing algorithm was presented [20] and later
extended to cover semimetric spaces [21]; an activized version also appeared [25]. As detailed in
[27], margin-regularized 1-NN methods enjoy a number of statistical and computational advantages
over the traditional k-NN classifier. Salient among these are explicit data-dependent generalization
bounds, and considerable runtime and memory savings. Sample compression affords additional
advantages, in the form of tighter generalization bounds and increased efficiency in time and space.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this work we study the Bayes-consistency of a compression-based 1-NN multiclass learning
algorithm, in both finite-dimensional and infinite-dimensional metric spaces. The algorithm is
essentially the passive component of the active learner proposed by Kontorovich, Sabato, and Urner
in [25], and we refer to it in the sequel as KSU; for completeness, we present it here in full (Alg. 1).
We show that in finite-dimensional metric spaces, KSU is both computationally efficient and Bayesconsistent. This is the first compression-based multiclass 1-NN algorithm proven to possess both of
these properties. We further exhibit a surprising phenomenon in infinite-dimensional spaces, where
we construct a distribution for which KSU is Bayes-consistent while k-NN is not.
Main results. Our main contributions consist of analyzing the performance of KSU in finite and
infinite dimensional settings, and comparing it to the classical k-NN learner. Our key findings are
summarized below.
? In Theorem 2, we show that KSU is computationally efficient and strongly Bayes-consistent
in metric spaces with a finite doubling dimension. This is the first (strong or otherwise)
Bayes-consistency result for an efficient sample compression scheme for a multiclass (or
even binary)1 1-NN algorithm. This result should be contrasted with the one in [27], where
margin-based regularization was employed, but not compression; the proof techniques
from [27] do not carry over to the compression-based scheme. Instead, novel arguments
are required, as we discuss below. The new sample compression technique provides a
Bayes-consistency proof for multiple (even countably many) labels; this is contrasted with
the multiclass 1-NN algorithm in [28], which is not compression-based, and requires solving
a minimum vertex cover problem, thereby imposing a 2-approximation factor whenever
there are more than two labels.
? In Theorem 4, we make the surprising discovery that KSU continues to be Bayes-consistent
in a certain infinite-dimensional setting, even though this setting violates the basic measuretheoretic conditions on which classic consistency proofs hinge, including Theorem 2. This
is all the more surprising, since it is known that k-NN is not Bayes-consistent for this
construction [9]. We are currently unaware of any separable2 metric probability space on
which KSU fails to be Bayes-consistent; this is posed as an intriguing open problem.
Our results indicate that in finite dimensions, an efficient, compression-based, Bayes-consistent
multiclass 1-NN algorithm exists, and hence can be offered as an alternative to k-NN, which is well
known to be Bayes-consistent in finite dimensions [12, 41]. In contrast, in infinite dimensions, our
results show that the condition characterizing the Bayes-consistency of k-NN does not extend to all
NN algorithms. It is an open problem to characterize the necessary and sufficient conditions for the
existence of a Bayes-consistent NN-based algorithm in infinite dimensions.
Related work. Following the pioneering work of [11] on nearest-neighbor classification, it was
shown by [13, 47, 14] that the k-NN classifier is strongly Bayes consistent in Rd . These results
made extensive use of the Euclidean structure of Rd , but in [41] a weak Bayes-consistency result was
shown for metric spaces with a bounded diameter and a bounded doubling dimension, and additional
distributional smoothness assumptions. More recently, some of the classic results on k-NN risk
decay rates were refined by [10] in an analysis that captures the interplay between the metric and the
sampling distribution. The worst-case rates have an exponential dependence on the dimension (i.e.,
the so-called curse of dimensionality), and Pestov [33, 34] examines this phenomenon closely under
various distributional and structural assumptions.
Consistency of NN-type algorithms in more general (and in particular infinite-dimensional) metric
spaces was discussed in [1, 5, 6, 9, 30]. In [1, 9], characterizations of Bayes-consistency were
given in terms of Besicovitch-type conditions (see Eq. (3)). In [1], a generalized ?moving window?
classification rule is used and additional regularity conditions on the regression function are imposed.
The filtering technique (i.e., taking the first d coordinates in some basis representation) was shown to
be universally consistent in [5]. However, that algorithm suffers from the cost of cross-validating
over both the dimension d and number of neighbors k. Also, the technique is only applicable in
1
An efficient sample compression algorithm was given in [20] for the binary case, but no Bayes-consistency
guarantee is known for it.
2
C?rou and Guyader [9] gave a simple example of a nonseparable metric on which all known nearest-neighbor
methods, including k-NN and KSU, obviously fail.
2
Hilbert spaces (as opposed to more general metric spaces) and provides only asymptotic consistency,
without finite-sample bounds such as those provided by KSU. The insight of [5] is extended to the
more general Banach spaces in [6] under various regularity assumptions.
None of the aforementioned generalization results for NN-based techniques are in the form of
fully empirical, explicitly computable sample-dependent error bounds. Rather, they are stated in
terms of the unknown Bayes-optimal rate, and some involve additional parameters quantifying the
well-behavedness of the unknown distribution (see [27] for a detailed discussion). As such, these
guarantees do not enable a practitioner to compute a numerical generalization error estimate for a
given training sample, much less allow for a data-dependent selection of k, which must be tuned via
cross-validation. The asymptotic expansions in [43, 37, 23, 40] likewise do not provide a computable
finite-sample bound. The quest for such bounds was a key motivation behind the series of works
[18, 28, 20], of which KSU [25] is the latest development.
The work of Devroye et al. [14, Theorem 21.2] has implications for 1-NN classifiers in Rd that
are defined based on data-dependent majority-vote partitions of the space. It is shown that under
some conditions, a fixed mapping from each sample size to a data-dependent partition rule induces a
strongly Bayes-consistent algorithm. This result requires the partition rule to have a bounded VC
dimension, and since this rule must be fixed in advance, the algorithm is not fully adaptive. Theorem
19.3 ibid. proves weak consistency for an inefficient compression-based algorithm, which selects
among all the possible compression sets of a certain size, and maintains a certain rate of compression
relative to the sample size. The generalizing power of sample compression was independently
discovered by [31], and later elaborated upon by [22]. In the context of NN classification, [14] lists
various condensing heuristics (which have no known performance guarantees) and leaves open the
algorithmic question of how to minimize the empirical risk over all subsets of a given size.
The first compression-based 1-NN algorithm with provable optimality guarantees was given in [20];
it was based on constructing ?-nets in spaces with a finite doubling dimension. The compression
size of this construction was shown to be nearly unimprovable by an efficient algorithm unless P=NP.
With ?-nets as its algorithmic engine, KSU inherits this near-optimality. The compression-based
1-NN paradigm was later extended to semimetrics in [21], where it was shown to survive violations
of the triangle inequality, while the hierarchy-based search methods that have become standard for
metric spaces (such as [4, 18] and related approaches) all break down.
It was shown in [27] that a margin-regularized 1-NN learner (essentially, the one proposed in [18],
which, unlike [20], did not involve sample compression) becomes strongly Bayes-consistent when the
margin is chosen optimally in an explicitly prescribed sample-dependent fashion. The margin-based
technique developed in [18] for the binary case was extended to multiclass in [28]. Since the algorithm
relied on computing a minimum vertex cover, it was not possible to make it both computationally
efficient and Bayes-consistent when the number of lables exceeds two. An additional improvement
over [28] is that the generalization bounds presented there had an explicit (logarithmic) dependence
on the number of labels, while our compression scheme extends seamlessly to countable label spaces.
Paper outline. After fixing the notation and setup in Sec. 2, in Sec. 3 we present KSU, the
compression-based 1-NN algorithm we analyze in this work. Sec. 4 discusses our main contributions
regarding KSU, together with some open problems. High-level proof sketches are given in Sec. 5 for
the finite-dimensional case, and Sec. 6 for the infinite-dimensional case. Full detailed proofs can be
found in [26].
2
Setting and Notation
Our instance space is the metric space (X , ?), where X is the instance domain and ? is the metric.
(See Appendix A in [26] for relevant background on metric measure spaces.) We consider a countable
label space Y. The unknown sampling distribution is a probability measure ?
? over X ? Y, with
marginal ? over X . Denote by (X, Y ) ? ?
? a pair drawn according to ?
?. The generalization error of a
classifier f : X ? Y is given by err?? (f ) := P?? (Y 6= f (X)),
P and its empirical error with respect to
a labeled set S 0 ? X ? Y is given by err(f,
c S 0 ) := |S10 | (x,y)?S 0 1[y 6= f (x)]. The optimal Bayes
risk of ?
? is R??? := inf err?? (f ), where the infimum is taken over all measurable classifiers f : X ? Y.
We say that ?
? is realizable when R??? = 0. We omit the overline in ?
? in the sequel when there is no
ambiguity.
3
For a finite labeled set S ? X ? Y and any x ? X , let Xnn (x, S) be the nearest neighbor of x with
respect to S and let Ynn (x, S) be the nearest neighbor label of x with respect to S:
(Xnn (x, S), Ynn (x, S)) := argmin ?(x, x0 ),
(x0 ,y 0 )?S
where ties are broken arbitrarily. The 1-NN classifier induced by S is denoted by hS (x) :=
Ynn (x, S). The set of points in S, denoted by X = {X1 , . . . , X|S| } ? X , induces
a Voronoi partition of X , V(X) := {V1 (X), . . . , V|S| (X)}, where each Voronoi cell is
Vi (X) := {x ? X : argminj?{1,...,|S|} ?(x, Xj ) = i}. By definition, ?x ? Vi (X), hS (x) = Yi .
A 1-NN algorithm is a mapping from an i.i.d. labeled sample Sn ? ?
?n to a labeled set Sn0 ? X ? Y,
yielding the 1-NN classifier hSn0 . While the classic 1-NN algorithm sets Sn0 := Sn , in this work we
study a compression-based algorithm which sets Sn0 adaptively, as discussed further below.
A 1-NN algorithm is strongly Bayes-consistent on ?
? if err(hSn0 ) converges to R? almost surely,
that is P[limn?? err(hSn0 ) = R? ] = 1. An algorithm is weakly Bayes-consistent on ?
? if err(hSn0 )
converges to R? in expectation, limn?? E[err(hSn0 )] = R? . Obviously, the former implies the
latter. We say that an algorithm is Bayes-consistent on a metric space if it is Bayes-consistent on all
distributions in the metric space.
A convenient property that is used when studying the Bayes-consistency of algorithms in metric
spaces is the doubling dimension. Denote the open ball of radius r around x by Br (x) := {x0 ?
?r (x) denote the corresponding closed ball. The doubling dimension of a
X : ?(x, x0 ) < r} and let B
metric space (X , ?) is defined as follows. Let n be the smallest number such that every ball in X can
be covered by n balls of half its radius, where all balls are centered at points of X . Formally,
n := min{n ? N : ?x ? X , r > 0, ?x1 , . . . , xn ? X s.t. Br (x) ? ?ni=1 Br/2 (xi )}.
Then the doubling dimension of (X , ?) is defined by ddim(X , ?) := log2 n.
For an integer n, let [n] := {1, . . . , n}. Denote the set of all index vectors of length d by In,d :=
[n]d . Given a labeled set Sn = (Xi , Yi )i?[n] and any i = {i1 , . . . , id } ? In,d , denote the subsample of Sn indexed by i by Sn (i) := {(Xi1 , Yi1 ), . . . , (Xid , Yid )}. Similarly, for a vector Y 0 =
{Y10 , . . . , Yd0 } ? Y d , denote by Sn (i, Y 0 ) := {(Xi1 , Y10 ), . . . , (Xid , Yd0 )}, namely the sub-sample
of Sn as determined by i where the labels are replaced with Y 0 . Lastly, for i, j ? In,d , we denote
Sn (i; j) := {(Xi1 , Yj1 ), . . . , (Xid , Yjd )}.
3
1-NN majority-based compression
In this work we consider the 1-NN majority-based compression algorithm proposed in [25], which
we refer to as KSU. This algorithm is based on constructing ?-nets at different scales; for ? > 0
and A ? X , a set X ? A is said to be a ?-net of A if ?a ? A, ?x ? X : ?(a, x) ? ? and for all
x 6= x0 ? X, ?(x, x0 ) > ?.3
The algorithm (see Alg. 1) operates as follows. Given an input sample Sn , whose set of points is
denoted Xn = {X1 , . . . , Xn }, KSU considers all possible scales ? > 0. For each such scale it
constructs a ?-net of Xn . Denote this ?-net by X(?) := {Xi1 , . . . , Xim }, where m ? m(?) denotes
its size and i ? i(?) := {i1 , . . . , im } ? In,m denotes the indices selected from Sn for this ?-net.
For every such ?-net, the algorithm attaches the labels Y 0 ? Y 0 (?) ? Y m , which are the empirical
majority-vote labels in the respective Voronoi cells in the partition V(X(?)) = {V1 , . . . , Vm }.
Formally, for i ? [m],
Yi0 ? argmax |{j ? [n] | Xj ? Vi , Yj = y}|,
(1)
y?Y
where ties are broken arbitrarily. This procedure creates a labeled set Sn0 (?) := Sn (i(?), Y 0 (?)) for
every relevant ? ? {?(Xi , Xj ) | i, j ? [n]} \ {0}. The algorithm then selects a single ?, denoted
? ? ? ?n? , and outputs hSn0 (? ? ) . The scale ? ? is selected so as to minimize a generalization error
bound, which upper bounds err(Sn0 (?)) with high probability. This error bound, denoted Q in the
algorithm, can be derived using a compression-based analysis, as described below.
3
For technical reasons, having to do with the construction in Sec. 6, we depart slightly from the standard
definition of a ?-net X ? A. The classic definition requires that (i) ?a ? A, ?x ? X : ?(a, x) < ? and (ii)
?x 6= x0 ? X : ?(x, x0 ) ? ?. In our definition, the relations < and ? in (i) and (ii) are replaced by ? and >.
4
Algorithm 1 KSU: 1-NN compression-based algorithm
Require: Sample Sn = (Xi , Yi )i?[n] , confidence ?
Ensure: A 1-NN classifier
1: Let ? := {?(Xi , Xj ) | i, j ? [n]} \ {0}
2: for ? ? ? do
3:
Let X(?) be a ?-net of {X1 , . . . , Xn }
4:
Let m(?) := |X(?)|
5:
For each i ? [m(?)], let Yi0 be the majority label in Vi (X(?)) as defined in Eq. (1)
6:
Set Sn0 (?) := (X(?), Y 0 (?))
7: end for
8: Set ?(?) := err(h
c Sn0 (?) , Sn )
9: Find ?n? ? argmin??? Q(n, ?(?), 2m(?), ?), where Q is, e.g., as in Eq. (2)
10: Set Sn0 := Sn0 (?n? )
11: return hSn0
m
We say that a mapping Sn 7? Sn0 is a compression scheme if there is a function C : ??
m=0 (X ?Y) ?
2X ?Y , from sub-samples to subsets of X ? Y, such that for every Sn there exists an m and a sequence
i ? In,m such that Sn0 = C(Sn (i)). Given a compression scheme Sn 7? Sn0 and a matching function
C, we say that a specific Sn0 is an (?, m)-compression of a given Sn if Sn0 = C(Sn (i)) for some
i ? In,m and err(h
c Sn0 , Sn ) ? ?. The generalization power of compression was recognized by [17]
and [22]. Specifically, it was shown in [21, Theorem 8] that if the mapping Sn 7? Sn0 is a compression
scheme, then with probability at least 1 ? ?, for any Sn0 which is an (?, m)-compression of Sn ? ?
?n ,
we have (omitting the constants, explicitly provided therein, which do not affect our analysis)
s
nm
? log(n) + log(1/?)
n
m log(n) + log(1/?)
err(hSn0 ) ?
? + O(
) + O( n?m
). (2)
n?m
n?m
n?m
Defining Q(n, ?, m, ?) as the RHS of Eq. (2) provides KSU with a compression bound. The following
proposition shows that KSU is a compression scheme, which enables us to use Eq. (2) with the
appropriate substitution.4
Proposition 1. The mapping Sn 7? Sn0 defined by Alg. 1 is a compression scheme whose output Sn0
is a (err(h
c Sn0 ), 2|Sn0 |)-compression of Sn .
? i , Y?i )i?[2m] ) = (X
? i , Y?i+m )i?[m] , and observe that for all
Proof. Define the function C by C((X
0
Sn , we have Sn = C(Sn (i(?); j(?))), where i(?) is the ?-net index set as defined above, and
j(?) = {j1 , . . . , jm(?) } ? In,m(?) is some index vector such that Yi0 = Yji for every i ? [m(?)].
Since Yi0 is an empirical majority vote, clearly such a j exists. Under this scheme, the output Sn0 of
this algorithm is a (err(h
c Sn0 ), 2|Sn0 |)-compression.
KSU is efficient, for any countable Y. Indeed, Alg. 1 has a naive runtime complexity of O(n4 ), since
O(n2 ) values of ? are considered and a ?-net is constructed for each one in time O(n2 ) (see [20,
Algorithm 1]). Improved runtimes can be obtained, e.g., using the methods in [29, 18]. In this work
we focus on the Bayes-consistency of KSU, rather than optimize its computational complexity. Our
Bayes-consistency results below hold for KSU, whenever the generalization bound Q(n, ?, m, ?n )
satisfies the following properties:
Property 1 For any integer n and ? ? (0, 1), with probability 1 ? ? over the i.i.d. random sample
Sn ? ?
?n , for all ? ? [0, 1] and m ? [n]: If Sn0 is an (?, m)-compression of Sn , then
err(hSn0 ) ? Q(n, ?, m, ?).
Property 2 Q is monotonically increasing in ? and in m.
Property 3 There is a sequence {?n }?
n=1 , ?n ? (0, 1) such that
lim
P?
n=1 ?n
< ? and for all m,
sup (Q(n, ?, m, ?n ) ? ?) = 0.
n?? ??[0,1]
4
In [25] the analysis was based on compression with side information, and does not extend to infinite Y.
5
The compression bound in Eq. (2) clearly
P?satisfies these properties. Note that Property 3 is satisfied
by Eq. (2) using any convergent series n=1 ?n < ? such that ?n = e?o(n) ; in particular, the decay
of ?n cannot be too rapid.
4
Main results
In this section we describe our main results. The proofs appear in subsequent sections. First, we show
that KSU is Bayes-consistent if the instance space has a finite doubling dimension. This contrasts
with classical 1-NN, which is only Bayes-consistent if the distribution is realizable.
Theorem 2. Let (X , ?) be a metric space with a finite doubling-dimension. Let Q be a generalization
bound that satisfies Properties 1-3, and let ?n be as stipulated by Property 3 for Q. If the input
confidence ? for input size n is set to ?n , then the 1-NN classifier hSn0 (?n? ) calculated by KSU is
strongly Bayes consistent on (X , ?): P(limn?? err(hSn0 ) = R? ) = 1.
The proof, provided in Sec. 5, closely follows the line of reasoning in [27], where the strong Bayesconsistency of an adaptive margin-regularized 1-NN algorithm was proved, but with several crucial
differences. In particular, the generalization bounds used by KSU are purely compression-based, as
opposed to the Rademacher-based generalization bounds used in [27]. The former can be much tighter
in practice and guarantee Bayes-consistency of KSU even for countably many labels. This however
requires novel technical arguments, which are discussed in detail in Appendix B.1 in [26]. Moreover,
since the compression-based bounds do not explicitly depend on ddim, they can be used even when
ddim is infinite, as we do in Theorem 4 below. To underscore the subtle nature of Bayes-consistency,
we note that the proof technique given here does not carry to an earlier algorithm, suggested in [20,
Theorem 4], which also uses ?-nets. It is an open question whether the latter is Bayes-consistent.
Next, we study Bayes-consistency of KSU in infinite dimensions (i.e., with ddim = ?) ? in particular, in a setting where k-NN was shown by [9] not to be Bayes-consistent. Indeed, a straightforward
application of [9, Lemma A.1] yields the following result.
Theorem 3 (C?rou and Guyader [9]). There exists an infinite dimensional separable metric space
(X , ?) and a realizable distribution ?
? over X ? {0, 1} such that no kn -NN learner satisfying
kn /n ? 0 when n ? ? is Bayes-consistent under ?
?. In particular, this holds for any space and
realizable distribution ?
? that satisfy the following condition: The set C of points labeled 1 by ?
?
satisfies
?r (x))
?(C ? B
?(C) > 0
and
?x ? C, lim
= 0.
(3)
?
r?0
?(Br (x))
Since ?(C) > 0, Eq. (3) constitutes a violation of the Besicovitch covering property. In doubling
spaces, the Besicovitch covering theorem precludes such a violation [15]. In contrast, as [35, 36]
show, in infinite-dimensional spaces this violation can in fact occur. Moreover, this is not an isolated
pathology, as this property is shared by Gaussian Hilbert spaces [45].
At first sight, Eq. (3) might appear to thwart any 1-NN algorithm applied to such a distribution.
However, the following result shows that this is not the case: KSU is Bayes-consistent on a distribution
with this property.
Theorem 4. There is a metric space equipped with a realizable distribution for which KSU is weakly
Bayes-consistent, while any k-NN classifier necessarily is not.
The proof relies on a classic construction of Preiss [35] which satisfies Eq. (3). We show that the
structure of the construction, combined with the packing and covering properties of ?-nets, imply that
the majority-vote classifier induced by any ?-net with a sufficienlty small ? approaches the Bayes
error. To contrast with Theorem 4, we next show that on the same construction, not all majority-vote
Voronoi partitions succeed. Indeed, if the packing property of ?-nets is relaxed, partition sequences
obstructing Bayes-consistency exist.
Theorem 5. For the example constructed in Theorem 4, there exists a sequence of Voronoi partitions
with a vanishing diameter such that the induced true majority-vote classifiers are not Bayes consistent.
The above result also stands in contrast to [14, Theorem 21.2], showing that, unlike in finite dimensions, the partitions? vanishing diameter is insufficient to establish consistency when ddim = ?. We
conclude the main results by posing intriguing open problems.
6
Open problem 1. Does there exist a metric probability space on which some k-NN algorithm is
consistent while KSU is not? Does there exist any separable metric space on which KSU fails?
Open problem 2. C?rou and Guyader [9] distill a certain Besicovitch condition which is necessary
and sufficient for k-NN to be Bayes-consistent in a metric space. Our Theorem 4 shows that the
Besicovitch condition is not necessary for KSU to be Bayes-consistent. Is it sufficient? What is a
necessary condition?
5
Bayes-consistency of KSU in finite dimensions
In this section we give a high-level proof of Theorem 2, showing that KSU is strongly Bayesconsistent in finite-dimensional metric spaces. A fully detailed proof is given in Appendix B in
[26].
Recall the optimal empirical error ?n? ? ?(?n? ) and the optimal compression size m?n ? m(?n? ) as
computed by KSU. As shown in Proposition 1, the sub-sample Sn0 (?n? ) is an (?n? , 2m?n )-compression
of Sn . Abbreviate the compression-based generalization bound used in KSU by
Qn (?, m) := Q(n, ?, 2m, ?n ).
To show Bayes-consistency, we start by a standard decomposition of the excess error over the optimal
Bayes into two terms:
err(hSn0 (?n? ) ) ? R? = err(hSn0 (?n? ) ) ? Qn (?n? , m?n ) + Qn (?n? , m?n ) ? R? =: TI (n) + TII (n),
and show that each term decays to zero with probability one. For the first term, Property 1 for Q,
together with the Borel-Cantelli lemma, readily imply lim supn?? TI (n) ? 0 with probability one.
The main challenge is showing that lim supn?? TII (n) ? 0 with probability one. We do so in
several stages:
1. Loosely speaking, we first show (Lemma 10) that the Bayes error R? can be well approximated using 1-NN classifiers defined by the true (as opposed to empirical) majority-vote
labels over fine partitions of X . In particular, this holds for any partition induced by a ?-net
of X with a sufficiently small ? > 0. This approximation guarantee relies on the fact that in
finite-dimensional spaces, the class of continuous functions with compact support is dense
in L1 (?) (Lemma 9).
2. Fix ?? > 0 sufficiently small such that any true majority-vote classifier induced by a ?? -net
has a true error close to R? , as guaranteed by stage 1. Since for bounded subsets of finitedimensional spaces the size of any ?-net is finite, the empirical error of any majority-vote
?-net almost surely converges to its true majority-vote error as the sample size n ? ?. Let
n(?
? ) sufficiently large such that Qn(?? ) (?(?
? ), m(?
? )) as computed by KSU for a sample of
0
size n(?
? ) is a reliable estimate for the true error of hSn(?
(?
?).
?)
3. Let ?? and n(?
? ) be as in stage 2. Given a sample of size n = n(?
? ), recall that KSU
selects an optimal ? ? such that Qn (?(?), m(?)) is minimized over all ? > 0. For margins
? ?? , which are prone to over-fitting, Qn (?(?), m(?)) is not a reliable estimate for
hSn0 (?) since compression may not yet taken place for samples of size n. Nevertheless, these
margins are discarded by KSU due to the penalty term in Q. On the other hand, for ?-nets
with margin ? ?? , which are prone to under-fitting, the true error is well estimated by
Qn (?(?), m(?)). It follows that KSU selects ?n? ? ?? and Qn (?n? , m?n ) ? R? , implying
lim supn?? TII (n) ? 0 with probability one.
As one can see, the assumption that X is finite-dimensional plays a major role in the proof. A simple
argument shows that the family of continuous functions with compact support is no longer dense
in L1 in infinite-dimensional spaces. In addition, ?-nets of bounded subsets in infinite dimensional
spaces need no longer be finite.
6
On Bayes-consistency of NN algorithms in infinite dimensions
In this section we study the Bayes-consistency properties of 1-NN algorithms on a classic infinitedimensional construction of Preiss [35], which we describe below in detail. This construction was
7
z1:k?2
?k?1
z1:k?1
?k
?k
z1:k
?k
?k
?k
?k
z
C = Z?
?? (z) for some z ? C.
Figure 1: Preiss?s construction. Encircled is the closed ball B
k?1
first introduced as a concrete example showing that in infinite-dimensional spaces the Besicovich
covering theorem [15] can be strongly violated, as manifested in Eq. (3).
Example 1 (Preiss?s construction). The construction (see Figure 1) defines an infinite-dimensional
metric space (X , ?) and a realizable measure ?
? over X ? Y with the binary label set Y = {0, 1}.
It relies on two sequences: a sequence of natural numbers {Nk }k?N and a sequence of positive
numbers {ak }k?N . The two sequences should satisfy the following:
P?
limk?? ak N1 . . . Nk+1 = ?; and limk?? Nk = ?. (4)
k=1 ak N1 . . . Nk = 1;
Q
These properties are satisfied, for instance, by setting Nk := k! and ak := 2?k / i?[k] Ni . Let Z0
be the set of all finite sequences (z1 , . . . , zk )k?N of natural numbers such that zi ? Ni , and let Z?
be the set of all infinite sequences (z1 , z2 , . . . ) of natural numbers such that zi ? Ni .
Define the example space X := Z0 ? Z? and denote ?k := 2?k , where ?? := 0. The metric ? over
X is defined as follows: for x, y ? X , denote by x ? y their longest common prefix. Then,
?(x, y) = (?|x?y| ? ?|x| ) + (?|x?y| ? ?|y| ).
It can be shown (see [35]) that ?(x, y) is a metric; in fact, it embeds isometrically into the square
norm metric of a Hilbert space.
To define ?, the marginal measure over X , let ?? be the uniform product distribution measure
over Z? , that is: for all i ? N, each zi in the sequence z = (z1 , z2 , . . . ) ? Z? is independently
drawn from a uniform distribution over [Ni ]. Let ?0 be an atomic measure on Z0 such that for all
z ? Z0 , ?0 (z) = a|z| . Clearly, the first condition in Eq. (4) implies ?0 (Z0 ) = 1. Define the marginal
probability measure ? over X by
?A ? Z0 ? Z? ,
?(A) := ??? (A) + (1 ? ?)?0 (A).
In words, an infinite sequence is drawn with probability ? (and all such sequences are equally likely),
or else a finite sequence is drawn (and all finite sequences of the same length are equally likely).
Define the realizable distribution ?
? over X ? Y by setting the marginal over X to ?, and by setting
the label of z ? Z? to be 1 with probability 1 and the label of z ? Z0 to be 0 with probability 1.
As shown in [35], this construction satisfies Eq. (3) with C = Z? and ?(C) = ? > 0. It follows
from Theorem 3 that no k-NN algorithm is Bayes-consistent on it. In contrast, the following theorem
shows that KSU is weakly Bayes-consistent on this distribution. Theorem 4 immediately follows
from the this result.
Theorem 6. Assume (X , ?), Y and ?
? as in Example 1. KSU is weakly Bayes-consistent on ?
?.
The proof, provided in Appendix C in [26], first characterizes the Voronoi cells for which the true
majority-vote yields a significant error for the cell (Lemma 15). In finite-dimensional spaces, the total
measure of all such ?bad? cells can be made arbitrarily close to zero by taking ? to be sufficiently
small, as shown in Lemma 10 of Theorem 2. However, it is not immediately clear whether this can
be achieved for the infinite dimensional construction above.
Indeed, we expect such bad cells, due to the unintuitive property that for any x ? C, we have
?? (x) ? C)/?(B
?? (x)) ? 0 when ? ? 0, and yet ?(C) > 0. Thus, if for example a significant
?(B
8
?? (x) with
portion of the set C (whose label is 1) is covered by Voronoi cells of the form V = B
x ? C, then for all sufficiently small ?, each one of these cells will have a true majority-vote 0. Thus
a significant portion of C would be misclassified. However, we show that by the structure of the
construction, combined with the packing and covering properties of ?-nets, we have that in any ?-net,
the total measure of all these ?bad? cells goes to 0 when ? ? 0, thus yielding a consistent classifier.
Lastly, the following theorem shows that on the same construction above, when the Voronoi partitions
are allowed to violate the packing property of ?-nets, Bayes-consistency does not necessarily hold.
Theorem 5 immediately follows from the following result.
Theorem 7. Assume (X , ?), Y and ?
? as in Example 1. There exists a sequence of Voronoi partitions
(Pk )k?N of X with maxV ?Pk diam(V ) ? ?k such that the sequence of true majority-vote classifiers
(hPk )k?N induced by these partitions is not Bayes consistent: lim inf k?? err(hPk ) = ? > 0.
The proof, provided in Appendix D, constructs a sequence of Voronoi partitions, where each partition
Pk has all of its impure Voronoi cells (those with both 0 and 1 labels) being bad. In this case, C is
incorrectly classified by hPk , yielding a significant error. Thus, in infinite-dimensional metric spaces,
the shape of the Voronoi cells plays a fundamental role in the consistency of the partition.
Acknowledgments. We thank Fr?d?ric C?rou for the numerous fruitful discussions and helpful
feedback on an earlier draft. Aryeh Kontorovich was supported in part by the Israel Science
Foundation (grant No. 755/15), Paypal and IBM. Sivan Sabato was supported in part by the Israel
Science Foundation (grant No. 555/15).
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11
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6,363 | 6,756 | A Scale Free Algorithm for Stochastic Bandits with
Bounded Kurtosis
Tor Lattimore?
[email protected]
Abstract
Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a
bound on the payoffs, or the knowledge of a variance or subgaussian parameter.
The notable exceptions are the analysis of Gaussian bandits with unknown mean
and variance by Cowan et al. [2015] and of uniform distributions with unknown
support [Cowan and Katehakis, 2015]. The results derived in these specialised
cases are generalised here to the non-parametric setup, where the learner knows
only a bound on the kurtosis of the noise, which is a scale free measure of the
extremity of outliers.
1
Introduction
SpaceBandits is a fictional company that specialises in optimising the power output of satellitemounted solar panels. The data science team wants to use a bandit algorithm to adjust the knobs
on a legacy satellite, but they don?t remember the units of the sensors, and have limited knowledge
about the noise distribution of the panel output or sensors. The SpaceBandits data science team
searches the literature for an algorithm that does not depend on the scale or location of the means of
the arms, and find this simple paper, in NIPS 2017.
It turns out that logarithmic regret is possible for finite-armed bandits with no assumptions on the
noise of the payoffs except for a known finite bound on the kurtosis, which corresponds to knowing
the likelihood/magnitude of outliers [DeCarlo, 1997]. Importantly, the kurtosis is independent of
the location of the mean and scale of the central tendency (the variance). This generalises the ideas
of Cowan et al. [2015] beyond the Gaussian case with unknown mean and variance to the nonparametric setting.
The setup is as follows. Let k ? 2 be the number of bandits (or arms). In each round 1 ? t ? n
the player should choose an action At ? {1, . . . , k} and subsequently receives a reward Xt ? ?At ,
where ?1 , . . . , ?k are a set of distributions that are not known in advance. Let ?i be the mean payoff
of the ith arm and ?? = maxi ?i and ?i = ?? ? ?i . The regret measures the expected deficit of the
player relative to the optimal choice of distribution:
" n
#
X
Rn = E
?A t .
(1)
t=1
The table below summarises many of the known results on the optimal achievable asymptotic regret
under different assumptions on (?i )i . A reference for each of the upper bounds is given in Table 1,
while the lower bounds are mostly due to Lai and Robbins [1985] and Burnetas and Katehakis
[1996]. An omission from the table is when the distributions are known to lie in a single-parameter
exponential family (which does not fit well with the columns). Details are by Capp?e et al. [2013].
?
Now at DeepMind, London.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1
Assumption
Known
Unknown
Bernoulli
Supp(?i ) ? {0, 1}
?i ? [0, 1]
Lai and Robbins [1985]
2
limn?? Rn / log(n)
X
1
i:?i >0
d(?i , ?? )
Supp(?i ) ? [0, 1]
distribution
it?s complicated
Supp(?i ) ? A
|A| < ?
distribution
it?s complicated
Burnetas and Katehakis [1996]
Semi-bounded
Supp(?i ) ? (??, 1]
distribution
it?s complicated
?i = N (?i , ?i2 )
?i ? R
?i = U(ai , bi )
ai , bi
Bounded
Honda and Takemura [2010]
3
4
Discrete
Honda and Takemura [2015]
5
Gaussian (known var.)
Katehakis and Robbins [1995]
6
Uniform
Cowan and Katehakis [2015]
7
log M?i (?) ?
Subgaussian
Bubeck and Cesa-Bianchi [2012]
?2 ?i2
2
??
distribution
X 2?i2
?i
i:?i >0
X
?i
2?i
i:?i >0 log 1 + b ?a
i
i
X 2?i2
?i
i:?i >0
?
8
Known variance
V[?i ] ? ?i2
distribution
?i = N (?i , ? 2 )
?i ? R, ?i2 > 0
Bubeck et al. [2013]
9
Gaussian
Cowan et al. [2015]
?
X ?i2
?
O?
?i
i:?i >0
X
2?i
i:?i >0
log (1 + ?2i /?i2 )
d(p, q) = p log(p/q) + (1 ? p) log((1 ? p)/(1 ? q)) and M? (?) = EX?? exp((X ? ?)?) with ? the
mean of ? is the centered moment generating function. All asymptotic results are optimal except for the
grey cells.
Table 1: Typical distributional assumptions and asymptotic regret
With the exception of rows 6 and 9 in Table 1, all entries essentially depend on some kind of scale
parameter. Missing is an entry for a non-parametric assumption that is scale free. This paper fills
that gap with the following assumption and regret guarantee.
Assumption 1. There exists a known ?? ? R such that for all 1 ? i ? k, the kurtosis of X ? ?i is
at most Kurt[X] = E[(X ? E[X])4 ]/V[X]2 ? ?? .
Theorem 2. If Assumption 1 holds, then the algorithm described in ?2 satisfies
X
?i2
Rn
lim sup
?C
?i ?? ? 1 + 2 ,
?i
n?? log(n)
i:?i >0
where ?i2 is the variance of ?i and C > 0 is a universal constant.
What are the implications of this result? The first point is that the algorithm in ?2 is scale and
translation invariant in the sense that its behaviour does not change if the payoffs are multiplied by
a positive constant or shifted. The regret also depends appropriately on the scale so that multiplying
the rewards of all arms by a positive constant factor also multiplies the regret by this factor. As far
as I know, this is the first scale free bandit algorithm for a non-parametric class. The assumption
on the boundedness of the kurtosis is much less restrictive than assuming an exact Gaussian model
(which has kurtosis 3) or uniform (kurtosis 9/5). See Table 2 for other examples.
As mentioned, the kurtosis is a measure of the likelihood/existence of outliers of a distribution,
and it makes intuitive sense that a bandit strategy might depend on some kind of assumption on
this quantity. How else to know whether or not to cease exploring an unpromising action? The
assumption can also be justified from a mathematical perspective. If the variance of an arm is not
assumed known, then calculating confidence intervals requires an estimate of the variance from the
data. Let X, X1 , X2 , . . . , Xn be a sequence of i.i.d. centered random variables with variance ? 2 and
2
kurtosis ?. A reasonable estimate of ? 2 is
n
?
?2 =
1X 2
X .
n t=1 t
(2)
Clearly this estimator is unbiased and has variance
? 4 (? ? 1)
E[X 4 ] ? E[X 2 ]2
=
.
V[?
?2 ] =
n
n
Therefore, if we are to expect good estimaDistribution Parameters
Kurtosis
tion of ? 2 , then the kurtosis should be fi2
nite. Note that if ? 2 is estimated by (2), then
Gaussian
? ? R, ? > 0 3
the central limit theorem combined with fi1?3?(1??)
Bernoulli
? ? [0, 1]
nite kurtosis is enough for an estimation er?(1??)
ror of O(? 2 ((? ? 1)/n)1/2 ) asymptotically.
Exponential
?>0
9
For bandits, however, finite-time bounds are
Laplace
? ? R, b > 0
9
required, which are not available using (2)
without additional moment assumptions (for
Uniform
a<b?R
9/5
example, on the moment generating function). An example demonstrating the necesTable 2: Kurtosis
sity of the limit in the standard central limit
theorem is as follows. Suppose that X1 , . . . , Xn are Bernoulli with bias p = 1/n, then for large n
the distribution of the sum is closely approximated by a Poisson distribution with parameter 1, which
is very different to a Gaussian. Finite kurtosis alone is enough if the classical empirical estimator is
replaced by a robust estimator such as the median-of-means estimator [Alon et al., 1996] or Catoni?s
estimator [Catoni, 2012]. Of course, if the kurtosis were not known, then you could try and estimate
it with assumptions on the eighth moment, and so on. Is there any justification to stop here? The
main reason is that this seems like a useful place to stop. Large classes of distributions have known
bounds on their kurtosis (see table) and the independence of scale is a satisfying property.
Contributions The main contribution is the new assumption, algorithm, and the proof of Theorem
2 (see ?2). The upper bound is also complemented by an asymptotic lower bound (?3) that applies
to all strategies with sub-polynomial regret and all bandit problems with bounded kurtosis.
Pt
Additional notation Let Ti (t) = s=1 1 {As = i} be the number of times arm i has been played
after round t. For measures P, Q on the same probability space, KL(P, Q) is the relative entropy
between P and Q and ?2 (P, Q) is the ?2 distance. The following lemma is well known.
Lemma 3. Let X1 , X2 be independent random variables with Xi having variance ?i2 and kurtosis
?i < ? and skewness ?i = E[(Xi ? E[Xi ])3 /?i3 ], then:
?
? 4 (?1 ? 3) + ?24 (?2 ? 3)
(a) Kurt[X1 + X2 ] = 3 + 1
(b) ?1 ? ?1 ? 1 .
2
2
2
(?1 + ?2 )
2
Algorithm and upper bound
Like the robust upper confidence bound algorithm by Bubeck et al. [2013], the new algorithm makes
use of the robust median-of-means estimator.
Median-of-means estimator Let Y1 , Y2 , . . . , Yn be a sequence of independent and identically
distributed random variables. The median-of-means estimator first partitions the data into m blocks
of equal size (up to rounding errors). The empirical mean of each block is then computed and the
estimate is the median of the means of each of the blocks. The number of blocks depends on the
desired confidence level and should be O(log(1/?)). The median-of-means estimator at confidence
level ? ? (0, 1) is denoted by [
MM? ((Yt )nt=1 ).
Lemma 4 (Bubeck et al. 2013). Let Y1 , Y2 , . . . , Yn be a sequence of independent and identically
distributed random variables with mean ? and variance ? 2 < ?.
s
!
?2
C2
[
n
P MM? ((Yt )t=1 ) ? ? ? C1
log
? ?,
n
?
3
where C1 =
?
12 ? 16 and C2 = exp(1/8) are universal constants.
Upper confidence bounds The new algorithm is a generalisation of UCB, but with optimistic
estimates of the mean and variance using confidence bounds about the median-of-means estimator.
Let ? ? (0, 1) and Y1 , Y2 , . . . , Yt be a sequence of independent and identically distributed random
variables with mean ?, variance ? 2 and kurtosis ? < ?? . Furthermore, let
s
(
)
?
?t2 ((Ys )ts=1 , ?, ?)
C2
t
t
[
?
?((Ys )s=1 , ?) = sup ? ? R : ? ? MM? (Ys )s=1 + C1
log
.
t
?
t
[
MM? (Ys ? ?)2 s=1
.
where ?
? 2 ((Ys )ts=1 , ?, ?) =
q
?? ?1
C2
max 0, 1 ? C1
log ?
t
Note that ?
?((Ys )ts=1 , ?) may be (positive) infinity if t is insufficiently large. The computation of
?
?(?) seems non-trivial and is discussed in the summary at the end of the paper where a roughly
equivalent and efficiently computable alternative is given. The following two lemmas show that ?
? is
indeed optimistic with high probability, and also that it concentrates with reasonable speed around
the true mean.
Lemma 5. P ?
?((Ys )ts=1 , ?) ? ? ? 2? .
Proof. By Lemma 4 and the fact that V[(Ys ? ?)2 ] = ? 4 (? ? 1) ? ? 4 (?? ? 1) it holds with
probability at least 1 ? ? that ?
? 2 ((Ys )ts=1 , ?, ?) ? ? 2 . Another application of Lemma 4 along with
a union bound ensures that with probability at least 1 ? 2?,
s
s
2
C
C2
?
?
?t2 ((Ys )ts=1 , ?, ?)
2
t
[
? C1
.
log
log
MM? ((Ys )s=1 ) ? C1
t
?
t
?
Therefore with probability at least 1 ? 2? the true mean ? is in the set of which ?
? is the supremum
and in this case ?
?((Ys )ts=1 , ?) ? ? as required.
Lemma 6. Let ?t be monotone decreasing and ?
?t = ?
?((Ys )ts=1 , ?t ). Then there exists a universal
constant C3 > 0 such that for any ? > 0,
n
n
X
X
?2
C2
P (?
?t ? ? + ?) ? C3 max ?? ? 1, 2 log
+2
?t .
?
?n
t=1
t=1
Proof. First, by Lemma 4
n
X
s
t
P [
MM?t (Ys )s=1 ? ? ? C1
t=1
?2
log
t
C2
?t
!
?
n
X
?t .
(3)
t=1
Similarly,
n
X
s
t
P [
MM?t (Ys ? ?)2 s=1 ? ? 2 ? C1 ? 2
t=1
?? ? 1
log
t
Suppose that t is a round where all of the following hold:
s
?2
log
t
C2
.
?t
s
?? ? 1
C2
t
[
2
2
2
(b) MM?t (Ys ? ?) s=1 ? ? < C1 ?
log
.
t
?t
C2
(c) t ? 16C12 (?? ? 1) log
.
?t
t
(a) [
MM?t (Ys )s=1 ? ? < C1
4
C2
?
!
?
n
X
t=1
?t .
(4)
t
Abbreviating ?
?t2 = ?
? 2 ((Ys )ts=1 , ?
?t , ?t ) and ?
?t = [
MM?t (Ys )s=1 ,
t
(Ys ? ?
?s )2 s=1
t
[
r
?
?t2 =
?t )2 s=1
? 2 MM?t (Ys ? ?
1 ? C1 ??t?1 log C?t2
t
? 4[
MM?t (Ys ? ?)2 s=1 + 4(?
?t ? ?)2
t
? 4[
MM?t (Ys ? ?)2 s=1 + 8(?
?t ? ?
?t )2 + 8(?
?t ? ?)2
s
?? ? 1
C2
8C12 (? 2 + ?
?t2 )(?? ? 1)
C2
11 2 ?
?2
2
2
< 4? + 4C1 ?
log
+
log
?
? + t ,
t
?t
t
?t
2
2
[
MM?t
where the first inequality follows from (c), the second since (x ? y)2 ? 2x2 + 2y 2 and the fact that
[
MM? ((aYs + b)ts=1 ) = a [
MM? ((Ys )ts=1 ) + b .
The third inequality again uses (x ? y)2 ? 2x2 + 2y 2 , while the last uses the definition of ?
?t and
(a,b). Therefore ?
?t2 ? 11? 2 , which means that if (a,b,c) and additionally
19C12 ? 2
log
(d) t ?
?2
1
?n
.
s
s
?
?t2
C2
?2
C2
Then |?
?t ? ?| ? |?
?t ? ?
?t | + |?
?t ? ?| < C1
log
log
+ C1
t
?n
t
?n
s
s
C2
C2
11? 2
?2
log
log
+ C1
? ?.
? C1
t
?n
t
?n
Combining this with (3) and (4) and choosing C3 = 19C12 completes the result.
Algorithm and Proof of Theorem 2 Let ?t = 1/(t2 log(1+t)) and ?
?i (t) = ?
?((Xs )s?[t],As =i , ?t ).
In each round the algorithm chooses At = arg maxi?[k] ?
?i (t ? 1), where ties are broken arbitrarily.
Proof of Theorem 2. Assume without loss of generality that ?1 = ?? . Then suboptimal arm i is
only played in round t if either ?
?1 (t ? 1) ? ?1 or ?
?i (t ? 1) ? ?1 . Therefore
E[Ti (n)] ?
n
X
P (?
?1 (t ? 1) ? ?1 ) +
t=1
n
X
P (?
?i (t ? 1) ? ?1 and At = i)
(5)
t=1
The two sums are bounded using Lemmas 5 and 6 respectively:
n
X
P (?
?1 (t ? 1) ? ?1 ) ?
t=1
n X
t
X
P (?
?1 (t ? 1) ? ?1 and T1 (t ? 1) = u)
t=1 u=1
n X
t
X
?2
n
X
t=1
P (?
?i (t ? 1) ? ?1 and At = i) ?
t=1 u=1
n
X
?t = 2
n
X
t?t = o(log(n)) .
(By Lem. 5)
t=1
P (?
?i (t ? 1) ? ?i ? ?i )
t=1
n
X
?2
C2
? C3 max ?? ? 1, i2 log
+2
?t = o(log(n)) .
?i
?n
t=1
(By Lem. 6)
And the result follows by substituting the above bounds into Eq. (5) and then into the regret decomPk
position Rn = i=1 ?i E[Ti (n)].
5
3
Lower bound
Let H?? = {? : ? has kurtosis less than ?? } be the class of all distributions with kurtosis bounded
by ?? . Following the nomenclature of Lai and Robbins [1985], a bandit strategy is called consistent
over H if Rn = o(np ) for all p ? (0, 1) and bandits (?i )i with ?i ? H?? for all i. The next theorem
shows that the upper bound derived in the previous section is nearly tight up to constant factors. Let
H be a family of distributions and let (?i )i be a bandit with ?i ? H for all i. Burnetas and Katehakis
[1996] showed that for any consistent strategy,
for all i ? [k] :
lim inf
n??
?1
E[Ti (n)]
? inf KL(?i , ?i0 ) : ?i0 ? H and EX??i0 [X] > ??
. (6)
log(n)
In parameterised families of distributions, the optimisation problem can often be evaluated analytically (eg., Bernoulli, Gaussian with known variance, Gaussian with unknown variance, Exponential). For non-parametric families the calculation is much more challenging. The following
theorem takes the first steps towards understanding this problem for the class of distributions H??
for ?? ? 7/2.
Theorem 7. Let ?? ? 7/2 and ? > 0 and ? ? H?? with mean ?, variance ? 2 > 0 and kurtosis ?.
Then for appropriately chosen universal constant C, C 0 > 0,
7
1 ?
0
0
inf {KL(?, ? ) : ? ? H? and EX?? 0 [X] > ? + ?} ? min
.
,
5
?? ?
If additionally it holds that ? + C 0 ??1/2 (? + 1) ? ?? , then
inf {KL(?, ? 0 ) : ? 0 ? H? and EX?? 0 [X] > ? + ?} ? C
?2
?2
Therefore provided that ? ? H?? is not too close to the boundary of H?? in the sense that its kurtosis
is not too close to ?? , then the lower bound derived from Theorem 7 and Eq. (6) matches the upper
bound up to constant factors. This condition is probably necessary because distributions like the
Bernoulli with kurtosis close to ?? have barely any wiggle room to increase the mean without also
increasing the kurtosis.
Proof of Theorem 7. Let ?? = ? + ? for small ? > 0. Assume without loss of generality that ? is
centered and has variance ? 2 = 1, which can always be achieved by shifting and scaling (neither
effects the kurtosis or the relative entropy). The first part of the claim is established by considering
the perturbed distribution obtained by adding a Bernoulli ?outlier?. Let X be a random variable
sampled from ? and B be a Bernoulli with parameter p = min {?? , 1/?? }. Let Z = X + Y where
Y = ?? B/p. Then E[Z] = ?? > ? and
2
(1?p)?2?
1?6p(1?p)
??3+
p
p(1?p)
? ? 3 + V[Y ]2 (Kurt[Y ] ? 3)
Kurt[Z] = 3 +
=3+
2
2
2
(1 + V[Y ])
(1?p)??
1+
p
2
2
(1?p)??
1?6p(1?p)
?? ? 3 +
p
p(1?p)
?3+
? ?? ,
2
2
(1?p)??
1+
p
where the first inequality used Lemma 3 and the final inequality follows from simple case-based
analysis, calculus and the assumption that ?? ? 7/2 (see Lemma 9 in the appendix). Let ? 0 = L(Y )
be the law of Y . Then
Z
Z
d?
1
1
p
7
1
0
KL(?, ? ) =
log 0 d? ?
log
d? = log
?
? min ?? ,
.
d?
1?p
1?p
1?p
5
??
R
R
Taking the limit as ? tends to 0 completes the proof of the first part of the theorem. Moving onto the
second claim and using C for a universal positive constant that changes
from equation to equation.
?
Let a > 0 be a constant to be chosen later and A = {x : |x| ? a?} and A? = R ? A. Define
6
R
alternative measure ? 0 (E) = E (1 + g)d? where g(x) = (? + ?x)1 {x ? A} for some constants ?
and ? chosen so that
Z
Z
Z
g(x)d?(x) = ?
d?(x) + ?
xd?(x) = 0 .
A
Z R
ZA
Z
g(x)xd?(x) = ?
xd?(x) + ?
x2 d?(x) = ?? .
R
A
A
Solving for ? and ? shows that
??
?=R
and
2
?=?
??
R
xd?(x)
2 .
R
?(A) A x2 d?(x) ? A xd?(x)
A
( xd?(x))
x2 d?(x) ? A ?(A)
R
R
This implies that R d? 0 (x) = 1 and R xd? 0 (x) = ?? > ?. It remains to show that ? 0 is a
probability measure with kurtosis bounded by ?? . That ? 0 is a probability measure will follow from
the positivity of 1 ? g(?). The first step is to control each of the terms appearing in the definitions of
? = ?(x2 ? a?) ? 1/(?a2 ) and
? and ?. By Cauchy-Schwarz and Chebyshev?s inequalities, ?(A)
Z
Z
q
? ?1? 1 .
x2 d?(x) ? 1 ? ??(A)
x2 d?(x) = 1 ?
a
?
A
A
R
A
R
Similarly, since ? is centered,
q
Z
Z
1
? ? ?
xd?(x) = xd?(x) ? ? 2 ?(A)
.
?
a
?
A
A
Therefore by choosing a = 2 and using the fact that the kurtosis is always larger than 1,
R
?
xd?(x)
?? / ?
4?
A
? ??
|?| = ??
?
2
R
R
1
1
1
?(A) x2 d?(x) ?
?
a
1
?
1
?
?
2
2
xd?(x)
?a
a
a ?
A
A
??
??
? 6?? .
|?| = R
R
2 ?
1
1
xd?(x)
)
1 ? a ? ?a2 1?
1
2 d?(x) ? ( A
x
(
)
2
a ?
?(A)
A
Now g(x) is a linear function supported on compact set A, so
?
?
?
?
4??
1
max |g(x)| = max |g( a?)|, |g(? a?)| ? |?| + a?|?| ? ? + 6?? 2? ? ,
x?R
2
?
?
?
?1/2
where the last inequality follows by assuming that ?? ? ?/(4(2 + 3 2?)) = O(?
), which
is reasonable without loss of generality, since if ?? is larger than this quantity, then we would prefer
the bound that depends on ?? derived in the first part of the proof. The relative entropy between ?
and ? 0 is bounded by
2
Z
Z
d?(x)
g(x)2
0
0
2
0
?
1
d?
(x)
=
d?(x)
KL(?, ? ) ? ? (?, ? ) =
d? 0 (x)
R
A 1 + g(x)
Z
Z
Z
?2
g(x)2 d?(x) ? 4
?2 d?(x) + 4
? 2 x2 d?(x) ? 4?2 + 4? 2
A
A
A
4 ? 16?2?
?
+ 4 ? 36?2? ? C?2? .
?
In order to bound the kurtosis we need to evaluate the moments:
Z
Z
Z
Z
Z
?
x2 d? 0 =
x2 d? +
g(x)x2 d? = 1 + ?
x2 d?(x) + ?
x3 d?(x) ? 1 + C?? ? .
A
A
ZR
ZR
ZA
?
x2 d? 0 =
x2 d? +
g(x)x2 d? ? 1 ? C?? ? .
R
R
A
Z
Z
Z
Z
Z
?
x4 d? 0 =
x4 d? +
g(x)x4 d? = ? + ?
x4 d?(x) + ?
x5 d?(x) ? ? 1 + C?? ? .
R
R
A
A
A
Z
sZ
Z
?
x3 d? 0 (x) ?
x2 d? 0 (x)
x4 d? 0 (x) ? C? .
R
R
R
7
Therefore if ?0 is the kurtosis of ? 0 , then
R 4 0
R
R
R
x d? (x) ? 3?4? + 6?2? R x2 d? 0 (x) ? 4?? R x3 d? 0 (x)
(x ? ?? )4 d? 0 (x)
0
R
R
? = R
2 =
2
R
R
x2 d? 0 (x) ? ?2?
1 ? ?2? + ? A x2 d?(x) + ? A x3 d?(x)
R
Therefore
R
R
x4 d? 0 (x) ? 3?4? + 6?2? R x2 d? 0 (x) ? 4?? R x3 d? 0 (x)
? =
2
R
x2 d? 0 (x) ? ?2?
R
? 1 + C?? ?1/2 + 6?2? (1 + C?? ?1/2 ) + C?? ?1/2
?
2
1 ? C?? ?1/2 ? ?2?
R
0
R
?
? + C?? ?1/2 (? + 1)
? ? + C?? ?1/2 (? + 1) .
1 ? C?? ?1/2
Therefore ?0 ? ?? provided ?? is sufficiently small, which after taking the limit as ? ? 0 completes
the proof.
4
Summary
The assumption of finite kurtosis generalises the parametric Gaussian assumption to a comparable
non-parametric setup with a similar basic structure. Of course there are several open questions.
Optimal constants The leading constants in the main results (Theorem 2 and Theorem 7) are
certainly quite loose. Deriving the optimal form of the regret is an interesting challenge, with both
lower and upper bounds appearing quite non-trivial. It may be necessary to resort to an implicit
analysis showing that (6) is (or is not) achievable when H is the class of distributions with kurtosis
bounded by some ?? . Even then, constructing an efficient algorithm would remain a challenge.
Certainly what has been presented here is quite far from optimal. At the very least the median-ofmeans estimator needs to be replaced, or the analysis improved. An excellent candidate is Catoni?s
estimator [Catoni, 2012], which is slightly more complicated than the median-of-means, but also
comes with smaller constants and could be plugged into the algorithm with very little effort. An
alternative approach is to use the theory of self-normalised processes [Pe?na et al., 2008], but even this
seems to lead to suboptimal constants. For the lower bound, there appears to be almost no work on
the explicit form of the lower bounds presented by Burnetas and Katehakis [1996] in interesting nonparametric classes beyond rewards with bounded or semi-bounded support [Honda and Takemura,
2010, 2015].
Absorbing other improvements There has recently been a range of improvements to the confidence level for the classical upper confidence bound algorithms that shave logarithmic terms from
the worst-case regret or improve the lower-order terms in the finite-time bounds [Audibert and
Bubeck, 2009, Lattimore, 2015]. Many of these enhancements can be incorporated into the algorithm presented here, which may lead to practical and theoretical improvements.
Computation complexity The main challenge is the computation of the index, which as written
seems challenging. The easiest solution is to change the algorithm slightly by estimating
?
?i (t) = [
MM?t ((Xs )s?[t],As =i )
?
?i2 (t) = [
MM?t ((Xs2 )s?[t],As =i ) ? ?
?i (t)2 .
Then an upper confidence bound on ?
?i (t) is easily derived from Lemma 4 and the rest of the analysis
goes through in about the same way. Naively the computational complexity of the above is ?(t) in
round t, which would lead to a running time over n rounds of ?(n2 ). Provided the number of buckets
used between rounds t and t + 1 is the same, then the median-of-means estimator can be updated
incrementally in O(Bt ) time, where Bt is the number of buckets. Now Bt = O(log(1/?t )) =
O(log(t)) so there are at most O(log(n)) changes over n rounds. Therefore the total computation is
O(nk + n log(n)).
8
Comparison to Bernoulli Table 2 shows that the kurtosis for a Bernoulli random variable
with mean ? is ? = O(1/(?(1 ? ?))), which is obviously not bounded as ? tends towards
the
P boundaries. The? optimal asymptotic regret for the Bernoulli case is limn?? Rn / log(n) =
i:?i >0 ?i /d(?i , ? ). The interesting differences occur near the boundary of the parameter space.
Suppose that ?i ? 0 for some arm i and ?? > 0 is close to zero. An easy calculation shows that
d(?i , ?? ) ? log(1/(1 ? ?i )) ? ?i . Therefore
lim inf
n??
E[Ti (n)]
1
1
?
?
.
log(n)
log(1/(1 ? ?i ))
?i
Here we see an algorithm is enjoying logarithmic regret on a class with infinite kurtosis! But this
is a special case and is not possible in general. The reason is that the structure of the hypothesis
class allows strategies to (essentially) estimate the kurtosis with reasonable accuracy and anticipate
outliers more/less depending on the data observed so far. Another way of saying it is that when the
kurtosis is actually small, the algorithms can learn this fact by examining the empirical mean.
A
Technical calculations
This section completes some of the calculations required in the proof of Theorem 7.
Lemma 8. Let ?? ? 7/2 and f (x) = 3 + (?? ? 3 + x)/(1 + x)2 . Then f (x) ? ?? for all x ? 0.
Proof. Clearly f (0) = ?? and for ?? ? 7/2 and x ? 0,
1
2(?? ? 3 + x)
1
?
? 0.
f 0 (x) =
(1 + x)2
1+x
Rx
Therefore f (x) = ?? + 0 f 0 (y)dy ? ?? .
Lemma 9. If ?? ? 7/2 and p = min {?, 1/?? }, then
2
2
1?6p(1?p)
?? ? 3 + (1?p)?
p
p(1?p)
3+
? ?? .
2
2
1 + (1?p)?
p
Proof. Suppose that p = ?. Then since ?? ? 7/2 ? 1, p ? 1. Therefore
2
2
1?6p(1?p)
?? ? 3 + (1?p)?
p
p(1?p)
?? ? 3 + ?(1 ? ?)(1 ? 6?(1 ? ?))
3+
=3+
2
2
(1?p)?2
(1 + ?(1 ? ?))
1+
p
?3+
?? ? 3 + ?(1 ? ?)
(1 + ?(1 ? ?))
2
? ?? ,
where the last inequality follows from Lemma 8. Now suppose that p = 1/?? . Then
2
2
2
1?6p(1?p)
(1?p)?2
1?6p(1?p)
?
?
3
+
?? ? 3 + (1?p)?
?
p
p(1?p)
p
p(1?p)
3+
?3+
2
2
2
2
1 + (1?p)?
1 + (1?p)?
p
p
(
)
??
? max ?? ,
?3
1 ? ?1?
? ?? ,
where the first inequality follows since (a+b)2 ? a2 +b2 for a, b ? 0. The second since the average
is less than the maximum. The third since ?? ? 7/2 > 4/3.
9
References
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10
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6,364 | 6,757 | Learning Multiple Tasks with Multilinear
Relationship Networks
Mingsheng Long, Zhangjie Cao, Jianmin Wang, Philip S. Yu
School of Software, Tsinghua University, Beijing 100084, China
{mingsheng,jimwang}@tsinghua.edu.cn
[email protected]
[email protected]
Abstract
Deep networks trained on large-scale data can learn transferable features to promote
learning multiple tasks. Since deep features eventually transition from general to
specific along deep networks, a fundamental problem of multi-task learning is how
to exploit the task relatedness underlying parameter tensors and improve feature
transferability in the multiple task-specific layers. This paper presents Multilinear
Relationship Networks (MRN) that discover the task relationships based on novel
tensor normal priors over parameter tensors of multiple task-specific layers in deep
convolutional networks. By jointly learning transferable features and multilinear
relationships of tasks and features, MRN is able to alleviate the dilemma of negativetransfer in the feature layers and under-transfer in the classifier layer. Experiments
show that MRN yields state-of-the-art results on three multi-task learning datasets.
1
Introduction
Supervised learning machines trained with limited labeled samples are prone to overfitting, while
manual labeling of sufficient training data for new domains is often prohibitive. Thus it is imperative
to design versatile algorithms for reducing the labeling consumption, typically by leveraging off-theshelf labeled data from relevant tasks. Multi-task learning is based on the idea that the performance
of one task can be improved using related tasks as inductive bias [4]. Knowing the task relationship
should enable the transfer of shared knowledge from relevant tasks such that only task-specific features
need to be learned. This fundamental idea of task relatedness has motivated a variety of methods,
including multi-task feature learning that learns a shared feature representation [1, 2, 6, 5, 23], and
multi-task relationship learning that models inherent task relationship [10, 14, 29, 31, 15, 17, 8].
Learning inherent task relatedness is a hard problem, since the training data of different tasks may be
sampled from different distributions and fitted by different models. Without prior knowledge on the
task relatedness, the distribution shift may pose a major difficulty in transferring knowledge across
different tasks. Unfortunately, if cross-task knowledge transfer is impossible, then we will overfit
each task due to limited amount of labeled data. One way to circumvent this dilemma is to use an
external data source, e.g. ImageNet, to learn transferable features through which the shift in the
inductive biases can be reduced such that different tasks can be correlated more effectively. This idea
has motivated some latest deep learning methods for learning multiple tasks [25, 22, 7, 27], which
learn a shared representation in feature layers and multiple independent classifiers in classifier layer.
However, these deep multi-task learning methods do not explicitly model the task relationships.
This may result in under-transfer in the classifier layer as knowledge can not be transferred across
different classifiers. Recent research also reveals that deep features eventually transition from general
to specific along the network, and feature transferability drops significantly in higher layers with
increasing task dissimilarity [28], hence the sharing of all feature layers may be risky to negativetransfer. Therefore, it remains an open problem how to exploit the task relationship across different
deep networks while improving the feature transferability in task-specific layers of the deep networks.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
This paper presents Multilinear Relationship Network (MRN) for multi-task learning, which discovers
the task relationships based on multiple task-specific layers of deep convolutional neural networks.
Since the parameters of deep networks are natively tensors, the tensor normal distribution [21] is
explored for multi-task learning, which is imposed as the prior distribution over network parameters
of all task-specific layers to learn find-grained multilinear relationships of tasks, classes and features.
By jointly learning transferable features and multilinear relationships, MRN is able to circumvent the
dilemma of negative-transfer in feature layers and under-transfer in classifier layer. Experiments show
that MRN learns fine-grained relationships and yields state-of-the-art results on standard benchmarks.
2
Related Work
Multi-task learning is a learning paradigm that learns multiple tasks jointly by exploiting the shared
structures to improve generalization performance [4, 19] and mitigate manual labeling consumption.
There are generally two categories of approaches: (1) multi-task feature learning, which learns a
shared feature representation such that the distribution shift across different tasks can be reduced
[1, 2, 6, 5, 23]; (2) multi-task relationship learning, which explicitly models the task relationship
in the forms of task grouping [14, 15, 17] or task covariance [10, 29, 31, 8]. While these methods
have achieved improved performance, they may be restricted by their shallow learning paradigm that
cannot embody task relationships by suppressing the task-specific variations in transferable features.
Deep networks learn abstract representations that disentangle and hide explanatory factors of variation
behind data [3, 16]. Deep representations manifest invariant factors underlying different populations
and are transferable across similar tasks [28]. Thus deep networks have been successfully explored
for domain adaptation [11, 18] and multi-task learning [25, 22, 32, 7, 20, 27], where significant
performance gains have been witnessed. Most multi-task deep learning methods [22, 32, 7] learn a
shared representation in the feature layers and multiple independent classifiers in the classifier layer
without inferring the task relationships. However, this may result in under-transfer in the classifier
layer as knowledge cannot be adaptively propagated across different classifiers, while the sharing of
all feature layers may still be vulnerable to negative-transfer in the feature layers, as the higher layers
of deep networks are tailored to fit task-specific structures and may not be safely transferable [28].
This paper presents a multilinear relationship network based on novel tensor normal priors to learn
transferable features and task relationships that mitigate both under-transfer and negative-transfer. Our
work contrasts from prior relationship learning [29, 31] and multi-task deep learning [22, 32, 7, 27]
methods in two key aspects. (1) Tensor normal prior: our work is the first to explore tensor normal
distribution as priors of network parameters in different layers to learn multilinear task relationships in
deep networks. Since the network parameters of multiple tasks natively stack into high-order tensors,
previous matrix normal distribution [13] cannot be used as priors of network parameters to learn task
relationships. (2) Deep task relationship: we define the tensor normal prior on multiple task-specific
layers, while previous deep learning methods do not learn the task relationships. To our knowledge,
multi-task deep learning by tensor factorization [27] is the first work that tackles multi-task deep
learning by tensor factorization, which learns shared feature subspace from multilayer parameter
tensors; in contrast, our work learns multilinear task relationships from multiplayer parameter tensors.
3
3.1
Tensor Normal Distribution
Probability Density Function
Tensor normal distribution is a natural extension of multivariate normal distribution and matrix-variate
normal distribution [13] to tensor-variate distributions. The multivariate normal distribution is order-1
tensor normal distribution, and matrix-variate normal distribution is order-2 tensor normal distribution.
Before defining tensor normal distribution, we first introduce the notations and operations of order-K
tensor. An order-K tensor is an element of the tensor product of K vector spaces, each of which
has its own coordinate system. A vector x ? Rd1 is an order-1 tensor with dimension d1 . A matrix
X ? Rd1 ?d2 is an order-2 tensor with dimensions (d1 , d2 ). A order-K tensor X ? Rd1 ?...?dK
with dimensions (d1 , . . . , dK ) has elements {xi1 ...iK : ik = 1, . . . , dk }. The vectorization of X is
unfolding the tensor into a vector, denoted by vec(X ). The matricization of X is a generalization of
vectorization, reordering the elements of X into a matrix. In this paper, to simply the notations and
2
describe the tensor relationships, we use the mode-k matricization and denote by X(k) the mode-k
matrix of tensor X , where row i of X(k) contains all elements of X having the k-th index equal to i.
QK
Consider an order-K tensor X ? Rd1 ?...?dK . Since we can vectorize X to a ( k=1 dk ) ? 1 vector,
the normal distributionQ
on a tensor X can be considered as a multivariate normal distribution on vector
K
vec(X ) of dimension k=1 dk . However, such an ordinary multivariate normal distribution ignores
the special structure of X as a d1 ? . . . ? dK tensor, and as a result, the covariance characterizing the
QK
QK
correlations across elements of X is of size ( k=1 dk ) ? ( k=1 dk ), which is often prohibitively
large for modeling and estimation. To exploit the structure of X , tensor normal distributions assume
QK
QK
that the ( k=1 dk ) ? ( k=1 dk ) covariance matrix ?1:K can be decomposed into the Kronecker
product ?1:K = ?1 ? . . . ? ?K , and elements of X (in vectorization) follow the normal distribution,
vec (X ) ? N (vec (M) , ?1 ? . . . ? ?K ) ,
(1)
dk ?dk
where ? is the Kronecker product, ?k ? R
is a positive definite matrix indicating
the covariance
Q
between the dk rows of the mode-k matricization X(k) of dimension dk ? ( k0 6=k dk0 ), and M is a
mean tensor containing the expectation of each element of X . Due to the decomposition of covariance
as the Kronecker product, the tensor normal distribution of an order-K tensor X , parameterized by
mean tensor M and covariance matrices ?1 , . . . , ?K , can define probability density function as [21]
!
K
Y
1
?d/2
?d/(2dk )
T ?1
(2)
p (x) = (2?)
|?k |
? exp ? (x ? ?) ?1:K (x ? ?) ,
2
k=1
where |?| is the determinant of a square matrix, and x = vec (X ) , ? = vec (M) , ?1:K = ?1 ?. . .?
QK
?K , d = k=1 dk . The tensor normal distribution corresponds to the multivariate normal distribution
with Kronecker decomposable covariance structure. X following tensor normal distribution, i.e.
vec (X ) following the normal distribution with Kronecker decomposable covariance, is denoted by
X ? T Nd1 ?...?dK (M, ?1 , . . . , ?K ) .
3.2
(3)
Maximum Likelihood Estimation
Consider a set of n samples {Xi }ni=1 where each Xi is an order-3 tensor generated by a tensor normal
distribution as in Equation (2). The maximum likelihood estimation (MLE) of the mean tensor M is
n
X
c= 1
M
Xi .
(4)
n i=1
b 1, . . . , ?
b 3 are computed by iteratively updating these equations:
The MLE of covariance matrices ?
b1 =
?
n
?1
1 X
b3 ? ?
b2
(Xi ? M)(1) ?
(Xi ? M)T(1) ,
nd2 d3 i=1
b2 =
?
n
?1
1 X
b3 ? ?
b1
(Xi ? M)(2) ?
(Xi ? M)T(2) ,
nd1 d3 i=1
b3 =
?
n
?1
1 X
b2 ? ?
b1
(Xi ? M)(3) ?
(Xi ? M)T(3) .
nd1 d2 i=1
(5)
This flip-flop algorithm [21] is efficient to solve by simple matrix manipulations and convergence is
b 1, . . . , ?
b 3 are not identifiable and the solutions to maximizing
guaranteed. Covariance matrices ?
density function (2) are not unique, while only the Kronecker product ?1 ?. . .??K (1) is identifiable.
4
Multilinear Relationship Networks
This work models multiple tasks by jointly learning transferable representations and task relationships.
t
Given T tasks with training data {Xt , Yt }Tt=1 , where Xt = {xt1 , . . . , xtNt } and Yt = {y1t , . . . , yN
}
t
are the Nt training examples and associated labels of the t-th task, respectively drawn from Ddimensional feature space and C-cardinality label space, i.e. each training example xtn ? RD and
ynt ? {1, . . . , C}. Our goal is to build a deep network for multiple tasks ynt = ft (xtn ) which learns
transferable features and adaptive task relationships to bridge different tasks effectively and robustly.
3
Task 1
TNPrior
TNPrior
Task T
input
conv1
conv2
conv3
conv4
conv5
fc6
fc7
fc8
output
Figure 1: Multilinear relationship network (MRN) for multi-task learning: (1) convolutional layers
conv1?conv5 and fully-connected layer f c6 learn transferable features, so their parameters are shared
across tasks; (2) fully-connected layers f c7?f c8 fit task-specific structures, so their parameters are
modeled by tensor normal priors for learning multilinear relationships of features, classes and tasks.
4.1
Model
We start with deep convolutional neural networks (CNNs) [16], a family of models to learn transferable
features that are well adaptive to multiple tasks [32, 28, 18, 27]. The main challenge is that in multitask learning, each task is provided with a limited amount of labeled data, which is insufficient to
build reliable classifiers without overfitting. In this sense, it is vital to model the task relationships
through which each pair of tasks can help with each other to enable knowledge transfer if they are
related, and can remain independent to mitigate negative transfer if they are unrelated. With this idea,
we design a Multilinear Relationship Network (MRN) that exploits both feature transferability and
task relationship to establish effective and robust multi-task learning. Figure 1 shows the architecture
of the proposed MRN model based on AlexNet [16], while other deep networks are also applicable.
We build the proposed MRN model upon AlexNet [16], which is comprised of convolutional layers
(conv1?conv5) and fully-connected
layers (f c6?f c8). The `-th f c layer learns a nonlinear mapping
`
t,` t,`?1
t
+ bt,` for task t, where ht,`
ht,`
n = a W hn
n is the hidden representation of each point xn ,
Wt,` and bt,` are the weight and bias parameters, and a` is the activation function, taken as ReLU
P|x|
a` (x) = max(0, x) for hidden layers or softmax units a` (x) = ex / j=1 exj for the output layer.
Denote by y = ft (x) the CNN classifier of t-th task, and the empirical error of CNN on {Xt , Yt } is
min
ft
Nt
X
J ft xtn , ynt ,
(6)
n=1
where J is the cross-entropy loss function, and ft (xtn ) is the conditional probability that CNN assigns
xtn to label ynt . We will not describe how to compute the convolutional layers since these layers can
learn transferable features in general [28, 18], and we will simply share the network parameters of
these layers across different tasks, without explicitly modeling the relationships of features and tasks
in these layers. To benefit from pre-training and fine-tuning as most deep learning work, we copy
these layers from a model pre-trained on ImageNet 2012 [28], and fine-tune all conv1?conv5 layers.
As revealed by the recent literature findings [28], the deep features in standard CNNs must eventually
transition from general to specific along the network, and the feature transferability decreases while
the task discrepancy increases, making the features in higher layers f c7?f c8 unsafely transferable
across different tasks. In other words, the f c layers are tailored to their original task at the expense
of degraded performance on the target task, which may deteriorate multi-task learning based on
deep neural networks. Most previous methods generally assume that the multiple tasks can be well
correlated given the shared representation learned by the feature layers conv1?f c7 of deep networks
[25, 22, 32, 27]. However, it may be vulnerable if different tasks are not well correlated under deep
features, which is common as higher layers are not safely transferable and tasks may be dissimilar.
Moreover, existing multi-task learning methods are natively designed for binary classification tasks,
which are not good choices as deep networks mainly adopt multi-class softmax regression. It remains
an open problem to explore the task relationships of multi-class classification for multi-task learning.
In this work, we jointly learn transferable features and multilinear relationships of features and tasks
for multiple task-specific layers L in a Bayesian framework. Based on the transferability of deep
4
T
networks discussed above, the task-specific layers L are set to {f c7, f c8}. Denote by X = {Xt }t=1 ,
`
`
T
Y = {Yt }t=1 the complete training data of T tasks, and by Wt,` ? RD1 ?D2 the network parameters
of the t-th task in the `-th layer, where D1` and D2` are the rows and columns of matrix Wt,` . In order
to capture the task relationship in the network parameters of all T tasks, we construct the `-th layer
`
`
parameter tensor as W ` = W1,` ; . . . ; WT,` ? RD1 ?D2 ?T . Denote by W = W ` : ` ? L the
set of parameter tensors of all the task-specific layers L = {f c7, f c8}. The Maximum a Posteriori
(MAP) estimation of network parameters W given training data {X , Y} for learning multiple tasks is
p ( W| X , Y) ? p (W) ? p (Y |X , W )
=
Y
Nt
T Y
Y
p W` ?
p ynt xtn , W ` ,
(7)
t=1 n=1
`?L
where we assume that for prior p (W), the parameter tensor of each layer W ` is independent on the
0
parameter tensors of the other layers W ` 6=` , which is a common assumption made by most feedforward neural network methods [3]. Finally, we assume when the network parameter is sampled
from the prior, all tasks are independent. These independence assumptions lead to the factorization of
the posteriori in Equation (7), which make the final MAP estimation in deep networks easy to solve.
The maximum likelihood estimation (MLE) part p (Y |X , W ) in Equation (7) is modeled by deep
CNN in Equation (6), which can learn transferable features in lower layers for multi-task learning.
We opt to share the network parameters of all these layers (conv1?f c6). This parameter sharing
strategy is a relaxation of existing deep multi-task learning methods [22, 32, 7], which share all the
feature layers except for the classifier layer. We do not share task-specific layers (the last feature
layer f c7 and classifier layer f c8), with the expectation to potentially mitigate negative-transfer [28].
The prior part p (W) in Equation (7) is the key to enabling multi-task deep learning since this prior
part should be able to model the multilinear relationship across parameter tensors. This paper, for the
first time, defines the prior for the `-th layer parameter tensor by tensor normal distribution [21] as
p W ` = T ND1` ?D2` ?T O, ?`1 , ?`2 , ?`3 ,
(8)
`
`
`
`
where ?`1 ? RD1 ?D1 , ?`2 ? RD2 ?D2 , and ?`3 ? RT ?T are the mode-1, mode-2, and mode-3
covariance matrices, respectively. Specifically, in the tensor normal prior, the row covariance matrix
?`1 models the relationships between features (feature covariance), the column covariance matrix ?`2
models the relationships between classes (class covariance), and the mode-3 covariance matrix ?`3
models the relationships between tasks in the `-th layer network parameters {W1,` , . . . , WT,` }. A
common strategy used by previous methods is to use identity covariance for feature covariance [31, 8]
and class covariance [2], which implicitly assumes independent features and classes and cannot
capture the dependencies between them. This work learns all feature covariance, class covariance,
task covariance and all network parameters from data to build robust multilinear task relationships.
We integrate the CNN error functional (6) and tensor normal prior (8) into MAP estimation (7)
and taking negative logarithm, which leads to the MAP estimation of the network parameters W, a
regularized optimization problem for Multilinear Relationship Network (MRN) formally writing as
min
Nt
T X
X
` K
ft |T
t=1 ,?k |k=1
J ft xtn , ynt
t=1 n=1
!
K
X
?1
1X
D` `
` T
`
`
+
vec(W ) (?1:K ) vec(W ) ?
ln |?k | ,
2
Dk`
`?L
k=1
(9)
QK
where D` = k=1 Dk` and K = 3 is the number of modes in parameter tensor W, which could be
K = 4 for the convolutional layers (width, height, number of feature maps, and number of tasks);
?`1:3 = ?`1 ? ?`2 ? ?`3 is the Kronecker product of the feature covariance ?`1 , class covariance ?`2 ,
and task covariance ?`3 . Moreover, we can assume shared task relationship across different layers as
?`3 = ?3 , which enhances connection between task relationships on features f c7 and classifiers f c8.
4.2
Algorithm
The optimization problem (9) is jointly non-convex with respect to the parameter tensors W as well as
feature covariance ?`1 , class covariance ?`2 , and task covariance ?`3 . Thus, we alternatively optimize
5
one set of variables with the others fixed. We first update Wt,` , the parameter of task-t in layer-`.
When training deep CNN by back-propagation, we only require the gradient of the objective function
(denoted by O) in Equation (10) w.r.t. Wt,` on each data point (xtn , ynt ), which can be computed as
?J (ft (xtn ) , ynt ) ` ?1
?O (xtn , ynt )
(10)
=
+ (?1:3 ) vec W ` ??t ,
t,`
t,`
?W
?W
where [(?`1:3 )?1 vec W ` ]??t is the (:, :, t) slice of a tensor folded from elements (?`1:3 )?1 vec(W ` )
that are corresponding to parameter matrix Wt,` . Since training a deep CNN requires a large amount
of labeled data, which is prohibitive for many multi-task learning problems, we fine-tune from an
AlexNet model pre-trained on ImageNet as in [28]. In each epoch, after updating W, we can update
the feature covariance ?`1 , class covariance ?`2 , and task covariance ?`3 by the flip-flop algorithm as
?1
1
`
`
`
(W
)
?
?
?
(W ` )T(1) + ID` ,
3
2
(1)
1
D2` T
?1
1
?`2 = ` (W ` )(2) ?`3 ? ?`1
(W` )T(2) + ID` ,
2
D1 T
?1
1
(W ` )T(3) + IT .
?`3 = ` ` (W ` )(3) ?`2 ? ?`1
D1 D2
?`1 =
(11)
where the last term of each update equation is a small penalty traded off by for numerical stability.
However, the above updating equations (11) are computationally prohibitive, due to the dimension
explosion of the Kronecker product, e.g. ?`2 ? ?`1 is of dimension D1` D2` ? D1` D2` . To speed up
?1
computation,
(A ? B) = A?1 ? B?1 and
we will use the following rules of Kronecker product:
`
T ?T
T
as an example, we have
B ? A vec (X) = vec (AXB). Taking the computation of ?3 ? R
(?`3 )ij =
=
1
D1` D2`
(W ` )(3),i? ?`2 ? ?`1
?1
(W ` )T
(3),j? + Iij
1
`
` ?1
`
` ?1
(W
)
vec
(?
)
W
(?
)
+ Iij ,
(3),i?
1
??j
2
D1` D2`
(12)
`
where (W ` )(3),i? denotes the i-th row of the mode-3 matricization of tensor W ` , and W??j
denotes
`
`
the (:, :, j) slice of tensor
W . We can derive that updating ?3 has a computational complexity of
O T 2 D1` D2` D1` + D2` , similarly for ?`1 and ?`2 . The total computational
complexity of updating
covariance matrices ?`k |3k=1 will be O D1` D2` T D1` D2` + D1` T + D2` T , which is still expensive.
A key to computation speedup is that the covariance matrices ?`k |3k=1 should be low-rank, since the
features and tasks are enforced to be correlated for multi-task learning. Thus, the inverses of ?`k |3k=1
do not exist in general and we have to compute the generalized inverses using eigendecomposition.
We perform eigendecomposition for each ?`k and maintain all eigenvectors with eigenvalues greater
`
than zero. The rank r of the eigen-reconstructed covariance matrices should be r ? min(D1` , D2
, T ).
` 3
` `
`
`
Thus, the total computational complexity for ?k |k=1 is reduced to O rD1 D2 T D1 + D2 + T . It
is straight-forward to see the computational complexity of updating the parameter tensor W is the cost
of back-propagation in standard CNNs plus the cost for computing
the gradient of regularization term
by Equation (10), which is O rD1` D2` T D1` + D2` + T given generalized inverses (?`k )?1 |3k=1 .
4.3
Discussion
The proposed Multilinear Relationship Network (MRN) is very flexible and can be easily configured
to deal with different network architectures and multi-task learning scenarios. For example, replacing
the network backbone from AlexNet to VGGnet [24] boils down to configuring task-specific layers
L = {f c7, f c8}, where f c7 is the last feature layer while f c8 is the classifier layer in the VGGnet.
The architecture of MRN in Figure 1 can readily cope with homogeneous multi-task learning where
all tasks share the same output space. It can cope with heterogeneous multi-task learning where
different tasks have different output spaces by setting L = {f c7}, by only considering feature layers.
The multilinear relationship learning in Equation (9) is a general framework that readily subsumes
many classical multi-task learning methods as special cases. Many regularized multi-task algorithms
can be classified into two main categories: learning with feature covariances [1, 2, 6, 5] and learning
6
with task relations [10, 14, 29, 31, 15, 17, 8]. Learning with feature covariances can be viewed
as a representative formulation in feature-based methods while learning with task relations is for
parameter-based methods [30]. More specifically, previous multi-task feature learning methods [1, 2]
can be viewed as a special case of Equation (9) by setting all covariance matrices but the feature
covariance to identity matrix, i.e. ?k = I|K
k=2 ; and previous multi-task relationship learning methods
[31, 8] can be viewed as a special case of Equation (9) by setting all covariance matrices but the
task covariance to identity matrix, i.e. ?k = I|K?1
k=1 . The proposed MRN is more general in the
architecture perspective in dealing with parameter tensors in multiple layers of deep neural networks.
It is noteworthy to highlight a concurrent work on multi-task deep learning using tensor decomposition
[27], which is feature-based method that explicitly learns the low-rank shared parameter subspace.
The proposed multilinear relationship across parameter tensors can be viewed as a strong alternative
to the tensor decomposition, with the advantage to explicitly model the positive and negative relations
across features and tasks. As a defense of [27], the tensor decomposition can extract finer-grained
feature relations (what to share and how much to share) than the proposed multilinear relationships.
5
Experiments
We compare MRN with state-of-the-art multi-task and deep learning methods to verify the efficacy of
learning transferable features and multilinear task relationships. Codes and datasets will be released.
5.1
Setup
Office-Caltech [12] This dataset is the standard benchmark for multi-task learning and transfer
learning. The Office part consists of 4,652 images in 31 categories collected from three distinct
domains (tasks): Amazon (A), which contains images downloaded from amazon.com, Webcam (W)
and DSLR (D), which are images taken by Web camera and digital SLR camera under different
environmental variations. This dataset is organized by selecting the 10 common categories shared by
the Office dataset and the Caltech-256 (C) dataset [12], hence it yields four multi-class learning tasks.
Real World
Product
Clipart
Art
Office-Home1 [26] This dataset is to evaluate
transfer learning algorithms using deep learning.
It consists of images from 4 different domains:
Artistic images (A), Clip Art (C), Product images (P) and Real-World images (R). For each
domain, the dataset contains images of 65 object
categories collected in office and home settings.
Spoon
Sink
Mug
Pen
Knife
Bed
Bike
Kettle
TV
Keyboard
Classes Alarm-Clock Desk-Lamp Hammer
Chair
Fan
Figure 2: Examples of the Office-Home dataset.
ImageCLEF-DA2 This dataset is the benchmark for ImageCLEF domain adaptation challenge,
organized by selecting the 12 common categories shared by the following four public datasets (tasks):
Caltech-256 (C), ImageNet ILSVRC 2012 (I), Pascal VOC 2012 (P), and Bing (B). All three datasets
are evaluated using DeCAF7 [9] features for shallow methods and original images for deep methods.
We compare MRN with standard and state-of-the-art methods: Single-Task Learning (STL), MultiTask Feature Learning (MTFL) [2], Multi-Task Relationship Learning (MTRL) [31], Robust MultiTask Learning (RMTL) [5], and Deep Multi-Task Learning with Tensor Factorization (DMTL-TF)
[27]. STL performs per-task classification in separate deep networks without knowledge transfer.
MTFL extracts the low-rank shared feature representations by learning feature covariance. RMTL
extends MTFL to further capture the task relationships using a low-rank structure and identify outlier
tasks using a group-sparse structure. MTRL captures the task relationships using task covariance of a
matrix normal distribution. DMTL-TF tackles multi-task deep learning by tensor factorization, which
learns shared feature subspace instead of multilinear task relationship in multilayer parameter tensors.
To go deep into the efficacy of jointly learning transferable features and multilinear task relationships,
we evaluate two MRN variants: (1) MRN8 , MRN using only one network layer f c8 for multilinear
relationship learning; (2) MRNt , MRN using only task covariance ?3 for single-relationship learning.
The proposed MRN model can natively deal with multi-class problems using the parameter tensors.
However, most shallow multi-task learning methods such as MTFL, RMTL and MTRL are formulated
1
2
http://hemanthdv.org/OfficeHome-Dataset
http://imageclef.org/2014/adaptation
7
Table 1: Classification accuracy on Office-Caltech with standard evaluation protocol (AlexNet).
Method
STL (AlexNet)
MTFL [2]
RMTL [6]
MTRL [31]
DMTL-TF [27]
MRN8
MRNt
MRN (full)
A
W
5%
D
C
Avg
A
W
10%
D
C
Avg
A
W
20%
D
C
Avg
88.9
90.0
91.3
86.4
91.2
91.7
91.1
92.5
73.0
78.9
82.3
83.0
88.3
96.4
96.3
97.5
80.4
90.2
88.8
95.1
92.5
96.9
97.4
97.9
88.7
86.9
89.1
89.1
85.6
86.5
86.1
87.5
82.8
86.5
87.9
88.4
89.4
92.9
92.7
93.8
92.2
92.4
92.6
91.1
92.2
92.7
92.5
93.6
80.9
85.3
85.2
87.1
91.9
97.1
97.7
98.6
88.2
89.5
93.3
97.0
97.4
97.3
96.6
98.6
88.9
89.2
87.2
87.6
86.8
86.6
86.7
87.3
87.6
89.1
89.6
90.7
92.0
93.4
93.4
94.5
91.3
93.5
94.3
90.0
92.6
93.2
91.9
94.4
83.3
89.0
87.0
88.8
97.6
96.9
96.6
98.3
93.7
95.2
96.7
99.2
94.5
99.4
95.9
99.9
94.9
92.6
93.4
94.3
88.4
82.8
90.0
89.1
90.8
92.6
92.4
93.1
93.3
94.4
93.6
95.5
Table 2: Classification accuracy on Office-Home with standard evaluation protocol (VGGnet).
Method
STL (VGGnet)
MTFL [2]
RMTL [6]
MTRL [31]
DMTL-TF [27]
MRN8
MRNt
MRN (full)
A
C
5%
P
R
Avg
A
C
10%
P
R
Avg
A
C
20%
P
R
Avg
35.8
40.1
42.3
42.7
49.2
52.7
52.0
53.3
31.2
30.4
32.8
33.3
34.5
34.7
34.0
36.4
67.8
61.5
62.3
62.9
67.1
70.1
69.9
70.5
62.5
59.5
60.6
61.3
62.9
67.6
66.8
67.7
49.3
47.9
49.5
50.1
53.4
56.3
55.7
57.0
51.0
50.3
49.7
51.6
57.2
59.1
58.6
59.9
40.7
35.0
34.6
36.3
42.3
42.7
42.6
42.7
75.0
66.3
65.9
67.7
73.6
75.1
74.9
76.3
68.8
65.0
64.6
66.3
69.9
72.8
72.4
73.0
58.9
54.2
53.7
55.5
60.8
62.4
62.1
63.0
56.1
55.2
55.2
55.8
58.3
58.4
57.7
58.5
54.6
38.8
39.2
39.9
56.1
55.6
54.8
55.6
80.4
69.1
69.6
70.2
79.3
80.4
80.2
80.7
71.8
70.0
70.5
71.2
72.1
72.4
71.6
72.8
65.7
58.3
58.6
59.3
66.5
66.7
66.1
66.9
only for binary-class problems, due to the difficulty in dealing with order-3 parameter tensors for
multi-class problems. We adopt one-vs-rest strategy to enable them working on multi-class datasets.
We follow the standard evaluation protocol [31, 5] for multi-task learning and randomly select 5%,
10%, and 20% samples from each task as training set and use the rest of the samples as test set. We
compare the average classification accuracy for all tasks based on five random experiments, where
standard errors are generally less than ?0.5%, which are not significant and thus are not reported for
space limitation. We conduct model selection for all methods using five-fold cross-validation on the
training set. For deep learning methods, we adopt AlexNet [16] and VGGnet [24], fix convolutional
layers conv1?conv5, fine-tune fully-connected layers f c6?f c7, and train classifier layer f c8 via
back-propagation. As the classifier layer is trained from scratch, we set its learning rate to be 10 times
that of the other layers. We use mini-batch stochastic gradient descent (SGD) with 0.9 momentum
1
and learning rate decaying strategy, and select learning rate between 10?5 and 10?2 by stepsize 10 2 .
5.2
Results
The multi-task classification results on the Office-Caltech, Office-Home and ImageCLEF-DA datasets
based on 5%, 10%, and 20% sampled training data are shown in Tables 1, 2 and 3, respectively. We
observe that the proposed MRN model significantly outperforms the comparison methods on most
multi-task problems. The substantial accuracy improvement validates that our multilinear relationship
networks through multilayer and multilinear relationship learning is able to learn both transferable
features and adaptive task relationships, which enables effective and robust multi-task deep learning.
We can make the following observations from the results. (1) Shallow multi-task learning methods
MTFL, RMTL, and MTRL outperform single-task deep learning method STL in most cases, which
confirms the efficacy of learning multiple tasks by exploiting shared structures. Among the shallow
multi-task methods, MTRL gives the best accuracies, showing that exploiting task relationship may
be more effective than extracting shared feature subspace for multi-task learning. It is worth noting
that, although STL cannot learn from knowledge transfer, it can be fine-tuned on each task to improve
performance, and thus when the number of training samples are large enough and when different
tasks are dissimilar enough (e.g. Office-Home dataset), STL may outperform shallow multi-task
learning methods, as evidenced by the results in Table 2. (2) Deep multi-task learning method
DMTL-TF outperforms shallow multi-task learning methods with deep features as input, which
confirms the importance of learning deep transferable features to enable knowledge transfer across
tasks. However, DMTL-TF only learns the shared feature subspace based on tensor factorization of
the network parameters, while the task relationships in multiple network layers are not captured. This
may result in negative-transfer in the feature layers [28] and under-transfer in the classifier layers.
Negative-transfer can be witnessed by comparing multi-task methods with single-task methods: if
multi-task learning methods yield lower accuracy in some of the tasks, then negative-transfer arises.
8
Table 3: Classification accuracy on ImageCLEF-DA with standard evaluation protocol (AlexNet).
C
I
5%
P
B
Avg
C
I
10%
P
B
Avg
C
I
20%
P
B
Avg
77.4
79.9
81.1
80.8
87.9
87.0
88.5
89.6
60.3
68.6
71.3
68.4
70.0
74.4
73.5
76.9
48.0
43.4
52.4
51.9
58.1
61.8
63.3
65.4
45.0
41.5
40.9
42.9
34.1
47.6
51.1
49.4
57.7
58.3
61.4
61.0
62.5
67.7
69.1
70.3
78.9
82.9
81.5
83.1
89.1
89.1
88.0
88.1
70.5
71.4
71.7
72.7
82.1
82.2
83.1
84.6
48.1
56.7
55.6
54.5
58.7
64.4
67.4
68.7
41.8
41.7
45.3
45.5
48.0
49.3
54.8
55.6
59.8
63.2
63.5
63.9
69.5
71.2
73.3
74.3
83.3
83.1
83.3
83.7
91.7
91.1
91.1
92.8
74.9
72.2
73.3
75.5
80.0
84.1
83.5
83.3
49.2
54.5
53.7
57.5
63.2
65.7
65.7
67.4
47.1
52.5
49.2
49.4
54.1
54.1
55.7
57.8
63.6
65.6
64.9
66.5
72.2
73.7
74.0
75.3
Method
STL (AlexNet)
MTFL [2]
RMTL [6]
MTRL [31]
DMTL-TF [27]
MRN8
MRNt
MRN (full)
We go deeper into MRN by reporting the results of the two MRN variants: MRN8 and MRNt , all
significantly outperform the comparison methods but generally underperform MRN (full), which
verify our motivation that jointly learning transferable features and multilinear task relationships can
bridge multiple tasks more effectively. (1) The disadvantage of MRN8 is that it does not learn the
task relationship in the lower layers f c7, which are not safely transferable and may result in negative
transfer [28]. (2) The shortcoming of MRNt is that it does not learn the multilinear relationship of
features, classes and tasks, hence the learned relationships may only capture the task covariance
without capturing the feature covariance and class covariance, which may lose some intrinsic relations.
A
A
W
W
D
D
C
C
A
W
D
C
A
W
D
C
A
W
D
C
(a) MTRL Relationship
A
W
D
C
(b) MRN Relationship
(c) DMTL-TF Features
(d) MRN Features
Figure 3: Hinton diagram of task relationships (a)(b) and t-SNE embedding of deep features (c)(d).
5.3
Visualization Analysis
We show that MRN can learn more reasonable task relationships with deep features than MTRL with
shallow features, by visualizing the Hinton diagrams of task covariances learned by MTRL and MRN
(?f3 c8 ) in Figures 3(a) and 3(b), respectively. Prior knowledge on task similarity in the Office-Caltech
dataset [12] describes that tasks A, W and D are more similar with each other while they are relatively
dissimilar to task C. MRN successfully captures this prior task relationship and enhances the task
correlation across dissimilar tasks, which enables stronger transferability for multi-task learning.
Furthermore, all tasks are positively correlated (green color) in MRN, implying that all tasks can
better reinforce each other. However, some of the tasks (D and C) are still negatively correlated (red
color) in MTRL, implying these tasks should be drawn far apart and cannot improve with each other.
We illustrate the feature transferability by visualizing in Figures 3(c) and 3(d) the t-SNE embeddings
[18] of the images in the Office-Caltech dataset with DMTL-TF features and MRN features, respectively. Compared with DMTL-TF features, the data points with MRN features are discriminated better
across different categories, i.e., each category has small intra-class variance and large inter-class
margin; the data points are also aligned better across different tasks, i.e. the embeddings of different
tasks overlap well, implying that different tasks reinforce each other effectively. This verifies that with
multilinear relationship learning, MRN can learn more transferable features for multi-task learning.
6
Conclusion
This paper presented multilinear relationship networks (MRN) that integrate deep neural networks
with tensor normal priors over the network parameters of all task-specific layers, which model the task
relatedness through the covariance structures over tasks, classes and features to enable transfer across
related tasks. An effective learning algorithm was devised to jointly learn transferable features and
multilinear relationships. Experiments testify that MRN yields superior results on standard datasets.
9
Acknowledgments
This work was supported by the National Key R&D Program of China (2016YFB1000701), National
Natural Science Foundation of China (61772299, 61325008, 61502265, 61672313) and TNList Fund.
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6,365 | 6,758 | Deep Hyperalignment
Muhammad Yousefnezhad, Daoqiang Zhang
College of Computer Science and Technology
Nanjing University of Aeronautics and Astronautics
{myousefnezhad,dqzhang}@nuaa.edu.cn
Abstract
This paper proposes Deep Hyperalignment (DHA) as a regularized, deep extension,
scalable Hyperalignment (HA) method, which is well-suited for applying functional alignment to fMRI datasets with nonlinearity, high-dimensionality (broad
ROI), and a large number of subjects. Unlink previous methods, DHA is not limited
by a restricted fixed kernel function. Further, it uses a parametric approach, rank-m
Singular Value Decomposition (SVD), and stochastic gradient descent for optimization. Therefore, DHA has a suitable time complexity for large datasets, and
DHA does not require the training data when it computes the functional alignment
for a new subject. Experimental studies on multi-subject fMRI analysis confirm
that the DHA method achieves superior performance to other state-of-the-art HA
algorithms.
1
Introduction
The multi-subject fMRI analysis is a challenging problem in the human brain decoding [1?7]. On
the one hand, the multi-subject analysis can verify the developed models across subjects. On the
other hand, this analysis requires authentic functional and anatomical alignments among neuronal
activities of different subjects, which these alignments can significantly improve the performance
of the developed models [1, 4]. In fact, multi-subject fMRI images must be aligned across subjects
in order to take between-subject variability into account. There are technically two main alignment
methods, including anatomical alignment and functional alignment, which can work in unison.
Indeed, anatomical alignment is only utilized in the majority of the fMRI studies as a preprocessing
step. It is applied by aligning fMRI images based on anatomical features of standard structural MRI
images, e.g. Talairach [2, 7]. However, anatomical alignment can limitedly improve the accuracy
because the size, shape and anatomical location of functional loci differ across subjects [1, 2, 7]. By
contrast, functional alignment explores to precisely align the fMRI images across subjects. Indeed, it
has a broad range of applications in neuroscience, such as localization of the Brain?s tumor [8].
As the widely used functional alignment method [1?7], Hyperalignment (HA) [1] is an ?anatomy free?
functional alignment method, which can be mathematically formulated as a multiple-set Canonical
Correlation Analysis (CCA) problem [2, 3, 5]. Original HA does not work in a very high dimensional
space. In order to extend HA into the real-world problems, Xu et al. developed the Regularized
Hyperalignment (RHA) by utilizing an EM algorithm to iteratively seek the regularized optimum
parameters [2]. Further, Chen et al. developed Singular Value Decomposition Hyperalignment
(SVDHA), which firstly provides dimensionality reduction by SVD, and then HA aligns the functional
responses in the reduced space [4]. In another study, Chen et al. introduced Shared Response Model
(SRM), which is technically equivalent to Probabilistic CCA [5]. In addition, Guntupalli et al.
developed SearchLight (SL) model, which is actually an ensemble of quasi-CCA models fits on
patches of the brain images [9]. Lorbert et al. illustrated the limitation of HA methods on the
linear representation of fMRI responses. They also proposed Kernel Hyperalignment (KHA) as
a nonlinear alternative in an embedding space for solving the HA limitation [3]. Although KHA
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
can solve the nonlinearity and high-dimensionality problems, its performance is limited by the
fixed employed kernel function. As another nonlinear HA method, Chen et al. recently developed
Convolutional Autoencoder (CAE) for whole brain functional alignment. Indeed, this method
reformulates the SRM as a multi-view autoencoder [5] and then uses the standard SL analysis [9] in
order to improve the stability and robustness of the generated classification (cognitive) model [6].
Since CAE simultaneously employs SRM and SL, its time complexity is so high. In a nutshell, there
are three main challenges in previous HA methods for calculating accurate functional alignments, i.e.
nonlinearity [3, 6], high-dimensionality [2, 4, 5], and using a large number of subjects [6].
As the main contribution of this paper, we propose a novel kernel approach, which is called Deep
Hyperalignment (DHA), in order to solve mentioned challenges in HA problems. Indeed, DHA
employs deep network, i.e. multiple stacked layers of nonlinear transformation, as the kernel function,
which is parametric and uses rank-m SVD [10] and Stochastic Gradient Descent (SGD) [13] for
optimization. Consequently, DHA generates low-runtime on large datasets, and the training data is
not referenced when DHA computes the functional alignment for a new subject. Further, DHA is not
limited by a restricted fixed representational space because the kernel in DHA is a multi-layer neural
network, which can separately implement any nonlinear function [11?13] for each subject to transfer
the brain activities to a common space.
The proposed method is related to RHA [2] and MVLSA [10]. Indeed, the main difference between
DHA and the mentioned methods lies in the deep kernel function. Further, KHA [3] is equivalent
to DHA, where the proposed deep network is employed as the kernel function. In addition, DHA
can be looked as a multi-set regularized DCCA [11] with stochastic optimization [13]. Finally,
DHA is related to DGCCA [12], when DGCCA is reformulated for functional alignment by using
regularization, and rank-m SVD [10].
The rest of this paper is organized as follows: In Section 2, this study briefly introduces HA method.
Then, DHA is proposed in Section 3. Experimental results are reported in Section 4; and finally, this
paper presents conclusion and pointed out some future works in Section 5.
2
Hyperalignment
n
o
(`)
As a training set, preprocessed fMRI time series for S subjects can be denoted by X(`) = xmn ?
RT ?V , ` = 1:S, m = 1:T, n = 1:V , where V denotes the number of voxels, T is the number of time
(`)
points in units of TRs (Time of Repetition), and xmn ? R denotes the functional activity for the `-th
subject in the m-th time point and the n-th voxel. For assuring temporal alignment, the stimuli in
the training set are considered time synchronized, i.e. the m-th time point for all subjects illustrates
the same simulation [2, 3]. Original HA can be defined based on Inter-Subject Correlation (ISC),
which is a classical metric in order to apply functional alignment: [1-4, 7]
S
S
S
S
X
X
X
X
max
ISC(X(i) R(i) , X(j) R(j) ) ? max
tr (X(i) R(i) )> X(j) R(j)
R(i) ,R(j)
R(i) ,R(j)
(1)
i=1 j=i+1
i=1 j=i+1
(`) (`) > (`) (`)
s.t.
X R
X R = I, ` = 1:S,
where tr() denotes the trace function, I is the identity matrix, R(`) ? RV ?V denotes the solution
for `-th subject. For avoiding overfitting, the constrains must be imposed in R(`) [2, 7]. If X(`) ?
N (0, 1), ` = 1:S are column-wise standardized, the ISC lies in [?1, +1]. Here, the large values
illustrate better alignment [2, 3]. In order to seek an optimum solution, solving (1) may not be the best
approach because there is no scale to evaluate the distance between current result and the optimum
(fully maximized) solution [2, 4, 7]. Instead, we can reformulate (1) as a minimization problem by
using a multiple-set CCA: [1?4]
S
S
2
>
X
X
(i) (i)
min
R ? X(j) R(j)
, s.t. X(`) R(`) X(`) R(`) = I, ` = 1:S, (2)
X
(i)
(j)
R
,R
i=1 j=i+1
F
where (2) approaches zero for an optimum result. Indeed, the main assumption in the original HA
is that the R(`) , ` = 1:S are noisy ?rotations? of a common template [1, 9]. This paper provides a
detailed description of HA methods in the supplementary materials (https://sourceforge.net/
projects/myousefnezhad/files/DHA/).
2
3
Deep Hyperalignment
Objective function of DHA is defined as follows:
S
S
2
X
X
(i)
(i) (i)
R ? fj X(j) ;?(j) R(j)
f
i X ;?
(i)
min
? (i) ,R
i=1 j=i+1
? (j) ,R(j)
s.t.
R(`)
>
f` X(`) ;?
F
(3)
>
(`)
f` X(`) ;?
(`)
+ I R(`) = I,
` = 1:S,
(`) (`)
where ?(`) = Wm , bm , m=2:C denotes all parameters in `-th deep network belonged to `-th
subject, R(`) ? RVnew ?Vnew is the DHA solution for `-th subject, Vnew ? V denotes the number of
features after transformation, the regularized
parameter is a small constant, e.g. 10?8 , and deep
(`) (`)
T ?Vnew
multi-layer kernel function f` X ;?
?R
is denoted as follows:
(`)
f` X(`) ;?(`) = mat hC , T, Vnew ,
(4)
where T denotes the number of time points, C ? 3 is number of deep network layers,
(`)
mat(x, m, n):Rmn ? Rm?n denotes the reshape (matricization) function, and hC ? RT Vnew
is the output layer of the following multi-layer deep network:
(`)
(`) (`)
(`)
h(`)
where h1 = vec X(`)
and m = 2:C.
(5)
m = g Wm hm?1 + bm ,
Here, g:R ? R is a nonlinear function applied componentwise, vec:Rm?n ? Rmn denotes the
(`)
vectorization function, consequently h1 = vec X(`) ? RT V . Notably, this paper considers both
vec() and mat() functions are linear transformations, where X ? Rm?n = mat vec(X), m, n for
any matrix X. By considering U (m) units in the m-th intermediate layer, parameters of distinctive
(C-1)
(`)
(`)
layers of f` X(`) ;?(`) are defined by following properties: WC ? RT Vnew ?U
and bC ?
(2)
(2)
(`)
(`)
RT Vnew for the output layer, W2 ? RU ?T V and b2 ? RU for the first intermediate layer,
(m)
(m-1)
(m)
(m)
(`)
(`)
(`)
and Wm ? RU ?U
, bm ? RU
and hm ? RU
for m-th intermediate layer (3 ? m ?
C ? 1).
Since (3) must be calculated for any new subject in the testing phase, it is not computationally
efficient. In other words, the transformed training data must be referenced by the current objective
function for each new subject in the testing phase.
Lemma 1. The equation (3) can be reformulated as follows where G ? RT ?Vnew is the HA template:
min
G,R(i) ,?
S
2
X
? fi X(i) ;?(i) R(i)
G
(i)
i=1
F
s.t.
G> G = I, where G =
S
1X
fj X(j) ;?(j) R(j) .
S j=1
(6)
Proof.
In a nutshell, both (3) and (6) can! be rewritten as ?S 2 tr G> G +
(`) >
(`)
PS
(`) (`)
(`) (`)
S `=1 tr f` X ;? R
f` X ;? R
. Please see supplementary materials for proof in details.
Remark 1. G is called DHA template, which can be used for functional alignment in the testing
phase.
Remark 2. Same as previous approaches for HA problems [1?7], a DHA solution is not unique. If
a DHA template G is calculated for a specific HA problem, then QG is another solution for that
specific HA problem, where Q ? RVnew ?Vnew can be any orthogonal matrix. Consequently, if two
independent templates
G1 , G2 are
trained for a specific dataset, the solutions can be mapped to each
other by calculating
G2 ? QG1
, where Q can be used as a coefficient for functional alignment in
the first solution in order to compare its results to the second one. Indeed, G1 and G2 are located in
different positions on the same contour line [5, 7].
3
3.1
Optimization
This section proposes an effective approach for optimizing the DHA objective function by using
rank-m SVD [10] and SGD [13]. This method seeks an optimum solution for the DHA objective
function (6) by using two different steps, which iteratively work in unison. By considering fixed
network parameters (?(`) ), a mini-batch of neural activities is firstly aligned through the deep network.
Then, back-propagation algorithm [14] is used to update the network parameters. The main challenge
for solving the DHA objective function is that we cannot seek a natural extension of the correlation
object to more than two random variables. Consequently, functional alignments are stacked in a
S ? S matrix and maximize a certain matrix norm for that matrix [10, 12].
As the first step, we consider network parameters are in an optimum state. Therefore, the mappings
(R(`) , ` = 1:S) and template (G) must be calculated to solve the DHA problem. In order to scale
DHA approach, this paper employs the rank-m SVD [10] of the mapped neural activities as follows:
SV D
>
(7)
f` X(`) ;?(`) = ?(`) ?(`) ?(`) ,
` = 1:S
where ?(`) ? Rm?m
denotes the diagonal matrix with m-largest singular values of the mapped
feature f` X(`) ;?(`) , ?(`) ? RT ?m and ?(`) ? Rm?Vnew are respectively the corresponding left
and right singular vectors. Based on (7), the projection matrix for `-th subject can be generated as
follows: [10]
?1
>
>
P(`) = f` X(`) ;?(`)
f` X(`) ;?(`)
f` X(`) ;?(`) + I
f` X(`) ;?(`)
(8)
?1
>
(`) (`) >
(`)
(`) >
(`)
(`) >
(`) (`)
(`) (`)
=? ?
? ?
+ I
? ?
=? D
? D
,
where P(`) ? RT ?T is symmetric and idempotent [10, 12], and diagonal matrix D(`) ? Rm?m is
?1
>
> (`) (`) >
(9)
D(`) D(`) = ?(`)
? ?
+ I
?(`) .
eA
e > is the Cholesky
Further, the sum of projection matrices can be defined as follows, where A
decomposition [10] of A:
S
X
e ? RT ?mS = ?(1) D(1) . . . ?(S) D(S) .
A
(10)
Lemma 2. Based on (10), the objective function of DHA (6) can be rewritten as follows:
S
X
min
G ? fi X(i) ;?(i) R(i)
? max tr G> AG .
(11)
A=
eA
e >,
P(i) = A
where
i=1
G
G,R(i) ,? (i) i=1
Proof. Since P(`) is idempotent, the trace form of (6) can be reformulated as maximizing the sum of
projections. Please see the supplementary materials for proof in details.
Based on Lemma 2, the first optimization
step of DHA problem can be expressed as eigendecomposition of AG = G?, where ? = ?1 . . . ?T and G respectively denote the eigenvalues and
eigenvectors of A. Further, the matrix G that we are interested in finding, can be calculated by the
e = G?
e?
e > , where G> G = I [10]. This paper utilizes Incremental SVD
left singular vectors of A
[15] for calculating these left singular vectors. Further, DHA mapping for `-th subject is denoted as
follows:
?1
>
>
(12)
R(`) =
f` X(`) ;?(`)
f` X(`) ;?(`) + I
f` X(`) ;?(`)
G.
PT
Lemma 3. In order to update network parameters as the second step, the derivative of Z = `=1 ?` ,
which is the sum of eigenvalues of A, over the mapped neural activities of `-th subject is defined as
follows:
>
>
?Z
(`) (`)
= 2R(`) G> ? 2R(`) R(`)
f
X
;?
.
(13)
`
?f` X(`) ;?(`)
Proof. This derivative can be solved by using the chain and product rules in the matrix derivative as
well as considering ?Z/?A = GG> [12]. Please see the supplementary materials for proof in details.
4
Algorithm 1 Deep Hyperalignment (DHA)
Input: Data X(i) , i = 1:S, Regularized parameter , Number of layers C, Number of units U (m)
b for testing phase (default ?), Learning rate ? (default 10?4 [13]).
for m = 2:C, HA template G
(`)
Output: DHA mappings R and parameters ?(`) , HA template G just from training phase
Method:
(`)
01. Initialize iteration counter:
` = 1:S.
m ? 1 and ? ? N (0, 1) for
(`) (`)
(`)
02. Construct f` X ;?
based on (4) and (5) by using ? , C, U (m) for ` = 1:S.
b
03. IF (G 6= ?) THEN
% The first step of DHA: fixed ?(`) and calculating G and R(`) ?
e by using (8) and (10).
04. Generate A
e = G?
e?
e >.
05. Calculate G by applying Incremental SVD [15] to A
06. ELSE
b
07. G = G.
08. END IF
09. Calculate mappings R(`) , ` = 1:S by using (12).
2
PS PS
10. Estimate error of iteration ?m = i=1 j=i+1
fi X(i) ;?(i) R(i) ? fj X(j) ;?(j) R(j)
.
F
11. IF (m > 3) and (?m ? ?m?1 ? ?m?2 ) THEN
% This is the finishing condition.
12. Return calculated G, R(`) , ?(`) (` = 1:S) related to (m-2)-th iteration.
13. END IF
% The secondstep of DHA: fixed G and R(`) and updating ?(`) ?
14. ??(`) ? backprop ?Z/?f` X(`) ;?(`) , ?(`) by using (13) for ` = 1:S.
15. Update ?(`) ? ?(`) ? ???(`) for ` = 1:S and then m ? m + 1
16. SAVE all DHA parameters related to this iteration and GO TO Line 02.
Algorithm 1 illustrates the DHA method for both training and testing phases. As depicted in this
algorithm, (12) is just needed as the first step in the testing phase because the DHA template G
is calculated for this phase based on the training samples (please see Lemma 1). As the second
step in the DHA method, the networks? parameters (?(`) ) must be updated. This paper employs
the back-propagation algorithm (backprop() function) [14] as well as Lemma 3 for this step. In
addition, finishing condition is defined by tackling errors in last three iterations, i.e. the average of the
difference between each pair correlations of aligned functional activities across subjects (?m for last
three iterations). In other words, DHA will be finished if the error rates in the last three iterations are
going to be worst. Further, a structure (nonlinear function for componentwise, and numbers of layers
and units) for the deep network can be selected based on the optimum-state error (?opt ) generated by
training samples across different structures (see Experiment Schemes in the supplementary materials).
In summary, this paper proposes DHA as a flexible deep kernel approach to improve the performance
of functional alignment in fMRI analysis. In order to seek an efficient functional alignment, DHA uses
a deep network (multiple stacked layers of nonlinear transformation) for mapping fMRI responses of
each subject to an embedded space (f` : RT ?V ? RT ?Vnew , ` = 1:S). Unlike previous methods
that use a restricted fixed kernel function, mapping functions in DHA are flexible across subjects
because they employ multi-layer neural networks, which can implement any nonlinear function [12].
Therefore, DHA does not suffer from disadvantages of the previous kernel approach. In order to
deal with high-dimensionality (broad ROI), DHA can also apply an optional feature selection by
considering Vnew < V for constructing the deep networks. The performance of the optional feature
selection will be analyzed in Section 4. Finally, DHA can be scaled across a large number of subjects
by using the proposed optimization algorithm, i.e. rank-m SVD, regularization, and mini-batch SGD.
4
Experiments
The empirical studies are reported in this section. Like previous studies [1?7, 9], this paper employs
the ?-SVM algorithms [16] for generating the classification model. Indeed, we use the binary ?-SVM
for datasets with just two categories of stimuli and multi-label ?-SVM [3, 16] as the multi-class
approach. All datasets are separately preprocessed by FSL 5.0.9 (https://fsl.fmrib.ox.ac.uk),
i.e. slice timing, anatomical alignment, normalization, smoothing. Regions of Interests (ROI) are
also denoted by employing the main reference of each dataset. In addition, leave-one-subject-out
5
Table 1: Accuracy of HA methods in post-alignment classification by using simple task datasets
?Algorithms, Datasets?
?-SVM [17]
HA [1]
RHA [2]
KHA [3]
SVD-HA [4]
SRM [5]
SL [9]
CAE [6]
DHA
DS005
71.65?0.97
81.27?0.59
83.06?0.36
85.29?0.49
90.82?1.23
91.26?0.34
90.21?0.61
94.25?0.76
97.92?0.82
DS105
22.89?1.02
30.03?0.87
32.62?0.52
37.14?0.91
40.21?0.83
48.77?0.94
49.86?0.4
54.52?0.80
60.39?0.68
DS107
38.84?0.82
43.01?0.56
46.82?0.37
52.69?0.69
59.54?0.99
64.11?0.37
64.07?0.98
72.16?0.43
73.05?0.63
DS116
67.26?1.99
74.23?1.40
78.71?0.76
78.03?0.89
81.56?0.54
83.31?0.73
82.32?0.28
91.49?0.67
90.28?0.71
DS117
73.32?1.67
77.93?0.29
84.22?0.44
83.32?0.41
95.62?0.83
95.01?0.64
94.96?0.24
95.92?0.67
97.99?0.94
Table 2: Area under the ROC curve (AUC) of different HA methods in post-alignment classification
by using simple task datasets
?Algorithms, Datasets?
?-SVM [17]
HA [1]
RHA [2]
KHA [3]
SVD-HA [4]
SRM [5]
SL [9]
CAE [6]
DHA
DS005
68.37?1.01
70.32?0.92
82.22?0.42
80.91?0.21
88.54?0.71
90.23?0.74
89.79?0.25
91.24?0.61
96.91?0.82
DS105
21.76?0.91
28.91?1.03
30.35?0.39
36.23?0.57
37.61?0.62
44.48?0.75
47.32?0.92
52.16?0.63
59.57?0.32
DS107
36.84?1.45
40.21?0.33
43.63?0.61
50.41?0.92
57.54?0.31
62.41?0.72
61.84?0.32
72.33?0.79
70.23?0.92
DS116
62.49?1.34
70.67?0.97
76.34?0.45
75.28?0.94
78.66?0.82
79.20?0.98
80.63?0.81
87.53?0.72
89.93?0.24
DS117
70.17?0.59
76.14?0.49
81.54?0.92
80.92?0.28
92.14?0.42
93.65?0.93
93.26?0.72
91.49?0.33
96.13?0.32
cross-validation is utilized for partitioning datasets to the training set and testing set. Different HA
methods are employed for functional aligning and then the mapped neural activities are used to
generate the classification model. The performance of the proposed method is compared with the
?-SVM algorithm as the baseline, where the features are used after anatomical alignment without
applying any hyperalignment mapping. Further, performances of the standard HA [1], RHA [2],
KHA [3], SVDHA [4], SRM [5], and SL [9] are reported as state-of-the-arts HA methods. In this
paper, the results of HA algorithm is generated by employing Generalized CCA proposed in [10].
In addition, regularized parameters (?, ?) in RHA are optimally assigned based on [2]. Further,
KHA algorithm is used by the Gaussian kernel, which is evaluated as the best kernel in the original
paper [3]. As another deep-learning-based alternative for functional alignment, the performance
of CAE [6] is also compared with the proposed method. Like the original paper [6], this paper
employs k1 = k3 = {5, 10, 15, 20, 25}, ? = {0.1, 0.25, 0.5, 0.75, 0.9}, ? = {0.1, 1, 5, 10}. Then,
aligned neural activities (by using CAE) are applied to the classification algorithm same as other
HA techniques. This paper follows the CAE setup to set the same settings in the proposed method.
Consequently, three hidden layers (C = 5) and the regularized parameters = {10?4 , 10?6 , 10?8 }
are employed in the DHA method. In addition, the number of units in the intermediate layers are
considered U (m) = KV , where m = 2:C-1, C is the number of layers, V denotes the number of
voxels and K is the number of stimulus categories in each dataset1 . Further, three distinctive activation
functions are employed, i.e. Sigmoid (g(x) = 1/1 + exp(?x)), Hyperbolic (g(x) = tanh(x)), and
Rectified Linear Unit or ReLU (g(x) = ln(1 + exp(x))). In this paper, the optimum parameters for
DHA and CAE methods are reported for each dataset. Moreover, all algorithms are implemented by
Python 3 on a PC with certain specifications2 by authors in order to generate experimental results.
Experiment schemes are also described in supplementary materials.
4.1
Simple Tasks Analysis
This paper utilizes 5 datasets, shared by Open fMRI (https://openfmri.org), for running empirical studies of this section. Further, numbers of original and aligned features are considered
1
Although we can use any settings for DHA, we empirically figured out this setting is acceptable to seek an
optimum solution. Indeed, we followed CAE setup in the network structure but used the number of categories
(K) rather than a series of parameters. In the current format of DHA, we just need to set the regularized constant
and the nonlinear activation function, while a wide range of parameters must be set in the CAE.
2
DEL, CPU = Intel Xeon E5-2630 v3 (8?2.4 GHz), RAM = 64GB, GPU = GeForce GTX TITAN X (12GB
memory), OS = Ubuntu 16.04.3 LTS, Python = 3.6.2, Pip = 9.0.1, Numpy = 1.13.1, Scipy = 0.19.1, Scikit-Learn
= 0.18.2, Theano = 0.9.0.
6
400
600
800
1000 1200
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
140
210
280
350
420
490
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
(e) Raiders
(TRs = 100)
400
600
800
1000 1200
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
(d) Forrest Gump
(TRs = 2000)
(f) Raiders
(TRs = 400)
70
65
60
55
50
45
40
35
30
70
140
210
280
350
420
490
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
(g) Raiders
(TRs = 800)
Classification Accuracy (%)
70
65
60
55
50
45
40
35
30
70
85
80
75
70
65
60
55
50
45
40
35
30
100 200
(c) Forrest Gump
(TRs = 800)
(b) Forrest Gump
(TRs = 400)
Classification Accuracy (%)
Classification Accuracy (%)
140
210
280
350
420
490
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
85
80
75
70
65
60
55
50
45
40
35
30
100 200
Classification Accuracy (%)
400
600
800
1000 1200
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
(a) Forrest Gump
(TRs = 100)
70
65
60
55
50
45
40
35
30
70
Classification Accuracy (%)
85
80
75
70
65
60
55
50
45
40
35
30
100 200
Classification Accuracy (%)
400
600
800
1000 1200
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
Classification Accuracy (%)
Classification Accuracy (%)
85
80
75
70
65
60
55
50
45
40
35
30
25
100 200
75
70
65
60
55
50
45
40
35
30
70
140
210
280
350
420
490
# of voxels per hemisphere
vSVM
HA
KHA
RHA
SL
SVDHA
SRM
CAE
DHA
(h) Raiders
(TRs = 2000)
Figure 1: Comparison of different HA algorithms on complex task datasets by using ranked voxels.
equal (V = Vnew ) for all HA methods. As the first dataset, ?Mixed-gambles task? (DS005) includes
S = 48 subjects. It also contains K = 2 categories of risk tasks in the human brain, where the
chance of selection is 50/50. In this dataset, the best results for CAE is generated by following
parameters k1 = k3 = 20, ? = 0.75, ? = 1 and for DHA by using = 10?8 and Hyperbolic
function. In addition, ROI is defined based on the original paper [17]. As the second dataset, ?Visual
Object Recognition? (DS105) includes S = 71 subjects. It also contains K = 8 categories of
visual stimuli, i.e. gray-scale images of faces, houses, cats, bottles, scissors, shoes, chairs, and
scrambles (nonsense patterns). In this dataset, the best results for CAE is generated by following
parameters k1 = k3 = 25, ? = 0.9, ? = 5 and for DHA by using = 10?6 and Sigmoid function. Please see [1, 7] for more information. As the third dataset, ?Word and Object Processing?
(DS107) includes S = 98 subjects. It contains K = 4 categories of visual stimuli, i.e. words,
objects, scrambles, consonants. In this dataset, the best results for CAE is generated by following
parameters k1 = k3 = 10, ? = 0.5, ? = 10 and for DHA by using = 10?6 and ReLU function.
Please see [18] for more information. As the fourth dataset, ?Multi-subject, multi-modal human
neuroimaging dataset? (DS117) includes MEG and fMRI images for S = 171 subjects. This paper
just uses the fMRI images of this dataset. It also contains K = 2 categories of visual stimuli, i.e.
human faces, and scrambles. In this dataset, the best results for CAE is generated by following
parameters k1 = k3 = 20, ? = 0.9, ? = 5 and for DHA by using = 10?8 and Sigmoid function.
Please see [19] for more information. The responses of voxels in the Ventral Cortex are analyzed
for these three datasets (DS105, DS107, DS117). As the last dataset, ?Auditory and Visual Oddball
EEG-fMRI? (DS116) includes EEG signals and fMRI images for S = 102 subjects. This paper only
employs the fMRI images of this dataset. It contains K = 2 categories of audio and visual stimuli,
including oddball tasks. In this dataset, the best results for CAE is generated by following parameters
k1 = k3 = 10, ? = 0.75, ? = 1 and for DHA by using = 10?4 and ReLU function. In addition,
ROI is defined based on the original paper [20]. This paper also provides the technical information of
the employed datasets in the supplementary materials. Table 1 and 2 respectively demonstrate the
classification Accuracy and Area Under the ROC Curve (AUC) in percentage (%) for the predictors.
As these tables demonstrate, the performances of classification analysis without HA method are
significantly low. Further, the proposed algorithm has generated better performance in comparison
with other methods because it provided a better embedded space in order to align neural activities.
4.2
Complex Tasks Analysis
This section uses two fMRI datasets, which are related to watching movies. The numbers of original
and aligned features are considered equal (V = Vnew ) for all HA methods. As the first dataset, ?A
high-resolution 7-Tesla fMRI dataset from complex natural stimulation with an audio movie? (DS113)
includes the fMRI data of S = 18 subjects, who watched ?Forrest Gump (1994)? movie during
the experiment. This dataset provided by Open fMRI. In this dataset, the best results for CAE is
generated by following parameters k1 = k3 = 25, ? = 0.9, ? = 10 and for DHA by using = 10?8
and Sigmoid function. Please see [7] for more information. As the second dataset, S = 10 subjects
watched ?Raiders of the Lost Ark (1981)?, where whole brain volumes are 48. In this dataset, the best
results for CAE is generated by following parameters k1 = k3 = 15, ? = 0.75, ? = 1 and for DHA
7
2
80
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78
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60
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SRM
CAE
DHA
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(B) DS107
(A) DS105
Figure 2: Classification by using feature selection.
A
H
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E
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C
SVDHA
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(A) DS105
60
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DHA
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CAE
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90
The Percentage of Selected Features
A
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The Percentage of Selected Features
1.5
0.5
62
40
100
Runtime (%)
60
Runtime (%)
Classification Accuracy
Classification Accuracy
82
65
(B) DS107
Figure 3: Runtime Analysis
by using = 10?4 and Sigmoid function. Please see [3-5] for more information. In these two datasets,
the ROI is defined in the ventral temporal cortex (VT). Figure 1 depicts the generated results, where
the voxels in ROI are ranked by the method proposed in [1] based on their neurological priorities same
as previous studies [1, 4, 7, 9]. Then, the experiments are repeated by using the different number of
ranked voxels per hemisphere, i.e. in Forrest: [100, 200, 400, 600, 800, 1000, 1200], and in Raiders:
[70, 140, 210, 280, 350, 420, 490]. In addition, the empirical studies are reported by using the first
T Rs = [100, 400, 800, 2000] in both datasets. Figure 1 shows that the DHA achieves superior
performance to other HA algorithms.
4.3
Classification analysis by using feature selection
In this section, the effect of features selection (Vnew < V ) on the performance of classification
methods will be discussed by using DS105 and DS107 datasets. Here, the performance of the
proposed method is compared with SVDHA [4], SRM [5], and CAE [6] as the state-of-the-art
HA techniques, which can apply feature selection before generating a classification model. Here,
multi-label ?-SVM [16] is used for generating the classification models after each of the mentioned
methods applied on preprocessed fMRI images for functional alignment. In addition, the setup of this
experiment is same as the previous sections (cross-validation, the best parameters, etc.). Figure 2
illustrates the performance of different methods by employing 100% to 60% of features. As depicted
in this figure, the proposed method has generated better performance in comparison with other
methods because it provides better feature representation in comparison with other techniques.
4.4
Runtime Analysis
In this section, the runtime of the proposed method is compared with the previous HA methods by
using DS105 and DS107 datasets. As mentioned before, all of the results in this experiment are
generated by a PC with certain specifications. Figure 3 illustrates the runtime of the mentioned
methods, where runtime of other methods are scaled based on the DHA (runtime of the proposed
method is considered as the unit). As depicted in this figure, CAE generated the worse runtime
because it concurrently employs modified versions of SRM and SL for functional alignment. Further,
SL also includes high time complexity because of the ensemble approach. By considering the
performance of the proposed method in the previous sections, it generates acceptable runtime. As
mentioned before, the proposed method employs rank-m SVD [10] as well as Incremental SVD [15],
which can significantly reduce the time complexity of the optimization procedure [10, 12].
5
Conclusion
This paper extended a deep approach for hyperalignment methods in order to provide accurate
functional alignment in multi-subject fMRI analysis. Deep Hyperalignment (DHA) can handle fMRI
datasets with nonlinearity, high-dimensionality (broad ROI), and a large number of subjects. We
have also illustrated how DHA can be used for post-alignment classification. DHA is parametric and
uses rank-m SVD and stochastic gradient descent for optimization. Therefore, DHA generates lowruntime on large datasets, and DHA does not require the training data when the functional alignment
is computed for a new subject. Further, DHA is not limited by a restricted fixed representational space
because the kernel in DHA is a multi-layer neural network, which can separately implement any
nonlinear function for each subject to transfer the brain activities to a common space. Experimental
studies on multi-subject fMRI analysis confirm that the DHA method achieves superior performance
to other state-of-the-art HA algorithms. In the future, we will plan to employ DHA for improving the
performance of other techniques in fMRI analysis, e.g. Representational Similarity Analysis (RSA).
8
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (61422204,
61473149, and 61732006), and NUAA Fundamental Research Funds (NE2013105).
References
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9
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6,366 | 6,759 | Online to Offline Conversions, Universality and
Adaptive Minibatch Sizes
Kfir Y. Levy
Department of Computer Science, ETH Z?rich.
[email protected]
Abstract
We present an approach towards convex optimization that relies on a novel scheme
which converts adaptive online algorithms into offline methods. In the offline
optimization setting, our derived methods are shown to obtain favourable adaptive
guarantees which depend on the harmonic sum of the queried gradients. We
further show that our methods implicitly adapt to the objective?s structure: in the
smooth case fast convergence rates are ensured without any prior knowledge of
the smoothness parameter, while still maintaining guarantees in the non-smooth
setting. Our approach has a natural extension to the stochastic setting, resulting in
a lazy version of SGD (stochastic GD), where minibathces are chosen adaptively
depending on the magnitude of the gradients. Thus providing a principled approach
towards choosing minibatch sizes.
1
Introduction
Over the past years data adaptiveness has proven to be crucial to the success of learning algorithms.
The objective function underlying ?big data" applications often demonstrates intricate structure:
the scale and smoothness are often unknown and may change substantially in between different
regions/directions, [1]. Learning methods that acclimatize to these changes may exhibit superior
performance compared to non adaptive procedures.
State-of-the-art first order methods like AdaGrad, [1], and Adam, [2], adapt the learning rate on the
fly according to the feedback (i.e. gradients) received during the optimization process. AdaGrad and
Adam are guaranteed to work well in the online convex optimization setting, where loss functions
may be chosen adversarially and change between rounds. Nevertheless, this setting is harder than the
stochastic/offline settings, which may better depict practical applications. Interestingly, even in the
offline convex optimization setting it could be shown that in several scenarios very simple schemes
may substantially outperform the output of AdaGrad/Adam. An example of such a simple scheme is
choosing the point with the smallest gradient norm among all rounds. In the first part of this work we
address this issue and design adaptive methods for the offline convex optimization setting. At heart of
our derivations is a novel scheme which converts adaptive online algorithms into offline methods with
favourable guarantees1 . Our shceme is inspired by standard online to batch conversions, [3].
A seemingly different issue is choosing the minibatch size, b, in the stochastic setting. Stochastic
optimization algorithms that can access a noisy gradient oracle may choose to invoke the oracle
b times in every query point, subsequently employing an averaged gradient estimate. Theory p
for
stochastic convex optimization suggests to use a minibatch of b = 1, and predicts a degradation of b
factor upon using larger minibatch sizes2 . Nevertheless in practice larger minibatch sizes are usually
found to be effective. In the second part of this work we design stochastic optimization methods in
1
For concreteness we concentrate in this work on converting AdaGrad, [1]. Note that our conversion scheme
applies more widely to other
p adaptive online methods.
2
A degradation by a b factor in the general case and by a b factor in the strongly-convex case.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
which minibatch sizes are chosen adaptively without any theoretical degradation. These are natural
extensions of the offline methods presented in the first part.
Our contributions:
Offline setting: We present two (families of) algorithms AdaNGD (Alg. 2) and SC-AdaNGD
(Alg. 3) for the convex/strongly-convex settings which achieve favourable adaptive guarantees
(Thms. 2.1, 2.2, 3.1, 3.2 ). The latter theorems also establish their universality, i.e., their ability to
implicitly take advantage of the objective?s smoothness and attain rates as fast as GD would have
achieved if the smoothness parameter was known. In contrast to other universal approaches such as
line-search-GD, [4], and universal gradient [5], we do so without any line search procedure. p
Concretely, without the knowledge of the smoothness parameter our algorithm ensures an O(1/ T )
rate in general convex case and an O(1/T ) rate if the objective is also smooth (Thms. 2.1, 2.2). In
the strongly-convex case our algorithm ensures an O(1/T ) rate in general and an O(exp( T )) rate
if the objective is also smooth (Thm. 3.2 ), where is the condition number.
Stochastic setting: We present Lazy-SGD (Algorithm 4) which is an extension of our offline
algorithms. Lazy-SGD employs larger minibatch sizes in points with smaller gradients, which
selectively reduces the variance in the ?more important" query points. Lazy-SGD guarantees are
comparable with SGD in the convex/strongly-convex settings (Thms. 4.2, 4.3).
On the technical side, our online to offline conversion schemes employ three simultaneous mechanisms: an adaptive online algorithm used in conjunction with gradient normalization and with
a respective importance weighting. To the best of our knowledge the combination of the above
techniques is novel, and we believe it might also find use in other scenarios.
This paper is organized as follows. In Sections 2,3, we present our methods for the offline
convex/strongly-convex settings. Section 4 describes our methods for the stochastic setting, and
Section 5 concludes. Extensions and a preliminary experimental study appear in the Appendix.
1.1
Related Work
The authors of [1] simultaneously to [6], were the first to suggest AdaGrad?an adaptive gradient
based method, and prove its efficiency in tackling online convex problems. AdaGrad was subsequently
adjusted to the deep-learning setting to yield the RMSprop, [7], and Adadelta, [8], heuristics. Adam,
[2], is a popular adaptive algorithm which is often the method of choice in deep-learning applications.
It combines ideas from AdaGrad together with momentum machinery, [9].
An optimization procedure is called universal if it implicitly adapts to the objective?s smoothness. In
[5], universal gradient methods are devised for the general convex setting. Concretely, without the
knowledge of the smoothness parameter, these methods
p attain the standard O(1/T ), an accelerated
2
O(1/T ) rates for smooth objectives, and an O(1/ T ) rate in the non-smooth case. The core
technique in this work is a line search procedure which estimates the smoothness parameter in
every iteration. For strongly-convex and smooth objectives, line search techniques, [4], ensure
linear convergence rate, without the knowledge of the smoothness parameter. However, line search
is not ?fully universal", in the sense that it holds no guarantees in the non-smooth case. For the
latter setting we present a method which is ?fully universal" (Thm. 3.2), nevertheless it requires the
strong-convexity parameter.
The usefulness of employing normalized gradients was demonstrated in several non-convex scenarios.
In the context of quasi-convex optimization, [10], and [11], established convergence guarantees for
the offline/stochastic settings. More recently, it was shown in [12], that normalized gradient descent
is more appropriate than GD for saddle-evasion scenarios.
In the context of stochastic optimization, the effect of minibatch size was extensively investigated
throughout the past years, [13, 14, 15, 16, 17, 18]. Yet, all of these studies: (i) assume a smooth
expected loss, (ii) discuss fixed minibatch sizes. Conversely, our work discusses adaptive minibatch
sizes, and applies to both smooth/non-smooth expected losses.
1.2
Preliminaries
Notation: k ? k denotes the `2 norm, G denotes a bound on the norm of the objective?s gradients, and
[T ] := {1, . . . , T }. For a set K 2 Rd its diameter is defined as D = supx,y2K kx yk. Next we
2
Algorithm 1 Adaptive Gradient Descent (AdaGrad)
Input: #Iterations T , x1 2 Rd , set K
Set: Q0 = 0
for t = 1 . . . T do
Calculate: gt = rft (xt )
Update:
p
Set: ?t = D/ 2Qt
Update: xt+1 = ?K (xt
end for
Qt = Qt
1
+ kgt k2
?t g t )
define H-strongly-convex/ -smooth functions,
f (x) + rf (x)> (y
x) +
f (y) ? f (x) + rf (x)> (y
x) +
f (y)
1.2.1
H
kx
2
yk2 ;
8x, y 2 K
(H-strong-convexity)
kx
yk2 ;
8x, y 2 K
( -smoothness)
2
AdaGrad
The methods presented in this paper lean on AdaGrad (Alg. 1), an online optimization method which
employs an adaptive learning rate. The following theorem states AdaGrad?s guarantees, [1],
Theorem 1.1. Let K be a convex set with diameter D. Let {ft }Tt=1 be an arbitrary sequence of
convex loss functions. Then Algorithm 1 guarantees the following regret;
v
u
T
T
T
u
X
X
X
ft (xt ) min
ft (x) ? t2D2
kgt k2 .
t=1
2
x2K
t=1
t=1
Adaptive Normalized Gradient Descent (AdaNGD)
In this section we discuss the convex optimization setting and introduce our AdaNGDk algorithm,
which depends on a parameter k 2 R. We first derive a general convergence rate which holds for a
general k. Subsequently, we elaborate on the k = 1, 2 cases which exhibit universality as well as
adaptive guarantees that may be substantially better compared to standard methods.
