Question: Represent the article for finding a single-sentence TLDR.

Question: It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons m required to approximate natural classes of multivariate polynomials of n variables grows only linearly with n for deep neural networks, but grows exponentially when merely a single hidden layer is allowed. We also provide evidence that when the number of hidden layers is increased from 1 to k, the neuron requirement grows exponentially not with n but with n^{1/k}, suggesting that the minimum number of layers required for practical expressibility grows only logarithmically with n. Deep learning has lately been shown to be a very powerful tool for a wide range of problems, from image segmentation to machine translation. Despite its success, many of the techniques developed by practitioners of artificial neural networks (ANNs) are heuristics without theoretical guarantees. Perhaps most notably, the power of feedforward networks with many layers (deep networks) has not been fully explained. The goal of this paper is to shed more light on this question and to suggest heuristics for how deep is deep enough. It is well-known BID7 BID11 BID15 BID1 BID23 that neural networks with a single hidden layer can approximate any function under reasonable assumptions, but it is possible that the networks required will be extremely large. Recent authors have shown that some functions can be approximated by deeper networks much more efficiently (i.e. with fewer neurons) than by shallower ones. Often, these admit one or more of the following limitations: "existence proofs" without explicit constructions of the functions in question; explicit constructions, but relatively complicated functions; or applicability only to types of network rarely used in practice. It is important and timely to extend this work to make it more concrete and actionable, by deriving resource requirements for approximating natural classes of functions using today's most common neural network architectures. BID17 recently proved that it is exponentially more efficient to use a deep network than a shallow network when Taylor-approximating the product of input variables. In the present paper, we move far beyond this in the following ways: (i) we use standard uniform approximation instead of Taylor approximation, (ii) we show that the exponential advantage of depth extends to all general sparse multivariate polynomials, and (iii) we address the question of how the number of neurons scales with the number of layers. Our apply to standard feedforward neural networks and are borne out by empirical tests. Our primary contributions are as follows:• It is possible to achieve arbitrarily close approximations of simple multivariate and univariate polynomials with neural networks having a bounded number of neurons (see §3).• Such polynomials are exponentially easier to approximate with deep networks than with shallow networks (see §4).• The power of networks improves rapidly with depth; for natural polynomials, the number of layers required is at most logarithmic in the number of input variables, where the base of the logarithm depends upon the layer width (see §5). Deeper networks have been shown to have greater representational power with respect to various notions of complexity, including piecewise linear decision boundaries BID22 and topological invariants BID2. Recently, and showed that the trajectories of input variables attain exponentially greater length and curvature with greater network depth. Work including BID8; BID10; BID23; BID24; shows that there exist functions that require exponential width to be approximated by a shallow network. BID1 provides bounds on the error in approximating general functions by shallow networks. BID20 and BID24 show that for compositional functions (those that can be expressed by recursive function composition), the number of neurons required for approximation by a deep network is exponentially smaller than the best known upper bounds for a shallow network. BID20 ask whether functions with tight lower bounds must be pathologically complicated, a question which we answer here in the negative. Various authors have also considered the power of deeper networks of types other than the standard feedforward model. The problem has also been posed for sum-product networks BID9 and restricted Boltzmann machines BID19. showed, using tools from tensor decomposition, that shallow arithmetic circuits can express only a measure-zero set of the functions expressible by deep circuits. A weak generalization of this to convolutional neural networks was shown in. In this paper, we will consider the standard model of feedforward neural networks (also called multilayer perceptrons). Formally, the network may be considered as a multivariate function DISPLAYFORM0.., A k are constant matrices and σ denotes a scalar nonlinear function applied element-wise to vectors. The constant k is referred to as the depth of the network. The neurons of the network are the entries of the vectors σ(A · · · σ(A 1 σ(A 0 x)) · · · ), for = 1,..., k − 1. These vectors are referred to as the hidden layers of the network. Two notions of approximation will be relevant in our : -approximation, also known as uniform approximation, and Taylor approximation. Definition 3.1. For constant > 0, we say that a network N (x) -approximates a multivariate function f (x) (for x in a specified domain DISPLAYFORM1 Definition 3.2. We say that a network N (x) Taylor-approximates a multivariate polynomial p(x) of degree d if p(x) is the dth order Taylor polynomial (about the origin) of N (x).The following proposition shows that Taylor approximation implies -approximation for homogeneous polynomials. The reverse implication does not hold. Proposition 3.3. Suppose that the network N (x) Taylor-approximates the homogeneous multivariate polynomial p(x). Then, for every, there exists a network N (x) that -approximates p(x), such that N (x) and N (x) have the same number of neurons in each layer. (This statement holds for x ∈ (−R, R) n for any specified R.) DISPLAYFORM2 is a Taylor series with each E i (x) homogeneous of degree i. Since N (x) is the function defined by a neural network, it converges for every x ∈ R n. Thus, E(x) converges, as does DISPLAYFORM3. By picking δ sufficiently small, we can make each term DISPLAYFORM4 d, and therefore: DISPLAYFORM5 We conclude that N (x) is an -approximation of p(x), as desired. For a fixed nonlinear function σ, we consider the total number of neurons (excluding input and output neurons) needed for a network to approximate a given function. Remarkably, it is possible to attain arbitrarily good approximations of a (not necessarily homogeneous) multivariate polynomial by a feedforward neural network, even with a single hidden layer, without increasing the number of neurons past a certain bound. (See also Corollary 1 in BID24 .) Theorem 3.4. Suppose that p(x) is a degree-d multivariate polynomial and that the nonlinearity σ has nonzero Taylor coefficients up to degree d. Let m k (p) be the minimum number of neurons in a depth-k network that -approximates p. Then, the limit lim →0 m k (p) exists (and is finite). (Once again, this statement holds for x ∈ (−R, R) n for any specified R.)Proof. We show that lim →0 m 1 (p) exists; it follows immediately that lim →0 m k (p) exists for every k, since an -approximation to p with depth k can be constructed from one with depth 1. DISPLAYFORM6. We claim that each p i (x) can be Taylor-approximated by a network N i (x) with one hidden layer. This follows, for example, from the proof in BID17 that products can be Taylor-approximated by networks with one hidden layer, since each monomial is the product of several inputs (with multiplicity); we prove a far stronger about N i (x) later in this paper (see Theorem 4.1).Suppose now that N i (x) has m i hidden neurons. By Proposition 3.3, we conclude that since p i (x) is homogeneous, it may be δ-approximated by a