Statistical Physics of Deep Neural Networks: Initialization toward Optimal Channels

TL;DR

This paper develops a statistical physics-based theory showing that initializing neural networks at dynamic isometry maximizes mutual information transfer, linking information theory and deep learning for optimal channel performance.

Contribution

It introduces a corrected mean-field framework and proves that mutual information is maximized at dynamic isometry, a novel insight for neural network initialization.

Findings

Mutual information maximization occurs at dynamic isometry.

The corrected mean-field theory accurately predicts information propagation.

Experimental validation confirms the theory's robustness.

Abstract

In deep learning, neural networks serve as noisy channels between input data and its representation. This perspective naturally relates deep learning with the pursuit of constructing channels with optimal performance in information transmission and representation. While considerable efforts are concentrated on realizing optimal channel properties during network optimization, we study a frequently overlooked possibility that neural networks can be initialized toward optimal channels. Our theory, consistent with experimental validation, identifies primary mechanics underlying this unknown possibility and suggests intrinsic connections between statistical physics and deep learning. Unlike the conventional theories that characterize neural networks applying the classic mean-filed approximation, we offer analytic proof that this extensively applied simplification scheme is not valid in studying neural networks as information channels. To fill this gap, we develop a corrected mean-field framework applicable for characterizing the limiting behaviors of information propagation in neural networks without strong assumptions on inputs. Based on it, we propose an analytic theory to prove that mutual information maximization is realized between inputs and propagated signals when neural networks are initialized at dynamic isometry, a case where information transmits via norm-preserving mappings. These theoretical predictions are validated by experiments on real neural networks, suggesting the robustness of our theory against finite-size effects. Finally, we analyze our findings with information bottleneck theory to confirm the precise relations among dynamic isometry, mutual information maximization, and optimal channel properties in deep learning.