Our method AdaNGDk is depicted in Alg. 2. This algorithm can be thought of as an online to offline
conversion scheme which utilizes AdaGrad (Alg. 1) as a black box and eventually outputs a weighted
sum of the online queries. Indeed, for a fixed k 2 R, it is not hard to notice that AdaNGDk is
equivalent to invoking AdaGrad with the following loss sequence {f?t (x) := gt> x/kgt kk }Tt=1 . And
eventually weighting each query point inversely proportional to the k?th power norm of its gradient.
The reason behind this scheme is that in offline optimization it makes sense to dramatically reduce
the learning rate upon uncountering a point with a very small gradient. For k 1, this is achieved by
invoking AdaGrad with gradients normalized by their k?th power norm. Since we discuss constrained
optimization, we use the projection operator defined as, ?K (y) := minx2K kx yk . The next lemma
states the guarantee of AdaNGD for a general k:
Lemma 2.1. Let k 2 R, K be a convex set with diameter D, and f be a convex function; Also let x
?T
be the output of AdaNGDk (Algorithm 2), then the following holds:
q
PT
2D2 t=1 1/kgt k2(k 1)
f (?
xT ) min f (x) ?
PT
k
x2K
t=1 1/kgt k
Proof sketch. Notice that the AdaNGDk algorithm is equivalent to applying AdaGrad to the following
loss sequence: {f?t (x) := gt> x/kgt kk }Tt=1 . Thus, applying Theorem 1.1, and using the definition of
x
?T together with Jensen?s inequality the lemma follows.
3
Algorithm 2 Adaptive Normalized Gradient Descent (AdaNGDk )
Input: #Iterations T , x1 2 Rd , set K , parameter k
Set: Q0 = 0
for t = 1 . . . T 1 do
Calculate: gt = rf (xt ), g?t = gt /kgt kk
Update:
Qt = Qt 1 + 1/kgt k2(k
p
Set ?t = D/ 2Qt
Update: xt+1 = ?K (xt ?t g?t )
end for
k
PT
tk
Return: x
?T = t=1 PT1/kg
x
1/kg kk t
? =1
1)
?
For k = 0, Algorithm 2 becomes AdaGrad (Alg. 1). Next we focus on the cases where k = 1, 2,
showing improved adaptive rates and universality compared to GD/AdaGrad. These improved rates
are attained thanks to the adaptivity of the learning rate: when query points with small gradients are
encountered, AdaNGDk (with k 1) reduces the learning rate, thus focusing on the region around
these points. The hindsight weighting further emphasizes points with smaller gradients.
2.1
AdaNGD1
p
Here we show that AdaNGD1 enjoys a rate of O(1/ T ) in the non-smooth convex setting, and a
fast rate of O(1/T ) in the smooth setting. We emphasize that the same algorithm enjoys these rates
simultaneously, without any prior knowledge of the smoothness or of the gradient norms.
From Algorithm 2 it can
p be noted that for k = 1 the learning rate becomes independent of the
gradients, i.e. ?t = D/ 2t, the update is made according to the direction of the gradients, and the
weighting is inversely proportional to the norm of the gradients. The following Theorem establishes
the guarantees of AdaNGD1 ,
Theorem 2.1. Let k = 1, K be a convex set with diameter D, and f be a convex function; Also let
x
?T be the outputs of AdaNGD1 (Alg. 2), then the following holds:
p
p
2D2 T
2GD
f (?
xT ) min f (x) ? PT
? p
.
x2K
T
t=1 1/kgt k
Moreover, if f is also -smooth and the global minimum x? = arg minx2Rn f (x) belongs to K, then:
p
D T
4 D2
f (?
xT ) min f (x) ? PT
?
.
x2K
T
t=1 1/kgt k
Proof sketch. The data dependent bound is a direct corollary of Lemma 2.1. The general case bound
p
PT
holds by using kgt k ? G. The bound for the smooth case is proven by showing t=1 kgt k ? O( T ).
PT
This translates to a lower bound t=1 1/kgt k ?(T 3/2 ), which concludes the proof.
The data dependent bound in Theorem 2.1 may be substantially better compared to the bound of
the GD/AdaGrad. As an example, assume that half of the gradients encountered during the run
of the algorithm are of O(1) norms, and the other gradient
norms decay proportionally to O(1/t).
p
In this case the guarantee of GD/AdaGrad is O(1/ T ), whereas AdaNGD1 guarantees a bound
that behaves like O(1/T 3/2 ). Note that the above example presumes that all algorithms encounter
the same gradient magnitudes, which might be untrue. Nevertheless in the smooth case AdaNGD1
provably benefits due to its adaptivity.
2.2
AdaNGD2
Here we show that AdaNGD2 enjoys comparable guarantees to AdaNGD1 in the general/smooth
case. Similarly to AdaNGD1 the same algorithm enjoys these rates simultaneously, without any
prior knowledge of the smoothness or of the gradient norms. The following Theorem establishes the
guarantees of AdaNGD2 ,
4
Algorithm 3 Strongly-Convex AdaNGD (SC-AdaNGDk )
Input: #Iterations T , x1 2 Rd , set K, strong-convexity H, parameter k
Set: Q0 = 0
for t = 1 . . . T 1 do
Calculate: gt = rf (xt ), g?t = gt /kgt kk
Update:
Qt = Qt 1 + 1/kgt kk
Set ?t = 1/HQt
Update: xt+1 = ?K (xt ?t g?t )
end for
k
PT
tk
Return: x
?T = t=1 PT1/kg
x
1/kg kk t
? =1
?
Theorem 2.2. Let k = 2, K be a convex set with diameter D, and f be a convex function; Also let
x
?T be the outputs of AdaNGD2 (Alg. 2), then the following holds:
p
p
2D2
2GD
f (?
xT ) min f (x) ? qP
? p
.
x2K
T
T
2
1/kg
k
t
t=1
Moreover, if f is also -smooth and the global minimum x? = arg minx2Rn f (x) belongs to K, then:
p
2D2
4 D2
f (?
xT ) min f (x) ? qP
?
.
x2K
T
T
2
1/kg
k
t
t=1
It is interesting to note that AdaNGD2 will have always performed better than AdaGrad, had both
algorithms encountered the same gradient norms. This is due to the well known inequality between
PT
PT1
arithmetic and harmonic means, [19], T1 t=1 at
, 8{at }Tt=1 ? R+ , which directly
1
t=1 1/at
T
qP
T
2
implies, pPT 1
? T1
t=1 kgt k .
2
t=1
3
1/kgt k
Adaptive NGD for Strongly Convex Functions
Here we discuss the offline optimization setting of strongly convex objectives. We introduce our
SC-AdaNGDk algorithm, and present convergence rates for general k 2 R. Subsequently, we
elaborate on the k = 1, 2 cases which exhibit universality as well as adaptive guarantees that may be
substantially better compared to standard methods.
Our SC-AdaNGDk algorithm is depicted in Algorithm 3. Similarly to its non strongly-convex
counterpart, SC-AdaNGDk can be thought of as an online to offline conversion scheme which utilizes
an online algorithm which we denote SC-AdaGrad (we elaborate on the latter in the appendix). The
next Lemma states its guarantees,
Lemma 3.1. Let k 2 R, and K be a convex set. Let f be an H-strongly-convex function; Also let x
?T
be the outputs of SC-AdaNGDk (Alg. 3), then the following holds:
f (?
xT )
min f (x) ?
x2K
2H
PT
1
t=1
kgt k
k
T
X
kgt k
Pt
t=1
? =1
2(k 1)
kg? k
k
.
Proof sketch. In the appendix we present and analyze SC-AdaGrad. This is an online first P
order algot
rithm for strongly-convex functions in which the learning rate decays according to ?t = 1/ ? =1 H? ,
where H? is the strong-convexity parameter of the loss function at time ? . Then we show that
SC-AdaNGDk is equivalent to applying SC-AdaGrad to the following loss sequence:
?
T
1
H
>
2
f?t (x) =
g
x
+
kx
x
k
.
t
kgt kk t
2kgt kk
t=1
The lemma follows by combining the regret bound of SC-AdaGrad together with the definition of x
?T
and with Jensen?s inequality.
5
For k = 0, SC-AdaNGD becomes the standard GD algorithm which uses learning rate of ?t = 1/Ht.
Next we focus on the cases where k = 1, 2.
3.1
SC-AdaNGD1
?
Here we show that SC-AdaNGD1 enjoys a rate of O(1/T
) for strongly-convex objectives, and a
2
?
faster rate of O(1/T ) assuming that the objective is also smooth. We emphasize that the same
algorithm enjoys these rates simultaneously, without any prior knowledge of the smoothness or of the
gradient norms. The following theorem establishes the guarantees of SC-AdaNGD1 ,
Theorem 3.1. Let k = 1, and K be a convex set. Let f be a G-Lipschitz and H-strongly-convex
function; Also let x
?T be the outputs of SC-AdaNGD1 (Alg. 3), then the following holds:
?
?P
??
T
G
G 1 + log
t=1 kgt k
G2 (1 + log T )
f (?
xT ) min f (x) ?
?
.
PT
x2K
2HT
2H t=1 1
kgt k
Moreover, if f is also -smooth and the global minimum x? = arg minx2Rn f (x) belongs to K, then,
2
f (?
xT )
3.2
min f (x) ?
x2K
( /H)G2 (1 + log T )
.
HT 2
SC-AdaNGD2
?
Here we show that SC-AdaNGD2 enjoys the standard O(1/T
) rate for strongly-convex objectives,
and a linear rate assuming that the objective is also smooth. We emphasize that the same algorithm
enjoys these rates simultaneously, without any prior knowledge of the smoothness or of the gradient
norms. In the case where k = 2 the guarantee of SC-AdaNGD is as follows,
Theorem 3.2. Let k = 2, K be a convex set, and f be a G-Lipschitz and H-strongly-convex function;
Also let x
?T be the outputs of SC-AdaNGD2 (Alg. 3), then the following holds:
PT
1 + log(G2 t=1 kgt k 2 )
G2 (1 + log T )
f (?
xT ) min f (x) ?
?
.
PT
x2K
2HT
2H t=1 kgt k 2
Moreover, if f is also -smooth and the global minimum x? = arg minx2Rn f (x) belongs to K, then,
?
?
3G2 H T
H
f (?
xT ) min f (x) ?
e
1+ T .
x2K
2H
Intuition: For strongly-convex objectives the appropriate GD algorithm utilizes two very extreme
learning rates of ?t / 1/t vs. ?t = 1/ for the general/smooth settings respectively. A possible
explanation to the universality of SCAdaNGD2 is that it implicitly interpolate between these rates.
k 2
Indeed the update rule of our algorithm can be written as follows, xt+1 = xt H1 Pt kgtkg
2 gt .
?k
? =1
Thus, ignoring the hindsight
weighting,
SCAdaNGD
is
equivalent
to
GD
with
an
adaptive
learning
2
Pt
rate ??t := kgt k 2 /H ? =1 kg? k 2 . Now, when all gradient norms are of the same magnitude, then
??t / 1/t, which boils down to the standard GD for strongly-convex objectives. Conversely, assume
that the gradients are exponentially decaying, i.e., that kgt k / q t for some q < 1. In this case ??t is
approximately constant. We believe that the latter applies for strongly-convex & smooth case.
4
Adaptive NGD for Stochastic Optimization
Here we show that using data-dependent minibatch sizes, we can adapt our (SC-)AdaNGD2 algorithms (Algs. 2, 3 with k = 2) to the stochastic setting, and achieve the well know convergence rates
for the convex/strongly-convex settings. Next we introduce the stochastic optimization setting, and
then we present and discuss our Lazy SGD algorithm.
Setup: We consider the problem of minimizing a convex/strongly-convex function f : K 7! R,
where K 2 Rd is a convex set. We assume that optimization lasts for T rounds; on each round
6
Algorithm 4 Lazy Stochastic Gradient Descent (LazySGD)
Input: #Oracle Queries T , x1 2 Rd , set K, ?0 , p
Set: t = 0, s = 0
while t ? T do
Update: s = s + 1
Set G = GradOracle(xs ), i.e., G generates i.i.d. noisy samples of rf (xs )
Get: (?
gs , ns ) = AE(G, T t) % Adaptive Minibatch
Update: t = t + ns
Calculate: g?s = ns g?s
Set: ?s = ?0 /tp
Update: xs+1 = ?K (xs ?s g?s )
end while
Ps
Return: x
?T = i=1 nTi xi . (Note that Psi=1 ni = T )
Algorithm 5 Adaptive Estimate (AE)
Input: random vectors generator G, sample budget Tmax , sample factor m0
Set: i = 0, N = 0, g?0 = 0
while N < Tmax do
Take ?i = min{2i , Tmax N } samples from G
Set N
N + ?i
Update: g?N
Average
of N samples received so far from G
p
If k?
gN k > 3m0 / N then return (?
gN , N )
Update i
i+1
end while
Return: (?
gN , N )
t = 1, . . . , T , we may query a point xt 2 K, and receive a feedback. After the last round, we choose
x
?T 2 K, and our performance measure is the expected excess loss, defined as,
E[f (?
xT )]
min f (x) .
x2K
Here we assume that our feedback is a first order noisy oracle G : K 7! Rd such that upon
querying G with a point xt 2 K, we receive a bounded and unbiased gradient estimate, G(xt ),
such E[G(xt )|xt ] = rf (xt ); kG(xt )k ? G. We also assume that the that the internal coin tosses
(randomizations) of the oracle are independent. It is well known that variants of Stochastic Gradient
Descent
?T such that the excess loss is bounded by
p (SGD) are ensured to output an estimate x
O(1/ T )/O(1/T ) for the setups of convex/strongly-convex stochastic optimization, [20], [21].
Notation: In this section we make a clear distinction between the number of queries to the gradient
oracle, denoted henceforth by T ; and between the number of iterations in the algorithm, denoted
henceforth by S. We care about the dependence of the excess loss in T .
4.1
Lazy Stochastic Gradient Descent
Data Dependent Minibatch sizes: The Lazy SGD (Alg. 4) algorithm that we present in this section,
uses a minibatch size that changes in between query points. Given a query point xs , Lazy SGD
2
3
?
invokes the noisy gradient oracle O(1/kg
s k ) times, where gs := rf (xs ) . Thus, in contrast to
SGD which utilizes a fixed number of oracle calls per query point, our algorithm tends to stall in
points with smaller gradients, hence the name Lazy SGD.
Here we give some intuition regarding our adaptive minibatch size rule: Consider the stochastic
optimization setting. However, imagine that instead of the noisy gradient oracle G, we may access an
improved (imaginary) oracle which provides us with unbiased estimates, g?(x), that are accurate up
to some multiplicative factor, e.g., E[?
g (x)|x] = rf (x), and 12 krf (x)k ? k?
g (x)k ? 2krf (x)k .
Then intuitively we could have used these estimates instead of the exact normalized gradients inside
our (SC-)AdaNGD2 algorithms (Algs. 2, 3 with k = 2), and still get similar (in expectation) data
3
Note that the gradient norm, kgs k, is unknown to the algorithm. Nevertheless it is estimated on the fly.
7
dependent bounds. Quite nicely, we may use our original noisy oracle G to generate estimates
2
?
from this imaginary oracle. This can be done by invoking G for O(1/kg
s k ) times at each query
point. Using this minibatch rule, the total number of calls to G (along all iterations) is equal to
PS
T = s=1 1/kgs k2 . Plugging this
pinto the data dependent bounds of (SC-)AdaNGD2 (Thms. 2.2,
?
?
3.2), we get the well known O(1/
T )/O(1/T
) rates for the stochastic convex settings.
The imaginary oracle: The construction of the imaginary oracle from the original oracle appears in
Algorithm 5 (AE procedure) . It receives as an input, G, a generator of independent random vectors
with an (unknown) expected value g 2 Rd . The algorithm outputs two variables: N which is an
estimate of 1/kgk2 , and g?N an average of N random vectors from G. Thus, it is natural to think of
N g?N as an estimate for g/kgk2 . Moreover, it can be shown that E[N (?
gN g)] = 0. Thus in a sense
we receive an unbiased estimate. The guarantees of Algorithm 5 appear below,
Lemma 4.1 (Informal). Let Tmax 1, 2 (0, 1). Suppose an oracle G : K 7! Rd that generates
G-bounded i.i.d. random vectors with an (unknown) expected value g 2 Rd . Then w.p. 1
,
invoking AE (Algorithm 5), with m0 = ?(G log(1/ )), it is ensured that:
N = ?(min{m0 /kgk2 , Tmax }), and E[N (?
gN
g)] = 0 .
Lazy SGD: Now, plugging the output of the AE algorithm into our offline algorithms (SC-)AdaNGD2 ,
we get their stochastic variants which appears in Algorithm 4 (Lazy SGD). This algorithm is equivalent
to the offline version of (SC-)AdaNGD2 , with the difference that we use ns instead of 1/krf (xs )k2
and ns g?s instead of rf (xs )/krf (xs )k2 .
Let T be a bound on the total number of queries to the the first order oracle G, and be the confidence
parameter used to set m0 in the AE procedure. Next we present the guarantees of LazySGD,
Lemma 4.2. Let = O(T 3/2 ); let K be a convex set with diameterp
D, and f be a convex function;
and assume kG(x)k ? G w.p.1. Then using LazySGD with ?0 = D/ 2G, p = 1/2, ensures:
?
?
GD log(T )
p
E[f (?
xT )] min f (x) ? O
.
x2K
T
Lemma 4.3. Let = O(T 2 ), let K be a convex set, and f be an H-strongly-convex convex function;
and assume kG(x)k ? G w.p.1. Then using LazySGD with ?0 = 1/H, p = 1, ensures:
? 2
?
G log2 (T )
E[f (?
xT )] min f (x) ? O
.
x2K
HT
Note that LazySGD uses minibatch
sizes that are adapted to the magnitude of the gradients, and still
p
maintains the optimal O(1/ T )/O(1/T ) rates. Inpcontrast
p using a fixed minibatch size b for SGD
might degrade the convergence rates, yielding O( b/ T )/O(b/T ) guarantees. This property of
LazySGD may be beneficial when considering distributed computations (see [13]).
5
Discussion
We have presented a new approach based on a conversion scheme, which exhibits universality and
new adaptive bounds in the offline convex optimization setting, and provides a principled approach
towards minibatch size selection in the stochastic setting. Among the many questions that remain
open is whether we can devise ?accelerated" universal methods. Furthermore, our universality results
only apply when the global minimum is inside the constraints. Thus, it is natural to seek for methods
that ensure universality when this assumption is violated. Moreover, our algorithms depend on
a parameter k 2 R, but only the cases where k 2 {0, 1, 2} are well understood. Investigating a
wider spectrum of k values is intriguing. Lastly, it is interesting to modify and test our methods in
non-convex scenarios, especially in the context of deep-learning applications.
Acknowledgments
I would like to thank Elad Hazan and Shai Shalev-Shwartz for fruitful discussions during the early
stages of this work.
This work was supported by the ETH Z?rich Postdoctoral Fellowship and Marie Curie Actions for
People COFUND program.
8
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6,367 | 676 | Destabilization and Route to Chaos
in Neural Networks
with Random Connectivity
Bernard Doyon
Unite INSERM 230
Service de Neurologie
CHUPurpan
F-31059 Toulouse Cedex, France
Bruno Cessac
Centre d'Etudes et de Recherches
de Toulouse
2, avenue Edouard Belin, BP 4025
F-31055 Toulouse Cedex, France
Mathias Quoy
Centre d'Etudes et de Recherches
de Toulouse
2, avenue Edouard Belin, BP 4025
F-31055 Toulouse Cedex, France
Manuel Samuelides
Ecole Nationale Superieure
de I'Aeronautique et de l'Espace
10, avenue Edouard Belin, BP 4032
F-31055 Toulouse Cedex, France
Abstract
The occurence of chaos in recurrent neural networks is supposed to
depend on the architecture and on the synaptic coupling strength. It is
studied here for a randomly diluted architecture. By normalizing the
variance of synaptic weights, we produce a bifurcation parameter,
dependent on this variance and on the slope of the transfer function but
independent of the connectivity, that allows a sustained activity and the
occurence of chaos when reaching a critical value. Even for weak
connectivity and small size, we find numerical results in accordance
with the theoretical ones previously established for fully connected
infinite sized networks. Moreover the route towards chaos is
numerically checked to be a quasi-periodic one, whatever the type of the
first bifurcation is (Hopf bifurcation, pitchfork or flip).
549
550
Doyon, Cessac, Quoy, and Samuelides
1
INTRODUCTION
Most part of studies on recurrent neural networks assume sufficient conditions of
convergence. Models with symmetric synaptic connections have dynamical properties
strongly connected with those of spin-glasses. In particular, they have relaxationnal
dynamics caracterised by the decreasing of a function which is analogous to the energy in
spin-glasses (or free energy for models submitted to thermal noise). Networks with
asymmetric synaptic connections lose this convergence property and can have more
complex dynamics. but searchers try to obtain such a convergence because the relaxation
to a stable network state is simply interpreted as a stored pattern.
However, as pointed out by Hirsch (1989), it might be very interesting, from an
engineering point of view. to investigate non convergent networks because their
dynamical possibilities are much richer for a given number of units. Moreover, the real
brain is a highly dynamic system. Recent neurophysiological findings have focused
attention on the rich temporal structures (oscillations) of neuronal processes (Gray et al.,
1989), which might play an important role in information processing. Chaotic behavior
has been found out in the nervous system (Gallez & Babloyantz, 1991) and might be
implicated in cognitive processes (Skarda & Freeman. 1987).
We have studied the emergent dynamics of a general class of non convergent networks.
Some results are already available in this field. Sompolinsky et al. (1988) established
strong theoretical results concerning the occurrence of chaos forfully connected networks
in the thermodynamic limit (N - 00) by using the Dynamic Mean Field Theory. Their
model is a continuous time. continuous state dynamical system with N fully connected
neurons. Each connection Jij is a gaussian random variable with zero mean and a
normalized variance fllN. As the Jij'S are independent. the constant term fl can be seen
as the variance of the sum of the weights connected to a given unit. Thus. the global
strength of coupling remains constant for each neuron as N increases. The output
function of each neuron is sigmoidal with a slope g. Sompolinsky et al. established that,
in the limit N - 00, there is a sharp transition from a stationary state to a chaotic flow.
The onset of chaos is given by the critical value gJ=l. For gJ<1 the system admits the
only fixed point zero, while for gJ >1 it is chaotic. The same authors performed
simulations on finite and large values of N and showed the existence of an intermediate
regime (nonzero stationary states or limit cycles) separating the stationary and the chaotic
phase. but the routes to chaos were not systematically explored. The range of gJ where
this intermediate behavior is observed shrinks as N increases.
2
THE MODEL
The hypothesis of a fully connected network being not biologically plausible. it could be
interesting to inspect how far these results could be extended as the dilution increases for
a general class of networks. The model we study is defined as follows: the number of
units is N. and K is the fixed number of connections received by one unit (K>I). There is
no connection from one unit to itself. The K connections are randomly selected (with an
uniform law) among the N-l' s. The state of each neuron i at time t is characterized by its
Destabilization and Route to Chaos in Neural Networks with Random Connectivity
output xi (t) which is a real variable varying between -1 and 1. The discrete and parallel
dynamics is given by:
Jij is the synaptic weight which couples the output of unit j to the input of unit i.
These weights are random independent variables chosen with a uniform law, with zero
menn and a normalized variance J 2 1K. Notice that, with such a normalization, the
standard deviation of the sum of the weights afferent to a given neuron is the constant J.
One has to distinguish two effects of coupling on the behavior of such a class of models.
The first effect is due to the strength of coupling, independent of the number of
connections. The second one is due to the architecture of coupling, which can be studied
by keeping constant the global synaptic effect of coupling. The genericity of our model
cancels the peculiar dynamic features which may occur due to geometrical effects.
Moreover it allows to study a model at different scales of dilution.
3
FIRST BIFURCATION
For such a system, zero is always a fixed point and for low bifurcation parameter value it
is the only fixed point and it is stable. Let us call Amax the eigenvalue of the matrix of
synaptic weights with the greatest modulus and p =IAmaxl the spectral radius of this
matrix. The loss of stability arises when the product gp is larger than 1. Our numerical
simulations allow us to state that p is approximately equal to J for sufficiently largesized networks. This statement can be derived rigorously for an approximate regularized
model in the thermodynamic limit (Doyon et aI., 1993).
Table 1: Mean Value of the Bifurcation Parameter gJ over 30 Networks.
Destabilization of the zero fixed point 1Onset of Chaos
Connectivity K
4
8
16
32
128
.954 / 1.337
.950 / 1.449
.951/1.434
.961 / 1.360
Number of neurons
256
.9651 1.298
.966 1 1.301
.9651 1.315
.958 1 1.333
512
.9701 1.258
.9781 1.233
.969/ 1.239
.972 I 1.246
We have studied by intensive simulations on a Cray I-XMP computer the statistical
spectral distribution for N ranging from 4 to 512 and for K ranging from 2 to 32. Figure
1 shows two examples of spectra (for convenience, J is set to 1). The apparent drawing of
a real axis is due to the real eigenvalue density but the distribution converges to a
uniform one over the J radius disk, as N increases. A similar result has been theoretically
551
552
Doyon, Cessac, Quay, and Samuelides
achieved for full gaussian matrices (Girko. 1985; Sommers et al.. 1988). Thus pquicldy
decreases to J, so the loss of stability arises for a mean gJ value that increases to 1 for
increasing size (Tab. 1). For a given N value, p is nearly independent of K .
.....:. ..
. . '.-'.
~
",-
Figure 1: Plot of the Unit Disk and of the Eigenvalues in the Complex Plane.
Left: 100 Spectra for N=64. K=4. Right: 10 Spectra for N=512. K=4.
Three types of first bifurcation can occur, depending on the eigenvalue Ama.'t :
a) Hopf Bifurcation: this corresponds to the appearance of oscillations. There are
two complex conjugate eigenvalues with maximal modulus p.
b) Pitchfork bifurcation: if Amax is real positive, the bifurcation arises when
gAmax = 1. Zero loses its stability and two branches of stable equilibria emerge.
c) Flip Bifurcation: for Amaxreal and negative a flip bifurcation occurs when
g Amax = - 1. This corresponds to the appearance of a period two oscillation.
As the network size increases, the proportion of Hopf bifurcations increases because the
proportion of real Amax decreases, nearly independent of K .
4
ROUTE TO CHAOS
To study the following bifurcations, we chose the global observable:
The value m(t) correctly characterizes all types of first bifurcation that can occur. Indeed
the route to chaos is qualitatively well described by this observable, as we checked it by
Destabilization and Route to Chaos in Neural Networks with Random Connectivity
studying simultaneously xi (I). The onset- of chaos was computed by testing the
sensitivity on initial conditions for m(1) . We observed the onset of chaos occurs for quite
low parameter values. The transient zone from fixed point to chaos shrinks slowly to
zero as the network size increases (fab. 1).
The qualitative study of the routes to chaos was made on a span of networks with various
connectivity and quite important size. The route towards chaos that was observed was a
quasi-periodic one in all cases with some variations due to the particular symmetry x- x. '~"he following figures are obtained by plotting m(l+l) versus m(1) after discarding the
transient (Fig. 2). They are not qualitatively different with a reconstruction in a higher
dimensional space. The dominant features are the following ones.
!
"'+J~
0.1-
a)
0.0
b)
0.0
-4.1,
~-4~.I----------------~o:o----------------~o.i----~-')-
~. II
~,--------~--------~--0.1
inti)
-4.1
0.0
r
",: ~_
..
d)
0.0
"r''<~
-4.1
. -.~~,;b :t"
0.0
0.1
"'(II
Figure 2: Example of route to chaos when the fust bifurcation is a Hopf one.
(N=128, K=16) .
a) After the first bifurcation, the zero fixed point has lost its stability.
The series of points (m(I), m(t+l) densely covers a cycle (gJ=l.O).
b) After the second Hopf bifurcation: projection of a T2 torus (gJ=l.23).
c) Frequency locking on the T2 torus (gJ=1.247).
d) Chaos (gJ=1.26).
553
554
Doyon, Cessac, Quoy, and Samuelides
When the first bifurcation is a Hopf one (Fig. 2a), it is followed by a second Hopf
bifurcation (Fig. 2b). Then there is a frequency locking occuring on the T2 torus born
from the second Hopf bifurcation (Fig. 2c), followed by chaos (Fig. 2d). This route is
then a quasi-periodic one (Ruelle & Takens, 1971 ; Newhouse et al., 1978). A slightly
different feature emerges when the first bifurcation is followed by a stable resonance due
to discrete time occuring before the second Hopf bifurcation. Then the limit cycle reduces
to periodic points. When the second bifurcation occurs, the resonance persists until chaos
is reached.
When the first bifurcation is a pitchfork, it is followed by a Hopf bifurcation for each
stable point of the pitchfork (due to the symmetry x - -x). Then a second Hopf
bifurcation occurs followed, via a frequency locking, by chaos. It follows then, despite
the pitchfork bifurcation, a quasi-periodicity route. Notice that in this case, we get two
symmetric strange attractors. When gJ increases, the two attractors fuse.
For a first bifurcation of flip type, the route followed is like the one described by Bauer
& Martienssen (1989). The flip bifurcation leads to an oscillatory system with two
states. A first Hopf bifurcation arises followed by a second one leading to a quasi-periodic
state, followed by a frequency locking preceeding chaos.
5
CONCLUSION
We have presented a type of neural network exhibiting a chaotic behavior when
increasing a bifurcation parameter. As in Sompolinsky's model, gJ is the control
parameter of the network dynamics. The variance of the synaptic weights being
normalized, the bifurcation values are nearly independent of the connectivity K. The
magnitude of dilution is not important for the behavior. The route to chaos by quasiperiodicity seems to be generic. It suggests that such high-dimensional networks behave
like low-dimensional dynamical systems. It could be much simpler to control such
networks than a priori expected.
From a biological point of view, we built our model to provide a tool that could be used
to investigate the influence of chaotic dynamics in the cognitive processes in the brain.
We clearly chose to simplify the biological complexity in order to understand a complex
dynamic. We think that, if chaos plays a role in cognitive processes, it does neither
depend on a specific architecture, nor on the exact internal modelling of the biological
neuron. However, it could be interesting to introduce some biological caracteristics in the
model. The next step will be to study the influence of non-zero entries on the behavior of
the system, leading to the modelling of learning in a chaotic network.
Acknowledgements
This research has been partly supported by the COGNISCIENCE research program of the
C.N.R.S. through PRESCOT, the Toulouse network of searchers in Cognitive Sciences.
Destabilization and Route to Chaos in Neural Networks with Random Connectivity
References
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D. Gallez & A. Babloyantz. (1991) Predictability of human EEG: a dynamical approach.
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V.l.. Girko. (1985) Circular Law. Theory Prob. Its Appl. (USSR) 29: 694-706.
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555
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6,368 | 6,760 | Stochastic Optimization with Variance Reduction
for Infinite Datasets with Finite Sum Structure
Alberto Bietti
Inria?
[email protected]
Julien Mairal
Inria?
[email protected]
Abstract
Stochastic optimization algorithms with variance reduction have proven successful
for minimizing large finite sums of functions. Unfortunately, these techniques are
unable to deal with stochastic perturbations of input data, induced for example by
data augmentation. In such cases, the objective is no longer a finite sum, and the
main candidate for optimization is the stochastic gradient descent method (SGD).
In this paper, we introduce a variance reduction approach for these settings when
the objective is composite and strongly convex. The convergence rate outperforms
SGD with a typically much smaller constant factor, which depends on the variance
of gradient estimates only due to perturbations on a single example.
1
Introduction
Many supervised machine learning problems can be cast as the minimization of an expected loss
over a data distribution with respect to a vector x in Rp of model parameters. When an infinite
amount of data is available, stochastic optimization methods such as SGD or stochastic mirror descent
algorithms, or their variants, are typically used (see [5, 11, 24, 34]). Nevertheless, when the dataset is
finite, incremental methods based on variance reduction techniques (e.g., [2, 8, 15, 17, 18, 27, 29])
have proven to be significantly faster at solving the finite-sum problem
n
n
o
1X
minp F (x) := f (x) + h(x) =
fi (x) + h(x) ,
x?R
n i=1
(1)
where the functions fi are smooth and convex, and h is a simple convex penalty that need not be
differentiable such as the ?1 norm. A classical setting is fi (x) = ?(yi , x? ?i ) + (?/2)kxk2 , where
(?i , yi ) is an example-label pair, ? is a convex loss function, and ? is a regularization parameter.
In this paper, we are interested in a variant of (1) where random perturbations of data are introduced,
which is a common scenario in machine learning. Then, the functions fi involve an expectation over
a random perturbation ?, leading to the problem
n
n
o
1X
minp F (x) :=
fi (x) + h(x) .
x?R
n i=1
with
fi (x) = E? [f?i (x, ?)].
(2)
Unfortunately, variance reduction methods are not compatible with the setting (2), since evaluating
a single gradient ?fi (x) requires computing a full expectation. Yet, dealing with random perturbations is of utmost interest; for instance, this is a key to achieve stable feature selection [23],
improving the generalization error both in theory [33] and in practice [19, 32], obtaining stable
and robust predictors [36], or using complex a priori knowledge about data to generate virtually
?
Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Table 1: Iteration complexity of different methods for solving the objective (2) in terms of number of
iterations required to find x such that E[f (x) ? f (x? )] ? ?. The complexity of N-SAGA [14] matches
the first term of S-MISO but is asymptotically biased. Note that we always have the perturbation
2
noise variance ?p2 smaller than the total variance ?tot
and thus S-MISO improves on SGD both in the
first term (linear convergence to a smaller ??) and in the second (smaller constant in the asymptotic
2
rate). In many application cases, we also have ?p2 ? ?tot
(see main text and Table 2).
Method
SGD
N-SAGA [14]
S-MISO
Asymptotic error
0
?0 = O
0
?p2
?
!
Iteration complexity
2
2
?tot
L
1
?tot
with ?? = O
O
log
+
?
??
??
?
1
L
O
n+
log
with ? > ?0
?
?
!
!
?p2
?p2
L
1
n+
O
log
+
with ?? = O
?
??
??
?
larger datasets [19, 26, 30]. Injecting noise in data is also useful to hide gradient information for
privacy-aware learning [10].
Despite its importance, the optimization problem (2) has been littled studied and to the best of our
knowledge, no dedicated optimization method that is able to exploit the problem structure has been
developed so far. A natural way to optimize this objective when h = 0 is indeed SGD, but ignoring the
finite-sum structure leads to gradient estimates with high variance and slow convergence. The goal of
this paper is to introduce an algorithm for strongly convex objectives, called stochastic MISO, which
exploits the underlying finite sum using variance reduction. Our method achieves a faster convergence
rate than SGD, by removing the dependence on the gradient variance due to sampling the data points i
in {1, . . . , n}; the dependence remains only for the variance due to random perturbations ?.
To the best of our knowledge, our method is the first algorithm that interpolates naturally between incremental methods for finite sums (when there are no perturbations) and the stochastic approximation
setting (when n = 1), while being able to efficiently tackle the hybrid case.
Related work. Many optimization methods dedicated to the finite-sum problem (e.g., [15, 29])
have been motivated by the fact that their updates can be interpreted as SGD steps with unbiased
estimates of the full gradient, but with a variance that decreases as the algorithm approaches the
optimum [15]; on the other hand, vanilla SGD requires decreasing step-sizes to achieve this reduction
of variance, thereby slowing down convergence. Our work aims at extending these techniques to the
case where each function in the finite sum can only be accessed via a first-order stochastic oracle.
Most related to our work, recent methods that use data clustering to accelerate variance reduction
techniques [3, 14] can be seen as tackling a special case of (2), where the expectations in fi are
replaced by empirical averages over points in a cluster. While N-SAGA [14] was originally not
designed for the stochastic context we consider, we remark that their method can be applied to (2).
Their algorithm is however asymptotically biased and does not converge to the optimum. On the other
hand, ClusterSVRG [3] is not biased, but does not support infinite datasets. The method proposed
in [1] uses variance reduction in a setting where gradients are computed approximately, but the
algorithm computes a full gradient at every pass, which is not available in our stochastic setting.
Paper organization. In Section 2, we present our algorithm for smooth objectives, and we analyze
its convergence in Section 3. For space limitation reasons, we present an extension to composite
objectives and non-uniform sampling in Appendix A. Section 4 is devoted to empirical results.
2
The Stochastic MISO Algorithm for Smooth Objectives
In this section, we introduce the stochastic MISO approach for smooth objectives (h = 0), which
relies on the following assumptions:
? (A1) global strong convexity: f is ?-strongly convex;
? (A2) smoothness: f?i (?, ?) is L-smooth for all i and ? (i.e., with L-Lipschitz gradients).
2
2
Table 2: Estimated ratio ?tot
/?p2 , which corresponds to the expected acceleration of S-MISO over
SGD. These numbers are based on feature vectors variance, which is closely related to the gradient
variance when learning a linear model. ResNet-50 denotes a 50 layer network [12] pre-trained on
the ImageNet dataset. For image transformations, the numbers are empirically evaluated from 100
2
2
different images, with 100 random perturbations for each image. Rtot
(respectively, Rcluster
) denotes
the average squared distance between pairs of points in the dataset (respectively, in a given cluster),
following [14]. The settings for unsupervised CKN and Scattering are described in Section 4. More
details are given in the main text.
Type of perturbation
Direct perturbation of
linear model features
Random image
transformations
Application case
Data clustering as in [3, 14]
Additive Gaussian noise N (0, ?2 I)
Dropout with probability ?
Feature rescaling by s in U (1 ? w, 1 + w)
ResNet-50 [12], color perturbation
ResNet-50 [12], rescaling + crop
Unsupervised CKN [22], rescaling + crop
Scattering [6], gamma correction
2
Estimated ratio ?tot
/?p2
2
2
?
Rtot /Rcluster
?
1 + 1/?2
?
1 + 1/?
?
1 + 3/w2
21.9
13.6
9.6
9.8
Note that these assumptions are relaxed in Appendix A by supporting composite objectives and
by exploiting different smoothness parameters Li on each example, a setting where non-uniform
sampling of the training points is typically helpful to accelerate convergence (e.g., [35]).
Complexity results. We now introduce the following quantity, which is essential in our analysis:
n
i
h
1X 2
?p2 :=
?i , with ?i2 := E? k?f?i (x? , ?) ? ?fi (x? )k2 ,
n i=1
where x? is the (unique) minimizer of f . The quantity ?p2 represents the part of the variance of the
gradients at the optimum that is due to the perturbations ?. In contrast, another quantity of interest is
2
the total variance ?tot
, which also includes the randomness in the choice of the index i, defined as
2
?tot
= Ei,? [k?f?i (x? , ?)k2 ] = ?p2 + Ei [k?fi (x? )k2 ]
(note that ?f (x? ) = 0).
2
The relation between ?tot
and ?p2 is obtained by simple algebraic manipulations.
2
The goal of our paper is to exploit the potential imbalance ?p2 ? ?tot
, occurring when perturbations
on input data are small compared to the sampling noise. The assumption is reasonable: given a data
point, selecting a different one should lead to larger variation than a simple perturbation. From a
theoretical point of view, the approach we propose achieves the iteration complexity presented in
Table 1, see also Appendix D and [4, 5, 24] for the complexity analysis of SGD. The gain over SGD
2
is of order ?tot
/?p2 , which is also observed in our experiments in Section 4. We also compare against
the method N-SAGA; its convergence rate is similar to ours but suffers from a non-zero asymptotic
error.
Motivation from application cases. One clear framework of application is the data clustering
scenario already investigated in [3, 14]. Nevertheless, we will focus on less-studied data augmentation
settings that lead instead to true stochastic formulations such as (2). First, we consider learning a
linear model when adding simple direct manipulations of feature vectors, via rescaling (multiplying
each entry vector by a random scalar), Dropout, or additive Gaussian noise, in order to improve the
generalization error [33] or to get more stable estimators [23]. In Table 2, we present the potential
gain over SGD in these scenarios. To do that, we study the variance of perturbations applied to
a feature vector ?. Indeed, the gradient of the loss is proportional to ?, which allows us to obtain
2
good estimates of the ratio ?tot
/?p2 , as we observed in our empirical study of Dropout presented
in Section 4. Whereas some perturbations are friendly for our method such as feature rescaling (a
rescaling window of [0.9, 1.1] yields for instance a huge gain factor of 300), a large Dropout rate
would lead to less impressive acceleration (e.g., a Dropout with ? = 0.5 simply yields a factor 2).
Second, we also consider more interesting domain-driven data perturbations such as classical image transformations considered in computer vision [26, 36] including image cropping, rescaling,
brightness, contrast, hue, and saturation changes. These transformations may be used to train a linear
3
Algorithm 1 S-MISO for smooth objectives
Input: step-size sequence
(?t )t?1 ;
P
initialize x0 = n1 i zi0 for some (zi0 )i=1,...,n ;
for t = 1, . . . do
Sample an index it uniformly at random, a perturbation ?t , and update
(
(1 ? ?t )zit?1 + ?t (xt?1 ? ?1 ?f?it (xt?1 , ?t )), if i = it
t
zi =
zit?1 ,
otherwise.
(3)
n
xt =
1X t
1
).
zi = xt?1 + (zitt ? zit?1
t
n i=1
n
(4)
end for
classifier on top of an unsupervised multilayer image model such as unsupervised CKNs [22] or
the scattering transform [6]. It may also be used for retraining the last layer of a pre-trained deep
neural network: given a new task unseen during the full network training and given limited amount
of training data, data augmentation may be indeed crucial to obtain good prediction and S-MISO
can help accelerate learning in this setting. These scenarios are also studied in Table 2, where the
experiment with ResNet-50 involving random cropping and rescaling produces 224 ? 224 images
from 256 ? 256 ones. For these scenarios with realistic perturbations, the potential gain varies from
10 to 20.
Description of stochastic MISO. We are now in shape to present our method, described in Algorithm 1. Without perturbations and with a constant step-size, the algorithm resembles the MISO/Finito
algorithms [9, 18, 21], which may be seen as primal variants of SDCA [28, 29]. Specifically, MISO
is not able to deal with our stochastic objective (2), but it may address the deterministic finite-sum
problem (1). It is part of a larger body of optimization methods that iteratively build a model of the
objective function, typically a lower or upper bound on the objective that is easier to optimize; for
instance, this strategy is commonly adopted in bundle methods [13, 25].
More precisely,
PnMISO assumes that each fi is strongly convex and builds a model using lower bounds
Dt (x) = n1 i=1 dti (x), where each dti is a quadratic lower bound on fi of the form
?
?
dti (x) = cti,1 + kx ? zit k2 = cti,2 ? ?hx, zit i + kxk2 .
(5)
2
2
These lower bounds are updated during the algorithm using strong convexity lower bounds at xt?1 of
the form lit (x) = fi (xt?1 ) + h?fi (xt?1 ), x ? xt?1 i + ?2 kx ? xt?1 k2 ? fi (x):
(1 ? ?t )dt?1
(x) + ?t lit (x), if i = it
t
i
di (x) =
(6)
t?1
di (x),
otherwise,
which corresponds to an update of the quantity zit :
(
(1 ? ?t )zit?1 + ?t (xt?1 ? ?1 ?fit (xt?1 )),
t
zi =
zit?1 ,
if i = it
otherwise.
The next iterate is then computed as xt = arg minx Dt (x), which is equivalent to (4). The original
MISO/Finito algorithms use ?t = 1 under a ?big data? condition on the sample size n [9, 21],
while the theory was later extended in [18] to relax this condition by supporting smaller constant
steps ?t = ?, leading to an algorithm that may be interpreted as a primal variant of SDCA (see [28]).
Note that when fi is an expectation, it is hard to obtain such lower bounds since the gradient
?fi (xt?1 ) is not available in general. For this reason, we have introduced S-MISO, which can
exploit approximate lower bounds to each fi using gradient estimates, by letting the step-sizes ?t
decrease appropriately as commonly done in stochastic approximation. This leads to update (3).
Separately, SDCA [29] considers the Fenchel conjugates of fi , defined by fi? (y) = supx x? y ?fi (x).
When fi is an expectation, fi? is not available in closed form in general, nor are its gradients, and in
fact exploiting stochastic gradient estimates is difficult in the duality framework. In contrast, [28]
gives an analysis of SDCA in the primal, aka. ?without duality?, for smooth finite sums, and our
work extends this line of reasoning to the stochastic approximation and composite settings.
4
Relationship with SGD in the smooth case. The link between S-MISO in the non-composite
setting and SGD can be seen by rewriting the update (4) as
1
?t
xt = xt?1 + (zitt ? zit?1
) = xt?1 + vt ,
t
n
n
where
1 ?
vt := xt?1 ? ?fit (xt?1 , ?t ) ? zit?1
.
(7)
t
?
Note that E[vt |Ft?1 ] = ? ?1 ?f (xt?1 ), where Ft?1 contains all information up to iteration t; hence,
the algorithm can be seen as an instance of the stochastic gradient method with unbiased gradients,
which was a key motivation in SVRG [15] and later in other variance reduction algorithms [8, 28]. It
= xt?1 ; hence
is also worth noting that in the absence of a finite-sum structure (n = 1), we have zit?1
t
our method becomes identical to SGD, up to a redefinition of step-sizes. In the composite case (see
Appendix A), our approach yields a new algorithm that resembles regularized dual averaging [34].
Memory requirements and handling of sparse datasets. The algorithm requires storing the
vectors (zit )i=1,...,n , which takes the same amount of memory as the original dataset and which
is therefore a reasonable requirement in many practical cases. In the case of sparse datasets, it is
fair to assume that random perturbations applied to input data preserve the sparsity patterns of the
original vectors, as is the case, e.g., when applying Dropout to text documents described with bag-ofwords representations [33]. If we further assume the typical setting where the ?-strong convexity
comes from an ?2 regularizer: f?i (x, ?) = ?i (x? ?i? ) + (?/2)kxk2 , where ?i? is the (sparse) perturbed
example and ?i encodes the loss, then the update (3) can be written as
(
?t ?t
(1 ? ?t )zit?1 ? ??t ??i (x?
t?1 ?i )?i , if i = it
t
zi =
zit?1 ,
otherwise,
which shows that for every index i, the vector zit preserves the same sparsity pattern as the examples ?i?
throughout the algorithm (assuming the initialization zi0 = 0), making the update (3) efficient. The
update (4) has the same cost since vt = zitt ? zit?1
is also sparse.
t
Limitations and alternative approaches. Since our algorithm is uniformly better than SGD in
terms of iteration complexity, its main limitation is in terms of memory storage when the dataset
cannot fit into memory (remember that the memory cost of S-MISO is the same as the input dataset).
In these huge-scale settings, SGD should be preferred; this holds true in fact for all incremental
methods when one cannot afford to perform more than one (or very few) passes over the data. Our
paper focuses instead on non-huge datasets, which are those benefiting most from data augmentation.
We note that a different approach to variance reduction like SVRG [15] is able to trade off storage
requirements for additional full gradient computations, which would be desirable in some situations.
However, we were not able to obtain any decreasing step-size strategy that works for these methods,
both in theory and practice, leaving us with constant step-size approaches as in [1, 14] that either
maintain a non-zero asymptotic error, or require dynamically reducing the variance of gradient
estimates. One possible way to explain this difficulty is that SVRG and SAGA [8] ?forget? past
gradients for a given example i, while S-MISO averages them in (3), which seems to be a technical
key to make it suitable to stochastic approximation. Nevertheless, the question of whether it is
possible to trade-off storage with computation in a setting like ours is open and of utmost interest.
3
Convergence Analysis of S-MISO
We now study the convergence properties of the S-MISO algorithm. For space limitation reasons,
all proofs are provided in Appendix B. We start by defining the problem-dependent quantities
zi? := x? ? ?1 ?fi (x? ), and then introduce the Lyapunov function
Ct =
n
1
?t X t
kxt ? x? k2 + 2
kz ? zi? k2 .
2
n i=1 i
(8)
Proposition 1 gives a recursion on Ct , obtained by upper-bounding separately its two terms, and
finding coefficients to cancel out other appearing quantities when relating Ct to Ct?1 . To this end, we
borrow elements of the convergence proof of SDCA without duality [28]; our technical contribution
is to extend their result to the stochastic approximation and composite (see Appendix A) cases.
5
Proposition 1 (Recursion on Ct ). If (?t )t?1 is a positive and non-increasing sequence satisfying
n
1
,
(9)
,
?1 ? min
2 2(2? ? 1)
with ? = L/?, then Ct obeys the recursion
? 2 ? 2
?t
t
p
E[Ct ] ? 1 ?
E[Ct?1 ] + 2
.
n
n
?2
(10)
We now state the main convergence result, which provides the expected rate O(1/t) on Ct based on
decreasing step-sizes, similar to [5] for SGD. Note that convergence of objective function values is
directly related to that of the Lyapunov function Ct via smoothness:
L
(11)
E[f (xt ) ? f (x? )] ? E kxt ? x? k2 ? L E[Ct ].
2
Theorem 2 (Convergence of Lyapunov function). Let the sequence of step-sizes (?t )t?1 be defined
2n
by ?t = ?+t
with ? ? 0 such that ?1 satisfies (9). For all t ? 0, it holds that
)
(
8?p2
?
(12)
where ? := max
, (? + 1)C0 .
E[Ct ] ?
?+t+1
?2
Choice of step-sizes in practice. Naturally, we would like ? to be small, in particular independent
of the initial condition C0 and equal to the first term in the definition (12). We would like the
dependence on C0 to vanish at a faster rate than O(1/t), as it is the case in variance reduction
algorithms on finite sums. As advised in [5] in the context of SGD, we can initially run the algorithm
with a constant step-size ?
? and exploit this linear convergence regime until we reach the level of
noise given by ?p , and then start decaying the step-size. It is easy to see that by using a constant
step-size ?
? , Ct converges near a value C? := 2?
??p2 /n?2 . Indeed, Eq. (10) with ?t = ?
? yields
?
? ? 1? ?
?
E[Ct ? C]
E[Ct?1 ? C].
n
n
Thus, we can reach a precision C0? with E[C0? ] ? ?? := 2C? in O( ?
?) iterations. Then, if we
? log C0 /?
start decaying step-sizes as in Theorem 2 with ? large enough so that ?1 = ?
? , we have
(? + 1) E[C0? ] ? (? + 1)?
? = 8?p2 /?2 ,
making both terms in (12) smaller than or equal to ? = 8?p2 /?2 . Considering these two phases, with
an initial step-size ?
? given by (9), the final work complexity for reaching E[kxt ? x? k2 ] ? ? is
!
?p2
C0
L
log
+O
.
(13)
O
n+
?
??
?2 ?
We can then use (11) in order to obtain the complexity for reaching E[f (xt ) ? f (x? )] ? ?. Note that
following this step-size strategy was found to be very effective in practice (see Section 4).
Acceleration by iterate averaging. When one is interested in the convergence in function values,
the complexity (13) combined with (11) yields O(L?p2 /?2 ?), which can be problematic for illconditioned problems (large condition number L/?). The following theorem presents an iterate
averaging scheme which brings the complexity term down to O(?p2 /??), which appeared in Table 1.
Theorem 3 (Convergence under iterate averaging). Let the step-size sequence (?t )t?1 be defined by
1
n
2n
for ? ? 1 s.t. ?1 ? min
,
.
?t =
?+t
2 4(2? ? 1)
We have
E[f (?
xT ) ? f (x? )] ?
where
16?p2
2??(? ? 1)C0
+
,
T (2? + T ? 1) ?(2? + T ? 1)
T
?1
X
2
(? + t)xt .
x
?T :=
T (2? + T ? 1) t=0
6
STL-10 ckn, ? = 10 ?3
10 -2
10 -3
10
10
10
10
10
10
10
10
10
0 50 100 150 200 250 300 350 400 450
epochs
?3
STL-10 scattering, ? = 10
-2
10
10
10
-4
10
-5
0
50
10
-3
-4
0
10
-3
STL-10 ckn,
0
10
100 150 200 250 300 350 400
? = 10 ?5
-1
10
-1
-2
10
-1
10
50 100 150 200 250 300 350 400 450
-2
-3
0
50
epochs
F - F*
F - F*
10
0
? = 10 ?4
10
10 -4
10 -5
STL-10 ckn,
0
f - f*
f - f*
10 -1
10
1
100 150 200 250 300 350 400
epochs
?4
STL-10 scattering, ? = 10
10
0
10
STL-10 scattering,
1
? = 10 ?5
0
-1
F - F*
S-MISO ? = 0. 1
S-MISO ? = 1. 0
N-SAGA ? = 0. 1
SGD ? = 0. 1
SGD ? = 1. 0
f - f*
10 0
-2
-3
10
10
10
-4
-5
0
50
epochs
10
100 150 200 250 300 350 400
-1
-2
-3
-4
0
50
epochs
100 150 200 250 300 350 400
epochs
Figure 1: Impact of conditioning for data augmentation on STL-10 (controlled by ?, where ? = 10?4
gives the best accuracy). Values of the loss are shown on a logarithmic scale (1 unit = factor 10).
? = 0.1 satisfies the theory for all methods, and we include curves for larger step-sizes ? = 1. We
omit N-SAGA for ? = 1 because it remains far from the optimum. For the scattering representation,
the problem we study is ?1 -regularized, and we use the composite algorithm of Appendix A.
f - f*
10 -3
10 -4
10 -5
10 -6
10
10
10
f - f*
S-MISO ? = 0. 1
S-MISO ? = 1. 0
N-SAGA ? = 0. 1
SGD ? = 0. 1
SGD ? = 1. 0
10
10
10
10
0
50 100 150 200 250 300 350 400
epochs
10
ResNet50,
0
? = 10 ?3
10
ResNet50,
0
? = 10 ?4
-1
10
-2
-3
f - f*
ResNet50, ? = 10 ?2
10 -2
-4
10
10
-1
-2
-3
-5
10
-6
-7
0
50
100 150 200 250 300 350 400
epochs
10
-4
-5
0
50
100 150 200 250 300 350 400
epochs
Figure 2: Re-training of the last layer of a pre-trained ResNet 50 model, on a small dataset with
random color perturbations (for different values of ?).
The proof uses a similar telescoping sum technique to [16]. Note that if T ? ?, the first term,
which depends on the initial condition C0 , decays as 1/T 2 and is thus dominated by the second
term. Moreover, if we start averaging after an initial phase with constant step-size ?
? , we can consider
C0 ? 4?
??p2 /n?2 . In the ill-conditioned regime, taking ?
? = ?1 = 2n/(? + 1) as large as allowed
by (9), we have ? of the order of ? = L/? ? 1. The full convergence rate then becomes
!
2
?
?
p
E[f (?
xT ) ? f (x? )] ? O
.
1+
?(? + T )
T
When T is large enough compared to ?, this becomes O(?p2 /?T ), leading to a complexity O(?p2 /??).
4
Experiments
We present experiments comparing S-MISO with SGD and N-SAGA [14] on four different scenarios,
in order to demonstrate the wide applicability of our method: we consider an image classification
dataset with two different image representations and random transformations, and two classification
tasks with Dropout regularization, one on genetic data, and one on (sparse) text data. Figures 1 and 3
show the curves for an estimate of the training objective using 5 sampled perturbations per example.
The plots are shown on a logarithmic scale, and the values are compared to the best value obtained
among the different methods in 500 epochs. The strong convexity constant ? is the regularization
parameter. For all methods, we consider step-sizes supported by the theory as well as larger step-sizes
that may work better in practice. Our C++/Cython implementation of all methods considered in this
section is available at https://github.com/albietz/stochs.
Choices of step-sizes. For both S-MISO and SGD, we use the step-size strategy mentioned in
Section 3 and advised by [5], which we have found to be most effective among many heuristics
7
10 -3
10
10
10 -4
10 0
10 -1
10
0
50 100 150 200 250 300 350 400
epochs
imdb dropout, = 0.30
?
S-MISO-NU = 1 0
S-MISO = 10 0
SGD-NU = 1 0
SGD = 10 0
N-SAGA = 10 0
?
?
f - f*
10 -2
.
.
?
?
.
.
?
.
f - f*
10 -5
10
10 -3
10 -4 0 50 100 150 200 250 300 350 400
epochs
10
gene dropout,
0
?
= 0.10
10
-1
10
-2
10
f - f*
10 -2
10
f - f*
f - f*
10
S-MISO ? = 0. 1
S-MISO ? = 1. 0
SGD ? = 0. 1
SGD ? = 1. 0
N-SAGA ? = 0. 1
N-SAGA ? = 1. 0
10 -1
-3
-4
10
10
10
-5
10
-6
0
50
100 150 200 250 300 350 400
epochs
imdb dropout, ? = 0.10
10 0
-1
10
10 -2
10 -3
10 -4
10 -5 0 50 100 150 200 250 300 350 400
epochs
f - f*
gene dropout, ? = 0.30
10 0
10
gene dropout,
0
?
= 0.01
-1
-2
-3
-4
-5
-6
-7
0
50
100 150 200 250 300 350 400
epochs
imdb dropout, ? = 0.01
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7 0 50 100 150 200 250 300 350 400
epochs
Figure 3: Impact of perturbations controlled by the Dropout rate ?. The gene data is ?2 -normalized;
hence, we consider similar step-sizes as Figure 1. The IMDB dataset is highly heterogeneous;
thus, we also include non-uniform (NU) sampling variants of Appendix A. For uniform sampling,
theoretical step-sizes perform poorly for all methods; thus, we show a larger tuned step-size ? = 10.
we have tried: we initially keep the step-size constant (controlled by a factor ? ? 1 in the figures)
for 2 epochs, and then start decaying as ?t = C/(? + t), where C = 2n for S-MISO, C = 2/?
for SGD, and ? is chosen large enough to match the previous constant step-size. For N-SAGA, we
maintain a constant step-size throughout the optimization, as suggested in the original paper [14].
The factor ? shown in the figures is such that ? = 1 corresponds to an initial step-size n?/(L ? ?)
? instead of L in
for S-MISO (from (19) in the uniform case) and 1/L for SGD and N-SAGA (with L
the non-uniform case when using the variant of Appendix A).
Image classification with ?data augmentation?. The success of deep neural networks is often
limited by the availability of large amounts of labeled images. When there are many unlabeled
images but few labeled ones, a common approach is to train a linear classifier on top of a deep
network learned in an unsupervised manner, or pre-trained on a different task (e.g., on the ImageNet
dataset). We follow this approach on the STL-10 dataset [7], which contains 5K training images
from 10 classes and 100K unlabeled images, using a 2-layer unsupervised convolutional kernel
network [22], giving representations of dimension 9 216. The perturbation consists of randomly
cropping and scaling the input images. We use the squared hinge loss in a one-versus-all setting. The
vector representations are ?2 -normalized such that we may use the upper bound L = 1 + ? for the
smoothness constant. We also present results on the same dataset using a scattering representation [6]
of dimension 21 696, with random gamma corrections (raising all pixels to the power ?, where ? is
chosen randomly around 1). For this representation, we add an ?1 regularization term and use the
composite variant of S-MISO presented in Appendix A.
Figure 1 shows convergence results on one training fold (500 images), for different values of ?,
allowing us to study the behavior of the algorithms for different condition numbers. The low variance
induced by data transformations allows S-MISO to reach suboptimality that is orders of magnitude
smaller than SGD after the same number of epochs. Note that one unit on these plots corresponds to
one order of magnitude in the logarithmic scale. N-SAGA initially reaches a smaller suboptimality
than SGD, but quickly gets stuck due to the bias in the algorithm, as predicted by the theory [14],
while S-MISO and SGD continue to converge to the optimum thanks to the decreasing step-sizes. The
best validation accuracy for both representations is obtained for ? ? 10?4 (middle column), and we
observed relative gains of up to 1% from using data augmentation. We computed empirical variances
of the image representations for these two strategies, which are closely related to the variance in
gradient estimates, and observed these transformations to account for about 10% of the total variance.
Figure 2 shows convergence results when training the last layer of a 50-layer Residual network [12]
that has been pre-trained on ImageNet. Here, we consider the common scenario of leveraging a deep
model trained on a large dataset as a feature extractor in order to learn a new classifier on a different
small dataset, where it would be difficult to train such a model from scratch. To simulate this setting,
we consider a binary classification task on a small dataset of 100 images of size 256x256 taken
from the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012, which we crop to
8
224x224 before performing random adjustments to brightness, saturation, hue and contrast. As in the
STL-10 experiments, the gains of S-MISO over other methods are of about one order of magnitude in
suboptimality, as predicted by Table 2.
Dropout on gene expression data. We trained a binary logistic regression model on the breast
cancer dataset of [31], with different Dropout rates ?, i.e., where at every iteration, each coordinate ?j
of a feature vector ? is set to zero independently with probability ? and to ?j /(1 ? ?) otherwise. The
dataset consists of 295 vectors of dimension 8 141 of gene expression data, which we normalize
in ?2 norm. Figure 3 (top) compares S-MISO with SGD and N-SAGA for three values of ?, as a
way to control the variance of the perturbations. We include a Dropout rate of 0.01 to illustrate the
impact of ? on the algorithms and study the influence of the perturbation variance ?p2 , even though
this value of ? is less relevant for the task. The plots show very clearly how the variance induced by
the perturbations affects the convergence of S-MISO, giving suboptimality values that may be orders
of magnitude smaller than SGD. This behavior is consistent with the theoretical convergence rate
established in Section 3 and shows that the practice matches the theory.
Dropout on movie review sentiment analysis data. We trained a binary classifier with a squared
hinge loss on the IMDB dataset [20] with different Dropout rates ?. We use the labeled part of
the IMDB dataset, which consists of 25K training and 250K testing movie reviews, represented as
89 527-dimensional sparse bag-of-words vectors. In contrast to the previous experiments, we do not
normalize the representations, which have great variability in their norms, in particular, the maximum
Lipschitz constant across training points is roughly 100 times larger than the average one. Figure 3
(bottom) compares non-uniform sampling versions of S-MISO (see Appendix A) and SGD (see
Appendix D) with their uniform sampling counterparts as well as N-SAGA. Note that we use a large
step-size ? = 10 for the uniform sampling algorithms, since ? = 1 was significantly slower for
all methods, likely due to outliers in the dataset. In contrast, the non-uniform sampling algorithms
required no tuning and just use ? = 1. The curves clearly show that S-MISO-NU has a much faster
convergence in the initial phase, thanks to the larger step-size allowed by non-uniform sampling, and
later converges similarly to S-MISO, i.e., at a much faster rate than SGD when the perturbations are
small. The value of ? used in the experiments was chosen by cross-validation, and the use of Dropout
gave improvements in test accuracy from 88.51% with no dropout to 88.68 ? 0.03% with ? = 0.1
and 88.86 ? 0.11% with ? = 0.3 (based on 10 different runs of S-MISO-NU after 400 epochs).
Finally, we also study the effect of the iterate averaging scheme of Theorem 3 in Appendix E.
Acknowledgements
This work was supported by a grant from ANR (MACARON project under grant number ANR14-CE23-0003-01), by the ERC grant number 714381 (SOLARIS project), and by the MSR-Inria
joint center.
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6,369 | 6,761 | Deep Learning with Topological Signatures
Roland Kwitt
Department of Computer Science
University of Salzburg, Austria
[email protected]
Christoph Hofer
Department of Computer Science
University of Salzburg, Austria
[email protected]
Marc Niethammer
UNC Chapel Hill, NC, USA
[email protected]
Andreas Uhl
Department of Computer Science
University of Salzburg, Austria
[email protected]
Abstract
Inferring topological and geometrical information from data can offer an alternative
perspective on machine learning problems. Methods from topological data analysis,
e.g., persistent homology, enable us to obtain such information, typically in the form
of summary representations of topological features. However, such topological
signatures often come with an unusual structure (e.g., multisets of intervals) that is
highly impractical for most machine learning techniques. While many strategies
have been proposed to map these topological signatures into machine learning
compatible representations, they suffer from being agnostic to the target learning
task. In contrast, we propose a technique that enables us to input topological
signatures to deep neural networks and learn a task-optimal representation during
training. Our approach is realized as a novel input layer with favorable theoretical
properties. Classification experiments on 2D object shapes and social network
graphs demonstrate the versatility of the approach and, in case of the latter, we
even outperform the state-of-the-art by a large margin.
1
Introduction
Methods from algebraic topology have only recently emerged in the machine learning community,
most prominently under the term topological data analysis (TDA) [7]. Since TDA enables us to
infer relevant topological and geometrical information from data, it can offer a novel and potentially
beneficial perspective on various machine learning problems. Two compelling benefits of TDA
are (1) its versatility, i.e., we are not restricted to any particular kind of data (such as images,
sensor measurements, time-series, graphs, etc.) and (2) its robustness to noise. Several works have
demonstrated that TDA can be beneficial in a diverse set of problems, such as studying the manifold
of natural image patches [8], analyzing activity patterns of the visual cortex [28], classification of 3D
surface meshes [27, 22], clustering [11], or recognition of 2D object shapes [29].
Currently, the most widely-used tool from TDA is persistent homology [15, 14]. Essentially1 ,
persistent homology allows us to track topological changes as we analyze data at multiple ?scales?.
As the scale changes, topological features (such as connected components, holes, etc.) appear and
disappear. Persistent homology associates a lifespan to these features in the form of a birth and
a death time. The collection of (birth, death) tuples forms a multiset that can be visualized as a
persistence diagram or a barcode, also referred to as a topological signature of the data. However,
leveraging these signatures for learning purposes poses considerable challenges, mostly due to their
1
We will make these concepts more concrete in Sec. 2.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Death
(2) Transform & Project
(?2 , ? 2 )
x = ?(p)
(?1 , ? 1 )
s?,?,? ((x0 , x1 ))
?
Output:
(y1 , y2 )> ? R0 + ?R0
(1) Rotate points in D by ?/4
Death-Birth
(persistence)
Input Layer
Input: D ? D
p = (b, d)
?1
Param.: ? = (?i , ? i )N
i=0
Birth
?
(x0 , x1 )
Birth+Death
?
Figure 1: Illustration of the proposed network input layer for topological signatures. Each signature, in the
form of a persistence diagram D ? D (left), is projected w.r.t. a collection of structure elements. The layer?s
learnable parameters ? are the locations ?i and the scales ? i of these elements; ? ? R+ is set a-priori and
meant to discount the impact of points with low persistence (and, in many cases, of low discriminative power).
The layer output y is a concatenation of the projections. In this illustration, N = 2 and hence y = (y1 , y2 )> .
unusual structure as a multiset. While there exist suitable metrics to compare signatures (e.g., the
Wasserstein metric), they are highly impractical for learning, as they require solving optimal matching
problems.
Related work. In order to deal with these issues, several strategies have been proposed. In [2] for
instance, Adcock et al. use invariant theory to ?coordinatize? the space of barcodes. This allows to
map barcodes to vectors of fixed size which can then be fed to standard machine learning techniques,
such as support vector machines (SVMs). Alternatively, Adams et al. [1] map barcodes to so-called
persistence images which, upon discretization, can also be interpreted as vectors and used with
standard learning techniques. Along another line of research, Bubenik [6] proposes a mapping
of barcodes into a Banach space. This has been shown to be particularly viable in a statistical
context (see, e.g., [10]). The mapping outputs a representation referred to as a persistence landscape.
Interestingly, under a specific choice of parameters, barcodes are mapped into L2 (R2 ) and the
inner-product in that space can be used to construct a valid kernel function. Similar, kernel-based
techniques, have also recently been studied by Reininghaus et al. [27], Kwitt et al. [20] and Kusano
et al. [19].
While all previously mentioned approaches retain certain stability properties of the original representation with respect to common metrics in TDA (such as the Wasserstein or Bottleneck distances),
they also share one common drawback: the mapping of topological signatures to a representation that
is compatible with existing learning techniques is pre-defined. Consequently, it is fixed and therefore
agnostic to any specific learning task. This is clearly suboptimal, as the eminent success of deep
neural networks (e.g., [18, 17]) has shown that learning representations is a preferable approach.
Furthermore, techniques based on kernels [27, 20, 19] for instance, additionally suffer scalability
issues, as training typically scales poorly with the number of samples (e.g., roughly cubic in case of
kernel-SVMs). In the spirit of end-to-end training, we therefore aim for an approach that allows to
learn a task-optimal representation of topological signatures. We additionally remark that, e.g., Qi et
al. [25] or Ravanbakhsh et al. [26] have proposed architectures that can handle sets, but only with
fixed size. In our context, this is impractical as the capability of handling sets with varying cardinality
is a requirement to handle persistent homology in a machine learning setting.
Contribution. To realize this idea, we advocate a novel input layer for deep neural networks that
takes a topological signature (in our case, a persistence diagram), and computes a parametrized
projection that can be learned during network training. Specifically, this layer is designed such that
its output is stable with respect to the 1-Wasserstein distance (similar to [27] or [1]). To demonstrate
the versatility of this approach, we present experiments on 2D object shape classification and the
classification of social network graphs. On the latter, we improve the state-of-the-art by a large
margin, clearly demonstrating the power of combining TDA with deep learning in this context.
2
Background
For space reasons, we only provide a brief overview of the concepts that are relevant to this work and
refer the reader to [16] or [14] for further details.
Homology. The key concept of homology theory is to study the properties of some object X by
means of (commutative) algebra. In particular, we assign to X a sequence of modules C0 , C1 , . . .
2
which are connected by homomorphisms ?n : Cn ? Cn?1 such that im ?n+1 ? ker ?n . A structure
of this form is called a chain complex and by studying its homology groups Hn = ker ?n / im ?n+1
we can derive properties of X.
A prominent example of a homology theory is simplicial homology. Throughout this work, it is
the used homology theory and hence we will now concretize the already presented ideas. Let K
be a simplicial complex and Kn its n-skeleton. Then we set Cn (K) as the vector space generated
(freely) by Kn over Z/2Z2 . The connecting homomorphisms ?n : Cn (K) ? Cn?1 (K) are
called
Pn boundary operators. For a simplex ? = [x0 , . . . , xn ] ? Kn , we define
P them as
P?n (?) =
?i ) = ?n (?i ).
i=0 [x0 , . . . , xi?1 , xi+1 , . . . , xn ] and linearly extend this to Cn (K), i.e., ?n (
Persistent homology. Let K be a simplicial complex and (K i )m
i=0 a sequence of simplicial complexes such that ? = K 0 ? K 1 ? ? ? ? ? K m = K. Then, (K i )m
i=0 is called a filtration of K. If we
use the extra information provided by the filtration of K, we obtain the following sequence of chain
complexes (left),
???
?3
C22
???
?3
?2
C11
?2
C12
?
C01
?1
C02
?
?
C2m
?1
C1m
?0
0
?
?1
C0m
C01 = [[v1 ], [v2 ]]Z2
K1
0
?
?
?2
?0
?0
Example
C21
C11 = 0
0
C02 = [[v1 ], [v2 ], [v3 ]]Z2
K2
?
?3
?
???
C12 = [[v1 , v3 ], [v2 , v3 ]]Z2
v2
K3
v3
v4
v1
C02 = [[v1 ], [v2 ], [v3 ], [v4 ]]Z2
C12 = [[v1 , v3 ], [v2 , v3 ], [v3 , v4 ]]Z2
C21 = 0
C22 = 0
C23 = 0
where Cni = Cn (Kni ) and ? denotes the inclusion. This then leads to the concept of persistent
homology groups, defined by
?ni,j
j
Hni,j = ker ?ni /(im ?n+1
? ker ?ni ) for
rank Hni,j ,
i?j .
The ranks,
=
of these homology groups (i.e., the n-th persistent Betti numbers),
capture the number of homological features of dimensionality n (e.g., connected components for
n = 0, holes for n = 1, etc.) that persist from i to (at least) j. In fact, according to [14, Fundamental
Lemma of Persistent Homology], the quantities
i,j?1
?i,j
? ?ni,j ) ? (?ni?1,j?1 ? ?ni?1,j ) for
n = (?n
i<j
(1)
encode all the information about the persistent Betti numbers of dimension n.
Topological signatures. A typical way to obtain a filtration of K is to consider sublevel sets of a
function f : C0 (K) ? R. This function can be easily lifted to higher-dimensional chain groups of
K by
f ([v0 , . . . , vn ]) = max{f ([vi ]) : 0 ? i ? n} .
?1
Given m = |f (C0 (K))|, we obtain (Ki )m
((??, ai ]) for
i=0 by setting K0 = ? and Ki = f
1 ? i ? m, where a1 < ? ? ? < am is the sorted sequence of values of f (C0 (K)). If we construct
a multiset such that, for i < j, the point (ai , aj ) is inserted with multiplicity ?i,j
n , we effectively
encode the persistent homology of dimension n w.r.t. the sublevel set filtration induced by f . Upon
adding diagonal points with infinite multiplicity, we obtain the following structure:
Definition 1 (Persistence diagram). Let ? = {x ? R2? : mult(x) = ?} be the multiset of the
diagonal R2? = {(x0 , x1 ) ? R2 : x0 = x1 }, where mult denotes the multiplicity function and let
R2? = {(x0 , x1 ) ? R2 : x1 > x0 }. A persistence diagram, D, is a multiset of the form
D = {x : x ? R2? } ? ? .
We denote by D the set of all persistence diagrams of the form |D \ ?| < ? .
For a given complex K of dimension nmax and a function f (of the discussed form), we can interpret
persistent homology as a mapping (K, f ) 7? (D0 , . . . , Dnmax ?1 ), where Di is the diagram of
dimension i and nmax the dimension of K. We can additionally add a metric structure to the space of
persistence diagrams by introducing the notion of distances.
Simplicial homology is not specific to Z/2Z, but it?s a typical choice, since it allows us to interpret n-chains
as sets of n-simplices.
2
3
Definition 2 (Bottleneck, Wasserstein distance). For two persistence diagrams D and E, we define
their Bottleneck (w? ) and Wasserstein (wqp ) distances by
! p1
X
,
(2)
w? (D, E) = inf sup ||x ? ?(x)||? and wqp (D, E) = inf
||x ? ?(x)||pq
? x?D
?
x?D
where p, q ? N and the infimum is taken over all bijections ? : D ? E.
Essentially, this facilitates studying stability/continuity properties of topological signatures w.r.t.
metrics in the filtration or complex space; we refer the reader to [12],[13], [9] for a selection of
important stability results.
Remark. By setting ?ni,? = ?ni,m ??ni?1,m , we extend Eq. (1) to features which never disappear, also
referred to as essential. This change can be lifted to D by setting R2? = {(x0 , x1 ) ? R ? (R ? {?}) :
x1 > x0 }. In Sec. 5, we will see that essential features can offer discriminative information.
3
A network layer for topological signatures
In this section, we introduce the proposed (parametrized) network layer for topological signatures
(in the form of persistence diagrams). The key idea is to take any D and define a projection w.r.t. a
collection (of fixed size N ) of structure elements.
In the following, we set R+ := {x ? R : x > 0} and R+
0 := {x ? R : x ? 0}, resp., and start by
rotating points of D such that points on R2? lie on the x-axis, see Fig. 1. The y-axis can then be
interpreted as the persistence of features. Formally, we let b0 and b1 be the unit vectors in directions
(1, 1)> and (?1, 1)> and define a mapping ? : R2? ? R2? ? R ? R+
0 such that x 7? (hx, b0 i, hx, b1 i).
This rotates points in R? ? R2? clock-wise by ?/4. We will later see that this construction is beneficial
for a closer analysis of the layers? properties.
Similar to [27, 19], we choose exponential functions as structure elements, but other choices are
possible (see Lemma 1). Differently to [27, 19], however, our structure elements are not at fixed
locations (i.e., one element per point in D), but their locations and scales are learned during training.
Definition 3. Let ? = (?0 , ?1 )> ? R ? R+ , ? = (?0 , ?1 ) ? R+ ? R+ and ? ? R+ . We define
s?,?,? : R ? R+
0 ?R
as follows:
s?,?,?
?
2
2
2
2
?
e??0 (x0 ??0 ) ??1 (x1 ??1 ) ,
x1 ? [?, ?)
?
?
?
?
x1
2
2
2
2
(x0 , x1 ) =
e??0 (x0 ??0 ) ??1 (ln( ? )+???1 ) , x1 ? (0, ?)
?
?
?
?
?
0,
x1 = 0
A persistence diagram D is then projected w.r.t. s?,?,? via
X
S?,?,? : D ? R,
D 7?
s?,?,? (?(x)) .
(3)
(4)
x?D
Remark. Note that s?,?,? is continuous in x1 as
x
+ ? and lim s?,?,? (x0 , x1 ) = 0 = s?,?,? (x0 , 0)
lim x = lim ln
x??
x??
x1 ?0
?
and e(?) is continuous. Further, s?,?,? is differentiable on R ? R+ , since
?x1
1 = lim+
(x) and
x?? ?x1
lim?
? ln
x1
?
?x1
x??
+?
(x) = lim?
x??
?
=1 .
x
Also note that we use the log-transform in Eq. (4) to guarantee that s?,?,? satisfies the conditions of
Lemma 1; this is, however, only one possible choice.
4
Remark. The intuition behind ? is the following. It is the threshold at which the log-transform starts
to operate. The log-transform, on the other hand, stretches the space between the x-axis and the line
drawn at x + ? to infinite length. As a consequence, s?,?,? = 0 for x ? R2? . This is necessary since
otherwise S?,?,? (D) = ? for D ? D (as each persistence diagram contains points at the diagonal
with infinite multiplicity).
Finally, given a collection of S?i ,?i ,? , we combine them to form the output of the network layer.
?1
+
+
+ N
Definition 4. Let N ? N, ? = (?i , ? i )N
and ? ? R+ . We define
i=0 ? (R ? R ) ? (R ? R )
N ?1
N
S?,? : D ? (R+
D 7? S?i ,?i ,? (D) i=0 .
0)
as the concatenation of all N mappings defined in Eq. (4).
Importantly, a network layer implementing Def. 4 is trainable via backpropagation, as (1) s?i ,?i ,? is
differentiable in ?i , ? i , (2) S?i ,?i ,? (D) is a finite sum of s?i ,?i ,? and (3) S?,? is just a concatenation.
4
Theoretical properties
In this section, we demonstrate that the proposed layer is stable w.r.t. the 1-Wasserstein distance wq1 ,
see Eq. (2). In fact, this claim will follow from a more general result, stating sufficient conditions on
q
functions s : R2? ? R2? ? R+
0 such that a construction in the form of Eq. (3) is stable w.r.t. w1 .
Lemma 1. Let
s : R2? ? R2? ? R+
0
have the following properties:
(i) s is Lipschitz continuous w.r.t. k ? kq and constant Ks
(ii) s(x = 0, for x ? R2?
Then, for two persistence diagrams D, E ? D, it holds that
X
X
s(x) ?
s(y) ? Ks ? wq1 (D, E) .
x?D
y?E
(5)
Proof. see Appendix B
Remark. At this point, we want to clarify that Lemma 1 is not specific to s?,?,? (e.g., as in Def. 3).
Rather, Lemma 1 yields sufficient conditions to construct a w1 -stable input layer. Our choice of
s?,?,? is just a natural example that fulfils those requirements and, hence, S?,? is just one possible
representative of a whole family of input layers.
With the result of Lemma 1 in mind, we turn to the specific case of S?,? and analyze its stability
properties w.r.t. wq1 . The following lemma is important in this context.
Lemma 2. s?,?,? has absolutely bounded first-order partial derivatives w.r.t. x0 and x1 on R ? R+ .
Proof. see Appendix B
Theorem 1. S?,? is Lipschitz continuous with respect to wq1 on D.
Proof. Lemma 2 immediately implies that s?,?,? from Eq. (3) is Lipschitz continuous w.r.t || ? ||q .
Consequently, s = s?,?,? ? ? satisfies property (i) from Lemma 1; property (ii) from Lemma 1 is
satisfied by construction. Hence, S?,?,? is Lipschitz continuous w.r.t. wq1 . Consequently, S?,? is
Lipschitz in each coordinate and therefore Liptschitz continuous.
Interestingly, the stability result of Theorem 1 is comparable to the stability results in [1] or [27]
(which are also w.r.t. wq1 and in the setting of diagrams with finitely-many points). However, contrary
to previous works, if we would chop-off the input layer after network training, we would then have a
mapping S?,? of persistence diagrams that is specifically-tailored to the learning task on which the
network was trained.
5
b5
shift due to noise
b8
S1
a2
a3
b1
b2,3,4
b6
b7
b7
b8
?
b5
b3
Artificially added noise
Filtration directions
Persistence diagram
(0-dim. features)
a3
b9
a2
a1
b9
b1
b2 a1 b4
b6
Figure 2: Height function filtration of a ?clean? (left, green points) and a ?noisy? (right, blue points) shape
along direction d = (0, ?1)> . This example demonstrates the insensitivity of homology towards noise, as the
added noise only (1) slightly shifts the dominant points (upper left corner) and (2) produces additional points
close to the diagonal, which have little impact on the Wasserstein distance and the output of our layer.
5
Experiments
To demonstrate the versatility of the proposed approach, we present experiments with two totally
different types of data: (1) 2D shapes of objects, represented as binary images and (2) social network
graphs, given by their adjacency matrix. In both cases, the learning task is classification. In each
experiment we ensured a balanced group size (per label) and used a 90/10 random training/test
split; all reported results are averaged over five runs with fixed ? = 0.1. In practice, points in input
diagrams were thresholded at 0.01 for computational reasons. Additionally, we conducted a reference
experiment on all datasets using simple vectorization (see Sec. 5.3) of the persistence diagrams in
combination with a linear SVM.
Implementation. All experiments were implemented in PyTorch3 , using DIPHA4 and Perseus [23].
Source code is publicly-available at https://github.com/c-hofer/nips2017.
5.1
Classification of 2D object shapes
We apply persistent homology combined with our proposed input layer to two different datasets of
binary 2D object shapes: (1) the Animal dataset, introduced in [3] which consists of 20 different
animal classes, 100 samples each; (2) the MPEG-7 dataset which consists of 70 classes of different
object/animal contours, 20 samples each (see [21] for more details).
Filtration. The requirements to use persistent homology on 2D shapes are twofold: First, we need
to assign a simplicial complex to each shape; second, we need to appropriately filtrate the complex.
While, in principle, we could analyze contour features, such as curvature, and choose a sublevel set
filtration based on that, such a strategy requires substantial preprocessing of the discrete data (e.g.,
smoothing). Instead, we choose to work with the raw pixel data and leverage the persistent homology
transform, introduced by Turner et al. [29]. The filtration in that case is based on sublevel sets of
the height function, computed from multiple directions (see Fig. 2). Practically, this means that we
directly construct a simplicial complex from the binary image. We set K0 as the set of all pixels
which are contained in the object. Then, a 1-simplex [p0 , p1 ] is in the 1-skeleton K1 iff p0 and p1
are 4?neighbors on the pixel grid. To filtrate the constructed complex, we denote by b the barycenter
of the object and with r the radius of its bounding circle around b. Finally, we define, for [p] ? K0
and d ? S1 , the filtration function by f ([p]) = 1/r ? hp ? b, di. Function values are lifted to K1 by
taking the maximum, cf. Sec. 2. Finally, let di be the 32 equidistantly distributed directions in S1 ,
starting from (1, 0)> . For each shape, we get a vector of persistence diagrams (Di )32
i=1 where Di is
the 0-th diagram obtained by filtration along di . As most objects do not differ in homology groups of
higher dimensions (> 0), we did not use the corresponding persistence diagrams.
Network architecture. While the full network is listed in the supplementary material, the key
architectural choices are: 32 independent input branches, i.e., one for each filtration direction. Further,
the i-th branch gets, as input, the vector of persistence diagrams from directions di?1 , di and di+1 .
This is a straightforward approach to capture dependencies among the filtration directions. We use
cross-entropy loss to train the network for 400 epochs, using stochastic gradient descent (SGD) with
mini-batches of size 128 and an initial learning rate of 0.1 (halved every 25-th epoch).
3
4
https://github.com/pytorch/pytorch
https://bitbucket.org/dipha/dipha
6
MPEG-7
Animal
Skeleton paths
Class segment sets
?
ICS
?
BCF
86.7
90.9
96.6
97.2
67.9
69.7
78.4
83.4
Ours
91.8
69.5
?
?
Figure 3: Left: some examples from the MPEG-7 (bottom) and Animal (top) datasets. Right: Classification
results, compared to the two best (?) and two worst (?) results reported in [30].
Results. Fig. 3 shows a selection of 2D object shapes from both datasets, together with the obtained
classification results. We list the two best (?) and two worst (?) results as reported in [30]. While,
on the one hand, using topological signatures is below the state-of-the-art, the proposed architecture
is still better than other approaches that are specifically tailored to the problem. Most notably, our
approach does not require any specific data preprocessing, whereas all other competitors listed in
Fig. 3 require, e.g., some sort of contour extraction. Furthermore, the proposed architecture readily
generalizes to 3D with the only difference that in this case di ? S2 . Fig. 4 (Right) shows an exemplary
visualization of the position of the learned structure elements for the Animal dataset.
5.2
Classification of social network graphs
In this experiment, we consider the problem of graph classification, where vertices are unlabeled
and edges are undirected. That is, a graph G is given by G = (V, E), where V denotes the set of
vertices and E denotes the set of edges. We evaluate our approach on the challenging problem of
social network classification, using the two largest benchmark datasets from [31], i.e., reddit-5k
(5 classes, 5k graphs) and reddit-12k (11 classes, ?12k graphs). Each sample in these datasets
represents a discussion graph and the classes indicate subreddits (e.g., worldnews, video, etc.).
Filtration. The construction of a simplicial complex from G = (V, E) is straightforward: we set
K0 = {[v] ? V } and K1 = {[v0 , v1 ] : {v0 , v1 } ? E}. We choose a very simple filtration based on
the vertex degree, i.e., the number of incident edges to a vertex v ? V . Hence, for [v0 ] ? K0 we get
f ([v0 ]) = deg(v0 )/ maxv?V deg(v) and again lift f to K1 by taking the maximum. Note that chain
groups are trivial for dimension > 1, hence, all features in dimension 1 are essential.
Network architecture. Our network has four input branches: two for each dimension (0 and 1) of
the homological features, split into essential and non-essential ones, see Sec. 2. We train the network
for 500 epochs using SGD and cross-entropy loss with an initial learning rate of 0.1 (reddit_5k), or
0.4 (reddit_12k). The full network architecture is listed in the supplementary material.
Results. Fig. 5 (right) compares our proposed strategy to state-of-the-art approaches from the
literature. In particular, we compare against (1) the graphlet kernel (GK) and deep graphlet kernel
(DGK) results from [31], (2) the Patchy-SAN (PSCN) results from [24] and (3) a recently reported
graph-feature + random forest approach (RF) from [4]. As we can see, using topological signatures
in our proposed setting considerably outperforms the current state-of-the-art on both datasets. This is
an interesting observation, as PSCN [24] for instance, also relies on node degrees and an extension of
the convolution operation to graphs. Further, the results reveal that including essential features is key
to these improvements.
5.3
Vectorization of persistence diagrams
Here, we briefly present a reference experiment we conducted following Bendich et al. [5]. The idea
is to directly use the persistence diagrams as features via vectorization. For each point (b, d) in a
persistence diagram D we calculate its persistence, i.e., d ? b. We then sort the calculated persistences
by magnitude from high to low and take the first N values. Hence, we get, for each persistence
diagram, a vector of dimension N (if |D \ ?| < N , we pad with zero). We used this technique
on all four data sets. As can be seen from the results in Table 4 (averaged over 10 cross-validation
runs), vectorization performs poorly on MPEG-7 and Animal but can lead to competitive rates on
reddit-5k and reddit-12k. Nevertheless, the obtained performance is considerably inferior to our
proposed approach.
7
MPEG-7
Animal
reddit-5k
reddit-12k
5
10
20
40
80
160
81.8
48.8
37.1
24.2
82.3
50.0
38.2
24.6
79.7
46.2
39.7
27.9
74.5
42.4
42.1
29.8
68.2
39.3
43.8
31.5
64.4
36.0
45.2
31.6
Ours
91.8
69.5
54.5
44.5
0.8
Death
1.0
N
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Birth
1.0
Figure 4: Left: Classification accuracies for a linear SVM trained on vectorized (in RN ) persistence diagrams
(see Sec. 5.3). Right: Exemplary visualization of the learned structure elements (in 0-th dimension) for the
Animal dataset and filtration direction d = (?1, 0)> . Centers of the learned elements are marked in blue.
1
1
1
G = (V, E)
f ?1 ((??, 2])
1
5
3
1
2
2
1
f ?1 ((??, 3])
f ?1 ((??, 5])
reddit-5k
reddit-12k
GK [31]
DGK [31]
PSCN [24]
RF [4]
41.0
41.3
49.1
50.9
31.8
32.2
41.3
42.7
Ours (w/o essential)
Ours (w/ essential)
49.1
54.5
38.5
44.5
Figure 5: Left: Illustration of graph filtration by vertex degree, i.e., f ? deg (for different choices of ai , see
Sec. 2). Right: Classification results as reported in [31] for GK and DGK, Patchy-SAN (PSCN) as reported in
[24] and feature-based random-forest (RF) classification. from [4].
Finally, we remark that in both experiments, tests with the kernel of [27] turned out to be computationally impractical, (1) on shape data due to the need to evaluate the kernel for all filtration directions
and (2) on graphs due the large number of samples and the number of points in each diagram.
6
Discussion
We have presented, to the best of our knowledge, the first approach towards learning task-optimal
stable representations of topological signatures, in our case persistence diagrams. Our particular
realization of this idea, i.e., as an input layer to deep neural networks, not only enables us to learn with
topological signatures, but also to use them as additional (and potentially complementary) inputs to
existing deep architectures. From a theoretical point of view, we remark that the presented structure
elements are not restricted to exponential functions, so long as the conditions of Lemma 1 are met.
One drawback of the proposed approach, however, is the artificial bending of the persistence axis (see
Fig. 1) by a logarithmic transformation; in fact, other strategies might be possible and better suited
in certain situations. A detailed investigation of this issue is left for future work. From a practical
perspective, it is also worth pointing out that, in principle, the proposed layer could be used to handle
any kind of input that comes in the form of multisets (of Rn ), whereas previous works only allow
to handle sets of fixed size (see Sec. 1). In summary, we argue that our experiments show strong
evidence that topological features of data can be beneficial in many learning tasks, not necessarily to
replace existing inputs, but rather as a complementary source of discriminative information.
Acknowledgements. This work was partially funded by the Austrian Science Fund FWF (KLI
project 00012) and the Spinal Cord Injury and Tissue Regeneration Center Salzburg (SCI-TReCS),
Paracelsus Medical University, Salzburg.
8
A
Technical results
Lemma 3. Let ? ? R+ and ? ? R. We have
i)
ln(x)
x?0 x
lim
2
1
x?0 x
? e??(ln(x)+?) = 0
ii) lim
2
? e??(ln(x)+?) = 0 .
Proof. We omit the proof for brevity (see supplementary material for details), but remark that only i)
needs to be shown as ii) follows immediately.
B
Proofs
Proof of Lemma 1. Let ? be a bijection between D and E which realizes wq1 (D, E) and let D0 =
D \ ?, E0 = E \ ?. To show the result of Eq. (5), we consider the following decomposition:
D = ??1 (E0 ) ? ??1 (?)
= (??1 (E0 ) \ ?) ? (??1 (E0 ) ? ?) ? (??1 (?) \ ?) ? (??1 (?) ? ?)
{z
} |
{z
} |
{z
} |
{z
}
|
A
B
C
(6)
D
Except
for the term D, all sets are finite. In fact, ? realizes the Wasserstein distance wq1 which implies
? D = id. Therefore, s(x) = s(?(x)) = 0 for x ? D since D ? ?. Consequently, we can ignore D
in the summation and it suffices to consider E = A ? B ? C. It follows that
X
X
X
X
X
X
=
s(x)
?
s(y)
s(x)
?
s(?(x))
=
s(x)
?
s(?(x))
x?D
y?E
x?D
x?D
x?E
x?E
X
X
=
s(x) ? s(?(x)) ?
|s(x) ? s(?(x))|
x?E
x?E
X
X
? Ks ?
||x ? ?(x)||q = Ks ?
||x ? ?(x)||q = Ks ? wq1 (D, E) .
x?E
x?D
Proof of Lemma 2. Since s?,?,? is defined differently for x1 ? [?, ?) and x1 ? (0, ?), we need to
distinguish these two cases. In the following x0 ? R.
(1) x1 ? [?, ?): The partial derivative w.r.t. xi is given as
?
? ??i2 (xi ??i )2
s?,?,? (x0 , x1 ) = C ?
e
(x0 , x1 )
?xi
?xi
=C ?e
??i2 (xi ??i )2
?
(?2?i2 )(xi
(7)
? ?i ) ,
where C is just the part of exp(?) which is not dependent on xi . For all cases, i.e., x0 ? ?, x0 ?
?? and x1 ? ?, it holds that Eq. (7) ? 0.
(2) x1 ? (0, ?): The partial derivative w.r.t. x0 is similar to Eq. (7) with the same asymptotic
behaviour for x0 ? ? and x0 ? ??. However, for the partial derivative w.r.t. x1 we get
?
? ??12 (ln( x1 )+???1 )2
?
s?,?,? (x0 , x1 ) = C ?
e
(x0 , x1 )
?x1
?x1
x
?
1
= C ? e( ... ) ? (?2?12 ) ? ln
+ ? ? ?1 ?
?
x1
(8)
x
1
1
1
= C 0 ? e( ... ) ? ln
?
+(? ? ?1 ) ? e( ... ) ?
.
?
x1
x
| {z 1}
|
{z
}
(a)
(b)
As x1 ? 0, we can invoke Lemma 3 i) to handle (a) and Lemma 3 ii) to handle (b); conclusively,
Eq. (8) ? 0. As the partial derivatives w.r.t. xi are continuous and their limits are 0 on R, R+ , resp.,
we conclude that they are absolutely bounded.
9
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11
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6,370 | 6,762 | Predicting User Activity Level In Point Processes
With Mass Transport Equation
Yichen Wang? , Xiaojing Ye? , Hongyuan Zha? , Le Song?
?
College of Computing, Georgia Institute of Technology
?
School of Mathematics, Georgia State University
{yichen.wang}@gatech.edu, [email protected]
{zha,lsong}@cc.gatech.edu
Abstract
Point processes are powerful tools to model user activities and have a plethora of
applications in social sciences. Predicting user activities based on point processes
is a central problem. However, existing works are mostly problem specific, use
heuristics, or simplify the stochastic nature of point processes. In this paper, we
propose a framework that provides an efficient estimator of the probability mass
function of point processes. In particular, we design a key reformulation of the
prediction problem, and further derive a differential-difference equation to compute
a conditional probability mass function. Our framework is applicable to general
point processes and prediction tasks, and achieves superb predictive and efficiency
performance in diverse real-world applications compared to the state of the art.
1
Introduction
Online social platforms, such as Facebook and Twitter, enable users to post opinions, share information, and influence peers. Recently, user-generated event data archived in fine-grained temporal
resolutions are becoming increasingly available, which calls for expressive models and algorithms
to understand, predict and distill knowledge from complex dynamics of these data. Particularly,
temporal point processes are well-suited to model the event pattern of user behaviors and have been
successfully applied in modeling event sequence data [6, 10, 12, 21, 23, 24, 25, 26, 27, 28, 33].
A fundamental task in social networks is to predict user activity levels based on learned point process
models. Mathematically, the goal is to compute E[f (N (t))], where N (?) is a given point process
that is learned from user behaviors, t is a fixed future time, and f is an application-dependent
function. A framework for doing this is critically important. For example, for social networking
services, an accurate inference of the number of reshares of a post enables the network moderator
to detect trending posts and improve its content delivery networks [13, 32]; an accurate estimate of
the change of network topology (the number of new followers of a user) facilitates the moderator to
identify influential users and suppress the spread of terrorist propaganda and cyber-attacks [12]; an
accurate inference of the activity level (number of posts in the network) allows us to gain fundamental
insight into the predictability of collective behaviors [22]. Moreover, for online merchants such as
Amazon, an accurate estimate of the number of future purchases of a product helps optimizing future
advertisement placements [10, 25].
Despite the prevalence of prediction problems, an accurate prediction is very challenging for two
reasons. First, the function f is arbitrary. For instance, to evaluate the homogeneity of user activities,
we set f (x) = x log(x) to compute the Shannon entropy; to measure the distance between a predicted
activity level and a target x? , we set f (x) = (x x? )2 . However, most works [8, 9, 13, 30, 31, 32]
are problem specific and only designed for the simple task with f (x) = x; hence these works are
not generalizable. Second, point process models typically have intertwined stochasticity and can
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Simulate point process on 0, #
&')( = {#) , #, , #-}
time
0
#) #, #-
#
&',( = {#) , #, , #-}
0
#)
#, #-
Construct conditional intensity
Compute conditional mass function on [0, #]
67) 8, 0
/ 0 &1)( ,
0 ? [0, #]
0
#) #, #-
0=0
1
#
0
/(0|&1,( )
Average
0 =#
x
1
67,(8, 0)
#
(a) Samples of Hawkes process
(c) Mass transport from 0 to #
(b) Intensity functions
(d) Unbiased estimator
Figure 1: An illustration of H YBRID using Hawkes process (Eq. 1). Our method first generates
two samples {Hti } of events; then it constructs intensity functions; with these inputs, it computes
conditional probability mass functions ?i (x, s) := P[N (s) = x|Hsi ] using a mass transport equation.
Panel (c) shows the transport of conditional mass at four different times (the initial probability mass
?(x, 0) is an indicator function I[x = 0], as there is no event with probability one). Finally, the
average of conditional mass functions yields our estimator of the probability mass.
co-evolve over time [12, 25], e.g., in the influence propagation problem, the information diffusion over
networks can change the structure of networks, which adversely influences the diffusion process [12].
However, previous works often ignore parts of the stochasticity in the intensity function [29] or make
heuristic approximations [13, 32]. Hence, there is an urgent need for a method that is applicable to an
arbitrary function f and keeps all the stochasticity in the process, which is largely nonexistent to date.
We propose H YBRID, a generic framework that provides an efficient estimator of the probability mass
of point processes. Figure 1 illustrates our framework. We also make the following contributions:
? Unifying framework. Our framework is applicable to general point processes and does not
depend on specific parameterization of intensity functions. It incorporates all stochasticity in point
processes and is applicable to prediction tasks with an arbitrary function f .
? Technical challenges. We reformulate the prediction problem and design a random variable with
reduced variance. To derive an analytical form of this random variable, we also propose a mass
transport equation to compute the conditional probability mass of point processes. We further
transform this equation to an Ordinary Differential Equation and provide a scalable algorithm.
? Superior performance. Our framework significantly reduces the sample size to estimate the
probability mass function of point processes in real-world applications. For example, to infer
the number of tweeting and retweeting events of users in the co-evolution model of information
diffusion and social link creation [12], our method needs 103 samples and 14.4 minutes, while
Monte Carlo needs 106 samples and 27.8 hours to achieve the same relative error of 0.1.
2
Background and preliminaries
Point processes. A temporal point process [1] is a random process whose realization consists of a set
of discrete events {tk }, localized in time. It has been successfully applied to model user behaviors
in social networks [16, 17, 19, 23, 24, 25, 28, 30]. It can be equivalently represented as a counting
process N (t), which records the number of events on [0, t]. The counting process is a right continuous
step function, i.e., if an event happens at t, N (t) N (t ) = 1.
Let Ht = {tk |tk < t} be the history of events happened up to time t. An important way to characterize point processes is via the conditional intensity function (t) := (t|Ht ), a stochastic model for
the time of the next event given the history. Formally, (t) is the conditional probability of observing
an event in [t, t + dt) given events on [0, t), i.e., P {event in [t, t + dt)|Ht } = E[dN (t)|Ht ] :=
(t)dt, where dN (t) 2 {0, 1}.
The intensity function is designed to capture the phenomena of interest. Some useful forms include
(i) Poisson process: the intensity is a deterministic function, and (ii) Hawkes process [15]: it captures
the mutual excitation phenomena between events and its intensity is parameterized as
X
(t) = ? + ?
?(t tk ),
(1)
tk 2Ht
2
where ? > 0 is the baseline intensity; the trigging kernel ?(t) = exp( t) models the decay of past
events? influence over time; ? > 0 quantifies the strength of influence from each past event. Here,
the occurrence of each historical event increases the intensity by a certain amount determined by ?(t)
and ?, making (t) history-dependent and a stochastic process by itself.
Monte Carlo (MC). To compute the probability mass of a point process, MC simulates n realizations
of history {Hti } using the thinning algorithm [20]. The number of events in sample i is defined
as N i (t) = |Hti |. Let (x, t) := P[N (t) = x], where x 2 N, be the probability mass. Then its
mc
?mc (x, t) and the estimator ?
?mc
estimator
n
P
P?n (t)i for ?(t) := E[f (N (t))] are defined as n (x, t) =
1
1
i
mc
?n (t) = n i f (N (t)). The root mean square error (RMSE) is defined as
i I[N (t) = x] and ?
n
p
p
mc
"(?
?n (t)) = E[?
?mc
?(t)]2 = VAR[f (N (t))]/n.
(2)
n (t)
3
Solution overview
Given an arbitrary point process N (t) that is learned from data, existing prediction methods for
computing E[f (N (t))] have three major limitations:
? Generalizability. Most methods [8, 9, 13, 30, 31, 32] only predict E[N (t)] and are not generalizable to an arbitrary function f . Moreover, they typically rely on specific parameterizations of the
intensity functions, such as the reinforced Poisson process [13] and Hawkes process [5, 32]; hence
they are not applicable to general point processes.
? Approximation and heuristics. These works also ignore parts of the stochasticity in the intensity
functions [29] or make heuristic approximations to the point process [13, 32]. Hence the accuracy
is limited by the approximations and heuristic corrections.
? Large sample size. The MC method overcomes the above limitations since it has an unbiased
estimator of the probability mass. However, the high stochasticity in point processes leads to a
large value of VAR[f (N (t))], which requires a large number of samples to achieve a small error.
To address these challenges, we propose a generic framework with a novel estimator of the probability
mass, which has a smaller sample size than MC. Our framework has the following key steps.
I. New random variable. We design a random variable g(Ht ), a conditional expectation given the
history. Its variance is guaranteed to be smaller than that of f (N (t)). For a fixed number of samples,
the error of MC is decided by the variance of the random variable of interest, as shown in (2). Hence,
to achieve the same error, applying MC to estimate the new objective EHt [g(Ht )] requires smaller
number of samples compared with the procedure that directly estimates E[f (N (t))].
II. Mass transport equation. To compute g(Ht ), we derive a differential-difference equation that
describes the evolutionary dynamics of the conditional probability mass P[N (t) = x|Ht ]. We
further formulate this equation as an Ordinary Differential Equation, and provide a scalable algorithm.
4
Hybrid inference machine with probability mass transport
In this section, we present technical details of our framework. We first design a new random variable
for prediction; then we propose a mass transport equation to compute this random variable analytically.
Finally, we combine the mass transport equation with the sampling scheme to compute the probability
mass function of general point processes and solve prediction tasks with an arbitrary function f .
4.1
New random variable with reduced variance
We reformulate the problem and design a new random variable g(Ht ), which has a smaller variance
than f (N (t)) and the same expectation. To do this, we express E[f (N (t))] as an iterated expectation
h
h
i
?
?i
E[f (N (t))] = EHt EN (t)|Ht f (N (t))|Ht
= EHt g(Ht ) ,
(3)
where EHt is w.r.t. the randomness of the history and EN (t)|Ht is w.r.t. the randomness of the
point process given the history. We design the random variable as a conditional expectation given the
history: g(Ht ) = EN (t)|Ht [f (N (t))|Ht ]. Theorem 1 shows that it has a smaller variance.
3
Theorem 1. For time t > 0 and an arbitrary function f , we have VAR[g(Ht )] < VAR[f (N (t))].
Theorem 1 extends the Rao-Blackwell (RB) theorem [3] to point processes. RB says that if ?? is an
? ]] 6 VAR[?],
? i.e., the
estimator of a parameter ? and T is a sufficient statistic for ?; then VAR[E[?|T
?
sufficient statistic reduces uncertainty of ?. However, RB is not applicable to point processes since it
studies a different problem (improving the estimator of a distribution?s parameter), while we focus on
the prediction problem for general point processes, which introduces two new technical challenges:
(i) Is there a definition in point processes whose role is similar to the sufficient statistic in RB?
Our first contribution shows that the history Ht contains all the necessary information in a point
process and reduces the uncertainty of N (t). Hence, g(Ht ) is an improved variable for prediction.
Moreover, in contrast to the RB theorem, the inequality in Theorem 1 is strict because the counting
process N (t) is right-continuous in time t and not predictable [4] (a predictable process is measurable
w.r.t. Ht , such as the processes that are left-continuous). Appendix C contains details on the proof.
(ii) Is g(Ht ) computable for general point processes and an arbitrary function f ? An efficient
computation will enable us to estimate EHt [g(Ht )] using the sampling method. Specifically, let
P
?
?n (t) = n1 i g(Hti ) be the estimator computed from n samples; then from the definition of RMSE
in (2), this estimator has smaller error than MC: "(?
?n (t)) < "(?
?mc
n (t)).
However, the challenge in our new formulation is that it seems very hard to compute this conditional
expectation, as one typically needs another round of sampling, which is undesirable as it will increase
the variance of the estimator. To address this challenge, next we propose a mass transport equation.
4.2
Transport equation for conditional probability mass function
We present a novel mass transport equation that computes the conditional probability mass ?(x, t) :=
P[N (t) = x|Ht ] of general point processes. With this definition, we derive an analytical expression
P
for the conditional expectation: g(Ht ) = x f (x) ?(x, t). The transport equation is as follows.
Theorem 2 (Mass Transport Equation for Point Processes). Let (t) := (t|Ht ) be the conditional
intensity function of the point process N (t) and ?(x, t) := P[N (t) = x|Ht ] be its conditional
probability mass function; then ?(x, t) satisfies the following differential-difference equation:
8
>
<
(t) ?(x, t)
?(x, t)
@
?t (x, t) :=
(t) ?(x, t)
=
|
{z
}
>
@t
"
:
rate of change in conditional mass
loss in mass, at rate (t)
+
(t) ?(x 1, t)
|
{z
}
gain in mass, at rate (t)
if x = 0
if x = 1, 2, 3, ? ? ?
(4)
Proof sketch. For the simplicity of notation, we set the right-hand-side of (4) to be F[ ?], where
F is a functional operator onP?. We also define the inner product between functions u : N ! R
and v : N ! R as (u, v) := x u(x)v(x). The main idea in our proof is to show that the equality
(v, ?t ) = (v, F[ ?]) holds for any test function v; then ?t = F[ ?] follows from the fundamental
lemma of the calculus of variations [14]. Specifically, the proof contains two parts as follows.
We first prove (v, ?t ) = (B[v], ?), where B[v] is a functional operator defined as B[v] = (v(x +
1) v(x)) (t). This equality can be proved by the property of point processes and the definition of
conditional mass. Second, we show (B[v], ?) = (v, F[ ?]) using a variable substitution technique.
Mathematically, this equality means B and F are adjoint operators on the function space. Combining
these two equalities yields the mass transport equation. Appendix A contains details on the proof.
Mass transport dynamics. This differential-difference equation describes the time evolution of the
conditional mass. Specifically, the differential term ?t , i.e., the instantaneous rate of change in the
probability mass, is equal to a first order difference equation on the right-hand-side. This difference
equation is a summation of two terms: (i) the negative loss of its own probability mass ?(x, t) at
rate (t), and (ii) the positive gain of probability mass ?(x 1, t) from last state x 1 at rate (t).
Moreover, since initially no event happens with probability one, we have ?(x, 0) = I[x = 0]. Solving
this transport equation on [0, t] essentially transports the initial mass to the mass at time t.
4
Algorithm 1: C ONDITIONAL M ASS F UNCTION Algorithm 2: H YBRID M ASS TRANSPORT
Input: Ht = {tk }K
? , set t = tK+1
k=1 ,
Output: Conditional probability mass function ?(t)
for k = 0, ? ? ? K do
Construct (s) and Q(s) on [tk , tk+1 ] ;
?(tk+1 ) = O DE 45[ ?(tk ), Q(s), ? )] (RK Alg);
end
Set ?(t) = ?(tK+1 )
4.3
Input: Sample size n, time t, ?
Output: ?
?n (t), ?n (x, t)
n
Generate n samples of point process: Hti i=1 ;
for i = 1, ? ? ? , n do
?i (x, t) = C OND -M ASS -F UNC(Hi , ? );
t
end
P
P
?n (x, t) = 1
?i (x, t), ?
?n (t) = x f (x) ?n (x, t)
i
n
Mass transport as a banded linear Ordinary Differential Equation (ODE)
To efficiently solve the mass transport equation, we reformulate it as a banded linear ODE. Specifically,
we set the upper bound for x to be M , and set ?(t) to be a vector that includes the value of ?(x, t) for
each integer x: ?(t) = ( ?(0, t), ?(1, t), ? ? ? , ?(M, t))> . With this representation of the conditional
mass, the mass transport equation in (4) can be expressed as a simple banded linear ODE:
?(t)0 = Q(t) ?(t),
(5)
where ?(t)0 = ( ?t (0, t), ? ? ? , ?t (M, t))> , and the matrix Q(t) is a sparse bi-diagonal matrix with
Qi,i =
(t) and Qi 1,i = (t). The following equation visualizes the ODE in (5) when M = 2.
0
1
1
!0 ?
?t (0, t)
(0, t)
(t)
@ ?t (1, t) A =
@ ?(1, t) A .
(t)
(t)
(6)
?t (2, t)
?(2, t)
(t)
(t)
This dynamic ODE is a compact representation of the transport equation in (4) and M decides
the dimension of the ODE in (5). In theory, M can be unbounded. However, the conditional
probability mass is tends to zero when M becomes large. Hence, in practice we choose a finite
support {0, 1, ? ? ? , M } for the conditional probability mass function. To choose a proper M , we
generate samples from the point process. Suppose the largest number of events in the samples
is L, we set M = 2L such that it is reasonably large. Next, with the initial probability mass
?(t0 ) = (1, 0, ? ? ? , 0)> , we present an efficient algorithm to solve the ODE.
4.4
Scalable algorithm for solving the ODE
We present the algorithm that transports the initial mass ?(t0 ) to ?(t) by solving the ODE.
Since the intensity function is history-dependent
and has a discrete jump when an event happens
at time tk , the matrix Q(t) in the ODE is discontinuous at tk . Hence we split [0, t] into intervals
[tk , tk+1 ]. On each interval, the intensity is continuous and we can use the classic numerical
Runge-Kutta (RK) method [7] to solve the ODE.
Figure 2 illustrates the overall algorithm.
$(
$&
$%
) * , * ? [$&, $% ]
) *
$&
1
!" 0
$&
!" $&
$%
$'
)(*)
$%
!" $%
$
)(*)
$'
$'
!" $'
$
!" $
Figure 2: Illustration of Algorithm 1 using Hawkes
process. The intensity is updated after each event
tk . Within [tk , tk+1 ], we use (tk ) and the intensity (s) to solve the ODE and obtain (tk+1 ).
Our algorithm works as follows. First, with the initial intensity on [0, t1 ] and ?(t0 ) as input, the RK
method solves the ODE on [0, t1 ] and outputs ?(t1 ). Since an event happens at t1 , the intensity is
updated on [t1 , t2 ]. Next, with the updated intensity and ?(t1 ) as the initial value, the RK method
solves the ODE on [t1 , t2 ] and outputs ?(t2 ). This procedure repeats for each [tk , tk+1 ] until time t.
Now we present the RK method that solves the ODE on each interval [tk , tk+1 ]. RK divides this
interval into equally-spaced subintervals [?i , ?i+1 ], for i = 0, ? ? ? , I and ? = ?i+1 ?i . It then
conducts linear extrapolation on each subinterval. It starts from ?0 = tk and uses ?(?0 ) and the
approximation of the gradient ?(?0 )0 to compute ?(?1 ). Next, ?(?1 ) is taken as the initial value and
the process is repeated until ?I = tk+1 . Appendix D contains details of this method.
The RK method approximates the gradient ?(t)0 with different levels of accuracy, called states s.
When s = 1, it is the Euler method, which uses the first order approximation ?(?i+1 ) ?(?i )/ ? .
5
We use the O DE 45 solver in MATLAB and choose the stage s = 4 for RK. Moreover, the main
computation in the RK method comes from the matrix-vector product. Since the matrix Q(t) is
sparse and bi-diagonal with O(M ) non-zero elements, the cost for this operation is only O(M ).
4.5
Hybrid inference machine with mass transport equation
With the conditional probability mass, we are now ready to express g(Ht ) in closed form and
estimate EHt [g(Ht )] using the MC sampling method. We present our framework H YBRID:
(i) Generate n samples {Hti } from a point process N (t) with a stochastic intensity (t).
(ii) For each sample Hti , we compute the value of intensity function (s|Hsi ), for each s 2
[0, t]; then we solve (5) to compute the conditional probability mass ?i (x, t).
(iii) We obtain the estimator of the probability mass function (x, t) and ?(t) by taking the
Pn
P
average: ?n (x, t) = n1 i=1 ?i (x, t), ?
?n (t) = x f (x) ?n (x, t)
Algorithm 2 summarizes the above procedure. Next, we discuss two properties of H YBRID.
First, our framework efficiently uses all event information in each sample. In fact, each event tk
influences the transport rate of the conditional probability mass (Figure 2). This feature is in sharp
contrast to MC that only uses the information of the total number of events and neglects the differences
in event times. For instance, the two samples in Figure 1(a) both have three events and MC treats them
equally; hence its estimator is an indicator function ?mc
n (x, t) = I[x = 3]. However, for H YBRID ,
these samples have different event information and conditional probability mass functions, and our
estimator in Figure 1(d) is much more informative than an indicator function.
Moreover, our estimator for the probability mass is unbiased if we can solve the mass transport
equation in (4) exactly. To prove this property, we show that the following equality holds for an
arbitrary function f : (f, ) = E[f (N (t))] = EHt [g(Ht )] = (f, EHt [ ?]). Then EHt [ ?n ] =
follows from the fundamental lemma of the calculus of variations [14]. Appendix B contains detailed
derivations. In practice, we choose a reasonable finite support for the conditional probability mass in
order to solve the mass transport ODE in (5). Hence our estimator is nearly unbiased.
5
Applications and extensions to multi-dimensional point processes
In this section, we present two real world applications, where the point process models have intertwined stochasticity and co-evolving intensity functions.
Predicting the activeness and popularity of users in social networks. The co-evolution model [12]
uses a Hawkes process Nus (t) to model information diffusion (tweets/retweets), and a survival process
Aus (t) to model the dynamics of network topology (link creation process). The intensity of Nus (t)
depends on the network topology Aus (t), and the intensity of Aus (t) also depends on Nus (t); hence
these processesP
co-evolve over time. We focus on two tasks in this model: (i) inferring the activeness
of a user by E[ u Nus (t)], which
P is the number of tweets and retweets from user s; and (ii) inferring
the popularity of a user by E[ u Aus (t)], which is the number of new links created to the user.
Predicting the popularity of items in recommender systems. Recent works on recommendation
systems [10, 25] use a point process Nui (t) to model user u?s sequential interaction with item i.
The intensity function ui (t) denotes user?s interest to the item. As users interact with items over
time, the user latent feature uu (t) and item latent feature iu (t) co-evolve over time, and are mutually
dependent [25]. The intensity is parameterized as ui (t) = ?ui +uu (t)> ii (t), where ?ui is a baseline
term representing the long-term preference, and the tendency for u to interact with i depends on the
>
compatibility of their instantaneous
P latent features uu (t) ii (t). With this model, we can infer an
item?s popularity by evaluating E[ u Nui (t)], which is the number of events happened to item i.
To solve these prediction tasks, we extend the
Ptransport equation to the multivariate case. Specifically,
we create a new stochastic process x(t) = u Nus (t) and compute its conditional mass function.
Theorem 3 (Mass Transport for Multidimensional Point Processes). Let Nus (t) be the point process
PU
with intensity us (t), x(t) = u=1 Nus (t), and ?(x, t) = P[x(t) = x|Ht ] be the conditional
P
?(x, t) + P us (t) ?(x 1, t).
probability mass of x(t); then ? satisfies: ?t =
u us (t)
u
To compute the conditional probability mass, we also solve the ODE in (5), where the diagonal and
off-diagonal of Q(t) is now the negative and positive summation of intensities in all dimensions.
6
0.8
0.4
0.6
0.4
0.2
2
3
4
5
0.6
0.6
Test time (half day)
0.65
0.7
0.75
0.4
0.2
2
3
4
5
0.6
Test time (half day)
Training data size in proportion
(a) MAPE vs. test time
0.6
0.4
1
0.8
HYBRID
MC-1e6
MC-1e3
FPE
0.8
0.2
0.2
1
HYBRID
MC-1e6
MC-1e3
FPE
0.8
MAPE
MAPE
MAPE
0.6
HYBRID
MC-1e6
MC-1e3
SEISMIC
RPP
FPE
0.8
MAPE
HYBRID
MC-1e6
MC-1e3
SEISMIC
RPP
FPE
(b) MAPE vs. train size
0.65
0.7
0.75
0.8
Training data size in proportion
(c) MAPE vs. test time
(d) MAPE vs. train size
Figure 3: Prediction results for user activeness and user popularity. (a,b) user activeness: predicting
the number of posts per user; (c,d) user popularity: predicting the number of new links per user. Test
times are the relative times after the end of train time. The train data is fixed with 70% of total data.
0.8
MAPE
MAPE
MAPE
0.4
0.6
0.4
2
3
4
Test time (week)
(a) MAPE vs. test time
5
0.6
0.8
0.4
0.2
0.2
0.2
1
HYBRID
MC-1e6
MC-1e3
SEISMIC
RPP
FPE
0.8
0.8
MAPE
HYBRID
MC-1e6
MC-1e3
SEISMIC
RPP
FPE
0.6
0.6
0.65
0.7
0.75
0.8
Training data size in proportion
0.6
0.4
0.2
2
4
6
8
10
0.6
Test time (day)
(b) MAPE vs. train size
(c) MAPE vs. test time
0.65
0.7
0.75
0.8
Training data size in proportion
(d) MAPE vs. train size
Figure 4: Prediction results for item popularity. (a,b) predicting the number of watching events per
program on IPTV; (c,d) predicting the number of discussions per group on Reddit.
6
Experiments
In this section, we evaluate the predictive performance of H YBRID in two real world applications in
Section 5 and a synthetic dataset. We use the following metrics:
(i) Mean Average Percentage Error (MAPE). Given a prediction time t, we compute the MAPE
|?
?n (t) ?(t)|/?(t) between the estimated value and the ground truth.
(ii) Rank correlation. For all users/items, we obtain two lists of ranks according to the true and
estimated value of user activeness/user popularity/item popularity. The accuracy is evaluated by
the Kendall-? rank correlation [18] between two lists.
6.1
Experiments on real world data
We show H YBRID has both accuracy and efficiency improvement in predicting the activeness and
popularity of users in social networks and predicting the popularity of items in recommender systems.
Competitors. We use 103 samples for H YBRID and compare it with the following the state of the art.
? S EISMIC [32]. It defines a self-exciting process with a post infectiousness factor. It uses the
branching property of Hawkes process and heuristic corrections for prediction.
? R PP [13]. It adds a reinforcement coefficient to Poisson process that depicts the self-excitation
phenomena. It sets dN (t) = (t)dt and solves a deterministic equation for prediction.
? F PE [29]. It uses a deterministic function to approximate the stochastic intensity function.
? M C -1 E 3. It is the MC sampling method with 103 samples (same as these for H YBRID), and
M C -1 E 6 uses 106 samples.
6.1.1
Predicting the activeness and popularity of users in social networks
We use a Twitter dataset [2] that contains 280,000 users with 550,000 tweet, retweet, and link creation
events during Sep. 21 - 30, 2012. This data is previously used to validate the network co-evolution
model [12]. The parameters for tweeting/retweeting processes and link creation process are learned
using maximum likelihood estimation [12]. S EISMIC and R PP are not designed for the popularity
prediction task since they do not consider the evolution of network topology. We use p proportion of
total data as the training data to learn parameters of all methods, and the rest as test data. We make
predictions for each user and report the averaged results.
7
10
3
4
600
1000
HYBRID
HYBRID
800
Time (s)
6
4
HYBRID
MC
2
Time (s)
8
Time (s)
4
HYBRID
MC
1
Time (s)
10
10
600
400
400
200
200
2
0
0.5
0.4
0.3
0.2
0
0.5
0.1
0.4
0.3
0.2
0
0.5
0.1
0.4
MAPE
MAPE
(a) User activeness
0.3
0.2
0
0.5
0.1
0.4
(b) Item popularity, IPTV
0.3
0.2
0.1
MAPE
MAPE
(c) User activeness
(d) Item popularity, IPTV
Figure 5: Scalability analysis: computation time as a function of error. (a,b) comparison between
H YBRID and M C in different problems; (c,d) scalability plots for H YBRID.
0.69
Methods
HYBRID
MC-1e6
FPE
MC-1e3
0.72
0.75
Rank correlation
Rank correlation
0.71
0.50
Methods
0.75
Rank correlation
HYBRID
MC-1e6
FPE
SEISMIC
RPP
MC-1e3
0.69
0.50
0.41
0.39
0.25
0.78
HYBRID
MC-1e6
FPE
SEISMIC
RPP
MC-1e3
0.76
0.50
0.44
0.25
0.50
0.51
0.41
0.25
0.75
0.58
0.51
0.31
0.21
0.15
0.11
0.00
Methods
HYBRID
MC-1e6
FPE
SEISMIC
RPP
MC-1e3
0.21
0.13
(a) User activeness
0.77
0.25
0.21
0.00
Methods
0.75
Rank correlation
Methods
0.75
0.00
Methods
0.00
Methods
(b) User popularity
Methods
(c) Item popularity, IPTV (d) Item popularity, Reddit
Figure 6: Rank correlation results in different problems. We vary the proportion p of training data
from 0.6 to 0.8, and the error bar represents the variance over different training sets.
Predictive performance. Figure 3(a) shows that MAPE increases as test time increases, since the
model?s stochasticity increases. H YBRID has the smallest error. Figure 3(b) shows that MAPE
decreases as training data increases since model parameters are more accurate. Moreover, H YBRID is
more accurate than S EISMIC and F PE with only 60% of training data, while these works need 80%.
Thus, we make accurate predictions by observing users in the early stage. This feature is important
for network moderators to identify malicious users and suppress the propagation undesired content.
Moreover, the consistent performance improvement shows two messages: (i) considering all the
randomness is important. H YBRID is 2? more accurate than S EISMIC and F PE because H YBRID
naturally considers all the stochasticity, but S EISMIC, F PE, and R PP need heuristics or approximations
that discard parts of the stochasticity; (ii) sampling efficiently is important. To consider all the
stochasticity, we need to use the sampling scheme, and H YBRID has a much smaller sample size.
Specifically, H YBRID uses the same 103 samples, but has 4? error reduction compared with M C -1 E 3.
M C -1 E 6 has a similar predictive performance as H YBRID, but needs 103 ? more samples.
Scalability. How does the reduction in sample size improve the speed? Figure 5(a) shows that as the
error decreases from 0.5 to 0.1, M C has higher computation cost, since it needs much more samples
than H YBRID to achieve the same error. We include the plots of H YBRID in (c). In particular, to
achieve the error of 0.1, M C needs 106 samples in 27.8 hours, but H YBRID only needs 14.4 minutes
with 103 samples. We use the machine with 16 cores, 2.4 GHz Intel Core i5 CPU and 64 GB memory.
Rank correlation. We rank all users according to the predicted level of activeness and level of
popularity separately. Figure 6(a,b) show that H YBRID performs the best with the accuracy around
80%, and it consistently identifies around 30% items more correctly than F PE on both tasks.
6.1.2
Predicting the popularity of items in recommender systems
In the recommendation system setting, we use two datasets from [25]. The IPTV dataset contains
7,100 users? watching history of 436 TV programs in 11 months, with around 2M events. The Reddit
dataset contains online discussions of 1,000 users in 1,403 groups, with 10,000 discussion events.
The predictive and scalability performance are consistent with the application in social networks.
Figure 4 shows that H YBRID is 15% more accurate than F PE and 20% than S EISMIC. Figure 5 also
shows that H YBRID needs much smaller amount of computation time than M C -1 E 6. To achieve the
error of 0.1, it takes 9.8 minutes for H YBRID and 7.5 hours for M C -1 E 6. Figure 6(c,d) show that
H YBRID achieves the rank correlation accuracy of 77%, with 20% improvement over F PE.
8
-3
10
10-2
10-3
101
102
103
104
105
101
102
10-2
103
104
105
106
101
number of samples
10
HYBRID
MC
-1
10-2
102
103
104
105
101
106
102
103
104
105
106
number of samples
number of samples
2
(b) f (x) = x log(x)
1
100
10-3
number of samples
(a) f (x) = x
10
HYBRID
MC
-1
MAPE
10
10
100
Hybrid
MC
-1
MAPE
10
-2
100
HYBRID
MC
MAPE
MAPE
10-1
(c) f (x) = x
(d) f (x) = exp(x)
Figure 7: Error of E[f (N (t))] as a function of sample size (loglog scale). (a-d) different choices of f .
0.03
0.02
0.025
0.05
0.025
0.01
0.02
Probability
0.01
Probability
Probability
Probability
0.015
0.015
0.01
0.005
0.005
0
0.04
0.015
0.02
0
85
Counts
(a) H YBRID, ?n (x, t)
160
0
0
80
160
0
85
Counts
Counts
(b) M C, ?mc
n (x, t)
0.02
0.01
0.005
0
0.03
(c) H YBRID, 1 sample
160
0
0
85
160
Counts
(d) H YBRID, 1 sample
Figure 8: Comparison of estimators of probability mass functions in H YBRID and M C. (a,b)
estimators with the same 1000 samples. (c,d) estimator with one sample in H YBRID.
6.2
Experiments on synthetic data
We compare H YBRID with MC in two aspects: (i) the significance of the reduction in the error and
sample size, and (ii) estimators of the probability mass function. We study a Hawkes process and set
the parameters of its intensity function as ? = 1.2, and ? = 0.5. We fix the prediction time to be
t = 30. The ground truth is computed with 108 samples from MC simulations.
Error vs. number of samples. In four tasks with different f , Figure 7 shows that given the same
number of samples, H YBRID has a smaller error. Moreover, to achieve the same error, H YBRID needs
100? less samples than M C. In particular, to achieve the error of 0.01, (a) shows H YBRID needs 103
and M C needs 105 samples; (b) shows H YBRID needs 104 and M C needs 106 samples.
Probability mass functions. We compare our estimator of the probability mass with M C. Figure 8(a,b) show that our estimator is much smoother than M C, because our estimator is the average of
conditional probability mass functions, which are computed by solving the mass transport equation.
Moreover, our estimator centers around 85, which is the ground truth of E[N (t)], while that of M C
centers around 80. Hence H YBRID is more accurate. We also plot two conditional mass functions in
(c,d). The average of 1000 conditional mass functions yields (a). Thus, this averaging procedure in
H YBRID adjusts the shape of the estimated probability mass. On the contrary, given one sample, the
estimator in M C is just an indicator function and cannot capture the shape of the probability mass.
7
Conclusions
We have proposed H YBRID, a generic framework with a new formulation of the prediction problem
in point processes and a novel mass transport equation. This equation efficiently uses the event
information to update the transport rate and compute the conditional mass function. Moreover,
H YBRID is applicable to general point processes and prediction tasks with an arbitrary function f .
Hence it can take any point process models as input, and the predictive performance of our framework
can be further improved with the advancement of point process models. Experiments on real world
and synthetic data demonstrate that H YBRID outperforms the state of the art both in terms of accuracy
and efficiency. There are many interesting lines for future research. For example, H YBRID can be
generalized to marked point processes [4], where a mark is observed along with the timing of each
event.
9
Acknowledgements. This project was supported in part by NSF IIS-1218749, NIH BIGDATA
1R01GM108341, NSF CAREER IIS-1350983, NSF IIS-1639792 EAGER, NSF CNS-1704701, ONR
N00014-15-1-2340, DMS-1620342, CMMI-1745382, IIS-1639792, IIS-1717916, NVIDIA, Intel
ISTC and Amazon AWS.
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11
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6,371 | 6,763 | Submultiplicative Glivenko-Cantelli and
Uniform Convergence of Revenues
Noga Alon
Tel Aviv University, Israel
and Microsoft Research
[email protected]
Moshe Babaioff
Microsoft Research
[email protected]
Yannai A. Gonczarowski
The Hebrew University of Jerusalem, Israel
and Microsoft Research
[email protected]
Shay Moran
Institute for Advanced Study, Princeton
[email protected]
Yishay Mansour
Tel Aviv University, Israel
and Google Research, Israel
[email protected]
Amir Yehudayoff
Technion ? IIT, Israel
[email protected]
Abstract
In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which
asserts uniform convergence of the empirical Cumulative Distribution Function
(CDF) to the CDF of the underlying distribution. Our variant allows for tighter
convergence bounds for extreme values of the CDF.
We apply our bound in the context of revenue learning, which is a well-studied
problem in economics and algorithmic game theory. We derive sample-complexity
bounds on the uniform convergence rate of the empirical revenues to the true
revenues, assuming a bound on the kth moment of the valuations, for any (possibly
fractional) k > 1.
For uniform convergence in the limit, we give a complete characterization and a
zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform
convergence almost never occurs.
1
Introduction
A basic task in machine learning is to learn an unknown distribution ?, given access to samples
from it. A natural and widely studied criterion for learning a distribution is approximating its
Cumulative Distribution Function (CDF). The seminal Glivenko-Cantelli Theorem [13, 6] addresses
this question when the distribution ? is over the real numbers. It determines the behavior of the
empirical distribution function as the number of samples grows: let X1 , X2 , . . . be a sequence of i.i.d.
random variables drawn from a distribution ? on R with Cumulative Distribution Function (CDF) F ,
and let x1 , x2 , . . . be their realizations. The empirical distribution ?n is
n
?n ,
1X
?x ,
n i=1 i
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
where
supported on xi . Let Fn denote the CDF of ?n , i.e., Fn (t) ,
?xi is the constant distribution
1
. The Glivenko-Cantelli Theorem formalizes the statement that ?n
?
{1
?
i
?
n
:
x
?
t}
i
n
converges to ? as n grows, by establishing that Fn (t) converges to F (t), uniformly over all t ? R:
Theorem 1.1 (Glivenko-Cantelli Theorem, [13, 6]). Almost surely,
lim sup Fn (t) ? F (t) = 0.
n??
t
Some twenty years after Glivenko [13] and Cantelli [6] discovered this theorem, Dvoretzky, Kiefer,
and Wolfowitz (DKW) [12] strengthened this result by giving an almost1 tight quantitative bound on
the convergence rate. In 1990, Massart [17] proved a tight inequality, confirming a conjecture due to
Birnbaum and McCarty [3]:
h
i
Theorem 1.2 ([17]). Pr supt Fn (t) ? F (t) > ? 2 exp(?2n2 ) for all > 0, n ? N.
The above theorems show that, with high probability, F and Fn are close up to some additive error.
We would have liked to prove a stronger, multiplicative bound on the error:
?t : F (t) ? Fn (t) ? ? F (t).
However, for some distributions, the above event has probability 0, no matter how large n is. For
example, assume that ? satisfies F (t) > 0 for all t. Since the empirical measure ?n has finite support,
there is t with Fn (t) = 0; for such a value of t, such a multiplicative approximation fails to hold.
So, the above multiplicative requirement is too strong to hold in general. A natural compromise is to
consider a submultiplicative bound:
?t : F (t) ? Fn (t) ? ? F (t)? ,
where 0 ? ? < 1. When ? = 0, this is the additive bound studied in the context of the GlivenkoCantelli Theorem. When ? = 1, this is the unattainable multiplicative bound. Our first main result
shows that the case of ? < 1 is attainable:
Theorem 1.3 (Submultiplicative Glivenko-Cantelli Theorem). Let > 0, ? > 0 and 0 ? ? < 1.
There exists n0 (, ?, ?) such that for all n > n0 , with probability 1 ? ?:
?t : F (t) ? Fn (t) ? ? F (t)? .
It is worth pointing out a central difference between Theorem 1.3 and other generalizations of
the Glivenko-Cantelli Theorem: for example, the seminal work of Vapnik and Chervonenkis [24]
shows that for every class of events F of VC dimension d, there
is n0 = n0 (, ?, d) such that for
every n ? n0 , with probability 1 ?
? it holds that ?A ? F : p(A) ? pn (A) ? . This yields
Glivenko-Cantelli by plugging F = (??, t] : t ? R , which has VC dimension 1. In contrast,
the submultiplicative
bound from Theorem 1.3 does not even extend to the VC dimension 1 class
F = {t} : t ? R . Indeed, pick any distribution p over R such
that p {t} = 0 for every
t, and
1/n, however p {x } = 0, and
observe that for every sample x1 , . . . , xn , it holds that
p
{x
}
?
n
i
i
?
therefore, as long as ? > 0, it is never the case that p {xi } ? pn {xi } ? p {xi } .
Our second main result gives an explicit upper bound on n0 (, ?, ?):
Theorem 1.4 (Submultiplicative Glivenko-Cantelli Bound). Let , ? ? 1/4, and ? < 1. Then
?
?
4?
! 1??
4?
? ln 6/? ? ? 1??
?
D+4
n0 (, ?, ?) ? max
,
(D
+
1)
10
?
ln
12
?
,
? 22
?
3
?(1 ? ?)
where D =
ln(6/?)
22
?
6
? ln
1+?
2?
4?
? 1??
.
1
Note that for fixed , ?, when ? ? 0 the above bound approaches the familiar O ln(2/?) bound
by DKW [12] and Massart [17] for ? = 0. On the other hand, when ? ? 1 the above bound tends
1
The inequality due to [12] has a larger constant C in front of the exponent on the right hand side.
2
to ?, reflecting the fact that the multiplicative variant of Glivenko-Cantelli (? = 1) does not hold.
Theorems 1.3 and 1.4 are proven in the supplementary material.
Note that the dependency of the above bound on the confidence parameter ? is polynomial. This
contrasts with standard uniform convergence rates, which, due to applications of concentration bounds
such as Chernoff/Hoeffding, achieve logarithmic dependencies on ?. These concentration bounds are
not applicable in our setting when the CDF values are very small, and we use Markov?s inequality
instead. The following example shows that a polynomial dependency on ? is indeed necessary and is
not due to a limitation of our proof.
Example 1.5. For large n, consider n independent samples x1 , . . . , xn from the uniform distribution
over [0, 1], and set ? = 1/2 and = 1. The probability of the event
?i : xi ? 1/n3
is roughly 1/n2 : indeed, the complementary event has probability (1?1/n3 )n ? exp(?1/n2 ) ? 1?
1/2
1/n2 . When this happens, we have: Fn (1/n3 ) ? 1/n >> 1/n3 +1/n3/2 = F (1/n3 )+ F (1/n3 ) .
Note that this happens with probability inverse polynomial in n (roughly 1/n2 ) and not inverse
exponential.
An application to revenue learning. We demonstrate an application of our Submultiplicative
Glivenko-Cantelli Theorem in the context of a widely studied problem in economics and algorithmic
game theory: the problem of revenue learning. In the setting of this problem, a seller has to decide
which price to post for a good she wishes to sell. Assume that each consumer draws her private
valuation for the good from an unknown distribution ?. We envision that a consumer with valuation v
will buy the good at any price p ? v, but not at any higher price. This implies that the expected
revenue at price p is simply r(p) , p ? q(p), where q(p) , PrV ?? [V ? p].
In the language of machine learning, this problem can be phrased as follows: the examples domain
Z , R+ is the set of all valuations v. The hypothesis space H , R+ is the set of all prices p. The
revenue (which is a gain, rather than loss) of a price p on a valuation v is the function p ? 1{p?v} .
The well-known revenue maximization problem is to find a price p? that maximizes the expected
revenue, given a sample of valuations drawn i.i.d. from ?. In this paper, we consider the more
demanding revenue estimation problem: the problem of well-approximating r(p), simultaneously for
all prices p, from a given sample of valuations. (This clearly also implies a good estimation of the
maximum revenue and of a price that yields it.) More specifically, we address the following question:
when do the empirical
revenues, rn (p) , p ? qn (p), where qn (p) , PrV ??n [V ? p] = n1 ? {1 ?
i ? n : xi ? t} , uniformly converge to the true revenues r(p)? More specifically, we would like to
show that for some n0 , for n ? n0 we have with probability 1 ? ? that
r(p) ? rn (p) ? .