network N DISPLAYFORM7 This theorem is perhaps surprising, since it is common for -approximations to functions to require ever-greater complexity, approaching infinity as → 0. For example, the function exp(| − x|) may be approximated on the domain (−π, π) by Fourier sums of the form m k=0 a m cos(kx). However, in order to achieve -approximation, we need to take m ∼ 1/ √ terms. By contrast, we have shown that a finite neural network architecture can achieve arbitrarily good approximations merely by altering its weights. Note also that the assumption of nonzero Taylor coefficients cannot be dropped from Theorem 3.4. For example, the theorem is false for rectified linear units (ReLUs), which are piecewise linear and do not admit a Taylor series. This is because -approximating a non-linear polynomial with a piecewise linear function requires an ever-increasing number of pieces as → 0. In this section, we compare the efficiency of shallow networks (those with a single hidden layer) and deep networks at approximating multivariate polynomials. Proofs of our main are included in the Appendix. Our first shows that uniform approximation of monomials requires exponentially more neurons in a shallow than a deep network. DISPLAYFORM0 Suppose that the nonlinearity σ has nonzero Taylor coefficients up to degree 2d. Then, we have: DISPLAYFORM1 where x denotes the smallest integer that is at least x. We can prove a comparable for m Taylor under slightly weaker assumptions on σ. Note that by setting r 1 = r 2 =... = r n = 1, we recover the of BID17 that the product of n numbers requires 2 n neurons in a shallow network but can be Taylor-approximated with linearly many neurons in a deep network. DISPLAYFORM2 It is worth noting that neither of Theorems 4.1 and 4.2 implies the other. This is because it is possible for a polynomial to admit a compact uniform approximation without admitting a compact Taylor approximation. It is natural now to consider the cost of approximating general polynomials. However, without further constraint, this is relatively uninstructive because polynomials of degree d in n variables live within a space of dimension n+d d, and therefore most require exponentially many neurons for any depth of network. We therefore consider polynomials of sparsity c: that is, those that can be represented as the sum of c monomials. This includes many natural functions. Theorem 4.3. Let p(x) be a multivariate polynomial of degree d and sparsity c, having monomials q 1 (x), q 2 (x),..., q c (x). Suppose that the nonlinearity σ has nonzero Taylor coefficients up to degree 2d. Then, we have: DISPLAYFORM3 These statements also hold if m uniform is replaced with m Taylor.As mentioned above with respect to ReLUs, some assumptions on the Taylor coefficients of the activation function are necessary for the we present. However, it is possible to loosen the assumptions of Theorem 4.1 and 4.2 while still obtaining exponential lower bounds on m DISPLAYFORM4 Hence, A is invertible, which means that multiplying its columns by nonzero values gives another invertible matrix. Suppose that we multiply the jth column of A by σ j to get A, where σ(x) = j σ j x j is the Taylor expansion of σ(x). Now, observe that the ith row of A is exactly the coefficients of σ(a i x), up to the degree-d term. Since A is invertible, the rows must be linearly independent, so the polynomials σ(a i x), restricted to terms of degree at most d, must themselves be linearly independent. Since the space of degree-d univariate polynomials is (d + 1)-dimensional, these d + 1 linearly independent polynomials must span the space. Hence, m Taylor 1 (p) ≤ d + 1 for any univariate degree-d polynomial p. In fact, we can fix the weights from the input neuron to the hidden layer (to be a 0, a 1, . . ., a d, respectively) and still represent any polynomial p with d + 1 hidden neurons. Proposition 4.6. Let p(x) = x d, and suppose that the nonlinearity σ(x) has nonzero Taylor coefficients up to degree 2d. Then, we have: DISPLAYFORM5 These statements also hold if m uniform is replaced with m Taylor.Proof. Part (i) follows from part (i) of Theorems 4.1 and 4.2 by setting n = 1 and r 1 = d. For part (ii), observe that we can Taylor-approximate the square x 2 of an input x with three neurons in a single layer: DISPLAYFORM6 We refer to this construction as a square gate, and the construction of Lin et al. FORMULA15 as a product gate. We also use identity gate to refer to a neuron that simply preserves the input of a neuron from the preceding layer (this is equivalent to the skip connections in residual nets BID14 We now consider how m uniform k (p) scales with k, interpolating between exponential in n (for k = 1) and linear in n (for k = log n). In practice, networks with modest k > 1 are effective at representing natural functions. We explain this theoretically by showing that the cost of approximating the product polynomial drops off rapidly as k increases. By repeated application of the shallow network construction in Lin et al. FORMULA15, we obtain the following upper bound on m uniform k (p), which we conjecture to be essentially tight. Our approach leverages the compositionality of polynomials, as discussed e.g. in BID20 and BID24, using a tree-like neural network architecture. Theorem 5.1. Let p(x) equal the product x 1 x 2 · · · x n, and suppose σ has nonzero Taylor coefficients up to degree n. Then, we have: DISPLAYFORM0 Proof. We construct a network in which groups of the n inputs are recursively multiplied up to Taylor approximation. The n inputs are first divided into groups of size b 1, and each group is multiplied in the first hidden layer using 2 b1 neurons (as described in Lin et al. FORMULA15). Thus, the first hidden layer includes a total of 2 b1 n/b 1 neurons. This gives us n/b 1 values to multiply, which are in turn divided into groups of size b 2. Each group is multiplied in the second hidden layer using 2 b2 neurons. Thus, the second hidden layer includes a total of 2 b2 n/(b 1 b 2) neurons. We continue in this fashion for b 1, b 2,..., b k such that b 1 b 2 · · · b k = n, giving us one neuron which is the product of all of our inputs. By considering the total number of neurons used, we conclude In fact, we can solve for the choice of b i such that the upper bound in is minimized, under the condition b 1 b 2 · · · b k = n. Using the technique of Lagrange multipliers, we know that the optimum occurs at a minimum of the function as n varies are shown for k = 1, 2, 3. Observe that the b i converge to n 1/k for large n, as witnessed by a linear fit in the log-log plot. The exact values are given by equations and. for n = 20 is shown in black. In the region above and to the right of the curve, it is possible to effectively approximate the product function (Theorem 5.1). DISPLAYFORM1 DISPLAYFORM2 Differentiating L with respect to b i, we obtain the conditions DISPLAYFORM3 Dividing by k j=i+1 b j and rearranging gives us the recursion DISPLAYFORM4 Thus, the optimal b i are not exactly equal but very slowly increasing with i (see FIG6).The following conjecture states that the bound given in Theorem 5.1 is (approximately) optimal. Conjecture 5.2. Let p(x) equal to the product x 1 x 2 · · · x n, and suppose that σ has all nonzero Taylor coefficients. Then, we have: DISPLAYFORM5 i.e., the exponent grows as n 1/k for n → ∞.We empirically tested Conjecture 5.2 by training ANNs to predict the product of input values x 1,..., x n with n = 20 (see FIG7 . The rapid interpolation from exponential to linear width aligns with our predictions. In our experiments, we used feedforward networks with dense connections between successive layers. In the figure, we show for σ(x) = tanh(x) (note that this behavior is even better than expected, since this function actually has numerous zero Taylor coefficients). Similar were also obtained for rectified linear units (ReLUs) as the nonlinearity, despite the fact that this function does not even admit a Taylor series. The