Entropy and mutual information in models of deep neural networks

We examine a class of stochastic deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) we show how entropies and mutual informations can be derived from heuristic statistical physics methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual informations throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.

Jean François Emmanuel Barbier, Florent Gérard Krzakala, Clément Dominique Luneau, Nicolas Macris, Lenka Zdeborová

We examine a class of stochastic deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) We show how entropies and mutual informations can be derived from heuristic statistical physics methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual informations throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2018

Statistical Physics Methods for Community Detection

Chun Lam Chan

This thesis is devoted to information-theoretic aspects of community detection. The importance of community detection is due to the massive amount of scientific data today that describes relationships between items from a network, e.g., a social network. Items from this network can be inherently partitioned into a known number of communities, but the partition can only be inferred from the data. To estimate the underlying partition, data scientists can apply any type of advanced statistical techniques; but the data could be very noisy, or the number of data is inadequate. A fundamental question here is about the possibility of weak recovery: does the data contain a sufficient amount of information that enables us to produce a non-trivial estimate of the partition? For the purpose of mathematical analysis, the above problem can be formulated as Bayesian inference on generative models. These models, including the stochastic block model (SBM) and censored block model (CBM), consider a random graph generated based on a hidden partition that divides the nodes in the graph into labelled groups. In the SBM, nodes are connected with a probability depending on the labels of the endpoints. Whereas, in the CBM, hidden variables are measured through a noisy channel, and the measurement outcomes form a weighted graph. In both models, inference is the task of recovering the hidden partition from the observed graph. The criteria for weak recovery can be studied via an information-theoretic quantity called mutual information. Once the asymptotic mutual information is computed, phase transitions for the weak recovery can be located. This thesis pertains to rigorous derivations of single-letter variational expressions for the asymptotic mutual information for models in community detection. These variational expressions, known as the replica predictions, come from heuristic methods of statistical physics. We present our development of new rigorous methods for confirming the replica predictions. These methods are based on extending the recently introduced adaptive interpolation method. We prove the replica prediction for the SBM in the dense-graph regime with two groups of asymmetric size. The existing proofs in the literature are indirect, as they involve mapping the model to an external problem whose mutual information is determined by a combination of methods. Here, on the contrary, we provide a self-contained and direct proof. Next, we extend this method to sparse models. Before this thesis, adaptive interpolation was known for providing a conceptually simple proof for replica predictions for dense graphs. Whereas, for a sparse graph, the replica prediction involves a more complicated variational expression, and rigorous confirmations are often lacking or obtained by rather complicated methods. Therefore, we focus on a simple version of CBM on sparse graphs, where hidden variables are measured through a binary erasure channel, for which we fully prove the replica prediction by the adaptive interpolation. The key for extending the adaptive interpolation to a broader class of sparse models is a concentration result for the so-called "multi-overlaps". This concentration forms the basis of the replica "symmetric" prediction. We prove this concentration result for a related sparse model in the context of physics. This provides inspiration for further development of the adaptive interpolation.

EPFL2019

Tractable Approaches to Learning and Planning in High Dimensions

Leonidas Lefakis

The continuous increase, witnessed in the last decade, of both the amount of available data and the areas of application of machine learning, has lead to a demand for both learning and planning algorithms that are capable of handling large-scale problems. Thus scalability has become an important characteristic of modern machine learning algorithms. In this thesis we concentrate on tractable approaches to both learning and planning, capable of solving problems in high-dimensional spaces. We investigate multiple forms of high-dimensionality, where dimensionality can refer to the number of samples to be processed, the number of samples available for training, the dimensionality of the feature space, or even the dimensionality of the state-space. We present work on classifier cascades, developing a novel framework for interpreting, training, and employing such structures of classifiers. A Boosting algorithm developed in this framework highlights its ability to jointly train the classifiers at the cascade nodes. The work presented on goal-planning also results in a novel framework, this time for automatic macro-action discovery in imitation learning, specifically addressing problems with large state-spaces as well as with observation signals of high-dimensionality. The proposed DPBoost algorithm is capable of solving very complex goal-planning tasks where previous state-of-the-art fails. Addressing computer hardware limitations, and in particular memory limitations, we propose a new learning framework, which we call reservoir learning, which attempts to directly account for this limited memory during the training process. In particular we propose to consider learning in the presence of a reservoir in which the learner may store samples. Based on Boosting we develop a novel strategy for populating this reservoir which avoids redundancy by taking into account the joint behavior of samples. Finally high-dimensionality is also addressed in its most typical setting, that is when the feature space is large. We propose novel feature selection algorithms in the context of classification. Based on information theoretic tools, these algorithms attempt to maximize the joint mutual information between the selected, continuous, variables and the, discrete, class label variable.