The revenue estimation problem is a basic instance of the more general problem of uniform convergence of empirical estimates. The main challenge in this instance is that the prices are unbounded
(and so are the private valuations that are drawn from the distribution ?).
Unfortunately, there is no (upper) bound on n0 that is only a function of and ?. Moreover, even if
we add the expectation of valuations, i.e., E[V ] where V is distributed according to ?, still there is no
bound on n0 that is a function of only those three parameters (see Section 2.3 for an example). In
contrast, when we consider higher moments of the distribution ?, we are able to derive bounds on the
value of n0 . These bounds are based on our Submultiplicative Glivenko-Cantelli Bound. Specifically,
assume that EV ?? [V 1+? ] ? C for some ? > 0 and C ? 1. Then, we show that for any , ? ? (0, 1),
we have
h
i
1
1+?
Pr ?v : r(v) ? rn (v) > ? Pr ?v : q(v) ? qn (v) >
.
1 q(v)
C 1+?
This essentially reduces uniform convergence bounds to our Submultiplicative Glivenko-Cantelli
variant. It then follows that there exists n0 (C, ?, , ?) such that for any n ? n0 , with probability at
least 1 ? ?,
?v : rn (v) ? r(v) ? .
3
1
We remark that when ? is large, our bound yields n0 ? O ln(2/?) , which recovers the standard
sample complexity bounds obtainable via DKW [12] and Massart [17].
When ? ? 0, our bound diverges to infinity, reflecting the fact (discussed above) that there is no
bound on n0 that depends only on , ?, and E[V ]. Nevertheless, we find that E[V ] qualitatively
determines whether uniform convergence occurs in the limit. Namely, we show that
? If E? [V ] < ?, then almost surely limn?? supv r(v) ? rn (v) = 0,
? Conversely, if E? [V ] = ?, then almost never limn?? supv r(v) ? rn (v) = 0.
1.1
Related work
Generalizations of Glivenko-Cantelli. Various generalizations of the Glivenko-Cantelli Theorem
were established. These include uniform convergence bounds for more general classes of functions as
well as more general loss functions (for example, [24, 23, 16, 2]). The results that concern unbounded
loss functions are most relevant to this work (for example, [9, 8, 23]). We next briefly discuss the
relevant results from Cortes et al. [8] in the context of this paper; more specifically, in the context of
Theorem 1.3. To ease presentation, set ? in this theorem to be 1/2. Theorem 1.3 analyzes the event
where the empirical quantile is bounded by2
p
qn (p) ? q(p) + q(p),
p
qn (p) ? q(p) ? q(p).
whereas, [8] analyzes the event where it is bounded it by:
p
? q(p) + q(p)/n + 1/n ,
qn (p) ? O
p
? q(p) ? qn (p)/n ? 1/n
qn (p) ? ?
Thus, the main difference is the additive 1/n term in the bound from [8]. In the context of uniform
convergence of revenues, it is crucial to use the upper bound on the empirical quantile as we do,
as it guarantees that large prices will not overfit, which is the main challenge in proving uniform
convergence in this context. In particular, the upper bound from [8] does not provide any guarantee
on the revenues of prices p >> n, as for such prices p ? 1/n >> 1.
It is also worth pointing out that our lower bound on the empirical quantile implies that with high
probability the quantile of the maximum sampled point is at least 1/n2 (or more generally, at least
1/n1/? when ? 6= 1/2), while the bound from [8] does not imply any non-trivial lower bound.
Another, more qualitative difference is that unlike the bounds in [8] that apply for general VC classes,
our bound is tailored for the class of thresholds (corresponding to CDF/quantiles), and does not
extend even to other classes of VC dimension 1 (see the discussion after Theorem 1.3).
Uniform convergence of revenues. The problem of revenue maximization is a central problem in
economics and Algorithmic Game Theory (AGT). The seminal work of Myerson [20] shows that
given a valuation distribution for a single good, the revenue-maximizing selling mechanism for this
good is a posted-price mechanism. In the recent years, there has been a growing interest in the case
where the valuation distribution is unknown, but the seller observes samples drawn from it. Most
papers in this direction assume that the distribution meets some tail condition that is considered
?natural? within the algorithmic game theory community, such as boundedness [18, 21, 19, 1, 14, 10]3 ,
such as a condition known as Myerson-regularity [11, 15, 7, 10], or such as a condition known as
monotone hazard rate [15].4 These papers then go on to derive computation- or sample-complexity
2
For consistency with the canonical statement of the Glivenko-Cantelli theorem, we stated our submultiplicative variants of this theorem with regard to the CDFs Fn and F . However, these results also hold when replacing
these CDFs with the respective quantiles (tail CDFs) qn and q. See Section 2.2 for details.
3
The analysis of [1] assumes a bound on the realized revenue (from any possible valuation profile) of any
mechanism/auction in the class that they consider. For the class of posted-price mechanisms, this is equivalent
to assuming a bound on the support of the valuation distribution. Indeed, for any valuation v, pricing at v
gives realized revenue v (from the valuation v), and so unbounded valuations (together with the ability to post
unbounded prices) imply unbounded realized revenues.
4
Both Myerson-regularity and monotone hazard rate are conditions on the second derivative of the revenue
as a function of the quantile of the underlying distribution. In particular, they impose restrictions on the tail of
the distribution.
4
bounds on learning an optimal price (or an optimal selling mechanism from a given class) for a
distribution that meets the assumed condition.
A recurring theme in statistical learning theory is that learnability guarantees are derived via a,
sometimes implicit, uniform convergence bound. However, this has not been the case in the context
of revenue learning. Indeed, while some papers that studied bounded distributions [18, 21, 19, 1]
did use uniform convergence bounds as part of their analysis, other papers, in particular those
that considered unbounded distributions, had to bypass the usage of uniform convergence by more
specialized arguments. This is due to the fact that many unbounded distributions do not satisfy
any uniform convergence bound. As a concrete example, the (unbounded, Myerson-regular) equal
revenue distribution5 has an infinite expectation and therefore, by our Theorem 2.3, satisfies no
uniform convergence, even in the limit. Thus, it turns out that the works that studied the popular
class of Myerson-regular distributions [11, 15, 7, 10] indeed could not have hoped to establish
learnability via a uniform convergence argument. For instance, the way [11, 7] establish learnability
for Myerson-regular distributions is by considering the guarded ERM algorithm
? (an algorithm that
chooses an empirical revenue maximizing price that is smaller than, say, the nth largest sampled
price), and proving
? a uniform convergence bound, not for all prices, but only for prices that are, say,
smaller than the nth largest sampled price, and then arguing that larger prices are likely to have a
small empirical revenue, compared to the guarded empirical revenue maximizer. This means that the
guarded ERM will output a good price, but it does not (and cannot) imply uniform convergence for
all prices.
We complement the extensive literature surveyed above in a few ways. The first is generalizing
the revenue maximization problem to a revenue estimation problem, where the goal is to uniformly
estimate the revenue of all possible prices, when no bound on the possible valuations is given (or
even exists). The problem of revenue estimation arises naturally when the seller has additional
considerations when pricing her good, such as regulations that limit the price choice, bad publicity
if the price is too high (or, conversely, damage to prestige if the price is too low), or willingness to
suffer some revenue loss for better market penetration (which may translate to more revenue in the
future). In such a case, the seller may wish to estimate the revenue loss due to posting a discounted
(or inflated) price.
The second, and most important, contribution to the above literature is that we consider arbitrary
distributions rather than very specific and limited classes of distributions (e.g., bounded, Myersonregular, monotone hazard rate, etc.). Third, we derive finite sample bounds in the case that the
expected valuation is bounded for some moment larger than 1. We further derive a zero-one law for
uniform convergence in the limit that depends on the finiteness of the first moment. Technically, our
bounds are based on an additive error rather than multiplicative ones, which are popular in the AGT
community.
1.2
Paper organization
The rest of the paper is organized as follows. Section 2 contains the application of our Submultiplicative Glivenko-Cantelli to revenue estimation, and Section 3 contains a discussion and possible
directions of future work. The proof of the Submultiplicative Glivenko-Cantelli variant, and some
extensions of it, appear in the supplementary material.
2
Uniform Convergence of Empirical Revenues
In this section we demonstrate an application of our Submultiplicative Glivenko-Cantelli variant by
establishing uniform convergence bounds for a family of unbounded random variables in the context
of revenue estimation.
2.1
Model
Consider a good g that we wish to post a price for. Let V be a random variable that models the
valuation of a random consumer for g. Technically, it is assumed that V is a nonnegative random
variable, and we denote by ? its induced distribution over R+ . A consumer who values g at a
5
This is a distribution that satisfies the special property that all prices have the same expected revenue.
5
valuation v is willing to buy the good at any price p ? v, but not at any higher price. This implies
that the realized revenue to the seller from a (posted) price p is the random variable p ? 1{p?V } . The
quantile of a value v ? R+ is
q(v) = q(v; ?) , ? {x : x ? v} .
This models the fraction of the consumers in the population that are willing to purchase the good if
priced at v. The expected revenue from a (posted) price p ? R+ is
r(p) = r(p; ?) , E p ? 1{p?V } = p ? q(p).
?
Let V1 , V2 , . . . be a sequence of i.i.d. valuations drawn from ?, and let v1 , v2 , . . . be their realizations.
The empirical quantile of a value v ? R+ is
qn (v) = q(v; ?n ) , 1 ? {1 ? i ? n : vi ? v} .
n
The empirical revenue from a price p ? R+ is
rn (p) = r(p; ?n ) , E p ? 1{p?V } = p ? qn (p).
?n
The revenue estimation error for a given sample of size n is
n , sup rn (p) ? r(p) .
p
It is worth highlighting the difference between revenue estimation and revenue maximization. Let p?
be a price that maximizes the revenue, i.e., p? ? arg supp r(p). The maximum revenue is r? = r(p? ).
The goal in many works in revenue maximization is to find a price p? such that r? ? r(?
p) ? , or
?
alternatively, to bound r /r(p)
?.
Given a revenue-estimation error n , one can clearly maximize the revenue within an additive error
of 2n by simply posting a price p?n ? arg maxp rn (p), thereby attaining revenue rn? = r(p?n ). This
follows since
rn? = r(p?n ) ? rn (p?n ) ? n ? rn (p? ) ? n ? r(p? ) ? 2n = r? ? 2n .
Therefore, good revenue estimation implies good revenue maximization.
We note that the converse does not hold. Namely, there are distributions for which revenue maximization is trivial but revenue estimation is impossible. One such case is the equal revenue distribution,
where all values in the support of ? have the same expected revenue. For such distributions, the
problem of revenue maximization becomes trivial, since any posted price is optimal. However, as
follows from Theorem 2.3, since the expected revenue of such distributions is infinite, almost never
do the empirical revenues uniformly converge to the true revenues.
2.2
Quantitative bounds on the uniform convergence rate
Recall that we are interested in deriving sample bounds that would guarantee uniform convergence
for the revenue estimation problem. We will show that given an upper bound on the kth moment of
V for some k > 1, we can derive a finite sample bound. To this end we utilize our Submultiplicative
Glivenko-Cantelli Bound (Theorem 1.4).
We also consider the case of k = 1, namely that E[V ] is bounded, and show that in this case there is
still uniform convergence in the limit, but that there cannot be any guarantees on the convergence
rate. Interestingly, it turns out that E[V ] < ? is not only sufficient but also necessary so that in the
limit, the empirical revenues uniformly converge to the true revenues (see Section 2.3).
We begin by showing that bounds on the kth moment for k > 1 yield explicit bounds on the
convergence rate. It is convenient to parametrize by setting k = 1 + ?, where ? > 0.
Theorem 2.1. Let EV ?? [V 1+? ] ? C for some ? > 0 and C ? 1, and let , ? ? (0, 1). Set6
4/? !
1
1/? )
1+?
2
6
?
C
ln(
?
n0 = O
C 1+?
.
(1)
2
? ln 1 + ?/2
For any n ? n0 , with probability at least 1 ? ?,
?v : rn (v) ? r(v) ? .
6
? conceals low order terms.
The O
6
1
Note that when ? is large, this bound approaches the standard O ln(2/?) sample complexity bound
of the additive Glivenko-Cantelli. For example, if all moments are uniformly bounded, then the
convergence is roughly as fast as in standard uniform convergence settings (e.g., VC-dimension based
bounds).
The proof of Theorem 2.1 follows from Theorem 1.4 and the next proposition, which reduces bounds
on the uniform convergence rate of the empirical revenues to our Submultiplicative Glivenko-Cantelli.
Proposition 2.2. Let EV ?? [V 1+? ] ? C for some ? > 0 and C ? 1, and let , ? ? (0, 1). Then,
h
i
1
1+? .
Pr ?v : r(v) ? rn (v) > ? Pr ?v : q(v) ? qn (v) >
q(v)
1
C 1+?
Thus, to prove Theorem 2.1, we first note that Theorem 1.4 (as well as Theorem 1.3) also holds
when Fn and F are respectively replaced in the definition of n0 with
qn and q (indeed, applying
Theorem 1.4 to the measure ?0 defined by ?0 (A) , ? {?a | a ? A} yields the required result with
1
regard to the measure ?). We then plug ? 1 and ? ? 1+?
into this variant of Theorem 1.4 to
C 1+?
yield a bound on the right-hand side of the inequality in Proposition 2.2, whose application concludes
the proof.
Proof of Proposition 2.2. By Markov?s inequality:
q(v) = Pr[V ? v] = Pr[V 1+? ? v 1+? ] ?
C
.
v 1+?
(2)
Now,
h
i
h
i
Pr ?v : r(v)?rn (v) > = Pr ?v : v ? q(v)?v ? qn (v) >
h
i
1
1+?
= Pr ?v : v ? q(v)?v ? qn (v) >
?q(v)) 1+?
1 (v
(v 1+? ? q(v)) 1+?
h
i
1
1+?
1+?
? Pr ?v : v ? q(v)?v ? qn (v) >
(v
?q(v))
1
C 1+? i
h
1
1+? .
= Pr ?v : q(v)?qn (v) >
1 q(v)
1+?
C
where the inequality follows from Equation (2).
2.3
A qualitative characterization of uniform convergence
The sample complexity bounds in Theorem 2.1 are meaningful as long as ? > 0, but deteriorate
drastically as ? ? 0. Indeed, as the following example shows, there is no bound on the uniform
convergence sample complexity that depends only on the first moment of V , i.e., its expectation.
Consider a distribution ?p so that with probability p we have V = 1/p and otherwise V = 0. Clearly,
E[V ] = 1. However, we need to sample mp = O(1/p) valuations to see a single nonzero value.
Therefore, there is no bound on the sample size mp as a function of the expectation, which is simply 1.
We can now consider the higher moments of ?p . Consider the kth moment, for k = 1 + ? and
1+?
? > 0, so k > 1. For this
] = p?/(1+?) , which implies that
moment, we have Ap,? = E[V
(1+?)/?
mp = O 1/(Ap,? )
. This does allow us to bound mp as a function of ? and E[V 1+? ], but for
small ? we have a huge exponent of approximately 1/?.
While the above examples show that there cannot be a bound on the sample size as a function of the
expectation of the value, it turns out that there is a very tight connection between the first moment
and uniform convergence:
Theorem 2.3. The following dichotomy holds for a distribution ? on R+ :
1. If E? [V ] < ?, then almost surely limn?? supv r(v) ? rn (v) = 0.
2. If E? [V ] = ?, then almost never limn?? supv r(v) ? rn (v) = 0.
7
That is, the empirical revenues uniformly converge to the true revenues if and only if E? [V ] < ?.
We use the following basic fact in the Proof of Theorem 2.3:
Lemma 2.4. Let X be a nonnegative random variable. Then
?
X
n=1
Proof. Note that:
?
X
?
X
Pr[X ? n] ? E[X] ?
Pr[X ? n].
n=0
1{X?n} = bXc ? X ? bXc + 1 =
n=1
?
X
1{X?n} .
n=0
The lemma follows by taking expectations.
Proof of Theorem 2.3. We start by proving item 2. Let ? be a distribution such that E? V = ?. If
supv v ? q(v) = ? then for every realization v1 , . . . , vn there is some v ? max{v1 , . . . , vn } such
that v ? q(v) ? 1, but v ? qn (v) = 0. So, we may assume supv v ? q(v) < ?. Without loss of generality
we may assume that supv v ? q(v) = 1/2 by rescaling the distribution if needed. Consider
the sequence
of events E1 , E2 , . . . where En denotes the event that Vn ? n. Since E? V = ?, Lemma 2.4
P?
implies that n=1 Pr[En ] = ?. Thus, since these events are independent, the second Borel-Cantelli
Lemma [4, 5] implies that almost surely, infinitely many of them occur and so infinitely often
Vn ? qn (Vn ) ? 1 ? Vn ? q(Vn ) + 12 .
Therefore, the probability that v ? qn (v) uniformly converge to v ? q(v) is 0.
Item 1 follows from the following monotone domination theorem:
Theorem 2.5. Let F be a family of nonnegative monotone functions, and let F be an upper envelope7
for F. If E? [F ] < ?, then almost surely:
lim sup E[f ] ? E [f ] = 0.
n?? f ?F ?
?n
Indeed, item 1 follows by plugging F = v ? 1x?v : v ? R+ , which is uniformly bounded by the
identity function F (x) = x. Now, by assumption E? [F ] < ?, and therefore, almost surely
lim sup r(v) ? rn (v) = lim sup E[f ] ? E [f ] = 0.
n?? v?R+
n?? f ?F ?
?n
Theorem 2.5 follows by known results in the theory of empirical processes (for example, with some
work it can be proved using Theorem 2.4.3 from Vaart and Wellner [22]). For completeness, we give
a short and basic proof in the supplementary material.
3
Discussion
Our main result is a submultiplicative variant of the Glivenko-Cantelli Theorem, which allows for
tighter convergence bounds for extreme values of the CDF. We show that for the revenue learning
setting our submultiplicative bound can be used to derive uniform convergence sample complexity
bounds, assuming a finite bound on the kth moment of the valuations, for any (possibly fractional)
k > 1. For uniform convergence in the limit, we give a complete characterization, where uniform
convergence almost surely occurs if and only if the first moment is finite.
It would be interesting to find other applications of our submultiplicative bound in other settings. A
potentially interesting direction is to consider unbounded loss functions (e.g., the squared-loss, or
log-loss). Many works circumvent the unboundedness in such cases by ensuring (implicitly) that
the losses are bounded, e.g., through restricting the inputs and the hypotheses. Our bound offers a
different perspective of addressing this issue. In this paper we consider revenue learning, and replace
the boundedness assumption by assuming bounds on higher moments. An interesting challenge is to
7
F is an upper envelope for F if F (v) ? f (v) for every v ? V and f ? F.
8
prove uniform convergence bounds for other practically interesting settings. One such setting might
be estimating the effect of outliers (which correspond to the extreme values of the loss).
In the context of revenue estimation, this work only considers the most na?ve estimator, namely of
estimating the revenues by the empirical revenues. One can envision other estimators, for example
ones which regularize the extreme tail of the sample. Such estimators may have a potential of
better guarantees or better convergence bounds. In the context of uniform convergence of selling
mechanism revenues, this work only considers the basic class of posted-price mechanisms. While
for one good and one valuation distribution, it is always possible to maximize revenue via a selling
mechanism of this class, this is not the case in more complex auction environments. While in many
more-complex environments, the revenue-maximizing mechanism/auction is still not understood well
enough, for environments where it is understood [7, 10, 14] (as well as for simple auction classes that
do not necessarily contain a revenue-maximizing auction [19, 1]) it would also be interesting to study
relaxations of the restrictive tail or boundedness assumptions currently common in the literature.
Acknowledgments
The research of Noga Alon is supported in part by an ISF grant and by a GIF grant. Yannai
Gonczarowski is supported by the Adams Fellowship Program of the Israel Academy of Sciences
and Humanities; his work is supported by ISF grant 1435/14 administered by the Israeli Academy
of Sciences and by Israel-USA Bi-national Science Foundation (BSF) grant number 2014389; this
project has received funding from the European Research Council (ERC) under the European Union?s
Horizon 2020 research and innovation programme (grant agreement No 740282). The research of
Yishay Mansour was supported in part by The Israeli Centers of Research Excellence (I-CORE)
program (Center No. 4/11), by a grant from the Israel Science Foundation, and by a grant from
United States-Israel Binational Science Foundation (BSF); the research was done while author was
co-affiliated with Microsoft Research. The research of Shay Moran is supported by the National
Science Foundations and the Simons Foundations; part of the research was done while author was
co-affiliated with Microsoft Research. The research of Amir Yehudayoff is supported by ISF grant
1162/15.
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6,372 | 6,764 | Deep Dynamic Poisson Factorization Model
Chengyue Gong
Department of Information Management
Peking University
[email protected]
Win-bin Huang
Department of Information Management
Peking University
[email protected]
Abstract
A new model, named as deep dynamic poisson factorization model, is proposed
in this paper for analyzing sequential count vectors. The model based on the
Poisson Factor Analysis method captures dependence among time steps by neural
networks, representing the implicit distributions. Local complicated relationship
is obtained from local implicit distribution, and deep latent structure is exploited
to get the long-time dependence. Variational inference on latent variables and
gradient descent based on the loss functions derived from variational distribution is
performed in our inference. Synthetic datasets and real-world datasets are applied
to the proposed model and our results show good predicting and fitting performance
with interpretable latent structure.
1
Introduction
There has been growing interest in analyzing sequentially observed count vectors x1 , x2 ,. . . , xT . Such
data appears in many real world applications, such as recommend systems, text analysis, network
analysis and time series analysis. Analyzing such data should conquer the computational or statistical
challenges, since they are often high-dimensional, sparse, and with complex dependence across the
time steps. For example, when analyzing the dynamic word count matrix of research papers, the
amount of words used is large and many words appear only few times. Although we know the trend
that one topic may encourage researchers to write papers about related topics in the following year,
the relationship among each time step and each topic is still hard to analyze completely.
Bayesian factor analysis model has recently reached success in modeling sequentially observed
count matrix. They assume the data is Poisson distributed, and model the data under Poisson
Factorize Analysis (PFA). PFA factorizes a count matrix, where ? ? RV+ ?K is the loading matrix and
? ? RT+?K is the factor score matrix. The assumption that ?t ? Gamma(?t?1 , ?t ) is then included
[1, 2] to smooth the transition through time. With property of the Gamma-Poisson distribution and
Gamma-NB process, inference via MCMC is used in these models. Considering the lack of ability to
capture the relationship between factors, a transition matrix is included in Poisson-Gamma Dynamical
System (PGDS) [2]. However, these models may still have some shortcomings in exploring the
long-time dependence among the time steps, as the independence assumption is made on ? t?1 and
? t+1 if ? t is given. In text analysis problem, temporal Dirichlet process [3] is used to catch the time
dependence on each topic using a given decay rate. This method may have weak points in analyzing
other data with different pattern long-time dependence, such as fanatical data and disaster data [3].
Deep models, which are also called hierarchical models in Bayesian learning field, are widely used in
Bayesian models to fit the deep relationship between latent variables. Examples of this include the
nested Chinese Restaurant Process [4], nest hierarchical Dirichlet process [5], deep Gaussian process
[6, 7] and so on. Some models based on neural network structure or recurrent structure is also used,
such as the Deep Exponential Families [8], the Deep Poisson Factor Analysis based on RBM or SBN
[9, 10], the Neural Autoregressive Density Estimator based on neural networks [11], Deep Poisson
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
?
0
?
1
?
3
?
2
?
4
?
5
?
0
?
1
?
2
(a)
?
3
?
4
?
5
(b)
h (0)
0
h (0)
1
h (0)
2
h (0)
3
h (0)
4
h (0)
5
h (0)
0
h (0)
1
h (0)
2
h (0)
3
h (0)
4
h (0)
5
?
0
?
1
?
2
?
3
?
4
?
5
?
0
?
1
?
2
?
3
?
4
?
5
(d)
(c)
Figure 1: The visual representation of our model. In (a), the structure of one-layer model is shown.
(b) shows the transmission of the posterior information. The prior and posterior distributions between
interfacing layers are shown in (c) and (d).
Factor Modeling with a recurrent structure based on PFA using a Bernoulli-Poisson link [12], Deep
Latent Dirichlet Allocation uses stochastic gradient MCMC [23]. These models capture the deep
relationship among the shallow models, and often outperform shallow models.
In this paper, we present the Deep Dynamic Poisson Factor Analysis (DDPFA) model. Based on PFA,
our model includes recurrent neural networks to represent implicit distributions, in order to learn
complicated relationship between different factors among short time. Deep structure is included in
order to capture the long-time dependence. An inference algorithm based on variational inference is
used for inferring the latent variables. Parameters in the neural networks are learnt according to a
loss function based on the variational distributions. Finally, the DDPFA model is used on several
synthetic and real-world datasets, and excellent results are obtained in prediction and fitting tasks.
2
Deep Dynamic Poisson Factorization Model
Assume that V -dimensional sequentially observed count data x1 , x2 ,. . . , xT are represented as a
V ? T count matrix X, a count data xvt ? {0, 1, . . . } is generated by the proposed DDPFA model as
follows:
PK
(1)
xvt ? P oisson( k=1 ?tk ?vk ?k ?v )
where the latent variables ?tk , ?vk , ?k and ?v are all positive variables. ?k represents the strength of
the k th component and is treated as factor. ?tk represents the strength of the k th component at the tth
time step. Feature-wise variable ?v captures the sparsity of the v th feature and ?k recognizes the
importance of the k th component. According to the regular setting in [2, 13-16], the factorization is
regarded as X ? P oisson(??T ). ? and ? can be absorbed into ?. In this paper, in order to extract
the sparsity of v th feature or k th component and impose a feature-wise or temporal smoothness
constraint, ? and ? are included in our model. The additive property of the Poisson distribution is
used to decompose the observed count of xvt as K latent counts xvtk , k ? {0, . . . , K}. In this way,
the model is rewritten as:
PK
(2)
xvt = k=1 xvtk and xvtk ? P oisson(?tk ?vk ?k ?v )
Capturing the complicated temporal dependence of ? is the major purpose in this paper. In the
previous work, transition via Gamma-Gamma-Poisson distribution structure is used, where ?t ?
Gamma(?t?1 , ?t ) [1]. Non-homogeneous Poisson process over time to model the stochastic
transition over different features is exploited in Poisson process models [17-19]. These models are
then trained via MCMC or variational inference. However, it is rough for these models to catch
complicated time dependence because of the weak points in their shallow structure in time dimension.
In order to capture the complex time dependence over ?, a deep and long-time dependence model
with a dynamic structure over time steps is proposed. The first layer over ? is as follows:
(0)
(0)
?t ? p(?t |ht?c , ..., ht )
2
(3)
where c is the size of a window for analysis, and the latent variables in the nth layer, n ? N , are
indicated as follows:
(n)
(n) (n+1)
(n+1)
(N )
(N ) (N )
(N )
(4)
ht ? p(ht |ht?c , ..., ht
) and ht ? p(ht |ht?c?1 , ..., ht?1 )
(n)
where the implicit probability distribution p(ht |?) is modeled as a recurrent neural network. Proba(n) (n?1)
(n?1)
bility AutoEncoder with an auxiliary posterior distribution p(ht |ht
, . . . , ht+c ), also modeled
(n)
as a neural network, is exploited in our training phase. ht is a K-dimensional latent variable in
(N )
the nth layer at the tth time step. Specially, in the nth layer, ht is generated from a Gamma
(N )
distribution with ht?c?1:t?1 as the prior information. This structure is illustrated in Figure 1.
Finally, prior parameters are placed over other latent variables for Bayesian inference. These variables
are generated as: ?vk ? Gamma(?? , ?? ) and?k ? Gamma(?? , ?? ) and ?v ? Gamma(?? , ?? ).
Although Dirichlet distribution is often used as prior distribution [13, 14, 15] over ?vk in previous
works, a Gamma distribution is exploited in our model due to the including of feature-wise parameter
?v and the purpose for obtaining feasible factor strength of ?k .
In real world applications, like recommendation systems, the observed binary count data can be
formulated by the proposed DDPFA model with a Bernoulli-Poisson link [1]. The distribution of b
given ? is called Bernoulli-Poisson distribution as: b = 1(x > 1), x ? P oisson(?) and the linking
b
distribution is rewritten as: f (b|x, ?) = e??(1?b) (1 ? e?? ) . The conditional posterior distribution
is then (x|b, ?) ? b ? P oisson+ (?), where P oisson+ (?) is a truncated Poisson distribution, so the
MCMC or VI methods can be used to do inference. Non-count real-valued matrix can also be linked
to a latent count matrix via a compound Poisson distribution or a Gamma belief network [20].
3
Inference
There are many classical inference approaches for Bayesian probabilistic model, such as Monte Carlo
methods and variational inference. In the proposed method, variational inference is exploited because
the implicit distribution is regarded as prior distribution over ?. Two stages of inference in our model
are adopted: the first stage updates latent variables by the coordinate-ascent method with a fixed
implicit distribution, and the parameters in neural networks are learned in the second one.
Mean-field Approximation: In order to obtain mean-field VI, all variables are independent and
governed by its own variational distribution. The joint distribution of the variational distribution is
written as:
Q
(n) (n)?
?
(5)
q(?, ?, ?, ?, H) = v,t,n,k q(?vk |??vk )q(?k |?k? )q(?tk |?tk
)q(?k |??k )q(htk |htk )
where y ? represents the prior variational parameter of the variable y. The variational parameters ?
are fitted to minimize the KL divergence:
? = argmin? ? KL(p(?, ?, ?, ?, H|X)||q(?, ?, ?, ?, H|?))
(6)
The variational distribution q(?|? ? ) is then used as a proxy for the posterior. The objective actually is
equal to maximize the evidence low bound (ELBO) [19]. The optimization can be performed by a
coordinate-ascent method or a variational-EM method. As a result, each variational parameter can be
optimized iteratively while the remaining parameters of the model are set to fixed value. Due to Eq.
2, the conditional distribution of (xvt1 , . . . , xvtk ) is a multinomial while its parameter is normalized
set of rates [19] and formulated as:
P
(xvt1 , . . . , xvtk )|?t , ?v , ?, ?v ? M ult(xvt? ; ?t ?v ??v / k ?tk ?vk ?k ?v )
(7)
Given the auxiliary variables xvtk , the Poisson factorization model is a conditional conjugate model.
The complete conditional of the latent variables is Gamma distribution and shown as:
?vk |?, ?, ?, ?, ?, X ? Gamma(?? + xv?k , ?? + ?k ?v ??k )
P
?k |?, ?, ?, ?, ?, X ? Gamma(?? + x??k , ?? + ??k v ?v ?vk )
P
?v |?, ?, ?, ?, ?, X ? Gamma(?? + xv?? , ?? + k ?k ?vk ??k )
3
(8)
Generally, these distributions are derived from conjugate properties between Poisson and Gamma
distribution. The posterior distribution of ?tk described in Eq. 3 can be a Gamma distribution while
(0)
the prior ht?c:t is given as:
P
(9)
?tk |?, ?, ?, h(0) , ?, X ? Gamma(??tk + xv?k , ?? + ?k v ?v ?vk )
(0)
(0)
where ??tk is calculated through a recurrent neural network with (ht?c , ..., ht ) as its inputs. Then
(0)
the posterior distribution of htk described in Eq. 4 is given as:
(0)
htk |?, h(1) , ?, X ? Gamma(?h(0) + ?h(0) , ?h )
tk
(10)
tk
where ?h(n) is the prior information given by the (n + 1)th layer, ?h(n) is the posterior information
tk
tk
(?1)
given by the (n ? 1)th layer. Here, the notation htk
tk
(n+1)
(n+1)
(n?1)
(n?1)
a recurrent neural network using (ht?c , ..., ht
is equal to ?tk . ?h(n) is calculated through
) as its inputs. ?h(n) is calculated through
tk
a recurrent neural network using (ht+c , ..., ht
) as its inputs. Therefore, the distribution
(0)
(0)
mentioned in Eq. 9 can be regarded as an implicit conditional distribution of ?tk given (ht?c , ..., ht ).
(n+1)
(n+1)
And the distribution in Eq. 10 is an implicit distribution of ?h(n) given (ht?c , ..., ht
) and
tk
(n?1)
(n?1)
(ht+c , ..., ht
).
Variational Inference: Mean field variational inference can approximate the latent variables while
all parameters of a neural network are given. If the observed data satisfies xvt > 0, the auxiliary
variables xvtk can be updated by:
shp
rte
rte
xvtk ? exp{? (?tk
) ? log?tk
+ ? (?shp
k ) ? log?k
(11)
rte
shp
rte
+ ? (?shp
vk ) ? log?vk + ? (?v ) ? log?v }
where ? (?) is the digamma function. Variables with the superscript ?shp? indicate the shape parameter
of Gamma distribution, and those with the superscript ?rte? are the rate parameter of it. This update
comes from the expectation of the logarithm of a Gamma variable as hlog?i = ? (?shp ) ? log(?rte ).
Here, ? is generated from a Gamma distribution and h?i represents the expectation of the variable.
Calculation of the expectation of the variable, obeyed Gamma distribution, is noted as h?i =
?shp /?rte . Variables can be updated by mean-field method as:
?vk ? Gamma(?? + hxv?k i, ?? + h?k ih?v ih??k i)
P
?k ? Gamma(?? + hx??k i, ?? + h??k i v h?v ih?vk i)
P
?v ? Gamma(?? + hxv?? i, ?? + k h?k ih?vk ih??k i)
(12)
The latent variables in the deep structure can also be updated by mean-field method:
P
?tk ? Gamma(??tk + hxv?k i, ?? + h?k i v h?v ih?vk i)
(13)
(n)
htk ? Gamma(?h(n) + ?h(n) , ?h )
tk
(14)
tk
n?1
where ?h(n) = ff eed (hhn+1 i), ?h(N ) = ff eed (hhN
i), ?h(N ) =
t?c?1:t?1 i) and ?h(n) = fback (hh
t
t
t
t
fback (hhN
t+c+1:t+1 i). ff eed (?) is the neural network constructing the prior distribution and fback (?)
is the neural network constructing the posterior distribution.
Probability AutoEncoder: This stage of the inference is to update the parameters of the neural
networks. The bottom layer is used by us as an example. Given all latent variables, these parameters
(0)
(0)
(0)
(n) (0)
(0)
can be approximated by p(?t |ht?c , ..., ht ) and p(ht |?t+c , ..., ?t ). p(?t |ht?c , ..., ht ) =
(0)
(0)
Gamma(??t , ?h ) is modeled by a RNN with the inputs (ht?c , ..., ht ) and the outputs, ??t . The
4
(0)
p(ht |?t+c , ..., ?t ) is also modeled as a RNN with the inputs (?t+c , ..., ?t ) and the outputs ,?h(0)
t
. With the posterior distribution from ? to H (0) and the prior distribution from H (0) to ?, the
probability of ? should be maximized. The loss function of these two neural networks is as follows:
R
max{ p(?|H (0) )p(H (0) |?)dH (0) }
(15)
W
where W represents the parameters in neural networks. Because the integration in Eq.
is intractable, a new loss function should include auxiliary variational variables H (0)0 .
sume that H (0)0 is generated by ?, the optimization can be regarded as maximizing
probability of ? with minimal difference between H (0)0 and H (0) as max{p(?|H (0) )}
W
15
Asthe
and
min{KL(p(H (0)0 |?)||p(H (0) |H (1) )}
W
Then approximating the variables generated from a distribution by its expectation, the loss function,
similar to variational AutoEncoder [21], can be simplified to:
min{khp(H (0)0 |?)i ? hp(H (0) |H (1) )ik2 + k? ? hp(?|H (0) )ik2 }
W
(16)
Since only a few samples are drawn from one certain distrbution, which means sampling all latent
variables is high-cost and useless, differentiable variational Bayes is not suitable. As a result, we
focus more on fitting data than generating data. In our objective, the first term, a regularization,
encourages the data to be reconstructed from the latent variables, and the second term encourages the
decoder to fit the data.
The parameters in the networks for nth and (n + 1)
min{khp(H
W
(n+1)0
|H
(n)
th
layer are trained by the loss function:
)i ? hp(H (n) |H (n+1) )ik2
+ kH (n) ? hp(H (n) |H (n+1) )ik2 }
(17)
In order to make the convergence more stable, the term of ? in the first layer is collapsed into X by
using the fixed latent variables approximated by mean-field VI, and the loss function is as follows:
min{khp(H (0)0 |?)i ? hp(H (0) |H (1) )ik2 + kX ? h?ih?ih?ihp(?|H (0) )ik2 }
W
(18)
After the layer-wise training, all the parameters in neural networks are jointly trained by the fine-tuning
trick in stacked AutoEncoder [22].
4
Experiments
In this section, four multi-dimensional synthetic datasets and five real-world datasets are exploited to
examine the performance of the proposed model. Besides, the results of three existed methods, PGDS,
LSTM, and PFA, are compared with results of our model. PGDS is a dynamic Poisson-Gamma
system mentioned in Section 1, and LSTM is a classical time sequence model. In order to prove the
deep relationship learnt by the deep structure can improve the performance, a simple PFA model is
also included as a baseline.
All hyperparameters of PGDS set in [2] are used in this paper. 1000 times gibbs sampling iterations
for PGDS is performed, 100 iterations used mean-field VI for PFA is performed, and 400 epochs is
executed for LSTM. The parameters in the proposed DDPFA model are set as follows:?(?,?,?) =
1, ?(?,?,?) = 2, ?(?,h) = 1, ?(?,h) = 1. The iterations is set to 100. The stochastic gradient
descent for the neural networks is executed 10 epochs in each iteration. The size of the window is 4.
Hyperparameters of PFA are set as the same to our model. Data in the last time step is exploited as
the predicting target in a prediction task. Mean squared error (MSE) between the ground truth and
the estimated value and the predicted mean squared error (PMSE) between the ground truth and the
predicted value in next time step are exploited to evaluate the performance of each model.
4.1
Synthetic Datasets
The multi-dimensional synthetic datasets are obtained by using the following functions where the
subscript stands for the index of dimension:
5
Data
SDS1
SDS2
SDS3
SDS4
Table 1: The result on the synthetic data
Measure
DDPFA
PGDS
LSTM
MSE
PMSE
MSE
PMSE
MSE
PMSE
MSE
PMSE
0.15 ? 0.01
2.07 ? 0.02
0.06 ? 0.01
2.01 ? 0.02
0.10 ? 0.02
2.14 ? 0.04
0.15 ? 0.03
1.48 ? 0.04
1.48 ? 0.00
5.96 ? 0.00
3.38 ? 0.00
3.50 ? 0.01
1.62 ? 0.00
4.33 ? 0.01
2.92 ? 0.00
6.41 ? 0.01
2.02 ? 0.23
2.94 ? 0.31
1.83 ? 0.04
2.41 ? 0.06
1.13 ? 0.06
3.03 ? 0.05
4.30 ? 0.26
4.67 ? 0.24
PFA
1.61 ? 0.00
4.42 ? 0.00
1.34 ? 0.00
0.25 ? 0.00
-
SDS1:f1 (t) = f2 (t) = t, f3 (t) = f4 (t) = t + 1 on the interval t = [1 : 1 : 6].
SDS2:f1 (t) = t (mod 2), f2 (t) = 2t (mod
2) + 2, f3 (t) = t on the interval t = [1 : 1 : 20].
SDS3:f1 (t) = f2 (t) = t, f3 (t) = f4 (t) = t + 1, f5 (t) = I(4|t) on the interval t = [1 : 1 : 20],
where I is an indicator function.
SDS4:f1 (t) = t (mod 2), f2 (t) = 2t (mod 2) + 2, f3 (t) = t (mod
t = [1 : 1 : 100].
10) on the interval
The number of factor is set to K = 3, and the number of the layers is 2. Both fitting and predicting
tasks are performed in each model. The hidden layer of LSTM is 4 and the size in each layer is 20. In
Table 1, it is obviously that DDPFA has the best performance in fitting and prediction task of all the
datasets. Note that the complex relationship learnt from the time steps helps the model catch more
time patterns according to the results of DDPFA, PGDS and PFA. LSTM performs worse in SDS4
because the noise in the synthetic data and the long time steps make the neural network difficult to
memorize enough information.
4.2
Real-world Datasets
Five real-world datasets are used as follows:
Integrated Crisis Early Warning System (ICEWS): ICEWS is an international relations event data set
extracted from news corpora used in [2]. We therefore treated undirected pairs of countries i ? j as
features and created a count matrix for the year 2003. The number of events for each pair during each
day time step is counted, and all pairs with fewer than twenty-five total events is discarded, leaving
T = 365, V = 6197, and 475646 events for the matrix.
NIPS corpus (NIPS): NIPS corpus contains the text of every NIPS conference paper from the year
1987 to 2003. We created a single count matrix with one column per year. The dataset is downloaded
from Gal?s page 1 , with T = 17, V = 14036, with 3280697 events for the matrix.
Ebola corpus (EBOLA)2 : EBOLA corpus contains the data for the 2014 Ebola outbreak in West
Africa every day from Mar 22th, 2014 to Jan 5th 2015, each column represents the cases or deaths in
a West Africa country. After data cleaning, the dataset is with T = 122, V = 16.
International Disaster(ID)3 : The International Disaster dataset contains essential core data on the
occurrence and effects of over 22,000 mass disasters in the world from 1900 to the present day. A
count matrix with T = 115 and V = 12 is built from the events of disasters occurred in Europe from
the year 1902 to 2016, classified according to their disaster types.
Annual Sheep Population(ASP)4 : The Annual Sheep Population contains the sheep population in
England & Wales from the year 1867 to 1939 yearly. The data matrix is with T = 73, V = 1.
1
http://ai.stanford.edu/gal/data.html
https://github.com/cmrivers/ebola/blob/master/country_timeseries.csv
3
http://www.emdat.be/
4
https://datamarket.com/data/list/?q=provider:tsdl
2
6
Data
ICEWS
NIPS
EBOLA
ID
ASP
Measure
MSE
PMSE
MSE
PMSE
MSE
PMSE
MSE
PMSE
MSE
PMSE
Table 2: The result on the real-world data
DDPFA
PGDS
LSTM
3.05 ? 0.02
0.96 ? 0.03
51.14 ? 0.03
289.21 ? 0.02
381.82 ? 0.13
490.32 ? 0.12
1.59 ? 0.01
5.18 ? 0.01
14.17 ? 0.02
21.23 ? 0.04
3.21 ? 0.01
0.97 ? 0.02
54.71 ? 0.08
337.60 ? 0.10
516.57 ? 0.01
1071.01 ? 0.01
3.45 ? 0.00
10.44 ? 0.00
2128.47 ? 0.02
760.42 ? 0.02
(a) PGDS
4.53 ? 0.04
6.30 ? 0.03
1053.12 ? 39.01
1728.04 ? 38.42
4892.34 ? 10.21
5839.26 ? 11.92
11.19 ? 1.32
10.37 ? 1.54
17962.47 ? 14.12
21324.72 ? 17.48
PFA
3.70 ? 0.01
69.05 ? 0.43
1493.32 ? 0.21
4.41 ? 0.01
388.02 ? 0.01
-
(b) DDPFA
Figure 2: The visual of the factor strength in each time step of the ICEWS data, the data is
normalized each time step. In (a), the result of PGDS shows the factors are shrunk to some local time
steps. In (b), the result of DDPFA shows the factors are not taking effects locally.
We set K = 3 for ID and ASP datasets, while set K = 10 for the others. The size of the hidden
layers of the LSTM is 40. The settings of remainder parameters here are the same as those in the
above experiment. The results of the experiment are shown in Table 2.
Table 2 shows the results of four different models on the five datasets, and the proposed model
DDPFA has satisfying performance in most experiments although the DDPFA?s result in ICEWS
prediction task is not good enough. While smoothed data obtained from the transition matrix in
PGDS performs well in this prediction task. However, In EBOLA and ASP datasets, PGDS fails in
catching complicated time dependence. And it is a tough challenge for LSTM network to memorize
enough useful patterns while its input data includes long-time patterns or the dimension of the data is
particular high.
According to the observation in Figure 2, it can be shown that the factors learnt by our model are
not activated locally compared to PGDS. Natrually, in real-world data, it is impossible that only
one factor happens in one time step. For example, in the ICEWS dataset, the connection between
Israel and Occupied Palestinian Territory still remains strong during the Iraq War or other accidents.
Figure 2(a) reveals that several factors at a certain time step are not captured by PGDS. In Figure 3,
the changes of two meaningful factors in ICEWS is shown. These two factor, respectively, indicate
Israel-Palestinian conflict and six-party talks. The long-time activation of factors is shown in thi
figure, since DDPFA model can capture weak strength along time.
In Table 3, we show the performance of our model with different sizes. From the table, performance
cannot be improved distinctly by adding more layers or adding more variables in upper layer. It is
also noticed that expanding the dimension in bottom layer is more useful than in upper layers. The
results reveal two problems of proposed DDPFA: "pruning" and uselessness of adding network layers.
7
(a)
(b)
Figure 3: The visual of the top two factors of the ICEWS data generated by DDPFA method. In (a),
?Japan?Russian Federation?, ?North Korea?United States?, ?Russian Federation?United States?,
?South Korea?United States?, and ?China?Russian Federation? are the largest features due to their
loading weights. This factor stands for six-party talks and other accidents about it. In (b),
?Israel?Occupied Palestinian Territory?, ?Israel?United States?, ?Occupied Palestinian
Territory?United States? are the largest features and it stands for the Israeli-Palestinian conflict.
Table 3: MSE on real datasets with different sizes.
Size
ICEWS NIPS EBOLA
10-10-10
10-10-10 (ladder structure)
10-10
32-32-32
32-32-32 (ladder structure)
32-64-64
64-32-32
2.94
2.88
3.05
2.95
2.86
2.93
2.90
51.24
49.81
51.14
50.12
49.26
50.18
50.04
382.17
379.08
381.82
379.64
377.81
380.01
378.87
[25] notices hierarchical latent variable models do not take advantage of the structure, and gives
such a conclusion that only using the bottom latent layer of hierarchical variational autoencoders
should be enough. In order to solve this problem, the ladder-like architecture, in which each layer
combines independent variables with latent variables depend on the upper layers, is used in our model.
It is noticed that using ladder architecture could reach much better results from Table 3. Another
problem, "pruning", is a phenomenon where the optimizer severs connections between most of the
latent variables and the data [24]. In our experiments, it is noticed that some dimenisions in the latent
layers only contain data noise. This problem is also found in differentiable variational Bayes and
solved by using auxiliary MCMC strcuture [24]. Therefore, we believe this problem is caused by
MF-variational inference used in our model and we hope it can be solved if we try other inference
methods.
5
Summary
A new model, called DDPFA, is proposed to obtain long-time and complicated dependence in time
series count data. Inference in DDPFA is based on variational method for estimating the latent
variables and approximating parameters in neural networks. In order to show the performance of the
proposed model, four multi-dimensional synthetic datasets and five real-world datasets, ICEWS, NIPS
corpus, EBOLA, International Disaster and Annual Sheep Population, are used, and the performance
of three existed methods, PGDS, LSTM, and PFA, are compared. According to our experimental
results, DDPFA has better effectivity and interpretability in sequential count analysis.
8
References
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AISTATS, 2015.
[2] A. Schein, M. Zhou, & H. Wallach. Poisson?Gamma Dynamical Systems. NIPS, 2016.
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Analysis and Machine Intelligence, 37(2):307?320, 2015.
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[17] S. A. Hosseini, K. Alizadeh, A. Khodadadi, & et al. Recurrent Poisson Factorization for Temporal
Recommendation. KDD, 2017.
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UAI, 2015.
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Research, 17(163):1?44, 2016.
[21] D. P. Kingma & W. Max. Auto-encoding variational Bayes. ICLR, 2014.
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2006.
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9
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6,373 | 6,765 | Positive-Unlabeled Learning with
Non-Negative Risk Estimator
Ryuichi Kiryo1,2 Gang Niu1,2 Marthinus C. du Plessis Masashi Sugiyama2,1
1
The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan
2
RIKEN, 1-4-1 Nihonbashi, Tokyo 103-0027, Japan
{ kiryo@ms., gang@ms., sugi@ }k.u-tokyo.ac.jp
Abstract
From only positive (P) and unlabeled (U) data, a binary classifier could be trained
with PU learning, in which the state of the art is unbiased PU learning. However,
if its model is very flexible, empirical risks on training data will go negative, and
we will suffer from serious overfitting. In this paper, we propose a non-negative
risk estimator for PU learning: when getting minimized, it is more robust against
overfitting, and thus we are able to use very flexible models (such as deep neural
networks) given limited P data. Moreover, we analyze the bias, consistency, and
mean-squared-error reduction of the proposed risk estimator, and bound the estimation error of the resulting empirical risk minimizer. Experiments demonstrate
that our risk estimator fixes the overfitting problem of its unbiased counterparts.
1
Introduction
Positive-unlabeled (PU) learning can be dated back to [1, 2, 3] and has been well studied since then.
It mainly focuses on binary classification applied to retrieval and novelty or outlier detection tasks
[4, 5, 6, 7], while it also has applications in matrix completion [8] and sequential data [9, 10].
Existing PU methods can be divided into two categories based on how U data is handled. The first
category (e.g., [11, 12]) identifies possible negative (N) data in U data, and then performs ordinary
supervised (PN) learning; the second (e.g., [13, 14]) regards U data as N data with smaller weights.
The former heavily relies on the heuristics for identifying N data; the latter heavily relies on good
choices of the weights of U data, which is computationally expensive to tune.
In order to avoid tuning the weights, unbiased PU learning comes into play as a subcategory of the
second category. The milestone is [4], which regards a U data as weighted P and N data simultaneously. It might lead to unbiased risk estimators, if we unrealistically assume that the class-posterior
probability is one for all P data.1 A breakthrough in this direction is [15] for proposing the first unbiased risk estimator, and a more general estimator was suggested in [16] as a common foundation.
The former is unbiased but non-convex for loss functions satisfying some symmetric condition; the
latter is always unbiased, and it is further convex for loss functions meeting some linear-odd condition [17, 18]. PU learning based on these unbiased risk estimators is the current state of the art.
However, the unbiased risk estimators will give negative empirical risks, if the model being trained
is very flexible. For the general estimator in [16], there exist three partial risks in the total risk (see
Eq. (2) defined later), especially it has a negative risk regarding P data as N data to cancel the bias
caused by regarding U data as N data. The worst case is that the model can realize any measurable
function and the loss function is not upper bounded, so that the empirical risk is not lower bounded.
This needs to be fixed since the original risk, which is the target to be estimated, is non-negative.
1
It implies the P and N class-conditional densities have disjoint support sets, and then any P and N data (as
the test data) can be perfectly separated by a fixed classifier that is sufficiently flexible.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
To this end, we propose a novel non-negative risk estimator that follows and improves on the stateof-the-art unbiased risk estimators mentioned above. This estimator can be used for two purposes:
first, given some validation data (which are also PU data), we can use our estimator to evaluate the
risk?for this case it is biased yet optimal, and for some symmetric losses, the mean-squared-error
reduction is guaranteed; second, given some training data, we can use our estimator to train binary
classifiers?for this case its risk minimizer possesses an estimation error bound of the same order
as the risk minimizers corresponding to its unbiased counterparts [15, 16, 19].
In addition, we propose a large-scale PU learning algorithm for minimizing the unbiased and nonnegative risk estimators. This algorithm accepts any surrogate loss and is based on stochastic optimization, e.g., [20]. Note that [21] is the only existing large-scale PU algorithm, but it only accepts
a single surrogate loss from [16] and is based on sequential minimal optimization [22].
The rest of this paper is organized as follows. In Section 2 we review unbiased PU learning, and in
Section 3 we propose non-negative PU learning. Theoretical analyses are carried out in Section 4,
and experimental results are discussed in Section 5. Conclusions are given in Section 6.
2
Unbiased PU learning
In this section, we review unbiased PU learning [15, 16].
Problem settings Let X ? Rd and Y ? {?1} (d ? N) be the input and output random variables.
Let p(x, y) be the underlying joint density of (X, Y ), pp (x) = p(x | Y = +1) and pn (x) = p(x |
Y = ?1) be the P and N marginals (a.k.a. the P and N class-conditional densities), p(x) be the U
marginal, ?p = p(Y = +1) be the class-prior probability, and ?n = p(Y = ?1) = 1 ? ?p . ?p is
assumed known throughout the paper; it can be estimated from P and U data [23, 24, 25, 26].
Consider the two-sample problem setting of PU learning [5]: two sets of data are sampled indepennp
u
dently from pp (x) and p(x) as Xp = {xpi }i=1
? pp (x) and Xu = {xui }ni=1
? p(x), and a classifier
n
needs to be trained from Xp and Xu .2 If it is PN learning as usual, Xn = {xni }ni=1
? pn (x) rather
than Xu would be available and a classifier could be trained from Xp and Xn .
Risk estimators Unbiased PU learning relies on unbiased risk estimators. Let g : Rd ? R be an
arbitrary decision function, and ` : R ? {?1} ? R be the loss function, such that the value `(t, y)
means the loss incurred by predicting an output t when the ground truth is y. Denote by Rp+ (g) =
Ep [`(g(X), +1)] and Rn? (g) = En [`(g(X), ?1)], where Ep [?] = EX?pp [?] and En [?] = EX?pn [?].
Then, the risk of g is R(g) = E(X,Y )?p(x,y) [`(g(X), Y )] = ?p Rp+ (g) + ?n Rn? (g). In PN learning,
thanks to the availability of Xp and Xn , R(g) can be approximated directly by
bpn (g) = ?p R
bp+ (g) + ?n R
bn? (g),
R
(1)
bp+ (g) = (1/np ) Pnp `(g(xp ), +1) and R
bn? (g) = (1/nn ) Pnn `(g(xn ), ?1). In PU learnwhere R
i
i=1
i
i=1
ing, Xn is unavailable, but Rn? (g) can be approximated indirectly, as shown in [15, 16]. Denote by
Rp? (g) = Ep [`(g(X), ?1)] and Ru? (g) = EX?p(x) [`(g(X), ?1)]. As ?n pn (x) = p(x) ? ?p pp (x),
we can obtain that ?n Rn? (g) = Ru? (g) ? ?p Rp? (g), and R(g) can be approximated indirectly by
bpu (g) = ?p R
bp+ (g) ? ?p R
bp? (g) + R
bu? (g),
R
(2)
bp? (g) = (1/np ) Pnp `(g(xp ), ?1) and R
bu? (g) = (1/nu ) Pnu `(g(xu ), ?1).
where R
i
i=1
i
i=1
The empirical risk estimators in Eqs. (1) and (2) are unbiased and consistent w.r.t. all popular loss
functions.3 When they are used for evaluating the risk (e.g., in cross-validation), ` is by default the
zero-one loss, namely `01 (t, y) = (1 ? sign(ty))/2; when used for training, `01 is replaced with a
surrogate loss [27]. In particular, [15] showed that if ` satisfies a symmetric condition:
`(t, +1) + `(t, ?1) = 1,
2
3
Xp is a set of independent data and so is Xu , but Xp ? Xu does not need to be such a set.
bpn (g) ? R(g) and R
bpu (g) ? R(g) as np , nn , nu ? ?.
The consistency here means for fixed g, R
2
(3)
we will have
bpu (g) = 2?p R
bp+ (g) + R
bu? (g) ? ?p ,
R
(4)
which can be minimized by separating Xp and Xu with ordinary cost-sensitive learning. An issue is
bpu (g) in (4) must be non-convex in g, since no `(t, y) in (3) can be convex in t. [16] showed that
R
bpu (g) in (2) is convex in g, if `(t, y) is convex in t and meets a linear-odd condition [17, 18]:
R
`(t, +1) ? `(t, ?1) = ?t.
(5)
Let g be parameterized by ?, then (5) leads to a convex optimization problem so long as g is linear
in ?, for which the globally optimal solution can be obtained. Eq. (5) is not only sufficient but also
necessary for the convexity, if ` is unary, i.e., `(t, ?1) = `(?t, +1).
Justification Thanks to the unbiasedness, we can study estimation error bounds (EEB). Let G be
b
b
the function class, and gbpn and gbpu be the empirical risk minimizers
? of Rpn (g)
? and Rpu (g). [19]
?
proved EEB of gbpu is tighter than EEB of gbpn when ?p / np + 1/ nu < ?n / nn , if (a) ` satisfies
?
(3) and is Lipschitz continuous; (b) the Rademacher complexity of G decays in O(1/ n) for data
4
of size n drawn from p(x), pp (x) or pn (x). In other ?
words, under
?mild conditions, PU learning is
?
likely to outperform PN learning when ?p / np + 1/ nu < ?n / nn . This phenomenon has been
observed in experiments [19] and is illustrated in Figure 1(a).
3
Non-negative PU learning
In this section, we propose the non-negative risk estimator and the large-scale PU algorithm.
3.1
Motivation
Let us look inside the aforementioned justification of unbiased PU (uPU) learning. Intuitively, the
advantage comes from the transformation ?n Rn? (g) = Ru? (g) ? ?p Rp? (g). When we approximate
?
n
?n Rn? (g) from N data {xni }ni=1
, the convergence rate is Op (?n / nn ), where Op denotes the order
np
u
and U data {xui }ni=1
,
in probability; when we approximate Ru? (g) ? ?p Rp? (g) from P data {xpi }i=1
?
?
the convergence rate becomes Op (?p / np + 1/ nu ). As a result, we might benefit from a tighter
?
?
?
uniform deviation bound when ?p / np + 1/ nu < ?n / nn .
However, the critical assumption on the Rademacher complexity is indispensable, otherwise it will
be difficult for EEB of gbpu to be tighter than EEB of gbpn . If G = {g | kgk? ? Cg } where Cg > 0
is a constant, i.e., it has all measurable functions with some bounded norm, then Rn,q (G) = O(1)
bpu (g)
for any n and q(x) and all bounds become trivial; moreover if ` is not bounded from above, R
becomes not bounded from below, i.e., it may diverge to ??. Thus, in order to obtain high-quality
gbpu , G cannot be too complex, or equivalently the model of g cannot be too flexible.
This argument is supported by an experiment as illustrated in Figure 1(b). A multilayer perceptron
was trained for separating the even and odd digits of MNIST hand-written digits [29]. This model is
so flexible that the number of parameters is 500 times more than the total number of P and N data.
From Figure 1(b) we can see:
(A) on training data, the risks of uPU and PN both decrease, and uPU is faster than PN;
(B) on test data, the risk of PN decreases, whereas the risk of uPU does not; the risk of uPU is
lower at the beginning but higher at the end than that of PN.
To sum up, the overfitting problem of uPU is serious, which evidences that in order to obtain highquality gbpu , the model of g cannot be too flexible.
3.2
Non-negative risk estimator
Nevertheless, we have no choice sometimes: we are interested in using flexible models, while labeling more data is out of our control. Can we alleviate the overfitting problem with neither changing
the model nor labeling more data?
4
Let ?1 , . . . , ?n be n Rademacher variables, the Rademacher
complexity of G for X of size n drawn from
P
q(x) is defined by Rn,q (G) = EX E?1 ,...,?n [supg?G n1 xi ?X ?i g(xi )] [28]. For any fixed G and q, Rn,q (G)
still depends on n and should decrease with n.
3
0.5
0.4
0.45
Risk w.r.t. surrogate loss
Risk w.r.t. surrogate loss
0.50
0.40
0.35
0.30
PN test
PN train
uPU test
uPU train
0.25
0.20
0.15
0
100
0.3
0.2
0.1
PN test
PN train
uPU test
uPU train
nnPU test
nnPU train
0.0
0.1
0.2
200
Epoch
300
400
500
0
(a) Plain linear model
100
200
Epoch
300
400
500
(b) Multilayer perceptron (MLP)
The dataset is MNIST; even/odd digits are regarded as the P/N class, and ?p ? 1/2; np = 100 and nn = 50
for PN learning; np = 100 and nu = 59, 900 for unbiased PU (uPU) and non-negative PU (nnPU) learning.
The model is a plain linear model (784-1) in (a) and an MLP (784-100-1) with ReLU in (b); it was trained by
Algorithm 1, where the loss ` is `sig , the optimization algorithm A is [20], with ? = 1/2 for uPU, and ? = 0
bpn (g) on test data where g ? {b
and ? = 1 for nnPU. Solid curves are R
gpn , gbpu , gepu }, and dashed curves are
b
b
e
Rpn (b
gpn ), Rpu (b
gpu ) and Rpu (e
gpu ) on training data. Note that nnPU is identical to uPU in (a).
Figure 1: Illustrative experimental results.
bpu (b
The answer is affirmative. Note that R
gpu ) keeps decreasing and goes negative. This should be
fixed since R(g) ? 0 for any g. Specifically, it holds that Ru? (g) ? ?p Rp? (g) = ?n Rn? (g) ? 0, but
bu? (g) ? ?p R
bp? (g) ? 0 is not always true, which is a potential reason for uPU to overfit. Based on
R
this key observation, we propose a non-negative risk estimator for PU learning:
n
o
epu (g) = ?p R
bp+ (g) + max 0, R
bu? (g) ? ?p R
bp? (g) .
R
(6)
epu (g) be the empirical risk minimizer of R
epu (g). We refer to the process
Let gepu = arg ming?G R
of obtaining gepu as non-negative PU (nnPU) learning. The implementation of nnPU will be given
epu (g) and gepu will be given in Section 4.
in Section 3.3, and theoretical analyses of R
Again, from Figure 1(b) we can see:
(A) on training data, the risk of nnPU first decreases and then becomes more and more flat, so
that the risk of nnPU is closer to the risk of PN and farther from that of uPU; in short, the
risk of nnPU does not go down with uPU after a certain epoch;
(B) on test data, the tendency is similar, but the risk of nnPU does not go up with uPU;
(C) at the end, nnPU achieves the lowest risk on test data.
In summary, nnPU works by explicitly constraining the training risk of uPU to be non-negative.
3.3
Implementation
A list of popular loss functions and their properties is shown in Table 1. Let g be parameterized by
?. If g is linear in ?, the losses satisfying (5) result in convex optimizations. However, if g needs to
be flexible, it will be highly nonlinear in ?; then the losses satisfying (5) are not advantageous over
bpu (g)
others, since the optimizations are anyway non-convex. In [15], the ramp loss was used and R
was minimized by the concave-convex procedure [30]. This solver is fairly sophisticated, and if we
bpu (g) with R
epu (g), it will be more difficult to implement. To this end, we propose to use
replace R
epu (g) can
the sigmoid loss `sig (t, y) = 1/(1 + exp(ty)): its gradient is everywhere non-zero and R
be minimized by off-the-shelf gradient methods.
bpu (g)
In front of big data, we should scale PU learning up by stochastic optimization. Minimizing R
e
b
epu (g)
is embarrassingly parallel while minimizing Rpu (g) is not, since Rpu (g) is point-wise but R
?
?
bu (g; Xu ) ? ?p R
bp (g; Xp )} is no greater
is not due to the max operator. That being said, max{0, R
PN
?
i
?
i
i
i
b
b
than (1/N ) i=1 max{0, Ru (g; Xu ) ? ?p Rp (g; Xp )}, where (Xp , Xu ) is the i-th mini-batch, and
epu (g) can easily be minimized in parallel.
hence the corresponding upper bound of R
4
Table 1: Loss functions for PU learning and their properties.
Name
Definition
(3)
(5)
Bounded
Lipschitz
`0 (z) 6= 0
Zero-one loss
Ramp loss
Squared loss
Logistic loss
Hinge loss
Double hinge loss
Sigmoid loss
(1 ? sign(z))/2
max{0, min{1, (1 ? z)/2}}
(z ? 1)2 /4
ln(1 + exp(?z))
max{0, 1 ? z}
max{0, (1 ? z)/2, ?z}
1/(1 + exp(z))
X
X
?
?
?
?
X
?
?
X
X
?
X
?
X
X
?
?
?
?
X
?
X
?
X
X
X
X
z=0
z ? [?1, +1]
z?R
z?R
z ? (??, +1]
z ? (??, +1]
z?R
All loss functions are unary, such that `(t, y) = `(z) with z = ty. The ramp loss comes from [15]; the double
hinge loss is from [16], in which the squared, logistic and hinge losses were discussed as well. The ramp and
squared losses are scaled to satisfy (3) or (5). The sigmoid loss is a horizontally mirrored logistic function; the
logistic loss is the negative logarithm of the logistic function.
Algorithm 1 Large-scale PU learning based on stochastic optimization
Input: training data (Xp , Xu );
hyperparameters 0 ? ? ? ?p supt maxy `(t, y) and 0 ? ? ? 1
Output: model parameter ? for gbpu (x; ?) or gepu (x; ?)
1: Let A be an external SGD-like stochastic optimization algorithm such as [20] or [31]
2: while no stopping criterion has been met:
3:
Shuffle (Xp , Xu ) into N mini-batches, and denote by (Xpi , Xui ) the i-th mini-batch
4:
for i = 1 to N :
bp? (g; Xpi ) ? ??:
bu? (g; Xui ) ? ?p R
5:
if R
bpu (g; Xpi , Xui )
6:
Set gradient ?? R
7:
Update ? by A with its current step size ?
8:
else:
bu? (g; Xui ))
bp? (g; Xpi ) ? R
9:
Set gradient ?? (?p R
10:
Update ? by A with a discounted step size ??
b? (g; X i ) ? ?p R
b? (g; X i ). In
The large-scale PU algorithm is described in Algorithm 1. Let ri = R
u
u
p
p
practice, we may tolerate ri ? ?? where 0 ? ? ? ?p supt maxy `(t, y), as ri comes from a single
mini-batch. The degree of tolerance is controlled by ?: there is zero tolerance if ? = 0, and we are
bpu (g) if ? = ?p supt maxy `(t, y). Otherwise if ri < ??, we go along ??? ri with a
minimizing R
step size discounted by ? where 0 ? ? ? 1, to make this mini-batch less overfitted. Algorithm 1 is
insensitive to the choice of ?, if the optimization algorithm A is adaptive such as [20] or [31].
4
Theoretical analyses
In this section, we analyze the risk estimator (6) and its minimizer (all proofs are in Appendix B).
4.1
Bias and consistency
epu (g) ? R
bpu (g) for any (Xp , Xu ) but R
bpu (g) is unbiased, which implies R
epu (g) is biased
Fix g, R
e
in general. A fundamental question is then whether Rpu (g) is consistent. From now on, we prove
bu? (g) ?
this consistency. To begin with, partition all possible (Xp , Xu ) into D+ (g) = {(Xp , Xu ) | R
?
?
?
?
b
b
b
?p Rp (g) ? 0} and D (g) = {(Xp , Xu ) | Ru (g) ? ?p Rp (g) < 0}. Assume there are Cg > 0 and
C` > 0 such that supg?G kgk? ? Cg and sup|t|?Cg maxy `(t, y) ? C` .
Lemma 1. The following three conditions are equivalent: (A) the probability measure of D? (g)
epu (g) differs from R
bpu (g) with a non-zero probability over repeated sampling of
is non-zero; (B) R
e
(Xp , Xu ); (C) the bias of Rpu (g) is positive. In addition, by assuming that there is ? > 0 such that
Rn? (g) ? ?, the probability measure of D? (g) can be bounded by
Pr(D? (g)) ? exp(?2(?/C` )2 /(?p2 /np + 1/nu )).
5
(7)
Based on Lemma 1, we can show the exponential?decay of the bias and also the consistency. For
?
convenience, denote by ?np ,nu = 2?p / np + 1/ nu .
Theorem 2 (Bias and consistency). Assume that Rn? (g) ? ? > 0 and denote by ?g the right-hand
epu (g) decays exponentially:
side of Eq. (7). As np , nu ? ?, the bias of R
epu (g)] ? R(g) ? C` ?p ?g .
0 ? EXp ,Xu [R
(8)
p
Moreover, for any ? > 0, let C? = C` ln(2/?)/2, then we have with probability at least 1 ? ?,
epu (g) ? R(g)| ? C? ? ?n ,n + C` ?p ?g ,
|R
p
u
(9)
and with probability at least 1 ? ? ? ?g ,
epu (g) ? R(g)| ? C? ? ?n ,n .
|R
p
u
(10)
epu (g) ? R(g) in Op (?p /?np + 1/?nu ).
Either (9) or (10) in Theorem 2 indicates for fixed g, R
This convergence rate is optimal according to the central limit theorem [32], which means the proposed estimator is a biased yet optimal estimator to the risk.
4.2
Mean squared error
epu (g) tends to overestimate R(g). It is not a shrinkage estimator [33,
After introducing the bias, R
bpu (g). However,
34] so that its mean squared error (MSE) is not necessarily smaller than that of R
we can still characterize this reduction in MSE.
epu (g)) < MSE(R
bpu (g)),5 if and only if
Theorem 3 (MSE reduction). It holds that MSE(R
Z
bpu (g) + R
epu (g) ? 2R(g))(R
bu? (g) ? ?p R
bp? (g)) dF (Xp , Xu ) > 0,
(R
(11)
(Xp ,Xu )?D? (g)
Qnp
Qnu
where dF (Xp , Xu ) = i=1
pp (xpi )dxpi ? i=1
p(xui )dxui . Eq. (11) is valid, if the following condi?
tions are met: (a) Pr(D (g)) > 0; (b) ` satisfies Eq. (3); (c) Rn? (g) ? ? > 0; (d) nu np , such
b? (g) ? 2? almost surely on D? (g). In fact, given these four conditions,
that we have Ru? (g) ? R
u
we have for any 0 ? ? ? C` ?p ,
bpu (g)) ? MSE(R
epu (g)) ? 3? 2 Pr{R
epu (g) ? R
bpu (g) > ?}.
MSE(R
(12)
The assumption (d) in Theorem 3 is explained as follows. Since U data can be much cheaper than
P data in practice, it would be natural to assume nu is much larger and grows much faster than np ,
bp? (g) ? Rp? (g) ? ?/?p } ? exp(np ? nu ) asymptotically.6
bu? (g) ? ?}/Pr{R
hence Pr{Ru? (g) ? R
This means the contribution of Xu is negligible for making (Xp , Xu ) ? D? (g) so that Pr(D? (g))
bu? (g) ? 2?} has stronger exponential
exhibits exponential decay mainly in np . As Pr{Ru? (g) ? R
?
?
b
decay in nu than Pr{Ru (g) ? Ru (g) ? ?} as well as nu np , we made the assumption (d).
4.3
Estimation error
While Theorems 2 and 3 addressed the use of (6) for evaluating the risk, we are likewise interested
in its use for training classifiers. In what follows, we analyze the estimation error R(e
gpu ) ? R(g ? ),
?
?
where g is the true risk minimizer in G, i.e., g = arg ming?G R(g). As a common practice [28],
assume that `(t, y) is Lipschitz continuous in t for all |t| ? Cg with a Lipschitz constant L` .
Theorem 4 (Estimation error bound). Assume that (a) inf g?G Rn? (g) ? ? > 0 and denote by ? the
right-hand side of Eq. (7); (b) G is closed under negation, i.e., g ? G if and only if ?g ? G. Then,
for any ? > 0, with probability at least 1 ? ?,
R(e
gpu ) ? R(g ? ) ? 16L` ?p Rnp ,pp (G) + 8L` Rnu ,p (G) + 2C?0 ? ?np ,nu + 2C` ?p ?,
(13)
p
where C?0 = C` ln(1/?)/2, and Rnp ,pp (G) and Rnu ,p (G) are the Rademacher complexities of G
for the sampling of size np from pp (x) and of size nu from p(x), respectively.
5
Here, MSE(?) is over repeated sampling of (Xp , Xu ).
This can be derived as np , nu ? ? by applying the central limit theorem to the two differences and then
L?H?pital?s rule to the ratio of complementary error functions [32].
6
6
Theorem 4 ensures that learning with (6) is also consistent: as np , nu ? ?, R(e
gpu ) ? R(g ? ) and
?
if ` satisfies (5), all optimizations are convex and gepu ? g . For ?
linear-in-parameter models with a
?
gpu ) ? R(g ? )
bounded norm, Rnp ,pp (G) = O(1/ np ) and Rnu ,p (G) = O(1/ nu ), and thus R(e
?
?
in Op (?p / np + 1/ nu ).
For comparison, R(b
gpu ) ? R(g ? ) can be bounded using a different proof technique [19]:
R(b
gpu ) ? R(g ? ) ? 8L` ?p Rnp ,pp (G) + 4L` Rnu ,p (G) + 2C? ? ?np ,nu ,
(14)
p
where C? = C` ln(2/?)/2. The differences of (13) and (14) are completely from the differences
of the corresponding uniform deviation bounds, i.e., the following lemma and Lemma 8 of [19].
Lemma 5. Under the assumptions of Theorem 4, for any ? > 0, with probability at least 1 ? ?,
epu (g) ? R(g)| ? 8L` ?p Rn ,p (G) + 4L` Rn ,p (G) + C?0 ? ?n ,n + C` ?p ?. (15)
supg?G |R
p
p
u
p
u
bpu (g) is point-wise while R
epu (g) is not due to the maximum, which makes Lemma 5
Notice that R
much more difficult to prove than Lemma 8 of [19]. The key trick is that after symmetrization, we
employ | max{0, z} ? max{0, z 0 }| ? |z ? z 0 |, making three differences of partial risks point-wise
(see (18) in the proof). As a consequence, we have to use a different Rademacher complexity with
the absolute value inside the supremum [35, 36], whose contraction makes the coefficients of (15)
doubled compared with Lemma 8 of [19]; moreover, we have to assume G is closed under negation
to change back to the standard Rademacher complexity without the absolute value [28]. Therefore,
the differences of (13) and (14) are mainly due to different proof techniques and cannot reflect the
intrinsic differences of empirical risk minimizers.
5
Experiments
In this section, we compare PN, unbiased PU (uPU) and non-negative PU (nnPU) learning experimentally. We focus on training deep neural networks, as uPU learning usually does not overfit if a
linear-in-parameter model is used [19] and nothing needs to be fixed.
Table 2 describes the specification of benchmark datasets. MNIST, 20News and CIFAR-10 have 10,
7 and 10 classes originally, and we constructed the P and N classes from them as follows: MNIST
was preprocessed in such a way that 0, 2, 4, 6, 8 constitute the P class, while 1, 3, 5, 7, 9 constitute
the N class; for 20News, ?alt.?, ?comp.?, ?misc.? and ?rec.? make up the P class, and ?sci.?, ?soc.? and
?talk.? make up the N class; for CIFAR-10, the P class is formed by ?airplane?, ?automobile?, ?ship?
and ?truck?, and the N class is formed by ?bird?, ?cat?, ?deer?, ?dog?, ?frog? and ?horse?. The dataset
epsilon has 2 classes and such a construction is unnecessary.
Three learning methods were set up as follows: (A) for PN, np = 1, 000 and nn = (?n /2?p )2 np ;
(B) for uPU, np = 1, 000 and nu is the total number of training data; (C) for nnPU, np and nu are
bpu (g) in
exactly same as uPU. For uPU and nnPU, P and U data were dependent, because neither R
epu (g) in Eq. (6) requires them to be independent. The choice of nn was motivated by
Eq. (2) nor R
[19] and may make nnPU potentially better than PN as nu ? ? (whether np < ? or np ? nu ).
The model for MNIST was a 6-layer multilayer perceptron (MLP) with ReLU [40] (more specifically, d-300-300-300-300-1). For epsilon, the model was similar while the activation was replaced
with Softsign [41] for better performance. For 20News, we borrowed the pre-trained word embeddings from GloVe [42], and the model can be written as d-avg_pool(word_emb(d,300))-300-300-1,
Table 2: Specification of benchmark datasets, models, and optimition algorithms.
Name
MNIST [29]
epsilon [37]
20News [38]
CIFAR-10 [39]
# Train
# Test
# Feature
?p
60, 000
400, 000
11, 314
50, 000
10, 000
100, 000
7, 532
10, 000
784
2, 000
61, 188
3, 072
0.49
0.50
0.44
0.40
Model g(x; ?)
Opt. alg. A
6-layer MLP with ReLU
6-layer MLP with Softsign
5-layer MLP with Softsign
13-layer CNN with ReLU
Adam [20]
Adam [20]
AdaGrad [31]
Adam [20]
See http://yann.lecun.com/exdb/mnist/ for MNIST, https://www.csie.ntu.edu.tw/~cjlin/
libsvmtools/datasets/binary.html for epsilon, http://qwone.com/~jason/20Newsgroups/ for
20Newsgroups, and https://www.cs.toronto.edu/~kriz/cifar.html for CIFAR-10.
7
0.50
0.4
0.45
0.40
0.3
Risk w.r.t. zero-one loss
Risk w.r.t. zero-one loss
0.5
0.2
0.1
0.0
PN test
PN train
uPU test
uPU train
nnPU test
nnPU train
0.1
0.2
0.3
0.4
0
25
50
75
100
Epoch
125
150
175
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
200
25
50
(a) MNIST
75
100
Epoch
125
150
175
200
125
150
175
200
(b) epsilon
0.4
0.5
0.4
Risk w.r.t. zero-one loss
Risk w.r.t. zero-one loss
0.3
0.2
0.1
0.0
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.1
0
25
50
75
100
Epoch
125
150
175
0.4
200
(c) 20News
0
25
50
75
100
Epoch
(d) CIFAR-10
Figure 2: Experimental results of training deep neural networks.
where word_emb(d,300) retrieves 300-dimensional word embeddings for all words in a document,
avg_pool executes average pooling, and the resulting vector is fed to a 4-layer MLP with Softsign.
The model for CIFAR-10 was an all convolutional net [43]: (32*32*3)-[C(3*3,96)]*2-C(3*3,96,2)[C(3*3,192)]*2-C(3*3,192,2)-C(3*3,192)-C(1*1,192)-C(1*1,10)-1000-1000-1, where the input is
a 32*32 RGB image, C(3*3,96) means 96 channels of 3*3 convolutions followed by ReLU, [ ? ]*2
means there are two such layers, C(3*3,96,2) means a similar layer but with stride 2, etc.; it is one
of the best architectures for CIFAR-10. Batch normalization [44] was applied before hidden layers.
Furthermore, the sigmoid loss `sig was used as the surrogate loss and an `2 -regularization was also
added. The resulting objectives were minimized by Adam [20] on MNIST, epsilon and CIFAR-10,
and by AdaGrad [31] on 20News; we fixed ? = 0 and ? = 1 for simplicity.
The experimental results are reported in Figure 2, where means and standard deviations of training
and test risks based on the same 10 random samplings are shown. We can see that uPU overfitted
training data and nnPU fixed this problem. Additionally, given limited N data, nnPU outperformed
PN on MNIST, epsilon and CIFAR-10 and was comparable to it on 20News. In summary, with the
proposed non-negative risk estimator, we are able to use very flexible models given limited P data.
We further tried some cases where ?p is misspecified, in order to simulate PU learning in the wild,
where we must suffer from errors in estimating ?p . More specifically, we tested nnPU learning by
replacing ?p with ?p0 ? {0.8?p , 0.9?p , . . . , 1.2?p } and giving ?p0 to the learning method, so that it
would regard ?p0 as ?p during the entire training process. The experimental setup was exactly same
as before except the replacement of ?p .
The experimental results are reported in Figure 3, where means of test risks of nnPU based on the
same 10 random samplings are shown, and the best test risks are identified (horizontal lines are the
best mean test risks and vertical lines are the epochs when they were achieved). We can see that on
MNIST, the more misspecification was, the worse nnPU performed, while under-misspecification
hurt more than over-misspecification; on epsilon, the cases where ?p0 equals to ?p , 1.1?p and 1.2?p
8
0.46
0.8
0.9
1.0
1.1
1.2
0.25
0.44
Risk w.r.t. zero-one loss
Risk w.r.t. zero-one loss
0.30
0.20
0.15
0.10
0.42
0.40
0.38
0.36
0.34
0.32
0.30
0.05
0
25
50
75
100
Epoch
125
150
175
0.28
200
0
25
50
(a) MNIST
100
Epoch
125
150
175
200
125
150
175
200
(b) epsilon
0.30
0.30
0.28
0.28
Risk w.r.t. zero-one loss
Risk w.r.t. zero-one loss
75
0.26
0.24
0.22
0.26
0.24
0.22
0.20
0.18
0.20
0
25
50
75
100
Epoch
125
150
175
200
0
(c) 20News
25
50
75
100
Epoch
(d) CIFAR-10
Figure 3: Experimental results given
?p0
? {0.8?p , 0.9?p , . . . , 1.2?p }.
were comparable, but the best was ?p0 = 1.1?p rather than ?p0 = ?p ; on 20News, these three cases
became different, such that ?p0 = ?p was superior to ?p0 = 1.2?p but inferior to ?p0 = 1.1?p ; at last
on CIFAR-10, ?p0 = ?p and ?p0 = 1.1?p were comparable again, and ?p0 = 1.2?p was the winner.
In all the experiments, we have fixed ? = 0, which may explain this phenomenon. Recall that uPU
overfitted seriously on all the benchmark datasets, and note that the larger ?p0 is, the more different
nnPU is from uPU. Therefore, the replacement of ?p with some ?p0 > ?p introduces additional bias
epu (g) in estimating R(g), but it also pushes R
epu (g) away from R
bpu (g) and then pushes nnPU
of R
away from uPU. This may result in lower test risks given some ?p0 slightly larger than ?p as shown
in Figure 3. This is also why under-misspecified ?p0 hurt more than over-misspecified ?p0 .
All the experiments were done with Chainer [45], and our implementation based on it is available
at https://github.com/kiryor/nnPUlearning.
6
Conclusions
We proposed a non-negative risk estimator for PU learning that follows and improves on the stateof-the-art unbiased risk estimators. No matter how flexible the model is, it will not go negative as
its unbiased counterparts. It is more robust against overfitting when being minimized, and training
very flexible models such as deep neural networks given limited P data becomes possible. We also
developed a large-scale PU learning algorithm. Extensive theoretical analyses were presented, and
the usefulness of our non-negative PU learning was verified by intensive experiments. A promising
future direction is extending the current work to semi-supervised learning along [46].
Acknowledgments
GN and MS were supported by JST CREST JPMJCR1403 and GN was also partially supported by
Microsoft Research Asia.
9
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11
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6,374 | 6,766 | Optimal Sample Complexity of M -wise Data for
Top-K Ranking
Minje Jang?
School of Electrical Engineering
KAIST
[email protected]
Sunghyun Kim?
Electronics and Telecommunications Research Institute
Daejeon, Korea
[email protected]
Changho Suh
School of Electrical Engineering
KAIST
[email protected]
Sewoong Oh
Industrial and Enterprise Systems Engineering Department
UIUC
[email protected]
Abstract
We explore the top-K rank aggregation problem in which one aims to recover a
consistent ordering that focuses on top-K ranked items based on partially revealed
preference information. We examine an M -wise comparison model that builds on
the Plackett-Luce (PL) model where for each sample, M items are ranked according
to their perceived utilities modeled as noisy observations of their underlying true
utilities. As our result, we characterize the minimax optimality on the sample size
for top-K ranking. The optimal sample size turns out to be inversely proportional to
M . We devise an algorithm that effectively converts M -wise samples into pairwise
ones and employs a spectral method using the refined data. In demonstrating
its optimality, we develop a novel technique for deriving tight `? estimation
error bounds, which is key to accurately analyzing the performance of top-K
ranking algorithms, but has been challenging. Recent work relied on an additional
maximum-likelihood estimation (MLE) stage merged with a spectral method to
attain good estimates in `? error to achieve the limit for the pairwise model. In
contrast, although it is valid in slightly restricted regimes, our result demonstrates
a spectral method alone to be sufficient for the general M -wise model. We run
numerical experiments using synthetic data and confirm that the optimal sample
size decreases at the rate of 1/M . Moreover, running our algorithm on real-world
data, we find that its applicability extends to settings that may not fit the PL model.
1
Introduction
Rank aggregation has been explored in a variety of contexts such as social choice [15, 6], web search
and information retrieval [20], recommendation systems [7], and crowd sourcing [16], to name a few.
It aims to bring a consistent ordering to a collection of items, given partial preference information.
?
Equal contribution.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Preference information can take various forms depending on the context. One such form, which we
examine in this paper, is ordinal; preferences for alternatives are represented as an ordering. Consider
crowd-sourced data collected by annotators asked to rank a few given alternatives based on their
preference. The aggregated data can be used to identify the most preferred. One example can be
a review process for conference papers (e.g., NIPS) where reviewers are asked to not only review
papers, but also order them based on how much they enjoy them. The collected data could be used to
highlight papers that may interest a large audience. Alternatively, consider sports (races or the like)
and online games where a number of players compete. One may wish to rank them according to skill.
Its broad range of applications has led to a volume of work done. Of numerous schemes developed,
arguably most dominant paradigms are spectral algorithms [14, 20, 37, 41, 47, 45] and maximum
likelihood estimation (MLE) [22, 28]. Postulating the existence of underlying real-valued preferences
of items, they aim to produce preference estimates consistent in a global sense, e.g., measured by low
squared loss. But such estimates do not necessarily guarantee optimal ranking accuracy. Accurate
ranking has more to do with how well the ordering of estimates matches that of the true preferences,
and less to do with how close the estimates are to the true preferences minimizing overall errors.
Moreover, in practice, what we expect from accurate ranking is an ordering that precisely separates
only a few items ranked highest from the rest, not an ordering that respects the entire items.
Main contributions. In light of it, we explore top-K ranking which aims to recover the correct set
of top-ranked items only. We examine the Plackett-Luce (PL) model which has been extensively
explored [24, 18, 5, 25, 38, 43, 33, 4]. It is a special case of random utility models [46] where true
utilities of items are presumed and a user?s revealed preference is a partial ordering according to
noisy manifestations of the utilities. It satisfies the ?independence of irrelevant alternatives? property
in social choice theory [34, 35] and is the most popular model in studying human choice behavior
given multiple alternatives (see Section 2). It is well-known that it subsumes as a special case the
Bradley-Terry-Luce (BTL) model [12, 32] which concerns two items. We consider an M -wise
comparison model where comparisons are given as a preference ordering of M items. In this setting,
we characterize the minimax limit on the sample size (i.e., sample complexity) needed to reliably
identify the set of top-K ranked items, which turns out to be inversely proportional to M . To the best
of our knowledge, it is the first result that characterizes the limit under an M -wise comparison model.
In achieving the limit, we propose an algorithm that consists of sample breaking and Rank Centrality
[37], one spectral method we choose among other variants [10, 9, 37, 33]. First, it converts M -wise
samples into many more pairwise ones, and in doing so, it carefully chooses only M out of all M
2
pairwise samples obtainable from each M -wise sample. This sample breaking (see Section 3.1)
extracts only the essential information needed to achieve the limit from given M -wise data. Next,
using the refined pairwise data, the algorithm runs a spectral method to identify top-ranked items.
A novel technique we develop to attain tight `? estimation error bounds has been instrumental to
our progress. Analyzing `? error bounds is a critical step to characterizing the minimax sample
complexity for top-K ranking as presented in [17], but has been technically challenging. Even after
decades of research since the introduction of spectral methods and MLE, two dominant approaches in
the field, we lack notable results for tight `? error bounds. This is largely because techniques proven
useful to obtain good `2 error bounds do not translate into attaining good `? error bounds. In this
regard, our result contributes to progress on `? error analysis (see Section 3.2 and the supplementary).
We can compare our result to that of [17] by considering M = 2. Although the two optimal sample
complexities match, the conditions under which they do differ; our result turns out to be valid under a
slightly restricted condition (see Section 3.3). In terms of achievability, the algorithm in [17] merges
an additional MLE stage with a spectral method, whereas we employ only a spectral method. From
numerical experiments, we speculate that the condition under which the result of [17] holds may not
be sufficient for spectral methods alone to achieve optimality (see Section 4.1).
We conduct numerical experiments to support our result. Using synthetic data, we show that the
minimax optimal sample size indeed decreases at the rate of 1/M . We run our algorithm on real-world
data collected from a popular online game (League of Legends) and find its applicability to extend to
settings that may not necessarily match the PL model. From the collected data, we extract M -wise
comparisons and rank top users in terms of skill. We examine its robustness aspect against partial
data and also evaluate its rank result with respect to the official rank League of Legends provides. In
both cases, we compare it with a counting-based algorithm [42, 11] and demonstrate its advantages.
2
Related work. To the best of our knowledge, [17] investigated top-K identification under the random
comparison model of interest for the first time. A key distinction here is that we examine the random
listwise comparison model based on the PL model. Rank Centrality was developed in [37] based on
which we devise our ranking scheme tailored for listwise comparison data.
In the PL model, some viewed ranking as parameter estimation. Maystre and Grossglauser [33]
developed an algorithm that shares a spirit of spectral ranking and showed its performance is the
same as MLE for estimating underlying preference scores. Hajek et al. [25] derived minimax lower
bounds of parameter estimation error, and examined gaps with upper bounds of MLE as well as MLE
with a rank-breaking scheme that decomposes partial rankings into pairwise comparisons.
Some works examined several sample breaking methods that convert listwise data into pairwise data
in the PL model. Azari Soufiani et al. [5] considered various methods to see if they sustain some
statistical
property in parameter estimation. It examined full breaking that converts an M -wise sample
into M
2 pairwise ones, and adjacent breaking that converts an ordinal M -wise sample into M ? 1
pairwise ones whose associated items are adjacent in the sample. Ashish and Oh [4] considered a
method that converts an M -wise sample into multiple pairwise ones and assigns different importance
weights to each, and examined the method on several types of comparison graphs.
There are a number of works that explored ranking problems in different models and with different
interests. Some works [43, 2] have adopted PAC (probably approximately correct) [44] or regret
[21, 8, 23] as their metric to allow some margin of error, in contrast to our work where 0/1 loss
(the most stringent criterion) is considered to investigate the worst-case scenario (see Section 2).
Rajkumar and Agarwal [40] put forth statistical assumptions that ensure the convergence of rank
aggregation methods including Rank Centrality and MLE to an optimal ranking. Active ranking
where samples are obtained adaptively has received attention as well. Jamieson and Nowak [29]
considered perfect total ranking and characterized the query complexity gain of adaptive sampling in
the noise-free case, and the works of [29, 1] explored the query complexity in the presence of noise
aiming at approximate total rankings. Recently, Braverman et al. [13] considered three noisy models,
examining if their algorithm can achieve reliable top-K ranking. Heckel et al. [27] considered a
model where noisy pairwise observations are given, with a goal to partition the items into sets of
pre-specified sizes based on their scores, which includes top-K ranking as a special case. Mohajer
et al. [36] considered a fairly general noisy model which subsumes as special cases various models.
They derived upper bounds on the sample size required for reliable top-K sorting as well as top-K
partitioning, and showed that active ranking can provide significant gains over passive ranking.
2
Problem Formulation
Notation. We denote by [n] to represent {1, 2, . . . , n}, and by G = ([n], E (M ) ) to represent an
M -wise comparison graph in which total n vertices reside and each hyper-edge is connected if there
is a comparison among M vertices, and di to represent the out-degree of vertex i.
Comparison model and assumptions. Suppose we perform a few evaluations on n items. We
assume the comparison outcomes are generated based on the PL model [39]. We consider M -wise
models where the comparison outcomes are obtained in the form of a preference ordering of M items.
Preference scores. The PL model assumes the existence of underlying preferences w :=
{w1 , w2 , . . . , wn }, where wi represents the preference score of item i. The outcome of each comparison depends solely on the latent scores of the items being compared. Without loss of generality, we
assume that w1 ? w2 ? ? ? ? ? wn > 0. We assume the range of scores to be fixed irrespective of n.
For some positive constants wmin and wmax , wi ? [wmin , wmax ], 1 ? i ? n. We note that the case
where the range wmax /wmin grows with n can be translated into the above fixed-range regime by
separating out those items with vanishing scores (e.g. via a voting method like Borda count [11, 3]).
Comparison model. We denote by G = ([n], E (M ) ) a comparison graph where a set of M items
I = {i1 , i2 , . . . , iM } are compared if and only if I belongs to the hyper-edge set E (M ) . We examine
random graphs, constructed in a similar manner according to the Erd?os-R?nyi random graph model;
each set of M vertices is connected by a hyper-edge independently with probability p. Notice that
when M = 2, such random graphs we consider follow precisely the Erd?os-R?nyi random model.
3
M -wise comparisons. We observe L samples for each I = {i1 , i2 , . . . , iM } ? E (M ) . Each
sample is an ordering of M items in order of preference. The outcome of the `th sample, de(`)
(`)
noted by
sI , is generated
according to the PL model: sI = (i1 , i2 , . . . , iM ) with probability
QM
PM
m=1 wim /
r=m wir , where item ia is preferred over item ib in I if ia appears to the left
(`)
of ib , which we also denote by ia ib . We assume that conditional on G, sI ?s are jointly independent over I and `. We denote the collection of all samples by s := {sI : I ? E (M ) }, where
(1) (2)
(L)
sI = {sI , sI , . . . , sI }.
Performance metric and goal. Given comparison data, one wishes to know whether or not the
top-K ranked items are identifiable. We consider the probability of error Pe in identifying the correct
set of the top-K ranked items: Pe (?) := P {?(s) 6= [K]}, where ? is any ranking scheme that
returns a set of K indices and [K] is the set of the first K indices. Our goal in this work is to
characterize the admissible region Rw of (p, L) in which top-K ranking is feasible for a given PL
parameter w, in other words, Pe can be vanishingly small as n grows. The admissible region Rw is
defined as Rw := {(p, L) : limn?? Pe (?(s)) = 0}. In particular, we are
interested
in the minimax
n
sample complexity of an estimator defined as S? :=
inf + sup M
pL : (p, L) ? Rv ,
p?[0,1],L?Z v???
where ?? = {v ? Rn : (vK ? vK+1 )/vmax ? ?}. Note that this definition shows that we
conservatively examine minimax scenarios where nature behaves adversely with the worst-case w.
3
Main Results
Separating the two items near the decision boundary (i.e., the K th and (K + 1)th ranked items) is
key in top-K ranking. Unless the gap is large enough, noise in the observations leads to erroneous
estimates which no ranking scheme can overcome. We pinpoint a separation measure as ?K :=
(wK ? wK+1 )/wmax , which turns out to be crucial in establishing the fundamental limit.
Noted in [22], if a comparison graph G is not connected, it is impossible to determine the relative preferences between two disconnected entities. Thus, we assume all comparison graphs to be connected.
n?1 2
To guarantee it, for a hyper-random graph with edge size M , we assume p > log n/ M
?1 .
Now, let us formally state our main results. First, for comparison graphs under M -wise observations,
we establish a necessary condition for top-K ranking.
(M )
Theorem 1. Fix ? (0, 21 ). Given
), if
anM -wise comparison graph G = ([n], E
n
n log n 1
,
pL ? c0 (1 ? )
M
?2K M
(1)
for some numerical constant c0 , then for any ranking scheme ?, there exists a preference score vector
w with separation measure ?K such that Pe (?) ? .
The proof is a generalization of Theorem 2 in [17], and we provide it in the supplementary. Next, for
comparison graphs under M -wise observations, we establish a sufficient condition for top-K ranking.
r
log n
(M )
Theorem 2. Given an M -wise comparison
graph
G
=
([n],
E
)
and
p
?
c
(M
?
1)
1
n?1 , if
(M
?1)
n
n log n 1
pL ? c2
,
(2)
M
?2K M
for some numerical constants c1 and c2 , then Rank Centrality correctly identifies the top-K ranked
1
items with probability at least 1 ? 2n? 15 .
We provide the proof of Theorem 2 in the supplementary. From below, we describe the algorithm we
use, sample breaking and Rank Centrality [37], and soon give an outline of the proof.
Note that Theorem 1 gives a necessary condition of the sample complexity S?K & n log n/M ?2K
and Theorem 2 gives a sufficient condition of it S?K . n log n/M ?2K , and they match. That is, we
establish the minimax optimality of Rank Centrality: n log n/M ?2K .
p > log n/ Mn?1 is derived in [19] as a sharp threshold for connectivity of hyper-graphs. We assume a
slightly more strict condition for ease of analysis. This does not make a big difference in our result, as the two
conditions are almost identical order-wise given M < n/2, a reasonable condition for regimes where n is large.
2
4
3.1
Algorithm description
Algorithm 1 Rank Centrality [37]
Input the collection of statistics s = sI : I ? E (M ) .
Convert the M -wise sample for each hyper-edge I into M pairwise samples:
1. Choose a circular permutation of the items in I uniformly at random,
2. Break it into the M pairs of adjacent items, and denote the set of pairs by ?(I),
3. Use the (pairwise) data of the pairs in ?(I).
? 1
if i 6= j;
? 2dmax yij
P
?
?
?
?
Compute the transition matrix P = [Pij ]1?i,j?n : Pij =
1 ? k:k6=j Pkj if i = j;
?
0
otherwise.,
where dmax is the maximum out-degree of vertices in E (M ) .
?.
Output the stationary distribution of matrix P
Rank Centrality aims to estimate rankings from pairwise comparison data. Thus, to make use of
M -wise comparison data for Rank Centrality, we apply a sample breaking method that converts
M -wise data into pairwise data. To be more specific, if there is a hyper-edge I = {1, 2, . . . , M }, we
choose a circular permutation of the items in I uniformly at random. Suppose we pick a circular
permutation (1, 2, . . . , M ? 1, M, 1). Then, we break it into M pairs of items in the order specified
by the permutation: {1, 2}, {2, 3}, . . . , {M ? 1, M }, {M, 1} (see Section 3.3 for a remark on why
we do not lose optimality by our sample breaking method). Let us denote by ?(I) this set of pairs.
We use the converted pairwise comparison data associated with the pairs in ?(I)3 :
L
X
1 X (`)
1 if {i, j} ? ?(I) and i j;
(`)
yij,I =
, yij :=
yij,I .
(3)
0 otherwise
L
I:{i,j}??(I)
`=1
In an ideal scenario where we obtain an infinite number of samples per M -wise comparison, i.e.,
PL (`)
L ? ?, sufficient statistics L1 l=1 yij,I converge to wi /(wi + wj ). Then, the constructed matrix
? defined in Algorithm 1 becomes a matrix P whose entries [Pij ]1?i,j?n are defined as
P
? 1 P
i
for I ? E (M ) ;
? 2dmaxP I:{i,j}??(I) wiw+w
j
Pij =
(4)
1 ? k:k6=j Pkj
if i = j;
?
0
otherwise.
The entries for observed item pairs represent the relative likelihood of item i being preferred over
item j. Intuitively, random walks of P in the long run visit some states more often, if they have been
preferred over other frequently-visited states and/or preferred over many other states. The random
walks are reversible as wi Pji = wj Pij holds, and irreducible under the connectivity assumption. Once
we obtain the unique stationary distribution, it is equal to w = {w1 , . . . , wn } up to some constant
? , a noisy version of P , will give us an approximation of w.
scaling. It is clear that random walks of P
3.2
Proof outline
We outline the proof of Theorem 2 by introducing Theorem 3, which we show leads to Theorem 2.
Theorem 3. When Rank Centrality is employed, with high probability, the `? norm estimation error
is upper-bounded by
s
r
? ? wk?
kw
n log n
1
.
,
(5)
n
kwk?
M
pL
M
r
n
where p ? c1 (M ? 1) log
n?1 , and c1 is some numerical constant.
(M
?1)
3
In comparison, the adjacent breaking method [5] directly follows the ordering evaluated in each sample; if it
is 1 ? 2 ? ? ? ? ? M ? 1 ? M , it is broken into pairs of adjacent items: 1 ? 2 up to M ? 1 ? M . Our method
Pr[y =1]
wi
turns out to be consistent, i.e., Pr[yij
= w
(see (4)), whereas the adjacent breaking method is not [5].
ji =0]
j
5
Let
q kwk? = pwmax = 1 for ease of presentation. Suppose ?K = wK ? wK+1 &
n
log n/ M
pL 1/M . Then, w
?i ? w
?j ? wi ? wj ? |w
?i ? wi | ? |w
?j ? wj | ? wK ? wK+1 ?
? ? wk? > 0, for all 1 ? i ?qK and j ? K + 1. That is, the top-K items are identified as
2kw
p
n
n
desired. Hence, as long as ?K & log n/ M
pL 1/M , i.e., M
pL & n log n/M ?2K , reliable
top-K ranking is achieved with the sample size of n log n/M ?2K .
Now, let us prove Theorem 3. To find an `? error bound, we first derive an upper bound on the
point-wise error between the score estimate of item i and its true score, which consists of three terms:
X
X
?
?
?
|w
? i ? wi | ? | w
?i ? wi | Pii +
|w
?j ? wj | Pij +
(wi + wj ) Pji ? Pji .
(6)
j:j6=i
j:j6=i
?w
? =P
? and w = P w. We then obtain upper bounds on the three terms:
We can obtain (6) from w
s
s
r
r
X
n log n
1 X
n log n
1
?
?
?
Pii < 1,
(wi + wj ) Pji ? Pji .
,
|w
?j ? wj | Pij .
,
n
n
M
M
M pL
M pL
j:j6=i
j:j6=i
(7)
with high probability (Lemmas 1, 2 and 3 in the supplementary). (7) ends the proof. We obtain the
firstqtwo from Hoeffding?s inequality. The last is key; this is where we sharply link an `2 error bound
p
n
of n log n/ M
pL 1/M (Theorem 4 in the supplementary) to the desired `? error bound (5).
On the left hand side of the third inequality, the point-wise error of item j which affects that of
item i as expressed in (6), may not be captured for some j, since there may be no hyper-edge that
includes items i and j. This makes it hard to draw a link from the obtained `2 error bound to the
inequality, since `2 errors can be seen as the sum of all point-wise errors. To include them all, we
recursively apply (6) to |w
?j ? wj | in the third inequality and then apply the rest two properly (for
detailed derivation, see the beginning of the proof of Lemma 3 in the supplementary). Then, we get
s
r
X
X X
n log n
1
?
?
?
|w
?j ? wj | Pij .
|w
?k ? wk | Pjk Pij +
.
(8)
n
M
pL
M
j:j6=i
j:j6=i k:k6=j
Manipulating the first term of the right hand side (for derivation, see the proof of Lemma 3), we get
v
X
2
uX
n
X
X
u n
?
?
?
?
t
? ? wk2
|w
?k ? wk |
Pjk Pij ? kw
Pjk Pij .
(9)
j:j ?{i,k}
/
k=1
k=1
j:j ?{i,k}
/
P
We show that j:j ?{i,k}
P?jk P?ij concentrates on the order of 1/n for all k?s in the proof of Lemma
?/
?
?
? ? wk2 / n ? kw
? ? wk2 /kwk2 . We derive this `2
3. Since kwk2 ? qnkwk? = n, we get kw
p
n
error bound to be n log n/ M pL 1/M (Theorem 4 in the supplementary), matching (5).
P
To describe the concentration of j:j ?{i,k}
P?jk P?ij , we need to consider dependencies in it. To see
/
them, we upper-bound it as follows (for details, see the proof of Lemma 3 in the supplementary).
X
j:j ?{i,k}
/
P?ij P?jk ?
1
X
X
4d2max
I1 :i,j?I1 ,I2 :j,k?I2
j:j ?{i,k}
/
XI1 I2 ,
(10)
where XI1 I2 := I [{i, j} ? ?(I1 )] I [{j, k} ? ?(I2 )] . For M > 2, there can exist ja and jb such that
{i, ja , jb } ? I1 , ja ? I2 and jb ?
/ I2 . Then, summing over j, XI1 I2 and XI1 I3 , where I3 is another
hyper-edge that includes jb and k, are dependent concerning the same hyper-edge I1 . To handle this,
we use Janson?s inquality [30], one of concentration inequalities that consider dependencies.
To derive a necessary condition matching our sufficient condition, we use a generalized version of
Fano?s inequality [26] as in the proof of Theorem 3 in [17] and complete combinatorial calculations.
6
3.3
Discussion
Optimality versus M ? intuition behind our sample breaking method: For each M -wise sample, we
form a circular permutation uniformly at random, and extract M pairwise samples each of which
concerns two adjacent items in it. Suppose we have an M -wise sample 1 ? 2 ? ? ? ? ? M , and
for simplicity we happen to form a circular permutation as (1, 2, . . . , M ? 1, M, 1); we extract M
pairwise samples as 1 ? 2, 2 ? 3, . . . , (M ? 1) ? M , 1 ? M . Let us provide the intuition behind
why this leads us to the optimal sample complexity. For the case of M = 2, Rank Centrality achieves
the optimal order-wise sample complexity of n log n/?2K as characterized in [17]. In addition, one
M -wise sample in the PL model can be broken into M ? 1 independent pairwise ones, since pairwise
data of two arbitrary items among the M items depend on the true scores of the two items only. In our
example, one can convert the M -wise sample into M ? 1 independent pairwise ones as 1 ? 2, 2 ? 3,
. . . , (M ? 1) ? M . From these, it is intuitive to see that we can achieve reliable top-K ranking with
an order-wise sample complexity of n log n/(M ? 1)?2K by converting each M -wise sample into
M ? 1 independent pairwise ones. Notice a close gap to the optimal sample complexity in Section 3.
Tight `? error bounds: As shown in 3.2, deriving a tight `? error bound is critical to analyzing the
performance of an algorithm for top-K rank aggregation. Recent work [17] has relied on combining
an additional stage of local refinement in series with Rank Centrality to derive it, and characterized
the optimal sample complexity for the pairwise model. In contrast, although it is valid in a slightly
restricted regime (see the next remark), we employ only Rank Centrality and still succeed in achieving
optimality for the M -wise model that includes the pairwise model. Deriving tight `? error bounds
being crucial, it is hard for one to attain this result without a fine analytical technique. It is our main
theoretical contribution to develop one. For details, see the proof of Lemma 3 in the supplementary
that sharply links an `? error bound (Theorem 3 therein) and an `2 error bound (Theorem 4 therein).
Rank Centrality has been shown to achieve the performance nearly as good as MLE in terms of `2
error, but little has been known in terms of `? error, until now. Our result has made clear progress.
Analytical technique: Our analysis is not limited to Rank Centrality. Whenever one wishes to compute
the difference between the leading eigenvector of any matrix and that of its noisy version, one can
obtain (6), (8) and (9). Thus, it can be adopted to link `2 and `? error bounds for any spectral method.
Dense regimes:
q Our main result concerns a slightly denser regime, indicated by the condition
n?1
p & (M ? 1) log n/ M
?1 , where many distinct item pairs are likely to be compared. One can see
that this dense regime condition is not necessary for top-K ranking; for the pairwise case M = 2, it is
p & log n/n as shown in [17]. However, it is not clear yet whether or not the dense regime condition
is required under our approach that employs only a spectral method. Our speculation
q from numerical
n?1
n?1
experiments is that the sparse regime condition, log n/ M
.
p
.
(M
?
1)
log n/ M
?1
?1 , may
not be sufficient for spectral methods to achieve reliable top-K ranking (see Section 4).
Experimental Results
Synthetic data simulation
1
?? norm of estimation errors
0.5
empirical success rate
Spectral MLE: p = 0.25
0.3
0.2
0.1
0
1
5
10
15
20
L: number of repeated comparisons
0.5
0.9
Rank Centrality: p = 0.25
0.4
25
0.8
0.4
0.7
0.6
Spectral MLE: p = 0.025
0.3
0.5
0.4
0.2
0.3
Rank Centrality: p = 0.25
0.2
Spectral MLE: p = 0.25
0.1
Borda Count: p = 0.25
0
1
1
0.9
Rank Centrality: p = 0.025
empirical success rate
4.1
?? norm of estimation errors
4
5
10
15
20
0.1
L: number of repeated comparisons
0.7
0.6
0.5
0.4
Rank Centrality: p = 0.025
0.3
Spectral MLE: p = 0.025
0.2
Borda Count: p = 0.025
0.1
0
10
25
0.8
50
100
150
200
L: number of repeated comparisons
250
0
10
50
100
150
200
250
L: number of repeated comparisons
Figure 1: Dense regime (pdense = 0.25, first two figures): empirical `? estimation error v.s. L (left);
empirical success rate v.s. L (right). Sparse regime (psparse = 0.025, last two figures): empirical `?
estimation error v.s. L (left); empirical success rate v.s. L (right).
First, we conduct a synthetic data experiment for M = 2, the pairwise comparison model,
p to compare
our result in Theorem 2 p
to that in recent work [17]. We consider both dense (p & log n/n) and
sparse (log n/n . p . log n/n) regimes. We set constant c1 = 2, and set pdense = 0.25 and
psparse = 0.025, to make each be in its proper range. We use n = 500, K = 10, and ?K = 0.1.
Each result in all numerical simulations is obtained by averaging over 10000 Monte Carlo trials.
7
In Figure 1, the first two figures show the experiments in the dense regime. We see that as L increases,
meaning as we obtain pairwise samples beyond the minimal sample complexity, (1) the `? error
of Rank Centrality decreases and meets that of Spectral MLE (left); (2) the success rate of Rank
Centrality increases and soon hits
p 100% along with Spectral MLE (right). The curves support our
result; in the dense regime p & log n/n, Rank Centrality alone can achieve reliable top-K ranking.
The last two figures show the experiments in the sparse regime. We see that as L increases, (1) the
`? error of Rank Centrality decreases but does not meet that of Spectral MLE (left); (2) the success
rate of Rank Centrality increases but does not reach that of Spectral MLE which hits nearly
p 100%
(right). The curves lead us to speculate that the sparse regime condition log n/n . p . log n/n
may not be sufficient for spectral methods to achieve reliable top-K ranking.
Empirical
10000
Curve ?tting: 1/M
8000
6000
4000
2000
0
3
4
5
6
7
8
9
10
11
12
13
14
?105
Empirical
2.5
Curve ?tting: 1/?2K
2
1.5
1
0.5
0
0.1
15
2
Minimal sample complexity
3
Minimal sample complexity
Minimal sample complexity
12000
0.15
0.2
0.25
?K
M : Size of hyper-edges
?105
Empirical
1.5
Curve ?tting: n log n
1
0.5
0
0.3
500
1000
1500
n
Figure 2: Empirical minimal sample complexity v.s. M (first), ?K (second), and n log n (third).
Next, we corroborate our optimal sample complexity result in Theorem 2. We examine whether the
empirical minimal sample complexity decreases at the rate of 1/M and 1/?2K , and increases at the
rate of n log n. To verify its reduction at the rate of 1/M , we run experiments for M ranging from
3 to 15. We increase the number of samples by increasing p until the success rate reaches 95% for
each M . The number of samples we use to achieve it is considered as the empirical minimal sample
complexity for each M . We set the other parameters as n = 100, L = 20, K = 5 and ?K = 0.3.
The result for each M in all simulations is obtained by averaging over 1000 Monte Carlo trials. To
verify the other two relations, we follow similar procedures. As for 1/?2K , we set n = 200, M = 2,
L = 20 and K = 5. As for n log n, we set M = 2, L = 4, K = 5 and ?K = 0.4.
The first figure in Figure 2 shows the reduction in empirical minimal sample complexity with a blue
solid curve. The red dashed curve is obtained by curve-fitting. We can see that the empirical minimal
sample complexity drops inversely proportional to M . From the second and third figures, we can see
that in terms of ?K and n log n, it also behaves as our result in Theorem 2 predicts.
1
1
0.7
0.6
0.5
0.4
Spectral MLE
Proposed
Least Square
Counting
0.3
0.2
0.1
15
20
25
30
L
35
40
45
Spectral MLE
0.9
80
Least Square
0.8
Counting
Percentile
Normalized overlap (K = 5)
Success rate
0.8
0
10
100
Proposed
0.9
0.7
0.6
60
40
Proposed
0.5
Spectral MLE
20
Least Square
0.4
Counting
50
1
0.9
0.8
0.7
f: Fraction of samples used
0.6
0.5
0
1
2
3
4
5
Top-5 users based on average league point per match
Figure 3: (First) Empirical success rates of four algorithms: our algorithm (blue circle), heuristic
Spectral MLE (red cross), least square (green plus), and counting (purple triangle); (Second) Top-5
ranked users: normalized overlap v.s. fraction of samples used; (Third) Top-5 users? (sorted by
average League of Legends points earned per match) percentile in the ranks by our algorithm, heuristic
Spectral MLE, least square, and counting. For instance, the user who earns largest points per match
(first entry) is at around the 80-th percentile according to our algorithm and heuristic Spectral MLE,
the 60-th percentile according to least square, and the 10-th percentile according to counting.
Last, we evaluate the success rates of various algorithms on M -wise comparison data. We consider
our proposed algorithm, Spectral MLE, least square (HodgeRank [31]), and counting. Since Spectral
MLE has been developed for pairwise data, we heuristically extend it. We apply our sample breaking
method to obtain pairwise data needed. For any parameters required to run Spectral MLE, we
heuristically find the best ones which give rise to the highest success rate. In the other two algorithms,
we first apply our sample breaking method as well. Then, for least square, we find a score vector
P
2
? such that the squared error (i,j)?E (log(w
w
? i /w
?j ) ? log(yij /yji )) , where E is the edge set for
the converted pairwise data, is minimized. For counting, we count each item?s number of wins in all
8
q
n?1
involved pairwise data. We use n = 100, M = 4, p = 0.0025 ? (M ? 1) log n/ M
?1 , K = 5 and
?K = 0.3. Each result in all simulations is obtained by averaging over 5000 Monte Carlo trials.
The first figure in Figure 3 shows that our algorithm and heuristic Spectral MLE perform best (the
latter being marginally better), achieving near-100% success rates for large L. It also shows that they
outperform the other two algorithms which do not achieve near-100% success rates even for large L.
4.2
Real-world data simulation
One natural setting where we can obtain M -wise comparison data is an online game. Users randomly
get together and play, and the results depend on their skills. We find League of Legends to be a proper
fit4 . In extracting M -wise data, we adopt a measure widely accepted as a factor that rates users? skill
in the user community5 . We incorporate this measure into our model as follows. For each match
(M -wise sample), we have 10 users, each associated with its measure. In breaking M -wise samples,
for each user pair (i, j), we compare their measures and declare user i wins if its measure is larger
(`)
than user j?s. This corresponds to yij in our model. We assign 1 if user i wins and 0 otherwise. They
PLij (`)
may play together in multiple, say Lij , matches. We can compute yij := ( `=1
yij )/Lij to use for
Rank Centrality. As M -wise data is extracted from team competitions, League of Legends does not
perfectly fit our model. Yet one main reason to run this experiment is to see whether our algorithm
works well in other settings that do not necessarily fit the PL model, being broadly applicable.
We first investigate the robustness aspect by evaluating the performance against partial information.
To this end, we use all collected data and obtain a ranking result for each algorithm which we consider
as its baseline. Then, for each algorithm, we reduce sample sizes by discarding some of the data, and
compare the results to the baseline to see how robust each algorithm is against partial information.
We conduct this experiment for four algorithms: our proposed algorithm, the heuristic extension of
Spectral MLE, least square and counting.
We choose our metric as a normalized overlap: |Scomp ? Spart |/K, where K = 5, Scomp is the set of
top-K users identified using the complete dataset and Spart is that identified using partial datasets.
In choosing partial data, we set f ? (0.5, 1), and discard each match result with probability f
independently. We compute the metric for each f by averaging over 1000 Monte Carlo trials.
The second figure of Figure 3 shows that over the range of f where overlaps above 60% are retained,
our algorithm, along with some others, demonstrates good robustness against partial information.
In addition, we compare the ranks estimated by the four algorithms to the rank provided by League of
Legends. By computing the average points earned per match for each user, we infer the rank of the
users determined by official standards. In the third figure of Figure 3, the x-axis indicates the top-5
users identified by computing average League of Legends points earned per match and sorting them
in descending order. The y-axis indicates the percentile of these top-5 users according to the ranks by
the algorithms of interest. Notice that the top-5 ranked users by League of Legends standards are also
placed at high ranks when ranked by our algorithm and heuristic Spectral MLE; they are all placed at
the 80-th percentile or above. On the other hand, most of them (4 users out of the top-5 users) are
placed at noticeably lower ranks when ranked by least square and counting.
5
Conclusion
We characterized the minimax (order-wise) optimal sample complexity for top-K rank aggregation
in the M -wise comparison model that builds on the PL model. We corroborated our result using
synthetic data experiments and verified the applicability of our algorithm on real-world data.
4
Two teams of 5 users compete. Each user kills an opponent, assists a mate to kill one, and dies from an
attack. At the end, one team wins, and different points are given to the users. We use users? kill/assist/death data
(non-negative integers), which can be considered as noisy measurements of their skill, and rank them by skill.
5
We define a measure as {(# of kills + # of assists)/(1 + # of deaths)}?weight. We adopt this measure
since it is similar to the one officially provided (called KDA statistics). We assign winning users a weight of 1.1
and losing users a weight of 1.0, to give extra credit (10%) to users who lead their team?s winning.
9
Acknowledgments
This work was supported by Institute for Information & communications Technology Promotion(IITP)
grant funded by the Korea government(MSIT) (2017-0-00694, Coding for High-Speed Distributed
Networks).
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11
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6,375 | 6,767 | Reliable Decision Support using
Counterfactual Models
Suchi Saria
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21211
[email protected]
Peter Schulam
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21211
[email protected]
Abstract
Making a good decision involves considering the likely outcomes under each
possible action. For example, would drug A or drug B lead to a better outcome
for this patient? Ideally, we answer these questions using an experiment, but this
is not always possible (e.g., it may be unethical). As an alternative, we can use
non-experimental data to learn models that make counterfactual predictions of
what we would observe had we run an experiment. To learn such models for
decision-making problems, we propose the use of counterfactual objectives in lieu
of classical supervised learning objectives. We implement this idea in a challenging
and frequently occurring context, and propose the counterfactual GP (CGP), a
counterfactual model of continuous-time trajectories (time series) under sequences
of actions taken in continuous-time. We develop our model within the potential
outcomes framework of Neyman [1923] and Rubin [1978]. The counterfactual GP
is trained using a joint maximum likelihood objective that adjusts for dependencies
between observed actions and outcomes in the training data. We report two sets
of experimental results. First, we show that the CGP?s predictions are reliable;
they are stable to changes in certain characteristics of the training data that are not
relevant to the decision-making problem. Predictive models trained using classical
supervised learning objectives, however, are not stable to such perturbations. In
the second experiment, we use data from a real intensive care unit (ICU) and
qualitatively demonstrate how the CGP?s ability to answer ?What if?? questions
offers medical decision-makers a powerful new tool for planning treatment.
1
Introduction
Making a good decision involves considering the likely outcomes under each possible action. Would
changing the color or text of an ad be more effective for increasing a firm?s revenue? Would drug A
or drug B lead to a better outcome for this patient? Ideally, we would run an experiment: clone the
patient and try each action on a different clone, compare the outcomes across scenarios, and choose
the one with the best result. Experiments, however, are not always feasible. An alternative is to
learn models from non-experimental data (i.e. data where we do not control actions) that can make
counterfactual predictions of the outcomes we would have observed had we run an experiment (see
e.g., Pearl 2009). The key challenge when learning counterfactual models from non-experimental
data is that it is difficult to distinguish between statistical dependence and causal relationships. For
instance, consider a drug that is often given to sicker patients who are also more likely to die. Without
accounting for this bias in the treatment policy, a statistical model would predict that the drug kills
patients even if it is actually beneficial.
This challenge is commonly addressed using the potential outcomes framework [Neyman, 1923,
1990, Rubin, 1978], which introduces a collection of counterfactual random variables {Y [a] : a ? C}
for an outcome Y and each action a from a set of choices C. The counterfactuals may be interpreted
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
History
?
(a)
H
(b)
120
?
100
?
80
100
??
?
?
?
?
?
?
60
?
80
??
?
?
?
?
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PFVC
100
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E[Y [
60
Drug A
Drug B
40
] | H]
5
10
Years Since First Symptom
15
?
80
??
?
?
?
E[Y [
] | H]
E[Y [
] | H]
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60
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0
(c)
120
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PFVC
Lung
Capacity
PFVC
120
40
0
5
10
Since First Symptom
Years Years
Since
First Symptom
15
0
5
10
15
Years Since First Symptom
Figure 1: Best viewed in color. An illustration of the counterfactual GP applied to health care. The red box in
(a) shows previous lung capacity measurements (black dots) and treatments (the history). Panels (b) and (c) show
the type of predictions we would like to make. We use Y [a] to represent the potential outcome under action a.
as probabilistic models of outcomes obtained by running experiments. Using the potential outcomes
framework, we can clearly state assumptions under which the distribution of the counterfactuals can
be learned from non-experimental data. When this is possible, learned counterfactual models can
make ?What if?? predictions to guide decisions.
In this paper, we use the potential outcomes framework to develop the counterfactual Gaussian
process (CGP), an approach to modeling the effects of sequences of actions on continuous-time
trajectories (i.e. time series). Figure 1 illustrates this idea. We show an individual with a lung
disease, and would like to predict how her lung capacity (y-axis) will progress in response to different
treatment plans. Panel (a) shows the history in the red box, which contains previous lung capacity
measurements (black dots) and previous treatments (green and blue bars). Panels (b) and (c) illustrate
the type of predictions we would like to make: given the history, what is the likely future trajectory
of lung capacity if we prescribe Drug B (green)? What if we prescribe two doses of drug A (blue)? A
physician might use these ?What if?? queries to decide that two doses of A is best.
Many authors have studied counterfactual models for discrete time series data. For instance, in health
care and epidemiology, Robins [1997] develops counterfactual models of a single outcome measured
after a sequence of actions in discrete time. Brodersen et al. [2015] build counterfactual models to
estimate the impact that a single, dicrete event has on a discrete time series of interest (e.g. daily sales
after a product launch). Others have modeled the effect of actions taken in continuous time on a single
outcome (e.g. Arjas and Parner 2004, Lok 2008). The CGP is unique in that it allows us to predict
how future trajectories in continuous-time will change in response to sequences of interventions.
Contributions. For problems in which learned predictive models are used to guide decisionmakers in choosing actions, this paper proposes the use of counterfactual objectives in lieu of
classical supervised learning objectives. We implement this idea in a challenging and frequently
occurring context; problems where outcomes are measured and actions are taken at discrete points in
continuous-time, and may be freely interwoven. Our key methodological contribution is an adjusted
maximum likelihood objective for Gaussian processes that allows us to learn counterfactual models
of continuous-time trajectories from observational traces: irregularly sampled sequences of actions
i
and outcomes denoted using D = {{(yij , aij , tij )}nj=1
}m
i=1 , where yij ? R ? {?}, aij ? C ? {?},
1
and tij ? [0, ? ]. We derive the technique by jointly modeling actions and outcomes using a marked
point process (MPP; see e.g., Daley and Vere-Jones 2007), where the GP models the conditional
distribution of the marks. When using potential outcomes, several assumptions are typically required
to show that the learned statistical model estimates the target counterfactuals. We describe one set of
assumptions sufficient for recovering the counterfactual GP from non-experimental data.
We report two sets of experimental results. First, we show that the CGP?s predictions are reliable;
they are stable to changes in certain characteristics of the training data that are not relevant to the
decision-making problem. Predictive models trained using classical supervised learning objectives,
however, are not stable to such perturbations. In this experiment, we use simulated data so that
(1) we can control characteristics of the training data, and (2) we have access to the ground truth
consequences of our actions on test data for evaluation. In our second experiment, we use data from a
real intensive care unit (ICU) to learn the CGP, and qualitatively demonstrate how its ability to answer
?What if?? questions offers medical decision-makers a powerful new tool for planning treatment.
1
yij and aij may be the null variable ? to allow for the possibility that an action is taken but no outcome is
observed and vice versa. [0, ? ] denotes a fixed period of time over which the trajectories are observed.
2
1.1
Related Work
The difference between counterfactual predictions of an outcome if an action had been taken and if it
had not been taken is defined as the causal effect of the action in the causal inference community (see
e.g., Pearl 2009 or Morgan and Winship 2014). Potential outcomes are commonly used to formalize
counterfactual predictions and obtain causal effect estimates [Neyman, 1923, 1990, Rubin, 1978]; we
will review them shortly. Potential outcomes are often applied to cross-sectional data (see examples in
Morgan and Winship 2014; recent examples from machine learning are Bottou et al. 2013, Johansson
et al. 2016), but have also been used to estimate the causal effect of a sequence of actions in discrete
time on a final outcome (e.g. Robins 1997, Taubman et al. 2009). Conversely, Brodersen et al. [2015]
estimate the effect that a single discrete intervention has on a discrete time series.
Recent work on optimal dynamic treatment regimes uses the sequential potential outcomes framework
proposed by Robins [1997] to learn lists of discrete-time treatment rules that optimize a scalar
outcome. Algorithms for learning these rules often use value-function approximations (Q-learning;
e.g., Nahum-Shani et al. 2012). Alternatively, A-learning directly learns the relative differences
between competing policies [Murphy, 2003].
Others have extended the potential outcomes framework in Robins [1997] to learn causal effects of
actions taken in continuous-time on a single final outcome using observational data. Lok [2008]
proposes an estimator based on structural nested models [Robins, 1992] that learns the instantaneous
effect of administering a single type of treatment. Arjas and Parner [2004] develop an alternative
framework for causal inference using Bayesian posterior predictive distributions to estimate the
effects of actions in continuous time on a final outcome. Xu et al. [2016] also learn effects of
actions in continuous-time on outcomes measured in continuous-time, but make one-step-ahead scalar
predictions instead of trajectory-valued predictions. Both Lok [2008] and Arjas and Parner [2004]
use marked point processes to formalize assumptions that make it possible to learn causal effects
from continuous-time observational data. We build on these ideas to learn causal effects of actions on
continuous-time trajectories instead of a single outcome. Cunningham et al. [2012] introduce the
Causal Gaussian Process, but their use of the term ?causal? is different from ours, and refers to a
constraint that holds for all samples drawn from the GP.
Causal effects in continuous-time have also been studied using differential equations. Mooij et al.
[2013] formalize an analog of Pearl?s ?do? operation for deterministic ordinary differential equations.
Sokol and Hansen [2014] make similar contributions for stochastic differential equations by studying
limits of discrete-time non-parametric structural equation models [Pearl, 2009].
Reinforcement learning (RL) algorithms learn from data where actions and observations are interleaved in discrete time (see e.g., Sutton and Barto 1998). In RL, however, the focus is on learning a
policy (a map from states to actions) that optimizes the expected reward, rather than a model that
predicts the effects of the agent?s actions on future observations. In model-based RL, a model of an
action?s effect on the subsequent state is produced as a by-product either offline before optimizing the
policy (e.g., Ng et al. 2006) or incrementally as the agent interacts with its environment. In most RL
problems, however, learning algorithms rely on active experimentation to collect samples. This is not
always possible, for example in health care we cannot actively experiment on patients, and so we
must rely on retrospective observational data.
In RL, a related problem known as off-policy evaluation also uses retrospective observational data (see
e.g., Dud?k et al. 2011, Jiang and Li 2016, P?aduraru et al. 2012). The goal is to use state-action-reward
sequences generated by an agent operating under an unknown policy to estimate the expected reward
of a target policy. Off-policy algorithms typically use value-function approximations, importance
reweighting, or doubly robust combinations of the two to estimate the expected reward.
2
Counterfactual Models from Observational Traces
Counterfactual GPs build on ideas from potential outcomes [Neyman, 1923, 1990, Rubin, 1978],
Gaussian processes [Rasmussen and Williams, 2006], and marked point processes [Daley and VereJones, 2007]. In the interest of space, we review potential outcomes and marked point processes, but
refer the interested reader to Rasmussen and Williams [2006] for background on GPs.
Background: Potential Outcomes. To formalize counterfactuals, we adopt the potential outcomes
framework [Neyman, 1923, 1990, Rubin, 1978], which uses a collection of random variables {Y [a] :
a ? C} to model the distribution over outcomes under each action a from a set of choices C. To make
3
counterfactual predictions, we must learn the distribution P (Y [a]) for each action a ? C. If we can
freely experiment by repeatedly taking actions and recording the effects, then it is straightforward to
learn a probabilistic model for each potential outcome Y [a]. Conducting experiments, however, may
not be possible. Alternatively, we can use observational data, where we have example actions A and
outcomes Y , but do not know how actions were chosen. Note the difference between the action a
and the random variable A that models the observed actions in our data. The notation Y [a] serves to
distinguish between the observed distribution P (Y | A = a) and the target distribution P (Y [a]).
In general, we can use observational data to estimate P (Y | A = a). Under two assumptions,
however, we can show that this conditional distribution is equivalent to the counterfactual model
P (Y [a]). The first is known as the Consistency Assumption.
Assumption 1 (Consistency). Let Y be the observed outcome, A ? C be the observed action, and
Y [a] be the potential outcome for action a ? C, then: ( Y , Y [a] ) | A = a.
Under consistency, we have that P (Y | A = a) = P (Y [a] | A = a). Now, the potential outcome
Y [a] may depend on the action A, so in general P (Y [a] | A = a) 6= P (Y [a]). The next assumption
posits that we have additional observed variables X known as confounders [Morgan and Winship,
2014] that are sufficient to d-separate Y [a] and A.
Assumption 2 (No Unmeasured Confounders (NUC)). Let Y be the observed outcome, A ? C be
the observed action, X = x be a vector containing potential confounders, and Y [a] be the potential
outcome under action a ? C, then: ( Y [a] ? A ) | X = x.
Under Assumptions 1 and 2, P (Y | A = a, X = x) = P (Y [a] | X = x). By marginalizing with
respect to P (X) we can estimate P (Y [a]). An extension of Assumption 2 introduced by Robins
[1997] known as sequential NUC allows us to estimate the effect of a sequence of actions in discrete
time on a single outcome. In continuous-time settings, where both the type and timing of actions may
be statistically dependent on the potential outcomes, Assumption 2 (and sequential NUC) cannot be
applied as-is. We will describe an alternative that serves a similar role for CGPs.
Background: Marked Point Processes. Point processes are distributions over sequences of timesN
tamps {Ti }i=1 , which we call points, and a marked point process (MPP) is a point process where
each point is annotated with an additional random variable Xi , called its mark. For example, a point
T might represent the arrival time of a customer, and X the amount that she spent at the store. We
emphasize that both the annotated points (Ti , Xi ) and the number of points N are random variables.
A point process can be characterized as a counting process {Nt : t ? 0} that counts the number of
PN
points that occured up to and including time t: Nt = i=1 I(Ti ?t) . By definition, this processes can
only take integer values, and Nt ? Ns if t ? s. In addition, it is commonly assumed that N0 = 0 and
that ?Nt = lim??0+ Nt ? Nt?? ? {0, 1}. We can parameterize a point process using a probabilistic
model of ?Nt given the history of the process Ht? up to but not including time t (we use t? to
denote the left limit of t). Using the Doob-Meyer decomposition [Daley and Vere-Jones, 2007], we
can write ?Nt = ?Mt + ??t , where Mt is a martingale, ?t is a cumulative intensity function, and
P (?Nt = 1 | Ht? ) = E [?Nt | Ht? ] = E [?Mt | Ht? ] + ??t (Ht? ) = 0 + ??t (Ht? ),
which shows that we can parameterize the point process using the conditional intensity function
?? (t) dt , ??t (Ht? ) The star superscript on the intensity function serves as a reminder that it
depends on the history Ht? . For example, in non-homogeneous Poisson processes ?? (t) is a function
of time that does not depend on the history. On the other hand, a Hawkes process is an example of
a point process where ?? (t) does depend on the history [Hawkes, 1971]. MPPs are defined by an
intensity that is a function of both the time t and the mark x: ?? (t, x) = ?? (t)p? (x | t). We have
written the joint intensity in a factored form, where ?? (t) is the intensity of any point occuring (that
is, the mark is unspecified), and p? (x | t) is the pdf of the observed mark given the point?s time. For
an MPP, the history Ht contains each prior point?s time and mark.
2.1
Counterfactual Gaussian Processes
Let {Yt : t ? [0, ? ]} denote a continuous-time stochastic process, where Yt ? R, and [0, ? ] defines
the interval over which the process is defined. We will assume that the process is observed at a
n
discrete set of irregular and random times {(yj , tj )}j=1 . We use C to denote the set of possible action
types, a ? C to denote the elements of the set, and define an action to be a 2-tuple (a, t) specifying
4
an action type a ? C and a time t ? [0, ? ] at which it is taken. To refer to multiple actions, we use
a = [(a1 , t1 ), . . . , (an , tn )]. Finally, we define the history Ht at a time t ? [0, ? ] to be a list of all
previous observations of the process and all previous actions. Our goal is to model the counterfactual:
m
P ({Ys [a] : s > t} | Ht ), where a = {(aj , tj ) : tj > t}j=1 .
(1)
n
i
To learn the counterfactual model, we will use traces D , {hi = {(tij , yij , aij )}j=1
}m
i=1 , where
yij ? R ? {?}, aij ? C ? {?}, and tij ? [0, ? ]. Our approach is to model D using a marked point
process (MPP), which we learn using the traces. Using Assumption 1 and two additional assumptions
defined below, the estimated MPP recovers the counterfactual model in Equation 1.
We define the MPP mark space as the Cartesian product of the outcome space R and the set of
action types C. To allow either the outcome or the action (but not both) to be the null variable ?, we
introduce binary random variables zy ? {0, 1} and za ? {0, 1} to indicate when the outcome y and
action a are not ?. Formally, the mark space is X = (R ? {?}) ? (C ? {?}) ? {0, 1} ? {0, 1}. We
can then write the MPP intensity as
?? (t, y, a, zy , za ) = ?? (t)p? (zy , za | t)
|
{z
}
[A] Event model
p? (y | t, zy )
|
{z
}
p? (a | y, t, za ),
|
{z
}
(2)
[B] Outcome model (GP) [C] Action model
where we have again used the ? superscript as a reminder that the hazard function and densities above
are implicitly conditioned on the history Ht? . The parameterization of the event and action models
can be chosen to reflect domain knowledge about how the timing of events and choice of action
depend on the history. The outcome model is parameterized using a GP (or any elaboration such as a
hierarchical GP, or mixture of GPs), and can be simply designed as a regression model that predicts
how the future trajectory will progress given the previous actions and outcome observations.
Learning. To learn the CGP, we maximize the likelihood of observational traces over a fixed
interval [0, ? ]. Let ? denote the model parameters, then the likelihood for a single trace is
Z ?
n
n
X
X
`(?) =
log p?? (yj | tj , zyj ) +
log ??? (t)p?? (aj , zyj , zaj | tj , yj ) ?
??? (s) ds. (3)
j=1
0
j=1
We assume that traces are independent, and so can learn from multiple traces by maximizing the
sum of the individual-trace log likelihoods with respect to ?. We refer to Equation 3 as the adjusted
maximum likelihood objective. We see that the first term fits the GP to the outcome data, and the
second term acts as an adjustment to account for dependencies between future outcomes and the
timing and types of actions that were observed in the training data.
Connection to target counterfactual. By maximizing Equation 3, we obtain a statistical model of
the observational traces D. In general, the statistical model may not recover the target counterfactual
model (Equation 1). To connect the CGP to Equation 1, we describe two additional assumptions. The
first assumption is an alternative to Assumption 2.
Assumption 3 (Continuous-Time NUC). For all times t and all histories Ht? , the densities ?? (t),
p? (zy , za | t), and p? (a | y, t, za ) do not depend on Ys [a] for all times s > t and all actions a.
The key implication of this assumption is that the policy used to choose actions in the observational
data did not depend on any unobserved information that is predictive of the future potential outcomes.
Assumption 4 (Non-Informative Measurement Times). For all times t and any history Ht? , the
following holds: p? (y | t, zy = 1) dy = P (Yt ? dy | Ht? ).
Under Assumptions 1, 3, and 4, we can show that Equation 1 is equivalent to the GP used to model
p? (y | t, zy = 1). In the interest of space, the argument for this equivalence is in Section A of the
supplement. Note that these assumptions are not statistically testable (see e.g., Pearl 2009).
3
Experiments
Counterfactual models that answer ?What if?? questions are more reliable decision support tools
than predictive models trained using classical supervised learning objectives because counterfactual
models are, by construction, invariant to the policy used to choose actions in the training data. In our
first experiments, we use simulated data so that (1) we can control characteristics of the training data,
and (2) we have access to the ground truth consequences of our actions on test data for evaluation. In
our second experiment, we use data from a real ICU to learn the CGP, and qualitatively demonstrate
how its ability to answer ?What if?? questions has the potential to give medical decision-makers a
powerful new tool for planning treatment.
5
Regime A
Baseline GP
CGP
Risk Score ? from A
Kendall?s ? from A
AUC
0.000
1.000
0.853
Regime B
Baseline GP
CGP
0.000
1.000
0.872
0.083
0.857
0.832
0.001
0.998
0.872
Regime C
Baseline GP
CGP
0.162
0.640
0.806
0.128
0.562
0.829
Table 1: Results measuring reliability for simulated data experiments. See Section 3.1 for details.
3.1
Reliable Decision-making with CGPs
We focus on a decision-making problem where the goal is to decide whether or not to treat a patient.
The decision-maker (a clinician), should treat a patient if the value of a severity marker is likely to fall
below a threshold in the future. A common approach to solving this problem is to build a predictive
model of the future outcome, and use the prediction to make the decision (i.e. if the predicted value is
below the threshold, then treat). We will use this approach as our baseline. If we could look into the
future and see what the marker value would be under different actions, then the best decision would
be to treat if the severity marker will fall below the threshold without treatment. It is precisely this
type of reasoning that we can perform with the CGP.
We simulate the value of a severity marker recorded over a period of 24 hours in the hospital; high
values indicate that the patient is healthy. For the baseline approach, we learn a GP that predicts the
future trajectory given clinical history up until time t; i.e. P ({Ys : s > t} | Ht ). Using the CGP,
we model the counterfactual ?What if we do not treat this patient??; i.e. P ({Ys [?] : s > t} | Ht ).