number of layers was varied, as was the number of neurons within a single layer. The networks were trained using the AdaDelta optimizer to minimize the absolute value of the difference between the predicted and actual values. Input variables x i were drawn uniformly at random from the interval, so that the expected value of the output would be of manageable size. Eq. provides a helpful rule of thumb for how deep is deep enough. Suppose, for instance, that we wish to keep typical layers no wider than about a thousand (∼ 2 10) neurons. Eq. then implies n 1/k ∼ < 10, i.e., that the number of layers should be at least k ∼ > log 10 n. It would be very interesting if one could show that general polynomials p in n variables require a superpolynomial number of neurons to approximate for any constant number of hidden layers. The analogous statement for Boolean circuits -whether the complexity classes T C 0 and T C 1 are equal -remains unresolved and is assumed to be quite hard. Note that the formulations for Boolean circuits and deep neural networks are independent statements (neither would imply the other) due to the differences between computation on binary and real values. Indeed, gaps in expressivity have already been proven to exist for real-valued neural networks of different depths, for which the analogous remain unknown in Boolean circuits (see e.g. BID21 BID4 BID3 ; Montufar et al. FORMULA15 ; ; Telgarsky FORMULA15). We have shown how the power of deeper ANNs can be quantified even for simple polynomials. We have proved that arbitrarily good approximations of polynomials are possible even with a fixed number of neurons and that there is an exponential gap between the width of shallow and deep networks required for approximating a given sparse polynomial. For n variables, a shallow network requires size exponential in n, while a deep network requires at most linearly many neurons. Networks with a constant number k > 1 of hidden layers appear to interpolate between these extremes, following a curve exponential in n 1/k. This suggests a rough heuristic for the number of layers required for approximating simple functions with neural networks. For example, if we want no layers to have more than 2 10 neurons, say, then the minimum number of layers required grows only as log 10 n. To further improve efficiency using the O(n) constructions we have presented, it suffices to increase the number of layers by a factor of log 2 10 ≈ 3, to log 2 n. The key property we use in our constructions is compositionality, as detailed in BID24. It is worth noting that as a consequence our networks enjoy the property of locality mentioned in, which is also a feature of convolutional neural nets. That is, each neuron in a layer is assumed to be connected only to a small subset of neurons from the previous layer, rather than the entirety (or some large fraction). In fact, we showed (e.g. Prop. 4.6) that there exist natural functions computable with linearly many neurons, with each neuron is connected to at most two neurons in the preceding layer, which nonetheless cannot be computed with fewer than exponentially many neurons in a single layer, no matter how may connections are used. Our construction can also be framed with reference to the other properties mentioned in: those of sharing (in which weights are shared between neural connections) and pooling (in which layers are gradually collapsed, as our construction essentially does with recursive combination of inputs). This paper has focused exclusively on the resources (neurons and synapses) required to compute a given function for fixed network depth. (Note also of BID18 ; BID13 ; BID12 for networks of fixed width.) An important complementary challenge is to quantify the resources (e.g. training steps) required to learn the computation, i.e., to converge to appropriate weights using training data -possibly a fixed amount thereof, as suggested in. There are simple functions that can be computed with polynomial resources but require exponential resources to learn . It is quite possible that architectures we have not considered increase the feasibility of learning. For example, residual networks (ResNets) BID14 and unitary nets (see e.g. BID0 BID16) are no more powerful in representational ability than conventional networks of the same size, but by being less susceptible to the "vanishing/exploding gradient" problem, it is far easier to optimize them in practice. We look forward to future work that will help us understand the power of neural networks to learn. Without loss of generality, suppose that r i > 0 for i = 1,..., n. Let X be the multiset in which x i occurs with multiplicity r i.We first show that n i=1 (r i + 1) neurons are sufficient to approximate p(x). Appendix A in demonstrates that for variables y 1,..., y N, the product y 1 · · · · · y N can be Taylorapproximated as a linear combination of the 2 N functions σ(±y 1 ± · · · ± y d).Consider setting y 1,..., y d equal to the elements of multiset X. Then, we conclude that we can approximate p(x) as a linear combination of the functions σ(±y 1 ± · · · ± y d). However, these functions are not all distinct: there are r i + 1 distinct ways to assign ± signs to r i copies of x i (ignoring permutations of the signs). Therefore, there are DISPLAYFORM0 We now show that this number of neurons is also necessary for approximating p(x). Suppose that N (x) is an -approximation to p(x) with depth 1, and let the Taylor series of N (x) be p(x)+E(x). Let E k (x) be the degree-k homogeneous component of E(x), for 0 ≤ k ≤ 2d. By the definition of -approximation, sup x E(x) goes to 0 as does, so by picking small enough, we can ensure that the coefficients of each E k (x) go to 0.Let m = m uniform 1 (p) and suppose that σ(x) has the Taylor expansion ∞ k=0 σ k x k. Then, by grouping terms of each order, we conclude that there exist constants a ij and w j such that DISPLAYFORM0 For each S ⊆ X, let us take the derivative of this equation by every variable that occurs in S, where we take multiple derivatives of variables that occur multiple times. This gives DISPLAYFORM1 DISPLAYFORM2 Observe that there are r ≡ n i=1 (r i + 1) choices for S, since each variable x i can be included anywhere from 0 to r i times. Define A to be the r × m matrix with entries A S,j = h∈S a hj. We claim that A has full row rank. This would show that the number of columns m is at least the number of rows r = n i=1 (r i + 1), proving the desired lower bound on m. Suppose towards contradiction that the rows A S,• admit a linear dependence: DISPLAYFORM3 where the coefficients c are all nonzero and the S denote distinct subsets of X. Let S * be such that |c * | is maximized. Then, take the dot product of each side of the above equation by the vector with entries (indexed by j) equal to w j (DISPLAYFORM4 We can use to simplify the first term and (with k = d + |S | − |S * |) to simplify the second term, giving us: DISPLAYFORM5 DISPLAYFORM6 Consider the coefficient of the monomial ∂ ∂S * p(x), which appears in the first summand with coefficient c * · |S * |! σ d ·d!. Since the S are distinct, this monomial does not appear in any other term ∂ ∂S p(x), but it could appear in some of the terms DISPLAYFORM7 By definition, |c * | is the largest of the values |c |, and by setting small enough, all coefficients of ∂ ∂S E k (x) can be made negligibly small for every k. This implies that the coefficient of the monomial ∂ ∂S * p(x) can be made arbitrarily close to c * · |S * |! σ d ·d!, which is nonzero since c * is nonzero. However, the left-hand side of equation FORMULA27 tells us that this coefficient should be zero -a contradiction. We conclude that A has full row rank, and therefore that m uniform 1 DISPLAYFORM8 This completes the proof of part (i).We now consider part (ii) of the theorem. It follows from Proposition 4.6, part (ii) that, for each i, we can Taylor-approximate x ri i using 7 log 2 (r i) neurons arranged in a deep network. Therefore, we can