# Baldi, Pierre

### On Neuronal Capacity

We define the capacity of a learning machine to be the logarithm of the number (or volume) of the functions it can implement. We review known results, and derive new results, estimating the capacity of several neuronal models: linear and polynomial threshold gates, linear and polynomial threshold gates with constrained weights (binary weights, positive weights), and ReLU neurons. We also derive capacity estimates and bounds for fully recurrent networks and layered feedforward networks. Papers published at the Neural Information Processing Systems Conference.

### Understanding Dropout

Dropout is a relatively new algorithm for training neural networks which relies on stochastically dropping out'' neurons during training in order to avoid the co-adaptation of feature detectors. We introduce a general formalism for studying dropout on either units or connections, with arbitrary probability values, and use it to analyze the averaging and regularizing properties of dropout in both linear and non-linear networks. For deep neural networks, the averaging properties of dropout are characterized by three recursive equations, including the approximation of expectations by normalized weighted geometric means. We provide estimates and bounds for these approximations and corroborate the results with simulations. We also show in simple cases how dropout performs stochastic gradient descent on a regularized error function."

### Searching for Higgs Boson Decay Modes with Deep Learning

Particle colliders enable us to probe the fundamental nature of matter by observing exotic particles produced by high-energy collisions. Because the experimental measurements from these collisions are necessarily incomplete and imprecise, machine learning algorithms play a major role in the analysis of experimental data. The high-energy physics community typically relies on standardized machine learning software packages for this analysis, and devotes substantial effort towards improving statistical power by hand crafting high-level features derived from the raw collider measurements. In this paper, we train artificial neural networks to detect the decay of the Higgs boson to tau leptons on a dataset of 82 million simulated collision events. We demonstrate that deep neural network architectures are particularly well-suited for this task with the ability to automatically discover high-level features from the data and increase discovery significance.

### Modeling Dynamic Functional Connectivity with Latent Factor Gaussian Processes

Dynamic functional connectivity, as measured by the time-varying covariance of neurological signals, is believed to play an important role in many aspects of cognition. While many methods have been proposed, reliably establishing the presence and characteristics of brain connectivity is challenging due to the high dimensionality and noisiness of neuroimaging data. We present a latent factor Gaussian process model which addresses these challenges by learning a parsimonious representation of connectivity dynamics. The proposed model naturally allows for inference and visualization of the time-varying connectivity. As an illustration of the scientific utility of the model, application to a data set of rat local field potential activity recorded during a complex non-spatial memory task provides evidence of stimuli differentiation.

### Curiosity-Driven Multi-Criteria Hindsight Experience Replay

Dealing with sparse rewards is a longstanding challenge in reinforcement learning. The recent use of hindsight methods have achieved success on a variety of sparse-reward tasks, but they fail on complex tasks such as stacking multiple blocks with a robot arm in simulation. Curiosity-driven exploration using the prediction error of a learned dynamics model as an intrinsic reward has been shown to be effective for exploring a number of sparse-reward environments. We present a method that combines hindsight with curiosity-driven exploration and curriculum learning in order to solve the challenging sparse-reward block stacking task. We are the first to stack more than two blocks using only sparse reward without human demonstrations.

### The capacity of feedforward neural networks

A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary logarithm of the number of functions it can compute, as the synaptic weights are varied. The capacity is an upper bound on the number of bits that can be "communicated" from the training data to the architecture over the learning channel. We study the capacity of layered, fully-connected, architectures of linear threshold neurons with $L$ layers of size $n_1,n_2, \ldots, n_L$ and show that in essence the capacity is given by a cubic polynomial in the layer sizes: $C(n_1,\ldots, n_L)=\sum_{k=1}^{L-1} \min(n_1,\ldots,n_k)n_kn_{k+1}$. In proving the main result, we also develop new techniques (multiplexing, enrichment, and stacking) as well as new bounds on the capacity of finite sets. We use the main result to identify architectures with maximal or minimal capacity under a number of natural constraints. This leads to the notion of structural regularization for deep architectures. While in general, everything else being equal, shallow networks compute more functions than deep networks, the functions computed by deep networks are more regular and "interesting".

### On Neuronal Capacity

We define the capacity of a learning machine to be the logarithm of the number (or volume) of the functions it can implement. We review known results, and derive new results, estimating the capacity of several neuronal models: linear and polynomial threshold gates, linear and polynomial threshold gates with constrained weights (binary weights, positive weights), and ReLU neurons. We also derive capacity estimates and bounds for fully recurrent networks and layered feedforward networks.

### On Neuronal Capacity

We define the capacity of a learning machine to be the logarithm of the number (or volume) of the functions it can implement. We review known results, and derive new results, estimating the capacity of several neuronal models: linear and polynomial threshold gates, linear and polynomial threshold gates with constrained weights (binary weights, positive weights), and ReLU neurons. We also derive capacity estimates and bounds for fully recurrent networks and layered feedforward networks.

### Efficient Neutrino Oscillation Parameter Inference with Gaussian Process

Neutrino oscillation study involves inferences from tiny samples of data which have complicated dependencies on multiple oscillation parameters simultaneously. This is typically carried out using the unified approach of Feldman and Cousins which is very computationally expensive, on the order of tens of millions of CPU hours. In this work, we propose an iterative method using Gaussian Process to efficiently find a confidence contour for the oscillation parameters and show that it produces the same results at a fraction of the computation cost.

### Solving the Rubik's Cube Without Human Knowledge

A generally intelligent agent must be able to teach itself how to solve problems in complex domains with minimal human supervision. Recently, deep reinforcement learning algorithms combined with self-play have achieved superhuman proficiency in Go, Chess, and Shogi without human data or domain knowledge. In these environments, a reward is always received at the end of the game, however, for many combinatorial optimization environments, rewards are sparse and episodes are not guaranteed to terminate. We introduce Autodidactic Iteration: a novel reinforcement learning algorithm that is able to teach itself how to solve the Rubik's Cube with no human assistance. Our algorithm is able to solve 100% of randomly scrambled cubes while achieving a median solve length of 30 moves -- less than or equal to solvers that employ human domain knowledge.
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