For all experiments, we consider a single decision time t = 12hrs. We create a risk score using the
negative of each model?s predicted value at the end of 24 hours, normalized to lie in [0, 1].
Data. We simulate data from three regimes. In regimes A and B, we simulate severity marker
trajectories that are treated by policies ?A and ?B respectively, that are both unknown to the baseline
model and CGP at train time. Both ?A and ?B are designed to satisfy Assumptions 1, 3, and 4. In
regime C, we use a policy that does not satisfy these assumptions. This regime will demonstrate the
importance of verifying whether the assumptions hold when applying the CGP. We train both the
baseline model and CGP on data simulated from all three regimes. In all regimes, we test decisions
on a common set of trajectories treated up until t = 12hrs with policy ?A .
Simulator. For each patient, we randomly sample outcome measurement times from a homogeneous Poisson process with with constant intensity ? over the 24 hour period. Given the measurement
times, outcomes are sampled from a mixture of three GPs. The covariance function is shared between
all classes, and is defined using a Mat?rn 3/2 kernel (variance 0.22 , lengthscale 8.0) and independent
Gaussian noise (scale 0.1) added to each observation. Each class has a distinct mean function
parameterized using a 5-dimensional, order-3 B-spline. The first class has a declining mean trajectory,
the second has a trajectory that declines then stabilizes, and the third has a stable trajectory.2 All
classes are equally likely a priori. At each measurement time, the treatment policy ? determines
a probability p of treatment administration (we use only a single treatment type). The treatments
increase the severity marker by a constant amount for 2 hours. If two or more actions occur within
2 hours of one another, the effects do not add up (i.e. it is as though only one treatment is active).
Additional details about the simulator and policies can be found in the supplement.
Model. For both the baseline predictive model and CGP outcome model, we use a mixture of
three GPs (as was used to simulate the data). We assume that the mean function coefficients, the
covariance parameters, and the treatment effect size are unknown and must be learned. We emphasize
that both the CGP and RGP have identical forms, but are trained using different objectives; the
RGP marginalizes over future actions, inducing an implicit dependence on the treatment policy in
the training data, while the CGP explicitly controls for them while learning. For both the baseline
model and CGP, we analytically sum over the mixture component likelihoods to obtain a closed
form expression for the likelihood, which we optimize using BFGS [Nocedal and Wright, 2006].3
Predictions for both models are made using the posterior predictive mean given data and interventions
up until 12 hours.
Results. When trained on data where actions were taken according to different policies, the baseline
model produces different risk scores for the same patient. In Table 1, the first row shows the average
2
3
The exact B-spline coefficients can be found in the simulation code included in the supplement.
Additional details can be found in the supplement.
6
Creatinine
Hours Since ICU Admission
Figure 2: Example factual (grey) and counterfactual (blue) predictions on real ICU data using the CGP.
difference in risk scores (calibrated to lie in [0, 1]) produced by the models trained in each regime and
produced by the models trained in regime A. In row 1, column B we see that the baseline GP?s risk
scores differ for the same person on average by around eight points (? = 0.083). From the perspective
of a decision-maker, this behavior could make the system appear less reliable. Intuitively, the risk for
a given patient should not depend on the policy used to determine treatments in retrospective data. On
the other hand, the CGP?s scores change very little when trained on different regimes (? = 0.001), as
long as Assumptions 1, 3, and 4 are satisfied. Note, however, that the scores do change for the CGP
in row 1, column C where the policy ?C does not satisfy these assumptions (? = 0.128). Although
we illustrate stability of the CGP comapred to the baseline GP using two regimes, this property is not
specific to the choice of policies used in regimes A and B. Rather, the issue persists as we generate
different training data by varying the distribution over the action choices.
A cynical reader might ask: even if the risk scores are unstable, perhaps it has no consequences on
the downstream decision-making task? In the second row of Table 1, we report Kendall?s ? computed
between each regime and regime A using the risk scores to rank the patient?s in the test data according
to severity (i.e. scores closer to 1 are more severe). In the third row, we report the AUC for both
models trained in each regime on the common test set. We label a patient as ?at risk? if the last
marker value in the untreated trajectory is below zero, and ?not at risk? otherwise.
In row 2, column B we see that the CGP has a high rank correlation (? = 0.998) between the two
regimes where the policies satisfy our key assumptions. The baseline GP model trained on regime
B, however, has a lower rank correlation of ? = 0.857 with the risk scores produced by the same
model trained on regime A. Similarly, in row three, columns A and B, we see that the CGP?s AUC
is unchanged (AUC = 0.872). The baseline GP, however, is unstable and creates a risk score with
poorer discrimination in regime B (AUC = 0.832) than in regime A (AUC = 0.853). Finally, we
see that in column C, where the policy ?C does not satisfy our key assumptions, the CGP?s rank
correlation degrades (? = 0.562), and the AUC degrades to 0.829 (note that the baseline GP?s rank
correlation and AUC also degrade). This further emphasizes the importance of verifying Assumptions
1, 3, and 4 when using the CGP.
These results have broad consequences for the practice of building predictive models from observational data for decision support. Classical supervised learning is commonly used to build predictive
models for assessing risk from non-experimental data. Our experiments show that the predictions
these models make and the decisions that they may lead to are highly dependent on action policies
in the training data. Intuitively, this is troubling because we do not expect that past action policies
should have any impact on the assessment of risk. As predictive models are becoming more widely
used in domains like health care (e.g., Li-wei et al. 2015, Schulam and Saria 2015, Alaa et al. 2016,
Wiens et al. 2016, Cheng et al. 2017) where safety is critical, the framework proposed here is increasingly pertinent. Others have noted this issue and studied the impact that action policies in the
training data have on models fit using supervised learning objectives (e.g., Dyagilev and Saria 2016).
Counterfactual GPs (and counterfactual models more broadly) make predictions that are independent
of policies in the training data, and offer a new more reliable way to train predictive models for
decision support.
3.2
CGPs for Medical Decision Support
CGPs offer a powerful new tool for decision-makers to evaluate different actions using data-driven
models. In health care, for instance, we can move closer towards the vision of evidence-based
medicine by allowing clinicians to answer ?What if?? questions at the point of care with predictions
tailored to the patient?s clinical history. When treatments are expensive or have potent side effects,
the CGP can estimate the effects, which can be weighed against the cost of its administration.
7
To demonstrate how the CGP can be used as a decision support tool, we extract observational
creatinine traces from the publicly available MIMIC-II database [Saeed et al., 2011]. Creatinine is a
compound produced as a by-product of the chemical reaction in the body that breaks down creatine
to fuel muscles. Healthy kidneys normally filter creatinine out of the body, which can otherwise be
toxic in large concentrations. During kidney failure, however, creatinine levels rise and the compound
must be extracted using a medical procedure called dialysis.
We extract patients in the database who tested positive for abnormal creatinine levels, which is a sign
of kidney failure. We also extract the times at which three different types of dialysis were given to
each individual: intermittent hemodialysis (IHD), continuous veno-venous hemofiltration (CVVH),
and continuous veno-venous hemodialysis (CVVHD). The data set includes a total of 428 individuals,
with an average of 34 (?12) creatinine observations each. We shuffle the data and use 300 traces for
training, 50 for validation and model selection, and 78 for testing.
Model. We parameterize the outcome model of the CGP using a mixture of GPs. We always
condition on the initial creatinine measurement and model the deviation from that initial value.
The mean for each class is zero (i.e. there is no deviation from the initial value on average). We
parameterize the covariance function using the sum of two non-stationary kernel functions. Let
? : t ? [1, t, t2 ]> ? R3 denote the quadratic polynomial basis, then the first kernel is k1 (t1 , t2 ) =
?> (t1 )??(t2 ), where ? ? R3?3 is a positive-definite symmetric matrix parameterizing the kernel.
The second kernel is the covariance function of the integrated Ornstein-Uhlenbeck (IOU) process
(see e.g., Taylor et al. 1994), which is parameterized by two scalars ? and ? and defined as
?2
kIOU (t1 , t2 ) = 2?
2?min(t1 , t2 ) + e??t1 + e??t2 ? 1 ? e??|t1 ?t2 | .
3
The IOU covariance corresponds to the random trajectory of a particle whose velocity drifts according
to an OU process. We assume that each creatinine measurement is observed with independent
Gaussian noise with scale ?. Each class in the mixture has a unique set of covariance parameters.
To model the treatment effects in the outcome model, we define a short-term function and longterm response function. If an action is taken at time t0 , the outcome ? = t ? t0 hours later will
be additively affected by the response function g(?; h1 , a, b, h2 , r) = gs (?; h1 , a, b) + g` (?; h2 , r),
where h1 , h2 ? R and a, b, r ? R+ . The
short-term and long-term response functions are defined
h1 a
as gs (?; h1 , a, b) = a?b
e?b?t ? e?a?t , and g` (? : h2 , r) = h2 ? (1.0 ? e?r?t ). The two response
functions are included in the mean function of the GP, and each class in the mixture has a unique set
of response function parameters. We assume that Assumptions 1, 3, and 4 hold, and that the event
and action models have separate parameters, so can remain unspecified when estimating the outcome
model. We fit the CGP outcome model using Equation 3, and select the number of classes in the
mixture using fit on the validation data (we choose three components).
Results. Figure 2 demonstrates how the CGP can be used for medical decision support. Each panel
in the figure shows data for an individual drawn from the test set. The green points show measurements
on which we condition to obtain a posterior distribution over mixture class membership and the
individual?s latent trajectory under each class. The red points are unobserved, future measurements.
In grey, we show predictions under the factual sequence of actions extracted from the MIMIC-II
database. Treatment times are shown using vertical bars marked with an ?x? (color indicates which
type of treatment was given). In blue, we show the CGP?s counterfactual predictions under an
alternative sequence of actions. The posterior predictive trajectory is shown for the MAP mixture
class (mean is shown by a solid grey/blue line, 95% credible intervals are shaded).
We qualitatively discuss the CGP?s counterfactual predictions, but cannot quantitatively evaluate them
without prospective experimental data from the ICU. We can, however, measure fit on the factual
data and compare to baselines to evaluate our modeling decisions. Our CGP?s outcome model allows
for heterogeneity in the covariance parameters and the response functions. We compare this choice
to two alternatives. The first is a mixture of three GPs that does not model treatment effects. The
second is a single GP that does model treatment effects. Over a 24-hour horizon, the CGP?s mean
absolute error (MAE) is 0.39 (95% CI: 0.38-0.40),4 , and for predictions between 24 and 48 hours
in the future the MAE is 0.62 (95% CI: 0.60-0.64). The pairwise mean difference between the first
baseline?s absolute errors and the CGP?s is 0.07 (0.06, 0.08) for 24 hours, and 0.09 (0.08, 0.10) for
24-48 hours. The mean difference between the second baseline?s absolute errors and the CGP?s is
0.04 (0.04, 0.05) for 24 hours and 0.03 (0.02, 0.04) for 24-48 hours. The improvements over the
4
95% confidence intervals computed using the pivotal bootstrap are shown in parentheses
8
baselines suggest that modeling treatments and heterogeneity with a mixture of GPs for the outcome
model are useful for this problem.
Figure 2 shows factual and counterfactual predictions made by the CGP. In the first (left-most)
panel, the patient is factually administered IHD about once a day, and is responsive to the treatment
(creatinine steadily improves). We query the CGP to estimate how the individual would have
responded had the IHD treatment been stopped early. The model reasonably predicts that we would
have seen no further improvement in creatinine. In the third panel, an individual with erratic creatinine
levels receives CVVHD for the last 100 hours and is responsive to the treatment. As before, the CGP
counterfactually predicts that she would not have improved had CVVHD not been given. Interestingly,
panel four shows the opposite situation: the individual did not receive treatment and did not improve
for the last 100 hours, but the CGP counterfactually predicts an improvement in creatinine similar to
that in panel 3 if daily CVVHD had been administered.
4
Discussion
Our key message is that predictive models used in decision-making problems should be trained
using counterfactual objectives (like the one shown in Equation 3). One reason this approach should
be preferred is because the models produced are stable to information in the training data that is
irrelevant to the downstream decision-making task (the action policies in the training data). We
studied this idea in the context of problems where outcomes are measured and actions are taken
at discrete points in continuous-time, and proposed the counterfactual Gaussian process (CGP) to
model the effects of sequences of actions taken on continuous-time trajectories (time series). The
CGP builds on previous ideas in continuous-time causal inference (e.g., Robins 1997, Arjas and
Parner 2004, Lok 2008), but is unique in that it can predict the full counterfactual trajectory; we
combined marked point processes (MPP) with GPs to model observational traces, and described
three assumptions that are sufficient to connect the statistical model to the target counterfactuals.
We presented two sets of experimental results. In the first, we used simulated data to show risk score
models fit using classical supervised learning objectives are sensitive to the action policies used in
the training data; information that is irrelevant to the downstream decision problem. We showed that
this sensitivity can alter risk assessments for the same individual, change relative risk assessments
across individuals, and can cause differences in AUC. On the other hand, the CGP is not sensitive
to the action policies in the training data, as long as they satisfy Assumptions 1, 3, and 4. In the
second set of experiments, we demonstrated how the CGP offers a powerful new tool for medical
decision support by learning the effects of dialysis on creatinine trajectories from real ICU data and
demonstrating the types of ?What if?? questions that it can be used to answer about patient prognosis
under various treatment plans.
These results suggest a number of new questions and directions for future work. First, the validity
of the CGP is conditioned upon a set of assumptions (this is true for all counterfactual models). In
general, these assumptions are not testable. The reliability of approaches using counterfactual models
therefore critically depends on the plausibility of those assumptions in light of domain knowledge.
Formal procedures, such as sensitivity analyses (e.g., Robins et al. 2000, Scharfstein et al. 2014), that
can identify when causal assumptions conflict with a data set will help to make these methods more
easily applied in practice. In addition, there may be other sets of structural assumptions beyond those
presented that allow us to learn counterfactual GPs from non-experimental data. For instance, the
back door and front door criteria are two separate sets of structural assumptions discussed by Pearl
[2009] in the context of estimating parameters of causal Bayesian networks from observational data.
More broadly, there are implications for recent pushes to introduce transparency, interpretability,
and accountability into machine learning systems embedded in decision-making processes. We have
characterized a notion of model stability relative to information in the training data that is not relevant
to its downstream application, and showed that models fit using counterfactual objectives achieve
stability. The framework can be further extended to incorporate recent ideas on model interpretability
and accountability in the context of supervised learning objectives (e.g., Caruana et al. 2015, Ribeiro
et al. 2016).
Acknowledgements
We thank the anonymous reviewers for their insightful feedback. This work was supported by
generous funding from DARPA YFA #D17AP00014 and NSF SCH #1418590. PS was also supported
by an NSF Graduate Research Fellowship. We thank Katie Henry and Andong Zhan for help with
the ICU data set.
9
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11
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6,376 | 6,768 | QSGD: Communication-Efficient SGD
via Gradient Quantization and Encoding
Dan Alistarh
IST Austria & ETH Zurich
[email protected]
Demjan Grubic
ETH Zurich & Google
[email protected]
Ryota Tomioka
Microsoft Research
[email protected]
Jerry Z. Li
MIT
[email protected]
Milan Vojnovic
London School of Economics
[email protected]
Abstract
Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to its excellent scalability properties. A fundamental
barrier when parallelizing SGD is the high bandwidth cost of communicating gradient updates between nodes; consequently, several lossy compresion heuristics have
been proposed, by which nodes only communicate quantized gradients. Although
effective in practice, these heuristics do not always converge.
In this paper, we propose Quantized SGD (QSGD), a family of compression
schemes with convergence guarantees and good practical performance. QSGD
allows the user to smoothly trade off communication bandwidth and convergence
time: nodes can adjust the number of bits sent per iteration, at the cost of possibly
higher variance. We show that this trade-off is inherent, in the sense that improving
it past some threshold would violate information-theoretic lower bounds. QSGD
guarantees convergence for convex and non-convex objectives, under asynchrony,
and can be extended to stochastic variance-reduced techniques.
When applied to training deep neural networks for image classification and automated speech recognition, QSGD leads to significant reductions in end-to-end
training time. For instance, on 16GPUs, we can train the ResNet-152 network to
full accuracy on ImageNet 1.8? faster than the full-precision variant.
1
Introduction
The surge of massive data has led to significant interest in distributed algorithms for scaling computations in the context of machine learning and optimization. In this context, much attention has
been devoted to scaling large-scale stochastic gradient descent (SGD) algorithms [33], which can be
briefly defined as follows. Let f : Rn ? R be a function which we want to minimize. We have access
to stochastic gradients ge such that E[e
g (x)] = ?f (x). A standard instance of SGD will converge
towards the minimum by iterating the procedure
xt+1 = xt ? ?t ge(xt ),
(1)
where xt is the current candidate, and ?t is a variable step-size parameter. Notably, this arises if
we are given i.i.d. data points X1 , . . . , Xm generated from an unknown distribution D, and a loss
function `(X, ?), which measures the loss of the model ? at data point X. We wish to find a model
?? which minimizes f (?) = EX?D [`(X, ?)], the expected loss to the data. This framework captures
many fundamental tasks, such as neural network training.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this paper, we focus on parallel SGD methods, which have received considerable attention recently
due to their high scalability [6, 8, 32, 13]. Specifically, we consider a setting where a large dataset is
partitioned among K processors, which collectively minimize a function f . Each processor maintains
a local copy of the parameter vector xt ; in each iteration, it obtains a new stochastic gradient update
(corresponding to its local data). Processors then broadcast their gradient updates to their peers, and
aggregate the gradients to compute the new iterate xt+1 .
In most current implementations of parallel SGD, in each iteration, each processor must communicate
its entire gradient update to all other processors. If the gradient vector is dense, each processor will
need to send and receive n floating-point numbers per iteration to/from each peer to communicate
the gradients and maintain the parameter vector x. In practical applications, communicating the
gradients in each iteration has been observed to be a significant performance bottleneck [35, 37, 8].
One popular way to reduce this cost has been to perform lossy compression of the gradients [11, 1,
3, 10, 41]. A simple implementation is to simply reduce precision of the representation, which has
been shown to converge under convexity and sparsity assumptions [10]. A more drastic quantization
technique is 1BitSGD [35, 37], which reduces each component of the gradient to just its sign
(one bit), scaled by the average over the coordinates of ge, accumulating errors locally. 1BitSGD
was experimentally observed to preserve convergence [35], under certain conditions; thanks to the
reduction in communication, it enabled state-of-the-art scaling of deep neural networks (DNNs) for
acoustic modelling [37]. However, it is currently not known if 1BitSGD provides any guarantees,
even under strong assumptions, and it is not clear if higher compression is achievable.
Contributions. Our focus is understanding the trade-offs between the communication cost of dataparallel SGD, and its convergence guarantees. We propose a family of algorithms allowing for lossy
compression of gradients called Quantized SGD (QSGD), by which processors can trade-off the
number of bits communicated per iteration with the variance added to the process.
QSGD is built on two algorithmic ideas. The first is an intuitive stochastic quantization scheme:
given the gradient vector at a processor, we quantize each component by randomized rounding to a
discrete set of values, in a principled way which preserves the statistical properties of the original.
The second step is an efficient lossless code for quantized gradients, which exploits their statistical
properties to generate efficient encodings. Our analysis gives tight bounds on the precision-variance
trade-off induced by QSGD.
?
At one extreme of this trade-off, we can guarantee that each processor transmits
? at most n(log n +
O(1)) expected bits per iteration, while increasing variance by at most a n multiplicative factor.
At the other extreme, we show that each processor can transmit ? 2.8n + 32 bits per iteration in
expectation, while increasing variance by a only a factor of 2. In particular, in the latter regime,
compared to full precision SGD, we use ? 2.8n bits of communication per iteration as opposed to
32n bits, and guarantee at most 2? more iterations, leading to bandwidth savings of ? 5.7?.
QSGD is fairly general: it can also be shown to converge, under assumptions, to local minima for nonconvex objectives, as well as under asynchronous iterations. One non-trivial extension we develop
is a stochastic variance-reduced [23] variant of QSGD, called QSVRG, which has exponential
convergence rate.
One key question is whether QSGD?s compression-variance trade-off is inherent: for instance, does
any algorithm guaranteeing at most constant variance blowup need to transmit ?(n) bits per iteration?
The answer is positive: improving asymptotically upon this trade-off would break the communication
complexity lower bound of distributed mean estimation (see [44, Proposition 2] and [38]).
Experiments. The crucial question is whether, in practice, QSGD can reduce communication cost
by enough to offset the overhead of any additional iterations to convergence. The answer is yes.
We explore the practicality of QSGD on a variety of state-of-the-art datasets and machine learning
models: we examine its performance in training networks for image classification tasks (AlexNet,
Inception, ResNet, and VGG) on the ImageNet [12] and CIFAR-10 [25] datasets, as well as on
LSTMs [19] for speech recognition. We implement QSGD in Microsoft CNTK [3].
Experiments show that all these models can significantly benefit from reduced communication when
doing multi-GPU training, with virtually no accuracy loss, and under standard parameters. For example, when training AlexNet on 16 GPUs with standard parameters, the reduction in communication
time is 4?, and the reduction in training to the network?s top accuracy is 2.5?. When training an
LSTM on two GPUs, the reduction in communication time is 6.8?, while the reduction in training
2
time to the same target accuracy is 2.7?. Further, even computationally-heavy architectures such as
Inception and ResNet can benefit from the reduction in communication: on 16GPUs, QSGD reduces
the end-to-end convergence time of ResNet152 by approximately 2?. Networks trained with QSGD
can converge to virtually the same accuracy as full-precision variants, and that gradient quantization
may even slightly improve accuracy in some settings.
Related Work. One line of related research studies the communication complexity of convex
optimization. In particular, [40] studied two-processor convex minimization in the same model,
provided a lower bound of ?(n(log n + log(1/))) bits on the communication cost of n-dimensional
convex problems, and proposed a non-stochastic algorithm for strongly convex problems, whose
communication cost is within a log factor of the lower bound. By contrast, our focus is on stochastic
gradient methods. Recent work [5] focused on round complexity lower bounds on the number of
communication rounds necessary for convex learning.
Buckwild! [10] was the first to consider the convergence guarantees of low-precision SGD. It gave
upper bounds on the error probability of SGD, assuming unbiased stochastic quantization, convexity,
and gradient sparsity, and showed significant speedup when solving convex problems on CPUs.
QSGD refines these results by focusing on the trade-off between communication and convergence.
We view quantization as an independent source of variance for SGD, which allows us to employ
standard convergence results [7]. The main differences from Buckwild! are that 1) we focus on the
variance-precision trade-off; 2) our results apply to the quantized non-convex case; 3) we validate
the practicality of our scheme on neural network training on GPUs. Concurrent work proposes
TernGrad [41], which starts from a similar stochastic quantization, but focuses on the case where
individual gradient components can have only three possible values. They show that significant
speedups can be achieved on TensorFlow [1], while maintaining accuracy within a few percentage
points relative to full precision. The main differences to our work are: 1) our implementation
guarantees convergence under standard assumptions; 2) we strive to provide a black-box compression
technique, with no additional hyperparameters to tune; 3) experimentally, QSGD maintains the same
accuracy within the same target number of epochs; for this, we allow gradients to have larger bit
width; 4) our experiments focus on the single-machine multi-GPU case.
We note that QSGD can be applied to solve the distributed mean estimation problem [38, 24] with an
optimal error-communication trade-off in some regimes. In contrast to the elegant random rotation
solution presented in [38], QSGD employs quantization and Elias coding. Our use case is different
from the federated learning application of [38, 24], and has the advantage of being more efficient to
compute on a GPU.
There is an extremely rich area studying algorithms and systems for efficient distributed large-scale
learning, e.g. [6, 11, 1, 3, 39, 32, 10, 21, 43]. Significant interest has recently been dedicated to
quantized frameworks, both for inference, e.g., [1, 17] and training [45, 35, 20, 37, 16, 10, 42]. In
this context, [35] proposed 1BitSGD, a heuristic for compressing gradients in SGD, inspired by
delta-sigma modulation [34]. It is implemented in Microsoft CNTK, and has a cost of n bits and two
floats per iteration. Variants of it were shown to perform well on large-scale Amazon datasets by [37].
Compared to 1BitSGD, QSGD can achieve asymptotically higher compression, provably converges
under standard assumptions, and shows superior practical performance in some cases.
2
Preliminaries
SGD has many variants, with different preconditions and guarantees. Our techniques are rather
portable, and can usually be applied in a black-box fashion on top of SGD. For conciseness, we will
focus on a basic SGD setup. The following assumptions are standard; see e.g. [7].
Let X ? Rn be a known convex set, and let f : X ? R be differentiable, convex, smooth, and
unknown. We assume repeated access to stochastic gradients of f , which on (possibly random) input
x, outputs a direction which is in expectation the correct direction to move in. Formally:
Definition 2.1. Fix f : X ? R. A stochastic gradient for f is a random function ge(x) so that
E[e
g (x)] = ?f (x). We say the stochastic gradient has second moment at most B if E[ke
g k22 ] ? B for
2
2
2
all x ? X . We say it has variance at most ? if E[ke
g (x) ? ?f (x)k2 ] ? ? for all x ? X .
Observe that any stochastic gradient with second moment bound B is automatically also a stochastic
gradient with variance bound ? 2 = B, since E[ke
g (x) ? ?f (x)k2 ] ? E[ke
g (x)k2 ] as long as
E[e
g (x)] = ?f (x). Second, in convex optimization, one often assumes a second moment bound
3
Data: Local copy of the parameter vector x
1 for each iteration t do
2
Let g
eti be an independent stochastic gradient ;
i
i
M ? Encode(e
g (x)) //encode gradients ;
3
broadcast M i to all peers;
for each peer ` do
receive M ` from peer `;
4
5
6
`
`
g
b ? Decode(M ) //decode gradients ;
7
8
9
10 end
Figure 1: An illustration of generalized
stochastic quantization with 5 levels.
end
P
xt+1 ? xt ? (?t /K) K
b` ;
`=1 g
Algorithm 1: Parallel SGD Algorithm.
when dealing with non-smooth convex optimization, and a variance bound when dealing with smooth
convex optimization. However, for us it will be convenient to consistently assume a second moment
bound. This does not seem to be a major distinction in theory or in practice [7].
Given access to stochastic gradients, and a starting point x0 , SGD builds iterates xt given by Equation
(1), projected onto X , where (?t )t?0 is a sequence of step sizes. In this setting, one can show:
Theorem 2.1 ([7], Theorem 6.3). Let X ? Rn be convex, and let f : X ? R be unknown, convex,
and L-smooth. Let x0 ? X be given, and let R2 = supx?X kx ? x0 k2 . Let T > 0 be fixed. Given
repeated, independent access to stochastic gradients with variance
bound ? 2 for f , SGD with initial
q
point x0 and constant step sizes ?t =
"
E f
T
1X
xt
T t=0
1
L+1/? ,
where ? =
R
?
!#
r
? min f (x) ? R
x?X
2
T
, achieves
LR2
2? 2
+
.
T
T
(2)
Minibatched SGD. A modification to the SGD scheme presented above often observed in practice
is a technique known as minibatching. In minibatched SGD, updates are of the form xt+1 =
e t (xt ) = 1 Pm get,i , and where each get,i is an independent stochastic
e t (xt )), where G
?X (xt ? ?t G
i=1
m
gradient for f at xt . It is not hard to see that if get,i are stochastic gradients with variance bound ? 2 ,
e t is a stochastic gradient with variance bound ? 2 /m. By inspection of Theorem 2.1, as
then the G
long as the first term in (2) dominates, minibatched SGD requires 1/m fewer iterations to converge.
Data-Parallel SGD. We consider synchronous data-parallel SGD, modelling real-world multi-GPU
systems, and focus on the communication cost of SGD in this setting. We have a set of K processors
p1 , p2 , . . . , pK who proceed in synchronous steps, and communicate using point-to-point messages.
Each processor maintains a local copy of a vector x of dimension n, representing the current estimate
of the minimizer, and has access to private, independent stochastic gradients for f .
In each synchronous iteration, described in Algorithm 1, each processor aggregates the value of x,
then obtains random gradient updates for each component of x, then communicates these updates
to all peers, and finally aggregates the received updates and applies them locally. Importantly, we
add encoding and decoding steps for the gradients before and after send/receive in lines 3 and 7,
respectively. In the following, whenever describing a variant of SGD, we assume the above general
pattern, and only specify the encode/decode functions. Notice that the decoding step does not
necessarily recover the original gradient ge` ; instead, we usually apply an approximate version.
When the encoding and decoding steps are the identity (i.e., no encoding / decoding), we shall refer
to this algorithm as parallel SGD. In this case, it is a simple calculation to see that at each processor,
if xt was the value of x that the processors held before iteration t, then the updated value of x by the
PK
end of this iteration is xt+1 = xt ? (?t /K) `=1 ge` (xt ), where each ge` is a stochatic gradient. In
particular, this update is merely a minibatched update of size K. Thus, by the discussion above, and
by rephrasing Theorem 2.1, we have the following corollary:
Corollary 2.2. Let X , f, L, x0 , and R be as in Theorem 2.1. Fix > 0. Suppose we run parallel
SGD on K processors, each with access to independent stochastic gradients with second moment
4
?
bound B, with step size ?t = 1/(L + K/?), where ? is as in Theorem 2.1. Then if
"
!#
T
1X
2B L
2
,
, then E f
xt
T = O R ? max
? min f (x) ? .
x?X
K2
T t=0
(3)
In most reasonable regimes, the first term of the max in (3) will dominate the number of iterations
necessary. Specifically, the number of iterations will depend linearly on the second moment bound B.
3
Quantized Stochastic Gradient Descent (QSGD)
In this section, we present our main results on stochastically quantized SGD. Throughout, log denotes
the base-2 logarithm, and the number of bits to represent a float is 32. For any vector v ? Rn , we
let kvk0 denote the number of nonzeros of v. For any string ? ? {0, 1}? , we will let |?| denote its
length. For any scalar x ? R, we let sgn (x) ? {?1, +1} denote its sign, with sgn (0) = 1.
3.1
Generalized Stochastic Quantization and Coding
Stochastic Quantization. We now consider a general, parametrizable lossy-compression scheme
for stochastic gradient vectors. The quantization function is denoted with Qs (v), where s ? 1 is
a tuning parameter, corresponding to the number of quantization levels we implement. Intuitively,
we define s uniformly distributed levels between 0 and 1, to which each value is quantized in a way
which preserves the value in expectation, and introduces minimal variance. Please see Figure 1.
For any v ? Rn with v 6= 0, Qs (v) is defined as
Qs (vi ) = kvk2 ? sgn (vi ) ? ?i (v, s) ,
(4)
where ?i (v, s)?s are independent random variables defined as follows. Let 0 ? ` < s be an
integer such that |vi |/kvk2 ? [`/s, (` + 1)/s]. That is, [`/s, (` + 1)/s] is the quantization interval
corresponding to |vi |/kvk2 . Then
(
|vi |
`/s
with probability 1 ? p kvk
,
s
;
2
?i (v, s) =
(` + 1)/s otherwise.
Here, p(a, s) = as ? ` for any a ? [0, 1]. If v = 0, then we define Q(v, s) = 0.
The distribution of ?i (v, s) has minimal variance over distributions with support {0, 1/s, . . . , 1}, and
its expectation satisfies E[?i (v, s)] = |vi |/kvk2 . Formally, we can show:
n
Lemma 3.1. For any
? vector v ? R , we have that (i) E[Qs (v)] = v (unbiasedness),?(ii) E[kQs (v)?
vk22 ] ? min(n/s2 , n/s)kvk22 (variance bound), and (iii) E[kQs (v)k0 ] ? s(s + n) (sparsity).
Efficient Coding of Gradients. Observe that for any vector v, the output of Qs (v) is naturally
expressible by a tuple (kvk2 , ?, ?), where ? is the vector of signs of the vi ?s and ? is the vector
of integer values s ? ?i (v, s). The key idea behind the coding scheme is that not all integer values
s ? ?i (v, s) can be equally likely: in particular, larger integers are less frequent. We will exploit this
via a specialized Elias integer encoding [14], presented in full in the full version of our paper [4].
Intuitively, for any positive integer k, its code, denoted Elias(k), starts from the binary representation
of k, to which it prepends the length of this representation. It then recursively encodes this prefix.
We show that for any positive integer k, the length of the resulting code has |Elias(k)| = log k +
log log k + . . . + 1 ? (1 + o(1)) log k + 1, and that encoding and decoding can be done efficiently.
Given a gradient vector represented as the triple (kvk2 , ?, ?), with s quantization levels, our coding
outputs a string S defined as follows. First, it uses 32 bits to encode kvk2 . It proceeds to encode
using Elias recursive coding the position of the first nonzero entry of ?. It then appends a bit denoting
?i and follows that with Elias(s ? ?i (v, s)). Iteratively, it proceeds to encode the distance from the
current coordinate of ? to the next nonzero, and encodes the ?i and ?i for that coordinate in the
same way. The decoding scheme is straightforward: we first read off 32 bits to construct kvk2 , then
iteratively use the decoding scheme for Elias recursive coding to read off the positions and values of
the nonzeros of ? and ?. The properties of the quantization and of the encoding imply the following.
Theorem 3.2. Let f : Rn ? R be fixed, and let x ? Rn be arbitrary. Fix s ? 2 quantization
levels. If ge(x) is a stochastic gradient for f at x with second moment bound B, then Qs (e
g (x)) is a
5
stochastic gradient for f at x with variance bound min
?
n
n
B.
s2 , s
Moreover, there is an encoding
scheme so that in expectation, the number of bits to communicate Qs (e
g (x)) is upper bounded by
2
?
3
2(s + n)
?
3+
s(s + n) + 32.
+ o(1) log
2
s(s + n)
Sparse
levels 0, 1, and ?1, the gradient density is
? Regime. For the case s = 1, i.e., quantization
?
O(?n), while the second-moment blowup is ? n. Intuitively, this means?that we will employ
O( n log n) bits per iteration, while the convergence time is increased by O( n).
?
Dense Regime. The variance blowup is minimized to at most 2 for s = n quantization levels; in
this case, we devise a more efficient encoding which yields an order of magnitude shorter codes
compared to the full-precision variant. The proof of this statement is not entirely obvious, as it
exploits both the statistical properties of the quantization and the guarantees of the Elias coding.
Corollary 3.3. Let f, x, and ge(x) be as in Theorem 3.2. There is an encoding scheme for Q?n (e
g (x))
which in expectation has length at most 2.8n + 32.
3.2
QSGD Guarantees
Putting the bounds on the communication and variance given above with the guarantees for SGD
algorithms on smooth, convex functions yield the following results:
Theorem 3.4 (Smooth Convex QSGD). Let X , f, L, x0 , and R be as in Theorem 2.1. Fix > 0.
Suppose we run parallel QSGD with s quantization levels on K processors accessing indepen?
dent stochastic gradients with second moment bound B, with stepsize ?t = 1/(L + K/?),
?
where ? is as in Theorem 2.1 with ? = B 0 , where B 0 = min sn2 , sn B. Then if T =
0
h P
i
T
L
2B
O R2 ? max K
, then E f T1 t=0 xt ? minx?X f (x) ? . Moreover, QSGD re2,
2
?
+n)
?
quires 3 + 23 + o(1) log 2(s
(s2 + n) + 32 bits of communication per round. In the
2
s + n
?
special case when s = n, this can be reduced to 2.8n + 32.
QSGD is quite portable, and can be applied to almost any stochastic gradient method. For illustration,
we can use quantization along with [15] to get communication-efficient non-convex SGD.
Theorem 3.5 (QSGD for smooth non-convex optimization). Let f : Rn ? R be a L-smooth
(possibly nonconvex) function, and let x1 be an arbitrary initial point. Let T > 0 be fixed, and
s > 0. Then there is a random stopping time R supported on {1, . . . , N } so that QSGD with
quantization level s, constant stepsizes ? = O(1/L) and access
? to stochastic gradients?of f with
L(f (x1 )?f ? )
min(n/s2 , n/s)B
1
second moment bound B satisfies L E k?f (x)k22 ? O
+
.
N
L
Moreover, the communication cost is the same as in Theorem 3.4.
3.3
Quantized Variance-Reduced SGD
Assume we are given K processors, and a parameter m > 0, where each processor i P
has access to
m
1
functions {fim/K , . . . , f(i+1)m/K?1 }. The goal is to approximately minimize f = m
i=1 fi . For
P
P
(i+1)m/K?1
K
1
processor i, let hi = m j=im/K
fi be the portion of f that it knows, so that f = i=1 hi .
A natural question is whether we can apply stochastic quantization to reduce communication for
parallel SVRG. Upon inspection, we notice that the resulting update will break standard SVRG. We
resolve this technical issue, proving one can quantize SVRG updates using our techniques and still
obtain the same convergence bounds.
?
e
= Q(v, n), where Q(v, s) is defined as in Section 3.1. Given
Algorithm Description. Let Q(v)
arbitrary starting point x0 , we let y (1) = x0 . At the beginning of epoch p, each processor broadcasts ?hi (y (p)
P),mthat is, the unquantized full gradient, from which the processors each aggregate
?f (y (p) ) = i=1 ?hi (y (p) ). Within each epoch, for each iteration t = 1, . . . , T , and for each
(p)
processor i = 1, . . . , K, we let ji,t be a uniformly random integer from [m] completely independent
from everything
else. Then, in iteration t in epochp, processor i broadcasts the update vector
(p)
(p)
e
ut,i = Q ?fj (p) (xt ) ? ?fj (p) (y (p) ) + ?f (y (p) ) .
i,t
i,t
6
Table 1: Description of networks, final top-1 accuracy, as well as end-to-end training speedup on 8GPUs.
Network
AlexNet
ResNet152
ResNet50
ResNet110
BN-Inception
VGG19
LSTM
Dataset
ImageNet
ImageNet
ImageNet
CIFAR-10
ImageNet
ImageNet
AN4
Params.
62M
60M
25M
1M
11M
143M
13M
Init. Rate
0.07
1
1
0.1
3.6
0.1
0.5
Top-1 (32bit)
59.50%
77.0%
74.68%
93.86%
81.13%
Top-1 (QSGD)
60.05% (4bit)
76.74% (8bit)
74.76% (4bit)
94.19% (4bit)
81.15 % (4bit)
Speedup (8 GPUs)
2.05 ?
1.56 ?
1.26 ?
1.10 ?
1.16? (projected)
2.25? (projected)
2? (2 GPUs)
PK
(p)
(p)
(p)
(p)
1
Each processor then computes the total update ut = K
i=1 ut,i , and sets xt+1 = xt ? ?ut .
P
(p)
T
At the end of epoch p, each processor sets y (p+1) = T1 t=1 xt . We can prove the following.
P
m
1
Theorem 3.6. Let f (x) = m
i=1 fi (x), where f is `-strongly convex, and fi are convex and
?
L-smooth, for all i. Let x be the unique minimizer of f over Rn . Then, if ? = O(1/L)
and T = O(L/`), then QSVRG with initial point y (1) ensures E f (y (p+1) ) ? f (x? ) ?
0.9p f (y (1) ) ? f (x? ) , for any epoch p ? 1. Moreover, QSVRG with T iterations per epoch
requires ? (F + 2.8n)(T + 1) + F n bits of communication per epoch.
Discussion. In particular, this allows us to largely decouple the dependence between F and the
condition number of f in the communication. Let ? = L/` denote the condition number of f . Observe
that whenever F ?, the second term is subsumed by the first and the per epoch communication
is dominated by (F + 2.8n)(T + 1). Specifically, for any fixed , to attain accuracy we must
take F = O(log 1/). As long as log 1/ ? ?(?), which is true for instance in the case when
? ? poly log(n) and ? poly(1/n), then the communication per epoch is O(?(log 1/ + n)).
Gradient Descent. The full version of the paper [4] contains an application of QSGD to gradient
descent. Roughly, in this case, QSGD can simply truncate the gradient to its top components, sorted
by magnitude.
4
QSGD Variants
Our experiments will stretch the theory, as we use deep networks, with non-convex objectives. (We
have also tested QSGD for convex objectives. Results closely follow the theory, and are therefore
omitted.) Our implementations will depart from the previous algorithm description as follows.
First, we notice that the we can control the variance the quantization by quantizing into buckets
of a fixed size d. If we view each gradient as a one-dimensional vector v, reshaping tensors if
necessary, a bucket will be defined as a set of d consecutive vector values. (E.g. the ith bucket is the
sub-vector v[(i ? 1)d + 1 : i ? d].) We will quantize each bucket independently, using QSGD. Setting
d = 1 corresponds to no quantization (vanilla SGD), and d = n corresponds to full quantization,
as described in the previous section. It is easy to see that, using bucketing, the guarantees from
Lemma 3.1 will be expressed in terms of d, as opposed to the full dimension n. This provides a
knob by which we can control variance, at the cost of storing an extra scaling factor on every d
bucket values. As an example, if we use a bucket ?
size of 512, and 4 bits, the variance increase
due to quantization will be upper bounded by only 512/24 ' 1.41. This provides a theoretical
justification for the similar convergence rates we observe in practice.
The second difference from the theory is that we will scale by the maximum value of the vector (as
opposed to the 2-norm). Intuitively, normalizing by the max preserves more values, and has slightly
higher accuracy for the same number of iterations. Both methods have the same baseline bandwidth
reduction because of lower bit width (e.g. 32 bits to 2 bits per dimension), but normalizing by the
max no longer provides any
? sparsity guarantees. We note that this does not affect our bounds in the
regime where we use ?( n) quantization levels per component, as we employ no sparsity in that
case. (However, we note that in practice max normalization also generates non-trivial sparsity.)
5
Experiments
Setup. We performed experiments on Amazon EC2 p2.16xlarge instances, with 16 NVIDIA K80
GPUs. Instances have GPUDirect peer-to-peer communication, but do not currently support NVIDIA
7
2.3x
1.6x
3.5x
Figure 2: Breakdown of communication versus computation for various neural networks, on 2, 4, 8, 16 GPUs,
2.0
1.5
Training loss
> 2x faster
2bit QSGD (d=128)
4bit QSGD (d=8192)
8bit QSGD (d=8192)
SGD
1.0
0.5
0.00
(a) AlexNet Accuracy versus Time.
300
600
900
Time (sec)
1200
(b) LSTM error vs Time.
1500
Test accuracy (%)
for full 32-bit precision versus QSGD 4-bit. Each bar represents the total time for an epoch under standard
parameters. Epoch time is broken down into communication (bottom, solid) and computation (top, transparent).
Although epoch time diminishes as we parallelize, the proportion of communication increases.
80
70
60
50
40
30
20
10
00
1bitSGD*
32bit
QSGD 4bit
QSGD 8bit
20
40
60
Epoch
80
100
120
(c) ResNet50 Accuracy.
Figure 3: Accuracy numbers for different networks. Light blue lines represent 32-bit accuracy.
NCCL extensions. We have implemented QSGD on GPUs using the Microsoft Cognitive Toolkit
(CNTK) [3]. This package provides efficient (MPI-based) GPU-to-GPU communication, and implements an optimized version of 1bit-SGD [35]. Our code is released as open-source [31].
We execute two types of tasks: image classification on ILSVRC 2015 (ImageNet) [12], CIFAR10 [25], and MNIST [27], and speech recognition on the CMU AN4 dataset [2]. For vision, we
experimented with AlexNet [26], VGG [36], ResNet [18], and Inception with Batch Normalization [22] deep networks. For speech, we trained an LSTM network [19]. See Table 1 for details.
Protocol. Our methodology emphasizes zero error tolerance, in the sense that we always aim to
preserve the accuracy of the networks trained. We used standard sizes for the networks, with hyperparameters optimized for the 32bit precision variant. (Unless otherwise stated, we use the default
networks and hyper-parameters optimized for full-precision CNTK 2.0.) We increased batch size
when necessary to balance communication and computation for larger GPU counts, but never past the
point where we lose accuracy. We employed double buffering [35] to perform communication and
quantization concurrently with the computation. Quantization usually benefits from lowering learning
rates; yet, we always run the 32bit learning rate, and decrease bucket size to reduce variance. We will
not quantize small gradient matrices (< 10K elements), since the computational cost of quantizing
them significantly exceeds the reduction in communication. However, in all experiments, more than
99% of all parameters are transmitted in quantized form. We reshape matrices to fit bucket sizes, so
that no receptive field is split across two buckets.
Communication vs. Computation. In the first set of experiments, we examine the ratio between
computation and communication costs during training, for increased parallelism. The image classification networks are trained on ImageNet, while LSTM is trained on AN4. We examine the cost
breakdown for these networks over a pass over the dataset (epoch). Figure 2 gives the results for
various networks for image classification. The variance of epoch times is practically negligible (<1%),
hence we omit confidence intervals.
Figure 2 leads to some interesting observations. First, based on the ratio of communication to
computation, we can roughly split networks into communication-intensive (AlexNet, VGG, LSTM),
and computation-intensive (Inception, ResNet). For both network types, the relative impact of
communication increases significantly as we increase the number of GPUs. Examining the breakdown
for the 32-bit version, all networks could significantly benefit from reduced communication. For
8
example, for AlexNet on 16 GPUs with batch size 1024, more than 80% of training time is spent on
communication, whereas for LSTM on 2 GPUs with batch size 256, the ratio is 71%. (These ratios
can be slightly changed by increasing batch size, but this can decrease accuracy, see e.g. [21].)
Next, we examine the impact of QSGD on communication and overall training time. (Communication
time includes time spent compressing and uncompressing gradients.) We measured QSGD with
2-bit quantization and 128 bucket size, and 4-bit and 8-bit quantization with 512 bucket size. The
results for these two variants are similar, since the different bucket sizes mean that the 4bit version
only sends 77% more data than the 2-bit version (but ? 8? less than 32-bit). These bucket sizes are
chosen to ensure good convergence, but are not carefully tuned.
On 16GPU AlexNet with batch size 1024, 4-bit QSGD reduces communication time by 4?, and
overall epoch time by 2.5?. On LSTM, it reduces communication time by 6.8?, and overall epoch
time by 2.7?. Runtime improvements are non-trivial for all architectures we considered.
Accuracy. We now examine how QSGD influences accuracy and convergence rate. We ran AlexNet
and ResNet to full convergence on ImageNet, LSTM on AN4, ResNet110 on CIFAR-10, as well as
a two-layer perceptron on MNIST. Results are given in Figure 3, and exact numbers are given in
Table 1. QSGD tests are performed on an 8GPU setup, and are compared against the best known
full-precision accuracy of the networks. In general, we notice that 4bit or 8bit gradient quantization
is sufficient to recover or even slightly improve full accuracy, while ensuring non-trivial speedup.
Across all our experiments, 8-bit gradients with 512 bucket size have been sufficient to recover or
improve upon the full-precision accuracy. Our results are consistent with recent work [30] noting
benefits of adding noise to gradients when training deep networks. Thus, quantization can be seen
as a source of zero-mean noise, which happens to render communication more efficient. At the
same time, we note that more aggressive quantization can hurt accuracy. In particular, 4-bit QSGD
with 8192 bucket size (not shown) loses 0.57% for top-5 accuracy, and 0.68% for top-1, versus full
precision on AlexNet when trained for the same number of epochs. Also, QSGD with 2-bit and 64
bucket size has gap 1.73% for top-1, and 1.18% for top-1.
One issue we examined in more detail is which layers are more sensitive to quantization. It appears
that quantizing convolutional layers too aggressively (e.g., 2-bit precision) can lead to accuracy loss
if trained for the same period of time as the full precision variant. However, increasing precision to
4-bit or 8-bit recovers accuracy. This finding suggests that modern architectures for vision tasks, such
as ResNet or Inception, which are almost entirely convolutional, may benefit less from quantization
than recurrent deep networks such as LSTMs.
Additional Experiments. The full version of the paper contains additional experiments, including a
full comparison with 1BitSGD. In brief, QSGD outperforms or matches the performance and final
accuracy of 1BitSGD for the networks and parameter values we consider.
6
Conclusions and Future Work
We have presented QSGD, a family of SGD algorithms which allow a smooth trade off between
the amount of communication per iteration and the running time. Experiments suggest that QSGD
is highly competitive with the full-precision variant on a variety of tasks. There are a number of
optimizations we did not explore. The most significant is leveraging the sparsity created by QSGD.
Current implementations of MPI do not provide support for sparse types, but we plan to explore
such support in future work. Further, we plan to examine the potential of QSGD in larger-scale
applications, such as super-computing. On the theoretical side, it is interesting to consider applications
of quantization beyond SGD.
The full version of this paper [4] contains complete proofs, as well as additional applications.
7
Acknowledgments
The authors would like to thank Martin Jaggi, Ce Zhang, Frank Seide and the CNTK team for their
support during the development of this project, as well as the anonymous NIPS reviewers for their
careful consideration and excellent suggestions. Dan Alistarh was supported by a Swiss National
Fund Ambizione Fellowship. Jerry Li was supported by the NSF CAREER Award CCF-1453261,
CCF-1565235, a Google Faculty Research Award, and an NSF Graduate Research Fellowship. This
work was developed in part while Dan Alistarh, Jerri Li and Milan Vojnovic were with Microsoft
Research Cambridge, UK.
9
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12
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6,377 | 6,769 | Convergent Block Coordinate Descent for Training
Tikhonov Regularized Deep Neural Networks
Ziming Zhang and Matthew Brand
Mitsubishi Electric Research Laboratories (MERL)
Cambridge, MA 02139-1955
{zzhang, brand}@merl.com
Abstract
By lifting the ReLU function into a higher dimensional space, we develop a smooth
multi-convex formulation for training feed-forward deep neural networks (DNNs).
This allows us to develop a block coordinate descent (BCD) training algorithm
consisting of a sequence of numerically well-behaved convex optimizations. Using
ideas from proximal point methods in convex analysis, we prove that this BCD
algorithm will converge globally to a stationary point with R-linear convergence
rate of order one. In experiments with the MNIST database, DNNs trained with
this BCD algorithm consistently yielded better test-set error rates than identical
DNN architectures trained via all the stochastic gradient descent (SGD) variants in
the Caffe toolbox.
1
Introduction
Feed-forward deep neural networks (DNNs) are function approximators wherein weighted combinations inputs are filtered through nonlinear activation functions that are organized into a cascade of
fully connected (FC) hidden layers. In recent years DNNs have become the tool of choice for many
research areas such as machine translation and computer vision.
The objective function for training a DNN is highly non-convex, leading to numerous obstacles
to global optimization [10], notably proliferation of saddle points [11] and prevalence of local
extrema that offer poor generalization off the training sample [8]. These observations have motivated
regularization schemes to smooth or simplify the energy surface, either explicitly such as weight
decay [23] or implicitly such as dropout [32] and batch normalization [19], so that the solutions are
more robust, i.e. better generalized to test data.
Training algorithms face many numerically difficulties that can make it difficult to even find a local
optimum. One of the well-known issues is so-called vanishing gradient in back propagation (chain
rule differentiation) [18], i.e. the long dependency chains between hidden layers (and corresponding
variables) tend to drive gradients to zero far from the optimum. This issue leads to very slow
improvements of the model parameters, an issue that becomes more and more serious in deeper
networks [16]. The vanishing gradient problem can be partially ameliorated by using non-saturating
activation functions such as rectified linear unit (ReLU) [25], and network architectures that have
shorter input-to-output paths such as ResNet [17]. The saddle-point problem has been addressed
by switching from deterministic gradient descent to stochastic gradient descent (SGD), which can
achieve weak convergence in probability [6]. Classic proximal-point optimization methods such as
the alternating direction method of multipliers (ADMM) have also shown promise for DNN training
[34; 41], but in the DNN setting their convergence properties remain unknown.
Contributions: In this paper,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1. We propose a novel Tikhonov regularized multi-convex formulation for deep learning, which
can be used to learn both dense and sparse DNNs;
2. We propose a novel block coordinate descent (BCD) based learning algorithm accordingly,
which can guarantee to globally converge to stationary points with R-linear convergence
rate of order one;
3. We demonstrate empirically that DNNs estimated with BCD can produce better representations than DNNs estimated with SGD, in the sense of yielding better test-set classification
rates.
Our Tikhonov regularization is motivated by the fact that the ReLU activation function is equivalent
to solving a smoothly penalized projection problem in a higher-dimensional Euclidean space. We use
this to build a Tikhonov regularization matrix which encodes all the information of the networks, i.e.
the architectures as well as their associated weights. In this way our training objective can be divided
into three sub-problems, namely, (1) Tikhonov regularized inverse problem [37], (2) least-square
regression, and (3) learning classifiers. Since each sub-problem is convex and coupled with the other
two, our overall objective is multi-convex.
Block coordinate descent (BCD) is often used for problems where finding an exact solution of a
sub-problem with respect to a subset (block) of variables is much simpler than finding the solution
for all variables simultaneously [27]. In our case, each sub-problem isolates block of variables which
can be solved easily (e.g. close-form solutions exist). One of the advantages of our decomposition
into sub-problems is that the long-range dependency between hidden layers is captured within a subproblem whose solution helps to propagate the information between inputs and outputs to stabilize
the networks (i.e. convergence). Therefore, it does not suffer from vanishing gradient at all. In our
experiments, we demonstrate the effectiveness and efficiency of our algorithm by comparing with
SGD based solvers.
1.1
Related Work
(1) Stochastic Regularization (SR) vs. Local Regularization vs. Tikhonov Regularization: SR
is a widely-used technique in deep learning to prevent the training from overfitting. The basic idea
in SR is to multiple the network weights with some random variables so that the learned network
is more robust and generalized to test data. Dropout [32] and its variants such like [22] are classic
examples of SR. Gal & Ghahramani [14] showed that SR in deep learning can be considered as
approximate variational inference in Bayesian neural networks.
Recently Baldassi et al. [2] proposed smoothing non-convex functions with local entropy, and latter
Chaudhari et al. [8] proposed Entropy-SGD for training DNNs. The idea behind such methods
is to locate solutions locally within large flat regions of the energy landscape that favors good
generalization. In [9] Chaudhari et al. provided the mathematical justification for these methods from
the perspective of partial differential equations (PDEs)
In contrast, our Tikhonov regularization tends to smooth the non-convex loss explicitly, globally, and
data-dependently. We deterministically learn the Tikhonov matrix as well as the auxiliary variables in
the ill-posed inverse problems. The Tikhonov matrix encodes all the information in the network, and
the auxiliary variables represent the ideal outputs of the data from each hidden layer that minimize
our objective. Conceptually these variables work similarly as target propagation [4].
(2) SGD vs. BCD: In [6] Bottou et al. proved weak convergence of SGD for non-convex optimization.
Ghadimi & Lan [15] showed that SGD can achieve convergence rates that scale as O t?1/2 for
non-convex loss functions if the stochastic gradient is unbiased with bounded variance, where t
denotes the number of iterations.
For non-convex optimization, the BCD based algorithm in [39] was proven to converge globally to
stationary points. For parallel computing another BCD based algorithm, namely Parallel Successive
Convex Approximation (PSCA), was proposed in [31] and proven to be convergent.
(3) ADMM vs. BCD: Alternating direction method of multipliers (ADMM) is a proximal-point
optimization framework from the 1970s and recently championed by Boyd [7]. It breaks a nearlyseparable problem into loosely-coupled smaller problems, some of which can be solved independently
and thus in parallel. ADMM offers linear convergence for strictly convex problems, and for certain
special non-convex optimization problems, ADMM can also converge [29; 36]. Unfortunately, thus
2
far there is no evidence or mathematical argument that DNN training is one of these special cases.
Therefore, even though empirically it has been successfully applied to DNN training [34; 41], it still
lacks of convergence guarantee.
Our BCD-based DNN training algorithm is also amenable to ADMM-like parallelization. More
importantly, as we prove in Sec. 4, it will converge globally to stationary points with R-linear
convergence.
2
Tikhonov Regularization for Deep Learning
2.1
Problem Setup
Key Notations: We denote xi ? Rd0 as the i-th training data, yi ? Y as its corresponding class label
from label set Y, ui,n ? Rdn as the output feature for xi from the n-th (1 ? n ? N ) hidden layer in
our network, Wn,m ? Rdn ?dm as the weight matrix between the n-th and m-th hidden layers, Mn
as the input layer index set for the n-th hidden layer, V ? RdN +1 ?dN as the weight matrix between
the last hidden layer and the output layer, U, V, W as nonempty closed convex sets, and `(?, ?) as a
convex loss function.
Network Architectures: In our networks we only consider ReLU as the activation functions. To
provide short paths through the DNN, we allow multi-input ReLU units which can take the outputs
from multiple previous layers as its inputs.
Fig. 1 illustrates a network architecture that we consider, where
the third hidden layers (with ReLU activations), for instance,
takes the input data and the outputs from the first and second
hidden layers as its inputs. Mathematically, we define our
multi-input ReLU function at layer n for data xi as:
input
hidden layers
output
xi , P
if n = 0
ui,n =
(1) Figure 1: Illustration of DNN architecmax 0, m?Mn Wn,m ui,m , otherwise
tures that we consider in the paper.
where max denotes the entry-wise max operator and 0 denotes a dn -dim zero vector. Note that
multi-input ReLUs can be thought of as conventional ReLU with skip layers [17] where W?s are set
to identity matrices accordingly.
Conventional Objective for Training DNNs with ReLU: We write down the general objective1 in
a recursive way as used in [41] as follows for clarity:
(
)
X
X
min
`(yi , Vui,N ), s.t. ui,n = max 0,
Wn,m ui,m , ui,0 = xi , ?i, ?n, (2)
?
V?V,W?W
i
m?Mn
? = {Wn,m }. Note that we separate the last FC layer (with weight matrix V) from the
where W
? intentionally, because V is for learning classifiers
rest hidden layers (with weight matrices in W)
? is for learning useful features. The network architectures we use in this paper are mainly for
while W
extracting features, on top of which any arbitrary classifier can be learned further.
Our goal is to optimize Eq. 2. To that end, we propose a novel BCD based algorithm which can solve
the relaxation of Eq. 2 using Tikhonov regularization with convergence guarantee.
2.2
Reinterpretation of ReLU
The ReLU, ordinarily defined as u = max{0, x} for x ? Rd , can be viewed as a projection onto a
convex set (POCS) [3], and thus rewritten as a simple smooth convex optimization problem,
max{0, x} ? arg min ku ? xk22 ,
u?U
(3)
where k ? k2 denotes the `2 norm of a vector and U here is the nonnegative closed half-space. This
non-negative least squares problem becomes the basis of our lifted objective.
1
For simplicity in this paper we always presume that the domain of each variable contains the regularization,
e.g. `2 -norm, without showing it in the objective explicitly.
3
2.3
Our Tikhonov Regularized Objective
We use Eq. 3 to lift and unroll the general training objective in Eq. 2 obtaining the relaxation:
2
X
X
X ?n
?
? V, W)
? =
Wn,m ui,m
,
min
f (U,
`(yi , Vui,N ) +
ui,n ?
?
?
2
U ?U ,V?V,W?W
i
i,n
m?Mn
s.t.
(4)
2
ui,n ? 0, ui,0 = xi , ?i, ?n ? 1,
where U? = {ui,n } and ?n ? 0, ?n denote predefined regularization constants. Larger ?n values
force ui,n , ?i to more closely approximate the output of ReLU at the n-th hidden layer. Arranging u
and ? terms into a matrix Q, we rewrite Eq. 4 in familiar form as a Tikhonov regularized objective:
X
1 T
?
?
?
min
f (U, V, W) ?
`(yi , VPui ) + ui Q(W)ui .
(5)
? ?U ,V?V,W?W
?
2
U
i
Here ui , ?i denotes the concatenating vector of all hidden outputs as well as the input data, i.e.
?
ui = [ui,n ]N
n=0 , ?i, P is a predefined constant matrix so that Pui = ui,N , ?i, and Q(W) denotes
?
another matrix constructed by the weight matrix set W.
? is positive semidefinite, leading to the following Tikhonov regularization:
Proposition 1. Q(W)
? i ? (?ui )T (?ui ) = k?ui k22 , ??, ?i,
uTi Q(W)u
where ? is the Tikhonov matrix.
Definition 1 (Block Multi-Convexity [38]). A function f is block multi-convex if for each block
variable xi , ?i, f is a convex function of xi while all the other blocks are fixed.
? V, W)
? is block multi-convex.
Proposition 2. f (U,
3
3.1
Block Coordinate Descent Algorithm
Training
Eq. 4 can be minimized using alternating optimization, which decomposes the problem into the
following three convex sub-problems based on Lemma 2:
? i , ?i.
? Tikhonov regularized inverse problem: minui ?U `(yi , VPui ) + 12 uTi Q(W)u
2
P
P
?n
? Least-square regression: min?Wn,m ?W
? 2
i ui,n ?
m?Mn Wn,m ui,m 2 ;
P
? Classification using learned features: minV?V i `(yi , VPui ).
All the three sub-problems can be solved efficiently due to their convexity. In fact the inverse subproblem alleviates the vanishing gradient issue in traditional deep learning, because it tries to obtain
the estimated solution for the output feature of each hidden layer, which are dependent on each other
through the Tikhonov matrix. Such functionality is similar to that of target (i.e. estimated outputs of
each layer) propagation [4], namely, propagating information between input data and output labels.
Unfortunately, a simple alternating optimization scheme cannot guarantee the convergence to stationary points for solving Eq. 4. Therefore we propose a novel BCD based algorithm for training
DNNs based on Eq. 4 as listed in Alg. 1. Basically we sequentially solve each sub-problem with an
extra quadratic term. These extra terms as well as the convex combination rule guarantee the global
convergence of the algorithm (see Sec. 4 for more details).
Our algorithm involves solving a sequence of quadratic programs (QP), whose computational complexity is cubic, in general, in the input dimension [28]. In this paper we focus on the theoretical
development of the algorithm, and consider fast implementations in future work.
3.2
Testing
?? ?
Given a test sample x and learned network weights W
n , V , based on Eq.o4 the ideal decision
?
? ? ) . This indicates that
function for classification should be y = arg miny?Y minu f (u, V? , W
4
Algorithm 1 Block Coordinate Descent (BCD) Algorithm for Training DNNs
Input :training data {(xi , yi )} and regularization parameters {?n }
?
Output :network weights W
(0)
?
? (0) ? W;
Randomly initialize U ? U , V(0) ? V, W
nP
o?
?
?
?k
Set sequence {?t }t=1 so that 0 ? ?t ? 1, ?t and sequence
converges to zero, e.g. ?t =
k=t 1??k
t=1
1
;
t2
for t = 1, 2, ? ? ? do
? (t?1) )ui + 1 (1 ? ?t )2 kui ? u(t?1) k22 , ?i;
u?i ? arg minui ?U `(yi , V(t?1) Pui ) + 21 uTi Q(W
i
2
(t)
(t?1)
(t?1)
?
ui ? ui
+ ?t (ui ? ui
), ?i;
P
(t)
V? ? arg minV?V i `(yi , VPui ) + 12 (1 ? ?t )2 kV ? V(t?1) k2F ;
(t)
(t?1)
?
(t?1)
V ?V
+ ?t (V ? V
);
P 1 (t) T
(t?1) 2
? ? ? arg min ?
? (t) + 1 (1 ? ?t )2 P P
[u
]
Q(
W)u
W
i
i
W?W
n
m?Mn kWn,m ? Wn,m kF
i 2
2
(t)
(t?1)
(t?1)
?
?
? ?;
Wn,m ? Wn,m + ?t (Wn,m
? Wn,m ), ?n, ?m ? Mn , Wn,m
?W
end
?
return W;
for each pair of test data and potential label we have to solve an optimization problem, leading to
unaffordably high computational complexity that prevents us from using it.
Recall that our goal is to train feed-forward DNNs using the BCD algorithm in Alg. 1. Considering
? ? to construct the network for extracting deep features. Since
this, we utilize the network weights W
these features are the approximation of U? in Eq. 4 (in fact this is a feasible solution of an extreme
case where ?n = +?, ?n), the learned classifier V? can never be reused at test time. Therefore, we
retain the architecture and weights of the trained network and replace the classification layer (i.e. the
last layer with weights V) with a linear support vector machine (SVM).
3.3
3.3.1
Experiments
MNIST Demonstration
To demonstrate the effectiveness and efficiency of our BCD
based algorithm in Alg. 1, we conduct comprehensive experiments on MNIST [26] dataset using its 28 ? 28 = 784 raw
w2,1
w1,0
w3,2
v
pixels as input features. We refer to our algorithm for learning
dense networks as ?BCD? and that for learning sparse networks
ui,1
ui,2
ui,3
as ?BCD-S?, respectively. For sparse learning, we define the
convex set W = {W | kWk k1 ? 1, ?k}, where Wk denotes
yi
the k-th row in matrix W and k ? k1 denotes the `1 norm of a xi
vector. All the comparisons are performed on the same PC. We
implement our algorithms using MATLAB GPU implementa- Figure 2: The network architecture for
algorithm/solver comparison.
tion without optimizing the code.
We compare our algorithms with the six SGD based solvers in Caffe [20], i.e. SGD [5], AdaDelta
[40], AdaGrad [12], Adam [21], Nesterov [33], RMSProp [35], which are coded in Python. The
network architecture that we implemented is illustrated in Fig. 2. This network has three hidden
layers (with ReLU) with 784 nodes per layer, four FC layers, and three skip layers inside. Therefore,
the mapping function from input xi to output yi defined by the network is:
f (xi ) = Vui,3 , ui,3 = max{0, xi + ui,1 + W3,2 ui,2 },
ui,2 = max{0, xi + W2,1 ui,1 }, ui,1 = max{0, W1,0 xi }.
For simplicity without loss of generality, we utilize MSE as the loss function, and learn the network
parameters using different solvers with the same inputs and random initial weights for each FC layer.
Without fine-tuning the regularization parameters, we simply set ?n = 0.1, ?n in Eq. 4 for both BCD
and BCD-S algorithms. For the Caffe solvers, we modify the demo code in Caffe for MNIST and run
the comparison with carefully tuning the parameters to achieve the best performance that we can. We
report the results within 100 epochs by averaging three trials, because at this point the training of
all the methods seems convergent already. For all competing algorithms, in each epoch the entire
5
2.5
?10
4
0.08
BCD
BCD-S
0.07
0.06
Test Error
Training Objective
2
1.5
1
0.05
0.04
0.03
0.02
0.5
0
0
0.01
0
10
20
30
40
50
60
70
80
90
100
Adadelta Adagrad Adam Nesterov Rmsprop
SGD
BCD
BCD-S
Solvers
# Epochs
(a)
(b)
2.5
1
0.9
Percentage of Nonzeros
Relative Running Time
2
1.5
1
0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Adadelta Adagrad Adam Nesterov Rmsprop
SGD
BCD
0
BCD-S
Solvers
BCD
BCD-S
1st
2nd
3rd
4th
Fully-connected Layer
(c)
(d)
Figure 3: (a) Illustration of convergence for BCD and BCD-S. (b) Test error comparison. (c) Running time
comparison. (d) Sparseness comparison for BCD and BCD-S.
training data is passed through once to update parameters. Therefore, for our algorithms each epoch
is equivalent to one iteration, and there are 100 iterations in total.
Convergence: Fig. 3(a) shows the change of training objective with increase of epochs for BCD and
BCD-S, respectively. As we see both curves decrease monotonically and become flatter and flatter
eventually, indicating that both algorithms converge. BCD-S converges much faster than BCD, but its
objective is higher than BCD. This is because BCD-S learns sparse models that may not fit data as
well as dense models learned by BCD.
Testing Error: As mentioned in Sec. 3.2, here we utilize linear SVMs and last-layer hidden
features extracted from training data to retrain the classifier. Based on the network in Fig. 2
the feature extraction function is ui,3 = max{0, xi + max{0, W1,0 xi } + W3,2 max{0, xi +
W2,1 max{0, W1,0 xi }}}. To conduct fair comparison, we retrain the classifiers for all the algorithms, and summarize the test-time results in Fig. 3(b) with 100 epochs. Our BCD algorithm
which learns dense architectures, same as the SGD based solvers, performs best, while our BCD-S
algorithm works still better than the SGD competitors, although it learns much sparser networks.
These results are consistent with the training objectives in Fig. 3(a) as well.
Computational Time: We compare the training time in Fig. 3(c). It seems that our BCD implementation is significantly faster than the Caffe solvers. For instance, our BCD achieves about 2.5 times
speed-up than the competitors, while achieving best classification performance at test time.
Sparseness: In order to compare the difference in terms of weights between the dense and sparse
networks learned by BCD and BCD-S, respectively, we compare the percentage of nonzero weights
in each FC layer, and show the results in Fig. 3(d). As we see, expect the last FC layer (corresponding
to parameter V as classifiers) BCD-S has the ability of learning much sparser networks for deep
feature extraction. In our case BCD-S learns a network with 2.42% nonzero weights2 , on average,
with classification accuracy 1.34% lower than that of BCD which learns a network with 97.15%
nonzero weights. Potentially this ability could be very useful in the scenarios such as embedding
systems where sparse networks are desired.
3.3.2
Supervised Hashing
To further demonstrate the usage of our approach, we compare with [41]3 for the application of
supervised hashing, which is the state-of-the-art in the literature. [41] proposed an ADMM based
2
3
Since we will retrain the classifiers after all, here we do not take the nonzeros in the last FC into account.
MATLAB code is available at https://zimingzhang.wordpress.com/publications/.
6
optimization algorithm to train DNNs with relaxed objective that is very related to ours. We train
the same DNN on MNIST as used in [41], i.e. with 48 hidden layers and 256 nodes per layer that
are sequentially and fully connected (see [41] for more details on the network). Using the same
image features, we consistently observe marginal improvement over the results (i.e. precision, recall,
mAP) reported in [41]. However, on the same PC we can finish training within 1 hour based on our
implementation, while using the MATLAB code for [41] the training needs about 9 hours. Similar
observations can be made on CIFAR-10 as used in [41] using a network with 16 hidden layers and
1024 nodes per layer.
4
4.1
Convergence Analysis
Preliminaries
Definition 2 (Lipschitz Continuity [13]). We say that function f is Lipschitz continuous with Lipschitz
constant Lf on X , if there is a (necessarily nonnegative) constant Lf such that
|f (x1 ) ? f (x2 )| ? Lf |x1 ? x2 |, ?x1 , x2 ? X .
Definition 3 (Global Convergence [24]). Let X be a set and x0 ? X a given point, Then an Algorithm,
A, with initial point x0 is a point-to-set map A : X ? P(X ) which generates a sequence {xk }?
k=1
via the rule xk+1 ? A(xk ), k = 0, 1, ? ? ? . A is said to be global convergent if for any chosen initial
point x0 , the sequence {xk }?
k=0 generated by xk+1 ? A(xk ) (or a subsequence) converges to a point
for which a necessary condition of optimality holds.
Definition 4 (R-linear Convergence Rate [30]). Let {xk } be a sequence in Rn that converges to
x? . We say that convergence is R-linear if there is a sequence of nonnegative scalars {vk } such that
kxk ? x? k ? vk , ?k, and {vk } converges Q-linearly to zero.
? = arg minw?Rd ?(w) + 12 kw ?
Lemma 1 (3-Point Property [1]). If function ?(w) is convex and w
2
d
w0 k2 , then for any w ? R ,
1
1
1
? ? w0 k22 ? ?(w) + kw ? w0 k22 ? kw ? wk
? 22 .
? + kw
?(w)
2
2
2
4.2
Theoretical Results
?
?
?
? =
? ?, ?), f2 (V) =
? =
Definition 5 (Assumptions on f in Eq. 4). Let f1 (U)
f (U,
f (?, V, ?), f3 (W)
? be the objectives of the three sub-problems, respectively. Then we assume that f is
f (?, ?, W)
lower-bounded and f1 , f2 , f3 are Lipschitz continuous with constants Lf1 , Lf2 , Lf3 , respectively.
2
Proposition 3. Let x, y, x
? ? X and y = (1 ? ?)x + ??
x. Then 12 k?
x ? yk22 = 12 (1 ? ?) k?
x ? xk22 .
Lemma 2. Let X be a nonempty closed convex set, function ? : X ? R is convex and Lipschitz
continuous with constant L, and scalar 0 ? ? ? 1. Suppose that ?x ? X , x
? = arg minz?X ?(z) +
1
2
x. Then we have
2 kz ? z0 k2 and z0 = y = (1 ? ?)x + ??
L?
1??
ky ? xk22 ? ?(x) ? ?(y) ? Lky ? xk2 ? ky ? xk2 ?
.
?
1??
Proof. Based on the convexity of ?, Prop. 3, and Lemma 1, we have
?(x) ? ?(y) ? ?(x) ? [(1 ? ?) ?(x) + ??(?
x)] = ? [?(x) ? ?(?
x)]
1
1
1
1??
? ? kx ? x
?k22 + k?
x ? z0 k22 ? kx ? z0 k22 = ? (1 ? ?) kx ? x
?k22 =
ky ? xk22 ,
2
2
2
?
where ky ? xk22 = 0 if and only if x
? = x (equivalently ?(x) = ?(y)); otherwise ky ? xk22 is
lower-bounded from 0 provided that ? 6= 1.
Based on Def. 2, we have ?(x) ? ?(y) ? Lky ? xk2 .
n
o?
? (t)
Theorem 1. Let U?(t) , V(t) , W
? U ?V ?W be an arbitrary sequence from a closed cont=1
nP
o?
?
?k
vex set that is generated by Alg. 1. Suppose that 0 ? ?t ? 1, ?t and the sequence
k=t 1??k
t=1
converges to zero. Then we have
7
1.
? (?) is a stationary point;
U?(?) , V(?) , W
n
o?
? (?) globally with R-linear
? (t)
will converge to U?(?) , V(?) , W
U?(t) , V(t) , W
t=1
convergence rate.
2.
? = f1 (U?(?) )
6 ? so that f1 (U?(?) + 4U)
Proof. 1. Suppose that for U?(?) there exists a 4U? =
(?)
?
(otherwise, it conflicts with the fact of U
being the limit point). From Lemma 2, f1 (U?(?) +
(?)
(?)
?
?
?
4U) = f1 (U
) is equivalent to U
+ 4U? = U?(?) , and thus 4U? = ?, which conflicts
?
with the assumption of 4U =
6 ?. Therefore, there is no direction that can decrease f1 (U?(?) ),
(?)
(?)
?
? (?) ) = 0. Therefore,
i.e.
) = 0.
?f1 (U
Similarly we have ?f2 (V ) = 0 and ?f3 (W
? (?) is a stationary point.
U?(?) , V(?) , W
2. Based on Def. 5 and Lemma 2, we have
v
2
u X
2
(t)
u
(?)
ui,n ? ui,n
+
V(t) ? V(?)
F +
t
2
?
ui,n ?U
?
F
?
Wn,m ?W
X
(t)
(?)
ui,n ? ui,n
+
V(t) ? V(?)
+
2
?
ui,n ?U
2
(t)
(?)
Wn,m ? Wn,m
X
F
X
(t)
(?)
Wn,m ? Wn,m
F
?
Wn,m ?W
?
?
?
X
X
X
X
X (k)
(k+1)
(k)
(k+1)
ui,n ? ui,n
+
V(k) ? V(k+1)
+
Wn,m
? Wn,m
=
? k=t
? k=t
k=t
2
F
F
Wn,m ?W
ui,n ?U
?
?
?
X
X
X
(k)
(k)
(k+1)
(k+1)
?
?
?
ui,n ? ui,n
+
V(k) ? V(k+1)
+
Wn,m ? Wn,m
k=t
?
?
X
k=t
2
?
ui,n ?U
F
?
X Lf ?k
Lf2 ?k
1
?
+
+
1 ? ?k
1 ? ?k
?
ui,n ?U
X
?
Wn,m ?W
F
?
Wn,m ?W
?
?
X
?k
Lf3 ?k ?
=O
1 ? ?k
1 ? ?k
!
.
k=t
By combining this with Def. 3 and Def. 4 we can complete the proof.
p
Corollary 1. Let ?t = 1t , ?t. Then when p > 1, Alg. 1 will converge globally with order one.
Proof.
?
X
k=t
?
Z
1
1
1 ? 1
1
d(x + 1) p =
(x + 1) p ?1 dx
p tp ?1 x
tp ?1 x
Z
?
?p>1 1
1
1
?
x p ?2 dx = (p ? 1)?1 (tp ? 1) p ?1 .
p tp ?1
X 1
?k
?
=
1 ? ?k
kp ? 1
k=t
Z
?
(6)
n
o?
1
Since the sequence (tp ? 1) p ?1
, ?p > 1 converges to zero sublinearly with order one, by
t=1
combining these with Def. 4 and Thm. 1 we can complete the proof.
5
Conclusion
In this paper we first propose a novel Tikhonov regularization for training DNNs with ReLU as the
activation functions. The Tikhonov matrix encodes the network architecture as well as parameterization. With its help we reformulate the network training as a block multi-convex minimization
problem. Accordingly we further propose a novel block coordinate descent (BCD) based algorithm,
which is proven to converge globally to stationary points with R-linear converge rate of order one.
Our empirical results suggest that our algorithm does converge, is suitable for learning both dense
and sparse networks, and may work better than traditional SGD based deep learning solvers.
8
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10
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6,378 | 677 | A Formal Model of the Insect Olfactory
Macroglomerulus: Simulations and
Analytical Results.
Christiane Linster
David Marsan
ESPCI, Laboratoire d'Electronique
10, Rue Vauquelin
75005 Paris, France
Claudine Masson
Laboratoire de Neurobiologie Comparee des
Invertebrees
INRA/CNRS (URA 1190)
91140 Bures sur Yvette, France
Michel Kerszberg
Institut Pasteur
CNRS (URA 1284)
Neurobiologie Moleculaire
25, Rue du Dr. Roux
75015 Paris, France
Gerard Dreyfus
Leon Personnaz
ESPCI, Laboratoire d'Electronique
10, Rue Vauquelin
75005 Paris, France
Abstract
It is known from biological data that the response patterns of
interneurons in the olfactory macroglomerulus (MGC) of insects are of
central importance for the coding of the olfactory signal. We propose an
analytically tractable model of the MGC which allows us to relate the
distribution of response patterns to the architecture of the network.
1. Introduction
The processing of pheromone odors in the antennallobe of several insect species relies on
a number of response patterns of the antennallobe neurons in reaction to stimulation with
pheromone components and blends. Antennallobe interneurons receive input from different
receptor types, and relay this input to antennal lobe projection neurons via excitatory as
well as inhibitory synapses. The diversity of the responses of the interneurons and
projection neurons as well the long response latencies of these neurons to pheromone
stimulation or electrical stimulation of the antenna, suggest a polysynaptic pathway
1022
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results
between the receptor neurons and these projection neurons (for a review see (Kaissling,
1990; Masson and Mustaparta, 1990)).
I.
Pf-EROMONE ce..ERALlSTS
A. Carnot Discrinilate Single Odors
AN)
Camot COde Ter1lXlI'aI Olanges
,. Excited Type
Stml'S
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ill'
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i. I 'I
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i
I
I
I
I
I
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Response
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. ._.-_. .'_'_' J1J~UlUlJ
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Stinhs
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Camot Code Terrporal Olanges
Response
(1) 00 (2)
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+
0
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?
stmt?
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?
B. can Discriri1ate 5ilgIe Odors
,L
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f>K)
Can Code
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T~a1 Olanges
8espoose
(1) 00 (2)
BAL
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+
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-/./- -/./-
Figure 1: With courtesy of John Hildebrand, by permission from Oxford University
Press, from: Christensen, Mustaparta and Hildebrand: Discrimination of sex
pheromone blends in the olfactory system of the moth, Chemical Senses, Vol 14,
no 3, pp 463-477, 1989.
1023
1024
Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz
In the MOC of Manduca sexta, antennal lobe interneurons respond in various ways to
antennal stimulation with single pheromone components or the blend: pheromone
generalists respond by either excitation or inhibition to both components and the blend:
they cannot discriminate the components; pheromone specialists respond (i) to one
component but not to the other by either excitation or inhibition, (ii) with different
response patterns to the presence of the single components or the blend, namely with
excitation to one component, with inhibition to the other component and with a mixed
response to the blend. These neurons can also follow pulsed stimulation up to a cut-off
frequency (Figure 1).
A model of the MOC (Linster et aI, 1993), based on biological data (anatomical and
physiological) has demonstrated that the full diversity of response patterns can be
reproduced with a random architecture using very simple ingredients such as spiking
neurons governed by a first order differential equation, and synapses modeled as simple
delay lines. In a model with uniform distributions of afferent, inhibitory and excitatory
synapses, the distribution of the response patterns depends on the following network
parameters: the percentage of afferent, inhibitory and excitatory synapses, the ratio of the
average excitation of any interneuron to its spiking threshold, and the amount of feedback
in the network.
In the present paper, we show that the behavior of such a model can be described by a
statistical approach, allowing us to search through parameter space and to make predictions
about the biological system without exhaustive simulations. We compare the results
obtained with simulation of the network model to the results obtained analytically by the
statistical approach, and we show that the approximations made for the statistical
descriptions are valid.
2. Simulations and comparison to biological data
In (Linster et aI, 1993), we have used a simple neuron model: all neurons are spiking
neurons, governed by a first order differential equation, with a membrane time constant and
a probabilistic threshold 9. The time constant represents the decay time of the membrane
potential of the neuron. The output of each neuron consists of an all-or-none action
potential with unit amplitude that is generated when the membrane potential of the cell
crosses a threshold, whose cumulative distribution function is a continuous and bounded
probabilistic function of the membrane potential. All sources of delay and signal
transformation from the presynaptic neuron to its postsynaptic site are modeled by a
synaptic time delay. These delays are chosen in a random distribution (gaussian), with a
longer mean value for inhibitory synapses than for excitatory synapses. We model two
main populations of olfactory neurons: receptor neurons which are sensitive to the main
pheromone component (called A) or to the minor pheromone component (called B) project
uniformly onto the network of interneurons; two types of interneurons (excitatory and
inhibitory) exist: each interneuron is allowed to make one synapse with any other
interneuron.
The model exhibits several behaviors that agree with biological data, and it allows us to
state several predictive hypotheses about the processing of the pheromone blend. We
observe two broad classes of intemeurons: selective (to one odor component) and nonselective neurons (in comparison to Figure 1). Selective neurons and non-selective neurons
exhibit a variety of response patterns, which fall into three classes: inhibitory, excitatory
and mixed (Figure 2). Such a classification has indeed been proposed for olfactory antennal
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results
lobe neurons (local interneurons and projection neurons) in the specialist olfactory system
in Manduca (Christensen and Hildebrand, 1987) and for the cockroach (Burrows et al, 1982;
Boeckh and Ernst, 1987).
Inhibitory response
Action
potenUals
Excitatory response
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Simple mixed response
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I
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,,-------,
,,-----"""',
Figure 2: Response patterns of interneurons in the model presented, in response to
stimulation with single components A and B, and with a blend with equal
component concentrations. Receptor neurons fIre at maximum frequency during the
stimulations. The interneuron in the upper row is inhibited by stimulus A, excited
by stimulus B, and has a mixed response (excitation followed by inhibition) to the
blend: in reference to Figure 1, this is a pheromone specialist receiving mixed input
from both types of receptor neurons. These types of simple and mixed responses can
be observed in the model at low connectivity, where the average excitation received
by an interneuron is low compared to its spiking threshold. The neuron in the middle
row responds with similar mixed responses to stimuli A, Band A+B. The neuron in
the lower row responds to all stimuli with the same oscillatory response, here the
average excitation received by an interneuron approaches or exceeds the spiking
threshold of the neurons. Network parameters: 15 receptor neurons; 35 interneurons;
40% excitatory interneurons; 60% inhibitory interneurons; afferent connectivity
10%; membrane time constant 25 ms; mean inhibitory synaptic delays 100 ms;
mean excitatory synaptic delays 25 ms, spiking threshold 4.0, synaptic weights +1
and -1.
1025
1026
Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz
In our model, as well as in biological systems (Christensen and Hildebrand 1988,
Christensen et ai., 1989) we observe a number of local interneurons that cannot follow
pulsed stimulation beyond a neuron-specific cut-off frequency. This frequency depends on
the neuron response pattern and on the duration of the interstimulus interval.
Therefore, the type of response pattern is of central importance for the coding of the
olfactory signal. Thus, in order to be able to relate the coding capabilities of a (model or
biological) network to its architecture, we have investigated the distribution of response
patterns both analytically and by simulations.
3. Analytical approach
In order to investigate these questions in a more rigorous way, some of us (C.L., D.M.,
G.D., L.P.) have designed a simplified, analytically tractable model.
We define two layers of interneurons: those which receive direct afferent input from the
receptor neurons (layer 1), and those which receive only input from other interneurons
(layer 2). In order to predict the response pattern of any interneuron as a function of the
network parameters, we make the following assumptions: (i) statistically, all interneurons
within a given layer receive the same synaptic input, (ii) the effect of feedback loops from
layer 2 can be neglected, (iii) the response patterns have the same distribution for
stimulations either by the blend or by pure components. Assumption (i) is correct because
of the uniform distribution of synapses in the network of interneurons. Assumption (ii) is
valid at low connectivity: if the average amount of excitation received by an interneuron is
low as compared to its spiking threshold, its firing probability is low; therefore, the effect
of the excitation from the receptors is vanishingly small beyond two interneurons: we thus
neglect the effect of signals sent from layer 2. Thus, feedback is present within layer 1, and
layer 2 receives only feed forward connections. Assumption (iii) is plausible if we suppose
that the natural pheromone blend is more relevant for the system than the single
components of the blend. We further assume in the analytical approach (as in the
simulations) that the synaptic delays are longer on the average for inhibitory synapses than
for excitatory synapses .
An interneuron can thus respond with four types of patterns: non-response, which means
that it does not have a presynaptic neuron (this response pattern can only occur in layer 2,
at low connectivity); excitation, meaning that an interneuron receives only afferent input
from receptor neurons or from excitatory interneurons; inhibition, meaning that an
interneuron receives only input from inhibitory interneurons (this can occur in layer 2
only); and mixed responses, covering all other combinations of presynaptic input.
We consider a network of N + N r neurons, N (number of interneurons) and N r (number of
receptor neurons) being random variables, N + Nr being fixed. We define the probability ni
that a neuron is an inhibitory interneuron, and the probability ne that it is an excitatory
interneuron. Any interneuron has a probability c to make one synapse (with synaptic
weight +1 or -1) with any other interneuron and a probability (1 - c) not to make a synapse
with this interneuron; Cr is the afferent connectivity: any receptor neuron has a probability
Cr to connect once to any interneuron, and a probability (1 - cr) not to connect to this
interneuron. Then na = 1 - (1 - cr)Nr is the probability that an interneuron belongs to layer
1, and the number of interneurons in layer I obeys a binomial distribution with expectation
value N nQ and variance N na (1 - na). In the following, the fixed number of interneurons in
layer 1 will be taken equal to its expectation value. Similarly, the number of interneurons
in layer 2 is taken to be N (1 - na).
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results
Because of the assumptions made above, in both layers, we take into account for each
interneuron the N na c synapses from presynaptic neurons of layerl. In layer 1, these
neurons respond with excitatory or mixed responses. P1= nena N C is the probability that
an interneuron in layer 1 responds with an excitation, and p~= 1 - neflaN c is the
probability that an interneuron in layer 1 receives mixed synaptic input.
In layer 2, we have to consider two cases: (i) at low connectivity, if
N c na < 1, P6 1 - N c na is the probability that an interneuron of layer 2 does not
receive a synapse, thus does not respond to stimulation, P; N c nane is the probability
that a neuron in layer 2 responds with excitation, p? =N c nam is the probability that an
interneuron responds with inhibition; (ii) at higher connectivity, N c na > 1, P6 =0,
P; =ne naN c and pl =m naN c. In both cases (i) and (ii), the probability that an
interneuron in layer 2 has a mixed response pattern is P; = 1 - P6 -Pe - Pl.
Thus, an interneuron in the model responds with excitation with probability
Pe =na P; + (1 - na) P;, with inhibition with probability Pi =na p/ + (1 - na) p? and has
a mixed response with probability Pm =na P~ + (1 - na) p;.
=
=
P
0 .80
Layer 1
0 .60
0 .40
0 .20
0 . 10
0.20
0 .30
0 .40
0.50
0 .60
0 .70 C
0 .60
0 .70 C
0.60
0 .70 C
P
0.80
Layer 2
0 .60
0 .40
0.20
0.20
0.30
0.40
0 .50
0 .80
Layers 1 & 2
0.60
0.40
0.20
0.10
0.20
0 .30
0 .40
0 .50
Figure 4: Analytically derived distribution of the response patterns in a typical
network (35 interneurons, 15 receptor neurons, 40% excitation, 60% inhibition,
spiking threshold 4.0); the curves show the percentage of interneurons in the model
which respond with a given pattern, as a function of the connectivity c. In this case,
the average excitation an interneuron receives from other interneurons is 3.15 at
c=O.3.
Figure 4 shows the distribution of the response patterns computed analytically for a typical
set of parameters. In order to perform comparisons between computed pattern distributions
and pattern distributions obtained from simulations with the model, we designed an
automatic classifier for the response patterns, based on the perceptron learning rule and the
pocket algorithm (Gallant 1986). The classifier is trained to classify the responses of
1027
1028
Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz
individual interneurons, based on their membrane potential, into 5 typical response classes:
non-response, pure excitation, pure inhibition, simple mixed response and oscillatory
responses. Figure 5 shows the simulation results for the same set of parameters as for
Figure 4. The agreement between the two curves shows that the approximations which we
have made in order to describe the analytical model are valid.
Figure 6 shows how the mixed responses in the simulations divide into simple mixed and
oscillatory responses. When the validity limit of the approximations made in the analytical
approach is reached, all neurons fire at maximum frequency and the network oscillates.
Therefore, the analytical model describes satisfactorily the whole range of connectivity in
which the pattern distribution does not reduce to oscillations. The oscillation frequency is
determined by the mean synaptic delays and by the membrane time constants; more detailed
results on the oscillatory behavior will be published in a future paper.
P
80
~
.................
../'4- Mixed
..-*
60
40
Layer 1
20
0.3
0.4
c
o~
P
80
60
Layer 2
40
20
80
60
40
Layers 1 & 2
20
0.4
Figure 5: Distribution of the response patterns obtained from simulations of the
model with the set of parameters described above. The curves show the percentages
of interneurons that respond with a given pattern, as a function of connectivity c.
For each value of c, 100 simulation runs with three different stimulation inputs have
been averaged.
pr-------------------7-=-=--------==~-----------------=------80
60
Layers 1 & 2
40
20
o .~
o.b
o.
c
Figure 6: Distribution of simple mixed and oscillatory responses in the simulation
model. With the set of parameters chosen, condition ne c :::: e is satisfied for c::::O.3.
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results
4. Conclusion
In the olfactory system of insects and mammals, a number of response patterns are
observed, which are of central importance for the coding of the olfactory signal. In the
present paper, we show that, under some constraints, an analytical model can predict the
existence and the distribution of these response patterns. We further show that the
transition between non-oscillatory and oscillatory regimes is governed by a single
parameter (ne c / E?. It is thus possible, to explore the parameter space without exhaustive
simulations, and to relate the coding capabilities of a model or biological network to its
architecture.
Acknowledgements
This work was supported in part by a grant from Ministere de la Recherche et de la
Technologie (Sciences de la Cognition). C. Linster has been supported by a research grant
(BFR91/051) from the Ministere des Affaires Culturelles, Grand-Duche de Luxembourg.
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1029
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6,379 | 6,770 | Train longer, generalize better: closing the
generalization gap in large batch training of neural
networks
Elad Hoffer1?,
Itay Hubara?,
Daniel Soudry2
1
Technion - Israel Institute of Technology, Haifa, Israel
2
Columbia University, New York, New York, USA
{elad.hoffer, itayhubara, daniel.soudry}@gmail.com
Abstract
Background: Deep learning models are typically trained using stochastic gradient
descent or one of its variants. These methods update the weights using their
gradient, estimated from a small fraction of the training data. It has been observed
that when using large batch sizes there is a persistent degradation in generalization
performance - known as the "generalization gap" phenomenon. Identifying the
origin of this gap and closing it had remained an open problem.
Contributions: We examine the initial high learning rate training phase. We
find that the weight distance from its initialization grows logarithmically with the
number of weight updates. We therefore propose a "random walk on a random
landscape" statistical model which is known to exhibit similar "ultra-slow" diffusion
behavior. Following this hypothesis we conducted experiments to show empirically
that the "generalization gap" stems from the relatively small number of updates
rather than the batch size, and can be completely eliminated by adapting the
training regime used. We further investigate different techniques to train models
in the large-batch regime and present a novel algorithm named "Ghost Batch
Normalization" which enables significant decrease in the generalization gap without
increasing the number of updates. To validate our findings we conduct several
additional experiments on MNIST, CIFAR-10, CIFAR-100 and ImageNet. Finally,
we reassess common practices and beliefs concerning training of deep models and
suggest they may not be optimal to achieve good generalization.
1
Introduction
For quite a few years, deep neural networks (DNNs) have persistently enabled significant improvements in many application domains, such as object recognition from images (He et al., 2016); speech
recognition (Amodei et al., 2015); natural language processing (Luong et al., 2015) and computer
games control using reinforcement learning (Silver et al., 2016; Mnih et al., 2015).
The optimization method of choice for training highly complex and non-convex DNNs, is typically
stochastic gradient decent (SGD) or some variant of it. Since SGD, at best, finds a local minimum of
the non-convex objective function, substantial research efforts are invested to explain DNNs ground
breaking results. It has been argued that saddle-points can be avoided (Ge et al., 2015) and that
"bad" local minima in the training error vanish exponentially (Dauphin et al., 2014; Choromanska
et al., 2015; Soudry & Hoffer, 2017). However, it is still unclear why these complex models tend to
generalize well to unseen data despite being heavily over-parameterized (Zhang et al., 2017).
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
?
Equal contribution
A specific aspect of generalization has recently attracted much interest. Keskar et al. (2017) focused
on a long observed phenomenon (LeCun et al., 1998a) ? that when a large batch size is used while
training DNNs, the trained models appear to generalize less well. This remained true even when the
models were trained "without any budget or limits, until the loss function ceased to improve" (Keskar
et al., 2017). This decrease in performance has been named the "generalization gap".
Understanding the origin of the generalization gap, and moreover, finding ways to decrease it,
may have a significant practical importance. Training with large batch size immediately increases
parallelization, thus has the potential to decrease learning time. Many efforts have been made to
parallelize SGD for Deep Learning (Dean et al., 2012; Das et al., 2016; Zhang et al., 2015), yet the
speed-ups and scale-out are still limited by the batch size.
In this study we suggest a first attempt to tackle this issue.
First,
? We propose that the initial learning phase can be described using a high-dimensional
"random walk on a random potential" process, with an "ultra-slow" logarithmic increase in
the distance of the weights from their initialization, as we observe empirically.
Inspired by this hypothesis, we find that
? By simply adjusting the learning rate and batch normalization the generalization gap can be
significantly decreased (for example, from 5% to 1% ? 2%).
? In contrast to common practices (Montavon et al., 2012) and theoretical recommendations
(Hardt et al., 2016), generalization keeps improving for a long time at the initial high learning
rate, even without any observable changes in training or validation errors. However, this
improvement seems to be related to the distance of the weights from their initialization.
? There is no inherent "generalization gap": large-batch training can generalize as well as
small batch training by adapting the number of iterations.
2
Training with a large batch
Training method. A common practice of training deep neural networks is to follow an optimization
"regime" in which the objective is minimized using gradient steps with a fixed learning rate and a
momentum term (Sutskever et al., 2013). The learning rate is annealed over time, usually with an
exponential decrease every few epochs of training data. An alternative to this regime is to use an
adaptive per-parameter learning method such as Adam (Kingma & Ba, 2014), Rmsprop (Dauphin
et al.) or Adagrad (Duchi et al., 2011). These methods are known to benefit the convergence rate of
SGD based optimization. Yet, many current studies still use simple variants of SGD (Ruder, 2016)
for all or part of the optimization process (Wu et al., 2016), due to the tendency of these methods to
converge to a lower test error and better generalization.
Thus, we focused on momentum SGD, with a fixed learning rate that decreases exponentially every
few epochs, similarly to the regime employed by He et al. (2016). The convergence of SGD is
also known to be affected by the batch size (Li et al., 2014), but in this work we will focus on
generalization. Most of our results were conducted on the Resnet44 topology, introduced by He et al.
(2016). We strengthen our findings with additional empirical results in section 6.
Empirical observations of previous work. Previous work by Keskar et al. (2017) studied the
performance and properties of models which were trained with relatively large batches and reported
the following observations:
? Training models with large batch size increase the generalization error (see Figure 1).
? This "generalization gap" seemed to remain even when the models were trained without
limits, until the loss function ceased to improve.
2
(a) Training error
(b) Validation error
Figure 1: Impact of batch size on classification error
? Low generalization was correlated with "sharp" minima2 (strong positive curvature), while
good generalization was correlated with "flat" minima (weak positive curvature).
? Small-batch regimes were briefly noted to produce weights that are farther away from the
initial point, in comparison with the weights produced in a large-batch regime.
Their hypothesis was that a large estimation noise (originated by the use of mini-batch rather than full
batch) in small mini-batches encourages the weights to exit out of the basins of attraction of sharp
minima, and towards flatter minima which have better generalization.In the next section we provide
an analysis that suggest a somewhat different explanation.
3
Theoretical analysis
Notation. In this paper we examine Stochastic Gradient Descent (SGD) based training of a Deep
Neural Network (DNN). The DNN is trained on a finite training set of N samples. We define w as
the vector of the neural network parameters, and Ln (w) as loss function on sample n. We find w by
minimizing the training loss.
N
1 X
L (w) ,
Ln (w) ,
N n=1
using SGD. Minimizing L (w) requires an estimate of the gradient of the negative loss.
g,
N
N
1 X
1 X
gn , ?
?Ln (w)
N n=1
N n=1
where g is the true gradient, and gn is the per-sample gradient. During training we increment the
? computed on some mini-batch B ? a set of M
parameter vector w using only the mean gradient g
randomly selected sample indices.
1 X
?,
g
gn .
M
n?B
In order to gain a better insight into the optimization process and the empirical results, we first
examine simple SGD training, in which the weights at update step t are incremented according to the
mini-batch gradient ?wt = ??
gt . With respect to the randomness of SGD,
E?
gt = g = ??L (wt ) ,
and the increments are uncorrelated between different mini-batches3 . For physical intuition, one
can think of the weight vector wt as a particle performing a random walk on the loss (?potential?)
landscape L (wt ). Thus, for example, adding momentum term to the increment is similar to adding
inertia to the particle.
2
It was later pointed out (Dinh et al., 2017) that certain "degenerate" directions, in which the parameters can
be changed without affecting the loss, must be excluded from this explanation. For example, for any c > 0 and
any neuron, we can multiply all input weights by c and divide the output weights by c: this does not affect the
loss, but can generate arbitrarily strong positive curvature.
3
Either exactly (with replacement) or approximately (without replacement): see appendix section A.
3
Motivation. In complex systems (such as DNNs) where we do not know the exact shape of the
loss, statistical physics models commonly assume a simpler description of the potential as a random
process. For example, Dauphin et al. (2014) explained the observation that local minima tend to have
low error using an analogy between L (w), the DNN loss surface, and the high-dimensional Gaussian
random field analyzed in Bray & Dean (2007), which has zero mean and auto-covariance
2
E (L (w1 ) L (w2 )) = f kw1 ? w2 k
(1)
for some function f , where the expectation now is over the randomness of the loss. This analogy
resulted with the hypothesis that in DNNs, local minima with high loss are indeed exponentially
vanishing, as in Bray & Dean (2007). Only recently, similar results are starting to be proved for
realistic neural network models (Soudry & Hoffer, 2017). Thus, a similar statistical model of the loss
might also give useful insights for our empirical observations.
Model: Random walk on a random potential. Fortunately, the high dimensional case of a particle
doing a ?random walk on a random potential? was extensively investigated already decades ago
(Bouchaud & Georges, 1990). The main result of that investigation was that the asymptotic behavior
of the auto-covariance of a random potential4 ,
?
E (L (w1 ) L (w2 )) ? kw1 ? w2 k
, ?>0
(2)
in a certain range, determines the asymptotic behavior of the random walker in that range:
2
4
E kwt ? w0 k ? (log t) ? .
(3)
This is called an ?ultra-slow diffusion? in which, typically kwt ? w0 k ? (log?t)2/? , in contrast
to standard diffusion (on a flat potential), in which we have kwt ? w0 k ? t . The informal
reason for this behavior (for any ? > 0), is that for a particle to move a distance d, it has to pass
potential barriers of height ? d?/2 , from eq. (2). Then, to climb (or go around) each barrier takes
exponentially long time in the height of the barrier: t ? exp(d?/2 ). Inverting this relation, we get eq.
d ? (log(t))2/? . In the high-dimensional case, this type of behavior was first shown numerically and
explained heuristically by Marinari et al. (1983), then rigorously proven for the case of a discrete
lattice by Durrett (1986), and explained in the continuous case by Bouchaud & Comtet (1987).
3.1
Comparison with empirical results and implications
To examine this prediction of ultra slow diffusion and find the value of ?, in Figure 2a, we examine
kwt ? w0 k during the initial training phase over the experiment shown in Figure 1. We found that
the weight distance from initialization point increases logarithmically with the number of training
iterations (weight updates), which matches our model with ? = 2:
kwt ? w0 k ? log t .
(4)
Interestingly, the value of ? = 2 matches the statistics of the loss estimated in appendix section B.
Moreover, in Figure 2a, we find that a very similar logarithmic graph is observed for all batch sizes.
Yet, there are two main differences. First, each graph seems to have a somewhat different slope
(i.e., it is multiplied by different positive constant), which peaks at M = 128 and then decreases
with the mini-batch size. This indicates a somewhat different diffusion rate for different batch sizes.
Second, since we trained all models for a constant number of epochs, smaller batch sizes entail more
training iterations in total. Thus, there is a significant difference in the number of iterations and the
corresponding weight distance reached at the end of the initial learning phase.
This leads to the following informal argument (which assumes flat minima are indeed important for
generalization). During the initial training phase, to reach a minima of "width" d the weight vector
wt has to travel at least a distance d, and this takes a long time ? about exp(d) iterations. Thus,
to reach wide ("flat") minima we need to have the highest possible diffusion rates (which do not
result in numerical instability) and a large number of training iterations. In the next sections we will
implement these conclusions in practice.
4
Note that this form is consistent with eq. (1), if f (x) = x?/2 .
4
(a) Before learning rate adjustment and GBN
(b) After learning rate adjustment and GBN
Figure 2: Euclidean distance of weight vector from initialization
4
Matching weight increment statistics for different mini-batch sizes
First, to correct the different diffusion rates observed for different batch sizes, we will aim to match
the statistics of the weights increments to that of a small batch size.
Learning rate. Recall that in this paper we investigate SGD, possibly with momentum, where the
weight updates are proportional to the estimated gradient.
?w ? ??
g,
(5)
where ? is the learning rate, and we ignore for now the effect of batch normalization.
In appendix section A, we show that the covariance matrix of the parameters update step ?w is,
!
N
?2
1 X
>
cov (?w, ?w) ?
gn gn
(6)
M N n=1
in the case of uniform sampling of the mini-batch indices (with or without replacement), when
M N . Therefore, a simple way to make sure that the covariance matrix stays the same for all
mini-batch sizes is to choose
?
(7)
?? M,
i.e., we should increase the learning rate by the square root of the mini-batch size.
We note that Krizhevsky (2014) suggested a similar learning rate scaling in order to keep the variance
in the gradient expectation constant, but chose to use a linear scaling heuristics as it reached better
empirical result in his setting. Later on, Li (2017) suggested the same.
Naturally, such an increase in the learning rate also increases the mean steps E [?w]. However,
we found that this effect is negligible since E [?w] is typically orders of magnitude lower than the
standard deviation.
Furthermore, we can match both the first and second order statistics by adding multiplicative noise to
the gradient estimate as follows:
N
1 X
?=
g
gn zn ,
M
n?B
where zn ? N 1, ? 2 are independent random Gaussian variables for which ? 2 ? M . This can
be verified by using similar calculation as in appendix section A. This method keeps the covariance
constant when we change the batch size, yet does not change the mean steps E [?w].
In both cases, for the first few iterations, we had to clip or normalize the gradients to prevent
divergence. Since both methods yielded similar performance 5 (due the negligible effect of the first
order statistics), we preferred to use the simpler learning rate method.
It is important to note that other types of noise (e.g., dropout (Srivastava et al., 2014), dropconnect
(Wan et al., 2013), label noise (Szegedy et al., 2016)) change the structure of the covariance matrix
and not just its scale, thus the second order statistics of the small batch increment cannot be accurately
5
a simple comparison can be seen in appendix (figure 3)
5
matched. Accordingly, we did not find that these types of noise helped to reduce the generalization
gap for large batch sizes.
Lastly, note that in our discussion above (and the derivations provided in appendix section A) we
assumed each per-sample gradient gn does not depend on the selected mini-batch. However, this
ignores the influence of batch normalization. We take this into consideration in the next subsection.
Ghost Batch Normalization. Batch Normalization (BN) (Ioffe & Szegedy, 2015), is known to
accelerate the training, increase the robustness of neural network to different initialization schemes
and improve generalization. Nonetheless, since BN uses the batch statistics it is bounded to depend
on the choosen batch size. We study this dependency and observe that by acquiring the statistics
on small virtual ("ghost") batches instead of the real large batch we can reduce the generalization
error. In our experiments we found out that it is important to use the full batch statistic as suggested
by (Ioffe & Szegedy, 2015) for the inference phase. Full details are given in Algorithm 1. This
modification by itself reduce the generalization error substantially.
Algorithm 1: Ghost Batch Normalization (GBN), applied to activation x over a large batch BL with
virtual mini-batch BS . Where BS < BL .
Require: Values of x over a large-batch: BL = {x1...m } size of virtual batch |BS |; Parameters to
be learned: ?, ?, momentum ?
Training Phase:
Scatter BL to {X 1 , X 2 , ...X |BL |/|BS | } = {x1...|BS | , x|BS |+1...2|BS | ...x|BL |?|BS |...m }
P|B |
?lB ? |B1S | i=1S Xil for l = 1, 2, 3 . . . {calculate ghost mini-batches means}
q
P|B |
l
? |B1S | i=1S (Xil ? ?B )2 + for l = 1, 2, 3 . . . {calculate ghost mini-batches std}
?B
P|B |/|BS |
?run = (1 ? ?)|BS | ?run + i=1L
(1 ? ?)i ? ? ? ?lB
P
|B
|/|B
|
S
l
?run = (1 ? ?)|BS | ?run + i=1L
(1 ? ?)i ? ? ? ?B
X l ??l
return ? ?l B + ?
B
Test Phase: l
X??run
+ ? {scale and shift}
return ? ?run
We note that in a multi-device distributed setting, some of the benefits of "Ghost BN" may already
occur, since batch-normalization is often preformed on each device separately to avoid additional
communication cost. Thus, each device computes the batch norm statistics using only its samples
(i.e., part of the whole mini-batch). It is a known fact, yet unpublished, to the best of the authors
knowledge, that this form of batch norm update helps generalization and yields better results than
computing the batch-norm statistics over the entire batch. Note that GBN enables flexibility in the
small (virtual) batch size which is not provided by the commercial frameworks (e.g., TensorFlow,
PyTorch) in which the batch statistics is calculated on the entire, per-device, batch. Moreover, in those
commercial frameworks, the running statistics are usually computed differently from "Ghost BN",
by weighting each update part equally. In our experiments we found it to worsen the generalization
performance.
Implementing both the learning rate and GBN adjustments seem to improve generalization performance, as we shall see in section 6. Additionally, as can be seen in Figure 6, the slopes of the
logarithmic weight distance graphs seem to better matched, indicating similar diffusion rates. We
also observe some constant shift, which we believe is related to the gradient clipping. Since this shift
only increased the weight distances, we assume it does not harm the performance.
5
Adapting number of weight updates eliminates generalization gap
According to our conclusions in section 3, the initial high-learning rate training phase enables the
model to reach farther locations in the parameter space, which may be necessary to find wider local
minima and better generalization. Examining figure 2b, the next obvious step to match the graphs for
different batch sizes is to increase the number of training iterations in the initial high learning rate
6
(a) Validation error
(b) Validation error - zoomed
Figure 3: Comparing generalization of large-batch regimes, adapted to match performance of smallbatch training.
regime. And indeed we noticed that the distance between the current weight and the initialization
point can be a good measure to decide upon when to decrease the learning rate.
Note that this is different from common practices. Usually, practitioners decrease the learning
rate after validation error appears to reach a plateau. This practice is due to the long-held belief
that the optimization process should not be allowed to decrease the training error when validation
error "flatlines", for fear of overfitting (Girosi et al., 1995). However, we observed that substantial
improvement to the final accuracy can be obtained by continuing the optimization using the same
learning rate even if the training error decreases while the validation plateaus. Subsequent learning
rate drops resulted with a sharp validation error decrease, and better generalization for the final model.
These observations led us to believe that "generalization gap" phenomenon stems from the relatively
small number of updates rather than the batch size. Specifically, using the insights from Figure 2 and
our model, we adapted the training regime to better suit the usage of large mini-batch. We "stretched"
the time-frame of the optimization process, where each time period of e epochs in the original regime,
L|
will be transformed to |B
|BS | e epochs according to the mini-batch size used. This modification ensures
that the number of optimization steps taken is identical to those performed in the small batch regime.
As can be seen in Figure 3, combining this modification with learning rate adjustment completely
eliminates the generalization gap observed earlier 6 .
6
Experiments
Experimental setting. We experimented with a set of popular image classification tasks:
? MNIST (LeCun et al., 1998b) - Consists of a training set of 60K and a test set of 10K
28 ? 28 gray-scale images representing digits ranging from 0 to 9.
? CIFAR-10 and CIFAR-100 (Krizhevsky, 2009) - Each consists of a training set of size 50K
and a test set of size 10K. Instance are 32 ? 32 color images representing 10 or 100 classes.
? ImageNet classification task Deng et al. (2009) - Consists of a training set of size 1.2M
samples and test set of size 50K. Each instance is labeled with one of 1000 categories.
To validate our findings, we used a representative choice of neural network models. We used the
fully-connected model, F1, as well as shallow convolutional models C1 and C3 suggested by Keskar
et al. (2017). As a demonstration of more current architectures, we used the models: VGG (Simonyan,
2014) and Resnet44 (He et al., 2016) for CIFAR10 dataset, Wide-Resnet16-4 (Zagoruyko, 2016) for
CIFAR100 dataset and Alexnet (Krizhevsky, 2014) for ImageNet dataset.
In each of the experiments, we used the training regime suggested by the original work, together with
a momentum SGD optimizer. We use a batch of 4096 samples as "large batch" (LB) and a small
batch (SB) of either 128 (F1,C1,VGG,Resnet44,C3,Alexnet) or 256 (WResnet). We compare the
original training baseline for small and large batch, as well as the following methods7 :
6
7
Additional graphs, including comparison to non-adapted regime, are available in appendix (figure 2).
Code is available at https://github.com/eladhoffer/bigBatch.
7
? Learning rate tuningq(LB+LR): Using a large batch, while adapting the learning rate to be
L|
larger so that ?L = |B
|BS | ?S where ?S is the original learning rate used for small batch, ?L
is the adapted learning rate and |BL |, |BS | are the large and small batch sizes, respectively.
? Ghost batch norm (LB+LR+GBN): Additionally using the "Ghost batch normalization"
method in our training procedure. The "ghost batch size" used is 128.
? Regime adaptation: Using the tuned learning rate as well as ghost batch-norm, but with
an adapted training regime. The training regime is modified to have the same number of
iterations for each batch size used - effectively multiplying the number of epochs by the
relative size of the large batch.
Results. Following our experiments, we can establish an empirical basis to our claims. Observing
the final validation accuracy displayed in Table 1, we can see that in accordance with previous works
the move from a small-batch (SB) to a large-batch (LB) indeed incurs a substantial generalization gap.
However, modifying the learning-rate used for large-batch (+LR) causes much of this gap to diminish,
following with an additional improvement by using the Ghost-BN method (+GBN). Finally, we can
see that the generalization gap completely disappears when the training regime is adapted (+RA),
yielding validation accuracy that is good-as or better than the one obtained using a small batch.
We additionally display results obtained on the more challenging ImageNet dataset in Table 2 which
shows similar impact for our methods.
Table 1: Validation accuracy results, SB/LB represent small and large batch respectively. GBN stands
for Ghost-BN, and RA stands for regime adaptation
Network
Dataset
SB
LB
+LR
+GBN
+RA
F1 (Keskar et al., 2017)
C1 (Keskar et al., 2017)
Resnet44 (He et al., 2016)
VGG (Simonyan, 2014)
C3 (Keskar et al., 2017)
WResnet16-4 (Zagoruyko, 2016)
MNIST
Cifar10
Cifar10
Cifar10
Cifar100
Cifar100
98.27%
87.80%
92.83%
92.30%
61.25%
73.70%
97.05%
83.95%
86.10%
84.1%
51.50%
68.15%
97.55%
86.15%
89.30%
88.6%
57.38%
69.05%
97.60%
86.4%
90.50%
91.50%
57.5%
71.20%
98.53%
88.20%
93.07%
93.03%
63.20%
73.57%
Table 2: ImageNet top-1 results using Alexnet topology (Krizhevsky, 2014), notation as in Table 1.
7
Network
LB size
Dataset
SB
LB8
+LR8
+GBN
+RA
Alexnet
Alexnet
4096
8192
ImageNet
ImageNet
57.10%
57.10%
41.23%
41.23%
53.25%
53.25%
54.92%
53.93%
59.5%
59.5%
Discussion
There are two important issues regarding the use of large batch sizes. First, why do we get worse
generalization with a larger batch, and how do we avoid this behaviour? Second, can we decrease the
training wall clock time by using a larger batch (exploiting parallelization), while retaining the same
generalization performance as in small batch?
This work tackles the first issue by investigating the random walk behaviour of SGD and the
relationship of its diffusion rate to the size of a batch. Based on this and empirical observations, we
propose simple set of remedies to close down the generalization gap between the small and large
batch training strategies: (1) Use SGD with momentum, gradient clipping, and a decreasing learning
rate schedule; (2) adapt the learning rate with batch size (we used a square root scaling); (3) compute
batch-norm statistics over several partitions ("ghost batch-norm"); and (4) use a sufficient number of
high learning rate training iterations.
Thus, the main point arising from our results is that, in contrast to previous conception, there is no
inherent generalization problem with training using large mini batches. That is, model training using
8
Due to memory limitation those experiments were conducted with batch of 2048.
8
large mini-batches can generalize as well as models trained using small mini-batches. Though this
answers the first issues, the second issue remained open: can we speed up training by using large
batch sizes?
Not long after our paper first appeared, this issue was also answered. Using a Resnet model on
Imagenet Goyal et al. (2017) showed that, indeed, significant speedups in training could be achieved
using a large batch size. This further highlights the ideas brought in this work and their importance
to future scale-up, especially since Goyal et al. (2017) used similar training practices to those we
described above. The main difference between our works is the use of a linear scaling of the learning
rate9 , similarly to Krizhevsky (2014), and as suggested by Bottou (2010). However, we found that
linear scaling works less well on CIFAR10, and later work found that linear scaling rules work less
well for other architectures on ImageNet (You et al., 2017).
We also note that current "rules of thumb" regarding optimization regime and explicitly learning rate
annealing schedule may be misguided. We showed that good generalization can result from extensive
amount of gradient updates in which there is no apparent validation error change and training error
continues to drop, in contrast to common practice. After our work appeared, Soudry et al. (2017)
suggested an explanation to this, and to the logarithmic increase in the weight distance observed in
Figure 2. We show this behavior happens even in simple logistic regression problems with separable
? + O(1)
data. In this case, we exactly solve the asymptotic dynamics and prove that w(t) = log(t)w
? is to the L2 maximum margin separator. Therefore, the margin (affecting generalization)
where w
improves slowly (as O(1/ log(t)), even while the training error is very low. Future work, based on
this, may be focused on finding when and how the learning rate should be decreased while training.
Conclusion. In this work we make a first attempt to tackle the "generalization gap" phenomenon.
We argue that the initial learning phase can be described using a high-dimensional "random walk
on a random potential" process, with a an "ultra-slow" logarithmic increase in the distance of the
weights from their initialization, as we observe empirically. Following this observation we suggest
several techniques which enable training with large batch without suffering from performance
degradation. Thus implying that the problem is not related to the batch size but rather to the amount
of updates. Moreover we introduce a simple yet efficient algorithm "Ghost-BN" which improves the
generalization performance significantly while keeping the training time intact.
Acknowledgments
We wish to thank Nir Ailon, Dar Gilboa, Kfir Levy and Igor Berman for their feedback on the
initial manuscript. The research was partially supported by the Intelligence Advanced Research
Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DoI/IBC) contract
number D16PC00003. The U.S. Government is authorized to reproduce and distribute reprints for
Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views
and conclusions contained herein are those of the authors and should not be interpreted as necessarily
representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/IBC,
or the U.S. Government.
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6,380 | 6,771 | Flexpoint: An Adaptive Numerical Format for
Efficient Training of Deep Neural Networks
Urs K?ster* , Tristan Webb* , Xin Wang* , Marcel Nassar* , Arjun Bansal, William Constable,
Oguz Elibol, Stewart Hall, Luke Hornof, Amir Khosrowshahi, Carey Kloss, Ruby Pai,
Naveen Rao
Artificial Intelligence Products Group, Intel Corporation
Abstract
Deep neural networks are commonly developed and trained in 32-bit floating point
format. Significant gains in performance and energy efficiency could be realized
by training and inference in numerical formats optimized for deep learning. Despite advances in limited precision inference in recent years, training of neural
networks in low bit-width remains a challenging problem. Here we present the
Flexpoint data format, aiming at a complete replacement of 32-bit floating point
format training and inference, designed to support modern deep network topologies
without modifications. Flexpoint tensors have a shared exponent that is dynamically adjusted to minimize overflows and maximize available dynamic range. We
validate Flexpoint by training AlexNet [1], a deep residual network [2, 3] and a
generative adversarial network [4], using a simulator implemented with the neon
deep learning framework. We demonstrate that 16-bit Flexpoint closely matches
32-bit floating point in training all three models, without any need for tuning of
model hyperparameters. Our results suggest Flexpoint as a promising numerical
format for future hardware for training and inference.
1
Introduction
Deep learning is a rapidly growing field that achieves state-of-the-art performance in solving many
key data-driven problems in a wide range of industries. With major chip makers? quest for novel
hardware architectures for deep learning, the next few years will see the advent of new computing
devices optimized for training and inference of deep neural networks with increasing performance at
decreasing cost.
Typically deep learning research is done on CPU and/or GPU architectures that offer native 64-bit,
32-bit or 16-bit floating point data format and operations. Substantial improvements in hardware
footprint, power consumption, speed, and memory requirements could be obtained with more efficient
data formats. This calls for innovations in numerical representations and operations specifically
tailored for deep learning needs.
Recently, inference with low bit-width fixed point data formats has made significant advancement,
whereas low bit-width training remains an open challenge [5, 6, 7]. Because training in low precision reduces memory footprint and increases the computational density of the deployed hardware
infrastructure, it is crucial to efficient and scalable deep learning applications.
In this paper, we present Flexpoint, a flexible low bit-width numerical format, which faithfully
maintains algorithmic parity with full-precision floating point training and supports a wide range of
deep network topologies, while at the same time substantially reduces consumption of computational
*Equal Contribution
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
resources, making it amenable for specialized training hardware optimized for field deployment of
already existing deep learning models.
The remainder of this paper is structured as follows. In Section 2, we review relevant work in literature.
In Section 3, we present the Flexpoint numerical format along with an exponent management
algorithm that tracks the statistics of tensor extrema and adjusts tensor scales on a per-minibatch
basis. In Section 4, we show results from training several deep neural networks in Flexpoint, showing
close parity to floating point performance: AlexNet and a deep residual network (ResNet) for
image classification, and the recently published Wasserstein GAN. In Section 5, we discuss specific
advantages and limitations of Flexpoint, and compare its merits to those of competing low-precision
training schemes.
2
Related Work
In 2011, Vanhoucke et al. first showed that inference and training of deep neural networks is
feasible with values of certain tensors quantized to a low-precision fixed point format [8]. More
recently, an increasing number of studies demonstrated low-precision inference with substantially
reduced computation. These studies involve, usually in a model-dependent manner, quantization of
specific tensors into low-precision fixed point formats. These include quantization of weights and/or
activations to 8-bit [8, 9, 10, 11], down to 4-bit, 2-bit [12, 13] or ternary [10], and ultimately all
binary [7, 14, 5, 6]. Weights trained at full precision are commonly converted from floating point
values, and bit-widths of component tensors are either pre-determined based on the characteristics
of the model, or optimized per layer [11]. Low-precision inference has already made its way into
production hardware such as Google?s tensor processing unit (TPU) [15].
On the other hand, reasonable successes in low-precision training have been obtained with binarized
[13, 16, 17, 5] or ternarized weights [18], or binarized gradients in the case of stochastic gradient
descent [19], while accumulation of activations and gradients is usually at higher precision. Motivated
by the non-uniform distribution of weights and activations, Miyashita et al. [20] used a logarithmic
quantizer to quantize the parameters and gradients to 6 bits without significant loss in performance.
XNOR-nets focused on speeding up neural network computations by parametrizing the activations and
weights as rank-1 products of binary tensors and higher precision scalar values [7]. This enables the
use of kernels composed of XNOR and bit-count operations to perform highly efficient convolutions.
However, additional high-precision multipliers are still needed to perform the scaling after each
convolution which limits its performance. Quantized Neural Networks (QNNs), and their binary
version (Binarized Nets), successfully perform low-precision inference (down to 1-bit) by keeping
real-valued weights and quantizing them only to compute the gradients and performing forward
inference [17, 5]. Hubara et al. found that low precision networks coupled with efficient bit shiftbased operations resulted in computational speed-up, from experiments performed using specialized
GPU kernels. DoReFa-Nets utilize similar ideas as QNNs and quantize the gradients to 6-bits to
achieve similar performance [6]. The authors also trained in limited precision the deepest ResNet (18
layers) so far.
The closest work related to this manuscript is by Courbariaux et al. [21], who used a dynamical
fixed point (DFXP) format in training a number of benchmark models. In their study, tensors are
polled periodically for the fraction of overflowed entries in a given tensor: if that number exceeds a
certain threshold the exponent is incremented to extend the dynamic range, and vice versa. The main
drawback is that this update mechanism only passively reacts to overflows rather than anticipating
and preemptively avoiding overflows; this turns out to be catastrophic for maintaining convergence of
the training.
3
3.1
Flexpoint
The Flexpoint Data Format
Flexpoint is a data format that combines the advantages of fixed point and floating point arithmetic.
By using a common exponent for integer values in a tensor, Flexpoint reduces computational and
memory requirements while automatically managing the exponent of each tensor in a user transparent
manner.
2
Figure 1: Diagrams of bit representations of different tensorial numerical formats. Red, green and
blue shading each signify mantissa, exponent, and sign bits respectively. In both (a) IEEE 754 32-bit
floating point and (b) IEEE 754 16-bit floating point a portion of the bit string are allocated to specify
exponents. (c) illustrates a Flexpoint tensor with 16-bit mantissa and 5-bit shared exponent.
Flexpoint is based on tensors with an N -bit mantissa storing an integer value in two?s complement
form, and an M -bit exponent e, shared across all elements of a tensor. This format is denoted as
flexN+M. Fig. 1 shows an illustration of a Flexpoint tensor with a 16-bit mantissa and 5-bit exponent,
i.e. flex16+5 compared to 32-bit and 16-bit floating point tensors. In contrast to floating point, the
exponent is shared across tensor elements, and different from fixed point, the exponent is updated
automatically every time a tensor is written.
Compared to 32-bit floating point, Flexpoint reduces both memory and bandwidth requirements in
hardware, as storage and communication of the exponent can be amortized over the entire tensor.
Power and area requirements are also reduced due to simpler multipliers compared to floating point.
Specifically, multiplication of entries of two separate tensors can be computed as a fixed point
operation since the common exponent is identical across all the output elements. For the same reason,
addition across elements of the same tensor can also be implemented as fixed point operations. This
essentially turns the majority of computations of deep neural networks into fixed point operations.
3.2
Exponent Management
These remarkable advantages come at the cost of added complexity of exponent management and
dynamic range limitations imposed by sharing a single exponent. Other authors have reported
on the range of values contained within tensors during neural network training: ?the activations,
gradients and parameters have very different ranges? and ?gradients ranges slowly diminish during
the training? [21]. These observations are promising indicators on the viability of numerical formats
based around tensor shared exponents. Fig. 2 shows histograms of values from different types of
tensors taken from a 110-layer ResNet trained on CIFAR-10 using 32-bit floating point.
In order to preserve a faithful representation of floating point, tensors with a shared exponent must
have a sufficiently narrow dynamic range such that mantissa bits alone can encode variability. As
suggested by Fig. 2, 16-bits of mantissa is sufficient to cover the majority of values of a single
tensor. For performing operations such as adding gradient updates to weights, there must be sufficient
mantissa overlap between tensors, putting additional requirements on number of bits needed to
represent values in training, as compared to inference. Establishing that deep learning tensors
conform to these requirements during training is a key finding in our present results. An alternative
solution to addressing this problem is stochastic rounding [22].
3
Finally, to implement Flexpoint efficiently in hardware, the output exponent has to be determined
before the operation is actually performed. Otherwise the intermediate result needs to be stored in
high precision, before reading the new exponent and quantizing the result, which would negate much
of the potential savings in hardware. Therefore, intelligent management of the exponents is required.
3.3
Exponent Management Algorithm
We propose an exponent management algorithm called Autoflex, designed for iterative optimizations,
such as stochastic gradient descent, where tensor operations, e.g. matrix multiplication, are performed
repeatedly and outputs are stored in hardware buffers. Autoflex predicts an optimal exponent for the
output of each tensor operation based on tensor-wide statistics gathered from values computed in
previous iterations.
The success of training in deep neural networks in Flexpoint hinges on the assumption that ranges
of values in the network change sufficiently slowly, such that exponents can be predicted with high
accuracy based on historical trends. If the input data is independently and identically distributed,
tensors in the network, such as weights, activations and deltas, will have slowly changing exponents.
Fig. 3 shows an example of training a deep neural network model.
The Autoflex algorithm tracks the maximum absolute value ?, of the mantissa of every tensor, by
using a dequeue to store a bounded history of these values. Intuitively, it is then possible to estimate
a trend in the stored values based on a statistical model, use it to anticipate an overflow, and increase
the exponent preemptively to prevent overflow. Similarly, if the trend of ? values decreases, the
exponent can be decreased to better utilize the available range.
We formalize our terminology as follows. After each kernel call, statistics are stored in the floating
point representation ? of the maximum absolute values of a tensor, obtained as ? = ??, by
multiplying the maximum absolute mantissa value ? with scale factor ?. This scale factor is related
to the exponent e by the relation ? = 2?e .
If the same tensor is reused for different computations in the network, we track the exponent e and
the statistics of ? separately for each use. This allows the underlying memory for the mantissa to be
shared across different uses, without disrupting the exponent management.
3.4
Autoflex Initialization
At the beginning of training, the statistics queue is empty, so we use a simple trial-and-error scheme
described in Algorithm 1 to initialize the exponents. We perform each operation in a loop, inspecting
the output value of ? for overflows or underutilization, and repeat until the target exponent is found.
Figure 2: Distributions of values for (a) weights, (b) activations and (c) weight updates, all during the
first epoch (blue) and last epoch (purple) of training a ResNet trained on CIFAR-10 for 165 epochs.
The horizontal axis covers the entire range of values that can be represented in 16-bit Flexpoint, with
the horizontal bars indicating the dynamic range covered by the 16-bit mantissa. All tensors have
a narrow peak close to the right edge of the horizontal bar, where values have close to the same
precision as if the elements had individual exponents.
4
Algorithm 1 Autoflex initialization algorithm. Scales are initialized by repeatedly performing the
operation and adjusting the exponent up in case of overflows or down if not all bits are utilized.
1: initialized ? False
2: ? = 1
3: procedure I NITIALIZE S CALE
4:
while not initialized do
5:
? ? returned by kernel call
6:
if ? ? 2N ?1 ? 1 then
. overflow: increase scale ?
N ?1
7:
? ? ? ? 2b 2 c
8:
else if ? < 2N ?2 then
. underflow: decrease scale ?
9:
? ? ? ? 2dlog2 max (?,1)e?(N ?2)
. Jump directly to target exponent
N ?1
10:
if ? > 2b 2 c?2 then
. Ensure enough bits for reliable jump
11:
initialized ? True
12:
else
. scale ? is correct
13:
initialized ? True
3.5
Autoflex Exponent Prediction
After the network has been initialized by running the initialization procedure for each computation
in the network, we train the network in conjunction with a scale update Algorithm 2 executed
twice per minibatch, once after forward activation and once after backpropagation, for each tensor /
computation in the network. We maintain a fixed length dequeue f of the maximum floating point
values encountered in the previous l iterations, and predict the expected maximum value for the next
iteration based on the maximum and standard deviation of values stored in the dequeue. If an overflow
is encountered, the history of statistics is reset and the exponent is increased by one additional bit.
Algorithm 2 Autoflex scaling algorithm. Hyperparameters are multiplicative headroom factor ? = 2,
number of standard deviations ? = 3, and additive constant ? = 100. Statistics are computed over a
moving window of length l = 16. Returns expected maximum ? for the next iteration.
1: f ? stats dequeue of length l
2: ? ? Maximum absolute value of mantissa, returned by kernel call
3: ? ? previous scale value ?
4: procedure A DJUST S CALE
5:
if ? ? 2N ?1 ? 1 then
. overflow: add one bit and clear stats
6:
clear f
7:
? ? 2?
8:
f ? [f , ??]
. Extend dequeue
9:
? ? ? [max(f ) + ?std(f ) + ??]
. Predicted maximum value for next iteration
10:
? ? 2dlog2 ?e?N +1
. Nearest power of two
3.6
Autoflex Example
We illustrate the algorithm by training a small 2-layer perceptron for 400 iterations on the CIFAR-10
dataset. During training, ? and ? values are stored at each iteration, as shown in Fig. 3, for instance,
a linear layer?s weight, activation, and update tensors. Fig. 3(a) shows the weight tensor, which
is highly stable as it is only updated with small gradient steps. ? slowly approaches its maximum
value of 214 , at which point the ? value is updated, and ? drops by one bit. Shown below is the
corresponding floating point representation of the statistics computed from ?, which is used to
perform the exponent prediction. Using a sliding window of 16 values, the predicted maximum is
computed, and used to set the exponent for the next iteration. In Fig. 3(a), the prediction crosses
the exponent boundary of 23 about 20 iterations before the value itself does, safely preventing an
overflow. Tensors with more variation across epochs are shown in Fig. 3(b) (activations) and Fig. 3(c)
(updates). The standard deviation across iterations is higher, therefore the algorithm leaves about half
a bit and one bit respectively of headroom. Even as the tensor fluctuates in magnitude by more than a
factor of two, the maximum absolute value of the mantissa ? is safely prevented from overflowing.
5
Weights
213
13
211
?9
?8
?
214
2
213
29
?16
2
?
2
Updates
Activations
214
2?17
2
2?10
2?10
2?11
2?12
25
24
?3
?
2
24
Raw value
Max
2?3.2
Estimate
23
0
100
200
300
0
100
200
300
(b)
(a)
22
20
0
100
200
300
(c)
Figure 3: Evolution of different tensors during training with corresponding mantissa and exponent
values. The second row shows the scale ?, adjusted to keep the maximum absolute mantissa values
(?, first row) at the top of the dynamic range without overflowing. As the product of the two (?,
third row) is anticipated to cross a power of two boundary, the scale is changed so as to keep the
mantissa in the correct range. (a) Shows this process for a weight tensor, which is very stable and
slowly changing. The black arrow indicates how scale changes are synchronized with crossings of
the exponent boundary. (b) shows an activation tensor with a noisier sequence of values. (c) shows
a tensor of updates, which typically displays the most frequent exponent changes. In each case the
Autoflex estimate (green line) crosses the exponent boundary (gray horizontal line) before the actual
data (red) does, which means that exponent changes are predicted before an overflow occurs.
The cost of this approach is that in the last example ? reaches 3 bits below the cutoff, leaving the top
bits zero and using only 13 of the 16 bits for representing data.
3.7
Simulation on GPU
The experiments described below were performed on Nvidia GPUs using the neon deep learning
framework1 . In order to simulate the flex16+5 data format we stored tensors using an int16 type.
Computations such as convolution and matrix multiplication were performed with a set of GPU
kernels which convert the underlying int16 data format to float32 by multiplying with ?, perform
operations in floating point, and convert back to int16 before returning the result as well as ?. The
kernels also have the ability to compute only ? without writing any outputs, to prevent writing invalid
data during exponent initialization. The computational performance of the GPU kernels is comparable
to pure floating point kernels, so training models in this Flexpoint simulator adds little overhead.
4
Experimental Results
4.1
Convolutional Networks
We trained two convolutional networks in flex16+5, using float32 as a benchmark: AlexNet [1],
and a ResNet [2, 3]. The ResNet architecture is composed of modules with shortcuts in the dataflow
graph, a key feature that makes effective end-to-end training of extremely deep networks possible.
These multiple divergent and convergent flows of tensor values at potentially disparate scales might
pose unique challenges for training in fixed point numerical format.
We built a ResNet following the design as described in [3]. The network has 12 blocks of residual
modules consisting of convolutional stacks, making a deep network of 110 layers in total. We trained
this model on the CIFAR-10 dataset [1] with float32 and flex16+5 data formats for 165 epochs.
Fig. 4 shows misclassification error on the validation set plotted over the course of training. Learning
curves match closely between float32 and flex16+5 for both networks. In contrast, models trained
1
Available at https://github.com/NervanaSystems/neon.
6
in float16 without any changes in hyperparameter values substantially underperformed those trained
in float32 and flex16+5.
4.2
Generative Adversarial Networks
Next, we validate training a generative adversarial network (GAN) in flex16+5. By virtue of an adversarial (two-player game) training process, GAN models provide a principled way of unsupervised
learning using deep neural networks. The unique characteristics of GAN training, namely separate
data flows through two components (generator and discriminator) of the network, in addition to feeds
of alternating batches of real and generated data of drastically different statistics to the discriminator
at early stages of the training, pose significant challenges to fixed point numerical representations.
We built a Wasserstein GAN (WGAN) model [4], which has the advantage of a metric, namely
the Wasserstein-1 distance, that is indicative of generator performance and can be estimated from
discriminator output during training. We trained a WGAN model with the LSUN [23] bedroom
dataset in float32, flex16+5 and float16 formats with exactly the same hyperparameter settings.
As shown in Fig. 5(a), estimates of the Wasserstein distance in flex16+5 training and in float32
training closely tracked each other. In float16 training the distance deviated significantly from
baseline float32, starting with an initially undertrained discriminator. Further, we found no differences in the quality of generated images between float32 and flex16+5 at specific stages of
the training 5(b), as quantified by the Fr?chet Inception Distance (FID) [24]. Generated images
from float16 training had lower quality (significantly higher FIDs, Fig. 5(b)) with noticeably more
saturated patches, examples illustrated in Fig. 5(c), 5(d) and 5(e).
5
Discussion
In the present work, we show that a Flexpoint data format, flex16+5, can adequately support training
of modern deep learning models without any modifications of model topology or hyperparameters,
achieving a numerical performance on par with float32, the conventional data format widely used
in deep learning research and development. Our discovery suggests a potential gain in efficiency and
performance of future hardware architectures specialized in deep neural network training.
Alternatives, i.e. schemes that more aggressively quantize tensor values to lower bit precisions, also
made significant progress recently. Here we list major advantages and limitations of Flexpoint, and
make a detailed comparison with competing methods in the following sections.
0.4
flex16+5
Top 1 Misclassification error
Top 5 Misclassification error
flex16+5
float32
float16
0.6
0.4
0.2
0
10
20
30
40
50
float32
float16
0.3
0.2
0.1
0
Epoch
50
100
150
Epoch
(a) ImageNet1k AlexNet
(b) CIFAR-10 ResNet
Figure 4: Convolutional networks trained in flex16+5 and float32 numerical formats. (a) AlexNet
trained on ImageNet1k, graph showing top-5 misclassification on the validation set. (b) ResNet of
110 layers trained on CIFAR-10, graph showing top-1 misclassification on the validation set.
7
Distinct from very low precision (below 8-bit) fixed point quantization schemes which significantly
alter the quantitative behavior of the original model and thus requires completely different training
algorithms, Flexpoint?s philosophy is to maintain numerical parity with the original network training
behavior in high-precision floating point. This brings about a number of advantages. First, all prior
knowledge of network design and hyperparameter tuning for efficient training can still be fully
leveraged. Second, networks trained in high-precision floating point formats can be readily deployed
in Flexpoint hardware for inference, or as component of a bigger network for training. Third, no
re-tuning of hyperparameters is necessary for training in Flexpoint?what works with floating point
simply works in Flexpoint. Fourth, the training procedure remains exactly the same, eliminating the
need of intermediate high-precision representations, with the only exception of intermediate higher
precision accumulation commonly needed for multipliers and adders. Fifth, all Flexpoint tensors are
managed in exactly the same way by the Autoflex algorithm, which is designed to be hidden from
the user, eliminating the need to remain cognizant of different type of tensors being quantized into
different bit-widths. And finally, the AutoFlex algorithm is robust enough to accommodate diverse
deep network topologies, without the need of model-specific tuning of its hyperparameters.