Taylor-approximate all of the x ri i using a total of i 7 log 2 (r i) neurons. From BID17, we know that these n terms can be multiplied using 4n additional neurons, giving us a total of i (7 log 2 (r i) +4). As above, suppose that r i > 0 for i = 1,..., n, and let X be the multiset in which x i occurs with multiplicity r i.It is shown in the proof of Theorem 4.1 that n i=1 (r i + 1) neurons are sufficient to Taylorapproximate p(x). We now show that this number of neurons is also necessary for approximating p(x). Let m = m Taylor 1 (p) and suppose that σ(x) has the Taylor expansion DISPLAYFORM9 Then, by grouping terms of each order, we conclude that there exist constants a ij and w j such that DISPLAYFORM10 For each S ⊆ X, let us take the derivative of equations FORMULA15 and FORMULA15 by every variable that occurs in S, where we take multiple derivatives of variables that occur multiple times. This gives DISPLAYFORM11 DISPLAYFORM12 for |S| ≤ k ≤ d − 1. Observe that there are r = n i=1 (r i + 1) choices for S, since each variable x i can be included anywhere from 0 to r i times. Define A to be the r × m matrix with entries A S,j = h∈S a hj. We claim that A has full row rank. This would show that the number of columns m is at least the number of rows r = n i=1 (r i + 1), proving the desired lower bound on m. Suppose towards contradiction that the rows A S,• admit a linear dependence: DISPLAYFORM13 where the coefficients c are nonzero and the S denote distinct subsets of X. Set s = max |S |. Then, take the dot product of each side of the above equation by the vector with entries (indexed by DISPLAYFORM14 We can use to simplify the first term and (with k = d + |S | − s) to simplify the second term, giving us: DISPLAYFORM15 Since the distinct monomials ∂ ∂S p(x) are linearly independent, this contradicts our assumption that the c are nonzero. We conclude that A has full row rank, and therefore that m Our proof in Theorem 4.1 relied upon the fact that all nonzero partial derivatives of a monomial are linearly independent. This fact is not true for general polynomials p; however, an exactly similar argument shows that m uniform 1 (p) is at least the number of linearly independent partial derivatives of p, taken with respect to multisets of the input variables. Consider the monomial q of p such that m uniform 1 (q) is maximized, and suppose that q(x) = x (q) is equal to the number n i=1 (r i + 1) of distinct monomials that can be obtained by taking partial derivatives of q. Let Q be the set of such monomials, and let D be the set of (iterated) partial derivatives corresponding to them, so that for d ∈ D, we have d(q) ∈ Q.Consider the set of polynomials P = {d(p) | d ∈ D}. We claim that there exists a linearly independent subset of P with size at least |D|/c. Suppose to the contrary that P is a maximal linearly independent subset of P with |P | < |D|/c. Since p has c monomials, every element of P has at most c monomials. Therefore, the total number of distinct monomials in elements of P is less than |D|. However, there are at least |D| distinct monomials contained in elements of P, since for d ∈ D, the polynomial d(p) contains the monomial d(q), and by definition all d(q) are distinct as d varies. We conclude that there is some polynomial p ∈ P \P containing a monomial that does not appear in any element of P. But then p is linearly independent of P, a contradiction since we assumed that P was maximal. We conclude that some linearly independent subset of P has size at least |D|/c, and therefore that the space of partial derivatives of p has rank at least |D|/c = m We will prove the desired lower bounds for m uniform 1 (p); a very similar argument holds for m Taylor 1 (p). As above, suppose that r i > 0 for i = 1,..., n. Let X be the multiset in which x i occurs with multiplicity r i.Suppose that N (x) is an -approximation to p(x) with depth 1, and let the degree-d Taylor polynomial of N (x) be p(x) + E(x). Let E d (x) be the degree-d homogeneous component of E(x). Observe that the coefficients of the error polynomial E d (x) can be made arbitrarily small by setting sufficiently small. Let m = m uniform 1 (p) and suppose that σ(x) has the Taylor expansion ∞ k=0 σ k x k. Then, by grouping terms of each order, we conclude that there exist constants a ij and w j such that DISPLAYFORM16 For each S ⊆ X, let us take the derivative of this equation by every variable that occurs in S, where we take multiple derivatives of variables that occur multiple times. This gives DISPLAYFORM17 Consider this equation as S ⊆ X varies over all C s multisets of fixed size s. The left-hand side represents a linear combination of the m terms (

represent this text to find a much much longer article about it with say hundreds-thousands of words
To give you a sense - "An approach to combine variational inference and Bayesian optimisation to solve complicated inverse problems" should be close to "Inverse problems are ubiquitous in natural sciences and refer to the challenging task of inferring complex and potentially multi-modal posterior distributions over hidden parameters given a set of observations. Typically, a model of the physical process in the form of differential equations is available but leads to intractable inference over its parameters. While the forward propagation of parameters through the model simulates the evolution of the system, the inverse problem of finding the parameters given the sequence of states is not unique. In this work, we propose a generalisation of the Bayesian optimisation framework to approximate inference. The ing method learns approximations to the posterior distribution by applying Stein variational gradient descent on top of estimates from a Gaussian process model. Preliminary demonstrate the method's performance on likelihood-free inference for reinforcement learning environments. We consider the problem of estimating parameters θ of a physical system according to observed data y. The forward model of the system is approximated by a computational model that generates dataŷ θ based on the given parameter settings θ. In many cases, the corresponding likelihood function p(ŷ θ |θ) is not available, and one resorts to likelihoodfree methods, such as approximate Bayesian computation (ABC) , conditional density estimation , etc. For certain applications in robotics and reinforcement learning, however, the number of simulations might be limited by resource constraints, imposing challenges to current approaches. Recent methods address the problem of efficiency in the use of simulations by either constructing conditional density estimators from joint data {θ i,ŷ i} N i=1, using, for example, mixture density networks , or by sequentially learning approximations to the likelihood function and then running Markov chain Monte Carlo (MCMC). In particular, derive an active learning approach using Bayesian optimisation (BO) to propose parameters for simulations. Their approach reduces the number of simulator runs from the typical thousands to a few hundreds. This paper investigates an approach to combine the flexible representative power of variational inference methods with the data efficiency of Bayesian optimisation. We present a Thompson sampling strategy to sequentially refine variational approximations to a black-box posterior. Parameters for new simulations are proposed by running Stein variational gradient descent (SVGD) over samples from a Gaussian process (GP) . The approach is also equipped with a method to optimally subsample the variational approximations for batch evaluations of the simulator models at each round. In the following, we present the derivation of our approach and preliminary experimental . Our goal is to estimate a distribution q that approximates a posterior distribution p(θ|y) over simulator parameters θ ∈ Θ ⊂ R d given observations y from a target system. We assume no access to a likelihood function p(y|θ), but only to a discrepancy measure 1 between simulator outputs and observations ∆ θ, as in. We take a Bayesian optimisation approach to find the optimal q * by minimising a discrepancy between q and the target p: where S represents the kernelised Stein discrepancy (KSD) . 