Despite these advantages, the same design philosophy of Flexpoint likely prescribes a potential
limitation in performance and efficiency, especially when compared to more aggressive quantization
schemes, e.g. Binarized Networks, Quantized Networks and the DoReFa-Net. However, we believe
Flexpoint strikes a desirable balance between aggressive extraction of performance and support
for a wide collection of existing models. Furthermore, potentials and implications for hardware
1.5
Fr?chet Inception Distance (FID)
300
flex16+5
float32
Wasserstein estimate
float16
1
0.5
200
100
0
flex16+5
float32
float16
0
0
50,000
100,000
150,000
200,000
0
10
15
20
25
Number of trained epochs
Generator iteration
(a) Training performance
(c) float32
5
(b) Quality of generated images
(d) flex16+5
(e) float16
Figure 5: Training performance of WGAN in flex16+5, float32 and float16 data formats.
(a) Learning curves, i.e. estimated Wasserstein distance by median filtered and down-sampled values
of the negative discriminator cost function, median filter kernel length 100 [4], and down-sampling by
plotting every 100th value. Examples of generated images by the WGAN trained with in (c) float32,
(d) flex16+5 and (e) float16 for 16 epochs. Fr?chet Inception Distance (FID) estimated from
5000 samples of the generator, as in [24].
8
architecture of other data formats in the Flexpoint family, namely flexN+M for certain (N , M ), are
yet to be explored in future investigations.
Low-precision data formats: TensorFlow provides tools to quantize networks into 8-bit for inference [9]. TensorFlow?s numerical format shares some common features with Flexpoint: each tensor
has two variables that encode the range of the tensor?s values; this is similar to Autoflex ? (although
it uses fewer bits to encode the exponent). Then an integer value is used to represent the dynamic
range with a dynamic precision.
The dynamic fixed point (DFXP) numerical format, proposed in [25], has a similar representation
as Flexpoint: a tensor consists of mantissa bits and values share a common exponent. This format
was used by [21] to train various neural nets in low-precision with limited success (with difficulty to
match CIFAR-10 maxout nets in float32). DFXP diverges significantly from Flexpoint in automatic
exponent management: DFXP only updates the shared exponent at intervals specified by the user
(e.g. per 100 minibatches) and solely based on the number of overflows occurring. Flexpoint is more
suitable for training modern networks where the dynamics of the tensors might change rapidly.
Low-precision networks: While allowing for very efficient forward inference, the low-precision
networks discussed in Section 2 share the following shortcomings when it comes to neural network
training. These methods utilize an intermediate floating point weight representation that is also
updated in floating point. This requires special hardware to perform these operations in addition to
increasing the memory footprint of the models. In addition, these low-precision quantizations render
the models so different from the exact same networks trained in high-precision floating point formats
that there is often no parity at the algorithmic level, which requires completely distinct training
algorithms to be developed and optimized for these low-precision training schemes.
6
Conclusion
To further scale up deep learning the future will require custom hardware that offers greater compute capability, supports ever-growing workloads, and minimizes memory and power consumption.
Flexpoint is a numerical format designed to complement such specialized hardware.
We have demonstrated that Flexpoint with a 16-bit mantissa and a 5-bit shared exponent achieved
numerical parity with 32-bit floating point in training several deep learning models without modifying
the models or their hyperparameters, outperforming 16-bit floating point under the same conditions.
Thus, specifically designed formats, like Flexpoint, along with supporting algorithms, such as
Autoflex, go beyond current standards and present a promising ground for future research.
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11
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6,381 | 6,772 | Model evidence from nonequilibrium simulations
Michael Habeck
Statistical Inverse Problems in Biophysics, Max Planck Institute for Biophysical Chemistry &
Institute for Mathematical Stochastics, University of G?ttingen, 37077 G?ttingen, Germany
email [email protected]
Abstract
The marginal likelihood, or model evidence, is a key quantity in Bayesian parameter estimation and model comparison. For many probabilistic models, computation
of the marginal likelihood is challenging, because it involves a sum or integral over
an enormous parameter space. Markov chain Monte Carlo (MCMC) is a powerful
approach to compute marginal likelihoods. Various MCMC algorithms and evidence estimators have been proposed in the literature. Here we discuss the use of
nonequilibrium techniques for estimating the marginal likelihood. Nonequilibrium
estimators build on recent developments in statistical physics and are known as
annealed importance sampling (AIS) and reverse AIS in probabilistic machine
learning. We introduce estimators for the model evidence that combine forward and
backward simulations and show for various challenging models that the evidence
estimators outperform forward and reverse AIS.
1
Introduction
The marginal likelihood or model evidence is a central quantity in Bayesian inference [1, 2], but
notoriously difficult to compute. If likelihood L(x) ? p(y|x, M ) models data y and prior ?(x) ?
p(x|M ) expresses our knowledge about the parameters x of the model M , the posterior p(x|y, M )
and the model evidence Z are given by:
Z
p(y|x, M ) p(x|M )
L(x) ?(x)
p(x|y, M ) =
=
, Z ? p(y|M ) = L(x) ?(x) dx .
(1)
p(y|M )
Z
Parameter estimation proceeds by drawing samples from p(x|y, M ), and different ways to model the
data are ranked by their evidence. For example, two models M1 and M2 can be compared via their
Bayes factor, which is proportional to the ratio of their marginal likelihoods p(y|M1 )/p(y|M2 ) [3].
Often the posterior (and perhaps also the prior) is intractable in the sense that it is not possible to
compute the normalizing constant and therefore also the evidence analytically. This makes it difficult
to compare different models via their posterior probability and model evidence. Markov chain Monte
Carlo (MCMC) algorithms [4] only require unnormalized probability distributions and are among the
most powerful and accurate methods to estimate the marginal likelihood, but they are computationally
expensive. Therefore, it is important to develop efficient MCMC algorithms that can sample from the
posterior and allow us to compute the marginal likelihood.
There is a close analogy between the marginal likelihood and the log-partition function or free energy
from statistical physics [5]. Therefore, many concepts and algorithms originating in statistical physics
have been applied to problems arising in probabilistic inference. Here we show that nonequilibrium
fluctuation theorems (FTs) [6, 7, 8] can be used to estimate the marginal likelihood from forward and
reverse simulations.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
2
Marginal likelihood estimation by annealed importance sampling
A common MCMC strategy to sample from the posterior and estimate the evidence is to simulate a
sequence of distributions pk that bridge between the prior and the posterior [9]. Samples can either
be generated in sequential order as in annealed importance sampling (AIS) [10] or in parallel as in
replica-exchange Monte Carlo or parallel tempering [11, 12]. Several methods have been proposed to
estimate the marginal likelihood from MCMC samples including thermodynamic integration (TI)
[9], annealed importance sampling (AIS) [10], nested sampling (NS) [13] or the density of states
(DOS) [14]. Most of these approaches (TI, DOS and NS) assume that we can draw exact samples
from the intermediate distributions pk , typically after a sufficiently large number of equilibration
steps has been simulated. AIS, on the other hand, does not assume that the samples are equilibrated
after each annealing step, which makes AIS very attractive for analyzing complex models for which
equilibration is hard to achieve.
AIS employs a sequence of K + 1 probability
R distributions pk and Markov transition operators Tk
whose stationary distributions are pk , i.e. Tk (x|x0 ) pk (x0 ) dx0 = pk (x). In a Bayesian setting,
p0 is the prior and pK the posterior. Typically, pk is intractable meaning that we only
R know an
unnormalized version fk , but not the normalizer Zk , i.e. pk (x) = fk (x)/Zk where Zk = fk (x) dx
and the evidence is Z = ZK /Z0 . Often, it is convenient to write fk as an energy based model
fk (x) = exp{?Ek (x)}. In Bayesian inference, a popular choice is
fk (x) = [L(x)]?k ?(x)
with prior ?(x) and likelihood L(x); ?k is an inverse temperature schedule that starts at ?0 = 0
(prior) and ends at ?K = 1 (posterior).
AIS samples paths x = [x0 , x1 , . . . , xK?1 ] according to the probability
Pf [x] = TK?1 (xK?1 |xK?2 ) ? ? ? T1 (x1 |x0 ) p0 (x0 )
(2)
where, following Crooks [8], calligraphic symbols and square brackets denote quantities that depend
on the entire path. The subscript indicates that the path is generated by a forward simulation, which
starts from p0 and propagates the initial state through a sequence of new states by the successive
action of the Markov kernels T1 , T2 , . . . , TK?1 .
The importance weight of a path is
w[x] =
K?1
Y
k=0
K?1
X
fk+1 (xk )
= exp ?
[ Ek+1 (xk ) ? Ek (xk ) ] .
fk (xk )
(3)
k=0
The average weight over many paths is a consistent and unbiased estimator of the model evidence
Z = ZK /Z0 , which follows from [15, 10] (see supplementary material for details):
Z
hwif = w[x] Pf [x] D[x] = Z
(4)
where the average h?if is an integral over all possible paths generated by the forward sequence
of transition kernels (D[x] = dx0 ? ? ? dxK?1
average weight of M forward paths x(i) is an
P ). The
1
(i)
estimate of the model evidence: Z ? M i w[x ]. This estimator is at the core of AIS and its
variants [10, 16]. To avoid overflow problems, it will be numerically more stable to compute log
weights. Logarithmic weights also naturally occur from a physical perspective where ? log w[x] is
identified as the work required to generate path x.
3
Nonequilibrium fluctuation theorems
Nonequilibrium fluctuations theorems (FTs) quantify the degree of irreversibility of a stochastic
process by relating the probability of generating a path by a forward simulation to the probability
of generating the exact same path by a time-reversed simulation. If the Markov kernels Tk satisfy
detailed balance, time reversal is achieved by applying the same sequence of kernels in reverse order.
For Markov kernels not satisfying detailed balance, the definition is slightly more general [7, 10].
Here we assume that all kernels Tk satisfy detailed balance, which is valid for Markov chains based
on the Metropolis algorithm and its variants [4].
2
Under these assumptions, the probability of generating the path x by a reverse simulation starting in
xK?1 is
Pr [x] = T1 (x0 |x1 ) ? ? ? TK?1 (xK?2 |xK?1 ) pK (xK?1 ) .
(5)
Averages over the reverse paths are indicated by h?ir . The detailed fluctuation theorem [6, 8] relates
the probabilities of generating x in a forward and reverse simulation (see supplementary material):
K?1
Pf [x]
ZK Y fk (xk )
Z
=
=
= exp{W[x] ? ?F }
Pr [x]
Z0
fk+1 (xk )
w[x]
(6)
k=0
where the physical analogs of theP
path weight and the marginal likelihood were introduced, namely
the work W[x] = ? log w[x] = k [ Ek+1 (xk ) ? Ek (xk ) ] and the free energy difference ?F =
? log Z = ? log(ZK /Z0 ). Various demonstrations of relation (6) exist in the physics and machine
learning literature [6, 7, 8, 10, 17].
Lower and upper bounds sandwiching the log evidence [17, 18, 16] follow directly from equation (6)
and the non-negativity of the relative entropy:
Z
DKL (Pf kPr ) = Pf [x] log(Pf [x]/Pr [x]) D[x] = hWif ? ?F ? 0 .
From DKL (Pr kPf ) ? 0 we obtain an upper bound on log Z, such that overall we have:
hlog wif = ?hWif ? log Z ? ?hWir = hlog wir .
(7)
Grosse et al. use these bounds to assess the convergence of bidirectional Monte Carlo [18].
Thanks to the detailed fluctuation theorem (Eq. 6), we can also relate the marginal distributions of
the work resulting from many forward and reverse simulations:
Z
pf (W ) = ?(W ? W[x]) Pf [x] D[x] = pr (W ) eW ??F
(8)
which is Crooks? fluctuation theorem (CFT) [7]. CFT tells us that the work distributions pf and pr
cross exactly at W = ?F . Therefore, by plotting histograms of the work obtained in forward and
backward simulations we can read off an estimate for the negative log evidence.
The Jarzynski equality (JE) [15] follows directly from CFT due to the normalization of pr :
Z
pf (W ) e?W dW = he?W if = e??F .
(9)
JE restates the AIS estimator hwif = Z (Eq. 4) in terms of the physical quantities. Remarkably, JE
holds for any stochastic dynamics bridging between the initial and target distribution. This suggests
to use fast annealing protocols to drag samples from the prior into the posterior. However, the JE
involves an exponential average in which paths requiring the least work contribute most strongly.
These paths correspond to work realizations that reside in the left tail of pf . With faster annealing,
the chance of generating a minimal work path decreases exponentially and becomes a rare event.
A key feature of CFT and JE is that they do not require exact samples from the stationary distributions
pk , which is needed in applications of TI or DOS. For complex probabilistic models, the states
generated by the kernels Tk will typically ?lag behind? due to slow mixing, especially near phase
transitions. The k-th state of the forward path will follow the intermediate distribution
qk (xk ) =
Z Y
k
Tl (xl |xl?1 ) p0 (x0 ) dx0 ? ? ? dxk?1 ,
q0 (x) = p0 (x) .
(10)
l=1
Unless the transition kernels Tk mix very rapidly, qk 6= pk for k > 0.
Consider the common case in Bayesian inference where Ek (x) = ?k E(x) and E(x) = ? log L(x).
Then, according to inequalities (7), we have the following lower bound on the marginal likelihood
hlog wif = ?hWif = ?
K?1
X
(?k+1 ? ?k ) hEiqk
k=0
3
(11)
0.3
pf(W)
pr(W)
logZ
0.2
0.1
0.0
30
B
20
pk
qk
10
work?W
A
parameter?x
work?distribution?p(W)
0.4
5
10 15 20
work?W
C
5
Wf
Wr
log e W f
logZ
10
0
0
0
0
5
10
index?k
0
50 100 150 200
K
Figure 1: Nonequilibrium analysis of a Gaussian toy model. (A) Work distributions pf and pr shown
in blue and green. The correct free energy difference (minus log evidence) is indicated by a dashed
line. (B) Comparison of stationary distribution pk and marginal distributions qk generated by the
transition kernels. Shown is a 1? band about the mean positions for pk (blue) and qk (green). (C)
Lower and upper bounds of the log evidence (Eq. 7) and logarithm of the exponential average over
the forward work distribution for increasingly slow annealing schedules.
and an analogous expression for the upper bound/reverse direction, in which the average energies
along the forward path hEiqk need to be replaced by the corresponding average energies along the
backward path. The difference between the forward and reverse averages is called ?hysteresis? in
physics. The larger the hysteresis, the more strongly will the marginal likelihood bounds disagree
and the more uncertain will our guess of the model evidence be. The opposite limiting case is slow
annealing and full equilibration where the bound (Eq. 11) approaches thermodynamic integration
(see supplementary material). So we expect that there is a tradeoff between switching fast in order to
save computation time, versus a desire to control the amount of hysteresis, which otherwise makes it
difficult to extract accurate evidence estimates from the simulations.
4
Illustration for a tractable model
Let us illustrate the nonequilibrium results for a tractable model where the initial, the target and
all intermediate distribution are Gaussians pk (x) = N x; ?k , ?k2 ) with means ?k and standard
deviations ?k > 0. The transition kernels are also Gaussian:
Tk (x|x0 ) = N x; (1 ? ?k )?k + ?k x0 , (1 ? ?k2 )?k2 )
with ?k ? [0, 1] controlling the speed of convergence: For ?k = 0 convergence is immediate, whereas
for ?k ? 1, the chain generated by Tk converges infinitely slowly. Note that the kernels Tk satisfy
detailed balance, therefore backward simulations apply the same kernels in reverse order. The
energies and exact log partition functions are Ek (x) = 2?1 2 (x ? ?k )2 and log Zk = log(2??k2 )/2.
k
We bridge between an initial distribution with mean ?0 = 20 and standard deviation ?0 = 10
and a target with ?K = 0, ?K = 1 using K = 10 intermediate distributions and compute work
distributions resulting from forward/backward simulations. Both distributions indeed cross at W =
? log Z = log(?0 /?K ) = log 10, as predicted by CFT (see Fig. 1A). Figure 1B illustrates the
difference between the marginal distribution of the samples after k annealing steps qk (Eq. 10) and
the stationary distribution pk . The marginal distributions qk are also Gaussian, but their means and
variances diverge from the means and variances of the stationary distributions pk . This divergence
results in hysteresis, if the annealing process is forced to progress very rapidly without equilibrating
the samples (quenching). Figure 1C confirms the validity of the JE (Eq. 9) and of the lower and
upper bounds (Eq. 7). For short annealing protocols, the bounds are very conservative, whereas
the Jarzynski equality gives the correct evidence even for fast protocols (small K). In realistic
applications, however, we cannot compute the work distribution pf over the entire range of work
values. In fast annealing simulations, it will become increasingly difficult to explore the left tail of the
work distribution, such that in practice the accuracy of the JE estimator deteriorates for too small K.
4
Algorithm 1 Bennett?s acceptance ratio (BAR)
(i)
(i)
Require: Work Wf , Wr from M forward and reverse nonequilibrium simulations, tolerance ?
(e.g. ? = 10?4 )
P
(i)
1
Z?M
i exp{?Wf } (Jarzynski estimator)
repeat
LHS ?
1
i 1+Z exp{W (i) } ,
f
LHS
RHS
P
Z?Z?
RHS ?
1
i 1+Z ?1 exp{?W (i) }
r
P
until | log(LHS/RHS)| < ?
return Z
5
Using the fluctuation theorem to estimate the evidence
To use the fluctuation theorem for evidence estimation, we run two sets of simulations. As in AIS,
forward simulations start from a prior sample which is successively propagated by the transition
(i)
kernels Tk . For each forward path x(i) the total work Wf is recorded. We also run reverse simulations starting from a posterior sample. For complex inference problems it is generally impossible
to generate an exact sample from the posterior. However, in some cases the mode of the posterior
is known or powerful methods for locating the posterior maximum exist. We can then generate
a posterior sample by applying the transition operator TK many times starting from the posterior
mode. The reverse simulations could also be started from the final states generated by the forward
(i)
(i)
simulations drawn according to their importance weights wf ? exp{?Wf }. Another possibility
to generate a posterior sample is to start from the data, if we want to evaluate an intractable generative
model such as a restricted Boltzmann machine. The posterior sample is then propagated by the
reverse chain of transition operators. Again, we accumulate the total work generated by the reverse
(i)
simulation Wr . The reverse simulation corresponds to reverse AIS proposed by Burda et al. [16].
5.1
Jarzynski and cumulant estimators
There are various options to estimate the evidence from forward and backward simulations. We can
(i)
(i)
apply the Jarzynski equality to Wf and Wr , which corresponds to the estimators used in AIS [10]
and reverse AIS [16]. According to Eq. (7) we can also compute an interval that likely contains the
log evidence, but typically this interval will be quite large. Hummer [19] has developed estimators
based on the cumulant expansion of pf and pr :
log Z ? ?hWif + varf (W)/2,
log Z ? ?hWir ? varr (W)/2
(12)
where varf (W) and varr (W) indicate the sample variances of the work generated during the forward
and reverse simulations. The cumulant estimators increase/decrease the lower/upper bound of the log
evidence (Eq. 7) by the sample variance of the work. The forward and reverse cumulant estimators
can also be combined into a single estimate [19]:
log Z ? ?
5.2
1
1
hWif + hWir +
varf (W) ? varr (W) .
2
12
(13)
Bennett?s acceptance ratio
Another powerful method is Bennett?s acceptance ratio (BAR) [20, 21], which is based on the
observation that according to CFT (Eq. 8)
Z
Z
?W
h(W ; ?F ) pf (W ) e
dW = h(W ; ?F ) pr (W ) e??F dW
for any function h. Therefore, any choice of h gives an implicit estimator for ?F . Bennett showed
[20, 9] that the minimum mean squared error is achieved for h ? (pf + pr )?1 , leading to the implicit
5
error
0.2
0.0
A
0.2
0
100
K
B
C
D
E
F
200
Figure 2: Performance of evidence estimators on the Gaussian toy model. M = 100 forward and
reverse simulations were run for schedules of increasing length K. This experiment was repeated
1000 times to probe the stability of the estimators. Shown is the difference between the log evidence
estimate and its true value ? log 10. The average over all repetitions is shown as red line; the light
band indicates one standard deviation over all repetitions. (A) Cumulant estimator (Eq. 12) based on
the forward simulation. (B) The combined cumulant estimator (Eq. 13). (C) Forward AIS estimator.
(D) Reverse AIS. (E) BAR. (F) Histogram estimator.
equation
1
X
1+Z
i
(i)
exp{Wf }
=
X
1
i
1 + Z ?1 exp{?Wr }
(i)
.
(14)
By numerically solving equation (14) for Z, we obtain an estimator of the evidence based on
Bennett?s acceptance ratio (BAR). A straightforward way to solve the BAR equation is to iterate over
the multiplicative update Z (t+1) ? Z (t) LHS(Z (t) )/RHS(Z (t) ) where LHS and RHS are the left
and right hand side of equation (14) and the superscript (t) indicates an iteration index. Algorithm (1)
provides pseudocode to compute the BAR estimator (further details are given in the supplementary
material).
5.3
Histogram estimator
Here we introduce yet another way of combining forward/backward simulations and estimating the
model evidence. According to CFT, we have:
(i)
Wr(i) ? pr (W ) = pf (W ) e?W /Z .
Wf ? pf (W ),
(i)
(i)
The idea is to combine all samples Wf and Wr to estimate pf , from which we can then obtain
the evidence by using the JE (Eq. 9). Thanks to the CFT, the samples from the reverse simulation
contribute most strongly to the integral in the JE. Therefore, if we can use the reverse paths to
estimate the forward work distribution, pf will be quite accurate in the region that is most relevant
for evaluating JE.
(i)
(i)
To estimate pf from Wf and Wr is mathematically equivalent to estimating the density of states
(DOS) (i.e. the marginal distribution of the log likelihood) from equilibrium simulations run at two
inverse temperatures ? = 0 and ? = 1. We can therefore directly apply histogram techniques [14, 22]
used to analyze equilibrium simulations to estimate pf from nonequilibrium simulations (details are
given in the supplementary material). Histogram techniques result in a non-parametric estimate of
the work distribution:
X
pf (W ) ?
pj ?(W ? Wj )
(15)
j
(i)
Wf
(i)
Wr ,
where all sampled work values,
and
were combined into a single set Wj and pj are
normalized weights associated with each Wj . Using the JE, we obtain
X
Z?
pj e?Wj
(16)
j
which is best evaluated in log space. The histogram iterations [14] used to determine pj and Z are
very similar to the multiplicative that solve the BAR equation (Eq. 14). After running the histogram
iterations, we obtain a non-parametric maximum likelihood estimate of pf (Eq. 15). It is also possible
6
A
0.75
pf(W)
pr(W)
logZ
0.50
0.25
0.00
1350 1300 1250
work?W
B
1.0
Ef
Er
E
1.5
2.0
0.4
0.5
0.6
0.7
inverse?temperature?
log?evidence?logZ
1e3
average?energy? E
work?distribution?p(W)
1.00 1e 1
1.35 1e3
C
1.34
1.33
103
104
equilibration?steps?N
Figure 3: Evidence estimation for the 32 ? 32 Ising model. (A) Work distributions obtained for
K = 1000 annealing and N = 1000 equilibration steps. (B) Average energy hEif and hEir at
different annealing steps k in comparison to the average energy of the stationary distribution hEi? .
Shown is a zoom into the inverse temperature range from 0.4 to 0.7; the average energies agree
quite well outside this interval. (C) Evidence estimates for increasing number of equilibration steps
N . Light/dark blue: lower/upper bounds hlog wif / hlog wir ; light/dark green: forward/reverse AIS
estimators loghwif / loghwir ; light red: BAR; dark red: histogram estimator. For N > 1000, BAR
and the histogram estimator produce virtually identical evidence estimates.
to carry out a Bayesian analysis, and derive a Gibbs sampler for pf , which does not only provide a
point estimate for log Z, but also quantifies its uncertainty (see supplementary material for details).
We studied the performance of the evidence estimators on forward/backward simulations of the
Gaussian toy model. The cumulant estimators (Figs. 2A, B) are systematically biased in case of rapid
annealing (small K). The combined cumulant estimator (Fig. 2B) is a significant improvement over
the forward estimator, which does not take the reverse simulation data into account. The forward
and reverse AIS estimators are shown in Figs. 2C and 2D. For this system, the evidence estimates
derived from the reverse simulation are systematically more accurate than the AIS estimate based on
the forward simulation, which is clear given that the work distribution from reverse simulations pr is
much more concentrated than the forward work distribution pf (see Fig. 1A). The most accurate,
least biased and most stable estimators are BAR (Fig. 2E) and the histogram estimator (Fig. 2F),
which both combine forward and backward simulations into a single evidence estimate.
6
Experiments
We studied the performance of the nonequilibrium marginal likelihood estimators on various challenging probabilistic models including Markov random fields and Gaussian mixture models. A
python package implementing the work simulations and evidence estimators can be downloaded from
https://github.com/michaelhabeck/paths.
6.1
Ising model
Our first test system is a 32 ? 32 Ising model for which the log evidence can be computed exactly:
log Z = 1339.27 [23]. A single configurationP
consists of 1024 spins xi = ?1. The energies of
the intermediate distributions are Ek (x) = ?k i?j xi xj where i ? j indicates nearest neighbors
on a 2D square lattice. We generate M = 1000 forward and reverse paths using a linear inverse
temperature schedule that interpolates between ?0 = 0 and ?K = 1 where K = 1000. Forward
simulations start from random spin configurations. For the reverse simulations, we start in one of the
two ground states with all spins either ?1 or +1. Tk are Metropolis kernels based on pk : a new spin
configuration is proposed by flipping a randomly selected spin and accepted or rejected according
to Metropolis? rule. The single spin-flip transitions are repeated N times at constant ?k , i.e. N is
the number of equilibration steps after a perturbation was induced by lowering the temperature. The
larger N , the more time we allow the simulation to equilibrate, and the closer will qk be to pk .
Figure 3A shows the work distributions obtained with N = 1000 equilibration steps per temperature
perturbation. Even though the forward and reverse work distributions overlap only weakly, the
7
logZ
50
40
30
103
A
N = 105
N = 106
N = 107
N = 108
B
6
4
logw f
logw r
log w f
log w r
BAR
Histogram
104
K
105
0.15
p(W)
+1.7e3
2
0
C
parallel
tempering
0.10
0.05
1
2
logZ
3
4
+1.74e3
0.00
475
logZ
450
Figure 4: Evidence estimation for the Potts model and RBM. (A) Estimated log evidence of the
Potts model for a fixed computational budget K ? N = 109 where M = 100 and ten repetitions
were computed. The reference value log Z = 1742 (obtained with parallel tempering) is shown as
dashed black line. (B) log Z distributions obtained with the Gibbs sampling version of the histogram
estimator for K = 1000 and varying number of equilibration steps. (C) Work distributions obtained
for a marginal and full RBM (light/dark blue: forward/reverse simulation of the marginal model;
light/dark green: forward/reverse simulation of the full model).
evidence estimates obtained with BAR and the histogram estimator are quite accurate with 1338.05
(BAR) and 1338.28 (histogram estimator), which differs only by approx. 1 nat from the true evidence
and corresponds to a relative error of ? 9 ? 10?4 (BAR) and 7 ? 10?4 (histogram estimator). Forward
and reverse AIS provide less accurate estimates of the log evidence: 1333.66 (AIS) and 1342.05
(RAISE). The lower and upper bounds are very broad hlog wif = 1290.5 and hlog wir = 1352.0,
which results from hysteresis effects. Figure 3B zooms into the average energies obtained during the
forward and reverse simulations and compares them with the average energy of a fully equilibrated
simulation. The average energies differ most strongly at inverse temperatures close to the critical
value ?crit ? 0.44 at which the Ising model undergoes a second-order phase transition. We also tested
the performance of the estimators as a function of the number of equilibration steps N . As already
observed for the Gaussian toy model, BAR and the histogram estimator outperform the Jarzynski
estimators (AIS and RAISE) also in case of the Ising model (see Fig. 3C).
6.2
Ten-state Potts model
Next we performed simulations of the ten-state Potts model defined over a 32 ? 32 lattice where the
spins of the Ising model are replaced by integer colors xi ? {1, . . . , 10} and an interaction energy
2?(xi , xj ). This model is significantly more challenging than the Ising model, because it undergoes a
first-order phase transition and has an astronomically larger number of states (101024 colorings rather
than 21024 spin configurations). We performed forward/backward simulations using a linear inverse
temperature schedule with ?0 = 0, ?K = 1 and a fixed computational budget K ? N = 109 . Figure
4A shows that there seems to be no advantage of increasing the number of intermediate distributions
at the cost of reducing the number of equilibration steps. Again, BAR and the histogram estimator
perform very similarly. The Gibbs sampling version of the histogram estimator also provides the
posterior of log Z (see Fig. 4B). For too few equilibration steps N , this distribution is rather broad or
even slightly biased, but for large N the log Z posterior concentrates around the correct log evidence.
6.3
Restricted Boltzmann machine
The restricted Boltzmann machine (RBM) is a common building block of deep learning hierarchies.
RBM is an intractable MRF with bipartite interactions: E(v, h) = ?(aT v + bT h + v T W h) where
a, b are the visible and hidden biases and W are the couplings between the visible and hidden units
vi and hj . Here we compare annealingP
of the full model Ek (v, h) = ?k E(v, h) against annealing of
the marginal model Ek (h) = ??k log v exp{?E(v, h)}. The full model can be simulated using a
Gibbs sampler, which is straightforward since the conditional distributions are Bernoulli. To sample
from the marginal model, we use a Metropolis kernel similar to the one used for the Ising model. To
start the reverse simulations, we randomly pick an image from the training set and generate an initial
8
hidden state by sampling from the conditional distribution p(h|v). We then run 100 steps of Gibbs
sampling with TK to obtain a posterior sample.
We ran tests on an RBM with 784 visible and 500 hidden units trained on the MNIST handwritten
digits dataset [24] with contrastive divergence using 25 steps [25]. Since the true log evidence is not
known, we use a reference value obtained with parallel tempering (PT): log Z ? 451.42. Figure 4C
compares evidence estimates based on annealing simulations of the full against the marginal model.
Both annealing approaches provide very similar evidence estimates 451.43 (full model) and 451.48
(marginal model) that are close to the PT result. However, simulation of the marginal model is three
times faster compared to the full model. Therefore, it seems beneficial to evaluate and train RBMs
based on sampling and annealing of the marginal model p(h) rather than the full model p(v, h).
6.4
Gaussian mixture model
Finally, we consider a sort of ?data annealing? strategy inP
which independent data points are added
one-by-one as in sequential Monte Carlo [10] Ek (x) = ? l<k log p(yl |x, M ). We applied thermal
and data annealing to a three-component Gaussian mixture model with means -5, 0, 5, standard
deviations 1, 3, 0.5 and equal weights. We generated K = 100 data points, and applied both types of
annealing to estimate the mixture parameters and marginal likelihood. Parallel tempering produced
a reference log evidence of -259.49. A Gibbs sampler utilizing cluster responsibilities as auxiliary
variables served as transition kernel. Forward simulations started from prior samples where conjugate
priors were used for the component means, widths and weights. The reverse simulations started from
a posterior sample obtained by running K-means followed by 100 Gibbs sampling iterations.
Thermal annealing with as many temperatures as data points and 10 Gibbs sampling steps per
temperature estimated a log evidence of -259.72 ? 0.60 (M = 100, 10 repetitions). For 100 Gibbs
steps, we obtain -259.47 ? 0.36. Data annealing with 10 Gibbs steps per addition of a data point
yields ?257.52 ? 0.97, which seems to be slightly biased. Increasing the number of Gibbs steps to
100 improves the accuracy of the log evidence estimate: -258.32 ? 1.21. This shows that there might
be some potential in a data annealing strategy, especially for larger datasets.
7
Summary
This paper applies nonequilibrium techniques to estimate the marginal likelihood of an intractable
probabilistic model. We outline the most important results from nonequilibrium statistical physics that
are relevant to marginal likelihood estimation and relate them to machine learning algorithms such as
AIS [10], RAISE [16] and bidirectional Monte Carlo [17, 18]. We introduce two estimators, BAR
and the histogram estimator, that are currently not used in the context of probabilistic inference. We
study the performance of the estimators on a toy system and various challenging probabilistic models
including Markov random fields and Gaussian mixture models. The two evidence estimators perform
very similarly and are superior to forward/reverse AIS and the cumulant estimators. Compared to
BAR, the histogram estimator has the additional advantage that it also quantifies the uncertainty of
the evidence estimate.
Acknowledgments
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) SFB 860, subproject
B09.
References
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10
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6,382 | 6,773 | Minimal Exploration
in Structured Stochastic Bandits
Richard Combes
Centrale-Supelec / L2S
[email protected]
Stefan Magureanu
KTH, EE School / ACL
[email protected]
Alexandre Proutiere
KTH, EE School / ACL
[email protected]
Abstract
This paper introduces and addresses a wide class of stochastic bandit problems
where the function mapping the arm to the corresponding reward exhibits some
known structural properties. Most existing structures (e.g. linear, Lipschitz, unimodal, combinatorial, dueling, . . . ) are covered by our framework. We derive an
asymptotic instance-specific regret lower bound for these problems, and develop
OSSB, an algorithm whose regret matches this fundamental limit. OSSB is not
based on the classical principle of ?optimism in the face of uncertainty? or on
Thompson sampling, and rather aims at matching the minimal exploration rates
of sub-optimal arms as characterized in the derivation of the regret lower bound.
We illustrate the efficiency of OSSB using numerical experiments in the case of
the linear bandit problem and show that OSSB outperforms existing algorithms,
including Thompson sampling.
1
Introduction
Numerous extensions of the classical stochastic MAB problem [30] have been recently investigated.
These extensions are motivated by applications arising in various fields including e.g. on-line
services (search engines, display ads, recommendation systems, ...), and most often concern structural
properties of the mapping of arms to their average rewards. This mapping can for instance be
linear [14], convex [2], unimodal [36], Lipschitz [3], or may exhibit some combinatorial structure
[10, 29, 35].
In their seminal paper, Lai and Robbins [30] develop a comprehensive theory for MAB problems
with unrelated arms, i.e., without structure. They derive asymptotic (as the time horizon grows large)
instance-specific regret lower bounds and propose algorithms achieving this minimal regret. These
algorithms have then been considerably simplified, so that today, we have a few elementary indexbased1 and yet asymptotically optimal algorithms [18, 26]. Developing a similar comprehensive
theory for MAB problems with structure is considerably more challenging. Due to the structure, the
rewards observed for a given arm actually provide side-information about the average rewards of
other arms2 . This side-information should be exploited so as to accelerate as much as possible the
process of learning the average rewards. Very recently, instance-specific regret lower bounds and
asymptotically optimal algorithms could be derived only for a few MAB problems with finite set of
arms and specific structures, namely linear [31], Lipschitz [32] and unimodal [12].
In this paper, we investigate a large class of structured MAB problems. This class extends the
classical stochastic MAB problem [30] in two directions: (i) it allows for any arbitrary structure;
(ii) it allows different kinds of feedback. More precisely, our generic MAB problem is as follows.
1
An algorithm is index-based if the arm selection in each round is solely made comparing the indexes of
each arm, and where the index of an arm only depends on the rewards observed for this arm.
2
Index-based algorithms cannot be optimal in MAB problems with structure.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In each round, the decision maker selects an arm from a finite set X . Each arm x ? X has
an unknown parameter ?(x) ? R, and when this arm is chosen in round t, the decision maker
observes a real-valued random variable Y (x, t) with expectation ?(x) and distribution ?(?(x)). The
observations (Y (x, t))x?X ,t?1 are independent across arms and rounds. If x is chosen, she also
receives an unobserved and deterministic3 reward ?(x, ?), where ? = (?(x))x?X . The parameter
? lies in a compact set ? that encodes the structural properties of the problem. The set ?, the
class of distributions ?, and the mapping (x, ?) 7? ?(x, ?) encode the structure of the problem,
are known to the decision maker, whereas ? is initially unknown. We denote by x? (t) the arm
selected in round t under algorithm ?; this selection is based on previously selected arms and
the corresponding observations. Hence the set ? of all possible arm selection rules consists in
algorithms ? such that for any t ? 1, x? (t) is Ft? -measurable where Ft? is the ?-algebra generated
by (x? (1), Y (x? (1), 1), . . . , x? (t ? 1), Y (x? (t ? 1), t ? 1). The performance of an algorithm ? ? ?
is defined through its regret up to round T :
R? (T, ?) = T max ?(x, ?) ?
x?X
T
X
E(?(x(t), ?)).
t=1
The above MAB problem is very generic, as any kind of structure can be considered. In particular,
our problem includes classical, linear, unimodal, dueling, and Lipschitz bandit problems as particular
examples, see Section 3 for details. Our contributions in this paper are as follows:
? We derive a tight instance-specific regret lower bound satisfied by any algorithm for our
generic structured MAB problem.
? We develop OSSB (Optimal Sampling for Structured Bandits), a simple and yet asymptotically optimal algorithm, i.e., its regret matches our lower bound. OSSB optimally exploits
the structure of the problem so as to minimize regret.
? We briefly exemplify the numerical performance of OSSB in the case of linear bandits.
OSSB outperforms existing algorithms (including Thompson Sampling [2], GLM-UCB
[16], and a recently proposed asymptotically optimal algorithm [31]).
As noticed in [31], for structured bandits (even for linear bandits), no algorithm based on the principle
of optimism (a la UCB) or on that of Thompson sampling can achieve an asymptotically minimal
regret. The design of OSSB does not follow these principles, and is rather inspired by the derivation of
the regret lower bound. To obtain this bound, we characterize the minimal rates at which sub-optimal
arms have to be explored. OSSB aims at sampling sub-optimal arms so as to match these rates. The
latter depends on the unknown parameter ?, and so OSSB needs to accurately estimate ?. OSSB
hence alternates between three phases: exploitation (playing arms with high empirical rewards),
exploration (playing sub-optimal arms at well chosen rates), and estimation (getting to know ? to
tune these exploration rates).
The main technical contribution of this paper is a finite-time regret analysis of OSSB for any generic
structure. In spite of the simplicity the algorithm, its analysis is involved. Not surprisingly, it uses
concentration-of-measure arguments, but it also requires to establish that the minimal exploration
rates (derived in the regret lower bound) are essentially smooth with respect to the parameter ?. This
complication arises due to the (additional) estimation phase of OSSB: the minimal exploration rates
should converge as our estimate of ? gets more and more accurate.
The remainder of the paper is organized as follows. In the next section, we survey recent results
on structured stochastic bandits. In Section 3, we illustrate the versatility of our MAB problem by
casting most existing structured bandit problems into our framework. Section 4 is devoted to the
derivation of the regret lower bound. In Sections 5 and 6, we present OSSB and provide an upper
bound of its regret. Finally Section 7 explores the numerical performance of OSSB in the case of
linear structures.
3
Usually in MAB problems, the reward is a random variable given as feedback to the decision maker. In our
model, the reward is deterministic (as if it was averaged), but not observed as the only observation is Y (x, t) if
x is chosen in round t. We will illustrate in Section 3 why usual MAB formulations are specific instances of our
model.
2
2
Related work
Structured bandits have generated many recent contributions since they find natural applications
in the design of computer systems, for instance: recommender systems and information retrieval
[28, 11], routing in networks and network optimization [22, 5, 17], and influence maximization
in social networks [8]. A large number of existing structures have been investigated, including:
linear [14, 34, 1, 31, 27] (linear bandits are treated here as a partial monitoring game), combinatorial
[9, 10, 29, 35, 13], Lipschitz [32], unimodal [36, 12]. The results in this paper cover all models
considered in the above body of work and are the first that can be applied to problems with any
structure in the set of allowed parameters.
Here, we focus on generic stochastic bandits with a finite but potentially large number of arms. Both
continuous as well as adversarial versions of the problem have been investigated, see survey [6].
The performance of Thompson sampling for generic bandit problems has appeared in the literature
[15, 20], however, the recent results in [31] prove that Thompson sampling is not optimal for all
structured bandits. Generic structured bandits were treated in [7, 21]. The authors show that the
regret of any algorithm must scale as C(?)ln T when T ? ? where C(?) is the optimal value of a
semi-infinite linear program, and propose asymptotically optimal algorithms. However the proposed
algorithms are involved and have poor numerical performance, furthermore their performance
guarantees are asymptotic, and no finite time analysis is available.
To our knowledge, our algorithm is the first which covers completely generic MAB problems, is
asymptotically optimal and is amenable to a finite-time regret analysis. Our algorithm is in the same
spirit as the DMED algorithm, presented in [24], as well as the algorithm in [31], but is generic
enough to be optimal in any structured bandit setting. Similar to DMED, our algorithm relies on
repeatedly solving an optimization problem and then exploring according to its solution, thus moving
away from the UCB family of algorithms.
3
Examples
The class of MAB problems described in the introduction covers most known bandit problems as
illustrated in the six following examples.
Classical Bandits. The classical MAB problem [33] with Bernoulli rewards is obtained by making
the following choices: ?(x) ? [0, 1]; ? = [0, 1]|X | ; for any a ? [0, 1], ?(a) is the Bernoulli
distribution with mean a; for all x ? X , ?(x, ?) = ?(x).
Linear Bandits. To get finite linear bandit problems [14],[31], in our framework we choose X as a
finite subset of Rd ; we pick an unknown vector ? ? Rd and define ?(x) = h?, xi for all x ? X ; the
set of possible parameters is ? = {? = (h?, xi)x?X , ? ? Rd }; for any a ? Rd , ?(a) is a Gaussian
distribution with unit variance and centered at a; for all x ? X , ?(x, ?) = ?(x). Observe that our
framework also includes generalized linear bandit problems as those considered in [16]: we just need
to define ?(x, ?) = g(?(x)) for some function g.
Dueling Bandits. To model dueling bandits [27] using our framework, the set of arms is X =
{(i, j) ? {1, . . . , d}2 }; for any x = (i, j) ? X , ?(x) ? [0, 1] denotes the probability that i is
better than j with the conventions that ?(i, j) = 1 ? ?(j, i) and that ?(i, i) = 1/2; ? = {? :
?i? : ?(i? , j) > 1/2, ?j 6= i? } is the set of parameters such there exists a Condorcet winner; for
any a ? [0, 1], ?(a) is the Bernoulli distribution with mean a; finally, we define the rewards as
?((i, j), ?) = 12 (?(i? , i) + ?(i? , j) ? 1). Note that the best arm is (i? , i? ) and has zero reward.
Lipschitz Bandits. For finite Lipschitz bandits [32], the set of arms X is a finite subset of a metric
space endowed with a distance `. For any x ? X , ?(x) is a scalar, and the mapping x 7? ?(x) is
Lipschitz continuous with respect to `, and the set of parameters is:
? = {? : |?(x) ? ?(y)| ? `(x, y) ?x, y ? X }.
As in classical bandits ?(x, ?(x)) = ?(x). The structure is encoded by the distance `, and is an
example of local structure so that arms close to each other have similar rewards.
Unimodal Bandits. Unimodal bandits [23],[12] are obtained as follows. X = {1, ..., |X |}, ?(x) is a
scalar, and ?(x, ?(x)) = ?(x). The added assumption is that x 7? ?(x) is unimodal. Namely, there
3
exists x? ? X such that this mapping is stricly incrasing on {1, ..., x? } and strictly decreasing on
{x? , ..., |X |}.
Combinatorial bandits. The combinatorial bandit problems with bandit feedback (see [9]) are
just particular instances of linear bandits where the set of arms X is a subset of {0, 1}d . Now
to model combinatorial problems with semi-bandit feedback, we need a slight extension of the
framework described in introduction. More precisely, the set of arms is still a subset of {0, 1}d .
The observation Y (x, t) is a d-dimensional r.v. with independent components, with mean ?(x)
and distribution ?(?(x)) (a product distribution). There is an unknown vector ? ? Rd such that
Pd
?(x) = (?(1)x(1), . . . , ?(d)x(d)), and ?(x, ?) = i=1 ?(i)x(i) (linear reward). With semi-bandit
feedback, the decision maker gets detailed information about the various components of the selected
arm.
4
Regret Lower Bound
To derive regret lower bounds, a strategy consists in restricting the attention to so-called uniformly
good algorithms [30]: ? ? ? is uniformly good if R? (T, ?) = o(T a ) when T ? ? for all a > 0
and all ? ? ?. A simple change-of-measure argument is then enough to prove that for MAB problems
without structure, under any uniformly good algorithm, the number of times that a sub-optimal
arm x should be played is greater than ln T /d(?(x), ?(x? )) as the time horizon T grows large, and
where x? denotes the optimal arm and d(?(x), ?(x? )) is the Kullback-Leibler divergence between
the distributions ?(?(x)) and ?(?(x? )). Refer to [25] for a direct and elegant proof.
For our structured MAB problems, we follow the same strategy, and derive constraints on the number
of times a sub-optimal arm x is played under any uniformly good algorithm. We show that this number
is greater than c(x, ?)ln T asymptotically where the c(x, ?)?s are the solutions of a semi-infinite
linear program [19] whose constraints directly depend on the structure of the problem.
Before stating our lower bound, we introduce the following notations. For ? ? ?, let x? (?) be the
optimal arm (we assume that it is unique), and define ?? (?) = ?(x? (?), ?). For any x ? X , we
denote by D(?, ?, x) the Kullback-Leibler divergence between distributions ?(?(x)) and ?(?(x)).
Assumption 1 The optimal arm x? (?) is unique.
Theorem 1 Let ? ? ? be a uniformly good algorithm. For any ? ? ?, we have:
R? (T, ?)
? C(?),
T ??
ln T
where C(?) is the value of the optimization problem:
X
minimize
?(x)(?? (?) ? ?(x, ?))
lim inf
?(x)?0 , x?X
subject to
(1)
(2)
x?X
X
?(x)D(?, ?, x) ? 1 , ?? ? ?(?),
(3)
x?X
where
?(?) = {? ? ? : D(?, ?, x? (?)) = 0, x? (?) 6= x? (?)}.
(4)
Let (c(x, ?))x?X denote the solutions of the semi-infinite linear program (2)-(3). In this program,
?(x)ln T indicates the number of times arm x is played. The regret lower bound may be understood as
follows. The set ?(?) is the set of ?confusing? parameters: if ? ? ?(?) then D(?, ?, x? (?)) = 0 so ?
and ? cannot be differentiated by only sampling the optimal arm x? (?). Hence distinguishing ? from
? requires to sample suboptimal arms x 6= x? (?). Further, since any uniformly good algorithm must
identify the best arm with high probability to ensure low regret and x? (?) 6= x? (?), any algorithm
must distinguish these two parameters. The constraint (3) states thatPfor any ?, a uniformly good
algorithm should perform a hypothesis test between ? and ?, and x?X ?(x)D(?, ?, x) ? 1 is
required to ensure there is enough statistical information to perform this test. In summary, for a
sub-optimal arm x, c(x, ?)lnT represents the asymptotically minimal number of times x should be
sampled. It is noted that this lower bound is instance-specific (it depends on ?), and is attainable
as we propose an algorithm which attains it. The proof of Theorem 1 is presented in appendix, and
leverages techniques used in the context of controlled Markov chains [21].
4
Next, we show that with usual structures as those considered in Section 3, the semi-infinite linear
program (2)-(3) reduces to simpler optimization problems (e.g. an LP) and can sometimes even be
solved explicitly. Simplifying (2)-(3) is important for us, since our proposed asymptotically optimal
algorithm requires to solve this program. In the following examples, please refer to Section 3 for
the definitions and notations. As mentioned already, the solutions of (2)-(3) for classical MAB is
c(x, ?) = 1/d(?(x), ?(x? )).
Linear bandits. For this class of problems, [31] recently proved that (2)-(3) was equivalent to the
following optimization problem:
X
minimize
?(x)(?(x? ) ? ?(x))
?(x)?0 , x?X
x?X
!
>
subject to x inv
X
?(z)zz
z?X
>
x?
(?(x? ) ? ?(x))2
,
2
?x 6= x? .
Refer to [31] for the proof of this result, and for insightful discussions.
Lipschitz bandits. It can be shown that for Bernoulli rewards (the reward of arm x is ?(x)) (2)-(3)
reduces to the following LP [32]:
X
minimize
?(x)(?(x? ) ? ?(x))
?(x)?0 , x?X
subject to
x?X
X
?(z)d(?(z), max{?(z), ?(x? ) ? `(x, z)}) ? 1 ,
z?X
?x 6= x? .
While the solution is not explicit, the problem reduces to a LP with |X | variables and 2|X | constraints.
Dueling bandits. The solution of (2)-(3) is as follows [27]. Assume to simplify that for any i 6= i? ,
?((i,j),?)
there exists a unique j minimizing d(?(i,j),1/2)
and such that ?(i, j) < 1/2. Let j(i) denote this
index. Then for any x = (i, j), we have
c(x, ?) =
1{j = j(i)}
.
d(?(i, j), 1/2)
Unimodal bandits. For such problems, it is shown in [12] that the solution of (2)-(3) is given by: for
all x ? X ,
1{|x ? x? | = 1}
.
c(x, ?) =
d(?(x), ?(x? ))
Hence, in unimodal bandits, under an asymptotically optimal algorithm, the sub-optimal arms
contributing to the regret (i.e., those that need to be sampled ?(ln T )) are neighbours of the optimal
arm.
5
The OSSB Algorithm
In this section we propose OSSB (Optimal Sampling for Structured Bandits), an algorithm that is
asymptotically optimal, i.e., its regret matches the lower bound of Theorem 1. OSSB pseudo-code is
presented in Algorithm 1, and takes as an input two parameters ?, ? > 0 that control the amount of
exploration performed by the algorithm.
The design of OSSB is guided by the necessity to explore suboptimal arms as much as prescribed
by the solution of the optimization problem (2)-(3), i.e., the sub-optimal arm x should be explored
c(x, ?)ln T times. If ? was known, then sampling arm x c(x, ?)ln T times for all x, and then selecting
the arm with the largest estimated reward should yield minimal regret.
Since ? is unknown, we have to estimate it. Define the empirical averages:
Pt
Y (x, s)1{x(s) = x}
m(x, t) = s=1
max(1, N (x, t))
5
Algorithm 1 OSSB(?,?)
s(0) ? 0, N (x, 1), m(x, 1) ? 0 , ?x ? X
{Initialization}
for t = 1, ..., T do
Compute the optimization problem (2)-(3) solution (c(x, m(t)))x?X where m(t) =
(m(x, t))x?X
if N (x, t) ? c(x, m(t))(1 + ?)ln t, ?x then
s(t) ? s(t ? 1)
x(t) ? x? (m(t))
{Exploitation}
else
s(t) ? s(t ? 1) + 1
N (x,t)
X(t) ? arg minx?X c(x,m(t))
X(t) ? arg minx?X N (x, t)
if N (X(t), t) ? ?s(t) then
x(t) ? X(t)
{Estimation}
else
x(t) ? X(t)
{Exploration}
end if
end if
{Update statistics}
Select arm x(t) and observe Y (x(t), t)
m(x, t + 1) ? m(x, t), ?x 6= x(t) ,
N (x, t + 1) ? N (x, t), ?x 6= x(t)
(x(t),t)
m(x(t), t + 1) ? Y (x(t),t)+m(x(t),t)N
N (x(t),t)+1
N (x(t), t + 1) ? N (x(t), t) + 1
end for
Pt
where x(s) is the arm selected in round s, and N (x, t) = s=1 1{x(s) = x} is the number of times
x has been selected up to round t. The key idea of OSSB is to use m(t) = (m(x, t))x?X as an
estimator for ?, and explore arms to match the estimated solution of the optimization problem (2)-(3),
so that N (x, t) ? c(x, m(t))ln t for all x. This should work if we can ensure certainty equivalence,
i.e. m(t) ? ?(t) when t ? ? at a sufficiently fast rate.
The OSSB algorithm has three components. More precisely, under OSSB, we alternate between
three phases: exploitation, estimation and exploration. In round t, one first attempts to identify the
optimal arm. We calculate x? (m(x, t)) the arm with the largest empirical reward. If N (x, t) ?
c(x, m(t))(1 + ?)ln t for all x, we enter the exploitation phase: we have enough information to infer
that x? (m(x, t)) = x? (?) w.h.p. and we select x(t) = x? (m(x, t)). Otherwise, we need to gather
more information to identify the optimal arm. We have two goals: (i) make sure that all components
of ? are accurately estimated and (ii) make sure that N (x, t) ? c(x, m(t))ln t for all x. We maintain
a counter s(t) of the number of times we have not entered the expoitation phase. We choose between
two possible arms, namely the least played arm X(t) and the arm X(t) which is the farthest from
satisfying N (x, t) ? c(x, m(t))ln t. We then consider the number of times X(t) has been selected.
If N (X(t), t) is much smaller than s(t), there is a possibility that X(t) has not been selected enough
times so that ?(X(t)) is not accurately estimated so we enter the estimation phase, where we select
X(t) to ensure that certainty equivalence holds. Otherwise we enter the exploration phase where
we select X(t) to explore as dictated by the solution of (2)-(3), since c(x, m(t)) should be close to
c(x, ?).
Theorem 2 states that OSSB is asymptotically optimal. The complete proof is presented in Appendix,
with a sketch of the proof provided in the next section. We prove Theorem 2 for Bernoulli or
Subgaussian observations, but the analysis is easily extended to rewards in a 1-parameter exponential
family of distributions. While we state an asymptotic result here, we actually perform a finite time
analysis of OSSB, and a finite time regret upper bound for OSSB is displayed at the end of next
section.
Assumption 2 (Bernoulli observations) ?(x) ? [0, 1] and ?(?(x)) =Ber(?(x)) for all x ? X .
Assumption 3 (Gaussian observations) ?(x) ? R and ?(?(x)) = N (?(x), 1) for all x ? X .
6
Assumption 4 For all x, the mapping (?, ?) 7? D(x, ?, ?) is continuous at all points where it is not
infinite.
Assumption 5 For all x, the mapping ? ? ?(x, ?) is continuous.
Assumption 6 The solution to problem (2)-(3) is unique.
Theorem 2 If Assumptions 1, 4, 5 and 6 hold and either Assumption 2 or 3 holds, then under the
algorithm ? =OSSB(?, ?) with ? < |X1 | we have:
R? (T )
? C(?)F (?, ?, ?),
T ?? ln T
lim sup
with F a function such that for all ?, we have F (?, ?, ?) ? 1 as ? ? 0 and ? ? 0.
We conclude this section by a remark on the computational complexity of the OSSB algorithm. OSSB
requires to solve the optimization problem (2)-(3) in each round. The complexity of solving this
problem strongly depends on the problem structure. For general structures, the complexity of this
problem is difficult to assess. However for problems exemplified in Section 3, this problem is usually
easy to solve. Note that the algorithm proposed in [31] for linear bandits requires to solve (2)-(3)
only once, and is hence simpler to implement; its performance however is much worse in practice
than that of OSSB as illustrated in Section 7.
6
Finite Time Analysis of OSSB
The proof of Theorem 2 is presented in Appendix in detail, and is articulated in four steps. (i) We
first notice that the probability of
Pselecting a suboptimal arm during the exploitation phase at some
round t is upper bounded by P( x?X N (x, t)D(m(t), ?, x) ? (1 + ?)ln t). Using a concentration
inequality on KL-divergences (Lemma 1 in Appendix), we show that this probability is small and
the regret caused by the exploitation phase is upper bounded by G(?, |X |) where G is finite and
depends solely on ? and |X |. (ii) The second step, which is the most involved, is to show Lemma 1
stating the solutions of (2)-(3) are continuous. The main difficulty is that the set ?(?) is not finite, so
that the optimization problem (2)-(3) is not a linear program. The proof strategy is similar to that
used to prove Berge?s maximal theorem, the additional difficulty being that the feasible set is not
compact, so that Berge?s theorem cannot be applied directly. Using Assumptions 1 and 5, both the
value ? 7? C(?) and the solution ? 7? c(?) are continuous.
Lemma 1 The optimal value of (2)-(3), ? 7? C(?) is continuous. If (2)-(3) admits a unique solution
c(?) = (c(x, ?))x?X at ?, then ? 7? c(?) is continuous at ?.
Lemma 1 is in fact interesting in its own right, since optimization problems such as (2)-(3) occur
in all bandit problems. (iii) The third step is to upper bound the number of times the solution to
(2)-(3) is not well estimated, so that C(m(t)) ? (1 + ?)C(?) for some ? > 0. From the previous
step this implies that ||m(t) ? ?||? ? ?(?) for some well-chosen ?(?) > 0. Using a deviation
result (Lemma 2 in Appendix), we show that the expected regret caused by such events is finite and
|
upper bounded by ??2|X
2 (?) . (vi) Finally a counting argument ensures that the regret incurred when
C(?) ? C(m(t)) ? (1 + ?)C(?) i.e. the solution (2)-(3)Pis well estimated is upper bounded by
(C(?)(1 + ?) + 2??(?))ln T , where ?(?) = |X |||c(?)||? x?X (?? (?) ? ?(x, ?)).
Putting everything together we obtain the finite-time regret upper bound:
2|X |
?
?
R (T ) ? ? (?) G(?, |X |) + 2
+ (C(?)(1 + ?) + 2??(?))(1 + ?)ln T.
?? (?)
This implies that:
R? (T )
? (C(?)(1 + ?) + 2??(?))(1 + ?).
T ?? ln T
lim sup
The above holds for all ? > 0, which yields the result.
7
7
Numerical Experiments
To assess the efficiency of OSSB, we compare its performance for reasonable time horizons to the
state of the art algorithms for linear bandit problems. We considered a linear bandit with Gaussian
rewards of unit variance, 81 arms of unit length, d = 3 and 10 parameters ? in [0.2, 0.4]3 , generated
uniformly at random. In our implementation of OSSB, we use ? = ? = 0 since ? is typically chosen
0 in the literature (see [18]) and the performance of the algorithm does not appear sensitive to the
choice ofp?. As baselines we select the extension of Thompson Sampling presented
p in [4](using
vt = R 0.5dln(t/?), we chose ? = 0.1, R = 1), GLM-UCB (using ?(t) = 0.5ln(t)), an
extension of UCB [16] and the algorithm presented in [31].
Thompson Sampling (Agrawal et al.)
GLM?UCB (Filippi et al.)
OSSB
Lattimore et al.
0
Average Regret
1000 2000 3000 4000 5000
Figure 1 presents the regret of the various algorithms averaged over the 10 parameters. OSSB clearly
exhibits the best performance in terms of average regret.
0e+00
2e+04
4e+04
6e+04
8e+04
1e+05
Time
Figure 1: Regret of various algorithms in the linear bandit setting with 81 arms and d = 3. Regret is
averaged over 10 randomly generated parameters and 100 trials. Colored regions represent the 95%
confidence intervals.
8
Conclusion
In this paper, we develop a unified solution to a wide class of stochastic structured bandit problems.
For the first time, we derive, for these problems, an asymptotic regret lower bound and devise OSSB,
a simple and yet asymptotically optimal algorithm. The implementation of OSSB requires that we
solve the optimization problem defining the minimal exploration rates of the sub-optimal arms. In
the most general case, this problem is a semi-infinite linear program, which can be hard to solve
in reasonable time. Studying the complexity of this semi-infinite LP depending on the structural
properties of the reward function is an interesting research direction. Indeed any asymptotically
optimal algorithm needs to learn the minimal exploration rates of sub-optimal arms, and hence needs
to solve this semi-infinite LP. Characterizing the complexity of the latter would thus yield important
insights into the trade-off between the complexity of the sequential arm selection algorithms and their
regret.
Acknowledgments
A. Proutiere?s research is supported by the ERC FSA (308267) grant. This work is supported by
the French Agence Nationale de la Recherche (ANR), under grant ANR-16-CE40-0002 (project
BADASS).
8
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9
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6,383 | 6,774 | Learned D-AMP: Principled Neural Network based
Compressive Image Recovery
Christopher A. Metzler
Rice University
[email protected]
Ali Mousavi
Rice University
[email protected]
Richard G. Baraniuk
Rice University
[email protected]
Abstract
Compressive image recovery is a challenging problem that requires fast and accurate algorithms. Recently, neural networks have been applied to this problem with
promising results. By exploiting massively parallel GPU processing architectures
and oodles of training data, they can run orders of magnitude faster than existing
techniques. However, these methods are largely unprincipled black boxes that are
difficult to train and often-times specific to a single measurement matrix.
It was recently demonstrated that iterative sparse-signal-recovery algorithms can
be ?unrolled? to form interpretable deep networks. Taking inspiration from this
work, we develop a novel neural network architecture that mimics the behavior of
the denoising-based approximate message passing (D-AMP) algorithm. We call
this new network Learned D-AMP (LDAMP).
The LDAMP network is easy to train, can be applied to a variety of different
measurement matrices, and comes with a state-evolution heuristic that accurately
predicts its performance. Most importantly, it outperforms the state-of-the-art
BM3D-AMP and NLR-CS algorithms in terms of both accuracy and run time. At
high resolutions, and when used with sensing matrices that have fast implementations, LDAMP runs over 50? faster than BM3D-AMP and hundreds of times
faster than NLR-CS.
1
Introduction
Over the last few decades computational imaging systems have proliferated in a host of different
imaging domains, from synthetic aperture radar to functional MRI and CT scanners. The majority of
these systems capture linear measurements y ? Rm of the signal of interest x ? Rn via y = Ax + ,
where A ? Rm?n is a measurement matrix and ? Rm is noise.
Given the measurements y and the measurement matrix A, a computational imaging system seeks to
recover x. When m < n this problem is underdetermined, and prior knowledge about x must be used
to recovery the signal. This problem is broadly referred to as compressive sampling (CS) [1; 2].
There are myriad ways to use priors to recover an image x from compressive measurements. In the
following, we briefly describe some of these methods. Note that the ways in which these algorithms
use priors span a spectrum; from simple hand-designed models to completely data-driven methods
(see Figure 1).
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: The spectrum of compressive signal recovery algorithms.
1.1
Hand-designed recovery methods
The vast majority of CS recovery algorithms can be considered ?hand-designed? in the sense that they
use some sort of expert knowledge, i.e., prior, about the structure of x. The most common signal prior
is that x is sparse in some basis. Algorithms using sparsity priors include CoSaMP [3], ISTA [4],
approximate message passing (AMP) [5], and VAMP [6], among many others. Researchers have also
developed priors and algorithms that more accurately describe the structure of natural images, such as
minimal total variation, e.g., TVAL3 [7], markov-tree models on the wavelet coefficients, e.g., ModelCoSaMP [8], and nonlocal self-similarity, e.g., NLR-CS [9]. Off-the-shelf denoising and compression
algorithms have also been used to impose priors on the reconstruction, e.g., Denoising-based AMP
(D-AMP) [10], D-VAMP [11], and C-GD [12]. When applied to natural images, algorithms using
advanced priors outperform simple priors, like wavelet sparsity, by a large margin [10].
The appeal of hand-designed methods is that they are based on interpretable priors and often have
well understood behavior. Moreover, when they are set up as convex optimization problems they often
have theoretical convergence guarantees. Unfortunately, among the algorithms that use accurate priors
on the signal, even the fastest is too slow for many real-time applications [10]. More importantly,
these algorithms do not take advantage of potentially available training data. As we will see, this
leaves much room for improvement.
1.2
Data-driven recovery methods
At the other end of the spectrum are data-driven (often deep learning-based) methods that use no
hand-designed models whatsoever. Instead, researchers provide neural networks (NNs) vast amounts
of training data, and the networks learn how to best use the structure within the data [13?16].
The first paper to apply this approach was [13], where the authors used stacked denoising autoencoders (SDA) [17] to recover signals from their undersampled measurements. Other papers in this
line of work have used either pure convolutional layers (DeepInverse [15]) or a combination of
convolutional and fully connected layers (DR2 -Net [16] and ReconNet [14]) to build deep learning
frameworks capable of solving the CS recovery problem. As demonstrated in [13], these methods can
compete with state-of-the-art methods in terms of accuracy while running thousands of times faster.
Unfortunately, these methods are held back by the fact that there exists almost no theory governing
their performance and that, so far, they must be trained for specific measurement matrices and noise
levels.
1.3
Mixing hand-designed and data-driven methods for recovery
The third class of recovery algorithms blends data-driven models with hand-designed algorithms.
These methods first use expert knowledge to set up a recovery algorithm and then use training
data to learn priors within this algorithm. Such methods benefit from the ability to learn more
realistic signal priors from the training data, while still maintaining the interpretability and guarantees
that made hand-designed methods so appealing. Algorithms of this class can be divided into two
subcategories. The first subcategory uses a black box neural network that performs some function
within the algorithm, such as the proximal mapping. The second subcategory explicitly unrolls and
iterative algorithm and turns it into a deep NN. Following this unrolling, the network can be tuned
with training data. Our LDAMP algorithm uses ideas from both these camps.
Black box neural nets. The simplest way to use a NN in a principled way to solve the CS problem
is to treat it as a black box that performs some function; such as computing a posterior probability.
2
(a) D-IT Iterations
(b) D-AMP Iterations
Figure 2: Reconstruction behavior of D-IT (left) and D-AMP (right) with an idealized denoiser.
Because D-IT allows bias to build up over iterations of the algorithm, its denoiser becomes ineffective
at projecting onto the set C of all natural images. The Onsager correction term enables D-AMP to
avoid this issue. Figure adapted from [10].
Examples of this approach include RBM-AMP and its generalizations [18?20], which use Restricted
Boltzmann Machines to learn non-i.i.d. priors; RIDE-CS [21], which uses the RIDE [22] generative
model to compute the probability of a given estimate of the image; and OneNet [23], which uses a
NN as a proximal mapping/denoiser.
Unrolled algorithms. The second way to use a NN in a principled way to solve the CS problem is
to simply take a well-understood iterative recovery algorithm and unroll/unfold it. This method is
best illustrated by the the LISTA [24; 25] and LAMP [26] NNs. In these works, the authors simply
unroll the iterative ISTA [4] and AMP [5] algorithms, respectively, and then treat parameters of
the algorithm as weights to be learned. Following the unrolling, training data can be fed through
the network, and stochastic gradient descent can be used to update and optimize its parameters.
Unrolling was recently applied to the ADMM algorithm to solve the CS-MRI problem [27]. The
resulting network, ADMM-Net, uses training data to learn filters, penalties, simple nonlinearities,
and multipliers. Moving beyond CS, the unrolling principle has been applied successfully in speech
enhancement [28], non-negative matrix factorization applied to music transcription [29], and beyond.
In these applications, unrolling and training significantly improve both the quality and speed of signal
reconstruction.
2
Learned D-AMP
2.1
D-IT and D-AMP
Learned D-AMP (LDAMP), is a mixed hand-designed/data-driven compressive signal recovery
framework that is builds on the D-AMP algorithm [10]. We describe D-AMP now, as well as the
simpler denoising-based iterative thresholding (D-IT) algorithm. For concreteness, but without loss
of generality, we focus on image recovery.
A compressive image recovery algorithm solves the ill-posed inverse problem of finding the image x
given the low-dimensional measurements y = Ax by exploiting prior information on x, such as fact
that x ? C, where C is the set of all natural images. A natural optimization formulation reads
argminx ky ? Axk22 subject to x ? C.
(1)
When no measurement noise is present, a compressive image recovery algorithm should return the
(hopefully unique) image xo at the intersection of the set C and the affine subspace {x|y = Ax} (see
Figure 2).
The premise of D-IT and D-AMP is that high-performance image denoisers D? , such as BM3D
[30], are high-quality approximate projections onto the set C of natural images.1,2 That is, suppose
1
The notation D? indicates that the denoiser can be parameterized by the standard deviation of the noise ?.
Denoisers can also be thought of as a proximal mapping with respect to the negative log likelihood of natural
images [31] or as taking a gradient step with respect to the data generating function of natural images [32; 33].
2
3
xo + ?z is a noisy observation of a natural image, with xo ? C and z ? N (0, I). An ideal denoiser
D? would simply find the point in the set C that is closest to the observation xo + ?z
D? (x) = argminx kxo + ?z ? xk22 subject to x ? C.
(2)
Combining (1) and (2) leads naturally to the D-IT algorithm, presented in (3) and illustrated in Figure
2(a). Starting from x0 = 0, D-IT takes a gradient step towards the {x|y = Ax} affine subspace and
then applies the denoiser D? to move to x1 in the set C of natural images . Gradient stepping and
denoising is repeated for t = 1, 2, . . . until convergence.
zt
D-IT Algorithm
x
t+1
= y ? Axt ,
= D?? t (xt + AH z t ).
(3)
Let ? t = xt + AH z t ? xo denote the difference between xt + AH z t and the true signal xo at each
iteration. ? t is known as the effective noise. At each iteration, D-IT denoises xt + AH z t = xo + ? t ,
i.e., the true signal plus the effective noise. Most denoisers are designed to work with ? t as additive
white Gaussian noise (AWGN). Unfortunately, as D-IT iterates, the denoiser biases the intermediate
solutions, and ? t soon deviates from AWGN. Consequently, the denoising iterations become less
effective [5; 10; 26], and convergence slows.
D-AMP differs from D-IT in that it corrects for the bias in the effective noise at each iteration
t = 0, 1, . . . using an Onsager correction term bt .
D-AMP Algorithm
bt
zt
?
?t
xt+1
z t?1 divD?? t?1 (xt?1 + AH z t?1 )
,
m
t
t
= y ? Ax + b ,
kz t k2
= ? ,
m
=
= D?? t (xt + AH z t ).
(4)
The Onsager correction term removes the bias from the intermediate solutions so that the effective
noise ? t follows the AWGN model expected by typical image denoisers. For more information on the
Onsager correction, its origins, and its connection to the Thouless-Anderson-Palmer equations [34],
t
k2
see [5] and [35]. Note that kz?m
serves as a useful and accurate estimate of the standard deviation of
t
? [36]. Typically, D-AMP algorithms use a Monte-Carlo approximation for the divergence divD(?),
which was first introduced in [37; 10].
2.2
Denoising convolutional neural network
NNs have a long history in signal denoising; see, for instance [38]. However, only recently have they
begun to significantly outperform established methods like BM3D [30]. In this section we review the
recently developed Denoising Convolutional Neural Network (DnCNN) image denoiser [39], which
is both more accurate and far faster than competing techniques.
The DnCNN neural network consists of 16 to 20 convolutional layers, organized as follows. The first
convolutional layer uses 64 different 3 ? 3 ? c filters (where c denotes the number of color channels)
and is followed by a rectified linear unit (ReLU) [40]. The next 14 to 18 convolutional layers each use
64 different 3 ? 3 ? 64 filters which are each followed by batch-normalization [41] and a ReLU. The
final convolutional layer uses c separate 3 ? 3 ? 64 filters to reconstruct the signal. The parameters
are learned via residual learning [42].
2.3
Unrolling D-IT and D-AMP into networks
The central contribution of this work is to apply the unrolling ideas described in Section 1.3 to D-IT
and D-AMP to form the LDIT and LDAMP neural networks. The LDAMP network, presented in (5)
and illustrated in Figure 3, consists of 10 AMP layers where each AMP layer contains two denoisers
4
Figure 3: Two layers of the LDAMP neural network. When used with the DnCNN denoiser, each
denoiser block is a 16 to 20 convolutional-layer neural network. The forward and backward operators
are represented as the matrices A and AH ; however function handles work as well.
with tied weights. One denoiser is used to update xl , and the other is used to estimate the divergence
using the Monte-Carlo approximation from [37; 10]. The LDIT network is nearly identical but does
not compute an Onsager correction term and hence, only applies one denoiser per layer. One of the
few challenges to unrolling D-IT and D-AMP is that, to enable training, we must use a denoiser
that easily propagates gradients; a black box denoiser like BM3D will not work. This restricts us to
denoisers such as DnCNN, which, fortunately, offers improved performance.
LDAMP Neural Network
l
l?1
z l?1 divDw
+ AH z l?1 )
l?1 (?
? l?1 ) (x
bl
=
zl
= y ? Axl + bl ,
kz l k2
= ? ,
m
?
?l
xl+1
m
l
l
H l
= Dw
l (?
? l ) (x + A z ).
,
(5)
l
Within (5), we use the slightly cumbersome notation Dw
l (?
? l ) to indicate that layer l of the network
l
uses denoiser D , that this denoiser depends on its weights/biases wl , and that these weights may be a
function of the estimated standard deviation of the noise ?
? l . During training, the only free parameters
1
L
we learn are the denoiser weights w , ...w . This is distinct from the LISTA and LAMP networks,
where the authors decouple and learn the A and AH matrices used in the network [24; 26].
3
Training the LDIT and LDAMP networks
We experimented with three different methods to train the LDIT and LDAMP networks. Here we
describe and compare these training methods at a high level; the details are described in Section 5.
? End-to-end training: We train all the weights of the network simultaneously. This is the
standard method of training a neural network.
? Layer-by-layer training: We train a 1 AMP layer network (which itself contains a 16-20
layer denoiser) to recover the signal, fix these weights, add an AMP layer, train the second
layer of the resulting 2 layer network to recover the signal, fix these weights, and repeat
until we have trained a 10 layer network.
? Denoiser-by-denoiser training: We decouple the denoisers from the rest of the network
and train each on AWGN denoising problems at different noise levels. During inference,
the network uses its estimate of the standard deviation of the noise to select which set of
denoiser weights to use. Note that, in selecting which denoiser weights to use, we must
discretize the expected range of noise levels; e.g., if ?
? = 35, then we use the denoiser for
noise standard deviations between 20 and 40.
5
End-to-end
Layer-by-layer
Denoiser-by-denoiser
LDIT
32.1
26.1
28.0
LDAMP
33.1
33.1
31.6
End-to-end
Layer-by-layer
Denoiser-by-denoiser
(a)
LDIT
8.0
-2.6
22.1
LDAMP
18.7
18.7
25.9
(b)
Figure 4: Average PSNRs4 of 100 40 ? 40 image reconstructions with i.i.d. Gaussian measurements
m
m
trained at a sampling rate of m
n = 0.20 and tested at sampling rates of n = 0.20 (a) and n = 0.05
(b).
Comparing Training Methods. Stochastic gradient descent theory suggests that layer-by-layer
and denoiser-by-denoiser training should sacrifice performance as compared to end-to-end training
[43]. In Section 4.2 we will prove that this is not the case for LDAMP. For LDAMP, layer-by-layer and
denoiser-by-denoiser training are minimum-mean-squared-error (MMSE) optimal. These theoretical
results are born out experimentally in Tables 4(a) and 4(b). Each of the networks tested in this section
consists of 10 unrolled DAMP/DIT layers that each contain a 16 layer DnCNN denoiser.
Table 4(a) demonstrates that, as suggested by theory, layer-by-layer training of LDAMP is optimal;
additional end-to-end training does not improve the performance of the network. In contrast, the table
demonstrates that layer-by-layer training of LDIT, which represents the behavior of a typical neural
network, is suboptimal; additional end-to-end training dramatically improves its performance.
Despite the theoretical result the denoiser-by-denoiser training is optimal, Table 4(a) shows that
LDAMP trained denoiser-by-denoiser performs slightly worse than the end-to-end and layer-by-layer
trained networks. This gap in performance is likely due to the discretization of the noise levels, which
is not modeled in our theory. This gap can be reduced by using a finer discretization of the noise
levels or by using deeper denoiser networks that can better handle a range of noise levels [39].
In Table 4(b) we report on the performance of the two networks when trained at a one sampling
rate and tested at another. LDIT and LDAMP networks trained end-to-end and layer-by-layer at a
m
sampling rate of m
n = 0.2 perform poorly when tested at a sampling rate of n = 0.05. In contrast, the
denoiser-by-denoiser trained networks, which were not trained at a specific sampling rate, generalize
well to different sampling rates.
4
Theoretical analysis of LDAMP
This section makes two theoretical contributions. First, we show that the state-evolution (S.E.), a
framework that predicts the performance of AMP/D-AMP, holds for LDAMP as well.5 Second, we
use the S.E. to prove that layer-by-layer and denoiser-by-denoiser training of LDAMP are MMSE
optimal.
4.1
State-evolution
In the context of LAMP and LDAMP, the S.E. equations predict the intermediate mean squared error
kx k2
(MSE) of the network over each of its layers [26]. Starting from ?0 = no 2 the S.E. generates a
sequence of numbers through the following iterations:
1
l
l
2
?l+1 (xo , ?, ?2 ) = E kDw
(6)
l (?) (xo + ? ) ? xo k2 ,
n
where (? l )2 = 1? ?l (xo , ?, ?2 ) + ?2 , the scalar ? is the standard deviation of the measurement noise
, and the expectation is with respect to ? N (0, I). Note that the notation ?l+1 (xo , ?, ?2 ) is used to
emphasize that ?l may depend on the signal xo , the under-determinacy ?, and the measurement noise.
Let xl denote the estimate at layer l of LDAMP. Our empirical findings, illustrated in Figure 5, show
that the MSE of LDAMP is predicted accurately by the S.E. We formally state our finding.
4
2
255
PSNR = 10 log10 ( mean((?
) when the pixel range is 0 to 255.
x?xo )2 )
5
For D-AMP and LDAMP, the S.E. is entirely observational; no rigorous theory exists. For AMP, the S.E. has
been proven asymptotically accurate for i.i.d. Gaussian measurements [44].
6
Figure 5: The MSE of intermediate reconstructions of the Boat test image across different layers for
the DnCNN variants of LDAMP and LDIT alongside their predicted S.E. The image was sampled
with Gaussian measurements at a rate of m
n = 0.1. Note that LDAMP is well predicted by the S.E.,
whereas LDIT is not.
Finding 1. If the LDAMP network starts from x0 = 0, then for large values of m and n, the
2
S.E. predicts the mean square error of LDAMP at each layer, i.e., ?l (xo , ?, ?2 ) ? n1
xl ? xo
2 , if
the following conditions hold: (i) The elements of the matrix A are i.i.d. Gaussian (or subgaussian)
with mean zero and standard deviation 1/m. (ii) The noise w is also i.i.d. Gaussian. (iii) The
denoisers Dl at each layer are Lipschitz continuous.6
4.2
Layer-by-layer and denoiser-by-denoiser training is optimal
The S.E. framework enables us to prove the following results: Layer-by-layer and denoiser-bydenoiser training of LDAMP are MMSE optimal. Both these results rely upon the following lemma.
Lemma 1. Suppose that D1 , D2 , ...DL are monotone denoisers in the sense that for l = 1, 2, ...L
l
2
1
1
inf wl EkDw
l (?) (xo + ?) ? xo k2 is a non-decreasing function of ?. If the weights w of D are set
to minimize Ex0 [?1 ] and fixed; and then the weights w2 of D2 are set to minimize Ex0 [?2 ] and fixed,
. . . and then the weights wL of DL are set to minimize Ex0 [?L ], then together they minimize Ex0 [?L ].
Lemma 1 can be derived using the proof technique for Lemma 3 of [10], but with ?l replaced by
Ex0 [?l ] throughout. It leads to the following two results.
Corollary 1. Under the conditions in Lemma 1, layer-by-layer training of LDAMP is MMSE optimal.
This result follows from Lemma 1 and the equivalence between Ex0 [?l ] and Ex0 [ n1 kxl ? xo k22 ].
Corollary 2. Under the conditions in Lemma 1, denoiser-by-denoiser training of LDAMP is MMSE
optimal.
l
This result follows from Lemma 1 and the equivalence between Ex0 [?l ] and Ex0 [ n1 E kDw
l (?) (xo +
l
2
? ) ? xo k2 ].
5
Experiments
Datasets Training images were pulled from Berkeley?s BSD-500 dataset [46]. From this dataset,
we used 400 images for training, 50 for validation, and 50 for testing. For the results presented in
Section 3, the training images were cropped, rescaled, flipped, and rotated to form a set of 204,800
overlapping 40 ? 40 patches. The validation images were cropped to form 1,000 non-overlapping
40 ? 40 patches. We used 256 non-overlapping 40 ? 40 patches for test. For the results presented in
this section, we used 382,464 50 ? 50 patches for training, 6,528 50 ? 50 patches for validation, and
seven standard test images, illustrated in Figure 6 and rescaled to various resolutions, for test.
Implementation. We implemented LDAMP and LDIT, using the DnCNN denoiser [39], in both
TensorFlow and MatConvnet [47], which is a toolbox for Matlab. Public implementations of both
versions of the algorithm are available at https://github.com/ricedsp/D-AMP_Toolbox.
6
A denoiser is said to be L-Lipschitz continuous if for every x1 , x2 ? C we have kD(x1 ) ? D(x2 )k22 ?
Lkx1 ? x2 k22 . While we did not find it necessary in practice, weight clipping and gradient norm penalization
can be used to ensure Lipschitz continuity of the convolutional denoiser [45].
7
(a) Barbara
(b) Boat
(c) Couple
(d) Peppers (e) Fingerprint (f) Mandrill
(g) Bridge
Figure 6: The seven test images.
Training parameters. We trained all the networks using the Adam optimizer [48] with a training
rate of 0.001, which we dropped to 0.0001 and then 0.00001 when the validation error stopped
improving. We used mini-batches of 32 to 256 patches, depending on network size and memory
usage. For layer-by-layer and denoiser-by-denoiser training, we used a different randomly generated
measurement matrix for each mini-batch. Training generally took between 3 and 5 hours per denoiser
on an Nvidia Pascal Titan X. Results in this section are for denoiser-by-denoiser trained networks
which consists of 10 unrolled DAMP/DIT layers that each contain a 20 layer DnCNN denoiser.
Competition. We compared the performance of LDAMP to three state-of-the-art image recovery
algorithms; TVAL3 [7], NLR-CS [9], and BM3D-AMP [10]. We also include a comparison with LDIT
to demonstrate the benefits of the Onsager correction term. Our results do not include comparisons
with any other NN-based techniques. While many NN-based methods are very specialized and only
work for fixed matrices [13?16; 27], the recently proposed OneNet [23] and RIDE-CS [21] methods
can be applied more generally. Unfortunately, we were unable to train and test the OneNet code
in time for this submission. While RIDE-CS code was available, the implementation requires the
measurement matrices to have orthonormalized rows. When tested on matrices without orthonormal
rows, RIDE-CS performed significantly worse than the other methods.
Algorithm parameters. All algorithms used their default parameters. However, NLR-CS was
initialized using 8 iterations of BM3D-AMP, as described in [10]. BM3D-AMP was run for 10
iterations. LDIT and LDAMP
? used 10 layers. LDIT had its per layer noise standard deviation estimate
?
? parameter set to 2kz l k2 / m, as was done with D-IT in [10].
Testing setup. We tested the algorithms with i.i.d. Gaussian measurements and with measurements
from a randomly sampled coded diffraction pattern [49]. The coded diffraction pattern forward
operator was formed as a composition of three steps; randomly (uniformly) change the phase, take a
2D FFT, and then randomly (uniformly) subsample. Except for the results in Figure 7, we tested the
algorithms with 128 ? 128 images (n = 1282 ). We report recovery accuracy in terms of PSNR. We
report run times in seconds. Results broken down by image are provided in the supplement.
Gaussian measurements. With noise-free Gaussian measurements, the LDAMP network produces
the best reconstructions at every sampling rate on every image except Fingerprints, which looks very
unlike the natural images the network was trained on. With noise-free Gaussian measurements, LDIT
and LDAMP produce reconstructions significantly faster than the competing methods. Note that,
despite having to perform twice as many denoising operations, at a sampling rate of m
n = 0.25 the
LDAMP network is only about 25% slower than LDIT. This indicates that matrix multiplies, not
denoising operations, are the dominant source of computation. Average recovery PSNRs and run
times are reported in Table 1. With noisy Gaussian measurements, LDAMP uniformly outperformed
the other methods; these results can be found in the supplement.
Coded diffraction measurements. With noise-free coded diffraction measurements, the LDAMP
network again produces the best reconstructions on every image except Fingerprints. With coded
diffraction measurements, LDIT and LDAMP produce reconstructions significantly faster than competing methods. Note that because the coded diffraction measurement forward and backward operator
can be applied in O(n log n) operations, denoising becomes the dominant source of computations:
LDAMP, which has twice as many denoising operations as LDIT, takes roughly 2? longer to complete. Average recovery PSNRs and run times are reported in Table 2. We end this section with a
visual comparison of 512 ? 512 reconstructions from TVAL3, BM3D-AMP, and LDAMP, presented
8
Table 1: PSNRs and run times (sec) of 128 ? 128 reconstructions with i.i.d. Gaussian measurements
and no measurement noise at various sampling rates.
Method
TVAL3
BM3D-AMP
LDIT
LDAMP
NLR-CS
m
n
= 0.10
m
n
= 0.15
m
n
= 0.20
m
n
= 0.25
PSNR
Time
PSNR
Time
PSNR
Time
PSNR
Time
21.5
23.1
20.1
23.7
23.2
2.2
4.8
0.3
0.4
85.9
22.8
25.1
20.7
25.7
25.2
2.9
4.4
0.4
0.5
104.0
24.0
26.6
21.1
27.2
26.8
3.6
4.2
0.4
0.5
124.4
25.0
27.9
21.7
28.5
28.2
4.3
4.1
0.5
0.6
146.3
Table 2: PSNRs and run times (sec) of 128?128 reconstructions with coded diffraction measurements
and no measurement noise at various sampling rates.
Method
TVAL3
BM3D-AMP
LDIT
LDAMP
NLR-CS
m
n
= 0.10
m
n
= 0.15
m
n
= 0.20
m
n
= 0.25
PSNR
Time
PSNR
Time
PSNR
Time
PSNR
Time
24.0
23.8
22.9
25.3
21.6
0.52
4.55
0.14
0.26
87.82
26.0
25.7
25.6
27.4
22.8
0.46
4.29
0.14
0.26
87.43
27.9
27.5
27.4
28.9
25.1
0.43
3.67
0.14
0.27
87.18
29.7
29.1
28.9
30.5
26.4
0.41
3.40
0.14
0.26
86.87
in Figure 7. At high resolutions, the LDAMP reconstructions are incrementally better than those of
BM3D-AMP yet computed over 60? faster.
(a) Original Image
(b) TVAL3 (26.4 dB, 6.85 (c) BM3D-AMP (27.2 dB, (d) LDAMP
sec)
75.04 sec)
1.22 sec)
(28.1
dB,
Figure 7: Reconstructions of 512 ? 512 Boat test image sampled at a rate of m
n = 0.05 using
coded diffraction pattern measurements and no measurement noise. LDAMP?s reconstructions are
noticeably cleaner and far faster than the competing methods.
6
Conclusions
In this paper, we have developed, analyzed, and validated a novel neural network architecture that
mimics the behavior of the powerful D-AMP signal recovery algorithm. The LDAMP network is
easy to train, can be applied to a variety of different measurement matrices, and comes with a stateevolution heuristic that accurately predicts its performance. Most importantly, LDAMP outperforms
the state-of-the-art BM3D-AMP and NLR-CS algorithms in terms of both accuracy and run time.
LDAMP represents the latest example in a trend towards using training data (and lots of offline
computations) to improve the performance of iterative algorithms. The key idea behind this paper
is that, rather than training a fairly arbitrary black box to learn to recover signals, we can unroll
a conventional iterative algorithm and treat the result as a NN, which produces a network with
well-understood behavior, performance guarantees, and predictable shortcomings. It is our hope this
paper highlights the benefits of this approach and encourages future research in this direction.
9
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12
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6,384 | 6,775 | Deliberation Networks: Sequence Generation
Beyond One-Pass Decoding ?
1
Yingce Xia, 2 Fei Tian, 3 Lijun Wu, 1 Jianxin Lin, 2 Tao Qin, 1 Nenghai Yu, 2 Tie-Yan Liu
1
University of Science and Technology of China, Hefei, China
2
3
Microsoft Research, Beijing, China
Sun Yat-sen University, Guangzhou, China
1
[email protected], [email protected], [email protected]
2
{fetia,taoqin,tie-yan.liu}@microsoft.com, 3 [email protected]
Abstract
The encoder-decoder framework has achieved promising progress for many sequence generation tasks, including machine translation, text summarization, dialog
system, image captioning, etc. Such a framework adopts an one-pass forward process while decoding and generating a sequence, but lacks the deliberation process:
A generated sequence is directly used as final output without further polishing.
However, deliberation is a common behavior in human?s daily life like reading
news and writing papers/articles/books. In this work, we introduce the deliberation
process into the encoder-decoder framework and propose deliberation networks for
sequence generation. A deliberation network has two levels of decoders, where the
first-pass decoder generates a raw sequence and the second-pass decoder polishes
and refines the raw sentence with deliberation. Since the second-pass deliberation
decoder has global information about what the sequence to be generated might
be, it has the potential to generate a better sequence by looking into future words
in the raw sentence. Experiments on neural machine translation and text summarization demonstrate the effectiveness of the proposed deliberation networks. On
the WMT 2014 English-to-French translation task, our model establishes a new
state-of-the-art BLEU score of 41.5.
1
Introduction
The neural network based encoder-decoder framework has been widely adopted for sequence generation tasks, including neural machine translation [1], text summarization [19], image captioning [27],
etc. In such a framework, the encoder encodes the source input x with length m into a sequence of
vectors {h1 , h2 , ? ? ? , hm }. The decoder, which is typically an RNN, generates an output sequence
word by word2 based on the source-side vector representations and previously generated words. The
attention mechanism [1, 35], which dynamically attends to different parts of x while generating
each target-side word, is integrated into the encoder-decoder framework to improve the quality of
generating long sequences [1].
Although the framework has achieved great success, one concern is that while generating one word,
one can only leverage the generated words but not the future words un-generated so far. That is,
when the decoder generates the t-th word yt , only y<t can be used, while the possible words y>t are
not explicitly considered. In contrast, in real-word human cognitive processes, global information,
including both the past and the future parts, is leveraged in an iterative polishing process. Here are
two examples: (1) Consider the situation that we are reading a sentence and meet an unknown word
?
2
This work was done when Yingce Xia, Lijun Wu and Jianxin Lin were interns at Microsoft Research.
Throughout this work, a word refers to the basic unit in a sequence.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
in the middle of the sentence. We do not stop here. Instead, we move forward until the end of the
sentence. Then we go back to the unknown word and try to understand it using its context, including
the words both preceding and after it. (2) To write a good document (or paragraph, article), we
usually first create a complete draft and then polish it based on global understanding of the whole
draft. When polishing a specific part, we take the whole picture of the draft into consideration to
evaluate how well the local element fits into the global environment rather than only looking back to
the preceding parts.
We call such a polishing process as deliberation. Motivated by such human cognitive behaviors, we
propose the deliberation networks, which leverage the global information with both looking back
and forward in sequence decoding through a deliberation process. Concretely speaking, to integrate
such a process into the sequence generation framework, we carefully design our architecture, which
consists of two decoders, a first-pass decoder D1 and a second-pass/deliberation decoder D2 , as
well as an encoder E. Given a source input x, the E and D1 jointly works like the standard encoderdecoder model to generate a coarse sequence y? as a draft and the corresponding representations
s? = {?
s1 , s?2 , ? ? ? , s?Ty? } used to generate y?, where Ty? is the length of y?. Afterwards, the deliberation
decoder D2 takes x, y? and s? as inputs and outputs the refined sequence y. When D2 generates the
t-th word yt , an additional
P attention model is used3 to assign an adaptive weight ?j to each y?j and s?j
yj ; s?j ] is fed into D2 . In this way, the global information of the target
for any j ? [Ty?], and ?j [?
sequence can be utilized to refine the generation process. We propose a Monte Carlo based algorithm
to overcome the difficulty brought by the discrete property of y? in optimizing the deliberation network.
To verify the effectiveness of our model, we work on two representative sequence generation tasks.
(1) Neural machine translation refers to using neural networks to translate sentences from a source
language to a target language [1, 33, 32, 34]. A standard NMT model consists of an encoder (used
to encode source sentences) and a decoder (used to generate target sentences), and thus can be
improved by our proposed deliberation network. Experimental results show that based on a widely
used single-layer GRU model [1], on the WMT?14 [29] English?French dataset, we can improve the
BLEU score [17], by 1.7 points compared to the model without deliberation. We also apply our model
on Chinese?English translations and improve BLEU by an averaged 1.26 points on four different
test sets. Furthermore, on the WMT?14 English?French translation task, by applying deliberation to
a deep LSTM model, we achieve a BLEU score 41.50, setting a new record for this task.
(2) Text summarization is a task that summarizes a long article into a short abstract. The encoderdecoder framework can also be used for such a task and thus could be refined by deliberation networks.
Experimental results on Gigaword dataset [6] show that deliberation network can improve ROUGE-1,
ROUGE-2, and ROUGE-L by 3.45, 1.70 and 3.02 points.
1.1
Related Work
Although there exist many works to improve the attention based encoder-decoder framework for
sequence generation, such as changing the training loss [28, 18, 22] or the decoding objective [14, 7],
not much attention has been paid to the structure of the encoder-decoder framework. Our work
changes the structure of the framework by introducing the second-pass decoder into it.
The idea of deliberation/refinement is not well explored for sequence generation tasks, especially
for the encoder-decoder based approaches [3, 23, 1] in neural machine translation. One related work
is post-editing [16, 2]: a source sentence e is first translated to f 0 , and then f 0 is refined by another
model. Different from our deliberation network, the two processes (i.e., generating and refining) in
post-editing are separated. As a comparison, what we build is a consistent model in which all the
components are coupled together and jointly optimized in an end-to-end way. As a result, deliberation
networks lead to better accuracies. Another related work is the review network [36]. The idea is to
review all the information encoded by the encoder to obtain thought vectors that are more compact
and abstractive. The thought vectors are then used in decoding. Different from our work, the review
steps are added on the encoder side, while the decoder side is unchanged and still adopts one-pass
decoding.
3
In this work, let [v1 ; v2 ; ? ? ? ; vn ] denote the long vector concatenated by the input vectors v1 , ? ? ? , vn . With
a little bit confusion, [m] with a single integer input m denotes the set {1, 2, ? ? ? , m}.
2
The rest of our paper is organized as follows. Our proposed deliberation network is introduced in
Section 2, including the model structure and the optimization process. Applications to neural machine
translation and text summarization are introduced in Section 3 and Section 4 respectively. Section 5
concludes the paper and discusses possible future directions.
2
The Framework
In this section, we first introduce the overall architecture of deliberation networks, then the details of
individual components, and finally propose an end-to-end Monte Carlo based algorithm to train the
deliberation networks.
2.1
Structure of Deliberation Networks
As shown in Figure 1, a deliberation network consists of an encoder E, a first-pass decoder D1 and
a second-pass decoder D2 . Deliberation happens at the second-pass decoder, which is also called
deliberation decoder alternatively. Briefly speaking, E is used to encode the source sequence into a
sequence of vector representations. D1 reads the encoder representations and generates a first-pass
target sequence as a draft, which is further provided as input to the deliberation decoder D2 for the
second-pass decoding. In the rest of this section, for simplicity of description, we use RNN as the
basic building block for both the encoder and decoders4 . All the W ?s and v?s in this section with
different superscripts or subscripts are the parameters to be learned. Besides, all the bias terms are
omitted to increase readability.
Figure 1: Framework of deliberation networks: Blue, yellow and green parts indicate encoder E,
first-pass decoder D1 and the second-pass decoder D2 respectively. The E-to-D1 attention model is
omitted for readability.
2.2
Encoder and First-pass Decoder
When an input sequence x is fed into the encoder E, it is encoded into Tx hidden states H =
{h1 , h2 , ? ? ? , hTx } where Tx is the length of x. Specifically, hi = RNN(xi , hi?1 ), where xi acts as
the representation (e.g., word embedding vector) for the i-th word in x and h0 is a zero vector.
The first-pass decoder D1 will generate a series of hidden states s?j ?j ? [Ty?], and a first-pass
sequence y?j ?j ? [Ty?], where Ty? is the length of the generated sequence. Next we show how they
are generated in detail.
4
The proposed deliberation networks are independent to the specific implementation of the recurrent units
and can be applied to simple RNN or its variants such as LSTM [11] or GRU [3].
3
Similar to the conventional encoder-decoder model, an attention model is included in D1 . At step j,
the attention model in D1 first generates a context ctxe defined as follows:
PTx
PTx
c
c
ctxe = i=1
?i hi ; ?i ? exp(v?T tanh(Watt,h
hi + Watt,?
?j?1 )) ?i ? [Tx ]; i=1
?i = 1. (1)
ss
Based on ctxe , s?j is calculated as s?j = RNN([?
yj?1 ; ctxe ], s?j?1 ). After obtaining s?j , another affine
transformation is applied on the concatenated vector [?
sj ; ctxe ; y?j?1 ]. Finally, the results of the
transformation are fed into a softmax layer, and the y?j is sampled out from the obtained multinomial
distribution.
2.3
Second-Pass Decoder
Once the first-pass target sequence y? is generated by the first-pass decoder D1 , it is fed into the
second-pass decoder D2 for further refinement. Based on the sequence y? and the hidden states s?j
?j ? [Ty?] provided by D1 , D2 eventually outputs the second-pass sequence y via the deliberation
process.
Specifically, at step t, D2 takes the previous hidden state st?1 generated by itself, previously decoded
word yt?1 , the source contextual information ctx0e and the first-pass contextual information ctxc as
inputs. Two detailed points are: (1) The computation of ctx0e is similar to that of ctxe shown in Eqn. (1)
with two differences: First, s?j?1 is replaced by st?1 ; second, the model parameters are different. (2)
To obtain ctxc , D2 has an attention model (i.e., the Ac in Figure 1) that can map the words y?j ?s and
the hidden states s?j ?s into a context vector. Mathematically speaking, in the refinement process at
t-th time step, the first-pass contextual information vector ctxc is computed as:
PTy?
PTy?
d
d
ctxc = j=1
?j [?
sj ; y?j ]; ?j ? exp(v?T tanh(Watt,
sj ; y?j ] + Watt,s
st?1 )) ?j ? [Ty?]; j=1
?j = 1.
sy
? [?
As can be seen from the above computation, the deliberation process at time step t in the second-pass
decoding uses the whole sequence generated by the first-pass decoder, including both the words
preceding and after t-th step in the first-pass sequence. That is, the first-pass contextual vector ctxc
aggregates the global information extracted from the first-pass sequence y?.
After receiving ctxc , we calculate st as st = RNN([yt?1 ; ctx0e ; ctxc ], st?1 ). Similar to sampling y?t in
D1 , [st ; ctx0e ; ctxc ; yt?1 ] will be further transformed to generate yt .
2.4
Algorithm
Let DXY = {(x(i) , y (i) )}ni=1 denote the training corpus with n paired sequences5 . Denote the
parameters of E, D1 and D2 as ?e , ?1 and ?2 respectively. The
Pn training of sequence-to-sequence
learning is usually to maximize the data log likelihood
(1/n) i=1 log P (yi |xi ). Under our setting,
P
this rule can be specialized to maximize (1/n) (x,y)?DXY J (x, y; ?e , ?1 , ?2 ), where
X
J (x, y; ?e , ?1 , ?2 ) = log
P (y|y 0 , E(x; ?e ); ?2 )P (y 0 |E(x; ?e ); ?1 ).
(2)
y 0 ?Y
In Eqn. (2), Y is the collection of all possible target sequences and E(x; ?e ) indicates a function that
maps x to its corresponding hidden states given by the encoder. One can verify that the first-order
derivative of J (x, y; ?e , ?1 , ?2 ) w.r.t ?1 is:
P
0
0
y 0 ?Y P (y|y , E(x; ?e ); ?2 )??1 P (y |E(x; ?e ); ?1 )
??1 J (x, y; ?e , ?1 , ?2 ) = P
,
0
0
y 0 ?Y P (y|y , E(x; ?e ); ?2 )P (y |E(x; ?e ); ?1 )
which is extremely hard to compute due to the large space of Y. Similarly, the gradients w.r.t. ?e and
?2 are also computationally intractable. To overcome such difficulties, we propose a Monte Carlo
based method to optimize the lower bound of J (x, y; ?e , ?1 , ?2 ). Note by the concavity of J w.r.t
y 0 , one can verify that J (x, y; ?e , ?1 , ?2 ) ? J?(x, y; ?e , ?1 , ?2 ), with the right-hand side acting as a
lower bound and defined as
X
J?(x, y; ?e , ?1 , ?2 ) =
P (y 0 |E(x; ?e ); ?1 ) log P (y|y 0 , E(x; ?e ); ?2 ).
(3)
y 0 ?Y
Let x(i) and y (i) denote i?th source input and target output in the training data. Let xi and yi denote the i-th
word in x and y.
5
4
? The gradients of J? w.r.t its parameters are:
Denote J?(x, y; ?e , ?1 , ?2 ) as J.
X
??1 J? =
P (y 0 |E(x; ?e ); ?1 ) log P (y|y 0 , E(x; ?e ); ?2 )??1 log P (y 0 |E(x; ?e ); ?1 );
|
{z
}
0
y ?Y
??2 J? =
X
y 0 ?Y
??e J? =
X
G1
0
0
P (y |E(x; ?e ); ?1 ) ??2 log P (y|y , E(x; ?e ); ?2 );
{z
}
|
(4)
G2
P (y 0 |E(x; ?e ); ?1 )Ge (x, y, y 0 ; ?e , ?1 , ?2 ), where Ge is defined as follows:
y 0 ?Y
Ge = ??e log P (y|y 0 , E(x; ?e ); ?2 ) + log P (y|y 0 , E(x; ?e ); ?2 )??e log P (y 0 |E(x; ?e ); ?1 ).
Let ? = [?1 ; ?2 ; ?e ] and G(x, y, y 0 ; ?) = [G1 ; G2 ; Ge ], where G1 , G2 and Ge are defined in Eqn. (4).
(For ease of reference, we assume that all the ?? ?s and G? ?s are flattened.) Obviously, if y 0 is sampled
from distribution P (y 0 |E(x; ?e ); ?1 ), G(x, y, y 0 ; ?) is an unbiased estimator of the gradient of J?
w.r.t. all model parameters ?. Based on that we propose our algorithm in Algorithm 1.
Algorithm 1: Algorithm to train the deliberation network
Input: Training data corpus DXY ; minibatch size m; optimizer Opt(? ? ? ) with gradients as input ;
while models not converged do
Randomly sample a mini-batch of m sequence pairs {x(i) , y (i) } ?i ? [m] from DXY ;
For any x(i) where i ? [m], sample y 0(i) according to distribution P (?|E(x(i) ; ?e ); ?1 );
Pm
1
(i) (i) 0(i)
Perform parameter update: ? ? ? + Opt( m
; ?)).
i=1 G(x , y , y
Discussions (1) The choice of Opt(...) is quite flexible. One can choose different optimizers such
as Adadelta [37], Adam [13], or SGD for different tasks, depending on common practice in the
specific task. (2) The Y space is usually extremely large in sequence generation tasks. To obtain
better sampled y 0 , we can use beam search instead of randomly sampling.
3
Application to Neural Machine Translation
We evaluate the deliberation networks with two different network structures: (1) the shallow model,
which is based on a widely-used single-layer GRU model named RNNSearch [1, 12]; (2) the deep
model, which is based on a deep LSTM model similar to GNMT [31]. Both of the two kinds of
models are implemented in Theano [24].
3.1
Shallow Models
3.1.1
Settings
Datasets We work on two translation tasks, English-to-French translation (denoted as En?Fr) and
Chinese-to-English translation (denoted as Zh?En). For En?Fr, we employ the standard filtered
WMT?14 dataset6 , which is widely used in NMT literature [1, 12]. There are 12M bilingual sentence
pairs in the dataset. We concatenate newstest2012 and newstest2013 together as the validation set and
use newstest2014 as the test set. For Zh?En, we choose 1.25M bilingual sentence pairs from LDC
dataset as training corpus, use NIST2003 as the validation set, and NIST2004, NIST2005, NIST2006,
NIST2008 as the test sets. Following the common practice [1, 12], we remove the sentences with
more than 50 words for both translation tasks. Furthermore, we limit the both the source words and
target words as 30k most-frequent ones. The out-of-vocabulary words are replaced by a special token
?UNK?.
Model We choose the most widely adopted NMT model RNNSearch [1, 12, 25] as the basic structure
to construct the deliberation network. To be specific, all of E, D1 and D2 are GRU networks [1] with
one hidden layer of 1000 neurons. The word embedding dimension is set as 620. For Zh?En, we
apply 0.5 dropout rate to the layer before softmax and no dropout is used in En?Fr translation.
6
http://www-lium.univ-lemans.fr/?schwenk/cslm_joint_paper/data/bitexts.tgz
5
Optimization All the models are trained on a single NVIDIA K40 GPU. We first pre-train two standard
encoder-decoder based NMT models (i.e., RNNSearch) until convergence, which take about two
weeks for En?Fr and one week for Zh?En using Adadelta [37]. For any deliberation network, (1)
the encoder is initialized by the encoder of the pre-trained RNNSearch model; (2) both the first-pass
and second-pass decoders are initialized by the decoder of the pre-trained RNNSearch model; (3)
the attention model used to compute the first-pass context vector is randomly initialized from a
uniform distribution on [?0.1, 0.1]. Then we train the deliberation networks by Algorithm 1 until
convergence, which takes roughly 5 days for both tasks. The minibatch size is fixed as 80 throughout
the optimization. Plain SGD is used as the optimizer in this process, with initial learning rate 0.2
and halving according to validation accuracy. To sample the intermediate translation output by the
first decoder, we use beam search with beam size 2, considering the tradeoff between accuracy and
efficiency.
Evaluation We use BLEU [17] as the evaluation metric for translation qualities. BLEU is the geometric
mean of n-gram precisions where n ? {1, 2, 3, 4}, weighted by sentence lengths. Following the
common practice in NMT, we use multi-bleu.pl7 to calculate case-sensitive BLEU scores for En?Fr,
while evaluating the translation qualities of Zh?En by case-insensitive BLEU scores. The larger the
BLEU score is, the better the translation quality is. For the baselines and deliberation networks, we
use beam search with beam size 12 to generate sentences.
Baselines We compare our proposed algorithms with the following baselines: (i) The standard NMT
algorithm RNNSearch [1, 12], denoted as Mbase ; (ii) The standard NMT model with two stacked
decoding layers, denoted as Mdec?2 ; (3) The review network proposed in [36]. We try both 4 and 8
reviewers and find the 4-reviewer model is slightly better. The review network in our experiment is
therefore denoted as Mreviewer?4 . We refer to our proposed algorithm as Mdelib .
3.1.2
Results
We compare our proposed algorithms with the following baselines: (i) The standard NMT algorithm,
denoted as Mbase ; (ii) The standard NMT model with two stacked decoding layers, denoted as
Mdec?2 ; (3) The review network proposed in [36]. We try both 4 and 8 reviewers and find the
4-reviewer model is slightly better. The review network in our experiment is therefore denoted as
Mreviewer?4 . We refer to our proposed algorithm as Mdelib . Table 1 shows the results of En?Fr
translation. We have several observations:
(1) Our proposed algorithm performs the best among all candidates, which validates the effectiveness
of the deliberation process. (2) Our method Mdelib outperforms the baseline algorithm Mbase . This
shows that further polishing the raw output indeed leads to better sequences. (3) Applying an
additional decoding layer, i.e., Mdec?2 , increases the translation quality, but it is still far behind
that of Mdelib . Clearly, the second decoder layer of Mdec?2 can still only leverage the previously
generated words but not unseen and un-generated future words, while the second-pass decoder of
Mdelib can leverage the richer information contained in all the words from the first-pass decoder. Such
a refinement process from the global view significantly improves the translation results. (4) Mdelib
outperforms Mreviewer?4 by 0.91 point, which shows that reviewing the possible future contextual
information from the source side is not enough. The ?future? information from the decoder side is
also very important, since it is directly related with the final output.
Table 1: BLEU scores of En?Fr translation
Algorithm
BLEU
Mbase
29.97
Mdec?2
30.40
Mreviewer?4
30.76
Mdelib
31.67
The translation results of Zh?En are summarized in Table 2. We have similar observations as those
for En?Fr translations: Mdelib outperforms all the baseline methods, particularly with an average
gain of 1.26 points over Mbase .
Apart from the quantitative analysis, we list two examples in Table 3 to better understand how a
deliberation network works. Each example contains five sentences, which are the source sentence
in Chinese, the reference sentence in English as ground truth translation, the translation generated
7
https://github.com/moses-smt/mosesdecoder/blob/master/scripts/generic/multi-bleu.perl
6
Table 2: BLEU scores of Zh?En translation
Algorithm
Mbase
Mdelib
NIST04
34.96
36.90
NIST05
34.57
35.57
NIST06
32.74
33.90
NIST08
26.21
27.13
by Mbase and the output translation by both the first-pass decoder and second-pass decoder (i.e., the
final translation by deliberation network Mdelib ).
Table 3: Case studies of Zh?En translations. Note the ?......? in the second example represents a common
sentence ?the two sides will discuss how to improve the implementation of the cease-fire agreement?.
[Source] Aiji shuo, zhongdong heping xieyi yuqi jiang you yige xinde jiagou .
[Reference] Egypt says a new framework is expected to come into being for the Middle East
peace agreement .
[Base] egypt ?s middle east peace agreement is expected to have a new framework , he said .
[First-pass] egypt ?s middle east peace agreement is expected to have a new framework , egypt said .
[Second-pass] egypt says the middle east peace agreement is expected to have a new framework .
[Source] Nuowei dashiguan zhichu, "shuangfang jiang taolun ruhe gaijin luoshi tinghuo xieyi, zhe
yeshi san nian lai shuangfang shouci zai ruci gao de cengji shang jinxing mianduimian tanpan"
[Reference] The Norwegian embassy pointed out that , " Both sides will discuss how to improve the
implementation of the cease-fire agreement , which is the first time for both sides to have
face-to-face negotiations at such a high level . "
[Base] " ...... , which is the first time for the two countries to conduct face-to-face talks on the basis
of a high level of three years , " it said .
[First-pass] " ...... , which is the first time for the two countries to conduct face-to-face talks on the
basis of a high level of three years , " the norwegian embassy said in a statement .
[Second-pass] " ...... , which is the first time in three years for the two countries to conduct
face-to-face talks at such high level , " the norwegian embassy said .
In the first example, the translation from both base model and first-pass decoder contains the phrase
egypt?s middle east peace agreement, which is odd and inaccurate, given that an agreement cannot
belong to a single country as Egypt. As a comparison, the second-pass decoder refines such phrase into
a more natural and accurate one. i.e., egypt says the middle east peace agreement, by looking forward
to the future translation phrase ?egypt said? output by the first-pass decoder. On the other hand, the
second-pass decoder outputs a sentence with correct tense, i.e., egypt says ... is .... However, the
two sentences output by Mbase and the first-pass decoder are inconsistent in tense, whose structures
are ?... is ..., egypt said ?. This problem is well addressed by the deliberation network, since the
second-pass decoder can access the global information contained in the draft sequence generated by
the first-pass decoder, and therefore output a more consistent sentence.
In the second example, as shown in bold fonts, the phrase ?conduct face-to-face talks on the basis
of a high level of three years? from both base model and first-pass decoder carries all necessary
information of its corresponding source segments, but apparently it is out-of-order and seems to be a
simple combination of words. The second-pass decoder refines such translation into a correct, and
more fluent one, by forwarding the sub phrase in three years to the position right after the first time.
At last we compare the decoding time of deliberation network with that of the RNNSearch. Based on
the Theano implementation, to translate 3003 English sentences to French, RNNSearch takes 964
seconds while the deliberation network takes 1924 seconds. Indeed, the deliberation network takes
roughly 2 times decoding time of RNNSearch, but can bring 1.7 points improvements in BLEU.
3.2
Deep Models
We work on a deep LSTM model to further evaluate deliberation networks through the WMT?14
En?Fr translation task. Compared to the shallow model, there are several different aspects: (1) We
use 34M sentence pairs from WMT?14 as training data, apply the BPE [21] techniques to split the
training sentences into sub-word units and restrict the source and target sentence lengths within 64
subwords. The encoder and decoder share a common vocabulary containing 36k subwords. (2) All of
E, D1 and D2 are 4-layer LSTMs with residual connections [9, 10]. The word embedding dimension
7
Table 4: Comparison between deliberation network and different deep NMT systems (En?Fr).
System
Configurations
BLEU
GNMT [31]
Stacked LSTM (8-layer encoder + 8 layer decoder) + RL finetune 39.92
FairSeq [4]
Convolution (15-layer) encoder and (15-layer) decoder
40.51
Transformer [26] Self-Attention + 6-layer encoder + 6-layer decoder
41.0
Stack LSTM (4-layer encoder and 4-layer decoder)
39.51
this work
Stack 4-layer NMT + Dual Learning
40.53
Stack 4-layer NMT + Dual Learning + Deliberation Network
41.50
and hidden node dimension are 512 and 1024 respectively. The dropout rate is set as 0.1. (3) We train
the standard encoder-decoder based deep model for about 25 days until convergence. Furthermore,
we leverage our recently proposed dual learning techniques [8, 33] to improve the model, which
takes another 7 days. We initialize the deliberation network in the same way in Section 3.1.1. Then,
we train the deliberation network by Algorithm 1 for 10 days. When generating translations, we use
beam search with beam size 8.
The experimental results of applying deliberation network to the deep LSTM model are shown
in Table 4. On En?Fr translation task, the baseline of our implemented NMT system is 39.51.
With dual learning, we achieve a 40.53 BLEU score. After applying deliberation techniques, the
BLEU score can be further improved to 41.50, which as far as we know, is a new single-model
state-of-the-art result for this task. This not only illustrates the effectiveness of deliberation network
again, but also shows that even if a model is good enough, it can still benefit from the deliberation
process.
4
Application to Text Summarization
We further verify the effectiveness of deliberation networks on text summarization, which is another
real-world application that encoder-decoder framework succeeds to help [19].
4.1
Settings
Text summarization refers to using a short and abstractive sentence to summarize the major points of
a sentence or paragraph, which is typically much longer. The training, validation and test sets for the
task are extracted from Gigaword Corpus [6]: For each selected article, the first sentence is used as
source-side input and the title used as target-side output. We process the data in the same way as that
proposed in [20, 30], and obtain training/validation/test sets with roughly 189k/18k/10k sentence
pairs respectively. There are roughly 42k unique words in the source input and 19k unique words in
the target output and we remain all of them as the vocabulary in the encoder-decoder models.
The model structure is the same as that used in Section 3.1 except that both word embedding
dimension and hidden node size are reduced to 128. We use Adadelta algorithm with gradient clip
value 5.0 to optimize deliberation network. The mini-batch size is fixed as 32.
The evaluation measures are chosen as ROUGE-1, ROUGE-2 and ROUGE-L, which are all widely
adopted evaluation metric for text summarization [15]. ROUGE-N (N = 1, 2 in our setting) is an
N-gram recall between a candidate summary and a set of reference summaries. ROUGE-L is a similar
statistic like ROUGE-N but based on longest common subsequences. When generating the titles, we
use beam search with beam size 10. For the thoroughness of comparison, similar to NMT, we add
another two baselines apart from the basic encoder-decoder model: the stacked-decoder model with 2
layers (Mdec?2 ), as well as the review net with 4 reviewers (Mreviewer?4 ).
4.2
Results
The experimental results of text summarization are listed in Table 5. Again, the deliberation network
achieves clear improvements over all the baselines. For example, in terms of ROUGE-2, it is 1.12 and
0.96 points better compared with stacked decoder model and review net respectively. Furthermore,
one may note that a significant difference between NMT and text summarization is that: In NMT, the
lengths of input and output sequence are very close; but in text summarization, the input is extremely
8
long while the output is very short. The better results brought by deliberation networks shows that
even if the output sentence is short, it is helpful to include the deliberation process which refines the
low-level draft in the first-pass decoder.
Table 5: ROUGE-{1, 2, L} scores of text summarization
Algorithm
Mbase
Mdec?2
Mreviewer?4
Mdelib
5
ROUGE-1
27.45
27.93
28.26
30.90
ROUGE-2
10.51
11.09
11.25
12.21
ROUGE-L
26.07
26.50
27.28
29.09
Conclusions and Future Work
In this work, we have proposed deliberation networks for sequence generation tasks, in which the
first-pass decoder is used for generating a raw sequence, and the second-pass decoder is used to
polish the raw sequence. Experiments show that our method achieves much better results than several
baseline methods in both machine translation and text summarization, and achieves a new single
model state-of-the-art result on WMT?14 English to French translation.
There are multiple promising directions to explore in the future. First, we will study how to apply
the idea of deliberation to tasks beyond sequence generation, such as improving the image qualities
generated by GAN [5]. Second, we will study how to refine/polish different levels of a neural network,
like the hidden states in an RNN, or feature maps in a CNN. Third, we are curious about whether
better sequences can be generated with more passes of decoders, i.e., refining a generated sequence
again and again. Fourth, we will study how to speed up the inference of deliberation networks and
reduce their inference time.
Acknowledgments
The authors would like to thank Yang Fan and Kaitao Song for implementing the deep neural machine
translation basic model. This work is partially supported by the National Natural Science Foundation
of China (Grant No. 61371192).
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6,385 | 6,776 | Adaptive Clustering through Semidefinite
Programming
Martin Royer
Laboratoire de Math?matiques d?Orsay, Univ. Paris-Sud, CNRS,
Universit? Paris-Saclay,
91405 Orsay, France
[email protected]
Abstract
We analyze the clustering problem through a flexible probabilistic model that
aims to identify an optimal partition on the sample X1 , ..., Xn . We perform
exact clustering with high probability using a convex semidefinite estimator that
interprets as a corrected, relaxed version of K-means. The estimator is analyzed
through a non-asymptotic framework and showed to be optimal or near-optimal in
recovering the partition. Furthermore, its performances are shown to be adaptive
to the problem?s effective dimension, as well as to K the unknown number of
groups in this partition. We illustrate the method?s performances in comparison
to other classical clustering algorithms with numerical experiments on simulated
high-dimensional data.
1
Introduction
Clustering, a form of unsupervised learning, is the classical problem of assembling n observations
X1 , ..., Xn from a p-dimensional space into K groups. Applied fields are craving for robust clustering techniques, such as computational biology with genome classification, data mining or image
segmentation from computer vision. But the clustering problem has proven notoriously hard when
the embedding dimension is large compared to the number of observations (see for instance the recent
discussions from [2, 21]).
A famous early approach to clustering is to solve for the geometrical estimator K-means [19, 13, 14].
The intuition behind its objective is that groups are to be determined in a way to minimize the total
intra-group variance. It can be interpreted as an attempt to "best" represent the observations by
K points, a form of vector quantization. Although the method shows great performances when
observations are homoscedastic, K-means is a NP-hard, ad-hoc method. Clustering with probabilistic
frameworks are usually based on maximum likelihood approaches paired with a variant of the EM
algorithm for model estimation, see for instance the works of Fraley & Raftery [11] and Dasgupta
& Schulman [9]. These methods are widespread and popular, but they tend to be very sensitive to
initialization and model misspecifications.
Several recent developments establish a link between clustering and semidefinite programming. Peng
& Wei [17] show that the K-means objective can be relaxed into a convex, semidefinite program,
leading Mixon et al. [16] to use this relaxation under a subgaussian mixture model to estimate the
cluster centers. Yan and Sarkar [24] use a similar semidefinite program in the context of covariate
clustering, when the network has nodes and covariates. Chr?tien et al. [8] use a slightly different form
of a semidefinite program to recover the adjacency matrix of the cluster graph with high probability.
Lastly in the different context of variable clustering, Bunea et al. [6] present a semidefinite program
with a correction step to produce non-asymptotic exact recovery results.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this work, we build upon the work and context of [6], and transpose and adapt their ideas for point
clustering: we introduce a semidefinite estimator for point clustering inspired by the findings of [17]
with a correction component originally presented in [6]. We show that it produces a very strong
contender for clustering recovery in terms of speed, adaptivity and robustness to model perturbations.
In order to do so we produce a flexible probabilistic model inducing an optimal partition of the data
that we aim to recover. Using the same structure of proof in a different context, we establish elements
of stochastic control (see for instance Lemma A.1 on the concentration of random subgaussian Gram
matrices in the supplementary material) to derive conditions of exact clustering recovery with high
probability and show optimal performances ? including in high dimensions, improving on [16], as
well as adaptivity to the effective dimension of the problem. We also show that our results continue
to hold without knowledge of the number of structures given one single positive tuning parameter.
Lastly we provide evidence of our method?s efficiency and further insight from simulated data.
Notation. Throughout this work we use the convention 0/0 := 0 and [n] = {1, ..., n}. We take
an . bn to mean that an is smaller than bn up to an absolute constant factor. Let Sd?1 denote the
0
unit sphere in Rd . For q ? N? ? {+?}, ? ? Rd , |?|q is the lq -norm and for M ? Rd?d , |M |q ,
|M |F and |M |op are respectively the entry-wise lq -norm, the Frobenius norm associated with scalar
product h., .i and the operator norm. |D|V is the variation semi-norm for a diagonal matrix D, the
difference between its maximum and minimum element. Let A < B mean that A ? B is symmetric,
positive semidefinite.
2
Probabilistic modeling of point clustering
Consider X1 , ..., Xn and let ?a = E [Xa ]. The variable Xa can be decomposed into
Xa = ?a + Ea ,
a = 1, ..., n,
(1)
p
with Ea stochastic centered variables in R .
Definition 1. For K > 1, ? = (?1 , ..., ?K ) ? (Rp )K , ? > 0 and G = {G1 , ..., GK } a partition of
[n], we say X1 , ..., Xn are (G, ?, ?)-clustered if ?k ? [K], ?a ? Gk , |?a ? ?k |2 6 ?. We then call
?(?) := min |?k ? ?l |2
k<l
(2)
the separation between the cluster means, and
?(G, ?, ?) := ?(?)/?
(3)
the discriminating capacity of (G, ?, ?).
In this work we assume that X1 , ..., Xn are (G, ?, ?)-clustered. Notice that this definition does not
impose any constraint on the data: for any given G, there exists a choice of ?, means and radius ?
important enough so that X1 , ..., Xn are (G, ?, ?)-clustered. But we are interested in partitions with
greater discriminating capacity, i.e. that make more sense in terms of group separation. Indeed remark
that if ?(G, ?, ?) < 2, the population clusters {?a }a?G1 , ..., {?a }a?GK are not linearly separable,
but a high ?(G, ?, ?) implies that they are well-separated from each other. Furthermore, we have the
following result.
?
Proposition 1. Let (GK
, ?? , ? ? ) ? arg max ?(G, ?, ?) for (G, ?, ?) such that X1 , ..., Xn are
?
?
(G, ?, ?)-clustered, and |G| = K. If ?(GK
, ?? , ? ? ) > 4 then GK
is the unique maximizer of
?(G, ?, ?).
?
So GK
is the partition maximizing the discriminating capacity over partitions of size K. Therefore
in this work, we will assume that there is a K > 1 such that X1 , ..., Xn is (G, ?, ?)-clustered with
|G| = K and ?(G, ?, ?) > 4. By Proposition 1, G is then identifiable. It is the partition we aim to
recover.
We also assume that X1 , ..., Xn are independent observations with subgaussian behavior. Instead of
the classical isotropic definition of a subgaussian random vector (see for example [20]), we use a
more flexible definition that can account for anisotropy.
Definition 2. Let Y be a random vector in Rd , Y has a subgaussian distribution if there exist
? ? Rd?d such that ?x ? Rd ,
h T
i
T
E ex (Y ?E Y ) 6 ex ?x/2 .
(4)
2
We then call ? a variance-bounding matrix of random vector Y , and write shorthand Y ? subg(?).
Note that Y ? subg(?) implies Cov(Y ) 4 ? in the semidefinite sense of the inequality. To sum-up
our modeling assumptions in this work:
Hypothesis 1. Let X1 , ..., Xn be independent, subgaussian, (G, ?, ?)-clustered with ?(G, ?, ?) > 4.
Remark that the modelization of Hypothesis 1 can be connected to another popular probabilistic
model: if we further ask that X1 , ..., Xn are identically-distributed within a group (and hence ? = 0),
the model becomes a realization of a mixture model.
3
Exact partition recovery with high probability
Let G = {G1 , ..., GK } and m := mink?[K] |Gk | denote the minimum cluster size. G can be
represented by its caracteristic matrix B ? ? Rn?n defined as ?k, l ? [K]2 , ?(a, b) ? Gk ? Gl ,
1/|Gk |
if k = l
?
Bab :=
0
otherwise.
In what follows, we will demonstrate the recovery of G through recovering its caracteristic matrix
B ? . We introduce the sets of square matrices
{0,1}
CK
:= {B ? Rn?n
: B T = B, tr(B) = K, B1n = 1n , B 2 = B}
+
Rn?n
+
CK := {B ?
[
C :=
CK .
T
: B = B, tr(B) = K, B1n = 1n , B < 0}
(5)
(6)
(7)
K?N
{0,1}
{0,1}
We have: CK
? CK ? C and CK is convex. Notice that B ? ? CK
? can be expressed as
(2007) [17] shows that the K-means estimator B
. A result by Peng, Wei
? = arg maxh?,
b Bi
B
(8)
{0,1}
B?CK
b := (hXa , Xb i)(a,b)?[n]2 ? Rn?n , the observed Gram matrix. Therefore a natural relaxation is
for ?
to consider the following estimator:
b := arg maxh?,
b Bi.
B
(9)
B?CK
b = ? + ? for ? := (h?a , ?b i)(a,b)?[n]2 ? Rn?n , and ? := E [hEa , Eb i]
Notice that E ?
(a,b)?[n]2 =
diag (tr(Var(Ea )))16a6n ? Rn?n . The following two results demonstrate that ? is the signal
structure that lead the optimizations of (8) and (9) to recover B ? , whereas ? is a bias term that can
hurt the process of recovery.
?
Proposition 2. There exist c0 > 1 absolute constant such that if ?2 (G, ?, ?) > c0 (6 + n/m) and
m?2 (?) > 8|?|V , then we have
arg maxh? + ?, Bi = B ? = arg maxh? + ?, Bi.
(10)
B?CK
{0,1}
B?CK
b estimator, as well as the K-means estimator, would recover partition
This proposition shows that the B
G on the population Gram matrix if the variation semi-norm of ? were sufficiently small compared to
the cluster separation. Notice that to recover the?
partition on the population version, we require the
discriminating capacity to grow as fast as 1 + ( n/m)1/2 instead of simply 1 from Hypothesis 1.
The following proposition demonstrates that if the condition on the variation semi-norm of ? is not
met, G may not even be recovered on the population version.
Proposition 3. There exist G, ?, ? and ? such that ?2 (G, ?, ?) = +? but we have m?2 (?) < 2|?|V
and
B? ?
/ arg maxh? + ?, Bi
and
B? ?
/ arg maxh? + ?, Bi.
B?CK
{0,1}
B?CK
3
(11)
So Proposition 3 shows that even if the population clusters are perfectly discriminated, there is a
configuration for the variances of the noise that makes it impossible to recover the right clustering by
K-means. This shows that K-means may fail when the random variable homoscedasticity assumption
is violated, and that it is important to correct for ? = diag(tr(Var(Ea )))16a6n .
b corr . Then substracting ?
b corr from ?
b can be interpreted as a
Suppose we produce such an estimator ?
b as an estimator of ?. Hence the previous results demonstrate
correcting term, i.e. a way to de-bias ?
the interest of studying the following semi-definite estimator of the projection matrix B ? , let
b corr := arg maxh?
b ??
b corr , Bi.
B
(12)
B?CK
In order to demonstrate the recovery of B ? by this estimator, we introduce different quantitative
measures of the "spread" of our stochastic variables, that affect the quality of the recovery. By
Hypothesis 1 there exist ?1 , ..., ?n such that ?a ? [n], Xa ? subg(?a ). Let
? 2 := max |?a |op ,
a?[n]
V 2 := max |?a |F ,
a?[n]
? 2 := max tr(?a )
a?[n]
(13)
b corr . Since there is no relation between the variances of the points in our model,
We now produce ?
there is very little hope of estimating Var(Ea ). As for our quantity of interest tr(Var(Ea )), a
form of volume, a rough estimation is challenging but possible. The estimator
from [6] can be
?Xd
adapted to our context. For (a, b) ? [n]2 let V (a, b) := max(c,d)?([n]\{a,b})2 hXa ? Xb , |XXcc?X
i,
d |2
bb1 := arg min
V (a, b) and bb2 := arg min
b V (a, b). Then for a ? [n], let
b?[n]\{a}
b?[n]\{a,b1 }
b corr := diag hXa ? Xb , Xa ? Xb ia?[n] .
?
b1
b2
(14)
Proposition 4. Assume that m > 2. For c6 , c7 > 0 absolute constants, with probability larger than
1 ? c6 /n we have
p
b corr ? ?|? 6 c7 ? 2 log n + (? + ? log n)? + ? 2 .
|?
(15)
So apart from the radius ? terms, that come from generous model assumptions, a proxy for ? is
produced at a ? 2 log n rate that we could not expect to improve on. Nevertheless, this control on ? is
key to attain the optimal rates below. It is general and completely independent of the structure of G,
as there is no relation between G and ?.
We are now ready to introduce this paper?s main result: a condition on the separation between the
cluster means sufficient for ensuring recovery of B ? with high probability.
Theorem 1. Assume that m > 2. For c1 , c2 > 0 absolute constants, if
p
p
?
m?2 (?) > c2 ? 2 (n + m log n) + V 2 ( n + m log n) + ?(? log n + ?) + ? 2 ( n + m) ,
(16)
b corr = B ? , and therefore Gbcorr = G.
then with probability larger than 1 ? c1 /n we have B
We call the right hand side of (16) the separating rate. Notice that we can read two kinds of
requirements coming from the separating rate: requirements on the radius ?, and?requirements
on ? 2 , V 2 , ? dependent on the distributions of observations. It appears as if ? + ? log n?can be
interpreted as a geometrical width of our problem. If we ask that ? is of the same order as ? log n,
a maximum gaussian deviation for n variables, then all conditions on ? from?(16) can be removed.
Thus for convenience of the following discussion we will now assume ? . ? log n.
How optimal is the result from Theorem 1? Notice that our result is adapted to anisotropy
in the noise,
?
but to discuss optimality it is easier to look at the isotropic scenario: V 2 = p? 2 and ? 2 = p? 2 .
Therefore ?2 (?)/? 2 represents a signal-to-noise ratio. For simplicity let us also assume that all
groups have equal size, that is |G1 | = ... = |GK | = m so that n = mK and the sufficient condition
(16) becomes
r
?2 (?)
pK
& K + log n + (K + log n)
.
(17)
?2
n
4
Optimality. To discuss optimality, we distinguish between low and high dimensional setups.
In the low-dimensional setup n ? m log n & p, we obtain the following condition:
?2 (?)
& K + log n .
2
?
(18)
Discriminating with high probability between n observations from two gaussians in dimension 1
would require a separating rate of at least ? 2 log n. This implies that when K . log n, our result is
minimax. Otherwise, to our knowledge the best clustering result on approximating mixture center
is from [16], and on the condition that ?2 (?)/? 2 & K 2 . Furthermore, the K & log n regime is
known in the stochastic-block-model community as a hard regime where a gap is surmised to exist
between the minimal information-theoretic rate and the minimal achievable computational rate (see
for example [7]).
In the high-dimensional setup n ? m log n . p, condition (17) becomes:
r
pK
?2 (?)
& (K + log n)
.
?2
n
(19)
There are few information-theoretic bounds for high-dimension clustering. Recently, Banks, Moore,
Vershynin, Verzelen and Xu (2017) [3] proved a lower
p bound for Gaussian mixture clustering
detection, namely they require a separation of order K(log K)p/n. When K . log n, our
condition is only different in that it replaces log(K) by log(n), a price to pay for going from detecting
the clusters to exactly recovering the clusters. Otherwise when K grows faster than log n there might
exist a gap between the minimal possible rate and the achievable, as discussed previously.
Adaptation to effective dimension. We can analyse further the condition (16) by introducing an
effective dimension r? , measuring the largest volume repartition for our variance-bounding matrices
?1 , ..., ?n . We will show that our estimator adapts to this effective dimension. Let
r? :=
maxa?[n] tr(?a )
?2
=
,
2
?
maxa?[n] |?a |op
(20)
r? can also be interpreted as a form of global effective rank of matrices ?a . Indeed, define Re(?) :=
tr(?)/|?|op , then we have r? 6 maxa?[n] Re(?a ) 6 maxa?[n] rank(?a ) 6 p.
?
?
Now using V 2 6 r? ? 2 and ? = r? ?, condition (16) can be written as
r
?2 (?)
r? K
& K + log n + (K + log n)
.
(21)
?2
n
By comparing this equation to (17), notice that r? is in place of p, indeed playing the role of an
effective dimension for the problem. This shows that our estimator adapts to this effective dimension,
without the use of any dimension reduction step. In consequence, equation (21) distinguishes between
an actual high-dimensional setup: n ? m log n . r? and a "low" dimensional setup r? . n ? m log n
under which, regardless of the actual value of p, our estimators recovers under the near-minimax
condition of (18).
b corr in the theorem above when n + m log n . r? .
This informs on the effect of correcting term ?
The un-corrected version of the semi-definite program (9) has a leading separating rate of ? 2 /m =
b corr correction on the other hand, (21) has leading separating factor smaller
? 2 r? /m,
pbut with the ?
?
?
2
than ? (K + log n)r? /m = ? 2 n + m log n ? r? /m. This proves
p that in a high-dimensional
setup, our correction enhances the separating rate of at least a factor (n + m log n)/r? .
4
Adaptation to the unknown number of group K
It is rarely the case that K is known, but we can proceed without it. We produce an estimator adaptive
to the number of groups K: let ?
b ? R+ , we now study the following adaptive estimator:
e corr := arg maxh?
b ??
b corr , Bi ? ?
B
b tr(B).
B?C
5
(22)
Theorem 2. Suppose that m > 2 and (16) is satisfied. For c3 , c4 , c5 > 0 absolute constants suppose
that the following condition on ?
b is satisfied
?
p
?
2
2
b < m?2 (?),
(23)
c4 V n + ? n + ?(? log n + ?) + ? 2 n < c5 ?
e corr = B ? with probability larger than 1 ? c3 /n
then we have B
Notice that condition (23) essentially requires ?
b to be seated between m?2 (?) and some components
of the right-hand side of (16). So under (23), the results from the previous section apply to the
e corr as well and this shows that it is not necessary to know K in order to
adaptive estimator B
perform well for recovering G. Finding an optimized, data-driven parameter ?
b using some form of
cross-validation is outside of the scope of this paper.
5
Numerical experiments
We illustrate our method on simulated Gaussian data in two challenging, high-dimensional setup
experiments for comparing clustering estimators. Our sample of n = 100 points are drawn from
K = 5 identically-sized, perfectly discriminated non-isovolumic clusters of Gaussians - that is we
have ?k ? [K], ?a ? Gk , Ea ? N (0, ?k ) such that |G1 | = ... = |GK | = 20. The distributions are
chosen to be isotropic, and the ratio between the lowest and the highest standard deviation is of 1 to 10.
We draw points of a Rp space in two different scenarii. In (S1 ), for a given dimension space p = 500
and a fixed isotropic noise level, we report the algorithm?s performances as the signal-to-noise ratio
?2 (?)/? 2 is increased from 1 to 15. In (S2 ) we impose a fixed signal to noise ratio and observe the
algorithm?s decay in performance as the space dimension p is increased from 102 to 105 (logarithmic
scale). All reported points of the simulated space represent a hundred simulations, and indicate a
median value with asymmetric standard deviations in the form of errorbars.
b corr is a hard problem as n grows. For this task we implemented an ADMM
Solving for estimator B
solver from the work of Boyd et al. [4] with multiple stopping criterions including a fixed number of
iterations of T = 1000. The complexity of the optimization is then roughly O(T n3 ). For reference,
we compare the recovering capacities of Gbcorr , labeled ?pecok? in Figure 1 with other classical
clustering algorithm. We chose three different but standard clustering procedures: Lloyd?s K-means
algorithm [13] with a thousand K-means++ initialization of [1] (although in scenario (S2 ), the
algorithm is too slow to converge as p grows so we do not report it), Ward?s method for Hierarchical
Clustering [23] and the low-rank clustering algorithm applied to the Gram matrix, a spectral method
appearing in McSherry [15]. Lastly we include the CORD algorithm from Bunea et al. [5].
We measure the performances of estimators by computing the adjusted mutual information (see for
instance [22]) between the truth and its estimate. In the two experiments, the results of Gbcorr are
markedly better than that of other methods. Scenario (S1 ) shows it can achieve exact recovery with a
lesser signal to noise ratio than its competitors, whereas scenario (S2 ) shows its performances start to
decay much later than the other methods as the space dimension is increased exponentially.
Table 1 summarizes the simulations in a different light: for different parameter value on each line, we
count the number of experiments (out of a hundred) that had an adjusted mutual information score
equal to 0.9 or higher. This accounts for exact recoveries, or approximate recoveries that reasonably
reflected the underlying truth. In this table it is also evident that Gbcorr performs uniformly better, be
it for exact or approximate recovery: it manages to recover the underlying truth much sooner in terms
of signal-to-noise ratio, and for a given signal-to-noise ratio it will represent the truth better as the
embedding dimension increases.
Lastly Table 1 provides the median computing time in seconds for each method over the entire
b corr is very costly to compute.
experiment. Gbcorr comes with important computation times because ?
Our method is computationally intensive but it is of polynomial order. The solving of a semidefinite
program is a vastly developing field of Operational Research and even though we used the classical
ADMM method of [4] that proved effective, this instance of the program could certainly have seen a
more profitable implementation in the hands of a domain expert. All of the compared methods have a
very hard time reaching high sample sizes n in the high dimensional context.
The P YTHON 3 implementation of the method used is found in open access here: martinroyer/pecok
[18]
6
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6,386 | 6,777 | Log-normality and Skewness of Estimated
State/Action Values in Reinforcement Learning
Liangpeng Zhang1,2 , Ke Tang3,1 , and Xin Yao3,2
1
School of Computer Science and Technology,
University of Science and Technology of China
2
University of Birmingham, U.K.
3
Shenzhen Key Lab of Computational Intelligence,
Department of Computer Science and Engineering,
Southern University of Science and Technology, China
[email protected], [email protected], [email protected]
Abstract
Under/overestimation of state/action values are harmful for reinforcement learning agents. In this paper, we show that a state/action value estimated using the
Bellman equation can be decomposed to a weighted sum of path-wise values that
follow log-normal distributions. Since log-normal distributions are skewed, the
distribution of estimated state/action values can also be skewed, leading to an
imbalanced likelihood of under/overestimation. The degree of such imbalance can
vary greatly among actions and policies within a single problem instance, making
the agent prone to select actions/policies that have inferior expected return and
higher likelihood of overestimation. We present a comprehensive analysis to such
skewness, examine its factors and impacts through both theoretical and empirical
results, and discuss the possible ways to reduce its undesirable effects.
1
Introduction
In reinforcement learning (RL) [1, 2], actions executed by the agent are decided by comparing relevant
state values V or action values Q. In most cases, the ground truth V and Q are not available to the
? instead. Therefore, whether or not an
agent, and the agent has to rely on estimated values V? and Q
?
?
RL algorithm yields sufficiently accurate V and Q is a key factor to its performance. Many researches
have proved that, for many popular RL algorithms such as Q-learning [3] and value iteration [4],
estimated values are guaranteed to converge in the limit to their ground truth values [5, 6, 7, 8].
Still, under/overestimation of state/action values occur frequently in practice. Such phenomena are
often considered as the result of insufficient sample size or the utilisation of function approximation
[9]. However, recent researches have pointed out that the basic estimators of V and Q derived
from the Bellman equation, which were considered unbiased and have been widely applied in RL
algorithms, are actually biased [10] and inconsistent [11]. For example, van Hasselt [10] showed that
the max operator in the Bellman equation and its transforms introduces bias to the estimated action
values, resulting in overestimation. New operators and algorithms have been proposed to correct such
biases [12, 13, 14], inconsistency [11] and other issues of value-based RL [15, 16, 17, 18].
This paper shows that, despite having great improvements in recent years, the value estimator
of RL can still suffer from under/overestimation. Specifically, we show that the distributions of
estimated state/action values are very likely to be skewed, resulting in imbalanced likelihood of
under/overestimation. Such skewness and likelihood can vary dramatically among actions/policies
within a single problem instance. As a result, the agent may frequently select undesirable actions/policies, regardless of its value estimator being unbiased.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
0.5
value 1
value 2
density
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
estimated value
Figure 1: Illustration of positive skewness (red distribution) and negative skewness (blue distribution).
Thick and thin vertical lines represent the corresponding expected values and medians, respectively.
Such phenomenon is illustrated in Figure 1. An estimated state/action value following the red
distribution has a mean 0.21 and a median ?0.61, thus tends to be underestimated. Another following
the blue distribution, on the other hand, has a mean ?0.92 and a median 0.61, thus likely to be
overestimated. Despite that the red expected return is noticeably greater than the blue, the probability
of an unbiased agent arriving at the opposite conclusion (blue is better) and thus selecting the inferior
action/policy is around 0.59, which is even worse than random guessing.
This paper also indicates that such skewness comes from the Bellman equation passing the dispersion
of transition dynamics to the state/action values. Therefore, as long as a value is estimated by
applying the Bellman equation to the observations of transition, it can suffer from the skewness
problem, regardless of the algorithm being used. Instead of proposing new algorithms, this paper
suggests two general ways to reduce the skewness. The first is to balance the impacts of positive and
negative immediate rewards to the estimated values. We show that positive rewards lead to positive
skewness and vice versa, and thus, a balance between the two may help neutralise the harmful effect
of skewness. The second way is to simply collect more observations of transitions. However, our
results in this paper indicate that the effectiveness of this approach diminishes quickly as the sample
size grows, and thus is recommended only when observations are cheap to obtain.
In the rest of this paper, we will elaborate our analysis to the distributions of state/action values
estimated by the Bellman equation. Specifically, we will show that an estimated value in a general
MDP can be decomposed to path-wise values in normalised single-reward Markov chains. The
path-wise values are shown to obey log-normal distributions, and thus the distribution of an estimated
value is the convolution of such log-normal distributions. To understand which factors have the most
impact to the skewness, we derive the expressions of the parameters of these log-normal distributions.
We then discuss whether the skewness of estimated values can be reduced in order to improve
learning performance. Finally, we provide our empirical results to complement our theoretical ones,
illustrating how substantial the undesirable effect of skewness can be, as well as to what degree such
effect can be reduced by obtaining more observations.
2
Preliminaries
The standard RL setup of [1] is followed in this paper. An environment is formulated as a finite
discounted Markov Decision Process (MDP) M = (S, A, P, R, ?), where S and A are finite sets
of states and actions, P (s0 |s, a) is a transition probability function, R(s, a, s0 ) is an immediate
reward function, and ? ? (0, 1) is a discount factor. A trajectory (s1 , a1 , s2 , r1 ), (s2 , a2 , s3 , r2 ), ...,
(st , at , st+1 , rt ) represents the interaction history between the agent and the MDP. The number of
occurrences of state-action pair (s, a) and transition (s, a, s0 ) in such trajectory are denoted Ns,a and
Ns,a,s0 , respectively.
A policy is denoted ?, and V ? (s) is the state value of ? starting from s. An action value Q? (s, a) is
essentially a state value following a non-stationary policy that selects a at the first step but follows ?
thereafter. It can be analysed in the same way as V ? , so it suffices to focus on V ? in the following
sections. For convenience, superscript ? in V ? will be dropped if it is clear from the context.
P
For any s ? S and policy ?, it holds that V ? (s) = s0 ?S P (s0 |s, ?(s))(R(s, ?(s), s0 ) + ?V ? (s0 )),
which is called the Bellman equation. Most model-based and model-free RL algorithms utilise this
equation, its equivalents, or its transforms to estimate state values. Since P and R are unknown to
? instead,
the agent, estimated values V? (s) are computed from estimated transitions P? and rewards R
2
? a, s0 ) = rt with (st , at , st+1 )=(s, a, s0 ). This is done
where P? (s0 |s, a) = Ns,a,s0 /Ns,a and R(s,
explicitly in model-based learning, and implicitly with frequencies of updates in model-free learning.
We will show in later section that the skewness of estimated values is decided by the dynamic effects
of the environment rather than the learning algorithm being used, and therefore, it suffices to focus
on the model-based case in order to evaluate such skewness.
p
The skewness in this paper refers to the Pearson 2 coefficient (E[X] ? median[X])/ Var[X]
[19, 20]. Following this definition, a distribution has a positive skewness if and only if its mean is
greater than its median, and vice versa. Assuming that the bias of V? is corrected or absent, we have
E[V? ] = V . Thus, a positive skewness of V? means Pr(V? <V ) > 0.5, indicating a higher likelihood
of underestimation, while a negative skewness indicates a higher likelihood of overestimation.
An informative indicator of skewness is CDFV? (V )?0.5 where CDFV? is the cumulative distribution
function of V? . The sign of this indicator is consistent with the Pearson 2 coefficient, while its absolute
value gives the extra probability of under/overestimation of V? compared to a zero-skew distribution.
A log-normal distribution with location parameter ? and scale parameter ? is denoted lnN (?, ? 2 ). A
random variable X follows lnN (?, ? 2 ) if and only if ln(X) follows normal distribution N (?, ? 2 ).
The parameters ? and ? of log-normal distribution can be calculated from its mean and variance
2
by ? = ln ? E[X]
, and ? 2 = ln 1 + Var[X]
E[X]2 , where E[X] and Var[X] are the mean and
2
E[X] +Var[X]
variance of X ? lnN (?, ? 2 ), respectively.
3
Log-normality of Estimated State Values
In this section, we elaborate our analysis to the distributions of estimated values V? . The analysis
is formed of three steps. First, state values in general MDPs are decomposed to the state values
in relevant normalised single-reward Markov chains. Second, they are further decomposed into
path-wise state values. Third, the path-wise state values are shown to obey log-normal distributions.
3.1
Decomposing into Normalised Single-reward Markov chains
Given an MDP M and a policy ?, the interaction between ? and M forms a Markov chain M ? , with
transition probability pi,j = P (sj |si , ?(si )) and reward ri,j = R(si , ?(si ), sj ) from arbitrary state
si to state sj . Let P ? be the transition matrix of M ? , V ? be the (column) vector of state values,
R? be the reward matrix, and J be a vector of 1 with the same size of V ? . Then Bellman equation
is equivalent to V ? = P ? ?R? J + ?P ? V ? = (I ? ?P ? )?1 (P ? ?R? J ), where I is an identity
matrix, and ? is Hadamard product.
This equation indicates that a state value is a weighted sum of dynamic effects, with rewards serving
? ?1
as the weights of summation. Precisely,
P let B = (I ? ?P ) , then the equation above becomes
?
?
?
?
V = B(P ?R J ), or V (si ) = j,k rj,k (bi,j pj,k ). Here, term (bi,j pj,k ) describes the joint
dynamic effect starting from si ending with transition sj sk , which will be elaborated in Section 3.2.
?
Let Mj,k
denote a normalised single-reward Markov chain (NSR-MC) of M ? , which has exactly the
?
same S, A, ? and P ? as M ? , but all rewards are trivially 0 except rj,k = 1. For an NSR-MC Mj,k
,
?
the equation above becomes VM ? (si ) = bi,j pj,k . Thus, a state value V of a general MDP M can
j,k
?
be rewritten as the weighted sum of state values of all |S|2 NSR-MCs {Mj,k
} of M , i.e.
P
?
? (si ).
VM
(si ) = j,k rj,k VMj,k
(1)
Therefore, the next step of analysis is to examine the state values in NSR-MCs.
3.2
Decomposing into Path-wise State Values
Seeing Markov chain M ? as a directed graph, a walk w of length |w| in such graph is a sequence
of |w| successive transitions through states s1 , s2 , s3 , ..., s|w|+1 .1 A path is a walk without repeated
states, with exception to the last state s|w|+1 , which can be either a visited or an unvisited one.
1
Superscripts here refer to the timestamps on w rather than the indices of specific states in S.
3
p1,1
p2,2
p1,2
1
p3,3
p2,3
2
p2,1
p3,4
3
4
p3,2
p3,1
Figure 2: Illustration of walks and a representative path. "Forward" and "backward" transitions are
drawn in thick and thin arrows, respectively, and pi,j denotes the transition probability from si to sj .
In an NSR-MC with unique non-zero reward rj,k = 1, a state value V ? (si ) = bi,j pj,k can be
expanded as a sum of the discounted occurrence probabilities of walks that start from si and end
with transition (sj , ?(sj ), sk ). Let Wi,j,k denotes the set of all possible walks w satisfying s1 =si ,
P
Q
s|w| =sj and s|w|+1 =sk . Then we have V (si ) = w?Wi,j,k (? |w|?1 (st ,st+1 ) on w pst ,st+1 ). Since
Wi,j,k is infinite, the walks in Wi,j,k need to be put into finite groups for further analysis.
Concretely, a step in a walk is considered "forward" if it arrives to a previously unvisited state, and
"backward" if the destination has already been visited before that step. The latter also includes the
cases where st+1 = st , that is, the agent stays at the same state after transition. The only exception
to this classification is the last transition of a walk, which is always considered a "forward" one,
regardless of if its destination having been visited or not. The start state s1 and all such "forward"
transitions of a walk w form a representative path of w, denoted w.
?
This is illustrated by Figure 2. In this example, all walks from s1 passing s2 ending with s3 s4 , such
as (s1 s1 s2 s3 s3 s4 ), (s1 s2 s3 s1 s2 s3 s4 ) and (s1 s2 s3 s2 s3 s2 s3 s4 ), are grouped with the representative
path (s1 s2 s3 s4 ). Note that transition s1 s3 will not happen within this group; rather, it belongs to the
groups that have s1 s3 in their representative paths.
As can be seen from Figure 2, all possible walks sharing one representative path w
? compose a chain
which has the same transition probability values with the original Markov chain M ? , but with only
two type of transitions: (forward) si to si+1 (i ? |w|);
? (backward) si to sj (j ? i ? |w|).
? We call
?
this chain the derived chain of w,
? denoted M (w),
? or simply M (w).
? Then the infinite sum becomes
P
V (s) = w?
(2)
? VM (w)
? (s),
? W
? is the set of all representative paths that start from s and end with the unique 1-reward
where W
?
transition of the relevant NSR-MC. Such VM (w)
? (s) are called path-wise state values of M .
Since the main concern of this paper is the skewness of V? , we do not provide a constructive method of
? is at most (|S|!), and thus an estimated
obtaining all M ? (w).
? Rather, we point out that the size of W
?
value V in NSR-MCs can be decomposed to finitely many estimated path-wise state values.
3.3
Log-normality of Estimated Path-wise State Values
Strictly speaking, derived chain M (w)
? of a representative path w
? is not necessarily a Markov chain,
because only part of the transitions in the original Markov chain M ? is included, allowing the
Pi+1
possibility of j=1 psi ,sj < 1. However, this does not make the path-wise state values violate
Bellman equation, and thus they can be treated as regular state values.
?
Since a representative path w
? has no repeated states (except for s|w|+1
which can either be a new state
k
or the same as some s ), the superscripts here can be treated as the indices of states for convenience.
i
Therefore, path-wise state value VM (w)
? (s ) is denoted Vi , and pi,j refers to psi ,sj in this section.
Given w,
? the most important path-wise value is V1 , which belongs to the start point of w.
?
Definition 3.1. Given a derived chain M (w)
? and discount factor ?, let pi,j be the transition probability
from si to sj on M (w).
? The joint dynamic effect of M (w)
? for i ? |w|
? is recursively defined as
Di =
?pi,i+1
.
Pi?1
Qi?1
1 ? ?(pi,i + j=1 pi,j k=j Dk )
Lemma 3.2. For all i < |w|,
? path-wise state values satisfy Vi = Di Vi+1 .
4
Proof. By Bellman equation, it holds that Vi =
P|w|+1
?
pi,j (ri,j +?Vj ). By definition of M (w)
?
Pi+1
we have pi,j = 0 for j > i+1 and ri,j = 0 for (i, j) 6= (|w|,
? |w|+1).
?
Thus Vi = ? j=1 pi,j Vj
?p1,2
for i < |w|.
? When i = 1, this becomes V1 = ?(p1,1 V1 +p1,2 V2 ) = 1??p
V2 = D1 V2 . SupQk 1,1
pose Vi = Di Vi+1 holds for all i ? k < |w|?1.
?
Then Vi = ( j=i Dj )Vk+1 for i ? k,
Pk+2
Pk+1
Qk
and therefore, Vk+1 = ? j=1 pk+1,j Vj = ?[ j=1 pk+1,j ( l=j Dl )Vk+1 + pk+1,k+2 Vk+2 ] =
?pk+1,k+2
P
Qk
V
= Dk+1 Vk+2 . Thus, by the principle of induction, Vi =
1??(p
+ k p
D ) k+2
k+1,k+1
j=1
k+1,j
l=j
j=1
l
Di Vi+1 holds for all i < |w|.
?
Lemma 3.3. For all i ? |w|,
? Vi =
1
?
Q|w|
?
j=i
Dj . Particularly, V1 =
1
?
Q|w|
?
j=1
Dj .
?
?
Proof. By definition of w,
? there are two possible cases of the last step from s|w|
to s|w|+1
:
|w|+1
?
1
|w|
?
|w|+1
?
k
(I) s
?
/ {s , ..., s }; (II) there exists k ? |w|
? such that s
=s .
?
(Case I) There is no transition starting from s|w|+1
in this case, thus V|w|+1
= 0. Therefore,
?
P|w|
P|w|
?
?
V|w|
(r|w|,|
+ ?V|w|+1
) + ? j=1 p|w|,j
+ ? j=1 p|w|,j
? = p|w|,|
? w|+1
?
? w|+1
?
?
? Vj = p|w|,|
? w|+1
?
? Vj =
Q
Q
p|w|,|
|w|?1
?
|w|
?
? w|+1
?
1
1
= ? D|w|
P|w|?1
Q|w|?1
? . Thus Vi = ( j=i Dj )V|w|
? = ?
?
?
j=i Dj .
1??(p|w|,|
? w|
? +
j=1
p|w|,j
?
k=j
Dk )
?
(Case II with s|w|+1
= sk ) In this case V|w|+1
= Vk and p|w|,|
= p|w|,k
?
? w|+1
?
? , thus V|w|
? =
P|w|
P|w|
?
?
p|w|,|
(r|w|,|
+ ?Vk ) + ? j=1,j6=k p|w|,j
+ ? j=1 p|w|,j
? w|+1
?
? w|+1
?
? Vj = p|w|,|
? w|+1
?
? Vj which is the
Q|w|
?
1
same expression as the first case, and therefore Vi = ? j=i Dj also holds for this case.
In both of the two cases above, V1 is the product of D1 , D2 , ..., D|w|
? given by Definition 3.1, and
P|w|
?
1
an additional factor ? . Thus we have ln(V1 ) = ? ln(?) + j=1 ln(Dj ). By replacing all pi,j in
? Then
Definition 3.1 with estimated transition p?i,j , we get the ?estimated? 2 joint dynamic effects D.
P
|
w|
?
? j ). Assuming D
? i ?s as independent
the equation above becomes ln(V?1 ) = ? ln(?) + j=1 ln(D
random variables, it can be shown by the central limit theorem that as |w|
? grows, ln(V?1 ) will tend to
a normal distribution, and therefore, V?1 approximates a log-normal distribution.
? are actually mutually dependent in most cases, thus the
The ?estimated? joint dynamic effects D
rigorous analysis of log-normality is more complicated. The main idea here is to first prove all
? i ? ?, and then show that the summation involving terms pi,j Qi?1 D
? k in Definition 3.1 diminish
D
k=j
? i is mostly decided by p?i,i and p?i,i+1 and thus the
quickly with the size of w,
? which indicates that D
?
dependency between any two D is relatively weak. As the focus here is to see the skewness of V?1 ,
such analysis is skipped, and we proceed to the study of parameters of log-normal distribution of V?1 .
? i , we provide the result on the most repreSince p?i,i and p?i,i+1 are the main factors that decide D
sentative case where pi,i + pi,i+1 = 1 and all other pi,j are 0 for i < |w|.
? Such M (w)
? is denoted
? i are mutually independent in such chains.
M0 (w)
? in the following text. It is easy to see that all D
The delta method [21, 22] below is used to obtain the expressions of parameters.
Lemma 3.4 (Delta method[21, 22]). Suppose X is a random variable with finite moments, E[X]
being its mean and Var[X] being its variance. Suppose f is a sufficiently differentiable function.
Then it holds that E[f (X)] ? f (E[X]), and Var[f (X)] ? f 0 (E[X])2 Var[X].
? j be Dj replacing all p with p?. Let Ni denotes the number of visits to the chain
Lemma 3.5. Let D
i
? j ] ? ?pj,j+1 , and
state s in a learning trajectory. In M0 (w)
? derived chains it holds that E[D
1??pj,j
?j] ?
Var[D
? 2 (1??)2
(1??pj,j )4
?
pj,j pj,j+1
.
Nj
Proof. It holds that Var[?
pj,j+1 ] = ( N1j )2 Nj pj,j pj,j+1 =
Definition 3.1.
2
pj,j pj,j+1
,
Nj
then by applying Lemma 3.4 to
Such ?estimation? is not done explicitly in actual algorithms, but implicitly when using Bellman equation.
5
Lemma 3.6. In M0 (w)
? derived chains it holds that
E[V?1 ] =
Var[V?1 ] ?
1
?2
Y
|w|
?
|w|
?
1Y ?
E[Dj ],
? j=1
? j ] + E[D
? j ]2 ) ?
(Var[D
j=1
|w|
?
Y
? j ]2 .
E[D
j=1
Qn
Proof. For independent X1 , X2 , ..., Xn it holds that Var[X1 ...Xn ] = j=1 (Var[Xj ] + E[Xj ]2 ) ?
Qn
2
?
? by applying this and Lemma 3.4 to Lemma
j=1 E[Xj ] . Since all D are independent in M0 (w),
3.3, the above results can be obtained.
Theorem 3.7. In M0 (w)
? with sufficiently large |w|,
? it holds that V?1 ?
? lnN (?, ? 2 ) with ? =
2
?
?
V1 ]
ln ? ?E[2V1 ] ? and ? 2 = ln 1+ Var[
, where E[V?1 ] and Var[V?1 ] are given by Lemma 3.6.
E[V? ]2
E[V1 ] +Var[V1 ]
1
Proof. By applying the equations on the parameters of log-normal (see Section 2) to V?1 .
4
Skewness of Estimated State Values, and Countermeasures
This section interprets the results presented in Section 3 in terms of skewness, and discuss how to
reduce the undesirable effects of skewness. The skewness is mainly decided by two factors: (a)
parameter ? of log-normal distributions; (b) non-zero immediate rewards.
4.1
Impact of Parameter ? of Log-normal Distributions
A regular log-normal distribution lnN (?, ? 2 ) has a positive skewness, which means a sampled value
from such distribution has more than 0.5 probability to be less than its expected value, resulting in a
higher likelihood of underestimation. Precisely, if X ? lnN (?, ? 2 ), then E[X] = exp(? + ? 2 /2)
and median[X] = exp(?), thus the Pearson 2 coefficient of X is greater than 0. Additionally, since
?
lnN (?, ? 2 ) has a CDF(x) = 0.5(1 + erf( ln(x)??
)) where erf(x) is the Gauss error function, our
2?
?
indicator CDF(E[X])?0.5 equals to 0.5 erf(?/ 8). This indicates that ? has a stronger impact than
? to the scale of the skewness in log-normal distributions.
Combining Lemma 3.6 and Theorem 3.7 shows that ? is decided by a complicated interaction
? j ?s. By Lemma 3.5, transition probabilities pj,? completely
between all observed dynamic effect D
?
? j ].
decide E[Dj ], and have substantial impacts to Var[D
This indicates that the main cause of skewness is the transition dynamics of MDPs rather than learning
algorithms. As an extreme case, if the forward transition of a state-action pair is deterministic (i.e.
? j ] = 0, resulting no contribution to the skewness. If an estimated
pj,j+1 = 1), then its Var[D
value consists of a large portion of such transitions, then the likelihoods of overestimation and
underestimation are both very low. On the other hand, if backward transition probability pj,j (or any
? j ] increases dramatically, resulting a noticeable skewness.
pj,k with k ? j) is close to 1, then Var[D
Real-world problems can be a mix of these two extremes, which leads to a great variety of skewness
among different actions/policies, making learning significantly more difficult.
By Lemma 3.5, ? is also dependent to the number of observations Nj . As Nj grows infinitely,
? j ] slowly decreases to 0, which reduces Var[V?1 ] in Lemma 3.6 and eventually leads ? to
Var[D
0. This indicates that running algorithms more steps does help reduce the skewness of estimated
? j ] in Lemma 3.5 also
values and improve the overall performance. However, the expression of Var[D
indicates that the degree of improvement diminishes quickly as Nj grows. Therefore, collecting more
observations is not always an efficient way to reduce the skewness.
6
0.8
0.8
positive
negative
density
0.4
1.5
positive
negative
convolution
0.6
density
density
0.6
0.4
positive
negative
convolution
1
0.5
0.2
0.2
0
-4
-2
0
2
0
-4
4
-2
estimated path-wise value
0
2
0
-4
4
estimated path-wise value
(a)
-2
0
2
4
estimated path-wise value
(b)
(c)
Figure 3: (a) Log-normals weighted by positive reward (red) and negative reward (blue). Thick/thin
vertical lines are means & medians. (b, c) Convolution of two log-normals, given by the purple curve.
1?p
1?p
p
1, rD
1
1?p
p
2
1
p
1
1
p
...
3
n
1, rG
1
Figure 4: A chain MDP with n states, forward probability p, goal reward rG and distraction reward
rD . Transitions under taking action a+ is drawn in solid arrows, and a? in dotted arrows.
4.2
Impact of Non-zero Immediate Rewards
Non-zero immediate rewards decide not only the scale of skewness, but also the direction of skewness.
By Equation 1 and 2 in Sections 3.1 and 3.2, path-wise values are weighted by their corresponding
immediate rewards before being summed into state values. If a path-wise state value is weighted by a
positive reward, then the resulting distribution is still a regular log-normal, which has a positive skewness and thus a higher likelihood of underestimation. However, if it is weighted by a negative reward,
then the result is a flipped log-normal, which has a negative skewness and thus a higher likelihood of
overestimation. This is illustrated in Figure 3 (a), where the red and blue distributions correspond to
the estimated path-wise values weighted by a positive and a negative reward, respectively.
In general cases, the sum of positively skewed random variables is not necessarily a positively skewed
random variable. However, the sum of regular log-normal random variables can be approximated by
another log-normal [23], thus is still positively skewed. Since path-wise state values are approximately
log-normal, it is clear that if an MDP only has positive immediate rewards, then all estimated values
are likely to be positively skewed and thus have higher likelihoods to be underestimated.
On the other hand, if an estimated value is composed of both positive and negative rewards, then the
skewness of regular and flipped log-normal distributions may partly be neutralised in their convolution.
The purple distribution in Figure 3 (b) shows the result of convolution of two skewed distributions
that lie symmetrically to x = 0. The skewness is perfectly neutralised in this case, resulting in a
symmetric distribution with a balanced likelihood of under/overestimation. In the case of Figure 3
(c), the convolution is still a skewed one, but the scale of this skewness is less than the original ones.
To make learning easier, one may hope to design the reward function such that the more desirable
actions/policies have both higher expected returns and higher likelihood of overestimation than the
less desirable ones. However, the former requires more positive rewards, while the latter calls for
more negative rewards, causing an unsolvable dilemma. Therefore, it is more realistic just to balance
the likelihood of under/overestimation, so that all actions/policies can compete fairly with each other.
Reward shaping [24, 25] can be a promising choice to achieve this goal, as it preserves the optimality
of policies. Since a better balance of positive and negative rewards directly reduces the impact of the
skewness of all relevant log-normal distributions, this approach might be more effective than simply
collecting more observations.
5
Experiments
In this section, we present our empirical results on the skewness of estimated values. There are two
purposes in these experiments: (a) to demonstrate how substantial the harm of the skewness can be;
(b) to see the improvement provided by collecting more observations, as mentioned in Section 4.1.
We conducted experiments in chain MDPs shown in Figure 4. There are n > 0 states s1 , s2 , ..., sn in
a chain MDP. At each state, the agent has two possible actions a+ and a? . By taking a+ at si with
7
sample distribution
theoretical distribution
0.15
density
probability of underestimation
0.2
0.1
0.05
0
0
5
10
15
20
0.8
empirical
theoretical
0.75
0.7
0.65
0.6
0.55
0.5
0
estimated value
100
200
300
400
#observations per state-action
(b)
(a)
+
Figure 5: (a) Distribution of V? ? (s1 ) at m = 200. (b) Underestimation probability curve.
i < n, the agent has probability p > 0 to be sent to si+1 , and 1 ? p to remain at si . Taking a+ at sn
yields a goal reward rG > 0, and the agent remains at sn . Taking a? , on the other hand, sends the
agent from si to si?1 (i > 1) or s1 (i = 1) with probability 1, and if a? is taken at s1 , then the agent
will be provided a distraction reward rD > 0.
The objective of the learning agent is to discover a policy that leads it to the goal sn and collects rG
as often as possible, rather than being distracted by rD . There are two policy of interest: ? + that
always take a+ , and ? ? that always take a? . Other policies can be proved to be always worse than
? + and ? ? in terms of V ? (s1 ) regardless of rG , rD , p, and discount factor ?.
Since using max operator may introduce bias [10], we modified the default value iteration algorithm
[4] to let it output the unbiased estimated state values by following predetermined policies rather than
using max operator. In each run of experiment, m observations were collected for each state-action
pair, resulting in a data set of size 2mn. Then, the observations were passed to the modified value
iteration algorithm to estimate the state values of ? + and ? ? under discount factor ? = 0.9.
+
?
The Markov chain M ? and M ? here are both single-path ones, and thus the corresponding
theoretical distributions of V? can be computed directly by applying Theorem 3.7. Further, since
?
?
transition probabilities in M ? are all 1, we have Var[V? ? ] = 0, and thus its estimated values
always equal trivially to the ground truth one (i.e. it will never be under/overestimated).
+
The empirical and theoretical distributions of estimated state value V? ? (s1 ) with m = 200, n = 20,
p = 0.1, rG = 1e6 in 1000 runs is shown in Figure 5 (a). One-sample Kolmogorov-Smirnov test was
conducted against the null hypotheses that the empirical data came from the theoretical log-normal
distributions. The resulting p-value was 0.1190, which failed to reject the null hypothesis at 5%
significance level, indicating no significant difference between the theoretical and sample distribution.
More importantly, Figure 5 (a) shows a clear positive skewness, indicating a higher likelihood of
underestimation. The empirical value of indicator CDF(E[V? ])?0.5 was +0.103, meaning that in
60.3% of runs, the state value was underestimated. This further indicates that, if the distraction reward
?
+
rD is set to a value such that V ? (s1 ) is slightly less than V ? (s1 ), then the agent will wrongly
?
select ? with probability close to 0.603, which is worse than random guess.
To see whether collecting more observations helps reduce skewness, the same experiments as above
were conducted with the number of observations per state-action m ranged from 20 to 400. Figure 5
+
+
(b) shows the theoretical and empirical probability of underestimation Pr(V? ? (s1 ) < EV? ? (s1 )).
At m = 20, 200 and 400, the empirical underestimate probability was 0.741, 0.603 and 0.563,
respectively. While from m = 20 to 200 there was an significant improvement of 0.138, or a 18.6%
relative improvement, from 200 to 400 it was only 0.040, or 6.6% relative. This result supports
the analysis in Section 4.1, demonstrating that the merit of collecting more observations is most
noticeable when the sample size is low, and diminishes quickly as the sample size grows.
We also conducted experiments in the complex maze domain [26] in the same manner as above. In
this domain, the task of the agent is to find a policy that can collect all flags and bring them to the
goal as often as possible, without falling into any traps. The maze used is given in Figure 6 (a).
The states in this domain is represented by the current position of the agent and the status of the three
flags. The agent starts at the start point indicated by S with no flag. At each time step, the agent can
8
1
T
T
T
T
X 3
T
2 T
m=10
lognormal fit
2
1
X X X
X
G
X
0.7
3
density
S
10-4
X
X
probability of underestimation
X
data
curve fit
0.65
0.6
0.55
0.5
0
2000
4000
6000
8000
estimated state value
(a)
(b)
10000
12000
0
40
80
120
160
200
#observations per state-action
(c)
Figure 6: (a) A complex maze. S, G, numbers, and circles stand for start, goal, flags, and traps,
?
respectively. (b) Distribution of V? ? (sstart ) at m = 10. (c) Underestimation probability curve.
select one of the four directions to move to. The agent is then sent to the adjacent grid at the chosen
direction with probability 0.7, and at each of the other three directions with probability 0.1, unless the
destination is blocked, in which case the agent remains at the current grid. Additionally, at the flag
grids (numbers in Figure 6 (a)), taking actions also provides the corresponding flag to the agent if
that flag has not been obtained yet. At the goal point (G), taking arbitrary action yields an immediate
reward equals to 1, 100, 1002 or 1003 if the agent holds 0, 1, 2 or 3 flags, respectively. Then the agent
is sent back to the start point, and all three flag are reset to their initial position. Finally, at any trap
grid (circles), taking actions sends the agent to S and resets all flags without yielding a goal reward.
The complex maze in Figure 6(a) has 440 states, 4 actions, 32 non-zero immediate rewards, and
complicated transition patterns, and thus is difficult to analyse manually. However, it is noticeable
that all non-zero immediate rewards are positive, and thus according to Section 4.2, estimated state
values are likely to have positive skew, resulting in greater likelihood of underestimation.
?
Figure 6 (b) shows the empirical distribution of estimated value V? ? (sstart, no flag ) under ? = 0.9 and
m = 10 in 1000 runs. Although it is not a path-wise state value, the distribution is approximately
log-normal with parameter ? ? 8.21, ? ? 0.480. In 67.6% of these 1000 runs, the optimal state
value at the start state was underestimated.
The effect of collecting a larger sample is show in Figure 6 (c). The probability of underestimation
decreased from 0.676 at m = 10 to 0.597 at m = 50, 0.563 at m = 100, and 0.556 at m = 200.
The data points approximated an exponential function y = 0.1725 exp(?0.04015x) + 0.5546, which
suggests that it can be very difficult to achieve underestimation probability lower than 0.55 by
collecting more data in this domain.
6
Conclusion and Future Work
This paper has shown that estimated state values computed using the Bellman equation can be
decomposed to the relevant path-wise state values, and the latter obey log-normal distributions.
Since log-normal distributions are skewed, the estimated state values also have skewed distributions,
resulting in imbalanced likelihood of under/overestimation, which can be harmful for learning.
We have also pointed out that the direction of such imbalance is decided by the immediate reward
associated to the log-normal distributions, and thus, by carefully balancing the impact of positive and
negative rewards when designing the MDPs, such undesirable imbalance can possibly be neutralised.
Collecting more observations, on the other hand, helps reduce the skewness to a degree, but such
effect becomes less significant when the sample size is already large.
It would be interesting to see how the skewness studied in this paper interacts with function approximation (e.g. neural networks [27, 28]), policy gradient [29, 30], or Monte-Carlo tree search [31, 32].
A reasonable guess is that these techniques introduce their own skewness, and the two different
skewness amplify each other, making learning even more difficult. On the other hand, reducing the
skewness discussed in this paper may improve learning performance even when such techniques
are used. Therefore, developing a concrete method of balancing positive and negative rewards (as
discussed in Section 4.2) can be very helpful, and will be investigated in the future.
9
Acknowledgements
This paper was supported by Ministry of Science and Technology of China (Grant No.
2017YFB1003102), the National Natural Science Foundation of China (Grant Nos. 61672478
and 61329302), the Science and Technology Innovation Committee Foundation of Shenzhen (Grant
No. ZDSYS201703031748284), EPSRC (Grant No. J017515/1), and in part by the Royal Society
Newton Advanced Fellowship (Reference No. NA150123).
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6,387 | 6,778 | Repeated Inverse Reinforcement Learning
Kareem Amin?
Google Research
New York, NY 10011
[email protected]
Nan Jiang?
Satinder Singh
Computer Science & Engineering,
University of Michigan, Ann Arbor, MI 48104
{nanjiang,baveja}@umich.edu
Abstract
We introduce a novel repeated Inverse Reinforcement Learning problem: the agent
has to act on behalf of a human in a sequence of tasks and wishes to minimize the
number of tasks that it surprises the human by acting suboptimally with respect to
how the human would have acted. Each time the human is surprised, the agent is
provided a demonstration of the desired behavior by the human. We formalize this
problem, including how the sequence of tasks is chosen, in a few different ways
and provide some foundational results.
1
Introduction
One challenge in building AI agents that learn from experience is how to set their goals or rewards.
In the Reinforcement Learning (RL) setting, one interesting answer to this question is inverse RL
(or IRL) in which the agent infers the rewards of a human by observing the human?s policy in a task
[2]. Unfortunately, the IRL problem is ill-posed for there are typically many reward functions for
which the observed behavior is optimal in a single task [3]. While the use of heuristics to select from
among the set of feasible reward functions has led to successful applications of IRL to the problem
of learning from demonstration [e.g., 4], not identifying the reward function poses fundamental
challenges to the question of how well and how safely the agent will perform when using the learned
reward function in other tasks.
We formalize multiple variations of a new repeated IRL problem in which the agent and (the same)
human face multiple tasks over time. We separate the reward function into two components, one
which is invariant across tasks and can be viewed as intrinsic to the human, and a second that is
task specific. As a motivating example, consider a human doing tasks throughout a work day, e.g.,
getting coffee, driving to work, interacting with co-workers, and so on. Each of these tasks has a
task-specific goal, but the human brings to each task intrinsic goals that correspond to maintaining
health, financial well-being, not violating moral and legal principles, etc. In our repeated IRL setting,
the agent presents a policy for each new task that it thinks the human would do. If the agent?s policy
?surprises? the human by being sub-optimal, the human presents the agent with the optimal policy.
The objective of the agent is to minimize the number of surprises to the human, i.e., to generalize the
human?s behavior to new tasks.
In addition to addressing generalization across tasks, the repeated IRL problem we introduce and
our results are of interest in resolving the question of unidentifiability of rewards from observations
in standard IRL. Our results are also of interest to a particular aspect of the concern about how
to make sure that the AI systems we build are safe, or AI safety. Specifically, the issue of reward
misspecification is often mentioned in AI safety articles [e.g., 5, 6, 7]. These articles mostly discuss
broad ethical concerns and possible research directions, while our paper develops mathematical
formulations and algorithmic solutions to a specific way of addressing reward misspecification.
*
?
This paper extends an unpublished arXiv paper by the authors [1].
Equal contribution.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In summary form, our contributions include: (1) an efficient reward-identification algorithm when the
agent can choose the tasks in which it observes human behavior; (2) an upper bound on the number of
total surprises when no assumptions are made on the tasks, along with a corresponding lower bound;
(3) an extension to the setting where the human provides sample trajectories instead of complete
behavior; and (4) identification guarantees when the agent can only choose the task rewards but is
given a fixed task environment.
2
Markov Decision Processes (MDPs)
An MDP is specified by its state space S, action space A, initial state distribution ? ? ?(S), transition
function (or dynamics) P : S ? A ? ?(S), reward function Y : S ? R, and discount factor
? ? [0, 1). We assume finite S and A, and ?(S) is the space of all distributions over S. A policy ? :
S ? A describes an agent?s behavior by specifying the action to take inPeach state. The (normalized)
?
value function or long-term utility of ? is defined as V ? (s)
(1??) E[ t=1 ? t?1 Y (st )|s0 = s; ?].2
P=
?
?
t?1
Similarly, the Q-value function is Q (s, a) = (1 ? ?) E[ t=1 ? Y (st )|s0 = s, a0 = a; ?]. Where
?
necessary we will use the notation VP,Y
to avoid ambiguity about the dynamics and the reward
?
function. Let ? : S ? A be an optimal policy, which maximizes V ? and Q? in all states (and
actions) simultaneously.
Given an initial distribution over states, ?, a scalar value that measures the goodness of ? is defined
as Es?? [V ? (s)]. We introduce some further notation to express Es?? [V ? (s)] in vector-matrix form.
?
Let ??,P
? R|S| be the normalized state occupancy under initial distribution ?, dynamics P , and
P?
policy ?, whose s-th entry is (1??) E[ t=1 ? t?1 I(st = s)|s0 ? ?; ?] (I(?) is the indicator function).
?1 >
?
This vector can be computed in closed-form as ??,P
= (1 ? ?) ?> P ? I|S| ? ?P ?
, where
P ? is an |S| ? |S| matrix whose (s, s0 )-th element is P (s0 |s, ?(s)), and I|S| is the |S| ? |S| identity
matrix. For convenience we will also treat the reward function Y as a vector in R|S| , and we have
?
Es?? [V ? (s)] = Y > ??,P
.
3
(1)
Problem setup
Here we define the repeated IRL problem. The human?s reward function ?? captures his/her safety
concerns and intrinsic/general preferences. This ?? is unknown to the agent and is the object of
interest herein, i.e., if ?? were known to the agent, the concerns addressed in this paper would be
solved. We assume that the human cannot directly communicate ?? to the agent but can evaluate the
agent?s behavior in a task as well as demonstrate optimal behavior. Each task comes with an external
reward function R, and the goal is to maximize the reward with respect to Y := ?? + R in each task.
As a concrete example, consider an agent for an autonomous vehicle. In this case, ?? represents the
cross-task principles that define good driving (e.g., courtesy towards pedestrians and other vehicles),
which are often difficult to explicitly describe. In contrast, R, the task-specific reward, could reward
the agent for successfully completing parallel parking. While R is easier to construct, it may not
completely capture what a human deems good driving. (For example, an agent might successfully
parallel park while still boxing in neighboring vehicles.)
More formally, a task is defined by a pair (E, R), where E = (S, A, ?, P, ?) is the task environment
(i.e., a controlled Markov process) and R is the task-specific reward function (task reward). We
assume that all tasks share the same S, A, ?, with |A| ? 2, but may differ in the initial distribution ?,
dynamics P , and task reward R; all of the task-specifying quantities are known to the agent. In any
task, the human?s optimal behavior is always with respect to the reward function Y = ?? + R. We
emphasize again that ?? is intrinsic to the human and remains the same across all tasks. Our use of
task specific reward functions R allows for greater generality than the usual IRL setting, and most
of our results apply equally to the case where R ? 0.
While ?? is private to the human, the agent has some prior knowledge on ?? , represented as a set of
possible parameters ?0 ? R|S| that contains ?? . Throughout, we assume that the human?s reward
has bounded and normalized magnitude, that is, k?? k? ? 1.
2
Here we differ (w.l.o.g.) from common IRL literature in assuming that reward occurs after transition.
2
A demonstration in (E, R) reveals ? ? , optimal for Y = ?? + R under environment E, to the agent. A
common assumption in the IRL literature is that the full mapping is revealed, which can be unrealistic
if some states are unreachable from the initial distribution. We address the issue by requiring only
??
the state occupancy vector ??,P
. In Section 7 we show that this also allows an easy extension to the
setting where the human only demonstrates trajectories instead of providing a policy.
Under the above framework for repeated IRL, we consider two settings that differ in how the sequence
of tasks are chosen. In both settings, we will want to minimize the number of demonstrations needed.
1. (Section 5) Agent chooses the tasks, observes the human?s behavior in each of them, and infers the
reward function. In this setting where the agent is powerful enough to choose tasks arbitrarily, we
will show that the agent will be able to identify the human?s reward function which of course implies
the ability to generalize to new tasks.
2. (Section 6) Nature chooses the tasks, and the agent proposes a policy in each task. The human
demonstrates a policy only if the agent?s policy is significantly suboptimal (i.e., a mistake). In this
setting we will derive upper and lower bounds on the number of mistakes our agent will make.
4
The challenge of identifying rewards
Note that it is impossible to identify ?? from watching human behavior in a single task. This is
because any ?? is fundamentally indistinguishable from an infinite set of reward functions that yield
exactly the policy observed in the task. We introduce the idea of behavioral equivalence below to
tease apart two separate issues wrapped up in the challenge of identifying rewards.
Definition 1. Two reward functions ?, ?0 ? R|S| are behaviorally equivalent in all MDP tasks, if for
any (E, R), the set of optimal policies for (R + ?) and (R + ?0 ) are the same.
We argue that the task of identifying the reward function should amount only to identifying the
(behavioral) equivalence class to which ?? belongs. In particular, identifying the equivalence class
is sufficient to get perfect generalization to new tasks. Any remaining unidentifiability is merely
representational and of no real consequence. Next we present a constraint that captures the reward
functions that belong to the same equivalence class.
Proposition 1. Two reward functions ? and ?0 are behaviorally equivalent in all MDP tasks if and
only if ? ? ?0 = c ? 1|S| for some c ? R, where 1|S| is an all-1 vector of length |S|.
The proof is elementary and deferred to Appendix A. For any class of ??s that are equivalent to
each other, we can choose a canonical element to represent this class. For example, we can fix an
arbitrary reference state sref ? S, and fix the reward of this state to 0 for ?? and all candidate ??s.
In the rest of the paper, we will always assume such canonicalization in the MDP setting, hence
?? ? ?0 ? {? ? [?1, 1]|S| : ?(sref ) = 0}.
5
Agent chooses the tasks
In this section, the protocol is that the agent chooses a sequence of tasks {(Et , Rt )}. For each task
(Et , Rt ), the human reveals ?t? , which is optimal for environment Et and reward function ?? + Rt .
Our goal is to design an algorithm which chooses {(Et , Rt )} and identifies ?? to a desired accuracy,
, using as few tasks as possible. Theorem 1 shows that a simple algorithm can identify ?? after only
O(log(1/)) tasks, if any tasks may be chosen. Roughly speaking, the algorithm amounts to a binary
search on each component of ?? by manipulating the task reward Rt .3 See the proof for the algorithm
specification. As noted before, once the agent has identified ?? within an appropriate tolerance, it can
compute a sufficiently-near-optimal policy for all tasks, thus completing the generalization objective
through the far stronger identification objective in this setting.
Theorem 1. If ?? ? ?0 ? {? ? [?1, 1]|S| : ?(sref ) = 0}, there exists an algorithm that outputs
? ? R|S| that satisfies k? ? ?? k? ? after O(log(1/)) demonstrations.
Proof. The algorithm chooses the following fixed environment in all tasks: for each s ? S \ {sref },
let one action be a self-loop, and the other action transitions to sref . In sref , all actions cause self-loops.
3
While we present a proof that manipulates Rt , an only slightly more complex proof applies to the setting
where all the Rt are exactly zero and the manipulation is limited to the environment [1].
3
The initial distribution over states is uniformly at random over S \ {sref }. Each task only differs in
the task reward Rt (where Rt (sref ) ? 0 always). After observing the state occupancy of the optimal
policy, for each s we check if the occupancy is equal to 0. If so, it means that the demonstrated optimal
policy chooses to go to sref from s in the first time step, and ?? (s) + Rt (s) ? ?? (sref ) + Rt (sref ) = 0;
if not, we have ?? (s) + Rt (s) ? 0. Consequently, after each task we learn the relationship between
?? (s) and ?Rt (s) on each s ? S \ {sref }, so conducting a binary search by manipulating Rt (s) will
identify ?? to -accuracy after O(log(1/)) tasks.
6
Nature chooses the tasks
While Theorem 1 yields a strong identification guarantee, it also relies on a strong assumption, that
{(Et , Rt )} may be chosen by the agent in an arbitrary manner. In this section, we let nature, who is
allowed to be adversarial for the purpose of the analysis, choose {(Et , Rt )}.
Generally speaking, we cannot obtain identification guarantees in such an adversarial setup. As an
example, if Rt ? 0 and Et remains the same over time, we are essentially back to the classical IRL
setting and suffer from the degeneracy issue. However, generalization to future tasks, which is our
ultimate goal, is easy in this special case: after the initial demonstration, the agent can mimic it to
behave optimally in all subsequent tasks without requiring further demonstrations. More generally, if
nature repeats similar tasks, then the agent obtains little new information, but presumably it knows
how to behave in most cases; if nature chooses a task unfamiliar to the agent, then the agent is likely
to err, but it may learn about ?? from the mistake.
To formalize this intuition, we consider the following protocol: the nature chooses a sequence of
tasks {(Et , Rt )} in an arbitrary manner. For every task (Et , Rt ), the agent proposes a policy ?t . The
human examines the policy?s value under ?t , and if the loss
h ?
i
h
i
?
lt = Es?? VEtt, ?? +Rt (s) ? Es?? VE?tt, ?? +Rt (s)
(2)
is less than some? then the human is satisfied and no demonstration
is needed; otherwise a mistake is
?t
?t?
counted and ??t ,Pt is revealed to the agent (note that ??t ,Pt can be computed by the agent if needed
from ?t? and its knowledge of the task). The main goal of this section is to design an algorithm that
has a provable guarantee on the total number of mistakes.
On human supervision Here we require the human to evaluate the agent?s policies in addition to
providing demonstrations. We argue that this is a reasonable assumption because (1) only a binary
signal I(lt > ) is needed as opposed to the precise value of lt , and (2) if a policy is suboptimal but
the human fails to realize it, arguably it should not be treated as a mistake. Meanwhile, we will also
provide identification guarantees in Section 6.4, as the human will be relieved from the supervision
duty once ?? is identified.
Before describing and analyzing our algorithm, we first notice that the Equation 2 can be rewritten as
??
lt = (?? + R)> (??tt ,Pt ? ???tt ,Pt ),
(3)
using Equation 1. So effectively, the given environment Et in each round induces a set of state
occupancy vectors {???t ,Pt : ? ? (S ? A)}, and we want the agent to choose the vector that has
the largest dot product with ?? + R. The exponential size of the set will not be a concern because
our main result (Theorem 2) has no dependence on the number of vectors, and only depends on the
dimension of those vectors. The result is enabled by studying the linear bandit version of the problem,
which subsumes the MDP setting for our purpose and is also a model of independent interest.
6.1
The linear bandit setting
In the linear bandit setting, D is a finite action space with size |D| = K. Each task is denoted as a
pair (X, R), where R is the task specific reward function as before. X = [x(1) ? ? ? x(K) ] is a d ? K
feature matrix, where x(i) is the feature vector for the i-th action, and kx(i) k1 ? 1. When we reduce
MDPs to linear bandits, each element of D corresponds to an MDP policy, and the feature vector is
the state occupancy of that policy.
As before, R, ?? ? Rd are the task reward and the human?s unknown reward, respectively. The initial
uncertainty set for ?? is ?0 ? [?1, 1]d . The value of the i-th action is calculated as (?? + R)> x(i) ,
4
Algorithm 1 Ellipsoid Algorithm for Repeated Inverse Reinforcement Learning
1: Input: ?0 .
2: ?1 ? MVEE(?0 ).
3: for t = 1, 2, . . . do
4:
Nature reveals (Xt , Rt ).
a
5:
Learner plays at = arg maxa?D c>
t xt , where ct is the center of ?t . ?t+1 ? ?t .
6:
if lt > then
a?
7:
Human reveals a?t . ?t+1 ? MVEE({? ? ?t : (? ? ct )> (xt t ? xat t ) ? 0}).
8:
end if
9: end for
and a? is the action that maximizes this value. Every round the agent proposes an action a ? D,
whose loss is defined as
?
lt = (?? + R)> (xa ? xa ).
We now show how to embed the previous MDP setting in linear bandits.
Example 1. Given an MDP problem with variables S, A, ?, ?? , sref , ?0 , {(Et , Rt )}, we can convert
it into a linear bandit problem as follows: (all variables with prime belong to the linear bandit problem,
and we use v \i to denote the vector v with the i-th coordinate removed)
\s
? D = {? : S ? A}, d = |S| ? 1, ??0 = ?? ref , ?00 = {?\sref : ? ? ?0 }.
\s
? x?t = (???t ,Pt )\sref . Rt0 = Rt ref ? Rt (sref ) ? 1d .
Note that there is a more straightforward conversion by letting d = |S|, ??0 = ?? , ?00 = ?0 , x?t =
???t ,Pt , Rt0 = Rt , which also preserves losses. We perform a more succinct conversion in Example 1
by canonicalizing both ?? (already assumed) and Rt (explicitly done here) and dropping the coordinate
for sref in all relevant vectors.
MDPs with linear rewards In IRL literature, a generalization of the MDP setting is often considered, that reward is linear in state features ?(s) ? Rd [2, 3]. In this new setting, ?? and R are reward
parameters, and the actual reward is (?? + R)> ?(s). This new setting can also be reduced to linear
bandits similarly to Example 1, except that the state occupancy is replaced by the discounted sum
of expected feature values. Our main result, Theorem 2, will still apply automatically, but now the
guarantee will only depend on the dimension of the feature space and has no dependence on |S|. We
include the conversion below but do not further discuss this setting in the rest of the paper.
Example 2. Consider an MDP problem with state features, defined by S, A, ?, d ? Z+ , ?? ?
Rd , ?0 ? [?1, 1]d , {(Et , ?t ? Rd , Rt ? Rd )}, where task reward and background reward in state s
are ??> ?t (s) and R> ?t (s) respectively, and ?? ? ?0 . Suppose k?t (s)k? ? 1 always holds, then we
can convert it into a linear
P? bandit problem as follows: D = {? : S ? A}. d, ?? , and Rt remain the
same. x?t = (1 ? ?) h=1 ? h?1 E[?(sh ) | ?t , Pt , ?]/d. Note that the division of d in x?t is for the
purpose of normalization, so that kx?t k1 ? k?k1 /d ? k?k? ? 1.
6.2
Ellipsoid Algorithm for Repeated Inverse Reinforcement Learning
We propose Algorithm 1, and provide the mistake bound in the following theorem.
Theorem 2. For ?0 = [?1, 1]d , the number of mistakes made by Algorithm 1 is guaranteed to be
O(d2 log(d/)).
To prove Theorem 2, we quote a result from linear programming literature in Lemma 1, which is
found in standard lecture notes (e.g., [8], Theorem 8.8; see also [9], Lemma 3.1.34).
Lemma 1 (Volume reduction in ellipsoid algorithm). Given any non-degenerate ellipsoid B in Rd
centered at c ? Rd , and any non-zero vector v ? Rd , let B + be the minimum-volume enclosing
1
ellipsoid (MVEE) of {u ? B : (u ? c)> v ? 0}. We have vol(B + )/vol(B) ? e? 2(d+1) .
a?
Proof of Theorem 2. Whenever a mistake is made, we can induce the constraint (Rt + ?? )> (xt t ?
a?
xat t ) > . Meanwhile, since at is greedy w.r.t. ct , we have (Rt + ct )> (xt t ? xat t ) ? 0, where ct is
5
the center of ?t as in Line 5. Taking the difference of the two inequalities, we obtain
a?
(?? ? ct )> (xt t ? xat t ) > .
(4)
Therefore, the update rule on Line 7 of Algorithm 1 preserves ?? in ?t+1 . Since the update makes
a central cut through the ellipsoid, Lemma 1 applies and the volume shrinks every time a mistake
is made. To prove the theorem, it remains to upper bound the initial volume and lower bound the
terminal volume of ?t . We first show that an update never eliminates B? (?? , /2), the `? ball
a?
centered at ?? with radius /2. This is because, any eliminated ? satisfies (? + ct )> (xt t ? xat t ) < 0.
Combining this with Equation 4, we have
a?
a?
< (?? ? ?)> (xt t ? xat t ) ? k?? ? ?k? kxt t ? xat t k1 ? 2k?? ? ?k? .
The last step follows from kxk1 ? 1. We conclude that any eliminated ? should be /2 far
away from
T ?? in `? distance. Hence, we can lower bound the volume of ?t for any t by that
of ?0 B? (?? , /2), which contains an `? ball with radius /4 at its smallest (when ?? is one of
?0 ?s vertices). To simplify calculation, we relax this lower bound (volume of the `? ball) to the
volume of the inscribed `2 ball.
Finally we put everything together: let MT be the number of mistakes made from round 1 to T , Cd
be the volume of the unit hypersphere in Rd (i.e., `2 ball with radius 1), and vol(?) denote the volume
of an ellipsoid, we have
?
? d
4 d
MT
d
? log(vol(?1 )) ? log(vol(?T +1 )) ? log(Cd ( d) ) ? log(Cd (/4) ) = d log
.
2(d + 1)
So MT ? 2d(d + 1) log
6.3
?
4 d
= O(d2 log d ).
Lower bound
In Section 5, we get an O(log(1/)) upper bound on the number of demonstrations, which has no
dependence on |S| (which corresponds to d + 1 in linear bandits). Comparing Theorem 2 to 1, one
may wonder whether the polynomial dependence on d is an artifact of the inefficiency of Algorithm 1.
We clarify this issue by proving a lower bound, showing that ?(d log(1/)) mistakes are inevitable
in the worst case when nature chooses the tasks. We provide a proof sketch below, and the complete
proof is deferred to Appendix E.
Theorem 3. For any randomized algorithm4 in the linear bandit setting, there always exists ?? ?
[?1, 1]d and an adversarial sequence of {(Xt , Rt )} that potentially adapts to the algorithm?s previous
decisions, such that the expected number of mistakes made by the algorithm is ?(d log(1/)).
Proof Sketch. We randomize ?? by sampling each element i.i.d. from Unif([?1, 1]). We will prove
that there exists a strategy of choosing (Xt , Rt ) such that any algorithm?s expected number of
mistakes is ?(d log(1/), which proves the theorem as max is no less than average.
In our construction, Xt = [0d , ejt ], where jt is some index to be specified. Hence, every round
the agent is essentially asked to decided whether ?(jt ) ? ?Rt (jt ). The adversary?s strategy goes in
phases, and Rt remains the same during each phase. Every phase has d rounds where jt is enumerated
over {1, . . . , d}.
The adversary will use Rt to shift the posterior on ?(jt ) + Rt (jt ) so that it is centered around the
origin; in this way, the agent has about 1/2 probability to make an error (regardless of the algorithm),
and the posterior interval will be halved. Overall, the agent makes d/2 mistakes in each phase, and
there will be about log(1/) phases in total, which gives the lower bound.
Applying the lower bound to MDPs The above lower bound is stated for linear bandits. In
principle, we need to prove lower bound for MDPs separately, because linear bandits are more general
than MDPs for our purpose, and the hard instances in linear bandits may not have corresponding
MDP instances. In Lemma 2 below, we show that a certain type of linear bandit instances can
always be emulated by MDPs with the same number of actions, and the hard instances constructed in
4
While our Algorithm 1 is deterministic, randomization is often crucial for online learning in general [10].
6
Theorem 3 indeed satisfy the conditions for such a type; in particular, we require the feature vectors
to be non-negative and have `1 norm bounded by 1. As a corollary, an ?(|S| log(1/)) lower bound
for the MDP setting (even with a small action space |A| = 2) follows directly from Theorem 3. The
proof of Lemma 2 is deferred to Appendix B.
Lemma 2 (Linear bandit to MDP conversion). Let (X, R) be a linear bandit task, and K be the
number of actions. If every xa is non-negative and kxa k1 ? 1, then there exists an MDP task
(E, R0 ) with d + 1 states and K actions, such that under some choice of sref , converting (E, R0 ) as
in Example 1 recovers the original problem.
6.4
On identification when nature chooses tasks
While Theorem 2 successfully controls the number of total mistakes, it completely avoids the
identification problem and does not guarantee to recover ?? . In this section we explore further
conditions under which we can obtain identification guarantees when Nature chooses the tasks.
The first condition, stated in Proposition 2, implies that if we have made all the possible mistakes,
then we have indeed identified the ?? , where the identification accuracy is determined by the tolerance
parameter that defines what is counted as a mistake. Due to space limit, the proof is deferred to
Appendix C.
Proposition 2. Consider the linear bandit setting. If there exists T0 such that for any round t ? T0 ,
no more mistakes can be ever made by the algorithm for any choice of (Et , Rt ) and any tie-braking
mechanism, then we have ?? ? B? (cT0 , ).
While the above proposition shows that identification is guaranteed if the agent exhausts the mistakes,
the agent has no ability to actively fulfill this condition when nature chooses tasks. For a stronger
identification guarantee, we may need to grant the agent some freedom in choosing the tasks.
Identification with fixed environment Here we consider a setting that fits in between Section 5
(completely active) and Section 6.1 (completely passive), where the environment E (hence the
induced feature vectors {x(1) , x(2) , . . . , x(K) }) is given and fixed, and the agent can arbitrarily
choose the task reward Rt . The goal is to obtain identification guarantee in this intermediate setting.
Unfortunately, a degenerate case can be easily constructed that prevents the revelation of any information about ?? . In particular, if x(1) = x(2) = . . . = x(K) , i.e., the environment is completely
uncontrolled, then all actions are equally optimal and nothing can be learned. More generally, if
for some v 6= 0 we have v > x(1) = v > x(2) = . . . = v > x(K) , then we may never recover ?? along
the direction of v. In fact, Proposition 1 can be viewed as an instance of this result where v = 1|S|
?
(recall that 1>
|S| ??,P ? 1), and that is why we have to remove such redundancy in Example 1 in
order to discuss identification in MDPs. Therefore, to guarantee identification in a fixed environment,
the feature vectors must have significant variation in all directions, and we capture this intuition
by defining a diversity score spread(X) (Definition 2) and showing that the identification accuracy
depends inversely on the score (Theorem 4).
Definition 2. Given the feature matrix X = x(1) x(2) ? ? ? x(K) whose size is d ? K, define
e := X(IK ? 1 1K 1> ).
spread(X) as the d-th largest singular value of X
K
K
Theorem 4. For a fixed feature matrix X, if spread(X) > 0, then there exists a sequence
R1 , R2 , . . . , RT with T = O(d2 log(d/)) and apsequence of tie-break choices of the algorithm,
such that after round T we have kcT ? ?? k? ? (K ? 1)/2/spread(X).
?
The proof is deferred to Appendix D. The K dependence in Theorem 4 may be of concern as K can
be exponentially large. However, Theorem 4 also holds if we replace X by any matrix that consists
of X?s columns, so we may choose a small yet most diverse set of columns as to optimize the bound.
7
Working with trajectories
In previous sections, we have assumed that the human evaluates the agent?s performance based on the
state occupancy of the agent?s policy, and demonstrates the optimal policy in terms of state occupancy
as well. In practice, we would like to instead assume that for each task, the agent rolls out a trajectory,
and the human shows an optimal trajectory if he/she finds the agent?s trajectory unsatisfying. We
7
Algorithm 2 Trajectory version of Algorithm 1 for MDPs
1: Input: ?0 , H, n.
? ? 0, Z? ? ? 0.
2: ?1 ? MVEE(?0 ), i ? 0, Z
3: for t = 1, 2, . . . do
4:
Nature reveals (Et , Rt ). Agent rolls-out a trajectory using ?t greedily w.r.t. ct + Rt .
5:
?t+1 ? ?t .
6:
if agent takes a in s with Q? (s, a) < V ? (s) ? then
7:
Human produces an H-step trajectory from s. Let the empirical state occupancy be z?i?,H .
8:
i ? i + 1, Z? ? ? Z? ? + z?i?,H .
9:
Let zi be the state occupancy of ?t from initial state s, and Z? ? Z? + zi .
10:
if i = n then
? ? 0}). i ? 0, Z? ? 0, Z? ? ? 0.
11:
?t+1 ? MVEE({? ? ?t : (? ? ct )> (Z? ? ? Z)
12:
end if
13:
end if
14: end for
are still concerned about upper bounding the number of total mistakes, and aim to provide a parallel
version of Theorem 2.
Unlike in traditional IRL, in our setting the agent is also acting, which gives rise to many subtleties.
First, the total reward on the agent?s single trajectory is a random variable, and may deviate from the
expected value of its policy. Therefore, it is generally impossible to decide if the agent?s policy is
near-optimal, and instead we assume that the human can check if each action that the agent takes
in the trajectory is near-optimal: when the agent takes a at state s, an error is counted if and only if
Q? (s, a) < V ? (s) ? . This criterion can be viewed as a noisy version of the one used in previous
sections, as taking expectation of V ? (s) ? Q? (s, ?(s)) over the occupancy induced by ? will recover
Equation 2.
While this resolves the issue on the agent?s side, how should the human provide his/her optimal
trajectory? The most straightforward protocol is that the human rolls out a trajectory from the
initial distribution of the task, ?t . We argue that this is not a reasonable protocol for two reasons:
(1) in expectation, the reward collected by the human may be less than that by the agent, because
conditioning on the event that an error is spotted may introduce a selection bias; (2) the human may
not encounter the problematic state in his/her own trajectory, hence the information provided in the
trajectory may be irrelevant.
To resolve this issue, we consider a different protocol where the human rolls out a trajectory using an
optimal policy from the very state where the agent errs.
Now we discuss how we can prove a parallel of Theorem 2 under this new protocol. First, let?s
assume that the demonstration were still given in the form a state occupancy vector starting at the
problematic state. In this case, we can reduce to the setting of Section 6 by changing ?t to a point
mass on the problematic state.5 To apply the algorithm and the analysis in Section 6, it remains
to show that the notion of error in this section (a suboptimal action) implies the notion of error in
Section 6 (a suboptimal policy): let s be the problematic state and ? be the agent?s policy, we have
V ? (s) = Q? (s, ?(s)) ? Q? (s, ?(s)) < V ? (s) ? . So whenever a suboptimal action is spotted in
state s, it indeed implies that the agent?s policy is suboptimal for s as the initial state. Hence, we can
run Algorithm 1 as-is and Theorem 2 immediately applies.
To tackle the remaining issue that the demonstration is in terms of a single trajectory, we will not
update ?t after each mistake as in Algorithm 1, but only make an update after every mini-batch of
mistakes, and aggregate them to form accurate update rules. See Algorithm 2. The formal guarantee
of the algorithm is stated in Theorem 5, whose proof is deferred to Appendix G.
5
At the first glance this might seem suspicious: the problematic state is random and depends on the learner?s
current policy, but in RL the initial distribution is usually fixed and the learner has no control over it. This
concern is removed thanks to our adversarial setup on (Et , Rt ) (of which ?t is a component).
8
Theorem 5. ?? ? (0, 1), with probability at least 1 ? ?, the number
of mistakes
made
' by Algorithm 2
&
?
m
l
4d(d+1) log 6 d
log(
)
?
, and n =
with parameters ?0 = [?1, 1]d , H = log(12/)
where d = |S|,6
1??
322
? d22 log( d )).7
is at most O(
?
8
Related work & Conclusions
Most existing work in IRL focused on inferring the reward function8 using data acquired from a fixed
environment [2, 3, 16, 17, 18, 19, 20]. There is prior work on using data collected from multiple ?
but exogenously fixed ? environments to predict agent behavior [21]. There are also applications
where methods for single-environment MDPs have been adapted to multiple environments [17].
Nevertheless, all these works consider the objective of mimicking an optimal behavior in the presented
environment(s), and do not aim at generalization to new tasks that is the main contribution of this
paper. Recently, Hadfield et. al. [22] proposed cooperative inverse reinforcement learning, where
the human and the agent act in the same environment, allowing the human to actively resolve the
agent?s uncertainty on the reward function. However, they only consider a single environment (or
task), and the unidentifiability issue of IRL still exists. Combining their interesting framework with
our resolution to unidentifiability (by multiple tasks) can be an interesting future direction.
Acknowledgement
This work was supported in part by NSF grant IIS 1319365 (Singh & Jiang) and in part by a Rackham
Predoctoral Fellowship from the University of Michigan (Jiang). Any opinions, findings, conclusions,
or recommendations expressed here are those of the authors and do not necessarily reflect the views
of the sponsors.
References
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[3] Pieter Abbeel and Andrew Y Ng. Apprenticeship Learning via Inverse Reinforcement Learning.
In Proceedings of the 21st International Conference on Machine learning, page 1. ACM, 2004.
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6
Here we use the simpler conversion explained right after Example 1. We can certainly improve the dimension
to d = |S| ? 1 by dropping the sref coordinate in all relevant vectors but that complicates presentation.
7
?
A log log(1/) term is suppressed in O(?).
8
While we do not discuss it here, in the economics literature, the problem of inferring an agent?s utility from
behavior-queries has long been studied under the heading of utility or preference elicitation [11, 12, 13, 14, 15].
While our result in Section 5 uses similar techniques to elicit the reward function, we do so purely by observing
the human?s behavior without external source of information (e.g., query responses).
9
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[17] Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy
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10
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6,388 | 6,779 | The Numerics of GANs
Lars Mescheder
Autonomous Vision Group
MPI T?bingen
[email protected]
Sebastian Nowozin
Machine Intelligence and Perception Group
Microsoft Research
[email protected]
Andreas Geiger
Autonomous Vision Group
MPI T?bingen
[email protected]
Abstract
In this paper, we analyze the numerics of common algorithms for training Generative Adversarial Networks (GANs). Using the formalism of smooth two-player
games we analyze the associated gradient vector field of GAN training objectives.
Our findings suggest that the convergence of current algorithms suffers due to two
factors: i) presence of eigenvalues of the Jacobian of the gradient vector field with
zero real-part, and ii) eigenvalues with big imaginary part. Using these findings,
we design a new algorithm that overcomes some of these limitations and has better
convergence properties. Experimentally, we demonstrate its superiority on training
common GAN architectures and show convergence on GAN architectures that are
known to be notoriously hard to train.
1
Introduction
Generative Adversarial Networks (GANs) [10] have been very successful in learning probability
distributions. Since their first appearance, GANs have been successfully applied to a variety of
tasks, including image-to-image translation [12], image super-resolution [13], image in-painting [27]
domain adaptation [26], probabilistic inference [14, 9, 8] and many more.
While very powerful, GANs are known to be notoriously hard to train. The standard strategy for
stabilizing training is to carefully design the model, either by adapting the architecture [21] or by
selecting an easy-to-optimize objective function [23, 4, 11].
In this work, we examine the general problem of finding local Nash-equilibria of smooth games. We
revisit the de-facto standard algorithm for finding such equilibrium points, simultaneous gradient
ascent. We theoretically show that the main factors preventing the algorithm from converging are
the presence of eigenvalues of the Jacobian of the associated gradient vector field with zero real-part
and eigenvalues with a large imaginary part. The presence of the latter is also one of the reasons that
make saddle-point problems more difficult than local optimization problems. Utilizing these insights,
we design a new algorithm that overcomes some of these problems. Experimentally, we show that
our algorithm leads to stable training on many GAN architectures, including some that are known to
be hard to train.
Our technique is orthogonal to strategies that try to make the GAN-game well-defined, e.g. by adding
instance noise [24] or by using the Wasserstein-divergence [4, 11]: while these strategies try to ensure
the existence of Nash-equilibria, our paper deals with their computation and the numerical difficulties
that can arise in practice.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In summary, our contributions are as follows:
? We identify the main reasons why simultaneous gradient ascent often fails to find local
Nash-equilibria.
? By utilizing these insights, we design a new, more robust algorithm for finding Nashequilibria of smooth two-player games.
? We empirically demonstrate that our method enables stable training of GANs on a variety of
architectures and divergence measures.
The proofs for the theorems in this paper can be found the supplementary material.1
2
Background
In this section we first revisit the concept of Generative Adversarial Networks (GANs) from a
divergence minimization point of view. We then introduce the concept of a smooth (non-convex)
two-player game and define the terminology used in the rest of the paper. Finally, we describe
simultaneous gradient ascent, the de-facto standard algorithm for finding Nash-equilibria of such
games, and derive some of its properties.
2.1
Divergence Measures and GANs
Generative Adversarial Networks are best understood in the context of divergence minimization:
assume we are given a divergence function D, i.e. a function that takes a pair of probability
distributions as input, outputs an element from [0, ?] and satisfies D(p, p) = 0 for all probability
distributions p. Moreover, assume we are given some target distribution p0 from which we can draw
i.i.d. samples and a parametric family of distributions q? that also allows us to draw i.i.d. samples. In
practice q? is usually implemented as a neural network that acts on a hidden code z sampled from
some known distribution and outputs an element from the target space. Our goal is to find ?? that
minimizes the divergence D(p0 , q? ), i.e. we want to solve the optimization problem
min D(p0 , q? ).
?
(1)
Most divergences that are used in practice can be represented in the following form [10, 16, 4]:
D(p, q) = max Ex?q [g1 (f (x))] ? Ex?p [g2 (f (x))]
f ?F
(2)
for some function class F ? X ? R and convex functions g1 , g2 : R ? R. Together with (1), this
leads to mini-max problems of the form
min max Ex?q? [g1 (f (x))] ? Ex?p0 [g2 (f (x))] .
?
f ?F
(3)
These divergences include the Jensen-Shannon divergence [10], all f-divergences [16], the Wasserstein
divergence [4] and even the indicator divergence, which is 0 if p = q and ? otherwise.
In practice, the function class F in (3) is approximated with a parametric family of functions,
e.g. parameterized by a neural network. Of course, when minimizing the divergence w.r.t. this
approximated family, we no longer minimize the correct divergence. However, it can be verified that
taking any class of functions in (3) leads to a divergence function for appropriate choices of g1 and
g2 . Therefore, some authors call these divergence functions neural network divergences [5].
2.2
Smooth Two-Player Games
A differentiable two-player game is defined by two utility functions f (?, ?) and g(?, ?) defined over a
common space (?, ?) ? ?1 ? ?2 . ?1 corresponds to the possible actions of player 1, ?2 corresponds
to the possible actions of player 2. The goal of player 1 is to maximize f , whereas player 2 tries to
maximize g. In the context of GANs, ?1 is the set of possible parameter values for the generator,
whereas ?2 is the set of possible parameter values for the discriminator. We call a game a zero-sum
game if f = ?g. Note that the derivation of the GAN-game in Section 2.1 leads to a zero-sum game,
1
The code for all experiments in this paper is available under https://github.com/LMescheder/
TheNumericsOfGANs.
2
Algorithm 1 Simultaneous Gradient Ascent (SimGA)
1: while not converged do
2:
v? ? ?? f (?, ?)
3:
v? ? ?? g(?, ?)
4:
? ? ? + hv?
5:
? ? ? + hv?
6: end while
whereas in practice people usually employ a variant of this formulation that is not a zero-sum game
for better convergence [10].
? ?)
? given by the two conditions
Our goal is to find a Nash-equilibrium of the game, i.e. a point x
? = (?,
? and ?? ? argmax g(?,
? ?).
?? ? argmax f (?, ?)
?
(4)
?
? ?)
? a local Nash-equilibrium, if (4) holds in a local neighborhood of (?,
? ?).
?
We call a point (?,
Every differentiable two-player game defines a vector field
?? f (?, ?)
v(?, ?) =
.
?? g(?, ?)
(5)
We call v the associated gradient vector field to the game defined by f and g.
For the special case of zero-sum two-player games, we have g = ?f and thus
?2? f (?, ?)
??,? f (?, ?)
0
v (?, ?) =
.
???,? f (?, ?) ??2? f (?, ?)
(6)
As a direct consequence, we have the following:
Lemma 1. For zero-sum games, v 0 (x) is negative (semi-)definite if and only if ?2? f (?, ?) is negative
(semi-)definite and ?2? f (?, ?) is positive (semi-)definite.
x) is negative semi-definite for any local Nash-equilibrium
Corollary 2. For zero-sum games, v 0 (?
x
?. Conversely, if x
? is a stationary point of v(x) and v 0 (?
x) is negative definite, then x
? is a local
Nash-equilibrium.
Note that Corollary 2 is not true for general two-player games.
2.3
Simultaneous Gradient Ascent
The de-facto standard algorithm for finding Nash-equilibria of general smooth two-player games
is Simultaneous Gradient Ascent (SimGA), which was described in several works, for example in
[22] and, more recently also in the context of GANs, in [16]. The idea is simple and is illustrated in
Algorithm 1. We iteratively update the parameters of the two players by simultaneously applying
gradient ascent to the utility functions of the two players. This can also be understood as applying the
Euler-method to the ordinary differential equation
d
x(t) = v(x(t)),
dt
(7)
where v(x) is the associated gradient vector field of the two-player game.
It can be shown that simultaneous gradient ascent converges locally to a Nash-equilibrium for a
zero-sum game, if the Hessian of both players is negative definite [16, 22] and the learning rate is
small enough. Unfortunately, in the context of GANs the former condition is rarely met. We revisit
the properties of simultaneous gradient ascent in Section 3 and also show a more subtle property,
namely that even if the conditions for the convergence of simultaneous gradient ascent are met, it
might require extremely small step sizes for convergence if the Jacobian of the associated gradient
vector field has eigenvalues with large imaginary part.
3
=(z)
=(z)
=(z)
<(z)
<(z)
<(z)
(a) Illustration how the eigenvalues (b) Example where h has to be cho- (c) Illustration how our method alleare projected into unit ball.
sen extremely small.
viates the problem.
Figure 1: Images showing how the eigenvalues of A are projected into the unit circle and what causes
problems: when discretizing the gradient flow with step size h, the eigenvalues of the Jacobian at a
fixed point are projected into the unit ball along rays from 1. However, this is only possible if the
eigenvalues lie in the left half plane and requires extremely small step sizes h if the eigenvalues are
close to the imaginary axis. The proposed method moves the eigenvalues to the left in order to make
the problem better posed, thus allowing the algorithm to converge for reasonable step sizes.
3
Convergence Theory
In this section, we analyze the convergence properties of the most common method for training
GANs, simultaneous gradient ascent2 . We show that two major failure causes for this algorithm
are eigenvalues of the Jacobian of the associated gradient vector field with zero real-part as well as
eigenvalues with large imaginary part.
For our theoretical analysis, we start with the following classical theorem about the convergence of
fixed-point iterations:
Proposition 3. Let F : ? ? ? be a continuously differential function on an open subset ? of Rn
and let x
? ? ? be so that
1. F (?
x) = x
?, and
2. the absolute values of the eigenvalues of the Jacobian F 0 (?
x) are all smaller than 1.
Then there is an open neighborhood U of x
? so that for all x0 ? U , the iterates F (k) (x0 ) converge
to x
?. The rate of convergence is at least linear. More precisely, the error kF (k) (x0 ) ? x
?k is in
O(|?max |k ) for k ? ? where ?max is the eigenvalue of F 0 (?
x) with the largest absolute value.
Proof. See [6], Proposition 4.4.1.
In numerics, we often consider functions of the form
F (x) = x + h G(x)
(8)
for some h > 0. Finding fixed points of F is then equivalent to finding solutions to the nonlinear
equation G(x) = 0 for x. For F as in (8), the Jacobian is given by
F 0 (x) = I + h G0 (x).
(9)
Note that in general neither F 0 (x) nor G0 (x) are symmetric and can therefore have complex eigenvalues.
The following Lemma gives an easy condition, when a fixed point of F as in (8) satisfies the
conditions of Proposition 3.
2
A similar analysis of alternating gradient ascent, a popular alternative to simultaneous gradient ascent, can
be found in the supplementary material.
4
Lemma 4. Assume that A ? Rn?n only has eigenvalues with negative real-part and let h > 0. Then
the eigenvalues of the matrix I + h A lie in the unit ball if and only if
h<
1
|<(?)|
2
1+
=(?)
<(?)
2
(10)
for all eigenvalues ? of A.
Corollary 5. If v 0 (?
x) only has eigenvalues with negative real-part at a stationary point x
?, then
Algorithm 1 is locally convergent to x
? for h > 0 small enough.
Equation 10 shows that there are two major factors that determine the maximum possible step size h:
(i) the maximum value of <(?) and (ii) the maximum value q of |=(?)/<(?)|. Note that as q goes to
infinity, we have to choose h according to O(q ?2 ) which can quickly become extremely small. This
is visualized in Figure 1: if G0 (?
x) has an eigenvalue with small absolute real part but big imaginary
part, h needs to be chosen extremely small to still achieve convergence. Moreover, even if we make h
small enough, most eigenvalues of F 0 (?
x) will be very close to 1, which leads by Proposition 3 to very
slow convergence of the algorithm. This is in particular a problem of simultaneous gradient ascent
for two-player games (in contrast to gradient ascent for local optimization), where the Jacobian G0 (?
x)
is not symmetric and can therefore have non-real eigenvalues.
4
Consensus Optimization
In this section, we derive the proposed method and analyze its convergence properties.
4.1
Derivation
Finding stationary points of the vector field v(x) is equivalent to solving the equation v(x) = 0. In
the context of two-player games this means solving the two equations
?? f (?, ?) = 0 and
?? g(?, ?) = 0.
(11)
A simple strategy for finding such stationary points is to minimize L(x) = 21 kv(x)k2 for x. Unfortunately, this can result in unstable stationary points of v or other local minima of 12 kv(x)k2 and in
practice, we found it did not work well.
We therefore consider a modified vector field w(x) that is as close as possible to the original vector
field v(x), but at the same time still minimizes L(x) (at least locally). A sensible candidate for such
a vector field is
w(x) = v(x) ? ??L(x)
(12)
for some ? > 0. A simple calculation shows that the gradient ?L(x) is given by
?L(x) = v 0 (x)T v(x).
(13)
This vector field is the gradient vector field associated to the modified two-player game given by the
two modified utility functions
f?(?, ?) = f (?, ?) ? ?L(?, ?) and
g?(?, ?) = g(?, ?) ? ?L(?, ?).
(14)
The regularizer L(?, ?) encourages agreement between the two players. Therefore we call the
resulting algorithm Consensus Optimization (Algorithm 2). 3 4
3
This algorithm requires backpropagation through the squared norm of the gradient with respect to the
weights of the network. This is sometimes called double backpropagation and is for example supported by the
deep learning frameworks Tensorflow [1] and PyTorch [19].
4
As was pointed out by Ferenc Huzs?r in one of his blog posts on www.inference.vc, naively implementing
this algorithm in a mini-batch setting leads to biased estimates of L(x). However, the bias goes down linearly
with the batch size, which justifies the usage of consensus optimization in a mini-batch setting. Alternatively,
it is possible to debias the estimate by subtracting a multiple of the sample variance of the gradients, see the
supplementary material for details.
5
Algorithm 2 Consensus optimization
1: while not converged do
2:
v? ? ?? (f (?, ?) ? ?L(?, ?))
3:
v? ? ?? (g(?, ?) ? ?L(?, ?))
4:
? ? ? + hv?
5:
? ? ? + hv?
6: end while
4.2
Convergence
For analyzing convergence, we consider a more general algorithm than in Section 4.1 which is given
by iteratively applying a function F of the form
F (x) = x + h A(x)v(x).
(15)
for some step size h > 0 and an invertible matrix A(x) to x. Consensus optimization is a special
case of this algorithm for A(x) = I ? ? v 0 (x)T . We assume that ?1 is not an eigenvalue of v 0 (x)T for
any x, so that A(x) is indeed invertible.
? is a fixed point of (15) if and only if
Lemma 6. Assume h > 0 and A(x) invertible for all x. Then x
it is a stationary point of v. Moreover, if x
? is a stationary point of v, we have
F 0 (?
x) = I + hA(?
x)v 0 (?
x).
(16)
Lemma 7. Let A(x) = I ? ?v 0 (x)T and assume that v 0 (?
x) is negative semi-definite and invertible5 .
Then A(?
x)v 0 (?
x) is negative definite.
As a consequence of Lemma 6 and Lemma 7, we can show local convergence of our algorithm to a
local Nash equilibrium:
Corollary 8. Let v(x) be the associated gradient vector field of a two-player zero-sum game and
A(x) = I ? ?v 0 (x)T . If x
? is a local Nash-equilibrium, then there is an open neighborhood U of x
? so
that for all x0 ? U , the iterates F (k) (x0 ) converge to x
? for h > 0 small enough.
Our method solves the problem of eigenvalues of the Jacobian with (approximately) zero real-part.
As the next Lemma shows, it also alleviates the problem of eigenvalues with a big imaginary-to-realpart-quotient:
Lemma 9. Assume that A ? Rn?n is negative semi-definite. Let q(?) be the maximum of |=(?)|
|<(?)|
(possibly infinite) with respect to ? where ? denotes the eigenvalues of A ? ?AT A and <(?) and
=(?) denote their real and imaginary part respectively. Moreover, assume that A is invertible with
|Av| ? ?|v| for ? > 0 and let
|?
v T (A + AT )v|
c = minn T
(17)
v (A ? AT )v|
v?S(C ) |?
where S(Cn ) denotes the unit sphere in Cn . Then
q(?) ?
1
.
c + 2?2 ?
(18)
Lemma 9 shows that the imaginary-to-real-part-quotient can be made arbitrarily small for an appropriate choice of ?. According to Proposition 3, this leads to better convergence properties near a local
Nash-equilibrium.
5
Experiments
Mixture of Gaussians In our first experiment we evaluate our method on a simple 2D-example
where our goal is to learn a mixture of 8 Gaussians with standard deviations equal to 10?2 and modes
Note that v 0 (?
x) is usually not symmetric and therefore it is possible that v 0 (?
x) is negative semi-definite and
invertible but not negative-definite.
5
6
(a) Simultaneous Gradient Ascent
(b) Consensus optimization
Figure 2: Comparison of Simultaneous Gradient Ascent and Consensus optimization on a circular
mixture of Gaussians. The images depict from left to right the resulting densities of the algorithm
after 0, 5000, 10000 and 20000 iterations as well as the target density (in red).
v 0 (x)
w0 (x)
Before
training
After
training
Figure 3: Empirical distribution of eigenvalues before and after training using consensus optimization.
The first column shows the distribution of the eigenvalues of the Jacobian v 0 (x) of the unmodified
vector field v(x). The second column shows the eigenvalues of the Jacobian w0 (x) of the regularized
vector field w(x) = v(x) ? ??L(x) used in consensus optimization. We see that v 0 (x) has eigenvalues close to the imaginary axis near the Nash-equilibrium. As predicted theoretically, this is not the
case for the regularized vector field w(x). For visualization purposes, the real part of the spectrum of
w0 (x) before training was clipped.
uniformly distributed around the unit circle. While simplistic, algorithms training GANs often fail to
converge even on such simple examples without extensive fine-tuning of the architecture and hyper
parameters [15].
For both the generator and critic we use fully connected neural networks with 4 hidden layers and
16 hidden units in each layer. For all layers, we use RELU-nonlinearities. We use a 16-dimensional
Gaussian prior for the latent code z and set up the game between the generator and critic using the
utility functions as in [10]. To test our method, we run both SimGA and our method with RMSProp
and a learning rate of 10?4 for 20000 steps. For our method, we use a regularization parameter of
? = 10.
The results produced by SimGA and our method for 0, 5000, 10000 and 20000 iterations are depicted
in Figure 2. We see that while SimGA jumps around the modes of the distribution and fails to
converge , our method converges smoothly to the target distribution (shown in red). Figure 3 shows
the empirical distribution of the eigenvalues of the Jacobian of v(x) and the regularized vector field
w(x). It can be seen that near the Nash-equilibrium most eigenvalues are indeed very close to the
7
(b) celebA
(a) cifar-10
Figure 4: Samples generated from a model where both the generator and discriminator are given as in
[21], but without batch-normalization. For celebA, we also use a constant number of filters in each
layer and add additional RESNET-layers.
(a) Discriminator loss
(b) Generator loss
(c) Inception score
Figure 5: (a) and (b): Comparison of the generator and discriminator loss on a DC-GAN architecture with 3 convolutional layers trained on cifar-10 for consensus optimization (without batchnormalization) and alternating gradient ascent (with batch-normalization). We observe that while
alternating gradient ascent leads to highly fluctuating losses, consensus optimization successfully
stabilizes the training and makes the losses almost constant during training. (c): Comparison of the
inception score over time which was computed using 6400 samples. We see that on this architecture
both methods have comparable rates of convergence and consensus optimization achieves slightly
better end results.
imaginary axis and that the proposed modification of the vector field used in consensus optimization
moves the eigenvalues to the left.
CIFAR-10 and CelebA In our second experiment, we apply our method to the cifar-10 and celebAdatasets, using a DC-GAN-like architecture [21] without batch normalization in the generator or the
discriminator. For celebA, we additionally use a constant number of filters in each layer and add
additional RESNET-layers. These architectures are known to be hard to optimize using simultaneous
(or alternating) gradient ascent [21, 4].
Figure 4a and 4b depict samples from the model trained with our method. We see that our method
successfully trains the models and we also observe that unlike when using alternating gradient ascent,
the generator and discriminator losses remain almost constant during training. This is illustrated
in Figure 5. For a quantitative evaluation, we also measured the inception-score [23] over time
(Figure 5c), showing that our method compares favorably to a DC-GAN trained with alternating
gradient ascent. The improvement of consensus optimization over alternating gradient ascent is even
more significant if we use 4 instead of 3 convolutional layers, see Figure 11 in the supplementary
material for details.
Additional experimental results can be found in the supplementary material.
6
Discussion
While we could prove local convergence of our method in Section 4, we believe that even more
insights can be gained by examining global convergence properties. In particular, our analysis from
8
Section 4 cannot explain why the generator and discriminator losses remain almost constant during
training.
Our theoretical results assume the existence of a Nash-equilibrium. When we are trying to minimize
an f-divergence and the dimensionality of the generator distribution is misspecified, this might not be
the case [3]. Nonetheless, we found that our method works well in practice and we leave a closer
theoretical investigation of this fact to future research.
In practice, our method can potentially make formerly instable stationary points of the gradient vector
field stable if the regularization parameter is chosen to be high. This may lead to poor solutions. We
also found that our method becomes less stable for deeper architectures, which we attribute to the fact
that the gradients can have very different scales in such architectures, so that the simple L2-penalty
from Section 4 needs to be rescaled accordingly.
Our method can be regarded as an approximation to the implicit Euler method for integrating the
gradient vector field. It can be shown that the implicit Euler method has appealing stability properties
[7] that can be translated into convergence theorems for local Nash-equilibria. However, the implicit
Euler method requires the solution of a nonlinear equation in each iteration. Nonetheless, we believe
that further progress can be made by finding better approximations to the implicit Euler method.
An alternative interpretation is to view our method as a second order method. We hence believe that
further progress can be made by revisiting second order optimization methods [2, 18] in the context
of saddle point problems.
7
Related Work
Saddle point problems do not only arise in the context of training GANs. For example, the popular
actor-critic models [20] in reinforcement learning are also special cases of saddle-point problems.
Finding a stable algorithm for training GANs is a long standing problem and multiple solutions have
been proposed. Unrolled GANs [15] unroll the optimization with respect to the critic, thereby giving
the generator more informative gradients. Though unrolling the optimization was shown to stabilize
training, it can be cumbersome to implement and in addition it also results in a big model. As was
recently shown, the stability of GAN-training can be improved by using objectives derived from
the Wasserstein-1-distance (induced by the Kantorovich-Rubinstein-norm) instead of f-divergences
[4, 11]. While Wasserstein-GANs often provide a good solution for the stable training of GANs, they
require keeping the critic optimal, which can be time-consuming and can in practice only be achieved
approximately, thus violating the conditions for theoretical guarantees. Moreover, some methods
like Adversarial Variational Bayes [14] explicitly prescribe the divergence measure to be used, thus
making it impossible to apply Wasserstein-GANs. Other approaches that try to stabilize training, try
to design an easy-to-optimize architecture [23, 21] or make use of additional labels [23, 17].
In contrast to all the approaches described above, our work focuses on stabilizing training on a wide
range of architecture and divergence functions.
8
Conclusion
In this work, starting from GAN objective functions we analyzed the general difficulties of finding
local Nash-equilibria in smooth two-player games. We pinpointed the major numerical difficulties that
arise in the current state-of-the-art algorithms and, using our insights, we presented a new algorithm
for training generative adversarial networks. Our novel algorithm has favorable properties in theory
and practice: from the theoretical viewpoint, we showed that it is locally convergent to a Nashequilibrium even if the eigenvalues of the Jacobian are problematic. This is particularly interesting
for games that arise in the context of GANs where such problems are common. From the practical
viewpoint, our algorithm can be used in combination with any GAN-architecture whose objective can
be formulated as a two-player game to stabilize the training. We demonstrated experimentally that
our algorithm stabilizes the training and successfully combats training issues like mode collapse. We
believe our work is a first step towards an understanding of the numerics of GAN training and more
general deep learning objective functions.
9
Acknowledgements
This work was supported by Microsoft Research through its PhD Scholarship Programme.
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11
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6,389 | 678 | Physiologically Based Speech Synthesis
~akoto
Hirayanaa
t ATR Human Information Processing Research Laboratories
2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Eric Vatikiotis-Bateson
tATR Auditory and Visual Perception Research Laboratories
Kiyoshi Hondat
Yasuharu Koiket
~itsuo
Kawatot*
Abstract
This study demonstrates a paradigm for modeling speech production based on neural networks. Using physiological data from
speech utterances, a neural network learns the forward dynamics
relating motor commands to muscles and the ensuing articulator
behavior that allows articulator trajectories to be generated from
motor commands constrained by phoneme input strings and global
performance parameters. From these movement trajectories, a second neural network generates PARCOR parameters that are then
used to synthesize the speech acoustics.
1
INTRODUCTION
Our group has attempted to model speech production computationally as a process
in which linguistic intentions are realized as speech through a causal succession of
patterned behavior. Our aim is to gain insight into the cognitive and neurophysiological mechanisms governing this complex skilled behavior as well as to provide
plausible models of speech synthesis and possibly recognition based on the physiology of speech production. It is the use of physiological data (EMG) representing
* Also, Laboratory of Parallel Distributed Processing, Research Institute for Electronic
Science, Hokkaido University, Sapporo, Hokkaido 060, Japan
658
Physiologically Based Speech Synthesis
motor commands to muscles that distinguishes our modeling effort from those of
others who use neural networks for articulation-based synthesis and/or inference
of the dynamical constraints on speech motor control (Jordan, 1986, Jordan, 1990,
Bailly, Laboissiere, and Schwalz, 1992, Saltzman, 1986, Bengio, Houde, and Jordan, 1992). This paper reports two areas in which implementation of the speech
production scheme shown in Figure 1 has progressed. Initially, we concentrated on
modeling the dynamics underlying articulation so that phoneme strings can specify motor commands to muscles, which then specify phoneme-specific articulator
behavior (Hirayama, Vatikiotis-Bateson, Kawato, and Jordan, 1992). A neural network learned the forward dynamics relating motor commands to muscles and the
ensuing articulator behavior associated with prosodic ally intact, but phonemic ally
simplified, reiterant speech utterances. Then, a cascade neural network (Kawato,
Maeda, Uno, and Suzuki, 1990) containing the forward dynamics model along with a
suitable smoothness criterion (Uno, Kawato, and Suzuki, 1989) was used to produce
continuous motor commands from a sequence of discrete articulatory targets corresponding to the phoneme input string. From this sequence of motor commands,
appropriate articulator trajectories were then generated.
Intention to Speak
Intended Phoneme
Global Performance
Sequence
Parameters
Articulator Movement
Figure 1: Conceptual scheme of speech production
Although the results of this early work were encouraging, there were two technical
limitations obstructing our effort to model real speech. First, using optoelectronic
transduction techniques, only simple speech samples whose primary articulators
were the lips and jaw could be recorded, hence the use of reiterant ba. Without dynamic tongue data, real speech could not be modeled. Also, the reiterant paradigm
introduced a degree of rhythmical movement behavior not observed in real speech.
The second limitation was that activity of only four muscles and generally only one
dimension of articulator motion could be recorded simultaneously. Thus, agonistantagonist muscle activity was not represented even for this limited set of articulators. Technical improvements in data acquisition and their consequences for
the subsequent dynamical modeling of real speech are presented in the next two
sections. The second area of progress has been to implement the transform from
model- generated articulator trajectories to acoustic output. A neural network is
659
660
Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato
used to acquire the mapping between articulation and acoustics in terms of PARCOR parameters (Itakura and Saito, 1969), which are correlated with vocal tract
area functions. Speech signals are then generated using a PARCOR synthesizer
from articulator input and appropriate glottal sources (currently, the residual of
the PARCOR analysis). The results of this modeling for real and reiterant speech
are reported in the final section of the paper.
2
EMPIRICAL DEVELOPMENTS
In order to acquire data more suitable for real speech modeling, two additional
experiments were run in which articulator position, EMG and acoustic data were
recorded while the same subject produced real and reiterant speech utterances 5-8
seconds long at different speaking rates and styles (e.g., casual vs. precise). In
the first of these, a sophisticated optoelectronic device, OPTOTRAK (Northern
Digital, Inc.), was used because it permitted simultaneous recording of numerous
3D articulator positions for the lips, jaw and head, ten EMG channels, the speech
acoustics, and even dynamic tongue-palate contact patterns. These data were used
for modeling of the forward dynamics (see Figure 2) and the forward acoustics.
Real speech utterances collected with this system were heavily loaded with labial
stops, /p,b,m/, and labiodental fricatives, /f,v /, as well as many low vowels /a,
ae/. Since surface EMG was used, it was difficult to obtain reliable recordings of
jaw opening (anterior belly of the digastric), and closing (medial pterygoid) muscles.
More recently, an electromagnetic position traking system, EMMA (Perkell, Cohen,
Svirsky, Matthies, Garabieta, and Jackson, 1992), was used to transduce midsagittal
motions of the tongue tip and tongue blade as well as the lips, jaw, and head.
Data were collected for the same speech utterances used in the OPTOTRAK and
original experiments as well as more natural utterances. Reiterant speech was also
recorded for tao For this experiment, surface and hooked-wire EMG techniques
were combined, which enabled nine orofacial and extrinsic tongue muscles to be
recorded for jaw opening and closing, lip opening and closing, and tongue raising
and lowering. The most important aspects of the signal processing for modeling
the forward dynamics concern the numerical differentiation of articulator position
to obtain velocity and acceleration, and the severe low-pass filtering (including
rectification and integration) of the EMG to from 2000 Hz to 20-40 Hz. Both of
these introduce spatiotemporal distortions, whose effects on the forward dynamics
model are currently being examined.
3
MODELING THE FORWARD DYNAMICS
The forward dynamics model was obtained using a 3-layer perceptron with back
propagation (Rumelhart, Hinton, and Williams, 1986). Inputs to the network were
instantaneous position and velocity for each dimension of articulator motion, and
the EMG signals of 9-10 related muscles, which serve as the record of motor commands to muscles; outputs were accelerations for each dimension of motion. Figure 2
shows an example of predicting lip and jaw accelerations from 10 orofacial muscles
for the 'natural' test utterance, "Pam put the bobbin in the frying pan and added
more puppy parts to the boiling potato soup." As shown by the generalization results in Figure 2, the acquired model produced appropriate acceleration trajectories
Physiologically Based Speech Synthesis
for real speech utterances, suggesting that utterance complexity is not a limiting
factor in this approach.
-Network Output
Upper Lip
....... ?Experimental Data
I---"-'~
Lower Lip
Jaw
o
200
400
600
800
1000
Figure 2: Estimated acceleration over time (5 ms samples) for vertical motion of the three
articulators is compared to that of the test sentence: "Pam put the bobbin in the frying
pan and added more puppy parts to the boiling potato soup".
Motor Commands
.Jln~ra.!!d EMG.L
r
-V-E~L~
l---j. . . .
,
Accelerlation
Predictor
P~S ( Forwa~d ) ACC
.~....-JI~
DynamiCs
Movement Trajectory
Figure 3: The musculo-skeletal forward dynamics model for producing articulator movement trajectories is implemented as a recurrent network. Continuous motor command
(EMG) input drives the network, which uses estimated acceleration at time tn, to predict
new velocity (integration) and position (double integration) values at the next time step
tn+l. D is a one-sample delay unit. The network is initialized with position and velocity
values taken from the test utterance at to.
Network training resulted in a one-step look-ahead predictor of the articulator dynamics, and was connected recurrently as shown in Figure 3. Using only initial
values of articulator position and velocity for the first sample and continuous EMG
input, estimated acceleration is looped back and summed with the velocities and
positions of the input layer to predict their values for each time step. This is perhaps an overly stringent test of the acquired model because errors are cumulative
over the entire utterance 5-8 second utterance. Yet the network outputs appropriate articulator trajectories for the entire utterance. Figure 4 shows the generated
trajectory for vertical motion of the jaw during reiterant production of ba (recorded
with the electromagnetometer). While the trajectory generated by the network
tends to underestimate movement amplitude and introduce a small DC offset, it
preserves the temporal properties of the test utterance very well everywhere except
before a phrasal pause. Although good results have been obtained for the analysis
661
662
Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato
-
Network Output
........ Experimental Data
JAW
(Vertical)
o
4
2
Time (5)
6
8
Figure 4: Jaw trajectories, generated by the forward dynamics network are compared
with experimental data.
of real speech using the larger sets of articulator and muscle inputs, network complexity has greatly increased. Performance of the full network has been poorer than
before in modeling simple reiterant speech, which suggests some form of modularity
should be introduced. Also, the addition of tongue data has increased the number
of apparent many-to-one mappings between muscle activity and articulator motion.
We are now incorporating as a boundary constraint the midsagittal profile of the
hard palate and alveolar ridge, against which tongue-tip articulations are made.
4
MODELING THE FORWARD ACOUSTICS
Articulator
Positions
Glottal Source
t----------......;-.:;~.)) )
"----------'
Acoustic wa;e
Figure 5: Forward acoustics network.
The final stage of our speech production model entails using a neural network to
acquire a model of the relation between articulator motion and the ensuing acoustics. As shown in Figure 5, a 3-layer perceptron, using articulator position as input,
was used to learn PARCOR analysis and generate appropriate 16-order PARCOR
parameters for subsequent speech synthesis (Itakura and Saito, 1969). We chose
PARCOR parameters, rather than more commonly used formant values, because
the parameters have some relation to specific cross-sections of the vocal tract e.g., the first PARCOR corresponds to the cross-sectional area closest to the lips
Physiologically Based Speech Synthesis
-
Network Output
.............
........ Experimental Data
? ?????
? ????
k1
k2
.-
k3
.
?
~
0,
. -"
~
k4
-,."
k5
.:
.1f":'_
-:
.....
,
k6
o
2
4
6
8
Time (s)
Figure 6: PARCOR parameter values (IS-order, 30ms Hanning window at 200Hz) for
reiterant ba are predicted by the network. Only the first six PARCOR parameters are
shown. The range of each parameter is -1 to 1 (small tick beside each wave label indicates
0). The value of kl is about 1 during vowels, and network output generally matches the
desire wave almost perfectly.
(Wakita, 1973). Also, PARCOR estimation errors do not have the radical consequences that formant estimation errors show. Finally, there is a unique mapping
from PARCOR to formant values, but not the reverse (Itakura and Saito, 1969).
Figure 6 shows the performance of the PARCOR estimation network for the first
6 parameters out of 16 parameters. Using the learned PARCOR coefficients and
a sound source, acoustic signals can be synthesized. Currently, we are investigating various models for controlling sound source as well as prosodic characteristics.
However, for this preliminary test of the network's ability to learn PARCOR parameters, the residual signal of PARCOR analysis served as the source waveform.
Figure 7 shows an example of the network-learned PARCOR synthesis for reiterant
ba. In this case, the training result is good as can be seen in the waveform (and
frequency spectrum), or by listening the synthesized sound. However, the results
have not been as good, so far, for real speech utterances containing a lot of abrupt
changes and variability in vocal tract shape. One reason for this may be that learning has not yet converged, because the number articulator input channels is still
too limited. So far, we have only two markers on tongue, which is not enough to
recover the full vocal tract shape. This situation, hopefully, will improve as data
for more tongue positions, or perhaps more functionally motivated placements, are
collected. Another reason may be the inherent weakness of PARCOR analysis for
modeling dynamic changes in vocal tract shape.
5
SUMMARY
This paper outlines two areas of progress in our effort to develop a computational
model of speech production. First, we extended our data acquisition to include more
663
664
Hirayama, Vatikiotis-Bateson, Honda, Koike, and Kawato
/ b a/
Experimental
Source
(Residual)
Synthesized
,_.".
............ .......
.t .......... -_.tl.
?? , ?? 1_'. II'
0
b
..
.
",
.....
tt ???
I
I
I
I
2
4
Time (s)
6
8
a
Experimental
Source
(Residual)
,.
Synthesized
3.00
3.02
3.04
3.06
3.08
3.10
Time (s)
Figure 7: Speech acoustics are synthesized by driving network-learned PARCOR parameters with a glottal source (the residual). The test sentence is reiterant speech using 00.
Top and bottom graphs differ only in time scale.
muscles and dimensions of motion for more articulators, especially the tongue, so
that we could begin modeling the articulatory dynamics of real speech. As hoped,
increasing the scope of the data demonstrated the applicability of our network approach to real speech. However, this also increases the size of the network, which has
introduced some interesting problems for modeling simple speech samples. We are
now considering modifications to the network architecture that will enable adaptive modeling of speech samples, whose complexity (e.g., number of physiological/ articulatory components) may vary. Second, we have employed a simple neural
network for modeling the articulatory-to-acoustic transform based on PARCOR
analysis, whose parameters are correlated with vocal tract shape. Although PARCOR can be used to synthesize speech, its main use for us is as a tool for assessing
empirical issues associated with articulatory-acoustic interface.
Acknowledgments
We thank Haskins Laboratories for use of their facilities (NIH grant DC-00121),
Vincent Gracco and Kiyoshi Ohsima for muscle insertions, M. I. Jordan for insightful
Physiologically Based Speech Synthesis
discussion, and Yoh'ichi Toh'kura for continuous encouragement. Further support
was provided by HFSP grants to M. Kawato.
References
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417-427.
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6,390 | 6,780 | Practical Bayesian Optimization for Model Fitting
with Bayesian Adaptive Direct Search
Luigi Acerbi?
Center for Neural Science
New York University
[email protected]
Wei Ji Ma
Center for Neural Science & Dept. of Psychology
New York University
[email protected]
Abstract
Computational models in fields such as computational neuroscience are often
evaluated via stochastic simulation or numerical approximation. Fitting these
models implies a difficult optimization problem over complex, possibly noisy
parameter landscapes. Bayesian optimization (BO) has been successfully applied
to solving expensive black-box problems in engineering and machine learning.
Here we explore whether BO can be applied as a general tool for model fitting.
First, we present a novel hybrid BO algorithm, Bayesian adaptive direct search
(BADS), that achieves competitive performance with an affordable computational
overhead for the running time of typical models. We then perform an extensive
benchmark of BADS vs. many common and state-of-the-art nonconvex, derivativefree optimizers, on a set of model-fitting problems with real data and models
from six studies in behavioral, cognitive, and computational neuroscience. With
default settings, BADS consistently finds comparable or better solutions than
other methods, including ?vanilla? BO, showing great promise for advanced BO
techniques, and BADS in particular, as a general model-fitting tool.
1
Introduction
Many complex, nonlinear computational models in fields such as behaviorial, cognitive, and computational neuroscience cannot be evaluated analytically, but require moderately expensive numerical
approximations or simulations. In these cases, finding the maximum-likelihood (ML) solution ?
for parameter estimation, or model selection ? requires the costly exploration of a rough or noisy
nonconvex landscape, in which gradients are often unavailable to guide the search.
Here we consider the problem of finding the (global) optimum x? = argminx?X E [f (x)] of a
possibly noisy objective f over a (bounded) domain X ? RD , where the function f can be intended
as the (negative) log likelihood of a parameter vector x for a given dataset and model, but is generally
a black box. With many derivative-free optimization algorithms available to the researcher [1], it is
unclear which one should be chosen. Crucially, an inadequate optimizer can hinder progress, limit
the complexity of the models that can be fit, and even cast doubt on the reliability of one?s findings.
Bayesian optimization (BO) is a state-of-the-art machine learning framework for optimizing expensive
and possibly noisy black-box functions [2, 3, 4]. This makes it an ideal candidate for solving difficult
model-fitting problems. Yet there are several obstacles to a widespread usage of BO as a general tool
for model fitting. First, traditional BO methods target very costly problems, such as hyperparameter
tuning [5], whereas evaluating a typical behavioral model might only have a moderate computational
cost (e.g., 0.1-10 s per evaluation). This implies major differences in what is considered an acceptable
algorithmic overhead, and in the maximum number of allowed function evaluations (e.g., hundreds vs.
?
Current address: D?partement des neurosciences fondamentales, Universit? de Gen?ve, CMU, 1 rue
Michel-Servet, 1206 Gen?ve, Switzerland. E-mail: [email protected].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
thousands). Second, it is unclear how BO methods would fare in this regime against commonly used
and state-of-the-art, non-Bayesian optimizers. Finally, BO might be perceived by non-practitioners
as an advanced tool that requires specific technical knowledge to be implemented or tuned.
We address these issues by developing a novel hybrid BO algorithm, Bayesian Adaptive Direct Search
(BADS), that achieves competitive performance at a small computational cost. We tested BADS,
together with a wide array of commonly used optimizers, on a novel benchmark set of model-fitting
problems with real data and models drawn from studies in cognitive, behaviorial and computational
neuroscience. Finally, we make BADS available as a free MATLAB package with the same user
interface as existing optimizers and that can be used out-of-the-box with no tuning.1
BADS is a hybrid BO method in that it combines the mesh adaptive direct search (MADS) framework
[6] (Section 2.1) with a BO search performed via a local Gaussian process (GP) surrogate (Section
2.2), implemented via a number of heuristics for efficiency (Section 3). BADS proves to be highly
competitive on both artificial functions and real-world model-fitting problems (Section 4), showing
promise as a general tool for model fitting in computational neuroscience and related fields.
Related work There is a large literature about (Bayesian) optimization of expensive, possibly
stochastic, computer simulations, mostly used in machine learning [3, 4, 5] or engineering (known
as kriging-based optimization) [7, 8, 9]. Recent work has combined MADS with treed GP models
for constrained optimization (TGP-MADS [9]). Crucially, these methods have large overheads and
may require problem-specific tuning, making them impractical as a generic tool for model fitting.
Cheaper but less precise surrogate models than GPs have been proposed, such as random forests [10],
Parzen estimators [11], and dynamic trees [12]. In this paper, we focus on BO based on traditional
GP surrogates, leaving the analysis of alternative models for future work (see Conclusions).
2
Optimization frameworks
2.1
Mesh adaptive direct search (MADS)
The MADS algorithm is a directional direct search framework for nonlinear optimization [6, 13].
Briefly, MADS seeks to improve the current solution by testing points in the neighborhood of the
current point (the incumbent), by moving one step in each direction on an iteration-dependent mesh.
In addition, the MADS framework can incorporate in the optimization any arbitrary search strategy
which proposes additional test points that lie on the mesh.
S
MADS defines the current mesh at the k-th iteration as Mk = x?Sk x + ?mesh
Dz : z ? ND ,
k
where Sk ? Rn is the set of all points evaluated since the start of the iteration, ?mesh
? R+ is the
k
mesh size, and D is a fixed matrix in RD?nD whose nD columns represent viable search directions.
We choose D = [ID , ?ID ], where ID is the identity matrix in dimension D.
Each iteration of MADS comprises of two stages, a SEARCH stage and an optional POLL stage. The
SEARCH stage evaluates a finite number of points proposed by a provided search strategy, with the
only restriction that the tested points lie on the current mesh. The search strategy is intended to inject
problem-specific information in the optimization. In BADS, we exploit the freedom of SEARCH to
perform Bayesian optimization in the neighborhood of the incumbent (see Section 2.2 and 3.3). The
POLL stage is performed if the SEARCH fails in finding a point with animproved objective value.
POLL constructs a poll set of candidate points, Pk , defined as Pk = xk + ?mesh
v : v ? Dk ,
k
where xk is the incumbent and Dk is the set of polling directions constructed by taking discrete linear
mesh
combinations of the set of directions D. The poll size parameter ?poll
defines the maximum
k ? ?k
poll
mesh
length of poll displacement vectors ?k v, for v ? Dk (typically, ?k ? ?mesh
||v||). Points in the
k
poll set can be evaluated in any order, and the POLL is opportunistic in that it can be stopped as soon
as a better solution is found. The POLL stage ensures theoretical convergence to a local stationary
point according to Clarke calculus for nonsmooth functions [6, 14].
If either SEARCH or POLL are a success, finding a mesh point with an improved objective value, the
incumbent is updated and the mesh size remains the same or is multiplied by a factor ? > 1. If neither
SEARCH or POLL are successful, the incumbent does not move and the mesh size is divided by ? . The
algorithm proceeds until a stopping criterion is met (e.g., maximum budget of function evaluations).
1
Code available at https://github.com/lacerbi/bads.
2
2.2
Bayesian optimization
The typical form of Bayesian optimization (BO) [2] builds a Gaussian process (GP) approximation
of the objective f , which is used as a relatively inexpensive surrogate to guide the search towards
regions that are promising (low GP mean) and/or unknown (high GP uncertainty), according to a rule,
the acquisition function, that formalizes the exploitation-exploration trade-off.
Gaussian processes GPs are a flexible class of models for specifying prior distributions over
unknown functions f : X ? RD ? R [15]. GPs are specified by a mean function m : X ? R and a
positive
or kernel function k : X ?X ? R. Given any finite collection of n points
definite covariance,
n
X = x(i) ? X i=1 , the value of f at these points is assumed to be jointly Gaussian with mean
(m(x(1) ), . . . , m(x(n) ))> and covariance matrix K, where Kij = k(x(i) , x(j) ) for 1 ? i, j ? n. We
assume i.i.d. Gaussian observation noise such that f evaluated at x(i) returns y (i) ? N f (x(i) ), ? 2 ,
and y = (y (1) , . . . , y (n) )> is the vector of observed values. For a deterministic f , we still assume a
small ? > 0 to improve numerical stability of the GP [16]. Conveniently, observation of such (noisy)
function values will produce a GP posterior whose latent marginal conditional mean ?(x; {X, y} , ?)
and variance s2 (x; {X, y} , ?) at a given point are available in closed form (see Supplementary
Material), where ? is a hyperparameter vector for the mean, covariance, and likelihood. In the
following, we omit the dependency of ? and s2 from the data and GP parameters to reduce clutter.
Covariance functions Our main choice of stationary (translationally-invariant) covariance function
is the automatic relevance determination (ARD) rational quadratic (RQ) kernel,
??
D
X
1
1 2
2
0
2
0
kRQ (x, x ) = ?f 1 +
r (x, x )
,
with r2 (x, x0 ) =
(xd ? x0d ) , (1)
2?
`2d
d=1
where ?f2
is the signal variance, `1 , . . . , `D are the kernel length scales along each coordinate direction,
and ? > 0 is the shape parameter. More common choices for Bayesian optimization include the
squared exponential (SE) kernel [9] or the twice-differentiable ARD Mat?rn 5/2 (M5/2 ) kernel [5],
but we found the RQ kernel to work best in combination with our method (see Section 4.2). We also
consider composite periodic kernels for circular or periodic variables (see Supplementary Material).
Acquisition function For a given GP approximation of f , the acquisition function, a : X ? R,
determines which point in X should be evaluated next via a proxy optimization xnext = argminx a(x).
We consider here the GP lower confidence bound (LCB) metric [17],
p
aLCB (x; {X, y} , ?) = ? (x) ? ??t s2 (x),
?t = 2 ln Dt2 ? 2 /(6?)
(2)
where ? > 0 is a tunable parameter, t is the number of function evaluations so far, ? > 0 is a
probabilistic tolerance, and ?t is a learning rate chosen to minimize cumulative regret under certain
assumptions. For BADS we use the recommended values ? = 0.2 and ? = 0.1 [17]. Another popular
choice is the (negative) expected improvement (EI) over the current best function value [18], and an
historical, less used metric is the (negative) probability of improvement (PI) [19].
3
Bayesian adaptive direct search (BADS)
We describe here the main steps of BADS (Algorithm 1). Briefly, BADS alternates between a series
of fast, local BO steps (the SEARCH stage of MADS) and a systematic, slower exploration of the
mesh grid (POLL stage). The two stages complement each other, in that the SEARCH can explore
the space very effectively, provided an adequate surrogate model. When the SEARCH repeatedly
fails, meaning that the GP model is not helping the optimization (e.g., due to a misspecified model,
or excess uncertainty), BADS switches to POLL. The POLL stage performs a fail-safe, model-free
optimization, during which BADS gathers information about the local shape of the objective function,
so as to build a better surrogate for the next SEARCH. This alternation makes BADS able to deal
effectively and robustly with a variety of problems. See Supplementary Material for a full description.
3.1
Initial setup
Problem specification The algorithm is initialized by providing a starting point x0 , vectors of hard
lower/upper bounds LB, UB, and optional vectors of plausible lower/upper bounds PLB, PUB, with the
3
Algorithm 1 Bayesian Adaptive Direct Search
Input: objective function f , starting point x0 , hard bounds LB, UB, (optional: plausible bounds PLB,
PUB, barrier function c, additional options)
1: Initialization: ?mesh
? 2?10 , ?poll
. Section 3.1
0
0 ? 1, k ? 0, evaluate f on initial design
2: repeat
3:
(update GP approximation at any step; refit hyperparameters if necessary)
. Section 3.2
4:
for 1 . . . nsearch do
. SEARCH stage, Section 3.3
5:
xsearch ? S EARCH O RACLE
. local Bayesian optimization step
6:
Evaluate f on xsearch , if improvement is sufficient then break
7:
if SEARCH is NOT successful then
. optional POLL stage, Section 3.3
8:
compute poll set Pk
9:
evaluate opportunistically f on Pk sorted by acquisition function
10:
if iteration k is successful then
11:
update incumbent xk+1
poll
12:
if POLL was successful then ?mesh
? 2?mesh
, ?poll
k
k
k ? 2?k
13:
else
1 poll
14:
?mesh
? 21 ?mesh
, ?poll
k
k
k ? 2 ?k
15:
k ?k+1
poll
16: until fevals > MaxFunEvals or ?k < 10?6 or stalling
. stopping criteria
17: return xend = arg mink f (xk ) (or xend = arg mink q? (xk ) for noisy objectives, Section 3.4)
requirement that for each dimension 1 ? d ? D, LBd ? PLBd < PUBd ? UBd .2 Plausible bounds
identify a region in parameter space where most solutions are expected to lie. Hard upper/lower
bounds can be infinite, but plausible bounds need to be finite. Problem variables whose hard bounds
are strictly positive and UBd ? 10 ? LBd are automatically converted to log space. All variables
are then linearly rescaled to the standardized box [?1, 1]D such that the box bounds correspond
to [PLB, PUB] in the original space. BADS supports bound or no constraints, and optionally other
constraints via a provided barrier function c (see Supplementary Material). The user can also specify
circular or periodic dimensions (such as angles); and whether the objective f is deterministic or noisy
(stochastic), and in the latter case provide a coarse estimate of the noise (see Section 3.4).
Initial design The initial design consists of the provided starting point x0 and ninit = D additional
points chosen via a space-filling quasi-random Sobol sequence [20] in the standardized box, and
forced to lie on the mesh grid. If the user does not specify whether f is deterministic or stochastic,
the algorithm assesses it by performing two consecutive evaluations at x0 .
3.2
GP model in BADS
The default GP model is specified by a constant mean function m ? R, a smooth ARD RQ kernel
(Eq. 1), and we use aLCB (Eq. 2) as a default acquisition function.
Hyperparameters The default GP has hyperparameters ? = (`1 , . . . , `D , ?f2 , ?, ? 2 , m). We
impose an empirical Bayes prior on the GP hyperparameters based on the current training set
(see Supplementary Material), and select ? via maximum a posteriori (MAP) estimation. We fit ?
via a gradient-based nonlinear optimizer, starting from either the previous value of ? or a weighted
draw from the prior, as a means to escape local optima. We refit the hyperparameters every 2D
to 5D function evaluations; more often earlier in the optimization, and whenever the current GP
is particularly inaccurate
at predicting new points, according to a normality test on the residuals,
p
z (i) = y (i) ? ?(x(i) ) / s2 (x(i) ) + ? 2 (assumed independent, in first approximation).
Training set The GP training set X consists of a subset of the points evaluated so far (the cache),
selected to build a local approximation of the objective in the neighborhood of the incumbent xk ,
constructed as follows. Each time X is rebuilt, points in the cache are sorted by their `-scaled distance
r2 (Eq. 1) from xk . First, the closest nmin = 50 points are automatically added to X. Second,
up to 10D additional points with r ? 3?(?) are included in the set, where ?(?) & 1 is a radius
A variable d can be fixed by setting (x0 )d = LBd = UBd = PLBd = PUBd . Fixed variables become
constants, and BADS runs on an optimization problem with reduced dimensionality.
2
4
? ?
function that depends on the decay of the kernel. For the RQ kernel, ?RQ (?) = ? e1/? ? 1 (see
Supplementary Material). Newly evaluated points are added incrementally to the set, using fast
rank-one updates of the GP posterior. The training set is rebuilt any time the incumbent is moved.
3.3
Implementation of the MADS framework
mesh
We initialize ?poll
= 2?10 (in standardized space), such that the initial poll steps can
0 = 1 and ?0
span the plausible region, whereas the mesh grid is relatively fine. We use ? = 2, and increase the
mesh size only after a successful POLL. We skip the POLL after a successful SEARCH.
Search stage We apply an aggressive, repeated SEARCH strategy that consists of up to nsearch =
max{D, b3 + D/2c} unsuccessful SEARCH steps. In each step, we use a search oracle, based on a
local BO with the current GP, to produce a search point xsearch (see below). We evaluate f (xsearch )
and add it to the training set. If the improvement in objective value is none or insufficient, that is less
3/2
than (?poll
, we continue searching, or switch to POLL after nsearch steps. Otherwise, we call it a
k )
success and start a new SEARCH from scratch, centered on the updated incumbent.
Search oracle We choose xsearch via a fast, approximate optimization inspired by CMA-ES [21].
2
We sample batches of points in the neighborhood of the incumbent xk , drawn ? N (xs , ?2 (?poll
k ) ?),
where xs is the current search focus, ? a search covariance matrix, and ? > 0 a scaling factor, and
we pick the point that optimizes the acquisition function (see Supplementary Material). We remove
from the SEARCH set candidate points that violate non-bound constraints (c(x) > 0), and we project
candidate points that fall outside hard bounds to the closest mesh point inside the bounds.
Across
SEARCH steps, we use both a diagonal matrix ?` with diagonal `21 /|`|2 , . . . , `2D /|`|2 , and a matrix
?WCM proportional to the weighted covariance matrix of points in X (each point weighted according
to a function of its ranking in terms of objective values yi ). We choose between ?` and ?WCM
probabilistically via a hedge strategy, based on their track record of cumulative improvement [22].
Poll stage We incorporate the GP approximation in the POLL in two ways: when constructing the
set of polling directions Dk , and when choosing the polling order. We generate Dk according to the
random LTMADS algorithm [6], but then rescale each vector coordinate 1 ? d ? D proportionally
to the GP length scale `d (see Supplementary Material). We discard poll vectors that do not satisfy
the given bound or nonbound constraints. Second, since the POLL is opportunistic, we evaluate points
in the poll set according to the ranking given by the acquisition function [9].
Stopping criteria We stop the optimization when the poll size ?poll
k goes below a threshold (default
10?6 ); when reaching a maximum number of objective evaluations (default 500D); or if there is no
significant improvement of the objective for more than 4 + bD/2c iterations. The algorithm returns
the optimum xend (transformed back to original coordinates) with the lowest objective value yend .
3.4
Noisy objective
In case of a noisy objective, we assume for the noise a hyperprior ln ? ? N (ln ?est , 1), with ?est
a base noise magnitude (default ?est = 1, but the user can provide an estimate). To account for
additional uncertainty, we also make the following changes: double the minimum number of points
added to the training set, nmin = 100, and increase the maximum number to 200; increase the initial
design to ninit = 20; and double the number of allowed stalled iterations before stopping.
Uncertainty handling Due to noise, we cannot simply use the output values yi as ground truth in
the SEARCH and POLL stages. Instead, we replace yi with the GP latent quantile function [23]
q? (x; {X, y} , ?) ? q? (x) = ? (x) + ??1 (?)s (x) ,
?1
? ? [0.5, 1),
(3)
where ? (?) is the quantile function of the standard normal (plugin approach [24]). Moreover, we
modify the MADS procedure by keeping an incumbent set {xi }ki=1 , where xi is the incumbent at the
end of the i-th iteration. At the end of each POLL we re-evaluate q? for all elements of the incumbent
set, in light of the new points added to the cache. We select as current (active) incumbent the point
with lowest q? (xi ). During optimization we set ? = 0.5 (mean prediction only), which promotes
exploration. We use a conservative ?end = 0.999 for the last iteration, to select the optimum xend
returned by the algorithm in a robust manner. Instead of yend , we return either ?(xend ) or an unbiased
estimate of E[f (xend )] obtained by averaging multiple evaluations (see Supplementary Material).
5
4
Experiments
We tested BADS and many optimizers with implementation available in MATLAB (R2015b, R2017a)
on a large set of artificial and real optimization problems (see Supplementary Material for details).
4.1
Design of the benchmark
Algorithms Besides BADS, we tested 16 optimization algorithms, including popular choices
such as Nelder-Mead (fminsearch [25]), several constrained nonlinear optimizers in the fmincon
function (default interior-point [26], sequential quadratic programming sqp [27], and active-set
actset [28]), genetic algorithms (ga [29]), random search (randsearch) as a baseline [30]; and
also less-known state-of-the-art methods for nonconvex derivative-free optimization [1], such as
Multilevel Coordinate Search (MCS [31]) and CMA-ES [21, 32] (cmaes, in different flavors). For
noisy objectives, we included algorithms that explicitly handle uncertainty, such as snobfit [33]
and noisy CMA-ES [34]. Finally, to verify the advantage of BADS? hybrid approach to BO, we also
tested a standard, ?vanilla? version of BO [5] (bayesopt, R2017a) on the set of real model-fitting
problems (see below). For all algorithms, including BADS, we used default settings (no fine-tuning).
Problem sets First, we considered a standard benchmark set of artificial, noiseless functions
(BBOB 09 [35], 24 functions) in dimensions D ? {3, 6, 10, 15}, for a total of 96 test functions. We
also created ?noisy? versions of the same set. Second, we collected model-fitting problems from six
published or ongoing studies in cognitive and computational neuroscience (CCN 17). The objectives
of the CCN 17 set are negative log likelihood functions of an input parameter vector, for specified
datasets and models, and can be deterministic or stochastic. For each study in the CCN 17 set we
asked its authors for six different real datasets (i.e., subjects or neurons), divided between one or two
main models of interest; collecting a total of 36 test functions with D ? {6, 9, 10, 12, 13}.
Procedure We ran 50 independent runs of each algorithm on each test function, with randomized
starting points and a budget of 500 ? D function evaluations (200 ? D for noisy problems). If an
algorithm terminated before depleting the budget, it was restarted from a new random point. We
consider a run successful if the current best (or returned, for noisy problems) function value is within a
given error tolerance ? > 0 from the true optimum fmin (or our best estimate thereof).3 For noiseless
problems, we compute the fraction of successful runs as a function of number of objective evaluations,
averaged over datasets/functions and over ? ? [0.01, 10] (log spaced). This is a realistic range for ?,
as differences in log likelihood below 0.01 are irrelevant for model selection; an acceptable tolerance
is ? ? 0.5 (a difference in deviance, the metric used for AIC or BIC, less than 1); larger ? associate
with coarse solutions, but errors larger than 10 would induce excessive biases in model selection. For
noisy problems, what matters most is the solution xend that the algorithm actually returns, which,
depending on the algorithm, may not necessarily be the point with the lowest observed function value.
Since, unlike the noiseless case, we generally do not know the solutions that would be returned by any
algorithm at every time step, but only at the last step, we plot instead the fraction of successful runs
at 200 ? D function evaluations as a function of ?, for ? ? [0.1, 10] (noise makes higher precisions
moot), and averaged over datasets/functions. In all plots we omit error bars for clarity (standard errors
would be about the size of the line markers or less).
4.2
Results on artificial functions (BBOB 09)
The BBOB 09 noiseless set [35] comprises of 24 functions divided in 5 groups with different properties:
separable; low or moderate conditioning; unimodal with high conditioning; multi-modal with adequate
/ with weak global structure. First, we use this benchmark to show the performance of different
configurations for BADS. Note that we selected the default configuration (RQ kernel, aLCB ) and
other algorithmic details by testing on a different benchmark set (see Supplementary Material). Fig 1
(left) shows aggregate results across all noiseless functions with D ? {3, 6, 10, 15}, for alternative
choices of kernels and acquisition functions (only a subset is shown, such as the popular M5/2 , EI
combination), or by altering other features (such as setting nsearch = 1, or fixing the search covariance
matrix to ?` or ?WCM ). Almost all changes from the default configuration worsen performance.
3
Note that the error tolerance ? is not a fractional error, as sometimes reported in optimization, because for
model comparison we typically care about (absolute) differences in log likelihoods.
6
1
bads (rq,lcb,default)
bads (search-wcm)
bads (m5/2,ei)
bads (search-?)
bads (se,pi)
bads ( n search =1)
0.5
0.25
BBOB09 with heteroskedastic noise
bads
fmincon (actset)
fmincon
fmincon (sqp)
cmaes (active)
mcs
fminsearch
cmaes
global
patternsearch
simulannealbnd
particleswarm
ga
randsearch
0.75
Fraction solved
Fraction solved
0.75
BBOB09 noiseless
1
0.5
Fraction solved at 200?D func. evals.
BBOB09 noiseless (BADS variants)
0.25
0
0
10
50
100
Function evaluations / D
500
10
50
100
Function evaluations / D
500
1
bads
cmaes (noisy,active)
cmaes (noisy)
snobfit
particleswarm
patternsearch
mcs
ga
simulannealbnd
fmincon (actset)
randsearch
fmincon
fmincon (sqp)
fminsearch
global
0.75
0.5
0.25
0
10
3
1
0.3
0.1
Error tolerance ?
Figure 1: Artificial test functions (BBOB 09). Left & middle: Noiseless functions. Fraction of
successful runs (? ? [0.01, 10]) vs. # function evaluations per # dimensions, for D ? {3, 6, 10, 15}
(96 test functions); for different BADS configurations (left) and all algorithms (middle). Right:
Heteroskedastic noise. Fraction of successful runs at 200 ? D objective evaluations vs. tolerance ?.
Noiseless functions We then compared BADS to other algorithms (Fig 1 middle). Depending on
the number of function evaluations, the best optimizers are BADS, methods of the fmincon family,
and, for large budget of function evaluations, CMA-ES with active update of the covariance matrix.
Noisy functions We produce noisy versions of the BBOB 09 set by adding i.i.d. Gaussian observation noise at each function evaluation, y (i) = f (x(i) ) + ?(x(i) )? (i) , with ? (i) ? N (0, 1). We
consider a variant with moderate homoskedastic (constant) noise (? = 1), and a variant with heteroskedastic noise with ?(x) = 1+0.1?(f (x)?fmin ), which follows the observation that variability
generally increases for solutions away from the optimum. For many functions in the BBOB 09 set, this
heteroskedastic noise can become substantial (? 10) away from the optimum. Fig 1 (right) shows
aggregate results for the heteroskedastic set (homoskedastic results are similar). BADS outperforms
all other optimizers, with CMA-ES (active, with or without the noisy option) coming second.
Notably, BADS performs well even on problems with non-stationary (location-dependent) features,
such as heteroskedastic noise, thanks to its local GP approximation.
4.3
Results on real model-fitting problems (CCN 17)
The objectives of the CCN 17 set are deterministic (e.g., computed via numerical approximation) for
three studies (Fig 2), and noisy (e.g., evaluated via simulation) for the other three (Fig 3).
The algorithmic cost of BADS is ? 0.03 s to 0.15 s per function evaluation, depending on D, mostly
due to the refitting of the GP hyperparameters. This produces a non-negligible overhead, defined as
100% ? (total optimization time / total function time ?1). For a fair comparison with other methods
with little or no overhead, for deterministic problems we also plot the effective performance of BADS
by accounting for the extra cost per function evaluation. In practice, this correction shifts rightward
the performance curve of BADS in log-iteration space, since each function evaluation with BADS has
an increased fractional time cost. For stochastic problems, we cannot compute effective performance
as easily, but there we found small overheads (< 5%), due to more costly evaluations (more than 1 s).
For a direct comparison with standard BO, we also tested on the CCN 17 set a ?vanilla? BO algorithm,
as implemented in MATLAB R2017a (bayesopt). This implementation closely follows [5], with
optimization instead of marginalization over GP hyperparameters. Due to the fast-growing cost of
BO as a function of training set size, we allowed up to 300 training points for the GP, restarting the
BO algorithm from scratch with a different initial design every 300 BO iterations (until the total
budget of function evaluations was exhausted). The choice of 300 iterations already produced a large
average algorithmic overhead of ? 8 s per function evaluation. In showing the results of bayesopt,
we display raw performance without penalizing for the overhead.
Causal inference in visuo-vestibular perception Causal inference (CI) in perception is the process whereby the brain decides whether to integrate or segregate multisensory cues that could arise
from the same or from different sources [39]. This study investigates CI in visuo-vestibular heading
7
0.5
0.25
0.5
0.25
0
50
100
Function evaluations / D
500
bads
bads [overhead-corrected, 14%]
fmincon
fmincon (sqp)
fmincon (actset)
cmaes (active)
cmaes
mcs
fminsearch
patternsearch
simulannealbnd
ga
global
particleswarm
randsearch
bayesopt
0.75
0.5
0.25
0
10
CCN17 neuronal selectivity
1
bads
bads [overhead-corrected, 68%]
fmincon
fmincon (sqp)
fmincon (actset)
cmaes (active)
cmaes
mcs
patternsearch
fminsearch
particleswarm
global
randsearch
simulannealbnd
ga
bayesopt
0.75
Fraction solved
Fraction solved
0.75
CCN17 Bayesian confidence
1
bads
bads [overhead-corrected, 24%]
cmaes (active)
cmaes
fminsearch
patternsearch
particleswarm
global
simulannealbnd
fmincon
fmincon (sqp)
mcs
ga
fmincon (actset)
randsearch
bayesopt
Fraction solved
CCN17 causal inference
1
0
10
50
100
Function evaluations / D
500
10
50
100
500
Function evaluations / D
Figure 2: Real model-fitting problems (CCN 17, deterministic). Fraction of successful runs (? ?
[0.01, 10]) vs. # function evaluations per # dimensions. Left: Causal inference in visuo-vestibular
perception [36] (6 subjects, D = 10). Middle: Bayesian confidence in perceptual categorization [37]
(6 subjects, D = 13). Right: Neural model of orientation selectivity [38] (6 neurons, D = 12).
perception across tasks and under different levels of visual reliability, via a factorial model comparison
[36]. For our benchmark we fit three subjects with a Bayesian CI model (D = 10), and another three
with a fixed-criterion CI model (D = 10) that disregards visual reliability. Both models include
heading-dependent likelihoods and marginalization of the decision variable over the latent space of
noisy sensory measurements (xvis , xvest ), solved via nested numerical integration in 1-D and 2-D.
Bayesian confidence in perceptual categorization This study investigates the Bayesian confidence hypothesis that subjective judgments of confidence are directly related to the posterior probability the observer assigns to a learnt perceptual category [37] (e.g., whether the orientation of a drifting
Gabor patch belongs to a ?narrow? or to a ?wide? category). For our benchmark we fit six subjects
to the ?Ultrastrong? Bayesian confidence model (D = 13), which uses the same mapping between
posterior probability and confidence across two tasks with different distributions of stimuli. This
model includes a latent noisy decision variable, marginalized over via 1-D numerical integration.
Neural model of orientation selectivity The authors of this study explore the origins of diversity of
neuronal orientation selectivity in visual cortex via novel stimuli (orientation mixtures) and modeling
[38]. We fit the responses of five V1 and one V2 cells with the authors? neuronal model (D = 12)
that combines effects of filtering, suppression, and response nonlinearity [38]. The model has one
circular parameter, the preferred direction of motion of the neuron. The model is analytical but still
computationally expensive due to large datasets and a cascade of several nonlinear operations.
Word recognition memory This study models a word recognition task in which subjects rated their
confidence that a presented word was in a previously studied list [40] (data from [41]). We consider
six subjects divided between two normative models, the ?Retrieving Effectively from Memory? model
[42] (D = 9) and a similar, novel model4 (D = 6). Both models use Monte Carlo methods to draw
random samples from a large space of latent noisy memories, yielding a stochastic log likelihood.
Target detection and localization This study looks at differences in observers? decision making
strategies in target detection (?was the target present??) and localization (?which one was the target??)
with displays of 2, 3, 4, or 6 oriented Gabor patches.5 Here we fit six subjects with a previously
derived ideal observer model [43, 44] (D = 6) with variable-precision noise [45], assuming shared
parameters between detection and localization. The log likelihood is evaluated via simulation due to
marginalization over latent noisy measurements of stimuli orientations with variable precision.
Combinatorial board game playing This study analyzes people?s strategies in a four-in-a-row
game played on a 4-by-9 board against human opponents ([46], Experiment 1). We fit the data of six
players with the main model (D = 10), which is based on a Best-First exploration of a decision tree
guided by a feature-based value heuristic. The model also includes feature dropping, value noise, and
lapses, to better capture human variability. Model evaluation is computationally expensive due to the
4
5
Unpublished; upcoming work from Aspen H. Yoo and Wei Ji Ma.
Unpublished; upcoming work from Andra Mihali and Wei Ji Ma.
8
0.75
0.5
0.25
0
10
3
1
0.3
Error tolerance ?
0.1
CCN17 target detection/localization
1
bads
noisy,
cmaes ( active
)
cmaes (noisy)
snobfit
bayesopt
particleswarm
mcs
patternsearch
fminsearch
simulannealbnd
ga
global
fmincon (actset)
randsearch
0.75
0.5
0.25
0
10
3
1
0.3
Error tolerance ?
0.1
CCN17 combinatorial game playing
Fraction solved at 200?D func. evals.
bads
cmaes (noisy,active)
cmaes (noisy)
fminsearch
patternsearch
particleswarm
fmincon (actset)
ga
mcs
simulannealbnd
randsearch
snobfit
global
bayesopt
Fraction solved at 200?D func. evals.
Fraction solved at 200?D func. evals.
CCN17 word recognition memory
1
1
bads
cmaes (noisy,active)
particleswarm
bayesopt
snobfit
mcs
patternsearch
fminsearch
0.75
0.5
0.25
0
10
3
1
0.3
0.1
Error tolerance ?
Figure 3: Real model-fitting problems (CCN 17, noisy). Fraction of successful runs at 200 ? D
objective evaluations vs. tolerance ?. Left: Confidence in word recognition memory [40] (6 subjects,
D = 6, 9). Middle: Target detection and localization [44] (6 subjects, D = 6). Right: Combinatorial
board game playing [46] (6 subjects, D = 10).
construction and evaluation of trees of future board states, and achieved via inverse binomial sampling,
an unbiased stochastic estimator of the log likelihood [46]. Due to prohibitive computational costs,
here we only test major algorithms (MCS is the method used in the paper [46]); see Fig 3 right.