2 We solve Equation 1 via a black-box approach which does not require gradients of the target distribution p nor its availability in closed form. The ing BO algorithm is composed of a GP model to form an approximate likelihood, a Thompson sampling acquisition function to select candidate distributions and a kernel herding procedure to optimally select samples of simulator parameters. A standard BO approach would place a GP to model the map from q's parameters to the corresponding KSD. However, such parameter space holds a weak connection with the original Θ and is possibly higher-dimensional. We choose to bypass this step by learning q directly via Stein variational gradient descent (SVGD) . Applying SVGD directly to Equation 1 would require gradients of the target log p. In our case, we have that: As p(y|θ) is unavailable, we use a GP to model g: θ → −∆ θ, which defines a synthetic likelihood function , i.e.: 1. The ABC literature offers a plenitude of choices for ∆ θ. For a review, we refer the reader to and. Our choice for experiments is given in Section 3. 2. Background details on the KSD are presented in the appendix (Section A.1). The simulations-observations discrepancy ∆ θ is possibly expensive to evaluate and not differentiable, due to the need of running a black-box simulator. The GP then provides an approximation which is cheap to evaluate and whose sample functions are differentiable for smooth kernels, allowing us to apply SVGD in the BO loop. We propose selecting candidate distributions q n ∈ Q based on a GP posterior sampling approach known as Thompson sampling , which has been successfully applied to BO problems in the case of selecting point candidates θ ∈ Θ (; ; Mutný and). Thompson sampling accounts for uncertainty in the model by sampling functions from the GP posterior. For models based on finite feature maps, such as sparse spectrum Gaussian processes (SSGPs) (Lázaro-), the Thompson sampling approach resumes to sampling weights w n from a multivariate Gaussian (Appendix A.2), so that: constitutes a sample from the posterior of a SSGP with mean function µ 0 and feature map φ. Recalling the objective in Equation 1, we can now define the acquisition function as: wherep n (θ) ∝ p(θ)e gn(θ) corresponds to an approximation to the target posterior p(θ|y) based on g n. SVGD represents the variational distribution q as a set of particles {θ i} M i=1 forming an empirical distribution. The particles are initialised as i.i.d. samples from the prior p(θ) and optimised via a sequence of smooth perturbations: where k(θ, θ) = φ(θ) T φ(θ) corresponds to the SSGP kernel, and η t is a small step size. Intuitively, the first term in the definition of ζ guides the particles to the local maxima of logp n, i.e. the modes ofp n, while the second term encourages diversification by repelling nearby particles. In contrast to the true posterior, the gradients of logp n are available as: Gradients of sample functions are always defined for SSGP models with differentiable mean functions, since the feature maps are smooth. For a uniform prior, which we use in experiments, also note that ∇ θ log p(θ) = 0 almost everywhere. Having selected a distribution q n, we need to run evaluations of ∆ θ from samples θ ∼ q n to update the GP model with. Representing q by a large number of particles M improves Algorithm 1: DBO Input: f, Q, N, S for n ∈ {1, . . . N} do q n ∈ argmax q∈Q h(q|D n−1) # Maximise acquisition function via SVGD {θ n,i} S i=1 ∼ Herding(q n, D n−1) # Sample simulator parameters for i ∈ {1, . . ., S} do z n,i:= −∆ θ n,i # Collect observation end end exploration of the approximate posterior surface, allowing SVGD to find distant modes. However, we should not use the large number of particles directly as sample parameters to run the simulator with, since simulations are expensive. Therefore, we select S M query parameters {θ n,j} S j=1 ⊂ Θ by optimally subsampling the candidate q n. Kernel herding constructs a set of samples which minimises the error on empirical estimates for expectations under a given distribution q. This error is bounded by the maximum mean discrepancy (MMD) between the kernel embedding of q and its subsampled version . In the case of SSGPs, the kernel herding procedure resumes to the following algorithm: for j ∈ {0, . . ., S − 1} and α 0 = ψ q = E θ∼q [φ(θ)]. However, instead of naively herding with the original feature map φ, we make use of the information encoded by the GP to select samples which will be the most informative for the model. Such information is encoded by the GP posterior kernel: where N is the covariance matrix of the GP weights posterior (defined in Appendix A.2). The posterior kernel provides an embedding for q given by: which accounts for the previously observed locations in the GP data. Replacing ψ q by ψ n q in Equation 8 yields the sampling scheme we use. The distributional Bayesian optimisation (DBO) algorithm is summarised in Algorithm 1. In this section, we present experimental evaluating DBO in synthetic data scenarios. As a baseline we compare the method against mixture density networks (MDNs), as in , which were learnt from a dataset of parameters sampled from the prior p(θ) and the corresponding simulator outputsŷ θ. The experiment evaluates the proposed method on OpenAI Gym's 3 cart-pole environment. We fix a given setting for its physics parameters θ real and generate a dataset y of 10 trajectories by executing randomly sampled actions. Summary statistics γ were the same as. The discrepancy was set to ∆ θ:= γ θ − γ real 2 /σ 2. We place a uniform prior p(θ) with bounds specific for the environment. Further details on the experimental setup are described in Appendix B. An open-source implementation can be found online 4. The in Figure 1 show that the mehtod is able to recover the target system's curve-shaped posterior and is able to obtain better approximations to the posterior when compared to the MDN approach. We can also see that in terms of MMD, DBO is able to provide a better overall approximation than the MDN. This paper presented a Bayesian optimisation approach to inverse problems on simulator parameters. Preliminary demonstrated the potential of the method for reinforcement learning applications. In particular, show that distributional Bayesian optimisation is able to provide a more sample-efficient approach than other likelihood-free inference methods when inferring parameters of a classical reinforcement learning environment. Future work includes further scalability and theoretical analysis of the method. 3. OpenAI Gym: https://gym.openai.com 4. Code available at: https://github.com/rafaol/dbo-aabi2019, instead., we can represent any function g sampled from the SSGP posterior as g(θ) = µ 0 (θ) + w T φ(θ), θ ∈ Θ, where: with Φ N = [φ(θ 1),..., φ(θ N)] ∈ R 2M ×N. The posterior over g is then determined by: µ N (θ):= µ 0 (θ) + φ(θ) where µ N and σ 2 N denote the GP posterior mean and variance functions, respectively. Fast incremental updates: To reduce the time complexity in the update of the GP posterior when given a new observation pair (θ N +1, z N +1), propose using the decomposition: To avoid recomputing A −1 N +1, one can instead keep track of its Cholesky factors. The latter allows us to update the GP posterior with time complexity O(M 2) , which is constant with respect to the number of data points N."

We prove that deep neural networks are exponentially more efficient than shallow ones at approximating sparse multivariate polynomials.

Represent the short summary for finding the corresponding scientific article (long)

We show shortcut connections should be placed in patterns that minimize between-layer distances during backpropagation, and design networks that achieve log L distances using L log(L) connections.