In all problems, BADS consistently performs on par with or outperforms all other tested optimizers,
even when accounting for its extra algorithmic cost. The second best algorithm is either some flavor
of CMA-ES or, for some deterministic problems, a member of the fmincon family. Crucially, their
ranking across problems is inconsistent, with both CMA-ES and fmincon performing occasionally
quite poorly (e.g., fmincon does poorly in the causal inference set because of small fluctuations
in the log likelihood landscape caused by coarse numerical integration). Interestingly, vanilla BO
(bayesopt) performs poorly on all problems, often at the level of random search, and always
substantially worse than BADS, even without accounting for the much larger overhead of bayesopt.
The solutions found by bayesopt are often hundreds (even thousands) points of log likelihood from
the optimum. This failure is possibly due to the difficulty of building a global GP surrogate for BO,
coupled with strong non-stationarity of the log likelihood functions; and might be ameliorated by more
complex forms of BO (e.g., input warping to produce nonstationary kernels [47], hyperparameter
marginalization [5]). However, these advanced approaches would substantially increase the already
large overhead. Importantly, we expect this poor perfomance to extend to any package which
implements vanilla BO (such as BayesOpt [48]), regardless of the efficiency of implementation.
5
Conclusions
We have developed a novel BO method and an associated toolbox, BADS, with the goal of fitting
moderately expensive computational models out-of-the-box. We have shown on real model-fitting
problems that BADS outperforms widely used and state-of-the-art methods for nonconvex, derivativefree optimization, including ?vanilla? BO. We attribute the robust performance of BADS to the
alternation between the aggressive SEARCH strategy, based on local BO, and the failsafe POLL stage,
which protects against failures of the GP surrogate ? whereas vanilla BO does not have such failsafe
mechanisms, and can be strongly affected by model misspecification. Our results demonstrate that
a hybrid Bayesian approach to optimization can be beneficial beyond the domain of very costly
black-box functions, in line with recent advancements in probabilistic numerics [49].
Like other surrogate-based methods, the performance of BADS is linked to its ability to obtain a fast
approximation of the objective, which generally deteriorates in high dimensions, or for functions
with pathological structure (often improvable via reparameterization). From our tests, we recommend
BADS, paired with some multi-start optimization strategy, for models with up to ? 15 variables,
a noisy or jagged log likelihood landscape, and when algorithmic overhead is . 75% (e.g., model
evaluation & 0.1 s). Future work with BADS will focus on testing alternative statistical surrogates
instead of GPs [12]; combining it with a smart multi-start method for global optimization; providing
support for tunable precision of noisy observations [23]; improving the numerical implementation;
and recasting some of its heuristics in terms of approximate inference.
9
Acknowledgments
We thank Will Adler, Robbe Goris, Andra Mihali, Bas van Opheusden, and Aspen Yoo for sharing
data and model evaluation code that we used in the CCN 17 benchmark set; Maija Honig, Andra
Mihali, Bas van Opheusden, and Aspen Yoo for providing user feedback on earlier versions of the
bads package for MATLAB; Will Adler, Andra Mihali, Bas van Opheusden, and Aspen Yoo for
helpful feedback on a previous version of this manuscript; John Wixted and colleagues for allowing us
to reuse their data for the CCN 17 ?word recognition memory? problem set; and the three anonymous
reviewers for useful feedback. This work has utilized the NYU IT High Performance Computing
resources and services.
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11
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6,391 | 6,781 | Learning Chordal Markov Networks
via Branch and Bound
Kari Rantanen
HIIT, Dept. Comp. Sci.,
University of Helsinki
Antti Hyttinen
HIIT, Dept. Comp. Sci.,
University of Helsinki
Matti J?rvisalo
HIIT, Dept. Comp. Sci.,
University of Helsinki
Abstract
We present a new algorithmic approach for the task of finding a chordal Markov
network structure that maximizes a given scoring function. The algorithm is
based on branch and bound and integrates dynamic programming for both domain
pruning and for obtaining strong bounds for search-space pruning. Empirically,
we show that the approach dominates in terms of running times a recent integer
programming approach (and thereby also a recent constraint optimization approach)
for the problem. Furthermore, our algorithm scales at times further with respect to
the number of variables than a state-of-the-art dynamic programming algorithm
for the problem, with the potential of reaching 20 variables and at the same time
circumventing the tight exponential lower bounds on memory consumption of the
pure dynamic programming approach.
1
Introduction
Graphical models offer a versatile and theoretically solid framework for various data analysis
tasks [1, 30, 17]. In this paper we focus on the structure learning task for chordal Markov networks
(or chordal/triangulated Markov random fields or decomposable graphs), a central class of undirected
graphical models [7, 31, 18, 17]. This problem, chordal Markov network structure learning (CMSL), is
computationally notoriously challenging; e.g., finding a maximum likelihood chordal Markov network
with bounded structure complexity (clique size) is known to be NP-hard [23]. Several Markov chain
Monte Carlo (MCMC) approaches have been proposed for this task in the literature [19, 27, 10, 11].
Here we take on the challenge of developing a new exact algorithmic approach for finding an optimal
chordal Markov network structure in the score-based setting. Underlining the difficulty of this
challenge, first exact algorithms for CMSL have only recently been proposed [6, 12, 13, 14], and
generally do no scale up to 20 variables. Specifically, the constraint optimization approach introduced
in [6] does not scale up to 10 variables within hours. A similar approach was also taken in [16] in the
form of a direct integer programming encoding for CMSL, but was not empirically evaluated in an
exact setting. Comparably better performance, scaling up to 10 (at most 15) variables, is exhibited
by the integer programming approach implemented in the GOBNILP system [2], extending the core
approach of GOBNILP to CMSL by enforcing additional constraints. The true state-of-the-art exact
algorithm for CMSL, especially when the clique size of the networks to be learned is not restricted, is
Junctor, implementing a dynamic programming approach [13]. The method is based on recursive
characterization of clique trees and storing in memory the scores of already-solved subproblems.
Due to its nature, the algorithm has to iterate through every single solution candidate, although its
effective memoization technique helps to avoid revisiting solution candidates [13]. As typical for
dynamic programming algorithms, the worst-case and best-case performance coincide: Junctor is
guaranteed to use ?(4n ) time and ?(3n ) space.
In this work, we develop an alternative exact algorithm for CMSL. While a number of branchand-bound algorithms have been proposed in the past for Bayesian network structure learning
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(BNSL) [25, 28, 20, 29, 26], to the best of our knowledge our approach constitutes the first non-trivial
branch-and-bound approach for CMSL. Our core search routine takes advantage of similar ideas
as a recently proposed approach for optimally solving BNSL [29], and, on the other hand, like
GOBNILP, uses the tight connection between BNSL and CMSL by searching over the space of
chordal Markov network structures via considering decomposable directed acyclic graphs. Central to
the efficiency of our approach is the integration of dynamic programming over Bayesian network
structures for obtaining strong bounds for effectively pruning the search space during search, as
well as problem-specific dynamic programming for efficiently implementing domain filtering during
search. Furthermore, we establish a condition which enables symmetry breaking for noticeably
pruning the search space over which we perform branch and bound. In comparison with Junctor,
a key benefit of our approach is the potential of avoiding worst-case behavior, especially in terms
of memory usage, based on using strong bounds to rule out provably non-optimal solutions from
consideration during search.
Empirically, we show the approach dominates the integer programming approach of GOBNILP [2],
and thereby also the constraint optimization approach [6, 12]. Furthermore, our algorithm scales
at times further in terms of the number of variables than the DP-based approach implemented in
Junctor [13], with the potential of reaching 20 variables within hours and at the same time circumventing the tight exponential lower bounds on memory consumption of the pure dynamic programming
approach, which is witnessed also in practice by noticeably lower memory consumption.1
2
Chordal Markov Network Structure Learning
A Markov network structure is represented by an undirected graph Gu = (V, E u ), where V =
{v1 , . . . , vn } is the set of vertices and E u the set of undirected edges. This structure represents
independencies vi ?
? vj |S according to the undirected separation property: vi and vj are separated
given set S if and only if all paths between them go through a vertex in set S. The undirected graph is
chordal iff every (undirected) cycle of length greater than three contains a chord, i.e., an edge between
two non-consecutive vertices in the cycle. Figure 1 a) shows an example. Here we focus on the task of
finding a chordal graph U that maximizes posterior probability P (Gu |D) = P (D|Gu )P (Gu )/P (D),
where D denotes the i.i.d. data set. As we assume a uniform prior over chordal graphs, this boils
down to maximizing the marginal likelihood P (D|Gu ).
Dawid et al. have shown that the marginal likelihood P (D|Gu ) for chordal Markov networks can be
calculated using a clique tree representation [7, 9]. A clique C is a fully connected subset of vertices.
A clique tree for an undirected graph Gu is an undirected tree such that
S
I.
i Ci = V ,
II. if {v` , vk } ? E u , then either {v` , vk } ? Ck or {v` , vk } ? C` , and
III. the running intersection property holds: whenever vk ? Ci and vk ? Cj , then vk is also in
every clique on the unique path between Ci and Cj .
The separators are the intersections of adjacent cliques in a clique tree. Figure 1Qb) shows anQexample.
The marginal likelihood factorizes according to the clique tree: P (D|U ) = i P (Ci )/ j P (Sj )
(assuming positivity and that the prior factorizes) [6]. The marginal likelihood P (S) for a set
S of random variables can be calculated with suitable priors; in this paper we consider discrete
data using a Dirichlet
prior. If we denote s(S) = log P (S), CMSL can be cast as maximizing
P
P
Ci s(Ci ) ?
Sj s(Sj ). For example, the marginal log-likelihood of the graph in Figure 1 a)
can be calculated using the clique tree presentation in Figure 1 b) as s({v1 , v6 }) + s({v1 , v5 }) +
s({v1 , v2 , v3 }) + s( v2 , v3 , v4 }) ? s({v1 }) ? s({v1 }) ? s({v2 , v3 }).
In this paper, we view the chordal Markov network structure learning problem from the viewpoint
of directed graphs, making use of the fact that for each chordal Markov network structure there are
equivalent directed graph structures [15, 7], which we call here decomposable DAGs. A decomposable
DAG is a DAG G = (V, E) such that the set of directed edges E ? V ? V does not include any
immoralities, i.e., structures of the form vi ? vk ? vj with no edges between vi and vj . Due
to lack of immoralities, the d-separation property on a decomposable DAG corresponds exactly
to the separation property on the chordal undirected graph (the skeleton of the decomposable
DAG). Thus, decomposable graphs represent distributions that are representable by Markov and by
1
Extended discussion and empirical results are available in [21].
2
v6
v2
v1
a)
v5
{v1 , v6 } {v1 }
{v2 , v3 }
{v1 , v2 , v3 }
{v2 , v3 , v4 }
v4
v3
b)
{v1 , v5 } {v1 }
c)
v6
v2
v1
v5
v4
v3
Figure 1: Three views on chordal Markov network structures: a) chordal undirected graph, b) clique
tree, (c) decomposable DAG.
Bayesian networks. Figure 1 c) shows a corresponding decomposable DAG for the chordal undirected
graph in a). Note that the decomposable DAG may not be unique; for example, v2 ? v3 can be
directed also in the opposite direction. The score of the decomposable DAG can be calculated as
s(v1 , ?) + s(v5 , {v1 }) + s(v6 , {v1 }) + s(v2 , {v1 }) + s(v3 , {v1 , v2 }) + s(v4 , {v2 , v3 }), where s(vi , S)
are the local scores for BNSL using e.g. a Dirichlet prior. Because these local scores s(?, ?) correspond
to s(?) through s(vi , S) = s({vi , S}) ? s(S) (and s(?) = 0), we find that this BNSL scoring gives
the same result as the clique tree based scoring rule.
Thus CMSL can also be cast as the optimization problem of finding a graph in
X
arg max
s(vi , paG (vi )),
G?G
vi ?V
where G denotes the class of decomposable DAGs. (This formulation is used also in the GOBNILP
system [2].) The optimal chordal Markov network structure is the skeleton of the optimal G. This
problem is notoriously computationally difficult in practice, emphasized by the fact that standard
score-pruning [3, 8] used for BNSL is not generally applicable in the context of CMSL as it will often
prevent finding the true optimum: pruning parent sets for vertices often circumvents other vertices
achieving high scoring parents sets (as immoralities would be induced).
3
Hybrid Branch and Bound for CMSL
In this section we present details on our branch-and-bound approach to CMSL. We start with an
overview of the search algorithm, and then detail how we apply symmetry breaking and make use of
dynamic programming to dynamically update variable domains, i.e., for computing parent set choices
during search, and to obtain tight bounds for pruning the search tree.
3.1
Branch and Bound over Ordered Decomposable DAGs
The search is performed over the space of ordered decomposable DAGs. While in general the order
of the vertices of a DAG can be ambiguous, this notion allows for differentiating the exact order of
the vertices, and allows for pruning the search space by identifying symmetries (see Section 3.2).
Definition 1. G = (V, E, ?) is an ordered decomposable DAG if and only if (V, E) is a decomposable
DAG and ? : {1...n} ? {1...n} a total order over V such that (vi , vj ) ? E only if ? ?1 (i) < ? ?1 (j)
for all vi , vj ? V .
Partial solutions during search are hence ordered decomposable DAGs, which are extended by adding
a parent set choice (v, P ), i.e., adding the new vertex v and edges from each of its parents in P to v.
Definition 2. Let G = (V, E, ?) be an ordered decomposable DAG. Given vk ?
/ V and P ? V , we
say that the ordered decomposable DAG G0 = (V 0 , E 0 , ? 0 ) is G with the parent set choice (vk , P ) if
the following conditions hold.
1. V 0 = V ? S
{vk }
2. E 0 = E ? v0 ?P {(v 0 , vk )}.
3. We have ? 0 (i) = ?(i) for all i = 1...|V |, and ? 0 (|V | + 1) = k.
Algorithm 1 represents the core functionality of the branch and bound. The recursive function
takes two arguments: the remaining vertices of the problem instance, U , and the current partial
solution G = (V, E, ?). In addition we keep stored a best lower bound solution G? , which is the
3
Algorithm 1 The core branch-and-bound search.
1: function B RANCH A ND B OUND(U, G = (V, E, ?))
2:
if U = ? and s(G? ) < s(G) then G? ? G
. Update LB if improved.
3:
if this branch cannot improve LB then return
. Backtrack
4:
for (vi , P ) ? PARENT S ET C HOICES(U, G) do
. Iterate the current parent set choices.
5:
Let G0 = (V 0 , E 0 , ? 0 ) be G with the parent set choice (vi , P ).
6:
B RANCH A ND B OUND(U \ {vi }, G0 )
. Continue the search.
highest-scoring solution that has been found so far. Thus, at the end of the search, G? is an optimal
solution. During the search we use G? for bounding as further detailed in Section 3.3.
In the loop on line 4 we branch with all the parent set choices that we have deemed necessary to try
during the search. The method PARENT S ET C HOICES(U, G) and the related symmetry breaking are
explained in Section 3.2. We sort the parent set choices into decreasing order based on their score, so
that (v, P ) is tried before (v 0 , P 0 ) if s(v, P ) > s(v 0 , P 0 ), where v, v 0 ? U and P, P 0 ? V . This is
done to focus the search first to the most promising branches for finding an optimal solution. When
U = ?, we have PARENT S ET C HOICES(U, G) = ?, and so the current branch gets terminated.
3.2
Dynamic Branch Selection, Parent Set Pruning, and Symmetry Breaking
We continue by proposing symmetry breaking for the space of ordered decomposable DAGs, and
propose a dynamic programming approach for dynamic parent set pruning during search. We start
with symmetry breaking.
In terms of our branch-and-bound approach to CMSL, symmetry breaking is a vital part of the search,
as there can be exponentially many decomposable DAGs which correspond to a single undirected
chordal graph; for example, the edges of a complete graph can be directed arbitrarily without the
resulting DAG containing any immoralities. Hence symmetry breaking in terms of pruning out
symmetric solution candidates during search has potential for noticeably speeding up search.
Chickering [4, 5] showed how so-called covered edges can be used to detect equivalencies between
Bayesian network structures. Later van Beek and Hoffmann [29] implemented covered edge based
symmetry breaking in their BNSL approach. Here we introduce the concept of preferred vertex
orders, which generalizes covered edges for CMSL based on the decomposability of the solution
graphs.
Definition 3. Let G = (V, E, ?) be an ordered decomposable DAG. A pair vi , vj ? V violates the
preferred vertex order in G if the following conditions hold.
1. i > j.
2. paG (vi ) ? paG (vj ).
3. There is a path from vi to vj in G.
Theorem 1 states that for any (partial) solution (i.e., an ordered decomposable DAG), there always
exists an equivalent solution that does not contain any violations of the preferred vertex order.
Mapping to practice, this theorem allows for very effectively pruning out all symmetric solutions but
the one not violating the preferred vertex order within our branch-and-bound approach. A detailed
proof is provided in Appendix A.
Theorem 1. Let G = (V, E, ?) be an ordered decomposable DAG. There exists an ordered decomposable DAG G0 = (V, E 0 , ? 0 ) that is equivalent to G, but where for all vi , vj ? V the pair (vi , vj )
does not violate the preferred vertex order in G0 .
It follows from Theorem 1 that for each solution (ordered decomposable DAG) there exists an
equivalent solution where the lexicographically smallest vertex is a source. Thus we can fix it as the
first vertex in the order at the beginning of the search.
Similarly as in [29] for BNSL, we define the depths of vertices as follows.
Definition 4. Let G = (V, E, ?) be an ordered decomposable DAG. The depth of v ? V in G is
(
0
if paG (v) = ?,
0
d(G, v) =
max
d(G,
v
)
+
1
otherwise.
0
v ?paG (v)
4
The depths of G are ordered if for all vi , vj ? V , where ? ?1 (i) < ? ?1 (j), the following hold.
1. d(G, vi ) ? d(G, vj ), and 2. If d(G, vi ) = d(G, vj ), then i < j.
Note that "violating the preferred vertex order" concerns the order in which the vertices are in
the underlying DAG, whereas "depths are ordered" concerns the order by which a solution was
constructed. We use the former to prune whole solution candidates from the search space, and the
latter to ensure that no solution candidate is seen twice during search.
We also propose a dynamic programming approach to branch selection and parent set pruning during
search, based on the following definition of valid parent sets.
Definition 5. Let G = (V, E, ?) be an ordered decomposable DAG. Given vk ?
/ V and P ? V , let
G0 = (V 0 , E 0 , ? 0 ) be G with the parent set choice (vk , P ). The parent set choice (vk , P ) is valid for
G if the following hold.
1. For all vi , vj ? P we have either (vi , vj ) ? E or (vj , vi ) ? E.
2. For all vi ? V , the pair (vi , vk ) does not violate the preferred vertex order in G0 .
3. The depths of G0 are ordered.
Given a partial solution G = (V, E, ?), a vertex v ?
/ V , and a subset P ? V , the function G ET S U PERSETS in Algorithm 2 represents a dynamic programming method for determining valid parent set
choices (v, P 0 ) for G where P 0 ? P . An advantage of this formulation is that invalidating conditions
for a parent set, such as immoralities or violations of the preferred vertex order, automatically hold
for all the supersets of the parent set; this is applied on line 6 to avoid unnecessary branching.
On line 8 we require that a parent set P is added to the list only if none of its valid supersets P 0 ? C
have a higher score. This pruning technique is based on the observation that P 0 provides all the same
moralizing edges as P , and therefore it is sufficient to only consider the parent set choice (v, P 0 ) in
the search when s(v, P ) ? s(v, P 0 ).
Given the set of remaining vertices U , the function PARENT S ET C HOICES in Algorithm 2 constructs
all the available parent set choices for the current partial solution G = (V, E, ?). The collection
M(G, vi ) contains the subset-minimal parent sets for vertex vi ? U that satisfy the 3rd condition
of Definition 5. If V = ?, then M(G, vi ) = {?}. Otherwise, let k be the maximum depth of the
vertices in G. Now M(G, vi ) contains the subset-minimal parent sets that would insert vi on depth
k + 1. In addition, if i > j for all vj ? V where d(G, vj ) = k, then M(G, vi ) also contains the
subset-minimal parent sets that would insert vi on depth k. Note that the cardinality of any parent set
in M(G, vi ) is at most one.
3.3
Computing Tight Bounds by Harnessing Dynamic Programming for BNSL
To obtain tight bounds during search, we make use of the fact that the score of the optimal BN
structures for the BNSL instance with same scores as in the CMSL instance at hand is guaranteed
to give an upper bound on the optimal solutions to the CMSL instance. To compute an optimal
BN structure, we use a variant of a standard dynamic programming algorithm by Silander and
Myllym?ki [22]. While there are far more efficient algorithms for BNSL [2, 32, 29], we use BNSL
DP for obtaining an upper bound during the branch-and-bound search under the current partial
Algorithm 2 Constructing parent set choices via dynamic programming.
1: function PARENT
S S ET
SC HOICES(U, G = (V, E, ?))
2:
return
G ET S UPERSETS(v, G, M )
v?U M ?M(G,v)
3: function G ET S UPERSETS(v, G = (V, E, ?), P )
4:
Let C = ?
5:
for v 0 ? V \ P \ {v} do
6:
if (v, P 0 ) is a valid parent set choice for G with some P 0 ? P ? {v 0 } then
7:
C ? C ? G ET S UPERSETS(v, G, P ? {v 0 })
8:
if (v, P ) is valid parent set choice for G and s(v, P ) > s(v, P 0 ) for all P 0 ? C then
9:
C ? C ? {(v, P )}
10:
return C
5
CMSL solution (i.e., under the current branch). Specifically, before the actual branch and bound, we
precompute a DP table which stores, for each subset of vertices V 0 ? V of the problem instance, the
score of the so-called BN extensions of V 0 , i.e., the optimal BN structures over U = V \ V 0 where
we additionally allow the vertices in U to also take parents from V 0 . This guarantees that the BN
extensions are compatible with the vertex order in the current branch of the branch-and-bound search
tree, and thereby the sum of the score of the current partial CMSL solution over V 0 and the score of
the optimal BN extensions of V 0 is a valid upper bound. By spending O(n ? 2n ) time in the beginning
of the branch and bound for computing the scores of optimal BN extensions of every V 0 ? V , we
can then look up these scores during branch and bound in O(1) time.
With the DP table, it takes only low polynomial time to construct the optimal BN structure over the
set of all vertices [22], i.e., a BN extension of ?. Thus, we can obtain an initial lower bound solution
G? for the branch and bound as follows.
1. Construct the optimal BN structure for the vertices of the problem instance
2. Try to make the BN decomposable by heuristically adding or removing edges.
3. Let G? be the highest-scoring decomposable DAG from step 2.
However, the upper bounds obtained via BNSL can be at times can be quite weak when the network
structures contain many immoralities. For this reason, in Algorithm 3, we introduce an additional
method for computing the upper bounds, taking immoralities ?relaxedly? into consideration. The
algorithm takes four inputs: A fixed partial solution G = (V, E, ?), a list of vertices A that we have
assigned during the upper bound computation, a list of remaining vertices U , and an integer d ? 0
which dictates the maximum recursion depth. As a fallback option, on line 3 we return the optimal
BN score for the remaining vertices if the maximum recursion depth is reached.
On line 4 we construct the collection of sets P that are the maximal sets that any vertex can take
as parent set during the upper bound computation. The sets in P take immoralities relaxedly into
consideration: For any vi , vj ? V , we have {vi , vj } ? P for some P ? P if and only if (vi , vj ) ? E
or (vj , vi ) ? E. That is, when choosing parent sets during the upper bound computation, we allow
immoralities to appear, as long as they are not between vertices of the fixed partial solution. In the
loop on line 6, we iterate through each vertex v ? U that is still remaining, and find its highestscoring relaxedly-moral parent set according to P. Note that given any P 0 ? P, we can find the
highest-scoring parent set P ? P 0 in O(1) time when the scores are stored in a segment tree. For
information about constructing such data structure, see [22]. Thus line 7 takes O(|V |) time to execute.
Finally, on line 8 of the loop, we split the problem into subproblems to see which parent set choice
(v, P ) provides the highest local upper bound u to be returned.
Algorithm 3 requires O((n ? m) ? m ? 2n?m ) time, where m = |V | is the number of vertices in the
partial solution and n the number of vertices in the problem instance, assuming that the BN extensions
and the segment trees have been precomputed. (In the empirical evaluation, the total runtimes of our
branch-and-bound approach include these computations.) The collections P can exist implicitly.
We use the upper bounds within branch and bound as follows. Let G = (V, E, ?) be the current
partial solution, let U be the set of remaining vertices, and let b be the score of optimal BN extensions
of V . We can close the current branch if s(G? ) ? s(G) + b. Otherwise, we can close the branch if
s(G? ) ? s(G) + U PPER B OUND(G, ?, U, d) for some d > 0. Our implementation uses d = 10.
Algorithm 3 Computing upper bounds for a partial solution via dynamic programming.
1: function U PPER B OUND(G = (V, E, ?), A, U, d)
2:
if U = ? then return 0
3:
if d = 0 then
S return the score of optimal BN extensions of V ? A
4:
Let P =
{{v} ? paG (v) ? A}
v?V
5:
6:
7:
Let u ? ??
for v ? U do
Let P = arg max s(v, P )
8:
9:
u ? max(u, s(v, P ) + U PPER B OUND(G, A ? {v}, U \ {v}, d ? 1))
return u
P ?P 0 ?P
6
4
Empirical Evaluation
We implemented the branch-and-bound algorithm in C++, and refer to this prototype as BBMarkov.
We compare the performance of BBMarkov to that of GOBNILP (the newest development version [24]
at the time of publication, using IBM CPLEX version 12.7.1 as the internal IP solver) as a state-ofthe-art BNSL system implementing a integer programming branch-and-cut approach to CMSL by
ruling out non-chordal graphs, and Junctor, implementing a state-of-the-art DP approach to CMSL.
We used a total of 54 real-world datasets used as standard benchmarks for exact approaches [32, 29].
For investigating scalability of the algorithms in terms of the number of variables n, we obtained
from each dataset several benchmark instances by restricting to the first n variables for increasing
values of n. We did not impose a bound on the treewidth of the chordal graphs of interest, i.e., the
size of candidate parent sets was not limited. We used the BDeu score with equivalent sample size 1.
As standard practice in benchmarking exact structure learning algorithms, we focus on comparing the
running times of the considered approaches on precomputed input CMSL instances. The experiments
were run under Debian GNU/Linux on 2.83-GHz Intel Xeon E5440 nodes with 32-GB RAM.
Figure 2 compares BBMarkov to GOBNILP and Junctor under a 1-h per-instance time limit, with
different numbers n of variables distinguished using different point styles. BBMarkov clearly
dominates GOBNILP in runtime performance (Fig. 2 left); instances for n > 15 are not shown as
GOBNILP was unable to solve them. Compared to Junctor (Fig. 2 middle, Table 1), BBMarkov
exhibits complementary performance. Junctor is noticeably strong on several datasets and lower
values of n, and exhibits fewer timeouts. For a fixed n, Junctor?s runtimes have a very low variance
independent of the dataset, which is due to the ?(4n ) (both worst-case and best-case) runtime
guarantee. However, BBMarkov shows potential for scaling up for larger n than Junctor: at n = 17
Junctor?s runtimes are very close to 1 h on all instances, while BBMarkov?s bounds rule out at times
very effectively non-optimal solutions, resulting in noticeable lower runtimes on specific datasets
with increasing n. This is show-cased in Table 1 on the right, highlighting some of the best-case
performance of BBMarkov using per-instance time limit of 24 h for both BBMarkov and Junctor.
Another benefit of BBMarkov compared to Junctor is the observed
lower memory consumption (Figure 3). Junctor?s ?(3n ) memory
1m
<1s
<1s
n=15
n=14
n=13
n=12
n=11
1m
Run time of BBMarkov
>1h
To find an optimal solution
>1h
Run time of Junctor
Run time of GOBNILP
>1h
1m
<1s
<1s
1m
Run time of BBMarkov
>1h
25GB
10GB
Memory usage
In terms of how the various search techniques implemented in BBMarkov contribute to the running times of BBMarkov, we observed
that the running times for obtaining BNSL-based bounds (via the
use of exact BN dynamic programming and segment trees) tend to
be only a small fraction of the overall running times. For example,
at n = 20, these computations take less than minute in total. Most
of the time in the search is typically used in the optimization loop
and in computing the tighter upper bounds that take immoralities
"relaxedly" into consideration. While computing the tighter bounds
is more expensive than computing the exact BNs at the beginning of
search, the tighter bounds often pay off in terms of overall running
times as branches can be closed earlier during search.
Junctor
BBMarkov
1GB
100MB
10MB
8 9
11 13 15 17 19
Number of variables
Figure 3: Memory usage
>1h
n=17
n=16
n=15
n=14
n=13
1m
<1s
<1s
1m
>1h
To find and prove the solution
Figure 2: Per-instance runtime comparisons. Left: BBMarkov vs GOBNILP. Middle: BBMarkov vs
Junctor. Right: BBMarkov time to finding vs BBMarkov time to proving an optimal solution.
7
Table 1: BBMarkov v Junctor. Left: smaller datasets and for different sample sizes on the Water
dataset. Right: Examples of best-case performance of BBMarkov. to: timeout, mo: memout.
Dataset
Wine
Adult
Letter
Voting
Zoo
Water100
Water1000
Water10000
Tumor
n
13
14
16
17
17
17
17
17
18
Running times (s)
BBMarkov
Junctor
<1
(<1)
6
58
(35)
29
>3600 (>3600)
592
281
(207)
3050
>3600 (>3600)
2690
100
(49)
2580
2731
(279)
2592
>3600 (>3600)
2928
610
(268)
12019
Dataset
Alarm
Heart
Hailfinder500
Water100
n
17
18
19
20
17
18
19
20
17
18
19
20
18
19
20
Running times (s)
BBMarkov
Junctor
268
(62)
2724
1462
(315)
12477
10274
(2028)
52130
49610
(50)
mo
41
(22)
3007
162
(85)
11179
1186
(698)
50296
15501 (13845)
mo
225
(108)
2588
2543
(1348)
12422
13749
(6418)
53108
33503 (25393)
mo
590
(244)
12244
6581
(6187)
52575
61152 (54806)
mo
usage results consistently in running out on memory for n ? 20. At n = 19, BBMarkov uses on
average approx. 1 GB of memory, while Junctor uses close to 30 GB. A further benefit of BBMarkov
is its ability to provide ?anytime? solutions during search. In fact, the bounds obtained during search
result at times in finding optimal solutions relatively fast: Figure 2 right shows the ratio of time
needed to find an optimal solution (x-axis) from time needed to terminate search, i.e., to find a
solution and prove its optimality (y-axis), and in Table 1, with the time needed to find an optimal
solution given in parentheses.
5
Conclusions
We introduced a new branch-and-bound approach to learning optimal chordal Markov network
structures, i.e., decomposable graphs. In addition to core branch-and-bound search, the approach
integrates dynamic programming for obtaining tight bounds and effective variable domain pruning
during search. In terms of practical performance, the approach has the potential of reaching 20
variables within hours of runtime, at which point the competing native dynamic programming
approach Junctor runs out of memory on standard modern computers. When approaching 20
variables, our approach is approximately 30 times as memory-efficient as Junctor. Furthermore, in
contrast to Junctor, the approach is ?anytime? as solutions can be obtained already before finishing
search. Efficient parallelization of the approach is a promising direction for future work.
Acknowledgments
The authors gratefully acknowledge financial support from the Academy of Finland under grants
251170 COIN Centre of Excellence in Computational Inference Research, 276412, 284591, 295673,
and 312662; and the Research Funds of the University of Helsinki.
A
Proofs
We give a proof for Theorem 1, central in enabling effective symmetric breaking in our branch-andbound approach. We start with a definition and lemma towards the proof.
Definition 6. Let V = {v1 , ..., vn } be a set of vertices and let ? and ? 0 be some total orders over V .
Let k = mini,?(i)6=?0 (i) i be the first difference between the orders. If no such difference exists, we
denote ? = ? 0 . Otherwise we denote ? < ? 0 if and only if ?(k) < ? 0 (k).
Lemma 1. Let G = (V, E, ?) be an ordered decomposable DAG. If there are vi , vj ? V such that
the pair (vi , vj ) violates the preferred vertex order in G, then there exists an ordered decomposable
DAG G0 = (V, E 0 , ? 0 ), where 1. G0 belongs to the same equivalence class with G, 2. the pair (vi , vj )
does not violate the preferred vertex order in G0 , and 3. ? < ? 0 .
8
Proof. We begin by defining a directed clique tree C = (V, E) over G.
Given vk ? V , let Ck = paG (vk ) ? {vk } be the clique defined by vk in G. The vertices of C are
these cliques; we also add an empty set as a clique to make sure the cliques form a tree (and not a
forest). Formally, V = {Ck | vk ? V } ? {?}.
Given vk ? V , where paG (vk ) 6= ?, let ?k = argmaxv` ?paG (vk ) ? ?1 (`) denote the parent of vk in
G that is in the least significant position in ?. Now, the edges of C are
E = {(?, Ck ) | Ck = {vk }, vk ? V } ? {(C` , Ck ) | v` = ?k , Ck 6= {vk }, vk ? V }.
In words, if vk ? V is a source vertex in G (i.e., Ck = {vk }), then the parent of Ck is ? in C.
Otherwise (i.e., Ck 6= {vk }) the parent of Ck is C` , where v` is the closest vertex to vk in order
?S
that satisfies C` ? paG (vk ) 6= ?. We see that all the requirements for clique trees hold for C:
I. C?V C = V , II. if {v` , vk } ? E, then either {v` , vk } ? Ck or {v` , vk } ? C` , and III. due to the
decomposability of G, we have Ca ? Cc ? Cb on any path from Ca to Cc through Cb (the running
intersection property).
Now assume that there are vi , vj ? V such that the pair (vi , vj ) violates the preferred vertex order in
G; that is, we have i > j, paG (vi ) ? paG (vj ) and a path from vi to vj in G. This means that there is
a path from Ci to Cj in C as well.
Let P ? V be the parent vertex of Ci in C. We see that Cj exists in a subtree T of C that is separated
from rest of C by P , and where Ci is the root vertex. Let T 0 be a new clique tree that is like T , but
redirected so that Cj is the root vertex of T 0 . Let C 0 be a new clique tree that is like C, but T is
replaced with T 0 .
We show that C 0 is a valid clique tree. First of all, the vertices (cliques) of C 0 are exactly the same as in
C, so C 0 clearly satisfies the requirements I and II. As for the requirement III, consider the non-trivial
case where Ca , Cb ? C have a path from Ca to Cb through Ci in C. This means vi ?
/ Ca (due to the
way C was constructed), and so we get
Ca ? Cb ? Ci
? Ca ? Cb ? Ci \ {vi }
? Ca ? Cb ? paG (vi )
?
paG (vj ) ? Cj .
Def. 3 (2)
Therefore the running intersection property holds for C 0 .
Let ?
? be the total order by which C 0 is ordered. Let G0 = (V, E 0 , ?
? ) be a new ordered decomposable
DAG that is equivalent to G, but where the edges E 0 are arranged to follow the order ?
?.
Finally, we see that G0 satisfies the conditions of the theorem: 1. The cliques of G0 are identical to
that of G, so G0 belongs to the same equivalence class with G. 2. We have ?
? ?1 (j) < ?
? ?1 (i), and
0
therefore there is no path from vi to vj in G . Thus the pair (vi , vj ) does not violate the preferred
vertex order in G0 . 3. Let o = ? ?1 (i). We have ?
? (o) = j < i = ?(o). Furthermore, the change from
T to T 0 in C 0 did not affect any vertex whose position was earlier than o. Therefore ?
? (k) = ?(k) for
all k = 1...(o ? 1). This implies ?
? < ?.
Proof of Theorem 1. Consider the following procedure for finding G0 .
1. Select vi , vj ? V where the pair (vi , vj ) violates the preferred vertex order in G. If there
are no such vertices, assign G0 ? G and terminate.
2. Let ? be the total order of the vertices of G. Construct an ordered decomposable DAG
? = (V, E,
? ? 0 ) such that I. the pair (vi , vj ) does not violate the preferred vertex order in G,
?
G
0
?
?
II. G belongs to the same equivalent class with G, and III. ? < ?. By Lemma 1, G can be
constructed from G.
? and return to step 1.
3. Assign G ? G
It is clear that when the procedure terminates, G0 belongs to same equivalence class with G and there
are no violations of the preferred vertex order in G0 . We also see that the total order of G (i.e., ?)
is lexicographically strictly decreasing every time the step 3 is reached. There are finite amount of
possible permutations (total orders) and therefore the procedure converges. The existence of this
procedure and its correctness proves that G0 exists.
9
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6,392 | 6,782 | Revenue Optimization with Approximate Bid
Predictions
? Medina
Andr?es Munoz
Google Research
76 9th Ave
New York, NY 10011
Sergei Vassilvitskii
Google Research
76 9th Ave
New York, NY 10011
Abstract
In the context of advertising auctions, finding good reserve prices is a notoriously
challenging learning problem. This is due to the heterogeneity of ad opportunity
types, and the non-convexity of the objective function. In this work, we show how
to reduce reserve price optimization to the standard setting of prediction under
squared loss, a well understood problem in the learning community. We further
bound the gap between the expected bid and revenue in terms of the average loss
of the predictor. This is the first result that formally relates the revenue gained to
the quality of a standard machine learned model.
1
Introduction
A crucial task for revenue optimization in auctions is setting a good reserve (or minimum) price. Set
it too low, and the sale may yield little revenue, set it too high and there may not be anyone willing
to buy the item. The celebrated work by Myerson [1981] shows how to optimally set reserves in
second price auctions, provided the value distribution of each bidder is known.
In practice there are two challenges that make this problem significantly more complicated. First,
the value distribution is never known directly; rather, the auctioneer can only observe samples drawn
from it. Second, in the context of ad auctions, the items for sale (impressions) are heterogeneous, and
there are literally trillions of different types of items being sold. It is therefore likely that a specific
type of item has never been observed previously, and no information about its value is known.
A standard machine learning approach addressing the heterogeneity problem is to parametrize each
impression by a feature vector, with the underlying assumption that bids observed from auctions
with similar features will be similar. In online advertising. these features encode, for instance, the
ad size, whether it?s mobile or desktop, etc.
The question is, then, how to use the features to set a good reserve price for a particular ad opportunity. On the face of it, this sounds like a standard machine learning question?given a set of features,
predict the value of the maximum bid. The difficulty comes from the shape of the loss function.
Much of the machine learning literature is concerned with optimizing well behaved loss functions,
such as squared loss, or hinge loss. The revenue function, on the other hand is non-continuous and
strongly non-concave, making a direct attack a challenging proposition.
In this work we take a different approach and reduce the problem of finding good reserve prices to
a prediction problem under the squared loss. In this way we can rely upon many widely available
and scalable algorithms developed to minimize this objective. We proceed by using the predictor to
define a judicious clustering of the data, and then compute the empirically maximizing reserve price
for each group. Our reduction is simple and practical, and directly ties the revenue gained by the
algorithm to the prediction error.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1.1
Related Work
Optimizing revenue in auctions has been a rich area of study, beginning with the seminal work
of Myerson [1981] who introduced optimal auction design. Follow up work by Chawla et al.
[2007] and Hartline and Roughgarden [2009], among others, refined his results to increasingly
more complex settings, taking into account multiple items, diverse demand functions, and weaker
assumptions on the shape of the value distributions.
Most of the classical literature on revenue optimization focuses on the design of optimal auctions
when the bidding distribution of buyers is known. More recent work has considered the computational and information theoretic challenges in learning optimal auctions from data. A long line
of work [Cole and Roughgarden, 2015, Devanur et al., 2016, Dhangwatnotai et al., 2015, Morgenstern and Roughgarden, 2015, 2016] analyzes the sample complexity of designing optimal auctions.
The main contribution of this direction is to show that under fairly general bidding scenarios, a
near-optimal auction can be designed knowing only a polynomial number of samples from bidders?
valuations. Other authors, [Leme et al., 2016, Roughgarden and Wang, 2016] have focused on the
computational complexity of finding optimal reserve prices from samples, showing that even for
simple mechanisms the problem is often NP-hard to solve directly.
Another well studied approach to data-driven revenue optimization is that of online learning. Here,
auctions occur one at a time, and the learning algorithm must compute prices as a function of the history of the algorithm. These algorithms generally make no distributional assumptions and measure
their performance in terms of regret: the difference between the algorithm?s performance and the
performance of the best fixed reserve price in hindsight. Kleinberg and Leighton [2003] developed
an online revenue optimization algorithm for posted-price auctions that achieves low regret. Their
work was later extended to second-price auctions by Cesa-Bianchi et al. [2015].
A natural approach in both of these settings is to attempt to predict an optimal reserve price, equivalently the highest bid submitted by any of the buyers. While the problem of learning this reserve
price is well understood for the simplistic model of buyers with i.i.d. valuations [Cesa-Bianchi et al.,
2015, Devanur et al., 2016, Kleinberg and Leighton, 2003], the problem becomes much more challenging in practice, when the valuations of a buyer also depend on features associated with the ad
opportunity (for instance user demographics, and publisher information).
This problem is not nearly as well understood as its i.i.d. counterpart. Mohri and Medina [2014]
provide learning guarantees and an algorithm based on DC programming to optimize revenue in
second-price auctions with reserve. The proposed algorithm, however, does not easily scale to large
auction data sets as each iteration involves solving a convex optimization problem. A smoother
version of this algorithm is given by [Rudolph et al., 2016]. However, being a highly non-convex
problem, neither algorithm provides a guarantee on the revenue attainable by the algorithm?s output.
Devanur et al. [2016] give sample complexity bounds on the design of optimal auctions with side
information. However, the authors consider only cases where this side information is given by
? ? [0, 1]. More importantly, their proposed algorithm only works under the unverifiable assumption
that the conditional distributions of bids given ? satisfy stochastic dominance.
Our results. We show that given a predictor of the bid with squared loss of ? 2 , we can construct
a reserve function r that extracts all but g(?) revenue, for a simple increasing function g. (See
Theorem 2 for the exact statement.) To the best of our knowledge, this is the first result that ties
the revenue one can achieve directly to the quality of a standard prediction task. Our algorithm for
computing r is scalable, practical, and efficient.
Along the way we show what kinds of distributions are amenable to revenue optimization via reserve
prices. We prove that when bids are drawn i.i.d. from a distribution F , the ratio between the mean bid
and the revenue extracted with the optimum monopoly reserve scales as O(log Var(F )) ? Theorem
5. This result refines the log h bound derived by Goldberg et al. [2001], and formalizes the intuition
that reserve prices are more successful for low variance distributions.
2
Setup
We consider a repeated posted price auction setup where every auction is parametrized by a feature
vector x ? X and a bid b ? [0, 1]. Let D be a distribution over X ? [0, 1]. Let h : X ? [0, 1], be a
bid prediction function and denote by ? 2 the squared loss incurred by h:
E[(h(x) ? b)2 ] = ? 2 .
2
We assume h is given, and make no assumption on the structure of h or how it is obtained. Notice
that while the existence of such h is not guaranteed for all values of ?, using historical data one
could use one of multiple readily available regression algorithms to find the best hypothesis h.
Let S = (x1 , b1 ), . . . , (xm , bm ) ? D be a set of m i.i.d. samples drawn from D and denote by
SX = (x1 , . . . , xm ) its projection on X . Given a price p let Rev(p, b) = p1b?p denote the revenue
obtained when the bidder bids b. For a reserve price function r : X ? [0, 1] we let:
1 X
b
R(r) = E
Rev(r(x), b) and R(r)
=
Rev(r(x), b)
m
(x,b)?D
(x,b)?S
denote the expected and empirical revenue of reserve price function r.
b = 1 Pm bi denote the population and empirical mean bid, and S(r) =
We also let B = E[b], B
i=1
m
b
b ? R(r)
b
B ? R(r), S(r)
= B
denote the expected and empirical separation between bid values
and the revenue. Notice that for a given reserve price function r, S(r) corresponds to revenue left
on the table. Our goal is, given S and h, to find a function r that maximizes R(r) or equivalently
minimizes S(r).
2.1
Generalization Error
Note that in our set up we are only given samples from the distribution, D, but aim to maximize the
expected revenue. Understanding the difference between the empirical performance of an algorithm
and its expected performance, also known as the generalization error, is a key tenet of learning
theory.
At a high level, the generalization error is a function of the training set size: larger training sets lead
to smaller generalization error; and the inherent complexity of the learning algorithm: simple rules
such as linear classifiers generalize better than more complex ones.
In this paper we characterize the complexity of a class G of functions by its growth function ?. The
growth function corresponds to the maximum number of binary labelings that can be obtained by
G over all possible samples SX . It is closely related to the VC-dimension when G takes values in
{0, 1} and to the pseudo-dimension [Morgenstern and Roughgarden, 2015, Mohri et al., 2012] when
G takes values in R.
We can give a bound on the generalization error associated with minimizing the empirical separation
over a class of functions G. The following theorem is an adaptation of Theorem 1 of [Mohri and
Medina, 2014] to our particular setup.
Theorem 1. Let ? > 0, with probability at least 1 ? ? over the choice of the sample S the following
bound holds uniformly for r ? G
r
r
log
1/?
2 log(?(G, m))
b +2
S(r) ? S(r)
+4
.
(1)
2m
m
Therefore, in order to minimize the expected separation S(r) it suffices to minimize the empirical
b over a class of functions G whose growth function scales polynomially in m.
separation S(r)
3
Warmup
In order to better understand the problem at hand, we begin by introducing a straightforward mechanism for transforming the hypothesis function h to a reserve price function r with guarantees on its
achievable revenue.
Lemma 1. Let r : X ? [0, 1] be defined by r(x) := max(h(x) ? ? 2/3 , 0). The function r then
satisfies S(r) ? ? 1/2 + 2? 2/3 .
The proof is a simple application of Jensen?s and Markov?s inequalities and it is deferred to Appendix B.
This surprisingly simple algorithm shows there are ways to obtain revenue guarantees from a simple
regressor. To the best of our knowledge these is the first guarantee of its kind. The reader may be
3
curious about the choice of ? 2/3 as the offset in our reserve price function. We will show that the
dependence on ? 2/3 is not a simple artifact of our analysis, but a cost inherent to the problem of
revenue optimization.
Moreover, observe that this simple algorithm fixes a static offset, and does not make a distinction
between those parts of the feature space, where the algorithm makes a low error, and those where the
error is relatively high. By contrast our proposed algorithm partitions the space appropriately and
calculates a different reserve for each partition. More importantly we will provide a data dependent
bound on the performance of our algorithm that only in the worst case scenario behaves like ? 2/3 .
4
Results Overview
In principle to maximize revenue we need to find a class of functions G with small complexity, but
that contains a function which approximately minimizes the empirical separation. The challenge
comes from the fact that the revenue function, Rev, is not continuous and highly non-concave?a
small change in the price, p, may lead to very large changes in revenue. This is the main reason why
simply using the predictor h(x) as a proxy for a reserve function is a poor choice, even if its average
error, ? 2 is small. For example a function h, that is just as likely to over-predict by ? as to under
predict by ? will have very small error, but lead to 0 revenue in half the cases.
A solution on the other end of the spectrum would simply memorize the optimum prices from the
sample S, setting r(xi ) = bi . While this leads to optimal empirical revenue, a function class G
containing r would satisfy ?(G, m) = 2m , making the bound of Theorem 1 vacuous.
In this work we introduce a family G(h, k) of classes parameterized by k ? N. This family admits
an approximate minimizer that can be computed in polynomial time, has low generalization error,
and achieves provable guarantees to the overall revenue.
More precisely, we show that given S, and a hypothesis h with expected squared loss of ? 2 :
? For every k ? 1 there exists a set of functions G(h, k) such that ?(G(h, k), m) = O(m2k ).
? For every k ? 1, there is a polynomial time algorithm that outputs rk ? G(h, k) such that
1
b k ) is bounded by O( 2/3
in the worst case scenario S(r
+ ? 2/3 + m11/6 ).
k
Effectively, we show how to transform any classifier h with low squared loss, ? 2 , to a reserve price
predictor that recovers all but O(? 2/3 ) revenue in expectation.
4.1
Algorithm Description
In this section we give an overview of the algorithm that uses both the predictor h and the set of
samples in S to develop a pricing function r. Our approach has two steps. First we partition the
set of feasible prices, 0 ? p ? 1, into k partitions, C1 , C2 , . . . , Ck . The exact boundaries between
partitions depend on the samples S and their predicted values, as given by h. For each partition
we find the price that maximizes the empirical revenue in the partition. We let r(x) return the
empirically optimum price in the partition that contains h(x).
For a more formal description, let Tk be the set of k-partitions of the interval [0, 1] that is:
Tk = {t = (t0 , t1 , . . . , tk?1 , tk ) | 0 = t0 < . . . < tk = 1}.
Pk?1
We define G(h, k) = {x 7? j=0 ri 1tj ?h(x)<tj+1 | rj ? [ti , tj+1 ] and t ? Tk }. A function in
G(h, k) chooses k level sets of h and k reserve prices. Given x, price rj is chosen if x falls on the
j-th level set.
It remains to define the function rk ? G(h, k). Given a partition vector t ? Tk , let the partition
C h = {C1h , . . . , Ckh } of X be given by Cjh = {x ? X |tj?1 < h(x) ? tj }. Let mj = |SX ? Cjh | be
the number of elements that fall into the j-th partition.
We define the predicted mean and variance of each group Cjh as
1 X
1 X
?hj =
h(xi )
and
(?jh )2 =
(h(xi ) ? ?j )2 .
mj
m
j
h
h
xi ?Cj
xi ?Cj
4
We are now ready to present algorithm RIC-h for computing rk ? Hk .
Algorithm 1. Reserve Inference from Clusters
Pk?1
1
h
Compute th ? Tk that minimizes m
j=0 mj ?j .
Let C h = C1h , C2h , . . . , Ckh be the induced partitions.
For each j ? 1, . . . , k, set rj = maxr r ? |{i|bi ? r ? xi ? Cjh }|.
Pk?1
Return x 7? j=0 rj 1h(x)?Cjh .
end
Our main theorem states that the separation of rk is bounded by the cluster variance of C h . For a
partition C = {C1 , . . . , Ck } of X let ?j denote the empirical variance of bids for auctions in Cj . We
define the weighted empirical variance by:
?(C) : =
k s
X
j=1
X
(bi ? bi0 )2 = 2
i,i0 :xi ,xi0 ?Ck
k
X
mj ?
bj
(2)
j=1
Theorem 2. Let ? > 0 and let rk denote the output of Algorithm 1 then rk ? G(h, k) and with
probability at least 1 ? ? over the samples S:
!
r
1/2 2/3
1
1
log
1/?
h
1/3
2
1/3
b
b k ) ? (3B)
b
.
?(C ) ? (3B)
+2 ? +
S(r
2m
2k
2m
Notice that our bound is data dependent and only in he worst case scenario it behaves like ? 2/3 . In
general it could be much smaller.
We also show that the complexity of G(h, k) admits a favorable bound. The proof is similar to that
in [Morgenstern and Roughgarden, 2015]; we include it in Appendix E for completness.
Theorem 3. The growth function of the class G(h, k) can be bounded as: ?(G(h, k), m) ?
m2k?1
.
kk
b in terms of B to conclude:
We can combine these results with Equation 1 and an easy bound on B
Corollary 1. Let ? > 0 and let rk denote the output of Algorithm 1 then rk ? G(h, k) and with
probability at least 1 ? ? over the samples S:
r k log m
1
log 1/? 1/6 r k log m
h
2 1/3
1/3 ?(C )
b
+
+O
? (12B? ) +O 2/3 +
.
S(rk ) ? (3B)
2m
m
2m
m
k
Since B ? [0, 1], this implies that when k = ?(m3/7 ), the separation is bounded by 2.28? 2/3 plus
? ?2/7 ).
additional error factors that go to 0 with the number of samples, m, as O(m
5
Bounding Separation
In this section we prove the main bound motivating our algorithm. This bound relates the variance
of the bid distribution and the maximum revenue that can be extracted when a buyer?s bids follow
such distribution. It formally shows what makes a distribution amenable to revenue optimization.
To gain intuition for the kind of bound we are striving for, consider a bid distribution F . If the
variance of F is 0, that is F is a point mass at some value v, then setting a reserve price to v leads to
no separation. On the other hand, consider the equal revenue distribution, with F (x) = 1? 1/x. Here
any reserve price leads to revenue of 1. However, the distribution has unbounded expected bid and
variance, so it is not too surprising that more revenue cannot be extracted. We make this connection
precise, showing that after setting the optimal reserve price, the separation can be bounded by a
function of the variance of the distribution.
Given any bid distribution F over [0, 1] we denote by G(r) = 1 ? limr0 ?r? F (r0 ) the probability
that a bid is greater than or equal to r. Finally, we will let R = maxr rG(r) denote the maximum
revenue achievable when facing a bidder whose bids are drawn from distribution F . As before we
denote by B = Eb?F [b] the mean bid and by S = B ? R the expected separation of distribution F .
5
S
Theorem 4. Let ? 2 denote the variance of F . Then ? 2 ? 2R2 e R ? B 2 ? R2 .
The proof of this theorem is highly technical and we present it in Appendix A.
Corollary 2. The following bound holds for any distribution F: S ? (3R)1/3 ? 2/3 ? (3B)1/3 ? 2/3
The proof of this corollary follows immediately by an application of Taylor?s theorem to the bound
of Theorem 4. It is also easy to show that this bound is tight (see Appendix D).
5.1
Approximating Maximum Revenue
In their seminal work Goldberg et al. [2001] showed that when faced with a bidder drawing values
distribution F on [1, M ] with mean B, an auctioneer setting the optimum monopoly reserve would
recover at least ?(B/ log M ) revenue. We show how to adapt the result of Theorem 4 to refine this
approximation ratio as a function of the variance of F . We defer the proof to Appendix B.
Theorem 5. For any distribution F with mean B and variance ? 2 , the maximum revenue with
?2
monopoly reserves, R, satisfies: B
R ? 4.78 + 2 log 1 + B 2
Note that since ? 2 ? M 2 this always leads to a tighter bound on the revenue.
5.2
Partition of X
Corollary 2 suggests clustering points in such a way that the variance of the bids in each cluster
bj =
is minimized. Given a partition C = {C1 , . . . , Ck } of X we denote by mj = |SX ? Cj |, B
P
P
1
1
2
2
b
bj = mj i:xi ?Cj (bi ? Bj ) . Let also rj = argmaxp>0 p|{bi > p|xi ? Cj }| and
i:xi ?Cj bi , ?
mj
bj = rj |{bi > rj |xi ? Cj }|.
R
1/3 P
2/3
Pk
k
1
b
b
Lemma 2. Let r(x) =
bj
3B
=
j=1 rj 1x?Cj then S(r) ?
j=1 mj ?
m
1/3
1
b
3B
2m ?(C) .
bj ?R
bj , Corollary 2 applied to the empirical bid distribution in Cj yields Sbj ?
Proof. Let Sbj = B
m
1/3 2/3
b
(3Bj ) ?
bj . Multiplying by mj , summing over all clusters and using H?older?s inequality gives:
k
k
k
k
X 3m
1/3 X m
2/3
1 X
1 X b 1/3 2/3
j b
j
b
S(r)
=
mj Sj ?
(3Bj ) ?
bj mj ?
Bj
?
bj
.
m j=1
m j=1
m
m
j=1
j=1
6
Clustering Algorithm
b is fixed, we can find a function minimizing the
In view of Lemma 2 and since the quantity B
expected separation by finding a partition of X that minimizes the weighted variance ?(C) defined
Section 4.1. From the definition of ?, this problem resembles a traditional k-means clustering
problem with distance function d(xi , xi0 ) = (bi ?bi0 )2 . Thus, one could use one of several clustering
algorithms to solve it. Nevertheless, in order to allocate a new point x ? X to a cluster, we would
require access to the bid b which at evaluation time is unknown. Instead, we show how to utilize the
predictions of h to define an almost optimal clustering of X .
For any partition C = {C1 , . . . , Ck } of X define
k s
X
X
?h (C) =
j=1
(h(xi ) ? h(xi0 ))2 .
i,i0 :xi ,xi0 ?Ck
1
Notice that 2m
?h (C) is the function minimized by Algorithm 1. The following lemma, proved in
Appendix B, bounds the cluster variance achieved by clustering bids according to their predictions.
6
Pm
1
2
b2 , and let C ? denote the partition
Lemma 3. Let h be a function such that m
i=1 (h(xi ) ? bi ) ? ?
h
h
?
that minimizes ?(C). If C minimizes ?h (C) then ?(C ) ? ?(C ) + 4mb
?.
Pm
1
Corollary 3. Let rk be the output of Algorithm 1. If m j=1 (h(xi ) ? bi )2 ? ?b2 then:
b k ) ? (3B)
b 1/3
S(r
1
2/3
2/3
b 1/3 Big( 1 ?(C ? ) + 2b
?(C h )
? (3B)
?
.
2m
2m
(3)
Proof. It is easy to see that the elements Cjh of C h are of the form Cj = {x|tj ? h(x) ? tj+1 } for
t ? Tk . Thus if rk is the hypothesis induced by the partition C h , then rk ? G(h, k). The result now
follows by definition of ? and lemmas 2 and 3.
The proof of Theorem 2 is now straightforward. Define a partition C by xi ? Cj if bi ?
Since (bi ? bi0 )2 ? k12 for bi , bi0 ? Cj we have
s
k
X
m2j
m
= .
?(C) ?
2
k
k
j=1
j?1
k
, kj .
(4)
Furthermore since E[(h(x) ? b)2 ] ? ? 2 , Hoeffding?s inequality implies that with probability 1 ? ?:
r
m
1 X
log 1/?
2
2
(h(xi ) ? bi ) ? ? +
.
(5)
m i=1
2m
In view of inequalities (4) and (5) as well as Corollary 3 we have:
!
!
r
r
1/2 2/3
1/2 2/3
1
1
log
1/?
log
1/?
1/3
2
1/3
2
b k ) ? (3B)
b
b
S(r
?(C)+2 ? +
? (3B)
+2 ? +
2m
2m
2k
2m
This completes the proof of the main result. To implement the algorithm, note that the problem
of minimizing ?h (C) reduces to finding a partition t ? Tk such that the sum of the variances
within the partitions is minimized. It is clear that it suffices to consider points tj in the set B =
{h(x1 ), . . . , h(xm )}. With this observation, a simple dynamic program leads to a polynomial time
algorithm with an O(km2 ) running time (see Appendix C).
7
Experiments
We now compare the performance of our algorithm against the following baselines:
1. The offset algorithm presented in Section 3, where instead P
of using the theoretical offset
m
? 2/3 we find the optimal t maximizing the empirical revenue i=1 h(xi )?t)1h(xi )?t?bi .
2. The DC algorithm introduced by Mohri and Medina [2014], which represents the state of
the art in learning a revenue optimal reserve price.
Synthetic data. We begin by running experiments on synthetic data to demonstrate the regimes
where each algorithm excels. We generate feature vectors xi ? R10 with coordinates sampled from
a mixture of lognormal distributions with means ?1 = 0, ?2 = 1, variance ?1 = ?2 = 0.5 and
mixture parameter p = 0.5. Let 1 ? Rd denote the vector with entries set to 1. Bids are generated
according to two different scenarios:
Linear Bids bi generated according to bi = max(x>
i 1 + ?i , 0) where ?i is a Gaussian random
variable with mean 0, and standard deviation ? ? {0.01, 0.1, 1.0, 2.0, 4.0}.
Bimodal Bids bi generated according to the following rule: let si = max(x>
i 1 + ?i , 0) if si > 30
then bi = 40 + ?i otherwise bi = si . Here ?i has the same distribution as ?i .
The linear scenario demonstrates what happens when we have a good estimate of the bids. The
bimodal scenario models a buyer, which for the most part will bid as a continuous function of
features but that is interested in a particular set of objects (for instance retargeting buyers in online
advertisement) for which she is willing to pay a much higher price.
7
(a)
(b)
(c)
Figure 1: (a) Mean revenue of the three algorithms on the linear scenario. (b) Mean revenue of the
three algorithms on the bimodal scenario. (c) Mean revenue on auction data.
For each experiment we generated a training dataset Strain , a holdout set Sholdout and a test set
Stest each with 16,000 examples. The function h used by RIC-h and the offset algorithm is found
by training a linear regressor over Strain . For efficiency, we ran RIC-h algorithm on quantizations
of predictions h(xi ). Quantized predictions belong to one of 1000 buckets over the interval [0, 50].
Finally, the choice of hyperparameters ? for the Lipchitz loss and k for the clustering algorithm was
done by selecting the best performing parameter over the holdout set. Following the suggestions in
[Mohri and Medina, 2014] we chose ? ? {0.001, 0.01, 0.1, 1.0} and k ? {2, 4, . . . , 24}.
Figure 1(a),(b) shows the average revenue of the three approaches across 20 replicas of the experiment as a function of the log of ?. Revenue is normalized so that the DC algorithm revenue is 1.0
when ? = 0.01. The error bars at one standard deviation are indistinguishable in the plot. It is
not surprising to see that in the linear scenario, the DC algorithm of [Mohri and Medina, 2014] and
the offset algorithm outperform RIC-h under low noise conditions. Both algorithms will recover a
solution close to the true weight vector 1. In this case the offset is minimal, thus recovering virtually
all revenue. On the other hand, even if we set the optimal reserve price for every cluster, the inherent
variance of each cluster makes us leave some revenue on the table. Nevertheless, notice that as the
noise increases all three algorithms seem to achieve the same revenue. This is due to the fact that
the variance in each cluster is comparable with the error in the prediction function h.
The results are reversed for the bimodal scenario where RIC-h outperforms both algorithms under
low noise. This is due to the fact that RIC-h recovers virtually all revenue obtained from high bids
while the offset and DC algorithms must set conservative prices to avoid losing revenue from lower
bids.
Auction data. In practice, however, neither of the synthetic regimes is fully representative of the
bidding patterns. In order to fully evaluate RIC-h, we collected auction bid data from AdExchange
for 4 different publisher-advertiser pairs. For each pair we sampled 100,000 examples with a set of
discrete and continuous features. The final feature vectors are in Rd for d ? [100, 200] depending
on the publisher-buyer pair. For each experiment, we extract a random training sample of 20,0000
points as well as a holdout and test sample. We repeated this experiment 20 times and present
the results on Figure 1 (c) where we have normalized the data so that the performance of the DC
algorithm is always 1. The error bars represent one standard deviation from the mean revenue
lift. Notice that our proposed algorithm achieves on average up to 30% improvement over the DC
algorithm. Moreover, the simple offset strategy never outperforms the clustering algorithm, and in
some cases achieves significantly less revenue.
8
Conclusion
We provided a simple, scalable reduction of the problem of revenue optimization with side information to the well studied problem of minimizing the squared loss. Our reduction provides the first
polynomial time algoritm with a quantifiable bound on the achieved revenue. In the analysis of our
algorithm we also provided the first variance dependent lower bound on the revenue attained by setting optimal monopoly prices. Finally, we provided extensive empirical evidence of the advantages
of RIC-h over the current state of theart.
8
References
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9
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6,393 | 6,783 | Solving Most Systems of Random
Quadratic Equations
Gang Wang?,?
?
Georgios B. Giannakis?
Yousef Saad?
Jie Chen?
Key Lab of Intell. Contr. and Decision of Complex Syst., Beijing Inst. of Technology
?
Digital Tech. Center & Dept. of Electrical and Computer Eng., Univ. of Minnesota
?
Department of Computer Science and Engineering, Univ. of Minnesota
{gangwang, georgios, saad}@umn.edu; [email protected].
Abstract
This paper deals with finding an n-dimensional solution x to a system of quadratic
equations yi = |hai , xi|2 , 1 ? i ? m, which in general is known to be NP-hard.
We put forth a novel procedure, that starts with a weighted maximal correlation
initialization obtainable with a few power iterations, followed by successive refinements based on iteratively reweighted gradient-type iterations. The novel
techniques distinguish themselves from prior works by the inclusion of a fresh
(re)weighting regularization. For certain random measurement models, the proposed procedure returns the true solution x with high probability in time proportional to reading the data {(ai ; yi )}1?i?m , provided that the number m of
equations is some constant c > 0 times the number n of unknowns, that is, m ? cn.
Empirically, the upshots of this contribution are: i) perfect signal recovery in the
high-dimensional regime given only an information-theoretic limit number of equations; and, ii) (near-)optimal statistical accuracy in the presence of additive noise.
Extensive numerical tests using both synthetic data and real images corroborate
its improved signal recovery performance and computational efficiency relative to
state-of-the-art approaches.
1
Introduction
One is often faced with solving quadratic equations of the form yi = |hai , xi|2 , or equivalently,
?i = |hai , xi|,
1?i?m
(1)
where x ? Rn /Cn (hereafter, symbol ?A/B? denotes either A or B) is the wanted unknown n ? 1
vector, while given observations ?i and feature vectors ai ? Rn /Cn that are collectively stacked in
the data vector ? := [?i ]1?i?m and the m ? n sensing matrix A := [ai ]1?i?m , respectively. Put
differently, given information about the (squared) modulus of the inner products of the signal vector
x and several known design vectors ai , can one reconstruct exactly (up to a global phase factor) x,
or alternatively, the missing phase of hai , xi? In fact, much effort has been devoted to determining
the number of such equations necessary and/or sufficient for the uniqueness of the solution x; see
e.g., [1, 8]. It has been proved that m ? 2n ? 1 (m ? 4n ? 4) generic 1 (which includes the case of
random vectors) real (complex) vectors ai are sufficient for uniquely determining an n-dimensional
real (complex) vector x [1, Theorem 2.8], [8], while in the real case m = 2n ? 1 is shown also
necessary [1]. In this sense, the number m = 2n ? 1 of equations as in (1) can be regarded as the
information-theoretic limit for such a quadratic system to be uniquely solvable.
1
It is out of the scope of the present paper to explain the meaning of generic vectors, whereas interested
readers are referred to [1].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In diverse physical sciences and engineering fields, it is impossible or very difficult to record phase
measurements. The problem of recovering the signal or phase from magnitude measurements only,
also commonly known as phase retrieval, emerges naturally [10, 11]. Relevant application domains
include e.g., X-ray crystallography, astronomy, microscopy, ptychography, and coherent diffraction
imaging [21]. In such setups, optical measurement and detection systems record solely the photon
flux, which is proportional to the (squared) magnitude of the field, but not the phase. Problem (1) in
its squared form, on the other hand, can be readily recast as an instance of nonconvex quadratically
constrained quadratic programming, that subsumes as special cases several well-known combinatorial
optimization problems involving Boolean variables, e.g., the NP-complete stone problem [2, Sec.
3.4.1]. A related task of this kind is that of estimating the mixture of linear regressions, where the
latent membership indicators can be converted into the missing phases [29]. Although of simple form
and practical relevance across different fields, solving systems of nonlinear equations is arguably the
most difficult problem in all of the numerical computations [19, Page 355].
Notation: Lower- (upper-) case boldface letters denote vectors (matrices), e.g., a ? Rn (A ? Rm?n ).
Calligraphic letters are reserved for sets. The floor operation bcc gives the largest integer no greater
than the given real quantity c > 0, the cardinality |S| counts the number of elements in set S, and
kxk denotes the Euclidean norm of x. Since for any phase ? ? R, vectors x ? Cn and ej? x are
indistinguishable given {?i } in (1), let dist(z, x) := min??[0,2?) kz ? xej? k be the Euclidean
distance of any estimate z ? Cn to the solution set {ej? x}0??<2? of (1); in particular, ? = 0/? in
the real case.
1.1
Prior contributions
Following the least-squares (LS) criterion (which coincides with the maximum likelihood (ML) one
assuming additive white Gaussian noise), the problem of solving quadratic equations can be naturally
recast as an empirical loss minimization
m
1 X
minimize
L(z)
:=
`(z; ?i /yi )
(2)
m i=1
z?Rn /Cn
2
where one can choose to work with the amplitude-based loss `(z; ?i ) := (?i ?|hai ,zi|) /2 [28, 30], or
2 2
the intensity-based one `(z; yi ) := (yi ?|hai ,zi| ) /2 [3], and its related Poisson likelihood `(z; yi ) :=
2
2
yi log(|hai , zi| ) ? |hai , zi| [7]. Either way, the objective functional L(z) is nonconvex; hence, it
is generally NP-hard and computationally intractable to compute the ML or LS estimate.
Minimizing the squared modulus-based LS loss in (2), several numerical polynomial-time algorithms
have been devised via convex programming for certain choices of design vectors ai [4, 25]. Such
convex paradigms first rely on the matrix-lifting technique to express all squared modulus terms into
linear ones in a new rank-1 matrix variable, followed by solving a convex semi-definite program (SDP)
after dropping the rank constraint. It has been established that perfect recovery and (near-)optimal
statistical accuracy are achieved in noiseless and noisy settings respectively with an optimal-order
number of measurements [4]. In terms of computational efficiency however, such lifting-based
convex approaches entail storing and solving for an n ? n semi-definite matrix from m general SDP
constraints, whose computational complexity in the worst case scales as n4.5 log 1/ for m ? n [25],
which is not scalable. Another recent line of convex relaxation [12], [13] reformulated the problem of
phase retrieval as that of sparse signal recovery, and solved a linear program in the natural parameter
vector domain. Although exact signal recovery can be established assuming an accurate enough
anchor vector, its empirical performance is in general not competitive with state-of-the-art phase
retrieval approaches.
Recent proposals advocate suitably initialized iterative procedures for coping with certain nonconvex formulations directly; see e.g., algorithms abbreviated as AltMinPhase, (R/P)WF, (M)TWF,
(S)TAF [16, 3, 7, 26, 28, 27, 30, 22, 6, 24], as well as a prox-linear algorithm [9]. These nonconvex
approaches operate directly upon vector optimization variables, thus leading to significant computational advantages over their convex counterparts. With random features, they can be interpreted as
performing stochastic optimization over acquired examples {(ai ; ?i /yi )}1?i?m to approximately
minimize the population risk functional L(z) := E(ai ,?i /yi ) [`(z; ?i /yi )]. It is well documented
that minimizing nonconvex functionals is generally intractable due to existence of multiple critical
points [17]. Assuming Gaussian sensing vectors however, such nonconvex paradigms can provably
locate the global optimum, several of which also achieve optimal (statistical) guarantees. Specifically,
2
starting with a judiciously designed initial guess, successive improvement is effected by means of a
sequence of (truncated) (generalized) gradient-type iterations given by
?t X
z t+1 := z t ?
?`i (z t ; ?i /yi ), t = 0, 1, . . .
(3)
m
t+1
i?T
t
where z denotes the estimate returned by the algorithm at the t-th iteration, ?t > 0 is learning rate
that can be pre-selected or found via e.g., the backtracking line search strategy, and ?`(z t , ?i /yi )
represents the (generalized) gradient of the modulus- or squared modulus-based LS loss evaluated at
z t . Here, T t+1 denotes some time-varying index set signifying the per-iteration gradient truncation.
Although they achieve optimal statistical guarantees in both noiseless and noisy settings, state-of-theart (convex and nonconvex) approaches studied under Gaussian designs, empirically require stable
recovery of a number of equations (several) times larger than the information-theoretic limit [7, 3, 30].
As a matter of fact, when there are numerously enough measurements (on the order of n up to some
polylog factors), the squared modulus-based LS functional admits benign geometric structure in
the sense that [23]: i) all local minimizers are also global; and, ii) there always exists a negative
directional curvature at every saddle point. In a nutshell, the grand challenge of tackling systems of
random quadratic equations remains to develop algorithms capable of achieving perfect recovery and
statistical accuracy when the number of measurements approaches the information limit.
1.2
This work
Building upon but going beyond the scope of the aforementioned nonconvex paradigms, the present
paper puts forward a novel iterative linear-time scheme, namely, time proportional to that required
by the processor to scan all the data {(ai ; ?i )}1?i?m , that we term reweighted amplitude flow, and
henceforth, abbreviate as RAF. Our methodology is capable of solving noiseless random quadratic
equations exactly, yielding an estimate of (near)-optimal statistical accuracy from noisy modulus
observations. Exactness and accuracy hold with high probability and without extra assumption on
the unknown signal vector x, provided that the ratio m/n of the number of equations to that of the
unknowns is larger than a certain constant. Empirically, our approach is shown able to ensure exact
recovery of high-dimensional unstructured signals given a minimal number of equations, where m/n
in the real case can be as small as 2. The new twist here is to leverage judiciously designed yet
conceptually simple (re)weighting regularization techniques to enhance existing initializations and
also gradient refinements. An informal depiction of our RAF methodology is given in two stages as
follows, with rigorous details deferred to Section 3:
S1) Weighted maximal correlation initialization: Obtain an initializer z 0 maximally correlated
with a carefully selected subset S ( M := {1, 2, . . . , m} of feature vectors ai , whose contributions
toward constructing z 0 are judiciously weighted by suitable parameters {wi0 > 0}i?S .
S2) Iteratively reweighted ?gradient-like? iterations: Loop over 0 ? t ? T :
z t+1 = z t ?
m
?t X t
w ?`(z t ; ?i )
m i=1 i
(4)
for some time-varying weighting parameters {wit ? 0}, each possibly relying on the current iterate
zt and the datum (ai ; ?i ).
Two attributes of the novel approach are worth highlighting next. First, albeit being a variant of
the spectral initialization devised in [28], the initialization here [cf. S1)] is distinct in the sense that
different importance is attached to each selected datum (ai ; ?i ). Likewise, the gradient flow [cf.
S2)] weighs judiciously the search direction suggested by each datum (ai ; ?i ). In this manner, more
robust initializations and more stable overall search directions can be constructed even based solely
on a rather limited number of data samples. Moreover, with particular choices of the weights wit ?s
(e.g., taking 0/1 values), the developed methodology subsumes as special cases the recently proposed
algorithms RWF [30] and TAF [28].
2
Algorithm: Reweighted Amplitude Flow
This section explains the intuition and basic principles behind each stage of the advocated RAF
algorithm in detail. For analytical concreteness, we focus on the real Gaussian model with x ? Rn ,
3
and independent sensing vectors ai ? Rn ? N (0, I) for all 1 ? i ? m. Nonetheless, the presented
approach can be directly applied when the complex Gaussian and the coded diffraction pattern (CDP)
models are considered.
2.1
Weighted maximal correlation initialization
A key enabler of general nonconvex iterative heuristics? success in finding the global optimum is
to seed them with an excellent starting point [14]. Indeed, several smart initialization strategies
have been advocated for iterative phase retrieval algorithms; see e.g., the spectral initialization [16],
[3] as well as its truncated variants [7], [28], [9], [30], [15]. One promising approach is the one
pursued in [28], which is also shown robust to outliers in [9]. To hopefully approach the informationtheoretic limit however, its performance may need further enhancement. Intuitively, it is increasingly
challenging to improve the initialization (over state-of-the-art) as the number of acquired data samples
approaches the information-theoretic limit.
In this context, we develop a more flexible initialization scheme based on the correlation property
(as opposed to the orthogonality in [28]), in which the added benefit is the inclusion of a flexible
weighting regularization technique to better balance the useful information exploited in the selected
data. Similar to related approaches of the same kind, our strategy entails estimating both the norm
kxk and the direction x/kxk of x. Leveraging the strong law of large numbers and the rotational
invariance of Gaussian ai vectors (the latter suffices to assume x = kxke1 , with e1 being the first
canonical vector in Rn ), it is clear that
m
m
m
2 1 X
1 X 2
1 X
?i =
hai , kxke1 i =
a2i,1 kxk2 ? kxk2
(5)
m i=1
m i=1
m i=1
Pm 2
?
whereby kxk can be estimated to be P
i=1 i /m. This estimate proves very accurate even with a
2
m
limited number of data samples because i=1 ai,1/m is unbiased and tightly concentrated.
The challenge thus lies in accurately estimating the direction of x, or seeking a unit vector maximally
aligned with x. Toward this end, let us first present a variant of the initialization in [28]. Note that
the larger the modulus ?i of the inner-product between ai and x is, the known design vector ai is
deemed more correlated to the unknown solution x, hence bearing useful directional information
of x. Inspired by this fact and having available data {(ai ; ?i )}1?i?m , one can sort all (absolute)
correlation coefficients {?i }1?i?m in an ascending order, yielding ordered coefficients 0 < ?[m] ?
? ? ? ? ?[2] ? ?[1] . Sorting m records takes time proportional to O(m log m).2 Let S $ M denote the
set of selected feature vectors ai to be used for computing the initialization, which is to be designed
next. Fix a priori the cardinality |S| to some integer on the order of m, say, |S| := b3m/13c. It is
then natural to define S to collect the ai vectors that correspond to one of the largest |S| correlation
coefficients {?[i] }1?i?|S| , each of which can be thought of as pointing to (roughly) the direction of x.
Approximating the direction of x therefore boils down to finding a vector to maximize its correlation
with the subset S of selected directional vectors ai . Succinctly, the wanted approximation vector can
be efficiently found as the solution of
1 X
2
1 X
maximize
hai , zi = z ?
ai a?i z
(6)
|S|
|S|
kzk=1
i?S
i?S
?
where the superscript represents the transpose or the conjugate transpose that will be P
clear from
2
m
the context. Upon scaling the unity-norm solution of (6) by the norm estimate obtained i=1 ?i /m
in (5), to match the magnitude of x, we will develop what we will henceforth refer to as maximal
correlation initialization.
As long as |S| is chosen on the order of m, the maximal correlation method outperforms the spectral
ones in [3, 16, 7], and has comparable performance to the orthogonality-promoting method [28].
Its performance around the information-limit however, is still not the best that we can hope for.
Recall from (6) that all selected directional vectors {ai }i?S are treated the same in terms of their
contributions to constructing the initialization. Nevertheless, according to our starting principle, this
ordering information carried by the selected ai vectors is not exploited by the initialization scheme
in (6) and [28]. In other words, if for i, j ? S, the correlation coefficient of ?i with ai is larger
2
f (m) = O(g(m)) means that there exists a constant C > 0 such that |f (m)| ? C|g(m)|.
4
than that of ?j with aj , then ai is deemed more correlated (with x) than aj is, hence bearing more
useful information about the direction of x. It is thus prudent to weigh more the selected ai vectors
associated with larger ?i values. Given the ordering information ?[|S|] ? ? ? ? ? ?[2] ? ?[1] available
from the sorting procedure, a natural way to achieve this goal is weighting each ai vector with simple
monotonically increasing functions of ?i , say e.g., taking the weights wi0 := ?i? , ?i ? S with the
0
0
0
exponent parameter ? ? 0 chosen to maintain the wanted ordering w|S|
? ? ? ? ? w[2]
? w[1]
. In a
nutshell, a more flexible initialization strategy, that we refer to as weighted maximal correlation, can
be summarized as follows
1 X
?0 := arg max z ?
z
?i? ai a?i z.
(7)
|S|
kzk=1
i?S
For any given > 0, the power method or the Lanczos algorithm can be called for to find an -accurate
solution to (7) in time proportional to O(n|S|) [20], assuming
P a positive eigengap between the largest
and the second largest eigenvalues of the matrix (1/|S|) i?S ?i? ai a?i , which is often true when
{ai } are sampled from continuous distribution. The proposed P
initialization can be obtained upon
?i2/m)?
?0 from (7) by the norm estimate in (5), to yield z0 := ( m
scaling z
z0 . By default, we take
i=1
p
0
1
? := /2 in all reported numerical implementations, yielding wi := |hai , xi| for all i ? S.
Regarding the initialization procedure in (7), we next highlight two features, whereas technical details
and theoretical performance guarantees are provided in Section 3:
F1) The weights {wi0 } in the maximal correlation scheme enable leveraging useful information that
each feature vector ai may bear regarding the direction of x.
F2) Taking wi0 := ?i? for all i ? S and 0 otherwise, problem (7) can be equivalently rewritten as
m
1 X
?0 := arg max z ?
z
wi0 ai a?i z
(8)
m i=1
kzk=1
which subsumes previous initialization schemes with particular selections of weights {wi0 }. For
instance, the spectral initialization in [16, 3] is recovered by choosing S := M, and wi0 := ?i2 for all
1 ? i ? m.
For comparison, define
dist(z, x)
1.5
Reweight. max. correlation
.
Spectral initialization
kxk
Trunc. spectral in TWF
1.4
Orthogonality promoting
Throughout the paper, all simulated results were
Trunc. spectral in RWF
averaged over 100 Monte Carlo (MC) realizations,
1.3
and each simulated scheme was implemented with
their pertinent default parameters. Figure 1 eval1.2
uates the performance of the developed initializa1.1
tion relative to several state-of-the-art strategies,
and also with the information limit number of
1
data benchmarking the minimal number of samples required. It is clear that our initialization
0.9
is: i) consistently better than the state-of-the-art;
1,000
2,000
3,000
4,000
5,000
n: signal dimension (m=2n-1)
and, ii) stable as n grows, which is in contrast to
the instability encountered by the spectral ones
[16, 3, 7, 30]. It is worth stressing that the more Figure 1: Relative initialization error for i.i.d.
than 5% empirical advantage (relative to the best) ai ? N (0, I1,000 ), 1 ? i ? 1, 999.
at the challenging information-theoretic benchmark is nontrivial, and is one of the main RAF upshots. This advantage becomes increasingly
pronounced as the ratio m/n grows.
Relative error
Relative error :=
2.2
Iteratively reweighted gradient flow
For independent data obeying the real Gaussian model, the direction that TAF moves along in stage
S2) presented earlier is given by the following (generalized) gradient [28]:
1 X
1 X ?
a? z
?`(z; ?i ) =
ai z ? ?i i?
ai
(9)
m
m
|ai z|
i?T
i?T
5
where the dependence on the iterate count t is neglected for notational brevity, and the convention
?
?
a?
i z/|ai z| := 0 is adopted when a z = 0.
i
Unfortunately, the (negative) gradient of the average in (9) generally may not point towards the true
solution x unless the current iterate z is already very close to x. Therefore, moving along such a
descent direction may not drag z closer to x. To see this, consider an initial guess z0 that has already
been in a basin of attraction (i.e., a region within which there is only a unique stationary point) of
a? z
x. Certainly, there are summands (a?i z ? ?i |a?i z| )ai in (9), that could give rise to ?bad/misleading?
i
a? z
a? x
gradient directions due to the erroneously estimated signs |a?i z| 6= |ai? x| [28], or (a?i z)(a?i x) < 0
i
i
[30]. Those gradients as a whole may drag z away from x, and hence out of the basin of attraction.
Such an effect becomes increasingly severe as m approaches the information-theoretic limit of
2n ? 1, thus rendering past approaches less effective in this case. Although this issue is somewhat
remedied by TAF with a truncation procedure, its efficacy is limited due to misses of bad gradients
and mis-rejections of meaningful ones around the information limit.
To address this challenge, reweighted amplitude flow effecting suitable gradient directions from all
data samples {(ai ; ?i )}1?i?m will be adopted in a (timely) adaptive fashion, namely introducing
appropriate weights for all gradients to yield the update
z t+1 = z t ? ?t ?`rw (z t ; ?i ), t = 0, 1, . . .
(10)
t
t
The reweighted gradient ?`rw (z ) evaluated at the current point z is given as
m
1 X
?`rw (z) :=
wi ?`(z; ?i )
(11)
m i=1
for suitable weights {wi }1?i?m to be designed next.
To that end, we observe that the truncation criterion [28]
|a? z|
(12)
T := 1 ? i ? m : ?i ? ?
|ai x|
with some given parameter ? > 0 suggests to include only gradient components associated with |a?i z|
?
of relatively large sizes. This is because gradients of sizable |ai z|/|a?i x| offer reliable and meaningful
?
directions pointing to the truth x with large probability [28]. As such, the ratio |ai z|/|a?i x| can be
somewhat viewed as a confidence score about the reliability or meaningfulness of the corresponding
gradient ?`(z; ?i ). Recognizing that confidence can vary, it is natural to distinguish the contributions
that different gradients make to the overall search direction. An easy way is to attach large weights
to the reliable gradients, and small weights to the spurious ones. Assume without loss of generality
that 0 ? wi ? 1 for all 1 ? i ? m; otherwise, lump the normalization factor achieving this into the
learning rate ?t . Building upon this observation and leveraging the gradient reliability confidence
?
score |ai z|/|a?i x|, the weight per gradient ?`(z; ?i ) in RAF is designed to be
1
wi :=
, 1?i?m
(13)
1 + ?i/(|a?i z|/|a?i x|)
in which {?i > 0}1?i?m are some pre-selected parameters.
Regarding the proposed weighting criterion in (13), three remarks are in order, followed by the RAF
algorithm summarized in Algorithm 1.
R1) The weights {wit }1?i?m are time adapted to z t . One can also interpret the reweighted gradient
t+1
flow
in (10) as performing a single gradient step to minimize the smooth reweighted loss
Pz
m
1
t
t
i=1 wi `(z; ?i ) with starting point z ; see also [4] for related ideas successfully exploited in the
m
iteratively reweighted least-squares approach to compressive sampling.
?
R2) Note that the larger |ai z|/|a?i x| is, the larger wi will be. More importance will be attached to
reliable gradients than to spurious ones. Gradients from almost all data points are are judiciously
accounted for, which is in sharp contrast to [28], where withdrawn gradients do not contribute the
information they carry.
R3) At the points {z} where a?i z = 0 for certain i ? M, the corresponding weight will be wi = 0.
That is, the losses `(z; ?i ) in (2) that are nonsmooth at points z will be eliminated, to prevent their
contribution to the reweighted gradient update in (10). Hence, the convergence analysis of RAF can
be considerably simplified because it does not have to cope with the nonsmoothness of the objective
function in (2).
6
2.3
Algorithmic parameters
To optimize the empirical performance and facilitate numerical implementations, choice of pertinent
algorithmic parameters of RAF is independently discussed here. It is obvious that the RAF algorithm
entails four parameters. Our theory and all experiments are based on: i) |S|/m ? 0.25; ii) 0 ? ?i ? 10
for all 1 ? i ? m; and, iii) 0 ? ? ? 1. For convenience, a constant step size ?t ? ? > 0 is suggested,
but other step size rules such as backtracking line search with the reweighted objective work as well.
As will be formalized in Section 3, RAF converges if the constant ? is not too large, with the upper
bound depending in part on the selection of {?i }1?i?m .
In the numerical tests presented in Sections 2 and 4, we take
|S| := b3m/13c,
?i ? ? := 10,
? := 0.5,
and ? := 2
(14)
while larger step sizes ? > 0 can be afforded for larger m/n values.
Algorithm 1 Reweighted Amplitude Flow
1: Input: Data {(ai ; ?i }1?i?m ; maximum number of iterations T ; step size ?t = 2/6 and
weighting parameter ?i = 10/5 for real/complex Gaussian model; |S| = b3m/13c, and ? = 0.5.
2: Construct S to include indices associated with the |S| largest entries among {?i }1?i?m .
pPm 2
? 0 with z
? 0 being the unit principal eigenvector of
3: Initialize z 0 :=
i=1 ?i /m z
m
Y :=
1 X 0
w ai a?i
m i=1 i
?i? , i ? S ? M
for all 1 ? i ? m.
0, otherwise
4: Loop: for t = 0 to T ? 1
m
?t X t ? t
a? z t
t+1
t
z
=z ?
wi ai z ? ?i i? t ai
m i=1
|ai z |
where
wi0
(15)
:=
where wit :=
t
|a?
i z |/?i
t
|a?
i z |/?i +?i
(16)
for all 1 ? i ? m.
T
5: Output: z .
3
Main results
Our main results summarized in Theorem 1 next establish exact recovery under the real Gaussian
model, whose proof is provided in the supplementary material. Our RAF approach however can be
generalized readily to the complex Gaussian and CDP models.
Theorem 1 (Exact recovery) Consider m noiseless measurements ? = |Ax| for an arbitrary
x ? Rn . If the data size m ? c0 |S| ? c1 n and the step size ? ? ?0 , then with probability at least
1 ? c3 e?c2 m , the reweighted amplitude flow?s estimates z t in Algorithm 1 obey
1
(1 ? ?)t kxk, t = 0, 1, . . .
(17)
10
where c0 , c1 , c2 , c3 > 0, 0 < ? < 1, and ?0 > 0 are certain numerical constants depending on the
choice of algorithmic parameters |S|, ?, ?, and ?.
dist(z t , x) ?
According to Theorem 1, a few interesting properties of our RAF algorithm are worth highlighting.
To start, RAF recovers the true solution exactly with high probability whenever the ratio m/n of
the number of equations to the unknowns exceeds some numerical constant. Expressed differently,
RAF achieves the information-theoretic optimal order of sample complexity, which is consistent
with the state-of-the-art including TWF [7], TAF [28], and RWF [30]. Notice that (17) also holds at
t = 0, namely, dist(z 0 , x) ? kxk/10, therefore providing performance guarantees for the proposed
initialization scheme (cf. Step 3 in Algorithm 1). Moreover, starting from this initial estimate, RAF
converges linearly to the true solution x. That is, to reach any -relative solution accuracy (i.e.,
dist(z T , x) ? kxk), it suffices to run at most T = O(log 1/) RAF iterations (cf. Step 4). This in
7
conjunction with the per-iteration complexity O(mn) confirms that RAF solves exactly a quadratic
system in time O(mn log 1/), which is linear in O(mn), the time required to read the entire data
{(ai ; ?i )}1?i?m . Given the fact that the initialization stage can be performed in time O(n|S|) and
|S| < m, the overall linear-time complexity of RAF is order-optimal.
Proof of Theorem 1 is provided in the supplementary material.
Simulated tests
Our theoretical findings about RAF have been corroborated with comprehensive numerical tests, a
sample of which are discussed next. Performance
of RAF is evaluated relative to the state-of-the-art
(T)WF, RWF, and TAF in terms of the empirical
success rate among 100 MC trials, where a success
will be declared for a trial if the returned estimate
incurs error
? ? |Az T |
? 10?5
kxk
30
25
! log10 L(zT )
4
20
15
10
5
where the modulus operator | ? | is understood
element-wise. The real Gaussian model and the
0
0
10
20
30
40
50
60
70
80
90
100
physically realizable CDPs were simulated in this
section. For fairness, all schemes were impleRealization number
mented with their suggested parameter values. The
true signal vector x was randomly generated using
T
x ? N (0, I), and the i.i.d. sensing vectors ai Figure 2: Function value L(z ) by RAF for
ai ? N (0, I). Each scheme obtained the initial 100 MC realizations when m = 2n ? 1.
guess based on 200 power iterations, followed by
a series of T = 2, 000 (truncated/reweighted) gradient iterations. All experiments were performed
using MATLAB on an Intel CPU @ 3.4 GHz (32 GB RAM) computer. For reproducibility, the Matlab
code of the RAF algorithm is publicly available at https://gangwg.github.io/RAF/.
To demonstrate the power of RAF in the high-dimensional regime, the function value L(z) in (2)
evaluated at the returned estimate z T for 100 independent trials is plotted (in negative logarithmic
scale) in Fig. 2, where m = 2n ? 1 = 9, 999. It is self-evident that RAF succeeded in all trials even
at this challenging information limit. To the best of our knowledge, RAF is the first algorithm that
empirically recovers any solution exactly from a minimal number of random quadratic equations. Left
panel in Fig. 3 further compares the empirical success rate of five schemes under the real Gaussian
model with n = 1, 000 and m/n varying by 0.1 from 1 to 5. Evidently, the developed RAF achieves
perfect recovery as soon as m is about 2n, where its competing alternatives do not work well. To
demonstrate the stability and robustness of RAF in the presence of additive noise, the right panel in
Fig. 3 depicts the normalized mean-square error
NMSE :=
dist2 (z T , x)
kxk2
as a function of the signal-to-noise ratio (SNR) for m/n taking values {3, 4, 5}. The noise model
?i = |hai , xi| + ?i ,
1?i?m
2
with ? := [?i ]1?i?m ? N (0, ? Im ) was employed, where ? 2 was set such that certain SNR :=
2
10 log10 (kAxk /m?2 ) values on the x-axis were achieved.
To examine the efficacy and scalability of RAF in real-world conditions, the last experiment entails
the Galaxy image 3 depicted by a three-way array X ? R1,080?1,920?3 , whose first two coordinates
encode the pixel locations, and the third the RGB color bands. Consider the physically realizable
CDP model with random masks [3]. Letting x ? Rn (n ? 2 ? 106 ) be a vectorization of a certain
band of X, the CDP model with K masks is
? (k) = |F D (k) x|,
3
1 ? k ? K,
Downloaded from http://pics-about-space.com/milky-way-galaxy.
8
10 0
m=3n
m=4n
m=5n
1
0.8
10 -2
NMSE
Empirical success rate
10 -1
0.6
10 -3
0.4
10 -4
RAF
TAF
TWF
RWF
WF
0.2
0
1
2
3
4
10 -5
10 -6
5
5
1,000
10
15
20
25
30
35
40
45
SNR (dB) for x2 R1,000
m/n for x2 R
Figure 3: Real Gaussian model: Empirical success rate (Left); and, Relative MSE vs. SNR (Right).
(k)
where F ? Cn?n is a DFT matrix, and diagonal matrices
their diagonal entries sampled
? D have (k)
uniformly at random from {1, ?1, j, ?j} with j := ?1. Each D represents a random mask
placed after the object to modulate the illumination patterns [5]. Implementing K = 4 masks, each
algorithm performs independently over each band 100 power iterations for an initial guess, which
was refined by 100 gradient iterations. Recovered images of TAF (left) and RAF (right) are displayed
in Fig. 4, whose relative errors were 1.0347 and 1.0715 ? 10?3 , respectively. WF and TWF returned
images of corresponding relative error 1.6870 and 1.4211, which are far away from the ground truth.
Figure 4: Recovered Galaxy images after 100 gradient iterations of TAF (Left); and of RAF (Right).
5
Conclusion
This paper developed a linear-time algorithm called RAF for solving systems of random quadratic
equations. Our procedure consists of two stages: a weighted maximal correlation initializer attainable
with a few power or Lanczos iterations, and a sequence of scalable reweighted gradient refinements
for a nonconvex nonsmooth LS loss function. It was demonstrated that RAF achieves the optimal
sample and computational complexity. Judicious numerical tests showcase its superior performance
over state-of-the-art alternatives. Empirically, RAF solves a set of random quadratic equations
with high probability so long as a unique solution exists. Promising extensions include studying
robust and/or sparse phase retrieval and matrix recovery via (stochastic) reweighted amplitude flow
counterparts, and in particular exploiting the power of (re)weighting regularization techniques to
enable more general nonconvex optimization such as training deep neural networks [18].
Acknowledgments
G. Wang and G. B. Giannakis were partially supported by NSF grants 1500713 and 1514056. Y.
Saad was partially supported by NSF grant 1505970. J. Chen was partially supported by the National
Natural Science Foundation of China grants U1509215, 61621063, and the Program for Changjiang
Scholars and Innovative Research Team in University (IRT1208).
9
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11
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6,394 | 6,784 | Unsupervised Learning of Disentangled and
Interpretable Representations from Sequential Data
Wei-Ning Hsu, Yu Zhang, and James Glass
Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
{wnhsu,yzhang87,glass}@csail.mit.edu
Abstract
We present a factorized hierarchical variational autoencoder, which learns disentangled and interpretable representations from sequential data without supervision.
Specifically, we exploit the multi-scale nature of information in sequential data by
formulating it explicitly within a factorized hierarchical graphical model that imposes sequence-dependent priors and sequence-independent priors to different sets
of latent variables. The model is evaluated on two speech corpora to demonstrate,
qualitatively, its ability to transform speakers or linguistic content by manipulating
different sets of latent variables; and quantitatively, its ability to outperform an
i-vector baseline for speaker verification and reduce the word error rate by as much
as 35% in mismatched train/test scenarios for automatic speech recognition tasks.
1
Introduction
Unsupervised learning is a powerful methodology that can leverage vast quantities of unannotated
data in order to learn useful representations that can be incorporated into subsequent applications in
either supervised or unsupervised fashions. One of the principle approaches to unsupervised learning
is probabilistic generative modeling. Recently, there has been significant interest in three classes of
deep probabilistic generative models: 1) Variational Autoencoders (VAEs) [23, 34, 22], 2) Generative
Adversarial Networks (GANs) [11], and 3) auto-regressive models [30, 39]; more recently, there are
also studies combining multiple classes of models [6, 27, 26]. While GANs bypass any inference of
latent variables, and auto-regressive models abstain from using latent variables, VAEs jointly learn an
inference model and a generative model, allowing them to infer latent variables from observed data.
Despite successes with VAEs, understanding the underlying factors that latent variables associate
with is a major challenge. Some research focuses on the supervised or semi-supervised setting using
VAEs [21, 17]. There is also research attempting to develop weakly supervised or unsupervised
methods to learn disentangled representations, such as DC-IGN [25], InfoGAN [1], and ?-VAE [13].
There is yet another line of research analyzing the latent variables with labeled data after the model
is trained [33, 15]. While there has been much research investigating static data, such as the
aforementioned ones, there is relatively little research on learning from sequential data [8, 3, 2, 9, 7,
18, 36]. Moreover, to the best of our knowledge, there has not been any attempt to learn disentangled
and interpretable representations without supervision from sequential data. The information encoded
in sequential data, such as speech, video, and text, is naturally multi-scaled; in speech for example,
information about the channel, speaker, and linguistic content is encoded in the statistics at the
session, utterance, and segment levels, respectively. By leveraging this source of constraint, we can
learn disentangled and interpretable factors in an unsupervised manner.
In this paper, we propose a novel factorized hierarchical variational autoencoder, which learns
disentangled and interpretable latent representations from sequential data without supervision by
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: FHVAE (? = 0) decoding results of three combinations of latent segment variables z1 and
latent sequence variables z2 from two utterances in Aurora-4: a clean one (top-left) and a noisy one
(bottom-left). FHVAEs learn to encode local attributes, such as linguistic content, into z1 , and encode
global attributes, such as noise level, into z2 . Therefore, by replacing z2 of a noisy utterance with
z2 of a clean utterance, an FHVAE decodes a denoised utterance (middle-right) that preserves the
linguistic content. Reconstruction results of the clean and noisy utterances are also shown on the
right. Audio samples are available at https://youtu.be/naJZITvCfI4.
explicitly modeling the multi-scaled information with a factorized hierarchical graphical model. The
inference model is designed such that the model can be optimized at the segment level, instead
of at the sequence level, which may cause scalability issues when sequences become too long. A
sequence-to-sequence neural network architecture is applied to better capture temporal relationships.
We evaluate the proposed model on two speech datasets. Qualitatively, the model demonstrates an
ability to factorize sequence-level and segment-level attributes into different sets of latent variables.
Quantitatively, the model achieves 2.38% and 1.34% equal error rate on unsupervised and supervised
speaker verification tasks respectively, which outperforms an i-vector baseline. On speech recognition
tasks, it reduces the word error rate in mismatched train/test scenarios by up to 35%.
The rest of the paper is organized as follows. In Section 2, we introduce our proposed model, and
describe the neural network architecture in Section 3. Experimental results are reported in Section 4.
We discuss related work in Section 5, and conclude our work as well as discuss future research plans
in Section 6. We have released the code for the model described in this paper.1
2
Factorized Hierarchical Variational Autoencoder
Generation of sequential data, such as speech, often involves multiple independent factors operating
at different time scales. For instance, the speaker identity affects fundamental frequency (F0) and
volume at the sequence level, while phonetic content only affects spectral contour and durations of
formants at the segmental level. This multi-scale behavior results in the fact that some attributes,
such as F0 and volume, tend to have a smaller amount of variation within an utterance, compared to
between utterances; while other attributes, such as phonetic content, tend to have a similar amount of
variation within and between utterances.
We refer to the first type of attributes as sequence-level attributes, and the other as segment-level
attributes. In this work, we achieve disentanglement and interpretability by encoding the two types of
attributes into latent sequence variables and latent segment variables respectively, where the former
is regularized by an sequence-dependent prior and the latter by an sequence-independent prior.
We now formulate a generative process for speech and propose our Factorized Hierarchical Variational
Autoencoder (FHVAE). Consider some dataset D = {X (i) }M
i=1 consisting of M i.i.d. sequences,
(i)
(i)
where X (i) = {x(i,n) }N
is
a
sequence
of
N
observed
variables. N (i) is referred to as the
n=1
1
https://github.com/wnhsu/FactorizedHierarchicalVAE
2
?
(i,n)
z1
(i)
(i,n)
x(i,n)
(i,n)
?2
z2
N (i)
?2
z2
x(i,n)
N (i)
M
(a) Generative Model
(i)
(i,n)
z1
M
(b) Inference Model
Figure 2: Graphical illustration of the proposed generative model and inference model. Grey nodes
denote the observed variables, and white nodes are the hidden variables.
number of segments for the i-th sequence, and x(i,n) is referred to as the n-th segment of the i-th
sequence. Note that here a ?segment? refers to a variable of smaller temporal scale compared to the
?sequence?, which is in fact a sub-sequence. We will drop the index i whenever it is clear that we are
referring to terms associated with a single sequence. We assume that each sequence X is generated
(n)
(n) N
from some random process involving the latent variables Z1 = {z1 }N
n=1 , Z2 = {z2 }n=1 , and
?2 . The following generation process as illustrated in Figure 2(a) is considered: (1) a s-vector
(n)
?2 is drawn from a prior distribution p? (?2 ); (2) N i.i.d. latent sequence variables {z2 }N
n=1
(n) N
and latent segment variables {z1 }n=1 are drawn from a sequence-dependent prior distribution
p? (z2 |?2 ) and a sequence-independent prior distribution p? (z1 ) respectively; (3) N i.i.d. observed
variables {x(n) }N
n=1 are drawn from a conditional distribution p? (x|z1 , z2 ). The joint probability
for a sequence is formulated in Eq. 1:
p? (X, Z1 , Z2 , ?2 ) = p? (?2 )
N
Y
(n)
(n)
(n)
(n)
p? (x(n) |z1 , z2 )p? (z1 )p? (z2 |?2 ).
(1)
n=1
Specifically, we formulate each of the RHS term as follows:
p? (x|z1 , z2 ) = N (x|f?x (z1 , z2 ), diag(f?x2 (z1 , z2 )))
p? (z1 ) = N (z1 |0, ?z21 I),
p? (z2 |?2 ) = N (z2 |?2 , ?z22 I),
2
p? (?2 ) = N (?2 |0, ??
I),
2
where the priors over the s-vectors ?2 and the latent segment variables z1 are centered isotropic
multivariate Gaussian distributions. The prior over the latent sequence variable z2 conditioned on ?2
is an isotropic multivariate Gaussian centered at ?2 . The conditional distribution of the observed
variable x is the multivariate Gaussian with a diagonal covariance matrix, whose mean and diagonal
variance are parameterized by neural networks f?x (?, ?) and f?x2 (?, ?) with input z1 and z2 . We use ?
to denote the set of parameters in the generative model.
This generative model is factorized in a way such that the latent sequence variables z2 within a
sequence are forced to be close to ?2 as well as to each other in Euclidean distance, and therefore are
encouraged to encode sequence-level attributes that may have larger variance across sequences, but
smaller variance within sequences. The constraint to the latent segment variables z1 is imposed globally, and therefore encourages encoding of residual attributes whose variation is not distinguishable
inter and intra sequences.
In the variational autoencoder framework, since the exact posterior inference is in(i)
(i)
(i)
tractable, an inference model, q? (Z1 , Z2 , ?2 |X (i) ), that approximates the true posterior,
(i)
(i)
(i)
p? (Z1 , Z2 , ?2 |X (i) ), for variational inference [19] is introduced. We consider the following
inference model as Figure 2(b):
(i)
(i)
(i)
q? (Z1 , Z2 , ?2 |X (i) )
=
(i)
q? (?2 )
(i)
N
Y
(i,n)
q? (z1
(i,n)
|x(i,n) , z2
(i,n)
)q? (z2
|x(i,n) )
n=1
(i)
q? (?2 )
=
(i)
2
N (?2 |g??2 (i), ??
? 2 I),
q? (z2 |x) = N (z2 |g?z2 (x), diag(g?z2 (x)))
2
q? (z1 |x, z2 ) = N (z1 |g?z1 (x, z2 ), diag(g?z2 (x, z2 ))),
1
3
where the posteriors over ?2 , z1 , and z2 are all multivariate diagonal Gaussian distributions. Note that
the mean of the posterior distribution of ?2 is not directly inferred from X, but instead is regarded as
part of inference model parameters, with one for each utterance, which would be optimized during
(i)
training. Therefore, g??2 (?) can be seen as a lookup table, and we use ?
? 2 = g??2 (i) to denote
the posterior mean of ?2 for the i-th sequence; we fix the posterior covariance matrix of ?2 for all
sequences. Similar to the generative model, g?z2 (?), g?z2 (?), g?z1 (?, ?), and g?z2 (?, ?) are also neural
2
1
networks whose parameters along with g??2 (?) are denoted collectively by ?. The variational lower
bound for this inference model on the marginal likelihood of a sequence X is derived as follows:
L(?, ?; X) =
N
X
L(?, ?; x(n) |?
?2 ) + log p? (?
?2 ) + const
n=1
L(?, ?; x
(n)
|?
?2 ) =Eq
(n)
(n)
(n) )
? (z1 ,z2 |x
? Eq
(n)
? (z2
|x(n) )
(n)
(n)
log p? (x(n) |z1 , z2 )
(n)
(n)
(n)
DKL (q? (z1 |x(n) , z2 )||p? (z1 ))
(n)
(n)
? DKL (q? (z2 |x(n) )||p? (z2 |?
?2 )).
The detailed derivation can be found in Appendix A. Because the approximated posterior of ?2 does
not depend on the sequence X, the sequence variational lower bound L(?, ?; X) can be decomposed
into the sum of L(?, ?; x(n) |?
?2 ), the conditional segment variational lower bounds, over segments,
plus the log prior probability of ?
? 2 and a constant. Therefore, instead of sampling a batch at the
sequence level to maximize the sequence variational lower bound, we can sample a batch at the
segment level to maximize the segment variational lower bound:
1
log p? (?
?2 ) + const.
(2)
N
This approach provides better scalability when the sequences are extremely long, such that computing
an entire sequence for a batched update is too computationally expensive.
L(?, ?; x(n) ) = L(?, ?; x(n) |?
?2 ) +
In this paper we only introduce two scales of attributes; however, one can easily extend this model
to more scales by simply introducing ?k for k = 2, 3, ? ? ? 2 that constrains the prior distribution of
latent variables at more scales, such as having session-dependent prior or dataset-dependent prior.
2.1
Discriminative Objective
The idea of having sequence-specific priors for each sequence is to encourage the model to encode
the sequence-level attributes and the segment-level attributes into different sets of latent variables.
However, when ?2 = 0 for all sequences, the prior probability of the s-vector is maximized, and the
KL-divergence of the inferred posterior of z2 is measured from the same conditional prior for all
sequences. This would result in trivial s-vectors ?2 , and therefore z1 and z2 would not be factorized
to encode sequence and segment attributes respectively.
(i,n)
To encourage z2 to encode sequence-level attributes, we use z2 , which is inferred from x(i,n) , to
infer the sequence index i of x(i,n) . We formulate the discriminative objective as:
(i,n)
log p(i|z2
(i,n)
) = log p(z2
|i) ? log
M
X
(i,n)
p(z2
|j)
(p(i) is assumed uniform)
j=1
(i,n)
:= log p? (z2
(i)
|?
?2 ) ? log
M
X
(i,n)
p? (z2
(j)
|?
?2 ) ,
j=1
Combining the discriminative objective using a weighting parameter ? with the segment variational
lower bound, the objective function to maximize then becomes:
(i,n)
Ldis (?, ?; x(i,n) ) = L(?, ?; x(i,n) ) + ? log p(i|z2
which we refer to as the discriminative segment variational lower bound.
2
The index starts from 2 because we do not introduce the hierarchy to z1 .
4
),
(3)
2.2
Inferring S-Vectors During Testing
?
? = {?
During testing, we may want to use the s-vector ?2 of an unseen sequence X
x(n) }N
n=1 as
the sequence-level attribute representation for tasks such as speaker verification. Since the exact
maximum a posterior estimation of ?2 is intractable, we approximate the estimation using the
conditional segment variational lower bound as follows:
? = argmax log p? (X,
? ?2 )
??2 = argmax log p? (?2 |X)
?2
?2
= argmax
?2
log p? (?
x(n) |?2 ) + log p? (?2 )
n=1
? argmax
?2
?
N
X
?
N
X
L(?, ?; x
?(n) |?2 ) + log p? (?2 ).
(4)
n=1
The closed form solution of ??2 can be derived by differentiating Eq. 4 w.r.t. ?2 (see Appendix B):
PN?
g?z (?
x(n) )
?
.
(5)
?2 = n=1 2 2
? + ?z2/??2
N
2
3
Sequence-to-Sequence Autoencoder Model Architecture
In this section, we introduce the detailed neural network architectures for our proposed FHVAE. Let
a segment x = x1:T be a sub-sequence of X that contains T time steps, and xt denotes the t-th time
step of x. We use recurrent network architectures for encoders that capture the temporal relationship
among time steps, and generate a summarized fixed-dimension vector after consuming an entire
sub-sequence. Likewise, we adopt a recurrent network architecture that generates a frame step by
step conditioned on the latent variables z1 and z2 . The complete network can be seen as a stochastic
sequence-to-sequence autoencoder that encodes x1:T stochastically into z1 , z2 , and stochastically
decodes from them back to x1:T .
Encoder
x1
x2
x3
xT
?
z2
q(z1|x1:T, z2)
x1
x2
p(x1|z1, z2)
p(x2|z1, z2)
xT
p(xT|z1, z2)
?
Decoder
q(z2|x1:T)
?
z1
Figure 3: Sequence-to-sequence factorized hierarchical variational autoencoder. Dashed lines indicate
the sampling process using the reparameterization trick [23]. The encoders for z1 and z2 are pink
and amber, respectively, while the decoder for x is blue. Darker colors denote the recurrent neural
networks, while lighter colors denote the fully-connected layers predicting the mean and log variance.
Figure 3 shows our proposed Seq2Seq-FHVAE architecture.3 Here we show the detailed formulation:
(hz2 ,t , cz2 ,t ) = LSTM(xt?1 , hz2 ,t?1 , cz2 ,t?1 ; ?LSTM,z2 )
q? (z2 |x1:T ) = N (z2 | MLP(hz2 ,T ; ?MLP? ,z2 ), diag(exp(MLP(hz2 ,T ; ?MLP?2 ,z2 ))))
(hz1 ,t , cz1 ,t ) = LSTM([xt?1 ; z2 ], hz1 ,t?1 , cz1 ,t?1 ; ?z1 )
q? (z1 |x1:T , z2 ) = N (z1 | MLP(hz1 ,T ; ?MLP? ,z1 ), diag(exp(MLP(hz1 ,T ; ?MLP?2 ,z1 ))))
(hx,t , cx,t ) = LSTM([z1 ; z2 ], hx,t?1 , cx,t?1 ; ?x )
p? (xt |z1 , z2 ) = N (xt | MLP(hx,t ; ?MLP? ,x ), diag(exp(MLP(hx,t ; ?MLP?2 ,x )))),
where LSTM refers to a long short-term memory recurrent neural network [14], and MLP refers to a
multi-layer perceptron, ?? are the related weight matrices. None of the neural network parameters
are shared. We refer to this model as Seq2Seq-FHVAE. Log-likelihood and qualitative comparison
with alternative architectures can be found in Appendix D.
3
Best viewed in color.
5
4
Experiments
We use speech, which inherently contains information at multiple scales, such as channel, speaker,
and linguistic content to test our model. Learning to disentangle the mixed information from the
surface representation is essential for a wide variety of speech applications: for example, noise robust
speech recognition [41, 38, 37, 16], speaker verification [5], and voice conversion [40, 29, 24].
The following two corpora are used for our experiments: (1) TIMIT [10], which contains broadband
16kHz recordings of phonetically-balanced read speech. A total of 6300 utterances (5.4 hours) are
presented with 10 sentences from each of 630 speakers, of which approximately 70% are male and
30% are female. (2) Aurora-4 [32], a broadband corpus designed for noisy speech recognition tasks
based on the Wall Street Journal corpus (WSJ0) [31]. Two microphone types, CLEAN/CHANNEL
are included, and six noise types are artificially added to both microphone types, which results in
four conditions: CLEAN, CHANNEL, NOISY, and CHANNEL + NOISY. Two 14 hour training sets are
used, where one is clean and the other is a mix of all four conditions. The same noise types and
microphones are used to generate the development and test sets, which both consist of 330 utterances
from all four conditions, resulting in 4,620 utterances in total for each set.
All speech is represented as a sequence of 80 dimensional Mel-scale filter bank (FBank) features
or 200 dimensional log-magnitude spectrum (only for audio reconstruction), computed every 10ms.
Mel-scale features are a popular auditory approximation for many speech applications [28]. We
consider a sample x to be a 200ms sub-sequence, which is on the order of the length of a syllable,
and implies T = 20 for each x. For the Seq2Seq-FHVAE model, all the LSTM and MLP networks
are one-layered, and Adam [20] is used for optimization. More details of the model architecture and
training procedure can be found in Appendix C.
4.1
Qualitative Evaluation of the Disentangled Latent Variables
Figure 4: (left) Examples generated by varying different latent variables. (right) An illustration
of harmonics and formants in filter bank images. The green block ?A? contains four reconstructed
examples. The red block ?B? contains ten original sequences on the first row with the corresponding
reconstructed examples on the second row. The entry on the i-th row and the j-th column in the blue
block ?C? is the reconstructed example using the latent segment variable z1 of the i-th row from block
?A? and the latent sequence variable z2 of the j-th column from block ?B?.
To qualitatively study the factorization of information between the latent segment variable z1 and the
latent sequence variable z2 , we generate examples x by varying each of them respectively. Figure 4
shows 40 examples in block ?C? of all the combinations of the 4 latent segment variables extracted
from block ?A? and the 10 latent sequence variables extracted from block ?B.? The top two examples
from block ?A? and the five leftmost examples from block ?B? are from male speakers, while the rest
are from female speakers, which show higher fundamental frequencies and harmonics.4
4
The harmonics corresponds to horizontal dark stripes in the figure; the more widely these stripes are spaced
vertically, the higher the fundamental frequency of the speaker is.
6
Figure 5: FHVAE (? = 0) decoding results of three combinations of latent segment variables z1
and latent sequence variables z2 from one male-speaker utterance (top-left) and one female-speaker
utterance (bottom-left) in Aurora-4. By replacing z2 of a male-speaker utterance with z2 of a femalespeaker utterance, an FHVAE decodes a voice-converted utterance (middle-right) that preserves the
linguistic content. Audio samples are available at https://youtu.be/VMX3IZYWYdg.
We can observe that along each row in block ?C?, the linguistic phonetic-level content, which manifests
itself in the form of the spectral contour and temporal position of formants, as well as the relative
position between formants, is very similar between elements; the speaker identity however changes
(e.g., harmonic structure). On the other hand, for each column we see that the speaker identity remains
consistent, despite the change of linguistic content. The factorization of the sequence-level attributes
and the segment-level attributes of our proposed Seq2Seq-FHVAE is clearly evident. In addition, we
also show examples of modifying an entire utterance in Figure 1 and 5, which achieves denoising
by replacing the latent sequence variable of a noisy utterance with those of a clean utterance, and
achieves voice conversion by replacing the latent sequence variable of one speaker with that of
another speaker. Details of the operations we applied to modify an entire utterance as well as more
larger-sized examples of different ? values can be found in Appendix E. We also show extra latent
space traversal experiments in Appendix H.
4.2
Quantitative Evaluation of S-Vectors ? Speaker Verification
To quantify the performance of our model on disentangling the utterance-level attributes from the
segment-level attributes, we present experiments on a speaker verification task on the TIMIT corpus
to evaluate how well the estimated ?2 encodes speaker-level information.5 As a sanity check, we
modify Eq. 5 to estimate an alternative s-vector based on latent segment variables z1 as follows:
PN?
? + ? 2 ). We use the i-vector method [5] as the baseline, which is
?1 =
x(n) )/(N
z1
n=1 g?z1 (?
the representation used in most state-of-the-art speaker verification systems. They are in a low
dimensional subspace of the Gaussian mixture model (GMM) mean supervector space, where the
GMM is the universal background model (UBM) that models the generative process of speech.
I-vectors, ?1 , and ?2 can all be extracted without supervision; when speaker labels are available
during training, techniques such as linear discriminative analysis (LDA) can be applied to further
improve the linear separability of the representation. For all experiments, we use the fast scoring
approach in [4] that uses cosine similarity as the similarity metric and compute the equal error rate
(EER). More details about the experimental settings can be found in Appendix F.
We compare different dimensions for both features as well as different ??s in Eq.3 for training
FHVAE models. The results in Table 1 show that the 16 dimensional s-vectors ?2 outperform i-vector
baselines in both unsupervised (Raw) and supervised (LDA) settings for all ??s as shown in the fourth
column; the more discriminatively the FHVAE model is trained (i.e., with larger ?), the better speaker
5
TIMIT is not a standard corpus for speaker verification, but it is a good corpus to show the utterance-level
attribute we learned via this task, because the main attribute that is consistent within an utterance is speaker
identity, while in Aurora-4 both speaker identity and the background noise are consistent within an utterance.
7
verification results it achieves. Moreover, with the appropriately chosen dimension, a 32 dimensional
?2 reaches an even lower EER at 1.34%. On the other hand, the negative results of using ?1 also
validate the success in disentangling utterance and segment level attributes.
Table 1: Comparison of speaker verification equal error rate (EER) on the TIMIT test set
4.3
Features
Dimension
?
Raw
LDA (12 dim)
LDA (24 dim)
i-vector
48
100
200
-
10.12%
9.52%
9.82%
6.25%
6.10%
6.54%
5.95%
5.50%
6.10%
?2
16
16
16
16
32
0
10?1
100
101
101
5.06%
4.91%
3.87%
2.38%
2.38%
4.02%
4.61%
3.86%
2.08%
2.08%
1.34%
?1
16
16
32
100
101
101
22.77%
27.68%
22.47%
15.62%
22.17%
16.82%
17.26%
Quantitative Evaluation of the Latent Segment Variables ? Domain Invariant ASR
Speaker adaptation and robust speech recognition in automatic speech recognition (ASR) can often
be seen as domain adaptation problems, where available labeled data is limited and hence the data
distributions during training and testing are mismatched. One approach to reduce the severity of this
issue is to extract speaker/channel invariant features for the tasks.
As demonstrated in Section 4.2, the s-vector contains information about domains. Here we evaluate
if the latent segment variables contains domain invariant linguistic information by evaluating on
an ASR task: (1) train our proposed Seq2Seq-FHVAE using FBank feature on a set that covers
different domains. (2) train an LSTM acoustic model [12, 35, 42] on the set that only covers partial
domains using mean and log variance of the latent segment variable z1 extracted from the trained
Seq2Seq-FHVAE. (3) test the ASR system on all domains. As a baseline, we also train the same ASR
models but use the FBank features alone. Detailed configurations are in Appendix G.
For TIMIT we assume that male and female speakers constitute different domains, and show the
results in Table 2. The first row of results shows that the ASR model trained on all domains (speakers)
using FBank features as the upper bound. When trained on only male speakers, the phone error rate
(PER) on female speakers increases by 16.1% for FBank features; however, for z1 , despite the slight
degradation on male speakers, the PER on the unseen domain, which are female speakers, improves
by 6.6% compared to FBank features.
Table 2: TIMIT test phone error rate of acoustic models trained on different features and sets
Train Set and Configuration
Test PER by Set
ASR
FHVAE
Features
Male
Female
All
Train All
-
FBank
20.1%
16.7%
19.1%
Train Male
Train All, ? = 10
FBank
z1
21.0%
22.0%
32.8%
26.2%
25.2%
23.5%
On Aurora-4, four domains are considered, which are clean, noisy, channel, and noisy+channel (NC
for short). We train the FHVAE on the development set for two purposes: (1) the FHVAE can be
considered as a general feature extractor, which can be trained on an arbitrary collection of data that
does not necessarily include the data for subsequent applications. (2) the dev set of Aurora-4 contains
the domain label for each utterance so it is possible to control which domain has been observed by the
FHVAE. Table 3 shows the word error rate (WER) results on Aurora-4, from which we can observe
that the FBank representation suffers from severe domain mismatch problems; specifically, the WER
8
increases by 53.3% when noise is presented in mismatched microphone recordings (NC). In contrast,
when the FHVAE is trained on data from all domains, using the latent segment variables as features
reduce WER from 16% to 35% compare to baseline on mismatched domains, with less than 2%
WER degradation on the matched domain. In addition, ?-VAEs [13] are trained on the same data as
the FHVAE to serve as the baseline feature extractor, from which we extract the latent variables z as
the ASR feature and show the result in the third to the sixth rows. The ?-VAE features outperform
FBank in all mismatched domains, but are inferior to the latent segment variable z1 from the FHVAE
in those domains. The results demonstrate the importance of learning not only disentangled, but also
interpretable representations, which can be achieved by our proposed FHVAE models. As a sanity
check, we replace z1 with z2 , the latent sequence variable and train an ASR, which results in terrible
WER performance as shown in the eighth row as expected.
Finally, we train another FHVAE on all domains excluding the combinatory NC domain, and shows
the results in the last row in Table 3. It can be observed that the latent segment variable still
outperforms the baseline feature with 30% lower WER on noise and channel combined data, even
though the FHAVE has only seen noise and channel variation independently.
Table 3: Aurora-4 test word error rate of acoustic models trained on different features and sets
Train Set and Configuration
ASR
{FH-,?-}VAE
Features
Clean
Noisy
Channel
NC
All
Train All
-
FBank
3.60%
7.06%
8.24%
18.49%
11.80%
Dev, ? = 1
Dev, ? = 2
Dev, ? = 4
Dev, ? = 8
Dev, ? = 10
Dev, ? = 10
FBank
z (?-VAE)
z (?-VAE)
z (?-VAE)
z (?-VAE)
z1 (FHVAE)
z2 (FHVAE)
3.47%
4.95%
3.57%
3.89%
5.32%
5.01%
41.08%
50.97%
23.54%
27.24%
24.40%
34.84%
16.42%
68.73%
36.99%
31.12%
30.56%
29.80%
36.13%
20.29%
61.89%
71.80%
46.21%
48.17%
47.87%
58.02%
36.33%
86.36%
55.51%
32.47%
34.75%
33.38%
42.76%
24.41%
72.53%
Dev\NC, ? = 10
z1 (FHVAE)
5.25%
16.52%
19.30%
40.59%
26.23%
Train Clean
5
Test WER by Set
Related Work
A number of prior publications have extended VAEs to model structured data by altering the underlying graphical model to dynamic Bayesian networks, such as SRNN [3] and VRNN [9], or to
hierarchical models, such as neural statistician [7] and SVAE [18]. These models have shown success
in quantitatively increasing the log-likelihood, or qualitatively generating reasonable structured data
by sampling. However, it remains unclear whether independent attributes are disentangled in the
latent space. Moreover, the learned latent variables in these models are not interpretable without
manually inspecting or using labeled data. In contrast, our work presents a VAE framework that
addresses both problems by explicitly modeling the difference in the rate of temporal variation of the
attributes that operate at different scales.
Our work is also related to ?-VAE [13] with respect to unsupervised learning of disentangled representations with VAEs. The boosted KL-divergence penalty imposed in ?-VAE training encourages
disentanglement of independent attributes, but does not provide interpretability without supervision.
We demonstrate in our domain invariant ASR experiments that learning interpretable representations
is important for such applications, and can be achieved by our FHVAE model. In addition, the idea
of boosting KL-divergence regularization is complimentary to our model, which can be potentially
integrated for better disentanglement.
6
Conclusions and Future Work
We introduce the factorized hierarchical variational autoencoder, which learns disentangled and
interpretable representations for sequence-level and segment-level attributes without any supervision.
We verify the disentangling ability both qualitatively and quantitatively on two speech corpora. For
future work, we plan to (1) extend to more levels of hierarchy, (2) investigate adversarial training for
disentanglement, and (3) apply the model to other types of sequential data, such as text and videos.
9
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12
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6,395 | 6,785 | Lookahead Bayesian Optimization
with Inequality Constraints
Remi R. Lam
Massachusetts Institute of Technology
Cambridge, MA
[email protected]
Karen E. Willcox
Massachusetts Institute of Technology
Cambridge, MA
[email protected]
Abstract
We consider the task of optimizing an objective function subject to inequality
constraints when both the objective and the constraints are expensive to evaluate.
Bayesian optimization (BO) is a popular way to tackle optimization problems
with expensive objective function evaluations, but has mostly been applied to
unconstrained problems. Several BO approaches have been proposed to address
expensive constraints but are limited to greedy strategies maximizing immediate
reward. To address this limitation, we propose a lookahead approach that selects
the next evaluation in order to maximize the long-term feasible reduction of the
objective function. We present numerical experiments demonstrating the performance improvements of such a lookahead approach compared to several greedy
BO algorithms, including constrained expected improvement (EIC) and predictive
entropy search with constraint (PESC).
1
Introduction
Constrained optimization problems are often challenging to solve, due to complex interactions between the goals of minimizing (or maximizing) the objective function while satisfying the constraints.
In particular, non-linear constraints can result in complicated feasible spaces, sometimes partitioned
in disconnected regions. Such feasible spaces can be difficult to explore for a local optimizer, potentially preventing the algorithm from converging to a global solution. Global optimizers, on the
other hand, are designed to tackle disconnected feasible spaces and optimization of multi-modal
objective functions. Such algorithms typically require a large number of evaluations to converge.
This can be prohibitive when the evaluation of the objective function or the constraints is expensive,
or when there is a finite budget of evaluations allocated for the optimization, as it is often the case
with expensive models. This evaluation budget typically results from resource scarcity such as the
restricted availability of a high-performance computer, finite financial resources to build prototypes,
or even time when working on a paper submission deadline.
Bayesian optimization (BO) [19] is a global optimization technique designed to address problems
with expensive function evaluations. Its constrained extension, constrained Bayesian optimization
(CBO), iteratively builds a statistical model for the objective function and the constraints. Based on
this model that leverages all the past evaluations, a utility function quantifies the merit of evaluating
any design under consideration. At each iteration, a CBO algorithm evaluates the expensive objective
function and constraints at the design which maximizes this utility function.
In most existing methods, the utility function only quantifies the reward obtained over the immediate
next step, and ignores the gains that could be collected at future steps. This results in greedy CBO
algorithms. However, quantifying long-term rewards may be beneficial. For instance, in the presence
of constraints, it could be valuable to learn the boundaries of the feasible space. In order to do so, it
is likely that an infeasible design would need to be evaluated, bringing no immediate improvement,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
but leading to long-term benefits. Such strategy requires planning over several steps. Planning is also
required to balance the so-called exploration-exploitation trade-off. Intuitively, in order to improve
the statistical model, the beginning of the optimization should mainly be dedicated to exploring the
design space, while the end of the optimization should focus on exploiting that statistical model to
find the best design. To balance this trade-off in a principled way, the optimizer needs to plan ahead
and be aware of the remaining evaluation budget.
To address the shortcomings of greedy algorithms, we propose a new lookahead formulation for
CBO with a finite budget. This approach is aware of the remaining budget and can balance the
exploration-exploitation trade-off in a principled way. In this formulation, the best optimization policy
sequentially evaluates the design yielding the maximum cumulated reward over multiple steps. This
optimal policy is the solution of an intractable dynamic programming (DP) problem. We circumvent
this issue by employing an approximate dynamic programming (ADP) algorithm: rollout, building on
the unconstrained BO algorithm in [17]. Numerical examples illustrate the benefits of the proposed
lookahead algorithm over several greedy ones, especially when the objective function is multi-modal
and the feasible space has a complex topology.
The next section gives an overview of CBO and discusses some of the related work (Sec. 2). Then, we
formulate the lookahead approach to CBO as a dynamic programming problem and demonstrate how
to approximately solve it by adapting the rollout algorithm (Sec. 3). Numerical results are provided
in Sec. 4. Finally, we present our conclusions in Sec. 5.
2
Constrained Bayesian Optimization
We consider the following optimization problem:
(OPc) x? = argmin f (x)
x?X
s.t.
(1)
gi (x) ? 0, ?i ? {1, . . . , I},
where x is a d-dimensional vector of design variables. The design space X is a bounded subset of
Rd , f : X 7? R is an objective function, I is the number of inequality constraints and gi : X 7? R
is the ith constraint function. The functions f and gi are considered expensive to evaluate. We are
interested in finding the minimizer x? of the objective function f subject to the non-linear constraints
gi ? 0 with a finite budget of N evaluations. We refer to this problem as the original constrained
problem (OPc).
Constrained Bayesian optimization (CBO) addresses the original constrained problem (OPc) by
modeling the objective function f and the constraints gi as realizations of stochastic processes.
Typically, each expensive-to-evaluate function is modeled with an independent Gaussian process
(GP). At every iteration n, new evaluations of f and gi become available and augment a training
set Sn = {(xj , f (xj ), g1 (xj ), ? ? ? , gI (xj ))}nj=1 . Using Bayes rule, the statistical model is updated
and the posterior quantities of the GP, conditioned on Sn , reflect the current representation of the
unknown expensive functions. In particular, for any design x, the posterior mean ?n (x; ?) and the
posterior variance ? 2n (x; ?) of the GP associated with the expensive function ? ? {f, g1 , ? ? ? , gI }
can be computed cheaply using a closed-form expression (see [24] for an overview of GP). CBO
leverages this statistical model to quantify, in a cheap-to-evaluate utility function Un , the usefulness of
any design under consideration. The next design to evaluate is then selected by solving the following
auxiliary problem (AP):
(AP)
xn+1 = argmax Un (x; Sn ).
(2)
x?X
The vanilla CBO algorithm is summarized in Algorithm 1.
Many utility functions have been proposed in the literature. To decide which design to evaluate next,
[27] proposed the use of constrained expected improvement EIc , which, in the case of independent
GPs, can be computed in closed-form as the product of the expected improvement (obtained by
considering the GP associated with the objective function) and the probability of feasibility associated
with each constraint. This approach was later applied to machine learning applications [6] and
extended to the multi-objective case [5]. Note that this method transforms an original constrained
optimization problem into an unconstrained auxiliary problem by modifying the utility function.
Other attempts to cast the constrained problem into an unconstrained one include [3]. That work uses
2
Algorithm 1 Constrained Bayesian Optimization
Input: Initial training set S1 , budget N
for n = 1 to N do
Construct GPs using Sn
Update hyper-parameters
Solve AP for xn+1 = argmaxx?X Un (x; Sn )
Evaluate f (xn+1 ), g1 (xn+1 ), ? ? ? , gI (xn+1 )
Sn+1 = Sn ? {(xn+1 , f (xn+1 ), g1 (xn+1 ), ? ? ? , gI (xn+1 ))}
end for
a penalty method to transform the original constrained problem into an unconstrained problem, to
which they apply a radial basis functions (RBF) method for global optimization (constrained RBF
methods exist as well [25]). Other techniques from local constrained optimization have been leveraged
in [10] where the utility function is constructed based on an augmented Lagrangian formulation. This
technique was recently extended in [22] where a slack-variables formulation allows the handling
of equality and mixed constraints. Another approach is proposed by [1]: at each iteration, a finite
set of candidate designs is first generated from a Latin hypercube, second, candidate designs with
expected constraint violation higher than a user-defined threshold are rejected. Finally, among the
remaining candidates, the ones achieving the best expected improvement are evaluated (several
designs can be selected simultaneously at each iteration in this formulation). Another method [26]
solves a constrained auxiliary optimization problem: the next design is selected to maximize the
expected improvement subject to approximated constraints (the posterior mean of the GP associated
with a constraint is used in lieu of the constraint itself). Note that the two previous methods solve a
constrained auxiliary problem.
Another method to address constrained BO is proposed by [11], who develop an integrated conditional
expected improvement criterion. Given a candidate design, this criterion quantifies the expected
improvement point-wise (conditioned on the fact that the candidate will be evaluated). This pointwise improvement is then integrated over the entire design space. In the unconstrained case, in the
integration phase, equal weight is given to designs throughout the design space. The constrained
case is addressed by defining a weight function that depends on the feasible probability of a design:
improvement at designs that are likely to be infeasible have low weight. The probability of a design
being feasible is calculated using a classification GP. The computation of this criterion is more
involved as there is no closed-form formulation available for the integration and techniques such as
Monte Carlo or Markov chain Monte Carlo must be employed. In a similar spirit, [21] introduces a
utility function which quantifies the benefit of evaluating a design by integrating its effect over the
design space. The proposed utility function computes the expected reduction of the feasible domain
below the best feasible value evaluated so far. This results in the expected volume of excursion
criteria which also requires approximation techniques to be computed.
The former approaches revolve around computing a quantity based on improvement and require
having at least one feasible design. Other strategies use information gain as the key element to
drive the optimization strategy. [7] proposed a two-step approach for constrained BO when the
objective and the constraints can be evaluated independently. The first step chooses the next location
by maximizing the constrained EI [27], the second step chooses whether to evaluate the objective or
a constraint using an information gain metric (i.e., entropy search [12]). [13, 14] developed a strategy
that simultaneously selects the design to be evaluated and the model to query (the objective or a
constraint). The criterion used, predictive entropy search with constraints (PESC), is an extension
of predictive entropy search (PES) [15]. One of the advantages of information gain-based methods
stems from the fact that one does not need to start with a feasible design.
All aforementioned methods use myopic utilities to select the next design to evaluate, leading
to suboptimal optimization strategies. In the unconstrained BO setting, multiple-steps lookahead
algorithms have been explored [20, 8, 18, 9, 17] and were shown to improve the performance of
BO. To our knowledge, such lookahead strategies for constrained optimization have not yet been
addressed in the literature and also have the potential to improve the performance of CBO algorithms.
3
3
Lookahead Formulation of CBO
In this section, we formulate CBO with a finite budget as a dynamic programming (DP) problem
(Sec. 3.1). This leads to an optimal but computationally challenging optimization policy. To mitigate
the cost of computing such a policy, we employ an approximate dynamic programming algorithm,
rollout, and demonstrate how it can be adapted to CBO with a finite budget (Sec. 3.2).
3.1
Dynamic Programming Formulation
We seek an optimization policy which leads, after consumption of the evaluation budget, to the
maximum feasible decrease of the objective function. Because the value of the expensive objective
function and constraints are not known before their evaluations, it is impossible to quantify such
long-term reward within a cheap-to-evaluate utility function Un . However, CBO endows the objective
function and the constraints with a statistical model that can be interrogated to inform the optimizer
of the likely values of f and gi at a given design. This statistical model can be leveraged to simulate
optimization scenarios over multiple steps and quantify their probabilities. Using this simulation
mechanism, it is possible to quantify, in an average sense, the long-term reward achieved under a
given optimization policy. The optimal policy is the solution of the DP problem that we formalize
now.
Let n be the current iteration number of the CBO algorithm, and N the total budget of evaluations,
or horizon. We refer to the future iterations of the optimization generated by simulation as stages.
For any stage k ? {n, ? ? ? , N }, all the information collected is contained in the training set Sk . The
function f and the I functions gi are modeled with independent GPs. Their posterior quantities,
conditioned on Sk , fully characterize our knowledge of f and gi . Thus, we define the state of our
knowledge at stage k to be the training set Sk ? Zk .
Based on the training set Sk , the simulation makes a decision regarding the next design xk+1 ? X to
evaluate using an optimization policy. An optimization policy ? = {?1 , ? ? ? , ?N } is a sequence of
rules, ?k : Zk 7? X for k ? {1, ? ? ? , N }, mapping a training set Sk to a design xk+1 = ?k (Sk ).
In the simulations, the values f (xk+1 ) and gi (xk+1 ) are unknown and are treated as uncertainties.
f
We model those I + 1 uncertain quantities with I + 1 independent Gaussian random variables Wk+1
gi
and Wk+1 based on the GPs:
f
Wk+1
? N (?k (xk+1 ; f ), ? 2k (xk+1 ; f )),
gi
Wk+1
? N (?k (xk+1 ; gi ), ? 2k (xk+1 ; gi )),
and ? 2k (xk+1 ; ?) are the posterior mean
(3)
(4)
where we recall that ?k (xk+1 ; ?)
and variance of the GP
associated with any expensive function ? ? {f, g1 , ? ? ? , gI }, conditioned on Sk , at xk+1 . Then, the
1
I
simulation generates an outcome. A simulated outcome wk+1 = (fk+1 , gk+1
, ? ? ? , gk+1
)?W?
f
g
gI
1
RI+1 is a sample of the (I + 1)-dimensional random variable Wk+1 = [Wk+1 , Wk+1
, ? ? ? , Wk+1
].
Note that simulating an outcome does not require evaluating the expensive f and gi . In particular,
i
fk+1 and gk+1
are not f (xk+1 ) and gi (xk+1 ).
1
I
Once an outcome wk+1 = (fk+1 , gk+1
, ? ? ? , gk+1
) is simulated, the system transitions to a new state
Sk+1 , governed by the system dynamic Fk : Zk ? X ? W 7? Zk+1 given by:
1
I
Sk+1 = Fk (Sk , xk+1 , wk+1 ) = Sk ? {(xk+1 , fk+1 , gk+1
, ? ? ? , gk+1
)).
(5)
Now that the simulation mechanism is defined, one needs a metric to assert the quality of a given
optimization policy. At stage k, a stage-reward function rk : Zk ? X ? W 7? R quantifies the merit
1
I
of querying a design if the outcome wk = (fk+1 , gk+1
, ? ? ? , gk+1
) occurs. This stage-reward is
defined as the reduction of the objective function satisfying the constraints:
n
o
Sk
rk (Sk , xk+1 , wk+1 ) = max 0, fbest
? fk+1 ,
(6)
Sk
i
if gk+1
? 0 for all i ? {1, ? ? ? , I}, and rk (?, ?, ?) = 0 otherwise, where fbest
is the best feasible value
at stage k. Thus, the expected (long-term) reward starting from training set Sn under optimization
policy ? is:
" N
#
X
J? (Sn ) = E
rk (Sk , ?k (Sk ), wk+1 ) ,
(7)
k=n
4
where the expectation is taken with respect to the (correlated) simulated values (wn+1 , ? ? ? , wN +1 ),
and the state evolution is governed by Eq. 5. An optimal policy, ? ? , is a policy maximizing this
long-term expected reward in the space of admissible policies ?:
J?? (Sn ) = max J? (Sn ).
(8)
???
The optimal reward J?? (Sn ) is given by Bellman?s principle of optimality and can be computed
using the DP recursive algorithm, working backward from k = N ? 1 to k = n,
JN (SN ) =
max E[rN (SN , xN +1 , wN +1 )] =
xN +1 ?X
max EIc (xN +1 ; SN )
xN +1 ?X
Jk (Sk ) = max E[rk (Sk , xk+1 , wk+1 ) + Jk+1 (Fk (Sk , xk+1 , wk+1 ))],
(9)
xk+1 ?X
where each expectation is taken with respect to one simulated outcome vector wk+1 , and we have
used the fact that E[rk (Sk , xk+1 , wk+1 )] = EIc (xk+1 ; Sk ) is the constrained expected improvement
known in closed-form [27]. The optimal reward is given by J?? (Sn ) = Jn (Sn ). Thus, at iteration n
of the CBO algorithm, the optimal policy select the next design xn+1 that maximizes Jn (Sn ) given
by Eqs. 9. In other words, the best decision to make at iteration n maximizes, on average, the sum of
the immediate reward rn and the future long-term reward Jn+1 (Sn+1 ) obtained by making optimal
subsequent decisions. This is illustrated in Fig. 1, left panel.
Sk wk+1 Sk+1
xk+1
xk+2
wk+2 Sk+2
xk+3
Sk wk+1 Sk+1 ?k+1 (Sk+1 )
xk+1
wk+2 Sk+2 ?k+2 (Sk+2 )
???
???
???
???
???
???
???
???
Figure 1: Left: Tree illustrating the intractable DP formulation. Each black circle represents a training
set and a design, each white circle is a training set. Dashed lines represent simulated outcomes
resulting in expectations. The double arrows represent designs selected with the (unknown) optimal
policy, leading to nested maximizations. Double arrows depict the bidirectional way information
propagates when the optimal policy is built: each optimal decision depends on the previous steps and
relies on the optimality of the future decisions. Right: Single arrows represent designs selected using
a heuristic. This illustrates the unidirectional propagation of information when a known heuristic
drives the simulations: each decision depends on the previous steps but is independent of the future
ones. The absence of nested maximization leads to a tractable formulation.
3.2
Rollout for Constrained Bayesian Optimization
The best optimization policy evaluates, at each iteration n of the CBO algorithm, the design xn+1
maximizing the optimal reward J?? (Sn ) (Eq. 8). This requires solving a problem with several nested
maximizations and expectations (Eqs. 9), which is computationally intractable. To mitigate the cost
of solving the DP algorithm, we employ an approximate dynamic programming (ADP) technique:
rollout (see [2, 23] for an overview). Rollout selects the next design by maximizing a (suboptimal)
long-term reward J? . The reward is computed by simulating optimization scenarios over several
future steps. However, the simulated steps are not controlled by the optimal policy ? ? . Instead,
rollout uses a suboptimal policy ?, i.e. a heuristic, to drive the simulation. This circumvents the
need for nested maximizations (as illustrated in Fig. 1, right panel) and simplifies the computation of
J? compared to J?? . We now formalize the rollout algorithm, propose a heuristic ? adapted to the
context of CBO with a finite budget, and detail further numerical approximations.
Let us consider the iteration n of the CBO algorithm. The long-term reward J? (Sn ) induced by
a (known) heuristic ? = {?1 , ? ? ? , ?N }, starting from state Sn , is defined by Eq. 7. This can be
rewritten as J? (Sn ) = Hn , where Hn is recursively defined, from k = N back to k = n, by:
HN +1 (SN +1 ) = 0
Hk (Sk ) = E[rk (Sk , ?k (Sk ), wk+1 ) + ?Hk+1 (Fk (Sk , ?k (Sk ), wk+1 ))],
5
(10)
where each expectation is taken with respect to one simulated outcome vector wk+1 , and ? ? [0, 1]
is a discount factor encouraging the early collection of reward. A discount factor ? = 0 leads to a
greedy policy, focusing on immediate reward. In that case, the reward J? simplifies to the constrained
expected improvement EIc . A discount factor ? = 1, on the other hand, is indifferent to when the
reward is collected.
The fundamental simplification introduced by the rollout algorithm lies in the absence of nested
maximizations in Eqs. 10. This is illustrated in Fig. 1, right panel. By applying a known heuristic,
information only propagates forward: every simulated step depends on the previous steps, but is
independent from the future simulated steps. This is in contrast to the DP algorithm, illustrated in
Fig. 1. Because the optimal policy is not known, it needs to be built by solving a sequence of nested
problems. Thus, information propagates both forward and backward.
While Hn is simpler to compute than Jn , it still requires computing nested expectations for which
there is no closed-form expression. To further alleviate the cost of computing the long-term reward,
we introduce two numerical simplifications. First, we use a rolling horizon h ? N to decrease the
? = min{N, n+h}.
number of future steps simulated. A rolling horizon h replaces the horizon N by N
Second, the expectations with respect to the (I + 1)-dimensional Gaussian random variables are
numerically approximated using Gauss-Hermite quadrature. We obtain the following formulation:
? ? (S ? ) = 0
H
N +1 N +1
? k (Sk ) = EIc (?k (Sk ); Sk ) + ?
H
Nq
X
(q)
? k+1 (Fk (Sk , ?k (Sk ), w
?(q) [H
k+1 ))],
(11)
q=1
(q)
where Nq is the number of quadrature weights ?(q) ? R and points wk+1 ? RI+1 .
For all iteration n ? {1, ? ? ? , N } and for all xn+1 ? X , we define the utility function of our rollout
algorithm for CBO with finite budget to be:
Un (xn+1 ; Sn ) = EIc (xn+1 ; Sn ) + ?
Nq
X
(q)
? n+1 (Fn (Sn , xn+1 , w
?(q) [H
n+1 ))].
(12)
q=1
The heuristic ? is problem-dependent. A desirable heuristic combines two properties: (1) it is
cheap to compute, (2) it is a good approximation of the optimal policy ? ? . In the case of CBO
with a finite budget, the heuristic ? ought to mimic the exploration-exploitation trade-off balanced
by the optimal policy ? ? . To do so, we propose using a combination of greedy CBO algorithms:
maximization of the constrained expected improvement (which has an exploratory behavior) and
a constrained optimization based on the posterior means of the GPs (which has an exploitative
behavior). For a given iteration n, we define the heuristic ? = {?n+1 , ? ? ? , ?N? } such that for stages
? ? 1}, the policy component ?k : Zk 7? X , maps a state Sk to the design xk+1
k ? {n + 1, ? ? ? , N
satisfying:
xk+1 = argmax EIc (x; Sk ).
(13)
x?X
The last policy component, ?N? : ZN? 7? X , maps a state SN? to xN? +1 such that:
xN? +1 = argmin ?N? (x; f ) s.t. P F (x; SN? ) ? 0.99,
(14)
x?X
where P F is the probability of feasibility known in closed-form. Every evaluation of the utility
function Un requires O Nqh applications of a heuristic component ?k . The heuristic that we propose
optimizes a quantity that requires O |Sk |2 of work.
To summarize, the proposed approach sequentially selects the next design to evaluate by maximizing
the long-term reward induced by a heuristic. This rollout algorithm is a one-step lookahead formulation (one maximization) and is easier to solve than the N -steps lookahead approach (N nested
maximizations) presented in Sec. 3.1. Rollout is a closed-loop approach where the information
collected at a given stage of the simulation is used to simulate the next stages. The heuristic used in
the rollout is problem-dependent, and we proposed using a combination of greedy CBO algorithms
to construct such a heuristic. The computation of the utility function is detailed in Algorithm 2.
6
Algorithm 2 Rollout Utility Function
Function: utility(x, h, S)
Construct GPs using S
if h = 0 then
U ? EIc (x; S)
else
U ? EIc (x; S)
Generate Nq Gauss-Hermite quadrature weights ?(q) and points w(q) associated with x
for q = 1 to Nq do
S 0 ? S ? {(x, w(q) )}
if h > 1 then
x0 ? ?(S 0 ) using Eq. 13
else
x0 ? ?(S 0 ) using Eq. 14
end if
U ? U + ??(q) utility(x0 , h ? 1, S 0 )
end for
end if
Output: U
4
Results
In this section, we numerically investigate the proposed algorithm and demonstrate its performance
on classic test functions and a reacting flow problem.
To compare the performance of the different CBO algorithms tested, we use the utility gap metric
[14]. At iteration n, the utility gap en measures the error between the optimum feasible value f ? and
the value of the objective function at a recommended design x?n :
|f (x?n ) ? f ? | if x?n is feasible,
en =
(15)
|? ? f ? |
else,
where ? is a user-defined penalty punishing infeasible recommendations. The recommended design,
x?n , differs from the design selected for evaluation xn . It is the design that the algorithm would
recommend to evaluate if the optimization were to be stopped at iteration n, without early notice. We
use the same system of recommendation as [14]:
x?n = argmin ?n (x; f ) s.t. P F (x; Sn ) ? 0.975.
(16)
x?X
Note that the utility gap en is not guaranteed to decrease because recommendations x?n are not
necessarily better with iterations. In particular, en is not the best error achieved in the training set Sn .
In the following numerical experiments, for the rollout algorithm, we use independent zero-mean GPs
with automatic relevance determination (ARD) square-exponential kernel to model each expensive-toevaluate function. In Algorithm. 1, when the GPs are constructed, the vector of hyper-parameters ? i
associated with the ith GP kernel is estimated by maximization of the marginal likelihood. However,
to reduce the cost of computing Un , the hyper-parameters are kept constant in the simulated steps (i.e.,
in Algorithm. 2). To compute the expectations of Eqs. 11-12, we employ Nq = 3I+1 Gauss-Hermite
quadrature weights and points and we set the discount factor to ? = 0.9. Finally, at iteration n, the
Sn
best value fbest
is set to the minimum posterior mean ?n (x; f ) over the designs x in the training
set Sn , such that the posterior mean of each constraint is feasible. If no such point can be found,
Sn
2
then fbest
is set to the maximum of {?n (x; f ) + 3?m } over the designs x in Sn , where ?m
is the
maximum variance of the GP associated with f . The EIC algorithm is computed as a special case of
the rollout with rolling horizon h = 0, and we use the Spearmint package1 to run the PESC algorithm.
We additionally run a CBO algorithm that selects the next design to evaluate based on the posterior
means of the GPs2 :
xn+1 = argmin ?n (x; f ) s.t. ?n (x; gi ) ? 0, ?i ? {1, . . . , I}.
(17)
x?X
1
2
https://github.com/HIPS/Spearmint/tree/PESC
As suggested by a reviewer.
7
0.0
log10 Median Utility Gap en
log10 Median Utility Gap en
1
0
?1
?2
?3
P ESC
PM
EIc
?4
Rollout, h = 1
Rollout, h = 2
Rollout, h = 3
?5
0
5
10
15
20
25
Iteration n
30
35
40
P ESC
PM
EIc
?0.5
Rollout, h = 1
Rollout, h = 2
Rollout, h = 3
?1.0
?1.5
?2.0
?2.5
?3.0
0
5
10
15
20
25
Iteration n
30
35
40
Figure 2: Left: Multi-modal objective and single constraint (P1). Right: Linear objective and multiple
non-linear constraints (P2). Shaded region indicates 95% confidence interval of the median statistic.
We refer to this algorithm as PM. We also compare the CBO algorithms to three local algorithms
(SLSQP, MMA and COBYLA) and to one global evolutionary algorithm (ISRES).
We now consider four problems with different design space dimensions d, several numbers of
constraints I, and various topologies of the feasible space. The three first problems, P1-3, are analytic
functions while the last one, P4, uses a reacting flow model that requires solving a set of partial
differential equations (PDEs) [4]. For P1 and P2, we use N = 40 evaluations (as in [6, 10]). For
P3 and P4, we use a small number of iterations N = 60, which corresponds to situations where the
functions are very expensive to evaluate (e.g. solving large systems of PDEs can take over a day on a
supercomputer). The full description of the problems is available in the appendix. In Figs. 2-3, we
show the median of the utility gap, the shadings represent the 95% confidence interval of the median
computed by bootstrap. Other statistics of the utility gap are shown in the appendix.
For P1, the median utility gap for EIC, PESC, PM and the rollout algorithm with h ? {1, 2, 3} is
shown in Fig. 2 (left panel). The PM algorithm does not improve its recommendations. This is not
surprising because PM focuses on exploitation (PM does not depends on posterior variance) which
can result in the algorithm failing to make further progress. Such behavior has already been reported
in [16] (Sec. 3). The three other CBO algorithms perform similarly in the first 10 iterations. PESC
is the first to converge to a utility gap ? 10?2.7 . The rollout performs better or similarly than EIC.
In the 15 first iterations, longer rolling horizons lead to slightly lower utility gaps. This is likely
to be due to the more exploratory behavior associated with lookahead, which helps differentiating
the global solution from the local ones. For the remaining iterations, the shorter rolling horizons
reduce the utility gap faster than longer rolling horizons before reaching a plateau. EIC and rollout
outperform PESC after 25 iterations. We note that EIC and rollout have essentially converged.
For P2, the median performance of EIC, PESC, PM and rollout with rolling horizon h ? {1, 2, 3} is
shown in Fig. 2 (right panel). The PM algorithm reduces the utility gap in the first 10 iterations, but
reaches a plateau at 10?1.7 . The three other CBO algorithms perform similarly up to iteration 15,
where PESC reaches a plateau 3 . This similarity may be explained by the fact that the local solutions
are easily differentiable from the global one, leading to no advantage for exploratory behavior. In
this example, the rollout algorithms reached the same plateau at 10?3 , with longer horizons h taking
more iterations to converge. EIC performs better than rollout h = 2 before its performance slightly
decreases, reaching a plateau at a larger utility gap 10?2.6 (note that the utility gap is not computed
with the best value observed so far and thus is not guaranteed to decrease). This increase of the
median utility gap can be explained by the fact that a few runs change their recommendation from
one local minimum to another one, resulting in the change in median utility function. This is also
reflected in the 95% confidence interval of the median, which further indicates that the statistic is
sensitive to a few runs.
For P3, the median utility gap for the four CBO algorithms is shown in Fig. 3 (left panel). PM is
rapidly outperformed by the other algorithms. The PESC algorithm is outperformed by EIC and
rollout after 25 iterations. Again, we note that rollout with h = 1 obtains a lower utility gap that EIC
at every iteration. The rollout with h ? {2, 3} exhibits a different behavior: it starts decreasing the
utility gap later in the optimization but achieves a better performance when the evaluation budget
3
Results obtained for PESC mean utility gap are consistent with [13].
8
3.5
log10 Median Utility Gap en
log10 Median Utility Gap en
2.2
2.0
1.8
1.6
P ESC
PM
EIc
1.4
Rollout, h = 1
Rollout, h = 2
Rollout, h = 3
1.2
0
10
20
30
Iteration n
40
50
2.5
Rollout, h = 1
Rollout, h = 2
Rollout, h = 3
2.0
1.5
1.0
0.5
0.0
?0.5
0
60
P ESC
PM
EIc
3.0
10
20
30
Iteration n
40
50
60
Figure 3: Left: Multi-modal 4-d objective and constraint (P3). Right: Reacting flow problem (P4).
The awareness of the remaining budget explains the sharp decrease in the last iterations for the rollout.
is consumed. Note that none of the algorithms has converged to the global solution, and the strong
multi-modality of the objective and constraint function seems to favor exploratory behaviors.
For the reacting flow problem P4, the median performances are shown in Fig. 3 (right panel). PM
rapidly reaches a plateau at en ? 101.3 . PESC reduces rapidly the utility gap, outperforming the
other algorithms after 15 iterations. EIC and rollout perform similarly and slowly decrease the utility
gap up to iteration 40, where EIC reaches a plateau and rollout continues to improve performance,
slightly outperforming PESC at the end of the optimization.
The results are summarized in Table. 1, and show that the rollout algorithm with different rolling
horizons h (R-h) performs similarly or favorably compared to the other algorithms.
Table 1: Log median utility gap log10 (eN ). Statistics computed over m independent runs.
Prob
d
N
I
m
SLSQP
MMA
COBYLA
ISRES
PESC
PM
EIC
R-1
R-2
R-3
P1
P2
P3
P4
2
2
4
4
40
40
60
60
1
2
1
1
500
500
500
50
0.59
-0.40
2.15
0.80
0.59
-0.40
3.06
0.80
-0.05
-0.82
3.06
0.80
-0.19
-0.70
1.68
0.13
-2.68
-2.43
1.66
0.09
0.30
-1.76
1.79
1.26
-4.45
-2.62
1.60
0.57
-4.59
-2.99
1.48
-0.10
-4.52
-2.99
1.31
-0.10
-4.42
-2.994
1.35
0.19
Based on the four previous examples, we notice that increasing the rolling horizon h does not
necessarily improve the performance of the rollout algorithm. One possible reason stems from the
fact that lookahead algorithms rely more on the statistical model that greedy algorithms. Because
this model is learned as the optimization unfolds, it is an imperfect model (in particular the hyperparameters of the GPs are updated after each iteration, but not after each stage of a simulated
scenario). By simulating too many steps with the GPs, one may be over-confidently using the model.
In some sense, the rolling horizon h, as well as the discount factor ?, can be interpreted as a form
of regularization. The effect of a larger rolling horizon is problem-dependent, and experiment P3
suggests that multimodal problems in higher dimension may benefits from longer rolling horizons.
5
Conclusions
We proposed a new formulation for constrained Bayesian optimization with a finite budget of
evaluations. The best optimization policy is defined as the one maximizing, in average, the cumulative
feasible decrease of the objective function over multiple steps. This optimal policy is the solution of
a dynamic programming problem that is intractable due to the presence of nested maximizations. To
circumvent this difficulty, we employed the rollout algorithm. Rollout uses a heuristic to simulate
optimization scenarios over several step, thereby computing an approximation of the long-term
reward. This heuristic is problem-dependent and, in this paper, we proposed to use a combination
of cheap-to-evaluate greedy CBO algorithms to construct such heuristic. The proposed algorithm
was numerically investigated and performed similarly or favorably compared to constrained expected
improvement (EIC) and predictive entropy search with constraint (PESC).
This work was supported in part by the AFOSR MURI on multi-information sources of multi-physics
systems under Award Number FA9550-15-1-0038, program manager Dr. Jean-Luc Cambier.
4
For cost reasons, the median for h = 3 was computed with m = 100 independent runs instead of 500.
9
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6,396 | 6,786 | Hierarchical Methods of Moments
Matteo Ruffini ?
Universitat Polit?cnica
de Catalunya
Guillaume Rabusseau ?
McGill University
Borja Balle ?
Amazon Research
Cambridge
Abstract
Spectral methods of moments provide a powerful tool for learning the parameters
of latent variable models. Despite their theoretical appeal, the applicability of
these methods to real data is still limited due to a lack of robustness to model
misspecification. In this paper we present a hierarchical approach to methods of
moments to circumvent such limitations. Our method is based on replacing the
tensor decomposition step used in previous algorithms with approximate joint
diagonalization. Experiments on topic modeling show that our method outperforms
previous tensor decomposition methods in terms of speed and model quality.
1
Introduction
Unsupervised learning of latent variable models is a fundamental machine learning problem. Algorithms for learning a variety of latent variable models, including topic models, hidden Markov
models, and mixture models are routinely used in practical applications for solving tasks ranging
from representation learning to exploratory data analysis. For practitioners faced with the problem of
training a latent variable model, the decades-old Expectation-Maximization (EM) algorithm [1] is
still the tool of choice. Despite its theoretical limitations, EM owes its appeal to (i) the robustness of
the maximum-likelihood principle to model misspecification, and (ii) the need, in most cases, to tune
a single parameter: the dimension of the latent variables.
On the other hand, method of moments (MoM) algorithms for learning latent variable models via
efficient tensor factorization algorithms have been proposed in the last few years [2?9]. Compared to
EM, moment-based algorithms provide a stronger theoretical foundation for learning latent variable
models. In particular, it is known that in the realizable setting the output of a MoM algorithm will
converge to the parameters of the true model as the amount of training data increases. Furthermore,
MoM algorithms only make a single pass over the training data, are highly parallelizable, and always
terminate in polynomial time. However, despite their apparent advantages over EM, the adoption of
MoM algorithms in practical applications is still limited.
Empirical studies indicate that initializing EM with the output of a MoM algorithm can improve
the convergence speed of EM by several orders of magnitude, yielding a very efficient strategy to
accurately learn latent variable models [8?10]. In the case of relatively simple models this approach
can be backed by intricate theoretical analyses [11]. Nonetheless, these strategies are not widely
deployed in practice either.
The main reason why MoM algorithms are not adopted by practitioners is their lack of robustness to
model misspecification. Even when combined with EM, MoM algorithms fail to provide an initial
estimate for the parameters of a model leading to fast convergence when the learning problem is too
far from the realizable setting. For example, this happens when the number of the latent variables
used in a MoM algorithm is too small to accurately represent the training data. In contrast, the model
?
[email protected]
[email protected]
?
[email protected]
?
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
obtained by standalone EM in this case is reasonable and desirable: when asked for a small number of
latent variables EM yields a model which is easy to interpret and can be useful for data visualization
and exploration. For example, an important application of low-dimensional learning can be found
in mixture models, where latent class assignments provided by a simple model can be used to split
the training data into disjoint datasets to which EM is applied recursively to produce a hierarchical
clustering [12, 13]. The tree produced by such clusterings procedure provides a useful aid in data
exploration and visualization even if the models learned at each branching point do not accurately
represent the training data.
In this paper we develop a hierarchical method of moments that produces meaningful results even in
misspecified settings. Our approach is different from previous attemps to design MoM algorithms
for misspecified models. Instead of looking for convex relaxations of existing MoM algorithms
like in [14?16] or analyzing the behavior of a MoM algorithm with a misspecified number of
latent states like in [17, 18], we generalize well-known simultaneous diagonalization approaches to
tensor decomposition by phrasing the problem as a non-convex optimization problem. Despite its
non-convexity, the hierarchical nature of our method allows for a fast accurate solution based on
low-dimensional grid search. We test our method on synthetic and real-world datasets on the topic
modeling task, showcasing the advantages of our approach and obtaining meaningful results.
2
Moments, Tensors, and Latent Variable Models
This section starts by recalling the basic ideas behind methods of moments for learning latent variable
models via tensor decompositions. Then we review existing tensor decomposition algorithms and
discuss the effect of model misspecification on the output of such algorithms.
For simplicity we consider first a single topic model with k topics over a vocabulary with d words.
A single topic model defines a generative process for text documents where first a topic Y ? [k] is
drawn from some discrete distribution P[Y = i] = ?i , and then each word Xt ? [d], 1 ? t ? T , in a
document of length T is independently drawn from some distribution P[Xt = j|Y = i] = ?i,j over
words conditioned on the document topic. The model is completely specified by the vector of topic
proportions ? ? Rk and the word distributions ?i ? Rd for each topic i ? [k]. We collect the word
distributions of the model as the columns of a matrix M = [?1 ? ? ? ?k ] ? Rd?k .
It is convenient to represent the words in a document using one-hot encodings so that Xt ? Rd is an
indicator vector. With this notation, the conditional expectation of any word in a document drawn
PT
from topic i is E[Xt |Y = i] = ?i , and the random vector X = t=1 Xt is conditionally distributed
as a multinomial random variable, with parameters ?i and T . Integrating
Pover topics drawn from ? we
obtain the first moment of the distribution over words M1 = E[Xt ] = i ?i ?i = M ?. Generalizing
this argument to pairs and triples of distinct words in a document yields the matrix of second order
moments and the tensor of third order moments of a single topic model:
X
M2 = E[Xs ? Xt ] =
?i ?i ? ?i ? Rd?d ,
(1)
i
M3 = E[Xr ? Xs ? Xt ] =
X
?i ?i ? ?i ? ?i ? Rd?d?d ,
(2)
i
where ? denotes the tensor (Kronecker) product between vectors. By defining the matrix ? = diag(?)
one also obtains the expression M2 = M ?M > .
A method of moments for learning single topic models proceeds by (i) using a collection of n
? 1, M
? 2, M
? 3 of the moments, and (ii) using matrix and
documents to compute empirical estimates M
tensor decomposition methods to (approximately) factor these empirical moments and extract the
model parameters from their decompositions. From the algorithmic point of view, the appeal of this
scheme resides in the fact that step (i) requires a single pass over the data which can be trivially
parallelized using map-reduce primitives, while step (ii) only requires linear algebra operations whose
running time is independent of n. The specifics of step (ii) will be discussed in Section 2.1.
? m from data with the property that E[M
? m ] = Mm for m ? {1, 2, 3} is the
Estimating moments M
essential requirement for step (i). In the case of single topic models, and more generally multi-view
models, such estimations are straightforward. For example, a simple consistent estimator takes a
? 3 = (1/n) Pn x(i) ? x(i) ? x(i) using the
collection of documents {x(i) }ni=1 and computes M
2
3
i=1 1
2
first three words from each document. More data-efficient estimators for datasets containing long
documents can be found in the literature [19].
For more complex models the method sketched above requires some modifications. Specifically,
it is often necessary to correct the statistics directly observable from data in order to obtain vectors/matrices/tensors whose expectation over a training dataset exhibits precisely the relation with the
parameters ? and M described above. For example, this is the case for Latent Dirichlet Allocation
and mixtures of spherical Gaussians [4, 6]. For models with temporal dependence between observations, e.g. hidden Markov models, the method requires a spectral projection of observables to obtain
moments behaving in a multi-view-like fashion [3, 20]. Nonetheless, methods of moments for these
models and many others always reduces to the factorization of a matrix and tensor of the form M2
and M3 given above.
2.1
Existing Tensor Decomposition Algorithms
Mathematically speaking, methods of moments attempt to solve the polynomial equations in ? and
? m into the expressions for their expectations
M arising from plugging the empirical estimates M
given above. Several approaches have been proposed to solve these non-linear systems of equations.
The tensor power method (TPM) [2] starts with a whitening step where, given the SVD M2 = U SU > ,
the whitening matrix E = U S 1/2 ? Rd?k is used to transform M3 into a symmetric orthogonally
Pk
decomposable tensor T = i=1 ?i E ? ?i ? E ? ?i ? E ? ?i ? Rk?k?k . The weights ?i and vectors
?i are then recovered from T using a tensor power method and inverting the whitening step.
The same whitening matrix is used in [3, 4], where the authors observe that the whitened slices
of M3 are simultaneously diagonalized by the Moore-Penrose pseudoinverse of M ?1/2 . Indeed,
since M2 = M ?M > = EE > , there exists a unique orthonormal matrix O ? Rk?k such that
M ?1/2 = EO. Writing M3,r ? Rd?d for the rth slice of M3 across its second mode and mr for the
rth row of M , it follows that
M3,r = M ?1/2 diag(mr )?1/2 M > = EOdiag(mr )O> E > .
Thus, the problem can be reduced to searching for the common diagonalizer O of the whitened slices
of M3 defined as
Hr = E ? M3,r E ?> = Odiag(mr )O> .
(3)
In the noiseless settings it is sufficient to diagonalize any of the slices M3,r . However, one can also
recover O as the eigenvectors of a random linear combination of the various Hr which is more robust
to noise [3].
Lastly, the method proposed in [21] consists in directly performing simultaneous diagonalization
of random linear combinations of slices of M3 without any whitening step. This method, which in
practice is slower than the others (see Section 4.1), under an incoherence assumption on the vectors
?i , can robustly recover the weights ?i and vectors ?i from the tensor M3 , even when it is not
orthogonally decomposable.
2.2
The Misspecified Setting
All the methods listed in the previous section have been analyzed in the case where the algorithm
only has access to noisy estimates of the moments. However, such analyses assume that the data was
generated by a model from the hypothesis class, that the matrix M has rank k, and that this rank is
known to the algorithm. In practice the dimension k of the latent variable can be cross-validated, but
in many cases this is not enough: data may come from a model outside the class, or from a model with
a very large true k. Besides, the moment estimates might be too noisy to provide reliabe estimates for
large number of latent variables. It is thus frequent to use these algorithms to estimate l < k latent
variables. However, existing algorithms are not robust in this setting, as they have not been designed
to work in this regime, and there is no theoretical explanation of what their outputs will be.
The methods relying on a whitening step [2?4], will perform the whitening using the matrix El?
obtained from the low-rank SVD truncated at rank l: M2 ? Ul Sl Ul> = El El> . TPM will use
El to whiten the tensor M3 to a tensor Tl ? Rl?l?l . However, when k > l, Tl may not admit a
3
symmetric orthogonal decomposition 4 . Consequently, it is not clear what TPM will return in this
case and there are no guarantees it will even converge. The methods from [3, 4] will compute the
matrices Hl,r = El? M3,r El?> for r ? [d] that may not be jointly diagonalizable, and in this case
there is no theoretical justification of what what the result of these algorithms will be. Similarly, the
simultaneous diagonalization method proposed in [21] produces a matrix that nearly diagonalizes the
slices of M3 , but no analysis is given for this setting.
3
Simultaneous Diagonalization Based on Whitening and Optimization
This section presents the main contribution of the paper: a simultaneous diagonalization algorithms
based on whitening and optimization we call SIDIWO (Simultaneous Diagonalization based on
Whitening and Optimization). When asked to produce l = k components in the noiseless setting,
SIDIWO will return the same output as any of the methods discussed in Section 2.1. However, in
contrast with those methods, SIDIWO will provide useful results with a clear interpretation even in a
misspecified setting (l < k).
3.1
SIDIWO in the Realizable Setting
To derive our SIDIWO algorithm we first observe that in the noiseless setting and when l = k, the
pair (M, ?) returned by all methods described in Section 2.1 is the solution of the optimization
problem given in the following lemma5 .
Lemma 3.1 Let M3,r be the r-th slice across the second mode of the tensor M3 from (2) with
parameters (M, ?). Suppose rank(M ) = k and let ? = diag(?). Then the matrix (M ?1/2 )? is the
unique optimum (up to column rescaling) of the optimization problem
!1/2
d
X X
> 2
min
(DM3,r D )i,j
,
(4)
D?Dk
i6=j
r=1
where Dk = {D : D = (EOk )? for some Ok s.t. Ok Ok> = Ik } and E is the whitening matrix
defined in Section 2.1.
Remark 1 (The role of the constraint) Consider the cost function of Problem (4): in an unconstrained setting, there may be several matrices minimizing that cost. A trivial example is the zero
matrix. A less trivial example is when the rows of D belong to the orthogonal complement of the
column space of the matrix M . The constraint D = (EOk )? for some orthonormal matrix Ok first
excludes the zero matrix from the set of feasible solutions, and second guarantees that all feasible
solutions lay in the space generated by the columns of M .
Algorithm 1 SIDIWO: Simultaneous Diagonalization based on Whitening and Optimization
Require: M1 , M2 , M3 , the number of latent states l
1: Compute a SVD of M2 truncated at the l-th singular vector: M2 ? Ul Sl Ul> .
1/2
2: Define the matrix El = Ul Sl ? Rd?l .
3: Find the matrix D ? Dl optimizing Problem (4).
??
? 1/2 = D?
M
?
4: Find (M , ?
? ) solving ? >
M?
? = M1
?,?
5: return (M
?)
Problem (4) opens a new perspective on using simultaneous diagonalization to learn the parameters
of a latent variable model. In fact, one could recover the pair (M, ?) from the relation M ?1/2 = D?
by first finding the optimal D and then individually retrieving M and ? by solving a linear system
using the vector M1 . This approach, outlined in Algorithm 1, is an alternative to the ones presented
in the literature up to now (even though in the noiseless, realizable setting, it will provide the same
4
5
See the supplementary material for an example corroborating this statement.
The proofs of all the results are provided in the supplementary material.
4
results). Similarly to existing methods, this approach requires to know the number of latent states.
We will however see in the next section that Algorithm 1 provides meaningful results even when a
misspecified number of latent states l < k is used.
3.2
The Misspecified Setting
Algorithm 1 requires as inputs the low order moments M1 , M2 , M3 along with the desired number
of latent states l to recover. If l = k, it will return the exact model parameters (M, ?); we will now
see that it will also provide meaningful results when l < k. In this setting, Algorithm 1 returns a pair
?,?
??
? 1/2 )? is optimal for the optimization problem
(M
? ) ? Rd?l ? Rl such that the matrix D = (M
!1/2
d
X X
> 2
min
(DM3,r D )i,j
.
(5)
D?Dl
i6=j
r=1
Analyzing the space of feasible solutions (Theorem 3.1) and the optimization function (Theorem
3.2), we will obtain theoretical guarantees on what SIDIWO returns when l < k, showing that the
trivial solutions are not feasible, and that, in the space of feasible solutions, SIDIWO?s optima will
approximate the true model parameters according to an intuitive geometric interpretation.
Remarks on the constraints. The first step consists in analyzing the space of feasible solutions
Dl when l < k. The observations outlined in Remark 1 still hold in this setting: the zero solution
and the matrices laying in the orthonormal complement of M are not be feasible. Furthermore, the
following theorem shows that other undesirable solutions will be avoided.
Theorem 3.1 Let D ? Dl with rows d1 , ..., dl , and let Ir,s denote the r ? s identity matrix. The
following facts hold under the hypotheses of Lemma 3.1:
1. For any row di , there exists at least one column of M such that hdi , ?j i =
6 0.
? satisfying M
??
? 1/2 = D? are a linear combination of those of M ,
2. The columns of any M
laying in the best-fit l-dimensional subspace of the space spanned by the columns of M .
3. Let ? be any permutation of {1, ..., d}, and let M? and ?? be obtained by permuting the
1/2
columns of M and ? according to ?. If h?i , ?j i =
6 0 for any i, j, then ((M? ?? )Ik,l )? ?
/
1/2 ?
Dl , and similarly Il,k (M? ?? ) ?
/ Dl .
The second point of Theorem 3.1 states that the feasible solutions will lay in the best l-dimensional
subspace approximating the one spanned by the columns of M . This has two interesting consequences:
? cannot simply be a sub-block of the
if the columns of M are not orthogonal, point 3 guarantees that M
original M , but rather a non-trivial linear combination of its columns laying in the best l-dimensional
subspace approximating its column space. In the single topic model case with k topics, when asked
to recover l < k topics, Algorithm 1 will not return a subset of the original k topics, but a matrix
? whose columns gather the original topics via a non trivial linear combination: the original topics
M
? with different weights. When the columns of M are
will all be represented in the columns of M
orthogonal, this space coincides with the space of the l columns of M associated with the l largest ?i ;
1/2
in this setting, the matrix (M? ?? )Ik,l (for some permutation ?) is a feasible solution and minimizes
Problem (5). Thus, Algorithm 1 will recover the top l topics.
??
? 1/2 )? ? Dl is a minimizer of Problem (5).
? be such that D = (M
Interpreting the optima. Let M
? and the original matrix M , we will show
In order to better understand the relation between M
that the cost function of Problem (5) can be written in an equivalent form, that unveils a geometric
interpretation.
Theorem 3.2 Let d1 , ..., dl denote the rows of D ? Dl and introduce the following optimization
problem
k
X
X
min
sup
hdi , ?h ihdj , ?h i?h vh
(6)
D?Dl
k
i6=j
v?VM
h=1
>
where VM = {v ? R : v = ? M, where k?k2 ? 1}. Then this problem is equivalent to (5).
5
First, observe
that the ?
cost function in Equation (6) prefers D?s such that the vectors ui =
?
[hdi , ?1 ?1 i, ..., hdi , ?k ?k i], i ? [l], have disjoint support. This is a consequence of the supv?VM ,
?
and requires that, for each j, the entries hdi , ?j ?j i are close zero for at least all but one of the
various di . Consequently, each center will be almost orthogonal to all but one row of the optimal D;
however the number of centers is greater than the number of rows of D, so the same row di may be
nonorthogonal to various centers.
For illustration, consider the single topic model: a solution D to Problem (6) would have rows that
should be as orthogonal as possible to some topics and as aligned as possible to the others; in other
?
words, for a given topic j, the optimization problem is trying to set hdi , ?j ?j i = 0 for all but one
? of Algorithm 1 should be in essence
of the various di . Consequently, each column of the output M
aligned with some of the topics and orthogonal to the others.
It is worth mentioning that the constraint set Dl forbids the trivial solutions such as the zero matrix, the
pseudo-inverse of any subset of l columns of M ?1/2 , and any subset of l rows of (M ?1/2 )? (which
all have an objective value of 0).
We remark that Theorem 3.2 doesn?t require the matrix M to be full rank k: we only need it to have
at least rank greater or equal to l, in order to guarantee that the constraint set Dl is well defined.
An optimal solution when l = 2. While Problem (4) can be solved in general using an extension
of the Jacobi technique [22, 23], we provide a simple and efficient method for the case l = 2.
This method will then be used to perform hierarchical topic modeling in Section 4. When l = 2,
Equation (5) can be solved optimally with few simple steps; in fact, the following theorem shows that
solving (5) is equivalent to minimizing a continuous function on the compact one-dimensional set
I = [?1, 1], which can easily be done by griding I. Using this in Step 3 of Algorithm 1, one can
efficiently compute an arbitrarily good approximation of the optimal matrix D ? D2 .
?
?
Theorem 3.3 Consider the continuous function F (x) = c1 x4 + c2 x3 1 ? x2 + c3 x 1 ? x2 +
c4 x2 + c5 , where the coefficients c1 , ..., c5 are functions of the entries of M2 and M3 . Let a be the
minimizer of F on [?1, 1], and consider the matrix
?
1 ? a2 ? a
.
Oa =
1 ? a2
?a
Then, the matrix D = (E2 Oa )? is a minimizer of Problem (5) when l = 2.
4
Case Study: Hierarchical Topic Modeling
In this section, we show how SIDIWO can be used to efficiently recover hierarchical representations
of latent variable models. Given a latent variable model with k states, our method allows to recover
?,?
? offer a
a pair (M
? ) from estimate of the moments M1 , M2 and M3 , where the l columns of M
synthetic representation of the k original centers. We will refer to these l vectors as pseudo-centers:
each pseudo-center is representative of a group of the original centers. Consider the case l = 2. A
dataset C of n samples can be split into two smaller subsets according to their similarity to the two
pseudo-centers. Formally, this assignment is done using Maximum A Posteriori (MAP) to find the
pseudo-center giving maximum conditional likelihood to each sample. The splitting procedure can
be iterated recursively to obtain a divisive binary tree, leading to a hierarchical clustering algorithm.
While this hierarchical clustering method can be applied to any latent variable model that can be
learned with the tensor method of moments (e.g. Latent Dirichlet Allocation), we present it here for
the single topic model for the sake of simplicity.
We consider a corpus C of n texts encoded as in Section 2 and we split C into two smaller corpora
according to their similarity to the two pseudo-centers in two steps: project the pseudo-centers on the
simplex to obtain discrete probability distributions (using for example the method described in [24]),
and use MAP assignment to assign each text x to a pseudo-center. This process is summarized in
Algorithm 2. Once the corpus C has been split into two subsets C1 and C2 , each of these subsets may
still contain the full set of topics but the topic distribution will differ in the two: topics similar to the
first pseudo-center will be predominant in the first subset, the others in the second. By recursively
iterating this process, we obtain a binary tree where topic distributions in the nodes with higher depth
are expected to be more concentrated on fewer topics.
6
Algorithm 2 Splitting a corpus into two parts
Require: A corpus of texts C = (x(1) , ..., x(n) ).
1: Estimate M1 , M2 and M3 .
2: Recover l = 2 pseudo-center with Algorithm 1 .
3: Project the Pseudo-center to the simplex
4: for i ? [n] do
? ], where
5:
Assign the text x(i) to the cluster Cluster(i) = arg maxj P[X = x(i) |Y = j, ?
?, M
?
P[X|Y = j, ?
? , M ] is the multinomial distr. associated to the j-th projected pseudo-center.
6: end for
7: return The cluster assignments Cluster.
(a)
(b)
(c)
Figure 1: Figure 1a provides a visualization of the topics used to generate the sample. Figure 1b
represents the hierarchy recovered with the proposed method. Table 1c reports the average and
standard deviation over 10 runs of the clustering accuracy for the various methods, along with average
running times.
In the next sections, we assess the validity of this approach on both synthetic and real-world data6 .
4.1
Experiment on Synthetic Data
In order to test the ability of SIDIWO to recover latent structures in data, we generate a dataset
distributed as a single topic model (with a vocabulary of 100 words) whose 8 topics have an intrinsic
hierarchical structure depicted in Figure 1a. In this figure, topics are on the x-axis, words on the
y-axis, and green (resp. red) points represents high (resp low) probability. We see for example that
the first 4 topics are concentrated over the 1st half of the vocabulary, and that topics 1 and 2 have
high probability on the 1st and 3rd fourth of the words while for the other two it is on the 1st and 4th.
We generate 400 samples according to this model and we iteratively run Algorithm 2 to create a
hierarchical binary tree with 8 leafs. We expect leafs to contain samples from a unique topic and
internal nodes to gather similar topics. Results are displayed in Figure 1b where each chart represents
a node of the tree (child nodes lay below their parent) and contains the heatmap of the samples
clustered in that node (x-axis corresponds to samples and y-axis to words, red points are infrequent
words and clear points frequent ones). The results are as expected: each leaf contains sample from
one of the topics and internal nodes group similar topics together.
We compare the clustering accuracy of SIDIWO with other methods using the Adjusted Rand Index
[26] of the partition of the data obtained at the leafs w.r.t the one obtained using the true topics;
comparisons are with the flat clustering on k = 8 topics with TPM, the method from [3] (SVD) and
the one from [21] (Rand. Proj.). We repeat the experiment 10 times with different random samples
and we report the average results in Table 1c; SIDIWO always recovers the original topic almost
perfectly, unlike competing methods. One intuition for this improvement is that each split in the
divisive clustering helps remove noise in the moments.
6
The experiments in this section have been performed in Python 2.7, using numpy [25] library for linear
algebra operations, with the exception of the implementation of the method from [21], for which we used
the author?s Matlab implementation: https://github.com/kuleshov/tensor-factorization. All the
experiments were run on a MacBook Pro with an Intel Core i5 processor.
7
Figure 2: Experiment on the NIPS dataset.
Figure 3: Experiment on the Wikipedia Mathematics Pages dataset.
4.2
Experiment on NIPS Conference Papers 1987-2015
We consider the full set of NIPS papers accepted between 1987 and 2015 , containing n = 11, 463
papers [27]. We assume that the papers are distributed according to a single topic model, we keep the
d = 3000 most frequent words as vocabulary and we iteratively run Algorithm 2 to create a binary
tree of depth 4. The resulting tree is shown in Figure 2 where each node contains the most relevant
words of the cluster, where the relevance [28] of a word w ? Cnode ? C is defined by
r(w, Cnode ) = ? log P[w|Cnode ] + (1 ? ?) log
P[w|Cnode ]
,
P[w|C]
where the weight parameter is set to ? = 0.7 and P[w|Cnode ] (resp. P[w|C]) is the empirical frequency
of w in Cnode (resp. in C). The leafs clustering and the whole hierarchy have a neat interpretation.
Looking at the leaves, we can easily hypothesize the dominant topics for the 8 clusters. From left to
right we have: [image processing, probabilistic models], [neuroscience, neural networks], [kernel
methods, algorithms], [online optimization, reinforcement learning]. Also, each node of the lower
levels gathers meaningful keywords, confirming the ability of the method to hierarchically find
meaningful topics. The running time for this experiment was 59 seconds.
4.3
Experiment on Wikipedia Mathematics Pages
We consider a subset of the full Wikipedia corpus, containing all articles (n = 809 texts) from the
following math-related categories: linear algebra, ring theory, stochastic processes and optimization.
We remove a set of 895 stop-words, keep a vocabulary of d = 3000 words and run SIDIWO to
perform hierarchical topic modeling (using the same methodology as in the previous section). The
resulting hierarchical clustering is shown in Figure 3 where we see that each leaf is characterized by
one of the dominant topics: [ring theory, linear algebra], [stochastic processes, optimization] (from
8
left to right). It is interesting to observe that the first level of the clustering has separated pure
mathematical topics from applied ones. The running time for this experiment was 6 seconds.
5
Conclusions and future works
We proposed a novel spectral algorithm (SIDIWO) that generalizes recent method of moments
algorithms relying on tensor decomposition. While previous algorithms lack robustness to model
misspecification, SIDIWO provides meaningful results even in misspecified settings. Moreover,
SIDIWO can be used to perform hierarchical method of moments estimation for latent variable
models. In particular, we showed through hierarchical topic modeling experiments on synthetic and
real data that SIDIWO provides meaningful results while being very computationally efficient.
A natural future work is to investigate the capability of the proposed hierarchical method to learn
overcomplete latent variable models, a task that has received significant attention in recent literature
[29, 30]. We are also interested in comparing the learning performance of SIDIWO the with those
of other existing methods of moments in the realizable setting. On the applications side, we are
interested in applying the methods developed in this paper to the healthcare analytics field, for
instance to perform hierarchical clustering of patients using electronic healthcare records or more
complex genetic data.
Acknowledgments
Guillaume Rabusseau acknowledges support of an IVADO postdoctoral fellowship. Borja Balle
completed this work while at Lancaster University.
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6,397 | 6,787 | Interpretable and Globally Optimal Prediction for
Textual Grounding using Image Concepts
Raymond A. Yeh,
Jinjun Xiong? ,
Minh N. Do,
Wen-mei W. Hwu,
Alexander G. Schwing
Department of Electrical Engineering, University of Illinois at Urbana-Champaign
?
IBM Thomas J. Watson Research Center
[email protected], [email protected], [email protected],
[email protected], [email protected]
Abstract
Textual grounding is an important but challenging task for human-computer interaction, robotics and knowledge mining. Existing algorithms generally formulate
the task as selection from a set of bounding box proposals obtained from deep
net based systems. In this work, we demonstrate that we can cast the problem of
textual grounding into a unified framework that permits efficient search over all
possible bounding boxes. Hence, the method is able to consider significantly more
proposals and doesn?t rely on a successful first stage hypothesizing bounding box
proposals. Beyond, we demonstrate that the trained parameters of our model can be
used as word-embeddings which capture spatial-image relationships and provide
interpretability. Lastly, at the time of submission, our approach outperformed the
current state-of-the-art methods on the Flickr 30k Entities and the ReferItGame
dataset by 3.08% and 7.77% respectively.
1
Introduction
Grounding of textual phrases, i.e., finding bounding boxes in images which relate to textual phrases, is
an important problem for human-computer interaction, robotics and mining of knowledge bases, three
applications that are of increasing importance when considering autonomous systems, augmented
and virtual reality environments. For example, we may want to guide an autonomous system by using
phrases such as ?the bottle on your left,? or ?the plate in the top shelf.? While those phrases are easy to
interpret for a human, they pose significant challenges for present day textual grounding algorithms,
as interpretation of those phrases requires an understanding of objects and their relations.
Existing approaches for textual grounding, such as [38, 15] take advantage of the cognitive performance improvements obtained from deep net features. More specifically, deep net models are
designed to extract features from given bounding boxes and textual data, which are then compared to
measure their fitness. To obtain suitable bounding boxes, many of the textual grounding frameworks,
such as [38, 15], make use of region proposals. While being easy to obtain, automatic extraction of
region proposals is limiting, because the performance of the visual grounding is inherently constrained
by the quality of the proposal generation procedure.
In this work we describe an interpretable mechanism which additionally alleviates any issues arising
due to a limited number of region proposals. Our approach is based on a number of ?image concepts?
such as semantic segmentations, detections and priors for any number of objects of interest. Based on
those ?image concepts? which are represented as score maps, we formulate textual grounding as a
search over all possible bounding boxes. We find the bounding box with highest accumulated score
contained in its interior. The search for this box can be solved via an efficient branch and bound
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
A woman in a green shirt is getting ready to throw her bowling
ball down the lane...
Two women wearing hats covered in flowers are posing.
Young man wearing a hooded
jacket sitting on snow in front
of mountain area.
painting next to the two on the
person all the way to the right
left
Figure 1: Results on the test set for grounding of textual phrases using our branch and bound based
algorithm. Top Row: Flickr 30k Entities Dataset. Bottom Row: ReferItGame Dataset (Groundtruth
box in green and predicted box in red).
second bike from right in front
scheme akin to the seminal efficient subwindow search of Lampert et al. [25]. The learned weights
can additionally be used as word embeddings. We are not aware of any method that solves textual
grounding in a manner similar to our approach and hope to inspire future research into the direction
of deep nets combined with powerful inference algorithms.
We evaluate our proposed approach on the challenging ReferItGame [20] and the Flickr 30k Entities
dataset [35], obtaining results like the ones visualized in Fig. 1. At the time of submission, our
approach outperformed state-of-the-art techniques on the ReferItGame and Flickr 30k Entities dataset
by 7.77% and 3.08% respectively using the IoU metric. We also demonstrate that the trained
parameters of our model can be used as a word-embedding which captures spatial-image relationships
and provides interpretability.
2
Related Work
Textual grounding: Related to textual grounding is work on image retrieval. Classical approaches
learn a ranking function using recurrent neural nets [30, 6], or metric learning [13], correlation
analysis [22], and neural net embeddings [9, 21]. Beyond work in image retrieval, a variety of
techniques have been considered to explicitly ground natural language in images and video. One of
the first models in this area was presented in [31, 24]. The authors describe an approach that jointly
learns visual classifiers and semantic parsers.
Gong et al. [10] propose a canonical correlation analysis technique to associate images with descriptive sentences using a latent embedding space. In spirit similar is work by Wang et al. [43], which
learns a structure-preserving embedding for image-sentence retrieval. It can be applied to phrase
localization using a ranking framework. In [11], text is generated for a set of candidate object regions
which is subsequently compared to a query. The reverse operation, i.e., generating visual features
from query text which is subsequently matched to image regions is discussed in [1].
In [23], 3D cuboids are aligned to a set of 21 nouns relevant to indoor scenes using a Markov random
field based technique. A method for grounding of scene graph queries in images is presented in [17].
Grounding of dependency tree relations is discussed in [19] and reformulated using recurrent nets
in [18]. Subject-Verb-Object phrases are considered in [40] to develop a visual knowledge extraction
system. Their algorithm reasons about the spatial consistency of the configurations of the involved
entities. In [15, 29] caption generation techniques are used to score a set of proposal boxes and
returning the highest ranking one. To avoid application of a text generation pipeline on bounding
box proposals, [38] improve the phrase encoding using a long short-term memory (LSTM) [12]
based deep net. Additional modeling of object context relationship were explored in [32, 14]. Video
2
9/5/2017
bbest_redraw
Input?
Features? ?
a
Query? :
"The?left?guy"
guy
Word?Prior
Energy?
left
Image? :
Detection
the
Output? :
youth
Segmentation
Figure 2: Overview of our proposed approach: We obtain word priors from the input query, take into
account geometric features, as well as semantic segmentation features computed from the provided
input image. We compute the three image cues to predict the four variables of the bounding box
y = (y1 , . . . , y4 ).
datasets, although not directly related to our work in this paper, were used for spatiotemporal language
grounding in [27, 47].
Common datasets for visual grounding are the ReferItGame dataset [20] and a newly introduced
Flickr 30k Entities dataset [34], which provides bounding box annotations for noun phrases of the
original Flickr 30k dataset [46].
In contrast to all of the aforementioned methods, which are largely based on region proposals, we
suggest usage of efficient subwindow search as a suitable inference engine.
Efficient subwindow search: Efficient subwindow search was proposed by Lampert et al. [25]
for object localization. It is based on an extremely effective branch and bound scheme that can
be applied to a large class of energy functions. The approach has been applied to very efficient
deformable part models [45], for object class detection [26], for weakly supervised localization [5],
indoor scene understanding [41], diverse object proposals [42] and also for spatio-temporal object
detection proposals [33].
3
Exact Inference for Grounding
We outline our approach for textual grounding in Fig. 2. In contrast to the aforementioned techniques
for textual grounding, which typically use a small set of bounding box proposals, we formulate our
language grounding approach as an energy minimization over a large number of bounding boxes.
The search over a large number of bounding boxes allows us to retrieve an accurate bounding-box
prediction for a given phrase and an image. Importantly, by leveraging efficient branch-and-bound1/1
techniques, we are able to find the global minimizer for a given energy function very effectively.
Our energy is based on a set of ?image concepts? like semantic segmentations, detections or image
priors. All those concepts come in the form of score maps which we combine linearly before searching
for the bounding box containing the highest accumulated score over the combined score map. It is
trivial to add additional information to our approach by adding additional score maps. Moreover,
linear combination of score maps reveals importance of score maps for specific queries as well as
similarity between queries such as ?skier? and ?snowboarder.? Hence the framework that we discuss
in the following is easy to interpret and extend to other settings.
General problem formulation: For simplicity we use x to refer to both given input data modalities,
i.e., x = (Q, I), with query text, Q, and image, I. We will differentiate them in the narrative. In
addition, we define a bounding box y via its top left corner (y1 , y2 ) and its bottom right corner (y3 , y4 )
Q4
and subsume the four variables of interest in the tuple y = (y1 , . . . , y4 ) ? Y = i=1 {0, . . . , yi,max }.
Every integral coordinate yi , i ? {1, . . . , 4} lies within the set {0, . . . , yi,max }, and Y denotes the
3
?left?
?center?
Algorithm 1 Branch and bound inference for
grounding
? Y, w), Y) into queue, set Y? =
1: put pair (E(x,
Y
2: repeat
3:
split Y? = Y?1 ? Y?2 with Y?1 ? Y?2 = ?
? Y1 , w), Y1 ) into queue
4:
put pair (E(x,
? Y2 , w), Y2 ) into queue
5:
put pair (E(x,
?
?
6:
retrieve Y having smallest E
?
7: until |Y| = 1
?right?
?floor?
(a)
(b)
Figure 3: Word priors in (a) and the employed inference algorithm in (b).
product space of all four coordinates. For notational simplicity only, we assume all images to be
scaled to identical dimensions, i.e., yi,max is not dependent on the input data x. We obtain a bounding
box prediction y? given our data x, by solving the energy minimization
y? = arg min E(x, y, w),
y?Y
(1)
to global optimality. Note that w refers to the parameters of our model. Despite the fact that we are
?only? interested in a single bounding box, the product space Y is generally too large for exhaustive
minimization of the energy specified in Eq. (1). Therefore, we pursue a branch-and-bound technique
in the following.
To apply branch and bound, we assume that the energy function E(x, y, w) depends on two sets
of parameters w = [wtT , wrT ]T , i.e., the top layer parameters wt of a neural net, and the remaining
parameters wr . In light of this decomposition, our approach requires the energy function to be of the
following form:
E(x, y, w) = wtT ?(x, y, wr ).
Note that the features ?(x, y, wr ) may still depend non-linearly on all but the top-layer parameters.
This assumption does not pose a severe restriction since almost all of the present-day deep net models
typically obtain the logits E(x, y, w) using a fully-connected layer or a convolutional layer with
kernel size 1 ? 1 as the last computation.
Energy Function Details: Our energy function E(x, y, w) is based on a set of ?image concepts,?
such as semantic segmentation of object categories, detections, or word priors, all of which we
subsume in the set C. Importantly, all image concepts c ? C are attached a parametric score map
??c (x, wr ) ? RW ?H following the image width W and height H. Note that those parametric score
maps may depend nonlinearly on some parameters wr . Given a bounding box y, we use the scalar
?c (x, y, wr ) ? R to refer to the score accumulated within the bounding box y of score map ??c (x, wr ).
To define the energy function we also introduce a set of words of interest, i.e., S. Note that this set
contains a special symbol denoting all other words not of interest for the considered task. We use the
given query Q, which is part of the data x, to construct indicators, ?s = ?(s ? Q) ? {0, 1}, denoting
for every token s ? S its existence in the query Q, where ? denotes the indicator function.
Based on this definition, we formulate the energy function as follows:
X X
E(x, y, w) =
ws,c ?c (x, y, wr ),
(2)
s?S:?s =1 c?C
where ws,c is a parameter connecting a word s ? S to an image concept c ? C. In other words,
wt = (ws,c : ?s ? S, c ? C). This energy function results in a sparse wt , which increases the speed
of inference.
Score maps: The energy is given by a linear combination of accumulated score maps ?c (x, y, wr ).
In our case, we use |C| = k1 + k2 + k3 of those maps, which capture three kinds of information:
(i) k1 word-priors; (ii) k2 geometric information cues; and (iii) k3 image based segmentations and
detections. We discuss each of those maps in the following.
4
Approach
Accuracy (%)
SCRC (2016) [15]
27.80
DSPE (2016) [44]
43.89
GroundeR (2016) [39]
47.81
CCA (2017) [36]
50.89
Ours (Prior + Geo + Seg + Det)
51.63
Ours (Prior + Geo + Seg + bDet)
53.97
Approach
Accuracy (%)
SCRC (2016) [15]
17.93
GroundeR (2016) [39]
23.44
GroundeR (2016) [39] +SPAT
26.93
Ours (Prior + Geo)
25.56
Ours (Prior + Geo + Seg)
33.36
Ours (Prior + Geo + Seg + Det)
34.70
Table 1: Phrase localization performance on
Flickr 30k Entities.
Table 2: Phrase localization performance on
ReferItGame.
people clothing body parts animals vehicles instruments scene other
# Instances
5,656 2,306
523
518
400
162
1,619 3,374
GroundeR(2016) [39] 61.00 38.12
10.33
62.55
68.75
36.42
58.18 29.08
CCA(2017) [36]
64.73 46.88
17.21
65.83
68.75
37.65
51.39 31.77
Ours
68.71 46.83
19.50
70.07
73.75
39.50
60.38 32.45
Table 3: Phrase localization performance over types on Flickr 30k Entities (accuracy in %).
For the top k1 words in the training set we construct word prior maps like the ones shown in Fig. 3 (a).
To obtain the prior for a particular word, we search a given training set for each occurrence of the
word. With the corresponding subset of image-text pairs and respective bounding box annotations at
hand, we compute the average number of times a pixel is covered by a bounding box. To facilitate
this operation, we scale each image to a predetermined size. Investigating the obtained word priors
given in Fig. 3 (a) more carefully, it is immediately apparent that they provide accurate location
information for many of the words.
The k2 = 2 geometric cues provide the aspect ratio and the area of the hypothesized bounding box y.
Note that the word priors and geometry features contain no information about the image specifics.
To encode measurements dedicated to the image at hand, we take advantage of semantic segmentation
and object detection techniques. The k3 image based features are computed using deep neural nets as
proposed by [4, 37, 2]. We obtain probability maps for a set of class categories, i.e., a subset of the
nouns of interest. The feature ? accumulates the scores within the hypothesized bounding box y.
Inference: The algorithm to find the bounding box y? with lowest energy as specified in Eq. (1) is
based on an iterative decomposition of the output space Y [25], summarized in Fig. 3 (b). To this end
we search across subsets of the product space Y and we define for every coordinate yi , i ? {1, . . . , 4}
a corresponding lower and upper bound, yi,low and yi,high respectively. More specifically, considering
the initial set of all possible bounding boxes Y, we divide it into two disjoint subsets Y?1 and Y?2 . For
example, by constraining y1 to {0, . . . , y1,max /2} and {y1,max /2 + 1, . . . , y1,max } for Y?1 and Y?2
respectively, while keeping all the other intervals unchanged. It is easy to see that we can repeat this
decomposition by choosing the largest among the four intervals and recursively dividing it into two
parts.
Given such a repetitive decomposition strategy for the output space, and since the energy E(x, y, w)
for a bounding box y is obtained using a linear combination of word priors and accumulated
segmentation masks, we can design an efficient branch and bound based search algorithm to exactly
solve the inference problem specified in Eq. (1). The algorithm proceeds by iteratively decomposing
a product space Y? into two subspaces Y?1 and Y?2 . For each subspace, the algorithm computes a lower
? Yj , w) for the energy of all possible bounding boxes within the respective subspace.
bound E(x,
Intuitively, we then know, that any bounding box within the subspace Y?j has a larger energy than
the lower bound. The algorithm proceeds by choosing the subspace with lowest lower-bound until
? = 1. We summarize this algorithm in Alg. 1
this subspace consists of a single element, i.e., until |Y|
(Fig. 3 (b)).
? Yj , w) on the energy for an
To this end, it remains to show how to compute a lower bound E(x,
output space, and to illustrate the conditions which guarantee convergence to the global minimum of
the energy function.
For the latter, we note that two conditions are required to ensure convergence to the optimum: (i) the
bound of the considered product space has to lower-bound the true energy for each of its bounding
5
The lady in the red car is
crossing the bridge.
A dog and a cow play together
inside the fence.
person on the left
black bottle front
A woman wearig the black
sunglasses and blue jean jacket
is smiling.
floor on the bottom
Figure 4: Results on the test set for grounding of textual phrases using our branch and bound based
algorithm. Top Row: Flickr 30k Entities Dataset. Bottom Row: ReferItGame Dataset (Groundtruth
box in green and predicted box in red).
? i.e., ??
? E(x,
? Y,
? w) ? E(x, y?, w); (ii) the bound has to be exact for all
box hypothesis y? ? Y,
y ? Y,
?
possible bounding boxes y ? Y, i.e., E(x, y, w) = E(x, y, w). Given those two conditions, global
convergence of the algorithm summarized in Alg. 1 is apparent: upon termination we obtain an
?interval? containing a single bounding box, and its energy is at least as low as the one for any other
interval.
For the former, we note that bounds on score maps for bounding box intervals can be computed by
considering either the largest or the smallest possible bounding box in the bounding box hypothesis,
? depending on whether the corresponding weight in wt is positive or negative and whether the
Y,
feature maps contain only positive or negative values. Intuitively, if the weight is positive and
? Y,
? w) by
the feature mask contains only positive values, we obtain the smallest lower bound E(x,
considering the content within the smallest possible bounding box. Note that the score maps do not
necessarily contain only positive or negative numbers. However we can split the given score maps
into two separate score maps (i.e., one with only positive values, and another with only negative
values) while applying the same weight.
? Y,
? w) has to be extremely effective for the
It is important to note that computation of the bound E(x,
algorithm to run at a reasonable speed. However, computing the feature mask content for a bounding
box is trivially possible using integral images. This results in a constant time evaluation of the bound,
which is a necessity for the success of the branch and bound procedure.
Learning the Parameters: With the branch and bound based inference procedure at hand, we
now describe how to formulate the learning task. Support-vector machine intuition can be applied.
Formally, we are given a training set D = {(x, y)} containing pairs of input data x and groundtruth
bounding boxes y. We want to find the parameters w of the energy function E(x, y, w) such that
the energy of the groundtruth is smaller than the energy of any other configuration. Negating this
statement results in the following desiderata when including an additional margin term L(y, y?), also
known as task-loss, which measures the loss between the groundtruth y and another configuration y?:
?E(x, y, w) ? ?E(x, y?, w) + L(?
y , y) ??
y ? Y.
Since we want to enforce this inequality for all configurations y? ? Y, we can reduce the number of
constraints by enforcing it for the highest scoring right hand side. We then design a cost function
which penalizes violation of this requirement linearly. We obtain the following structured support
vector machine based surrogate loss minimization:
X
C
min
kwk22 +
max (?E(x, y?, w) + L(?
y , y)) + E(x, y, w)
(3)
w
y??Y
2
(x,y)?D
where C is a hyperparameter adjusting the squared norm regularization to the data term. For the task
loss L(?
y , y) we use intersection over union (IoU).
6
her shoes
a red shirt
a dirt bike
Figure 5: Flickr 30k Failure Cases. (Green box: ground-truth, Red box:predicted)
By fixing the parameters wr and only learning the top layer parameters wt , Eq. (3) is equivalent to the
problem of training a structured SVM. We found the cutting-plane algorithm [16] to work well in our
context. The cutting-plane algorithm involves solving the maximization task. This maximization over
the output space Y is commonly referred to as loss-augmented inference. Loss augmented inference
is structurally similar to the inference task given in Eq. (1). Since maximization is identical to
negated minimization, the computation of the bounds for the energy E(x, y?, w) remains identical. To
bound the IoU loss, we note that a quotient can be bounded by bounding nominator and denominator
independently. To lower bound the intersection of the groundtruth box with the hypothesis space we
use the smallest hypothesized bounding box. To upper bound the union of the groundtruth box with
the hypothesis space we use the largest bounding box.
Further, even though not employed to obtain the results in this paper, we mention that it is possible
to backpropagate through the neural net parameters wr that influence the energy non-linearly. This
underlines that our initial assumption is merely a construct to design an effective inference procedure.
4
Experimental Evaluation
In the following we first provide additional details of our implementation before discussing the results
of our approach.
Language processing: In order to process free-form textual phrases efficiently, we restricted the
vocabulary size to the top 200 most frequent words in the training set for the ReferItGame, and to the
top 1000 most frequent training set words for Flickr 30k Entities; both choices cover about 90% of
all phrases in the training set. We map all the remaining words into an additional token. We don?t
differentiate between uppercase and lower case characters and we also ignore punctuation.
Segmentation and detection maps: We employ semantic segmentation, object detection, and poseestimation. For segmentation, we use the DeepLab system [4], trained on PASCAL VOC-2012 [8]
semantic image segmentation task, to extract the probability maps for 21 categories. For detection,
we use the YOLO object detection system [37], to extract 101 categories, 21 trained on PASCAL
VOC-2012, and 80 trained on MSCOCO [28]. For pose estimation, we use the system from [2] to
extract the body part location, then post-process to get the head, upper body, lower body, and hand
regions.
For the ReferItGame, we further fine-tuned the last layer of the DeepLab system to include the
categories of ?sky,? ?ground,? ?building,? ?water,? ?tree,? and ?grass.? For the Flickr 30k Entities, we
also fine-tuned the last layer of the DeepLab system using the eight coarse-grained types and eleven
colors from [36].
Preprocessing and post-processing: For word prior feature maps and the semantic segmentation maps, we take an element-wise logarithm to convert the normalized feature counts into logprobabilities. The summation over a bounding box region then retains the notion of a joint logprobability. We also centered the feature maps to be zero-mean, which corresponds to choosing
an initial decision threshold. The feature maps are resized to dimension of 64 ? 64 for efficient
computation, and the predicted box is scaled back to the original image dimension during evaluation.
We re-center the prediction box by a constant amount determined using the validation set, as resizing
truncate box coordinates to an integer.
Efficient sub-window search implementation: In order for the efficient subwindow search to run
at a reasonable speed, the lower bound on E needs to be computed as fast as possible. Observe
that, E(x, y, w), is a weighted sum of the feature maps over the region specified by a hypothesized
bounding box. To make this computation efficient, we pre-compute integral images. Given an integral
7
Query word, s
Query word, s
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
bicycle
bike
camera
cellphone
coffee
cup
drink
man
skier
snowboarder
woman
0.90
0.75
0.60
0.45
0.30
0.15
0.00
air
cht
ca
r
ca
s
bu le
tt
boat
bo d
bir le
yc e
bic plan
ro
ae
womanoarder
nowb
sskier
man
drink
cup
coffee ne
cellphoa
camer
bike
bicycle
plane
bike
bird
ship
bottle
bus
car
cat
chair
Query word, s 0
Concept, c
(a)
(b)
Figure 6: (a) Trained weight, ws,c , visualization on words, s and segmentation concepts, c, on Flicker
30k. (b) Cosine similairty visualization between words vector, ws and ws0 on Flicker 30k.
image, the computation for each of the bounding box is simply a look-up operation. This trick can
similarly be applied for the geometric features. Since we know the range of the ratio and areas of the
bounding boxes ahead of time, we cache the results in a look up table as well.
The ReferItGame dataset consists of more than 99,000 regions from 20,000 images. Bounding
boxes are assigned to natural language expressions. We use the same bounding boxes as [38] and
the same training test set split, i.e., 10,000 images for testing, 9,000 images for training and 1,000
images for validation.
The Flickr 30k Entities dataset consists of more than 275k bounding boxes from 31k image, where
each bounding box is annotated with the corresponding natural language phrase. We us the same
training, validation and testing split as in [35].
Quantitative evaluation: In Tab. 1 and Tab. 2 we quantitatively compare the results of our approach
to recent state-of-the-art baselines, where Prior = word priors, Geo = geometric information, Seg
= Segmentation maps, Det = Detection maps, bDet = Detection maps + body parts detection. An
example is considered as correct, if the predicted box overlaps with the ground-truth box by more
than 0.5 IoU. We observe our approach to outperform competing methods by around 3% on the Flickr
30k Entities dataset and by around 7% on the ReferItGame dataset.
We also provide an ablation study of the word and image information as shown in Tab. 1 and Tab. 2.
In Tab. 3 we analyze the results for each ?phrase type? provided by Flicker30k Entities dataset. As
can be seen, our system outperforms the state-of-the-art in all phrase types except for clothing.
We note that our results have been surpassed by [3, 7], where they fine-tuned the entire network
including the feature extractions; CCA, GroundeR and our approach uses a fixed pre-trained network
for extracting image features.
Qualitative evaluation: Next we evaluate our approach qualitatively. In Fig. 1 and Fig. 4 we show
success cases. We observe that our method successfully captures a variety of objects and scenes. In
Fig. 5 we illustrate failure cases. We observe that for a few cases word prior may hurt the prediction
(e.g., shoes are typically on the bottom half of the image.) Also our system may fail when the energy
is not a linear combination of the feature scores. For example, the score of ?dirt bike? should not be
the score of ?dirt? + the score of ?bike.? We provide additional results in the supplementary material.
Learned parameters + word embedding: Recall, in Eq. (2), our model learns a parameter per
phrase word and concept pair, ws,c . We visualize its magnitude in Fig. 6 (a) for a subset of words and
concepts. As can be seen, ws,c is large, when the phrase word and the concept are related, (e.g. s =
ship and c = boat). This demonstrates that our model successfully learns the relationship between
phrase words and image concepts. This also means that the ?word vector,? ws = [ws,1 , ws,2 , ...ws,|C| ],
can be interpreted as a word embedding. Therefore, in Fig. 6 (b), we visualize the cosine similarity
between pairs of word vectors. Expected groups of words form, for example (bicycle, bike), (camera,
cellphone), (coffee, cup, drink), (man woman), (snowboarder, skier). The word vectors capture
8
image-spatial relationship of the words, meaning items that can be ?replaced? in an image are similar;
(e.g., a ?snowboarder? can be replaced with a ?skier? and the overall image would still be reasonable).
Computational Efficiency: Overall, our method?s inference speed is comparable to CCA and much
faster than GroundeR. The inference speed can be divided into three main parts, (1) extracting image
features, (2) extracting language features, and (3) computing scores. For extracting image features,
GroundeR requires a forward pass on VGG16 for each image region, where CCA and our approach
requires a single forward pass which can be done in 142.85 ms. For extracting language features, our
method requires index lookups, which takes negligible amount of time (less than 1e-6 ms). CCA,
uses Word2vec for processing the text, which takes 0.070 ms. GroundeR uses a Long-Short-Term
Memory net, which takes 0.7457 ms. Computing the scores with our C++ implementation takes
1.05ms on a CPU. CCA needs to compare projections of the text and image features, which takes
13.41ms on a GPU and 609ms on a CPU. GroundeR uses a single fully connected layer, which takes
0.31 ms on a GPU.
5
Conclusion
We demonstrated a mechanism for grounding of textual phrases which provides interpretability, is
easy to extend, and permits globally optimal inference. In contrast to existing approaches which are
generally based on a small set of bounding box proposals, we efficiently search over all possible
bounding boxes. We think interpretability, i.e., linking of word and image concepts, is an important
concept, particularly for textual grounding, which deserves more attention.
Acknowledgments: This material is based upon work supported in part by the National Science
Foundation under Grant No. 1718221. This work is supported by NVIDIA Corporation with the
donation of a GPU. This work is supported in part by IBM-ILLINOIS Center for Cognitive Computing
Systems Research (C3SR) - a research collaboration as part of the IBM Cognitive Horizons Network.
9
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11
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6,398 | 6,788 | Revisit Fuzzy Neural Network:
Demystifying Batch Normalization and ReLU with
Generalized Hamming Network
Lixin Fan
[email protected]
Nokia Technologies
Tampere, Finland
Abstract
We revisit fuzzy neural network with a cornerstone notion of generalized hamming distance, which provides a novel and theoretically justified framework to
re-interpret many useful neural network techniques in terms of fuzzy logic. In particular, we conjecture and empirically illustrate that, the celebrated batch normalization (BN) technique actually adapts the ?normalized? bias such that it approximates
the rightful bias induced by the generalized hamming distance. Once the due bias
is enforced analytically, neither the optimization of bias terms nor the sophisticated
batch normalization is needed. Also in the light of generalized hamming distance,
the popular rectified linear units (ReLU) can be treated as setting a minimal hamming distance threshold between network inputs and weights. This thresholding
scheme, on the one hand, can be improved by introducing double-thresholding on
both positive and negative extremes of neuron outputs. On the other hand, ReLUs
turn out to be non-essential and can be removed from networks trained for simple
tasks like MNIST classification. The proposed generalized hamming network
(GHN) as such not only lends itself to rigorous analysis and interpretation within
the fuzzy logic theory but also demonstrates fast learning speed, well-controlled
behaviour and state-of-the-art performances on a variety of learning tasks.
1
Introduction
Since early 1990s the integration of fuzzy logic and computational neural networks has given birth to
the fuzzy neural networks (FNN) [1]. While the formal fuzzy set theory provides a strict mathematical
framework in which vague conceptual phenomena can be precisely and rigorously studied [2, 3, 4, 5],
application-oriented fuzzy technologies lag far behind theoretical studies. In particular, fuzzy neural
networks have only demonstrated limited successes on some toy examples such as [6, 7]. In order to
catch up with the rapid advances in recent neural network developments, especially those with deep
layered structures, it is the goal of this paper to demonstrate the relevance of FNN, and moreover, to
provide a novel view on its non-fuzzy counterparts.
Our revisiting of FNN is not merely for the fond remembrances of the golden age of ?soft computing?
[8]. Instead it provides a novel and theoretically justified perspective of neural computing, in which
we are able to re-examine and demystify some useful techniques that were proposed to improve
either effectiveness or efficiency of neural networks training processes. Among many others, batch
normalization (BN) [9] is probably the most influential yet mysterious trick, that significantly
improved the training efficiency by adapting to the change in the distribution of layers? inputs (coined
as internal covariate shift). Such kind of adaptations, when viewed within the fuzzy neural network
framework, can be interpreted as rectifications to the deficiencies of neuron outputs with respect to the
rightful generalized hamming distance (see definition 1) between inputs and neuron weights. Once
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
the appropriate rectification is applied , the ill effects of internal covariate shift are automatically
eradicated, and consequently, one is able to enjoy the fast training process without resorting to a
sophisticated learning method used by BN.
Another crucial component in neural computing, Rectified linear unit (ReLU), has been widely used
due to its strong biological motivations and mathematical justifications [10, 11, 12]. We show that
within the generalized hamming group endowed with generalized hamming distance, ReLU can be
regarded as setting a minimal hamming distance threshold between network input and neuron weights.
This novel view immediately leads us to an effective double-thresholding scheme to suppress fuzzy
elements in the generalized hamming group.
The proposed generalized hamming network (GHN) forms its foundation on the cornerstone notion
of generalized hamming distance (GHD), which is essentially defined as h(x, w) := x + w ? 2xw
for any x, w ? R (see definition 1). Its connection with the inferencing rule in neural computing is
obvious: the last term (?2xw) corresponds to element-wise multiplications of neuron inputs and
weights, and since we aim to measure the GHD between inputs x and weights w, the bias term then
should take the value x + w. In this article we define any network that has its neuron outputs fulfilling
this requirement (3) as a generalized hamming network. Since the underlying GHD induces a fuzzy
XOR logic, GHN lends itself to rigorous analysis within the fuzzy logics theory (see definition 4).
Apart from its theoretical appeals, GHN also demonstrates appealing features in terms of fast learning
speed, well-controlled behaviour and simple parameter settings (see Section 4).
1.1
Related Work
Fuzzy logic and fuzzy neural network: the notion of fuzzy logic is based on the rejection of the
fundamental principle of bivalence of classical logic i.e. any declarative sentence has only two
possible truth values, true and false. Although the earliest connotation of fuzzy logic was attributed
to Aristotle, the founder of classical logic [13], it was Zadeh?s publication in 1965 that ignited the
enthusiasm about the theory of fuzzy sets [2]. Since then mathematical developments have advanced
to a very high standard and are still forthcoming to day [3, 4, 5]. Fuzzy neural networks were proposed
to take advantages of the flexible knowledge acquiring capability of neural networks [1, 14]. In theory
it was proved that fuzzy systems and certain classes of neural networks are equivalent and convertible
with each other [15, 16]. In practice, however, successful applications of FNNs are limited to some
toy examples only [6, 7].
Demystifying neural networks: efforts of interpreting neural networks by means of propositional
logic dated back to McCulloch & Pitts? seminial paper [17]. Recent research along this line include
[18] and the references therein, in which First Order Logic (FOL) rules are encoded using soft logic
on continuous truth values from the interval [0, 1]. These interpretations, albeit interesting, seldom
explain effective neural network techniques such as batch normalization or ReLU. Recently [19]
provided an improvement (and explanation) to batch normalization by removing dependencies in
weight normalization between the examples in a minibatch.
Binary-valued neural network: Restricted Boltzmann Machine (RBM) was used to model an ?ensemble of binary vectors? and rose to prominence in the mid-2000s after fast learning algorithms
were demonstrated by Hinton et. al. [20, 21]. Recent binarized neural network [22, 23] approximated
standard CNNs by binarizing filter weights and/or inputs, with the aim to reduce computational
complexity and memory consumption. The XNOR operation employed in [23] is limited to binary
hamming distance and not readily applicable to non-binary neuron weights and inputs.
Ensemble of binary patterns: the distributive property of GHD described in (1) provides an intriguing
view on neural computing ? even though real-valued pattens are involved in the computation, the
computed GHD is strictly equivalent to the mean of binary hamming distances across two ensembles
of binary patterns! This novel view illuminates the connection between generalized hamming
networks and efficient binary features, that have long been used in various computer vision tasks,
for instance, the celebrated Adaboost face detection[24], numerous binary features for key-point
matching [25, 26] and binary codes for large database hashing [27, 28, 29, 30].
2
h(a, b)
F(h(a, b))
??
?(h(a, b))
?a
(h(a, b))
0
20
?1
0
1
2
3
(a)
4
?1
?2
?3
0
1
2
3
0.010
0.005
0.6
?600
0
0.000
0.4
?800
?10
?2
0.8
?400
?0.005
?0.010
0.2
?1000
?20
?3
0.015
?200
10
?0.015
?1200
4
?3 ?2
?1 0
1
2
3
4
?1
?2
?3
0
1
2
3
4
?3
(b)
?2
?1
0
1
2
3
4
?1
?2
?3
0
1
2
3
4
(c)
?3
?2
?1
0
1
2
3
4
?1
?2
?3
0
1
2
3
4
(d)
Figure 1: (a) h(a, b) has one fuzzy region near the identity element 0.5 (in white), two positively
confident (in red) and two negatively confident (in blue) regions from above and below, respectively.
(b) Fuzziness F (h(a, b)) = h(a, b) ? h(a, b) has its maxima along a = 0.5 or b = 0.5. (c)
?(h(a, b)) : U ? I where ?(h) = 1/(1+exp(0.5?h)) is the logistic function to assign membership
to fuzzy set elements (see definition 4). (d) partial derivative of ?(h(a, b)). Note that magnitudes of
gradient in the fuzzy region is non-negligible.
2
Generalized Hamming Distance
Definition 1. Let a, b, c ? U ? R, and a generalized hamming distance (GHD), denoted by ?, be a
binary operator h : U ? U ? U ; h(a, b) := a ? b = a + b ? 2 ? a ? b . Then
(i) for U = {0, 1} GHD de-generalizes to binary hamming distance with
0 ? 0 = 0; 0 ? 1 = 1; 1 ? 0 = 1; 1 ? 1 = 0;
(ii) for U = [0.0, 1.0] the unitary interval I, a ? b ? I (closure);
Remark: this case is referred to as the ?restricted? hamming distance, in the sense that inverse
of any elements in I are not necessarily contained in I (see below for definition of inverse).
(iii) for U = R, H := (R, ?) is a group satisfying five abelian group axioms, thus is referred to as
the generalized hamming group or hamming group:
? a ? b = (a + b ? 2 ? a ? b) ? R (closure);
? a ? b = (a + b ? 2 ? a ? b) = b ? a (commutativity);
? (a ? b) ? c = (a + b ? 2 ? a ? b) + c ? 2(a + b ? 2 ? a ? b)c
= a + (b + c ? 2 ? b ? c) ? 2 ? a ? (b + c ? 2 ? b ? c) = a ? (b ? c) (associativity);
? ?e = 0 ? R such that e ? a = a ? e = (0 + a ? 2 ? 0 ? a) = a (identity element);
a
a
? for each a ? R \ {0.5}, ?a?1 := a/(2 ? a ? 1) s.t. a ? a?1 = (a + 2?a?1
? 2a ? 2?a?1
)
?1
= 0 = e; and we define ? := (0.5) (inverse element).
Remark: note that 1 ? a = 1 ? a which complements a. ?0.5? is a fixed point since ?a ?
R, 0.5 ? a = 0.5, and 0.5 ? ? = 0 according to definition1 .
(iv) GHD naturally leads to a measurement of fuzziness: F (a) := a ? a, R ? (??, 0.5] :
F (a) ? 0, ?a ? [0, 1]; F (a) < 0 otherwise. Therefore [0, 1] is referred to as the fuzzy
region in which F (0.5) = 0.5 has the maximal fuzziness and F (0) = F (1) = 0 are two
boundary points. Outer regions (??, 0] and [1, ?) are negative and positive confident regions
respectively. See Figure 1 (a) for the surface of h(a, b) which has one central fuzzy region, two
positive confident and two negative confident regions.
(v) The direct sum of hamming group is still a hamming group HL := ?l?L Hl : let x =
{x1 , . . . , xL }, y = {y1 , . . . , yL } ? HL be two group members, then the generalized hamming distance is defined as the arithmetic mean of element-wise GHD: G L (x ?L y) :=
1
L (x1 ? y1 + . . . + xL ? yL ).
And let x
? = (x1 + . . . xL )/L, y? = (y1 + . . . yL )/L be arithmetic means of respective elements,
PL
2
then G L (x ?L y) = x
? + y? ? (x ? y) , where x ? y = l=1 xl ? yl is the dot product.
L
1
By this extension, it is R = R ? {??, +?} instead of R on which we have all group members.
3
? M = (x1 + . . . xM )/M ? HL be element-wise arithmetic mean
(vi) Distributive property: let X
m
? N be defined in the same vein. Then GHD is distributive:
of a set of members x ? HL , and Y
L
M
N
L
X
1 1 1 X XX m
? M ?L Y
?N) = 1
G L (X
x
?l ? y?l =
xl ? yln
L
M N L m=1 n=1
l=1
l=1
1
=
MN
M X
N
X
(1)
G L (xm ?L yn ).
m=1 n=1
n
Remark: in case that xm
l , yl ? {0, 1} i.e. for two sets of binary patterns, the mean of binary
hamming distance between two sets can be efficiently computed as the GHD between two real?M, Y
? N . Conversely, a real-valued pattern can be viewed as the element-wise
valued patterns X
average of an ensemble of binary patterns.
3
Generalized Hamming Network
Despite the recent progresses in deep learning, artificial neural networks has long been criticized
for its ?black box? nature: ?they capture hidden relations between inputs and outputs with a highly
accurate approximation, but no definitive answer is offered for the question of how they work? [16].
In this section we provide an interpretation on neural computing by showing that, if the condition
specified in (3) is fulfilled, outputs of each neuron can be strictly defined as the generalized hamming
distance between inputs and weights. Moreover, the computations of GHD induces fuzzy implication
of XOR connective, and therefore, the inferencing of entire network can be regarded as a logical
calculus in the same vein as described in McCulloch & Pitts? seminial paper [17].
3.1
New perspective on neural computing
The bearing of generalized hamming distance on neural computing is elucidated by looking at the
negative of generalized hamming distance, (GHD, see definition 1), between inputs x ? HL and
weights w ? HL in which L denotes the length of neuron weights e.g. in convolution kernels:
?G L (w ?L x) =
Divide (2) by the constant
2
L
L
L
l=1
l=1
1X
2
1X
wl ?
xl
w?x?
L
L
L
(2)
L
L
X
1 X
wl +
xl
2
(3)
and let
b=?
l=1
l=1
then it becomes the familiar form (w ? x + b) of neuron outputs save the non-linear activation
function. By enforcing the bias term to take the given value in (3), standard neuron outputs measure
PL
negatives of GHD between inputs and weights. Note that, for each layer, the bias term l=1 xl is
PL
averaged over neighbouring neurons in individual input image. The bias term l=1 wl is computed
separately for each filter in fully connected or convolution layers. When weights are updated during
PL
the optimization, l=1 wl changes accordingly to keep up with weights and maintain stable neuron
outputs. We discuss below (re-)interpretations of neural computing in terms of GHD.
Fuzzy inference: As illustrated in definition 4 GHD induces a fuzzy XOR connective. Therefore the
negative of GHD quantifies the degree of equivalence between inputs x and weights w (see definition
4 of fuzzy XOR), i.e. the fuzzy truth value of the statement ?x ? w? where ? denotes a fuzzy
equivalence relation. For GHD with multiple layers stacked together, neighbouring neuron outputs
from the previous layer are integrated to form composite statements e.g. ?(x11 ? w11 , . . . , x1i ?
wi1 ) ? wj2 ? where superscripts correspond to two layers. Thus stacked layers will form more
complex, and hopefully more powerful, statements as the layer depth increases.
4
Y: Mean outputs
X:epochs( 100)
0.00
BN
XOR
WO_BN
Y: Max outputs
X:epochs(x100)
8
BN
XOR
WO_BN
?0.02
7
?0.04
6
?2
?0.06
5
?3
?0.08
4
?0.10
3
?0.12
2
?0.14
1
?0.16
0
0
5
10
15
20
25
30
Y: Min outputs
X:epochs(x100)
0
BN
XOR
WO_BN
?1
?4
?5
?6
?7
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Figure 2: Left to right: mean, max and min of neuron outputs, with/without batch normalized (BN,
WO_BN) and generalized hamming distance (XOR). Outputs are averaged over all 64 filters in the
first convolution layer and plotted for 30 epochs training of a MNIST network used in our experiment
(see Section 4).
Batch normalization demystified: When a mini-batch of training samples X = {x1 , . . . , xM } is
involved in the computation, due to the distributive property of GHD, the data-dependent bias term
L
P
xl equals the arithmetic mean of corresponding bias terms computed for each sample in the
l=1
mini-batch i.e.
1
M
M P
L
P
m=1 l=1
xm
l . It is almost impossible to maintain a constant scalar b that fulfils
this requirement when mini-batch changes, especially at deep layers of the network whose inputs
are influenced by weights of incoming layers. The celebrated batch normalization (BN) technique
therefore proposed a learning method to compensate for the input vector change, with additional
parameters ?, ? to be learnt during the training [9]. It is our conjecture that batch normalization is
approximating these rightful bias through optimization, and this connection is empirically revealed
in Figure 2 with very similar neuron outputs obtained by BN and GHD. Indeed they are highly
correlated during the course of training (with Pearson correlation coefficient=0.97), confirming our
view that BN is attempting to influence the bias term according to (3).
Once b is enforced to follow (3), neither the optimization of bias terms nor the sophisticated learning
method of BN is needed. In the following section we will illustrate a rectified neural network designed
as such.
Rectified linear units (ReLU) redesigned: Due to its strong biological motivations [10] and mathematical justifications [11], rectified linear unit (ReLu) is the most popular activation function used for
deep neural network [31]. If neuron outputs are rectified as the generalized hamming distances, the
activation function max(0, 0.5 ? h(x, w)) then simply sets a minimal hamming distance threshold
of 0.5 (see Figure 1). Astute readers may immediately spot two limitations of this activation function:
a) it only takes into account the negative confidence region while disregards positive confidence
regions; b) it allows elements in the fuzzy regime near 0.5 to misguide the optimization with their
non-negligible gradients.
A straightforward remedy to ReLU is to suppress elements within the fuzzy region by setting outputs
between [0.5 ? r, 0.5 + r] to 0.5, where r is a parameter to control acceptable fuzziness in neuron
outputs. In particular, we may set thresholds adaptively e.g. [0.5 ? r ? O, 0.5 + r ? O] where O
is the maximal magnitude of neuron outputs and the threshold ratio r is adjusted by the optimizer.
This double-thresholding strategy effectively prevents noisy gradients of fuzzy elements, since
0.5 is a fixed point and x ? 0.5 = 0.5 for any x. Empirically we found this scheme, in tandem
with the rectification (3), dramatically boosts the training efficiency for challenging tasks such as
CIFAR10/100 image classification. It must be noted that, however, the use of non-linear activation as
such is not essential for GHD-based neural computing. When the double-thresholding is switched-off
(by fixing r = 0), the learning is prolonged for challenging CIFAR10/100 image classification but its
influence on the simple MNIST classification is almost negligible (see Section 4 for experimental
results).
3.2
Ganeralized hamming network with induced fuzzy XOR
Definition 2. A generalized hamming network (GHN) is any networks consisting of neurons, whose
outputs h ? HL are related to neuron inputs x ? HL and weights w ? HL by h = x ?L w .
5
Remark: In case that the bias term is computed directly from (3) such that h = x ?L w is fulfilled
strictly, the network is called a rectified GHN or simply a GHN. In other cases where bias terms are
approximating the rightful offsets (e.g. by batch normalization [9]), the trained network is called an
approximated GHN.
Compared with traditional neural networks, the optimization of bias terms is no longer needed in
GHN. Empirically, it is shown that the proposed GHN benefits from a fast and robust learning process
that is on par with that of the batch-normalization approach, yet without resorting to sophisticated
learning process of additional parameters (see Section 4 for experimental results). On the other hand,
GHN also benefits from the rapid developments of neural computing techniques, in particular, those
employing parallel computing on GPUs. Due to this efficient implementation of GHNs, it is the first
time that fuzzy neural networks have demonstrated state-of-the-art performances on learning tasks
with large scale datasets.
Often neuron outputs are clamped by a logistic activation function to within the range [0, 1], so
that outputs can be compared with the target labels in supervised learning. As shown below, GHD
followed by such a non-linear activation actually induces a fuzzy XOR connective. We briefly review
basic notion of fuzzy set used in our work and refer readers to [2, 32, 13] for thorough treatments and
review of the topic.
Definition 3. Fuzzy Set: Let
X be an universal set of elements x ? X, then a fuzzy set A is a set
of pairs: A := { x, ?A (x) |x ? X, ?A (x) ? I}, in which ?A : X ? I is called the membership
function (or grade membership).
Remark: In this work we let X be a Cartesian product of two sets X = P ? U where P are (2D or
3D) collection of neural nodes and U are real numbers in ? I or ? R. We define the membership
function ?X (x) := ?U (xp ), ?x = (p, xp ) ? X such that it is dependent on xp only. For the sake of
brevity we abuse the notation and use ?(x), ?X (x) and ?U (xp ) interchangeably.
Definition 4. Induced fuzzy XOR: let two fuzzy set elements a, b ? U be assigned with respective
grade or membership by a membership function ? : U ? I : ?(a) = i, ?(b) = j, then the
generalized hamming distance h(a, b) : U ? U ? U induces a fuzzy XOR connective E : I ? I ? I
whose membership function is given by
?R (i, j) = ?(h(??1 (i), ??1 (j))).
(4)
Remark: For the restricted case U = I the membership function can be trivially defined as the identity
function ? = idI as proved in [4].
Remark: For the generalized case where U = R, the fuzzy membership ? can be defined by a sigmoid
function such as logistic, tanh or any function : U ? I. In this work we adopt the logistic function
1
?(a) = 1+exp(0.5?a)
and the resulting fuzzy XOR connective is given by following membership
function:
1
,
?R (i, j) =
(5)
1 + exp 0.5 ? ??1 (i) ? ??1 (j)
where ??1 (a) = ? ln( a1 ? 1) + 21 is the inverse of ?(a). Following this analysis, it is possible to
rigorously formulate neuron computing of the entire network according to inference rules of fuzzy
logic theory (in the same vein as illustrated in [17]). Nevertheless, research along this line is out of
the scope of the present article and will be reported elsewhere.
4
4.1
Performance evaluation
A case study with MNIST image classification
Overall performance: we tested a simple four-layered GHN (cv[1,5,5,16]-pool-cv[16,5,5,64]-poolfc[1024]-fc[1024,10]) on the MNIST dataset with 99.0% test accuracy obtained. For this relatively
simple dataset, GHN is able to reach test accuracies above 0.95 with 1000 mini-batches and a
learning rate 0.1. This learning speed is on par with that of the batch normalization (BN), but without
resorting to the learning of additional parameters in BN. It was also observed a wide range of large
learning rates (from 0.01 to 0.1) all resulted in similar final accuracies (see below). We ascribe this
well-controlled robust learning behaviour to rectified bias terms enforced in GHNs.
6
Accuracy
Accuracy
0.95
0.90
0.80
3.0
rate0.05 (98.86%)
rate0.025 (98.96%)
rate0.01 (98.69%)
3.5
4.0
4.5
log(#mini_batch)
1.00
0.95
0.95
0.90
0.90
0.85
rate0.1 (98.91%)
rate0.1 (98.97%)
0.85
1.00
Accuracy
1.00
5.0
0.75
3.0
3.5
4.0
4.5
log(#mini_batch)
0.85
rate0.1 (98.98%)
rate0.05 (99.01%)
rate0.025 (98.86%)
rate0.01 (98.65%)
0.80
rate0.05 (98.83%)
rate0.025 (98.84%)
rate0.01 (98.63%)
0.80
5.0
0.75
3.0
3.5
4.0
4.5
log(#mini_batch)
5.0
Figure 3: Test accuracies of MNIST classification with Generalized Hamming Network (GHN). Left:
test accuracies without using non-linear activation (by setting r = 0). Middle: with r optimized for
each layer. Right: with r optimized for each filter. Four learning rates i.e. {0.1, 0.05, 0.025, 0.01} are
used for each case with the final accuracy reported in brackets. Note that the number of mini-batch
are in logarithmic scale along x-axis.
Influence of learning rate: This experiment compares performances with different learning rates
and Figure 3 (middle,right) show that a very large learning rate (0.1) leads to much faster learning
without the risk of divergences. A small learning rate (0.01) suffice to guarantee the comparable final
test accuracy. Therefore we set the learning rate to a constant 0.1 for all experiments unless stated
otherwise.
Influence of non-linear double-thresholding: The non-linear double-thresholding can be turned off by
setting the threshold ratio r = 0 (see texts in Section 3.1). Optionally the parameter r is automatically
optimized together with the optimization of neuron weights. Figure 3 (left) shows that the GHN
without non-linear activation (by setting r = 0) performs equally well as compared with the case
where r is optimized (in Figure 3 left, right). There are no significant differences between two settings
for this relative simple task.
4.2
CIFAR10/100 image classification
In this experiment, we tested a six-layered GHN (cv[3,3,3,64]-cv[64,5,5,256]-pool-cv[256,5,5,256]pool-fc[1024]-fc[1024,512]-fc[1024,nclass]) on both CIFAR10 (nclass=10) and CIFAR100
(nclass=100) datasets. Figure 4 shows that the double-thresholding scheme improves the learning efficiency dramatically for these challenging image classification tasks: when the parameter r
is optimized for each feature filter the numbers of iterations required to reach the same level of test
accuracy are reduced by 1 to 2 orders of magnitudes. It must be noted that performances of such a
simple generalized hamming network (89.3% for CIFAR10 and 60.1% for CIFAR100) are on par
with many sophisticated networks reported in [33]. In our view, the rectified bias enforced by (3) can
be readily applied to these sophisticated networks, although resulting improvements may vary and
remain to be tested.
4.3
Generative modelling with Variational Autoencoder
In this experiment, we tested the effect of rectification in GHN applied to a generative modelling
setting. One crucial difference is that the objective is now to minimize reconstruction error instead of
classification error. It turns out the double-thresholding scheme is no longer relevant for this setting
and thus not used in the experiment.
The baseline network (784-400-400-20) used in this experiment is an improved implementation [34]
of the influential paper [35], trained on the MNIST dataset of images of handwritten digits. We have
rectified the outputs following (3) and, instead of optimizing the lower bound of the log marginal
likelihood as in [35], we directly minimize the reconstruction error. Also we did not include weights
regularization terms for the optimization as it is unnecessary for GHN. Figure 5 (left) illustrates
the reconstruction error with respect to number of training steps (mini-batches). It is shown that
the rectified generalized hamming network converges to a lower minimal reconstruction error as
compared to the baseline network, with about 28% reduction. The rectification also leads to a faster
convergence, which is in accordance with our observations in other experiments.
7
0.9
0.6
0.5
0.7
Accuracy
Accuracy
0.8
0.4
0.3
0.6
0.5
0.2
OPT_THRES (89.26%)
0.4
3.0
0.1
WO_THRES (84.63%)
3.5
4.0
4.5
5.0
log(#mini_batch)
5.5
6.0
OPT_THRES (60.05%)
WO_THRES (51.71%)
3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75
log(#mini_batch)
Figure 4: Left: GHN test accuracies of CIFAR10 classification (OPT THRES: parameter r optimized; WO THRES: without nonlinear activation). Right: GHN test accuracies of CIFAR100
classification(OPT THRES: parameter r optimized; WO THRES: without non-linear activation).
accuracy
Reconstruction error
GHN
VAE
5000
0.75
4500
0.70
4000
0.65
3500
3000
0.60
2500
0.55
2000
1500
GHN
CNN
0.50
0
20000
40000
60000
#mini_batch
80000
0
100000
1000
2000
3000
#mini_batch
4000
5000
Figure 5: Left: Reconstruction errors of convolution VAE with and w/o rectification. Right: Evaluation accuracies of Sentence classification with GHN rectification and w/o rectification).
4.4
Sentence classification
A simple CNN has been used for sentence-level classification tasks and excellent results were
demonstrated on multiple benchmarks [36]. The baseline network used in this experiment is a
re-implementation of [36] made available from [37]. Figure 5 (right) plots accuracy curves from both
networks. It was observed that the rectified GHN did improve the learning speed, but did not improve
the final accuracy as compared with the baseline network: both networks yielded the final evaluation
accuracy around 74% despite that the training accuracy were almost 100%. The over-fitting in this
experiment is probably due to the relatively small Movie Review dataset size with 10,662 example
review sentences, half positive and half negative.
5
Conclusion
In summary, we proposed a rectified generalized hamming network (GHN) architecture which materializes a re-emerging principle of fuzzy logic inferencing. This principle has been extensively studied
from a theoretic fuzzy logic point of view, but has been largely overlooked in the practical research
of ANN. The rectified neural network derives fuzzy logic implications with underlying generalized
hamming distances computed in neuron outputs. Bearing this rectified view in mind, we proposed to
compute bias terms analytically without resorting to sophisticated learning methods such as batch
normalization. Moreover, we have shown that, the rectified linear units (ReLU) was theoretically
non-essential and could be skipped for some easy tasks. While for challenging classification problems,
the double-thresholding scheme did improve the learning efficiency significantly.
The simple architecture of GHN, on the one hand, lends itself to being analysed rigorously and this
follow up research will be reported elsewhere. On the other hand, GHN is the first fuzzy neural
network of its kind that has demonstrated fast learning speed, well-controlled behaviour and stateof-the-art performances on a variety of learning tasks. By cross-checking existing networks against
GHN, one is able to grasp the most essential ingredient of deep learning. It is our hope that this kind
of comparative study will shed light on future deep learning research and eventually open the ?black
box? of artificial neural networks [16].
8
Acknowledgement
I am grateful to anonymous reviewers for their constructive comments to improve the quality of this
paper. I greatly appreciate valuable discussions and supports from colleagues at Nokia Technologies.
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6,399 | 6,789 | Speeding Up Latent Variable Gaussian Graphical
Model Estimation via Nonconvex Optimization
Pan Xu
Department of Computer Science
University of Virginia
Charlottesville, VA 22904
[email protected]
Jian Ma
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Quanquan Gu
Department of Computer Science
University of Virginia
Charlottesville, VA 22904
[email protected]
Abstract
We study the estimation of the latent variable Gaussian graphical model (LVGGM),
where the precision matrix is the superposition of a sparse matrix and a low-rank
matrix. In order to speed up the estimation of the sparse plus low-rank components,
we propose a sparsity constrained maximum likelihood estimator based on matrix
factorization, and an efficient alternating gradient descent algorithm with hard
thresholding to solve it. Our algorithm is orders of magnitude faster than the
convex relaxation based methods for LVGGM. In addition, we prove that our
algorithm is guaranteed to linearly converge to the unknown sparse and low-rank
components up to the optimal statistical precision. Experiments on both synthetic
and genomic data demonstrate the superiority of our algorithm over the state-ofthe-art algorithms and corroborate our theory.
1
Introduction
For a d-dimensional Gaussian graphical model (i.e., multivariate Gaussian distribution) N (0, ?? ),
the inverse of covariance matrix ?? = (?? ) 1 (also known as the precision matrix or concentration
matrix) measures the conditional dependence relationship between marginal random variables [19].
When the number of observations is comparable to the ambient dimension of the Gaussian graphical
model, additional structural assumptions are needed for consistent estimation. Sparsity is one of
the most common structures imposed on the precision matrix in Gaussian graphical models (GGM),
because it gives rise to a sparse graph, which characterizes the conditional dependence of the marginal
variables. The problem of estimating the sparse precision matrix in Gaussian graphical models has
been studied by a large body of literature [23, 29, 12, 28, 6, 34, 37, 38, 33]. However, the real world
data may not follow a sparse GGM, especially when some of the variables are unobservable.
To alleviate this problem, the latent variable Gaussian graphical model (LVGGM) [9, 24] has been
studied, where the precision matrix of the observed variables is conditionally sparse given the latent
variables (i.e., unobserved) , but marginally not sparse. It is well-known that in LVGGM, the precision
matrix ?? can be represented as the superposition of a sparse matrix S? and a low-rank matrix L? ,
where the latent variables contribute to the low rank component in the precision matrix. In other
words, we have ?? = S? + L? .
In the learning problem of LVGGM, the goal is to estimate both the unknown sparse component
S? and the low-rank component L? of the precision matrix simultaneously. In the seminal work,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Chandrasekaran et al. [9] proposed a maximum-likelihood estimator based on `1 norm penalty on
the sparse matrix and nuclear norm penalty on the low-rank matrix, and proved the model selection
consistency for LVGGM estimation. Meng et al. [24] studied a similar penalized estimator, and
derived Frobenius norm error bounds based on the restricted strong convexity [26] and the structural
Fisher incoherence condition between the sparse and low-rank components. Both of these two
methods for LVGGM estimation are based on a penalized convex optimization problem, which can
be solved by log-determinant proximal point algorithm [32] and alternating direction method of
multipliers [22]. Due to the nuclear norm penalty, these convex optimization algorithms need to do
full singular value decomposition (SVD) to solve the proximal mapping of nuclear norm at each
iteration, which results in an extremely high time complexity of O(d3 ). When d is large as often in
the high dimensional setting, the convex relaxation based methods are computationally intractable. It
is worth noting that full SVD cannot be accelerated by power method [13] or other randomized SVD
algorithms [15], hence the O(d3 ) is unavoidable whenever nuclear norm regularization is employed.
In this paper, in order to speed up learning LVGGM, we propose a novel sparsity constrained
maximum likelihood estimator for LVGGM based on matrix factorization. Specifically, inspired by
the recent work on matrix factorization [18, 16, 44, 45, 11, 30], we propose to reparameterize the
low-rank component L in the precision matrix as the product of smaller matrices, i.e., L = ZZ> ,
where Z 2 Rd?r and r ? d is the number of latent variables. This factorization captures the
intrinsic low-rank structure of L, and automatically ensures its low-rankness. We propose an
alternating gradient descent with hard thresholding to solve the new estimator. We prove that the
output of our algorithm is guaranteed to linearly converge to the unknown parameters up to the
statistical precision. In detail, our algorithm enjoys O(d2 r) per-iteration time complexity, which
outperforms the O(d3 ) per-iteration complexity of state-of-the-art LVGGM estimators based on
nuclear norm
24]. In addition, the estimators from our algorithm for LVGGM attain
p penalty [9, 22, p
max{Op ( s? log d/n), Op ( rd/n)} statistical rate of convergence in terms of Frobenius norm,
where s? is the conditional sparsity of the precision matrix (i.e., sparsity of S? ), and r is the number
of latent variables (i.e., rank of L? ). This matches the minimax optimal convergence rate for LVGGM
estimation [9, 1, 24]. Thorough experiments on both synthetic and breast cancer genomic datasets
show that our algorithm is orders of magnitude faster than existing methods.
It is also worth noting that, although our estimator and algorithm is designed for LVGGM, it is
directly applicable to the Gaussian graphical model where the precision matrix is the sum of a sparse
matrix and a low-rank matrix. And the theoretical guarantees of our algorithm still hold.
The remainder of this paper is organized as follows: In Section 2, we briefly review existing work
that is relevant to our study. We present our estimator and algorithm in detail in Section 3, and the
main theory in Section 4. In Section 5, we compare the proposed algorithm with the state-of-the-art
algorithms on both synthetic data and real-world breast cancer data. Finally, we conclude this paper
in Section 6.
Notation For matrices A, B with commensurate dimensions, we use hA, Bi = tr(A> B) to denote
their inner product and A ? B denote their Kronecker product. For a matrix A 2 Rd?d , we denote
its (ordered) singular values by 1 (A)
...
0. We denote by A 1 the
2 (A)
d (A)
inverse of A, and denote by |A| its determinant. We use the notation k ? k for various types of matrix
norms, including the spectral norm kAk2 and the Frobenius norm kAkF . We also use the following
P
Pd
norms kAk0,0 = i,j 1(Aij 6= 0), kAk1,1 = max1?i,j?d |Aij |, and kAk1,1 = i,j=1 |Aij |. A
constant is called absolute constant if it does not depend on the parameters of the problem, e.g.,
dimension and sample size. We denote a . b if a is less than b up to a constant.
2
Additional Related Work
Precision matrix estimation in sparse Gaussian graphical models (GGM) is commonly formulated
as a penalized maximum likelihood estimation problem with `1,1 norm regularization [12, 29, 28]
(graphical Lasso) or regularization on diagonal elements of Cholesky decomposition for precision
matrix [17]. Due to the complex dependency among marginal variables in many applications, sparsity
assumption on the precision matrix often does not hold. To relax this assumption, the conditional
Gaussian graphical model (cGGM) was proposed in [41, 5] and the partial Gaussian graphical model
(pGGM) was proposed in [42], both of which impose blockwise sparsity on the precision matrix
and estimate multiple blocks therein. Despite a good interpretation of these models, they need to
access both the observed variables as well as the latent variables for estimation. Another alternative
2
is the latent variable Gaussian graphical model (LVGGM), which was proposed in [9], and later
investigated in [22, 24]. Compared with cGGM and pGGM, the estimation of LVGGM does not need
to access the latent variables and therefore is more flexible.
Another line of research related to ours is low-rank matrix estimation based on alternating minimization and gradient descent [18, 16, 44, 45, 11, 30, 3, 35, 43]. However, extending them to low-rank
and sparse matrix estimation as in LVGGM turns out to be highly nontrivial. The most related work
to ours includes [14] and [40], which studied nonconvex optimization for low-rank plus sparse matrix
estimation. However, they are limited to robust PCA [8] and multi-task regression [1] in the noiseless
setting. Due to the square loss in RPCA, the sparse matrix S can be calculated by subtracting the
low-rank matrix L from the observed data matrix. Nevertheless, in LVGGM, there is no closed-form
solution for the sparse matrix due to the log-determinant term, and we need to use gradient descent
to update S. On the other hand, both the algorithm in [40] and our algorithm have an initialization
stage. Yet our initialization algorithm is new and different from the initialization algorithm in [40] for
RPCA. Furthermore, our analysis of the initialization algorithm is built on the spikiness condition,
which is also different from that for RPCA.
The last but not least related work is expectation maximization (EM) algorithm [2, 36], which shares a
similar bivariate structure as our estimator. However, the proof technique used in [2, 36] is not directly
applicable to our algorithm, due to the matrix factorization structure in our estimator. Moreover, to
overcome the dependency issue between consecutive iterations in the proof, sample splitting strategy
[18, 16] was employed in [2, 36, 39], i.e., dividing the whole dataset into T pieces and using a fresh
piece of data in each iteration. Unfortunately,
p the sample splitting technique results in a suboptimal
statistical rate, incurring an extra factor of T in the rate. In sharp contrast, our proof technique does
not rely on sample splitting, because we are able to prove a uniform convergence result over a small
neighborhood of the unknown parameters, which directly resolves the dependency issue.
3
The Proposed Estimator and Algorithm
In this section, we present a new estimator for LVGGM estimation, together with a new algorithm.
3.1
Latent Variable GGMs
Let XO be the d-dimensional random vector with observed variables and XL be the r-dimensional
random vector with latent variables. We assume that the concatenated random vector X =
>
e and sparse
(XO
, XL> )> follows a multivariate Gaussian distribution with covariance matrix ?
e =?
e 1 . It is proved in [10] that the observed data XO follows a normal disprecision matrix ?
e OO , which is the top-left block matrix in ?
e
tribution with marginal covariance matrix ?? = ?
corresponding to XO . The precision matrix of XO is then given by Schur complement [13]:
e OO )
?? = (?
1
e OO
=?
e OL ?
e 1?
e
?
LL LO .
(3.1)
Since we can only observe XO , the marginal precision matrix ? is generally not sparse. We define
?
e OO and L? := ?
e OL ?
e 1?
e
e
S? := ?
LL LO . Then S is sparse due to the sparsity of ?. We do not
e
e LO can be
impose any dependency restriction on XO and XL , and thus the matrices ?OL and ?
potentially dense. We assume that the number of latent variables is smaller than that of the observed.
Therefore, L? is low-rank and may be dense. Thus, the precision matrix of LVGGM can be written as
?
?? = S ? + L? ,
(3.2)
where kS k0,0 = s and rank(L ) = r. We refer to [9] for a detailed discussion of LVGGM.
?
3.2
?
?
The Proposed Estimator
Suppose that we observe i.i.d. samples X1 , . . . , Xn from N (0, ?? ). Our goal is to estimate the
sparse component S? and the low-rank component L? of the unknown precision matrix ?? in (3.2).
The negative log-likelihood of the Gaussian graphical model is proportional to the following sample
loss function up to a constant
?
?
b S+L
pn (S, L) = tr ?
log |S + L|,
(3.3)
P
b = 1/n n Xi X > is the sample covariance matrix, and |S + L| is the determinant of
where ?
i
i=1
? = S + L. We employ the maximum likelihood principle to estimate S? and L? , which is equivalent
to minimizing the negative log-likelihood in (3.3).
3
The low-rank structure of the precision matrix, i.e., L poses a great challenge for computation. A
typical way is to use nuclear-norm regularized estimator, or rank constrained estimator to estimate
L. However, such kind of estimators involve singular value decomposition at each iteration, which
is computationally very expensive. To overcome this computational obstacle, we reparameterize
L as the product of smaller matrices. More specifically, due to the symmetry of L, it can be
reparameterized by L = ZZ> , where Z 2 Rd?r and r > 0 is the number of latent variables and
is a tuning parameter. This kind of reparametrization has recently been used in low-rank matrix
estimation [18, 16, 44, 45, 11, 30] based on matrix factorization. Then we can rewrite the sample
loss function in (3.3) as the following objective function
?
?
b S + ZZ>
qn (S, Z) = tr ?
log |S + ZZ> |.
(3.4)
Based on (3.4), we propose a nonconvex estimator using sparsity constrained maximum likelihood:
min
S,Z
qn (S, Z)
subject to kSk0,0 ? s,
(3.5)
where s > 0 is a tuning parameter that controls the sparsity of S.
3.3
The Proposed Algorithm
Due to the matrix factorization based reparameterization L = ZZ> , the objective function in (3.5)
is nonconvex. In addition, the sparsity based constraint in (3.5) is nonconvex as well. Therefore,
the estimation in (3.5) is essentially a nonconvex optimization problem. We propose to solve it by
alternately performing gradient descent with respect to one parameter matrix with the other one fixed.
The detailed algorithm is displayed in Algorithm 1, which consists of two stages.
b (0) , Z
b (0) , which, we will show later,
In the initialization stage (Stage I), it outputs initial points S
?
?
are guaranteed to fall in the small neighborhood of S and Z respectively. Note that we need to do
inversion in Line 3, whose complexity is O(d3 ). Nevertheless, we only need to do inversion once. In
sharp contrast, convex relaxation approaches need to do full SVD with O(d3 ) complexity at each
iteration, which is much more time consuming than ours.
In the alternating gradient descent stage (Stage II), we iteratively estimate S while fixing Z, and
then estimate Z while fixing S. Instead of solving each subproblem exactly, we propose to perform
one-step gradient descent for S and Z alternately, using step sizes ? and ? 0 . In Lines 6 and 8 of
Algorithm 1, rS qn (S, Z) and rZ qn (S, Z) denote the partial gradient of qn (S, Z) with respect to S
and Z respectively. The choice of the step sizes will be clear according to our theory. In practice,
one can also use line search to choose the step sizes. Due to the sparsity constraint kSk0,0 ? s,
we apply hard thresholding [4] right after the gradient descent step for S, in Line 7 of Algorithm
1. For a matrix S 2 Rd?d and an integer s, the hard thresholding operator HTs (S) preserves the s
largest magnitudes in S and sets the rest entries to zero. Algorithm 1 does not involve singular value
decomposition in each iteration, neither solve an exact optimization problem, which makes it much
faster than the convex relaxation based algorithms [9, 24]. The computational overhead of Algorithm
1 mainly comes from the calculation of the partial gradient with respect to Z, whose time complexity
is O(rd2 ). Therefore, our algorithm has a per-iteration complexity of O(rd2 ).
4
Main Theory
We present our main theory in this section, which characterizes the convergence rate of Algorithm
1, and the statistical rate of its output. We begin with some definitions and assumptions, which are
necessary for our theoretical analysis.
Assumption 4.1. There is a constant ? > 0 such that 0 < 1/? ? min (?? ) ? max (?? ) ? ? < 1,
where min (?? ) and max (?? ) are the minimal and maximal eigenvalues of ?? respectively.
Assumption 4.1 requires the eigenvalues of true covariance matrix ?? to be finite and bounded below
from a positive number, which is a standard assumption for Gaussian graphical models [29, 21, 28].
The relation between the covariance matrix and the precision matrix ?? = (?? ) 1 immediately
yields 1/? ? min (?? ) ? max (?? ) ? ?.
It is well understood that the estimation problem of the decomposition ?? = S? +L? can be ill-posed,
where identifiability issue arises when the low-rank matrix L? is also sparse [10, 7]. The concept of
incoherence condition, which was originally proposed for matrix completion [7], has been adopted in
[9, 10], which ensures the low-rank matrix not to be too sparse by restricting the degree of coherence
4
Algorithm 1 Alternating Thresholded Gradient Descent (AltGD) for LVGGM
1: Input: i.i.d. samples X1 , . . . , Xn from LVGGM, max number of iterations T , and parameters
?, ? 0 , r, s.
Stage I: Initialization
b = 1 P n Xi X > .
2: ?
i
i=1
n
b (0) = HTs (?
b 1 ), which preserves the s largest magnitudes of ?
b 1.
3: S
1
(0)
>
b
b
b (0) = UD1/2
4: Compute SVD: ?
S = UDU , where D is a diagonal matrix. Let Z
r ,
where Dr is the first r columns of D.
Stage II: Alternating Gradient Descent
5: for t = 0, . . . , T
1 do
b (t+0.5) = S
b (t) ?rS qn S
b (t) , Z
b (t) ;
6:
S
?
?
b (t+1) = HTs S
b (t+0.5) , which preserves the s largest magnitudes of S
b (t+0.5) ;
S
b (t+1) = Z
b (t) ? 0 rZ qn S
b (t) , Z
b (t) ;
8:
Z
9: end for
b (T ) , Z
b (T ) .
10: output: S
7:
between singular vectors and the standard basis. Later work such as [1, 25] relaxed this condition to
a constraint on the spikiness ratio, and showed that spikiness condition is milder than incoherence
condition. In our theory, we use the notion of spikiness as follows.
Assumption 4.2 (Spikiness Condition [25]). For a matrix L 2 Rd?d , the spikiness ratio is defined
as ?sp (L) := dkLk1,1 /kLkF . For the low-rank matrix L? in (3.2), we assume that there exists a
constant ?? > 0 such that
?sp (L? ) ? kL? kF
??
kL? k1,1 =
?
.
(4.1)
d
d
Since rank(L? ) = r, we define max = 1 (L? ) > 0 and min = r (L? ) > 0 to be the maximal and
minimal nonzero singular value of L? respectively. We observe that the decomposition of low-rank
matrix L? in Section 3.2 is not unique, since we have L? = (Z? U)(Z? U)> for any r ? r orthogonal
matrix U. Thus, we define the following solution set for Z:
e 2 Rd?r |Z
e = Z? U for some U 2 Rr?r with UU> = U> U = Ir .
U= Z
e = p max and r (Z)
e = p min for any Z
e 2 U.
Note that 1 (Z)
(4.2)
To measure the closeness between our estimator for Z and the unknown parameter Z? , we use
the following distance d(?, ?), which is invariant to rotation. Similar definition has been used in
[45, 30, 40].
e F , where U
Definition 4.3. Define the distance between Z and Z? as d(Z, Z? ) = minZ2U
kZ Zk
e
is the solution set defined in (4.2).
At the core of our proof technique is the first-order stability condition on the population loss function.
In detail, the population loss function is defined as the expectation of sample loss function in (3.3):
p(S, L) = tr(?? (S + L))
(4.3)
log S + L .
For the ease of presentation, we define two balls around S and Z respectively: BF (S , R) = {S 2
Rd?d : kS S? kF ? R}, Bd (Z? , R) = {Z 2 Rd?r : d(Z, Z? ) ? R}. Then the first-order stability
condition is stated as follows.
Condition 4.4 (First-order Stability). Suppose S 2 BF (S? , R), Z 2 Bd (Z? , R) for some R > 0;
by definition we have L = ZZ> and L? = Z? Z?> . The gradient of population loss function with
respect to S satisfies
?
rS p(S, L)
rS p(S, L? )
F
rL p(S, L)
rL p(S? , L)
F
?
?
2
?
1
?
? kL
L ? kF .
? kS
S ? kF ,
The gradient of the population loss function with respect to L satisfies
where
1, 2
> 0 are constants.
5
Condition 4.4 requires the population loss function has a variant of Lipschitz continuity for the
gradient. Note that the gradient is taken with respect to one variable (S or L), while the Lipschitz
continuity is with respect to the other variable. Also, the Lipschitz property is defined only between
the true parameters S? , L? and arbitrary elements S 2 BF (S? , R) and L = ZZ> such that Z 2
Bd (Z? , R). It should be noted that Condition 4.4, as is verified in the appendix, is inspired by a
similar condition originally introduced in [2]. We extend it to the loss function of LVGMM with both
sparse and low-rank structures, which plays an important role in the analysis.
The following theorem characterize the theoretical properties of Algorithm 1.
Theorem 4.5. Suppose Assumptions 4.1 and 4.2 hold. Assume that the sample size satisfies
n
484k?? k1,1 ? 2 rs? log d/(25R2 min ) and the sparsity of the unknown sparse matrix satisp
p
?
2
fies s ? 25d2 R2 min /(121r??2 ), where R = min{1/4 max , 1/(2?),
min /(6.5? )}. Then
(0)
(0)
b ,Z
b obtained by the initialization stage of
with probability at least 1 C/d, the initial points S
Algorithm 1 satisfies
b (0)
S
S?
F
? R,
and
b (0) , Z? ? R,
d Z
(4.4)
where C > 0 is an absolute constant. Furthermore, suppose Condition 4.4 holds. Let the step
sizes satisfy ? ? C0 /( max ? 2 ) and ? 0 ? C0 min /( max ? 4 ), and the sparsity parameter satisfies
p
s
4(1/(2 ?) 1)2 + 1 s? , where C0 > 0 is a constant that can be chosen arbitrarily small. Let
? and ? be
?
?
2
?
? 0 min
48C02 s? log d 32C02 min
rd
? = max 1
,
1
,
?
=
max
,
.
2
4
6 n
?2
?2
?
n
?
max
max
Then for any t
1, with probability at least 1 C1 /d, the output of Algorithm 1 satisfies
n
o
p
?
b (t+1) S? 2 , d2 (Z
b (t+1) , Z? ) ?
max S
+
?t+1 ? R ,
p
F
| {z }
1
?
| {z }
optimization error
(4.5)
statistical error
where C1 > 0 is an absolute constant.
In Theorem 4.5, ? is the contraction parameter of linear convergence rate, and it depends on the step
size ?. Therefore, we can always choose a sufficiently small step size by choosing a small enough
C0 , such that ? is strictly between 0 and 1.
Remark 4.6. (4.4) suggests that, in order to ensure that the initial points returned by the initialization
stage of Algorithm 1 fall in small neighborhoods of S? and Z? , we require n = O(s? log d), which
essentially attains the optimal sample complexity for LVGGM estimation. In addition, we require
s? . d2 /(r??2 ), which means the unknown sparse matrix cannot be too dense.
Remark 4.7. (4.5) suggests that the estimation error of the output of Algorithm 1 consists of two
terms: the first term is the statistical error, andp
the second term is the
The statistical
p optimization error. p
?
error comes from ? and scales as max Op ( s log d/n), Op ( rd/n) , where Op ( s? log d/n)
p
corresponds to the statistical error of S? , and Op ( rd/n) corresponds to the statistical error of L? 1 .
This matches the minimax optimal rate of estimation errors in Frobenius norm for LVGGM estimation
[9, 1, 24]. For the optimization error, note that max and min are fixed constants. For a sufficiently
small constant C0 , we can always ensure ? < 1, and this establishes the linear convergence rate for
Algorithm 1. Actually, after T max{O(log(? 4 n/(s? log d))), O(log(? 6 n/(rd)))} iterations, the
total estimation error of our algorithm achieves the same order as the statistical error.
Remark 4.8. Our statistical rate is sharp, because our theoretical analysis is conducted uniformly
over the neighborhood of true parameters S? and Z? , rather than doing sample splitting. This is
another big advantage of our approach over existing algorithms which are also built upon first-order
stability [2, 36] but rely on sample splitting technique.
5
Experiments
In this section, we present numerical results on both synthetic and real datasets to verify the theoretical
properties of our algorithm, and compare it with the state-of-the-art methods. Specifically, we
1
b (t) , it is in the same order as the error bound of L
b (t) by
While the derived error bound in (4.5) is for Z
definition.
6
compare our method, denoted by AltGD, with two convex relaxation based methods for estimating
LVGGM: (1) LogdetPPA [9, 32] for solving log-determinant semidefinite programs, denoted by
PPA, and (2) the alternating direction method of multipliers in [22, 24], denoted by ADMM. We
also considered alternatives of the convex methods which use the randomized SVD method [15]
in each iteration. However, the randomized SVD method still needs to compute a full SVD for
nuclear norm regularization and in our experiments, we found that it is slower than the full SVD
method implemented in [22]. Thus, we only report the results of the orignial convex relaxations
in [9, 32, 22, 24]. The implementation of these two methods were downloaded from the authors?
website. All numerical experiments were run in MATLAB R2015b on a laptop with Intel Core i5 2.7
GHz CPU and 8GB of RAM.
5.1
Synthetic Data
In the synthetic experiment, we first validate the performance of our method on the latent variable
GGM. Then we show that our method also performs well on a more general GGM where the precision
matrix is the sum of an arbitrary sparse matrix S? and arbitrary low rank matrix L? . Specifically, we
generated data according to the following two schemes:
? Scheme I: we generated data from the latent variable GGM defined in Section 3.1. In detail, the
dimension of observed data is d and the number of latent variables is r. We randomly generated
e 2 R(d+r)?(d+r) , with sparsity s? = 0.02d2 . According to
a sparse positive definite matrix ?
e 1:d;1:d and the low-rank component
(3.1), the sparse component of the precision matrix is S? := ?
?
1e
e
e
is L := ?1:d;(d+1):(d+r) [?(d+1):(d+r);(d+1):(d+r) ] ?(d+1):(d+r);1:d . Then we sampled data
X1 , . . . , Xn from distribution N (0, (?? ) 1 ), where ?? = S? + L? is the true precision matrix.
? Scheme II: the dimension of observed data is d and the number of latent variables is r. S? is a
symmetric positive definite matrix with entries randomly generated from [ 1, 1] with sparsity
s? = 0.05d2 . L? = Z? Z?> , where Z? 2 Rd?r with entries randomly generated from [ 1, 1].
Then we sampled data X1 , . . . , Xn from multivariate normal distribution N (0, (?? ) 1 ) with
?? = S? + L? being the true precision matrix.
Table 1: Scheme I: estimation errors of sparse and low-rank components S? and L? as well as the true
precision matrix ?? in terms of Frobenius norm on different synthetic datasets. Data were generated
from LVGGM and results were reported on 10 replicates in each setting.
b (T )
kS
S? k F
b (T )
kL
L? kF
b (T )
k?
?? k F
Time (s)
0.7350?0.0359
0.7563?0.0298
0.6236?0.0669
1.1610
1.1120
0.0250
0.0195?0.0046
0.0294?0.0041
0.0125?0.0000
0.9813?0.0192
1.0610?0.0134
0.8210?0.0143
35.7220
25.8010
0.4800
1.1620?0.0177
1.1867?0.0253
0.9016?0.0245
0.0224?0.0034
0.0356?0.0033
0.0167?0.0030
1.1639?0.0179
1.1869?0.0254
0.9021?0.0244
356.7360
156.5550
7.4740
1.4822?0.0302
1.5010?0.0240
1.3449?0.0073
0.0371?0.0052
0.0442?0.0068
0.0208?0.0014
1.4824?0.0120
1.5012?0.0240
1.3449?0.0084
33522.0200
21090.7900
445.6730
Setting
Method
d = 100, r = 2, n =
2000
PPA
ADMM
AltGD
0.7335?0.0352
0.7521?0.0288
0.6241?0.0668
0.0170?0.0125
0.0224?0.0115
0.0113?0.0014
d = 500, r = 5, n =
10000
PPA
ADMM
AltGD
0.9803?0.0192
1.0571?0.0135
0.8212?0.0143
d = 1000, r = 8, n =
2.5 ? 104
PPA
ADMM
AltGD
d = 5000, r = 10, n =
2 ? 105
PPA
ADMM
AltGD
In both schemes, we conducted experiments in different settings of d, n, s? and r. The step sizes
of our method were set as ? = c1 /( max ? 2 ) and ? 0 = c1 min /( max ? 4 ) according to Theorem 4.5,
where c1 = 0.25. The thresholding parameter s is set as c2 s? , where c2 > 1 was selected by 4-fold
cross-validation. The regularization parameters for `1,1 -norm and nuclear norm in PPA and ADMM
and the tuning parameter r in our algorithm were selected by 4-fold cross-validation. Under both
schemes, we repeatedly generated 10 datasets for each setting of d, n, s? and r? , and calculated the
mean and standard error of the estimation error. We summarize the results of Scheme I over 10
replications in Table 1. Note that our algorithm AltGD outputs a slightly better estimator in terms of
estimation errors compared with PPA and ADMM. It should also be noted that they do not differ
too much because their statistical rates should be in the same order. To demonstrate the efficiency
of our algorithm, we also reported the mean CPU time in the last column of Table 1. We observe
7
significant speed-ups brought by our algorithm, which is almost 50 times faster than the convex
ones. In particular, when the dimension d scales up to several thousands, the computation of SVD
in PPA and ADMM takes enormous time and therefore the computational time of them increases
dramatically. We report the averaged results of Scheme II over 10 repetitions in Table 2. Again,
it can be seen that our method AltGD achieves comparable or slightly better estimators in terms
of estimation errors in Frobenius norm compared against PPA and ADMM. Our method AltGD is
nearly 50 times faster than the other two methods based on convex algorithms.
Table 2: Scheme II: estimation errors of sparse and low-rank components S? and L? as well as
the true precision matrix ?? in terms of Frobenius norm on different synthetic datasets. Data were
generated from multivariate distribution where the precision matrix is the sum of an arbitrary sparse
matrix and an arbitrary low-rank matrix. Results were reported on 10 replicates in each setting.
b (T )
kS
b (T )
kL
S? k F
b (T )
k?
L? kF
?? k F
Time (s)
0.8912?0.0356
0.8588?0.0375
0.7483?0.0742
1.6710
1.2790
0.0460
0.7802?0.0104
0.7803?0.0104
0.7594?0.0111
1.1363?0.0131
1.1363?0.0131
0.9718?0.0146
43.8000
25.8980
0.8690
0.9235?0.0193
0.9209?0.0212
0.7249?0.0158
1.1985?0.0084
1.2131?0.0084
0.9651?0.0093
1.4913?0.0162
1.4975?0.0171
1.2029?0.0141
487.4900
163.9350
7.1360
1.1883?0.0091
1.2846?0.0089
1.0681?0.0034
1.0970?0.0022
1.1568?0.0023
1.0685?0.0023
1.3841?0.0083
1.5324?0.0085
1.2068?0.0032
44098.6710
20393.3650
287.8630
Setting
Method
d = 100, r = 2, n =
2000
PPA
ADMM
AltGD
0.5710?0.0319
0.6198?0.0361
0.5639?0.0905
0.6231?0.0261
0.5286?0.0308
0.4824?0.0323
d = 500, r = 5, n =
10000
PPA
ADMM
AltGD
0.8140?0.0157
0.8140?0.0157
0.6139?0.0198
d = 1000, r = 8, n =
2.5 ? 104
PPA
ADMM
AltGD
d = 5000, r = 10, n =
2 ? 105
PPA
ADMM
AltGD
In addition, we illustrate the convergence rate of our algorithm in Figure 1(a) and 1(b), where the
x-axis is iteration number and y-axis is the estimation errors in Frobenius norm. We can see that
our algorithm converges in dozens of iterations, which confirms our theoretical guarantee on linear
convergence rate. We plot the overall estimation errors against the scaled statistical errors of S(T ) and
L(T ) under different configurations of d, n, s? and r in Figure 1(c) and 1(d). According to Theorem
b (t) S? kF and kL
b (t) L? kF will converge to the statistical errors as the number of iterations
4.5, kS
p
p
t goes up, which are in the order of O( s? log d/n) and O( rd/n) respectively. We can see that
the estimation errors grow linearly with the theoretical rate, which validates our theoretical guarantee
on the minimax optimal statistical rate.
1.18
d = 100; n = 1000; r = 2
d = 500; n = 10000; r = 5
d = 1000; n = 25000; r = 8
1.14
1.04
1.5
0.7
0.65
0.6
1
1.06
2
0.75
1.2
1.08
2.5
0.8
1.6
kS(t) ! S$ kF
kL(t) ! L$ kF
1.1
3
1.8
1.4
1.12
3.5
0.55
r
r
r
r
0.8
1.02
0.6
1
0.5
d = 100; n = 1000; r = 2
d = 500; n = 10000; r = 5
d = 1000; n = 25000; r = 8
1.16
kS(t) ! S$ kF
4
1
0
2
4
6
8
10
12
14
Number of iteration (t)
16
18
0.98
kL(t) ! L$ kF
5
4.5
0
5
10
15
20
25
30
35
0.4
0.5
40
Number of iteration (t)
p
1
1.5
s$ log d=n
=2
=5
=7
= 10
s$
s$
s$
s$
0.5
0.45
2
0.4
0.35
0.4
0.45
0.5
0.55
0.6
p
rd=n
0.65
0.7
= 300
= 400
= 500
= 600
0.75
0.8
(a) Estimation error for S? (b) Estimation error for L? (c) r fixed and varying (d) s? fixed and varying
n, d and s?
n, d and r
Figure 1: (a)-(b): Evolution of estimation errors with number of iterations t going up with the sparsity
b (T ) S? kF and
parameter s? set as 0.02 ? d2 and varying d, npand r. (c)-(d): Estimation
errors kS
p
(T )
?
?
b
kL
L kF versus scaled statistical errors s log d/n and rd/n.
5.2
Genomic Data
In this subsection, we apply our method to TCGA breast cancer gene expression data to infer
regulatory network. We downloaded the gene expression data from cBioPortal2 . Here we focused on
299 breast cancer related transcription factors (TFs) and estimated the regulatory relationships for
each pair of TFs over two breast cancer subtypes: luminal and basal. We compared our method AltGD
2
http://www.cbioportal.org/
8
with ADMM and PPA which are all based on LVGGM. We also compared it with the graphical
Lasso (GLasso) which only considers the sparse structure of precision matrix and ignores the latent
variables; we chose QUIC3 to solve the GLasso estimator. Regarding the benchmark standard, we
used the ?regulatory potential scores? between a pair of genes (a TF and a target gene) for these two
breast cancer subtypes compiled based on both co-expression and TF ChIP-seq binding data from the
Cistrome Cancer Database4 .
For luminal subtype, there are n = 601 samples
Table 3: Summary of CPU time of different meth- and d = 299 TFs. The regularization parameods on luminal subtype breast cancer dataset.
ters for `1,1 norm in GLasso, for `1,1 norm and
Method
GLasso
PPA
ADMM AltGD nuclear norm in PPA and ADMM were tuned
by grid search. The step sizes of AltGD were
Time (s) 38.6310 85.0100
7.6700
0.1500 set as ? = 0.1/b
? 2 and ? 0 = 0.1/b
? 4 , where ?b is
the maximal eigenvalue of sample covariance
matrix. The thresholding parameter s and number of latent variables r were tuned by grid search. In
Table 3, we present the CPU time of each method. Importantly, we can see that AltGD is the fastest
among all the methods and is even more than 50 times faster than the second fastest method ADMM.
HDAC2
?
H2AFX
HDAC2
?
?
H2AFX
ELF5?
ELF5?
ATF4
?
?
?
POLR2B
?
?
MXI1
ELF5?
?
ATF4
?
?
SUMO2
IRF4
?
SF1
(a) GLasso
IRF4
H2AFX
?
?
?
POLR2B
?
?
SF1
?
ATF4
?
?
SUMO2
?
POLR2B
?
MXI1
?
IRF4
(b) PPA
SREBF2
ELF5?
ATF4
?
MXI1
?
?
?
SUMO2
?
POLR2B
?
HDAC2
?
SREBF2
MXI1
?
H2AFX
SREBF2
?
SUMO2
HDAC2
?
SREBF2
?
SF1
(c) ADMM
?
IRF4
?
SF1
(d) AltGD
Figure 2: An example of subnetwork in the transcriptional regulatory network of luminal breast
cancer. Here gray edges are the interactions from the Cistrome Cancer Database; red edges are the
ones inferred by the respective methods; green edges are incorrectly inferred interactions.
To demonstrate the performances of different methods on recovering the overall transcriptional regulatory network, we randomly selected 10 TFs in the benchmark network and plotted the subnetwork
in Figure 2 which has 70 edges with nonzero regulatory potential scores. Specifically, the gray edges
form the benchmark network, the red edges are those identified correctly and the green edges are
those incorrectly inferred by each method. We can observe from Figure 2 that the methods based on
LVGGMs are able to recover more edges accurately than graphical Lasso because of the intervention
of latent variables. We remark that all the methods were not able to completely recover the entire
regulatory network partly because we only used the gene expression data for TFs (instead of all
genes) and the regulatory potential scores from the Cistome Cancer Database also used TF binding
information. Due to space limit, we postpone additional experimental results to the appendix.
6
Conclusions
In this paper, to speed up the learning of LVGGM, we proposed a sparsity constrained maximum likelihood estimator based on matrix factorization. We developed an efficient alternating gradient descent
algorithm, and proved that the proposed algorithm is guaranteed to converge to the unknown sparse
and low-rank matrices with a linear convergence rate up to the optimal statical error. Experiments on
both synthetic and real world genomic data supported our theory.
Acknowledgements
We would like to thank the anonymous reviewers for their helpful comments. This research was
sponsored in part by the National Science Foundation IIS-1652539, IIS-1717205 and IIS-1717206.
The views and conclusions contained in this paper are those of the authors and should not be
interpreted as representing any funding agencies.
3
4
http://www.cs.utexas.edu/~sustik/QUIC/
http://cistrome.org/CistromeCancer/
9
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