Question: Represent the text to find a very short summary (1-2 sentences max)

Question: Convolutional neural networks (CNNs) in recent years have made a dramatic impact in science, technology and industry, yet the theoretical mechanism of CNN architecture design remains surprisingly vague. The CNN neurons, including its distinctive element, convolutional filters, are known to be learnable features, yet their individual role in producing the output is rather unclear. The thesis of this work is that not all neurons are equally important and some of them contain more useful information to perform a given task. Hence, we propose to quantify and rank neuron importance, and directly incorporate neuron importance in the objective function under two formulations: a game theoretical approach based on Shapley value which computes the marginal contribution of each filter; and a probabilistic approach based on what-we-call, the importance switch using variational inference. Using these two methods we confirm the general theory that some of the neurons are inherently more important than the others. Various experiments illustrate that learned ranks can be readily useable for structured network compression and interpretability of learned features. Neural networks have achieved state-of-the art in various cognition tasks, including image and speech recognition, machine translation, reinforcement learning (; ;). Many of these applications involved CNNs which excel in particular in the vision tasks due to its ability to capture visual by means of convolution filters. Although the effectiveness of convolutional networks is unquestionable, the details of the architecture design and what particularly makes neural network work in detail remain highly uncertain. The experimental roughly confirm that the accuracy of the network and representational capacity is correlated with the depth of the network; ). Interestingly, the deeper architecture also become wider, although the link between width and network expressivity is questionable and the choice of the number of neurons is rather discretionary. As a the discussion about the network architecture often revolves around the numbers of filters and layers and their relative positioning, putting aside the conversation about the quality of the information that it contains. The increasing size of the network architectures have faced scrutiny that made claims that the networks are overparametrized raising two main concerns: heavy computational load and potential overfitting. In response to the need to build networks that are smaller yet accurate, a stream of research attempted to remove redundant units, compress the networks and design lighter architectures (; . A widespread approach to network reduction has been removing weights that are small or even close to zero . This line of research implicitly discerns that nodes with larger weights are more significant for learning task than the small weights. As a , broadly speaking, this approach divides features between those that are useful which are kept and those which are insignificant and therefore discarded, forming a sort of binary approach. In this work, we would like to scrutinize the individual filters and form an explicit theory that states that the units in the network (both convolutional filters and nodes in fully connected layers) are not equally important when it comes to performing an inference task. The corollary of this thesis is that CNNs learn features in a discriminative way so that some of them carry more significance than others, and the knowledge about the input is not uniformly distributed among the CNN features. This theory is in line of research that adding more filters does not make the network more expressive since learning relevant information to the network has already been addressed by other filters. Given the proposed theory, we would like to make a step forward in gaining insight what the CNN learns and propose to extend the binary approach to form a quantifiable ranking of features. In other words, we attempt to estimate the importance of each feature compared to the others with particular focus on convolutional filters, which may be visualized. We introduce a theoretical framework to quantify how important each feature is through proposing a feature ranking method based on two different approaches. The first approach derives from the game theoretical concept of Shapley value , which assesses the importance of an individual in a group of neurons based on its marginal contribution to the group. The second method takes a probabilistic approach and introduces additional learnable parameters, which we call importance switches, that take real values and are trained by means of variational inference to give more weight to the important features. The extensive experimental using these approaches indicate that some features are inherently more significant than others. The theoretical underpinnings of the feature rankings have further direct practical implications we explore. Firstly, the knowledge of the ranking allows to know which features directly impact the score of our method and consequently a more informed way of building an effective model. Thus, we are able to build a network around the the relevant features and discard the less relevant ones, effectively compressing the network achieving state-of-the-art . Secondly and perhaps more significantly, the feature ranking of convolutional features provides more interpretable information about the network and places meaning on particular features in the context of a given task, thus casting light on the black box models. To achieve human interpretability, we visualize the most significant features which significantly show the significance of repeated and complementary features. In early years of CNN development, the networks were limited to a few layers . Recently, the architectures have become deeper and wider (; . The emergence of GPU implementability and regularization algorithms has allowed to use large architectures which train and generalize well. Nevertheless, the trend towards building larger neural networks ironically opposed the research about the nature, interpretability and knowledge extraction from within the neural network models, which we are interested in this work. Therefore, we will compare our method to existing ones in terms of compression ability and interpretability, and then frame the idea in terms of neuron ranking. Compression. The early work on compression largely focused on non-Bayesian approaches, e.g., and mostly centered around non-structured pruning methods, e.g., removing single weights from the architectures of CNNs . Then, hardware-oriented structured pruning techniques made more practical speed-ups (; ; ;). More recently Bayesian methods using the network weights' uncertainties have achieved impressive compression rates, e.g., using sparsity inducing prior on scale parameter , using Gaussian mixture priors, and using the grouping of weights through a group Horseshoe prior, among many. However, none of these methods prune neurons based on the direct optimization for extracting the importance of each neuron. Interpretability. Broadly there are three lines of work done for intepretability of CNNs. The first line of work, the early and used-to-be very popular work, focused on visualization of neurons in CNNs to understand how information propagates within the network . Another line of work focused on probing trained CNNs to obtain local & pixel level explanations either layer-wise or class-wise using gradient information of querying points (; ;). Last line of work focused on mapping semantic concepts to latent representations of CNNs . Other somewhat related work for interpretability using Shapley value also exist (but not in the context of CNNs) . Compared to existing methods, our method provides a global view of a trained model in terms of the importance of learned features. In what follows, we introduce the two methods to perform neuron ranking. The first approach derives from the game theoretical concept of Shapley value . The concept allows to compute the importance score (payoff) of an individual based on the payoffs given to collections of individuals. We subsequently adapt the concept to rank the neurons in terms of their predictive utility. Assuming that an important feature allows for task generalization, the feature importance further translates into finding features that carry most information and usefulness in the prediction task that lead to achieving higher accuracy. A coalitional game is a game where utility is given to a group of players (in our case, nodes or neurons) instead of each agent individually. Let N be the number of agents, which in this work are CNN features (also referred as neurons or nodes). To be specific, let N l to be the number of neurons in a layer l (in unambiguous cases, for clarity we omit the subscript). For every group of players, a coalitional game specifies the payoff the members receive as a group or a coalition. We define a coalition of the neurons N of a layer L as a subset of neurons, C ⊆ N. To assess quantitatively the performance of a group of agents, each coalition is assigned to a real number, which is interpreted as a payoff that a coalition receives from being together. Mathematically, the value of a coalition is given by a characteristic function, which assigns a real number to a set of nodes. Formally, a characteristic function ν: 2 N → R maps each coalition (subset) C ⊆ N to a real number ν(C). Therefore, a coalitional game is defined by a tuple (N, ν), where N is a set of players and ν is a function that assigns payoffs to every coalition of N. A critical component of a coalitional game is specifying the choice of characteristic function that is assigned to a given subset of features. In the case of CNN, the test metric is accuracy which assesses whether the (argmax of) the network output is the correct label averaged over the number of examples. As a , we choose the accuracy on a validation set as the characteristic function, that is, ν(C) = acc(C) and ν(N) = acc(N). The question now remains how to assess the importance of a single feature given the information about the payoffs for each subset of nodes. To this end, we employ the concept of Shapley value about the normative payoff of the total reward/cost, that is a division scheme that allows to distribute the total payoff uniquely and in a fair way. Shapley proposes to evaluate each player by the marginal contribution that the player makes to every coalition averaged over all the coalitions. The marginal contribution of an agent n is the difference between the value of a coalition C that contains n and the coalition C \ n. For example, when a coalition has no members, i.e. is empty, and the neuron n 1 joins the coalition, the value of its marginal contribution is equal to the value of the one-member coalition as the value of the empty coalition is equal to 0, ν({n 1}) − ν({∅}) = ν({n 1}) where {n 1} = C. Subsequently, when another agent n 2 joins this coalition, its marginal contribution is equal to ν({n 1, n 2}) − ν({n 1}). The process continues until all the nodes join the coalition. The coalition of all the nodes is called the grand coalition. The order of nodes, which builds subsequent coalitions to finally the grand coalition, can be represented as a permutation of nodes. For example, in the case of permutation n 5 n 3 n 7...n N...n 2, the neuron n 5 creates the first non-empty coalition on it own, and we measure the accuracy of the pretrained model which includes only one neuron, n 5, in the given layer of the original pre-trained model. Then two-element coalition n 5 n 3 is formed corresponding to the two-neuron layer, and so on. All the subsequent nodes join the coalition in the order given by the permutation. There are N! permutations of N nodes, meaning that there are N! different ways to form a coalition. To compute the Shapley value of the node n, we compare the accuracy of the architecture before and after adding the node n, that is the marginal contributions of n (which may be negative) for each of the N! permutations and divide the sum by all the permutations. The Shapley value of n is then the averaged marginal contribution of n. Formally, let π denote a permutation, π(i) a place of the neuron n i in the permutation π, and C π (i) the coalition formed by the predecessors of n i such that C π (i) = {n j ∈ π : π(j) before π(i)}. For example, in the permutation n 5 n 3 n 7...n N...n 2, π = n 7 and C π = {n 5, n 3}. The Shapley value (SV i) of the node n i is thus defined as follows: This formula can also be written in a form that considers sets instead of permutations: PRACTICAL CONSIDERATIONS First, Shapley value is a mathematically rigorous division scheme and, strictly speaking, it has been proposed as the only measure that satisfies four normative criteria regarding the fair payoff distribution. These criteria are efficiency where the total gain is distributed among the agents, symmetry; if i and j are agents such that ν(C ∪ i) = ν(C ∪ j) for each coalition C of N, then SV (i) = SV (j), null player payoff such that an agent who contributes nothing to every coalition obtains zero individual payoff and linearity; ν(C) = ν 1 (C) + ν 2 (C) for every coalition implies SV ν1 (i) + SV ν2 (i) = SV ν (i). Nevertheless, the choice of a characteristic function which satisfies these criteria is not feasible in case of our application due to the fact that we do not have control over the output of the model. As a , the characteristic function may not be monotone which violates the first criterion. However, the payoff produced by the Shapley value, although may not be unique, is a valid cost division which works well in practice. Second, computing the characteristic function for every subset is combinatorial and takes exponential time complexity. Hence, for large networks, computing Shapley value is computationally infeasible. We propose the following solutions to approximate the optimal solution and obtain a sensible ranking metric based on Shapley value. The first solution entails computing the Shapley value for the subsets no larger than arbitrary k. As a we only compute the synergies that are no larger than k. Intuitively, we assume that that the larger the coalition, the less information is to be obtained from computing the large subsets. The second solution is based on sampling and sampling provides an unbiased estimate of the optimal . Thus, we first sample the characteristic function and then sample the permutations needed for the computations of the Shapley value. What comes next describes our proposal to improve the speed of computation for identifying the neuron ranking in a continuous manner. To infer the neuron ranking in each layer, we propose to make a slight modification in the existing neural network architecture. We introduce a component, the importance switch, denoted by s l for each layer l. Each importance switch is a probability vector of length D l (the output dimension of the lth layer) and D l j s l,j = 1, where s l,j is the jth element of the vector. With this addition, we rewrite the forward pass under a deep neural network model, where the function f (W l, x i) can be the convolution operation for CNNs or simple matrix multiplication for MLPs between the weights W l and the unit x i, Pre-activation followed by a switch s l: Input to the next layer after going through a nonlinearity σ: where • is an element-wise product. Introducing a switch operation between layers in a neural network model was also presented in, although in their case, the switch is a binary random variable (called a gate). The output probability under such networks with L hidden layers for solving classification problems can be written as where g is the softmax operation. A natural choice to model the distribution over the switch is the Dirichlet distribution, which defines a probability distribution over a probability vector. We model each switch as a vector of independent Dirichlet distributed random variables When there is no prior knowledge, i.e., a priori we don't know which feature would be more important for prediction, so we treat them all equally important features by setting the same value to each parameter, i.e., α 0 = α 0 * 1 D l where 1 D l is a vector of ones of length D l. When we apply the same parameter to each dimension, this special case of Dirichlet distribution is called symmetric Dirichlet distribution. In this case, if we set α 0 < 1, this puts the probability mass toward a few components, ing in only a few components that are non-zero, i.e., inducing sparse probability vector. If we set α 0 > 1, all components become similar to each other. We model the posterior over s l as the Dirichlet distribution as well but with asymmetric form to learn a different probability on different elements of the switch (or neurons), using a set of variational parameters (the parameters for the posterior). We denote the variational parameters by φ l, where each element of the vector can choose any values above 0. Our posterior distribution over the switch is, hence, defined by With this parametric form of prior and posterior, we optimize the variational parameters φ l over each layer's importance switch by maximizing the variational lower bound with freezing all the weights to the pre-trained values, We do this variational learning for each layer's importance switch sequentially from the input layer to the last layer before the output layer. Computing the gradient of equation 8 with respect to φ l requires computing the gradients of the integral (the first term on RHS) and also the KL divergence term (the second term on RHS), as both depends on the value of φ l. The KL divergence between two Dirichlet distributions can be wrttien in closed form. However, the first term is tricky. As described in , the usual reparameterization trick, i.e., replacing a probability distribution with an equivalent parameterization of it by using a deterministic and differentiable transformation of some fixed base distribution 1, does not work. For instance, in an attempt to find a reparameterization, one could adopt the representation of a k-dimensional Dirichlet random variable as a weighted sum of Gamma random variables, s l,j = y j /(, where the shape parameter of Gamma is φ l,j and the scale parameter is 1. However, this does not allow us to detach the randomness from the parameters as the parameter still appears in the Gamma distribution, hence one needs to sample from the posterior every time the variational parameters are updated, which is costly and time-consuming. Existing methods suggest either explicitly or implicitly computing the gradients of the inverse CDF of the Gamma distribution during training to decrease the variance of the gradients (e.g., and among many). The length of the importance switch we consider is mostly less than on the order of 100s, in which case the variance of gradients does not affect the speed of convergence as significantly as in other cases such as Latent Dirichlet Allocation (LDA). Hence, when training for the importance switch in each layer, we use the analytic mean of the Dirichlet random variable to make a point estimate of the integral q φ l (s l) log p(D|s l)ds l ≈ log p(D|s l), wherẽ s l,j = φ l,j / D l j =1 φ l,j, which allows us to directly compute the gradient of the quantity without sampling from the posterior. As illustrated in section 4, this approximation performs well with relatively low dimensional switches. Game-theoretic vs. probabilistic neuron ranking How are the game-theoretic and the probabilistic neuron ranking methods related? Consider a vector of random variables r that describes a certain ranking for a certain number of neurons. The predictive distribution on the test data D *, in this case, can be obtained by integrating out a plausible distribution over the neuron rankings, which we denote by f (r), where the second line is a reasonable approximation if the distribution is highly peaked at around the optimal rankingr, meaning that there is indeed such an optimal ranking with a high confidence. This predictive distribution specifies the likelihood of the test data given that optimal ranking. In the multi-class classification, this predictive distribution is the likelihood of true lables given a classifier's predictions. When we compute the Shapley value, we use an "approximate" version of this predictive likelihood, namely, we introduce a max operation for choosing a single label that has the maximum probability for that class, and then see if the label matches the true label, ing in the frequency of correct labeling as an accuracy measure. Hence, both methods attempt to find the best ranking in order to "maximize the likelihood of data". While the Shapley optimization attempts to maximize the test data likelihood approximately in a combinatorial manner, the switch optimization attempts to maximize the training data likelihood with a regularization as in equation 8 in a continuous manner. In this section we present experimental based on the two proposed approaches for CNN features ranking, the Shapley value and the importance switch methods. The tests have been performed on LeNet-5 trained on MNIST and FashionMNIST, and VGG-16 trained on CIFAR-10. To compute the rankings for both methods the same pretrained model is used. To compute the Shapley value of each neuron in the trained model, we remove the subsets of features (both weights and biases) and test the network on a validation set. As mentioned, the accuracy is the payoff for a given group of features. The computation of the complete set of payoffs is of combinatorial nature and therefore we compute the power set for layers up to 25 nodes. To account for this limitation and to illustrate better the proposed method, we choose to limit the number of nodes in the pretrained LeNet-5 architecture to 10-20-100-25. When using the trained VGG-16, we use the same number of filters in each layer as in the original architecture. For the layers with larger number of features, we use one of the two methods to compute marginal contributions. The first method uses equation 1 and only limits the number of coalitions we consider to compute SV. The second method uses equation 2 the accuracy change between two subsets which differ by a single node. Both node and the first combination were sampled uniformly at random. When we learn the importance switches, we load the same train model which has been used to compute the Shapley value and then only add parameters for switches and trained them per layer with fixing all the other network parameters to the trained values. We run the training of the importance switches for 300 epochs, however, in practice, even a few iterations is sufficient to distinguish important nodes from the rest. We start with comparing the learnt ranks of the two methods. As summarized in Table 1, the first observation is that for the model pretrained both on MNIST and FashionMNIST both methods have identified similar nodes to be the most important. The similarity is more significant for smaller layers where over 50% of top nodes (here we consider top-5 nodes for clarity and top-10 nodes for the large fc1 layer) and in three out of six cases the top two nodes are the same. Significantly for conv2 on MNIST the group of four nodes are the same, and as far as fc2 on FashionMNIST is concerned, the top five nodes chosen from the set of 25 nodes are the same (the probability to select this subset at random is 6 · 10 −5), showing that the methods agree when it comes to both convolutional and fully connected layers. For brevity, please look at the Appendix for the rankings of the less significant nodes but what is notable is that both methods also identified similar groups of unimportant nodes, particularly in fc2 where every node indexed higher than 9 (as compared to nodes indexed lower than 9) scored very low for both methods. When it comes to larger layers, the methods however are more discrepant (yet still significantly the common nodes are found as seen in the case of fc1 layer). The differences may also come from the inexact computation of the Shapley value. FashionMNIST MNIST conv1 SH 0, 7, 6, 5, 1 1, 8, 7, 4, 6 IS 0, 7, 5, 9, 6 8, 1, 3, 9, 6 conv2 SH 5, 10, 0, 13, 9 2, 8, 9, 19, 4 IS 5, 8, 13, 14, 15 9, 2, 8, 19, 6 fc1 SH 60, 13, 43, 88, 94, 20, 70, 44, 32, 64 56, 86, 25, 64, 33, 17, 23, 96, 52, 81 IS 94, 7, 50, 92, 13, 25, 60, 40, 75, 45 25, 96, 58, 56, 88, 52, 23, 43, 30, 4 fc2 SH 5, 1, 8, 9, 7 1, 7, 2, 3, 0 IS 1, 7, 9, 5, 8 7, 1, 4, 6, 9 Table 1: Rankings of filters for the Shapley value (SH) and the importance switches (IS) methods on a four-layer network, 10-20-100-25. For each layer the top five neurons are shown, the numbers in bold indicate the common top neurons across both the methods. Interpretability: One of the main aims of this work has been to understand better the process of learning of convolutional neural networks. Building on the previous works which visualized CNN filters, we want to add an extra component and interpret that visual features by means of the filter rankings. In figure 1, we visualize feature maps produced by the first convolution layer of filters. Knowing the important filters allows to ponder over what features the network learns and deems useful. For instance, in the MNIST digits, the learnt filters identify local parts of the image (such as lower and upper parts of the digit '2' and opposite parts of the digit '0'). The interesting observation is that the most important features, on the one hand, complement each other (such as complementing parts of the digit '0' or the dog in CIFAR-10) but, on the other and, overlap to seemingly reinforce its importance. Finally, the important features appear smoother as compared to unimportant ones, which outline the object with no particular focus. Compression: The consequence of the feature ranking is that some of the nodes within each layer are less significant than others and, as argued in network compression literature, the network may do as well without them. The compression experiments procedure follows from the previous experiments. Given the rankings, we prune the neurons from the bottom of the ranking and then we retrain the network. We run the tests for both of the methods on several different architectures. In all the trainings we use SGD with decreasing learning rate from 0.1 to 0.001, momentum, 0.9, weight decay, 5e-4, and early stopping. LeNetVGG presents the for LeNet-5 as trained on MNIST, and VGG-16 as trained on CIFAR-10. For LeNet-5, the compressed architecture has 17K parameters which is less than all the other methods, and 137K FLOPs which is second to FDOO , which however has over three times more parameters. The method fares relatively well also on VGG producing an architecture which is smaller than others in the earlier layers but larger in later layers (the second proposed architecture has overall the least number of parameters at the cost of the performance, though). We hope to test a larger set of possible architectures in the future and devise a way to combine both rankings for a more optimal compression. Nevertheless, the show that the neuron ranking method is adequate for condensing both small and large architectures. 8.3% 142M 1.0M BC-GHS 51-62-125-128-228-129-38-13-9-6-5-6-6-20 8.3% 122M 0.8M Table 2: The structured pruning of LeNet-5 and VGG-16 The final experiment demonstrates how our method compares to magnitude pruning commonly done in compression literature. In figure 2, our method (blue trace) outperforms magnitude pruning methods (L1 and L2 norm over weights). No retraining is used in this case to show how the proposed method retains the relevant neurons that affect the predictive accuracy. We would like to emphasize that the magnitude approaches may be more appropriate for the unstructured pruning where single weights are removed based on its magnitude. However, in the the case of pruning entire channels, considering a norm of weights may be too simplistic as the interactions between the weights within a channel are rather complex. The proposed new paradigm treats the channels as whole units that directly contribute to the task generalization. In summary, this work suggests a theory that the learnable CNN features contain inherent hierarchy where some of the features are more significant than others. This multidisciplinary work which builds on top of probability and game theoretical concepts proposes two methods to produce feature ranking and select most important features in the CNN network. The striking observation is that the different methods lead to similar and allow to distinguish important nodes with larger confidence. The ranking methods allow to build an informed way to build a slim network architecture where the significant nodes remain and unimportant nodes are discarded. A future search for further methods which allow to quantify the neuron importance is the next step to develop the understanding of the feature importance in CNNs.

Represent the short summary for finding the corresponding scientific article (long)

We propose CNN neuron ranking with two different methods and show their consistency in producing the result which allows to interpret what network deems important and compress the network by keeping the most relevant nodes.

Represent the TLDR for finding the corresponding scientific article

We introduce the NLC, a metric that is cheap to compute in the networks randomly initialized state and is highly predictive of generalization, at least in fully-connected networks.