Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks

TL;DR

This paper introduces a new theoretical framework based on adaptive symmetries to understand how deep neural networks self-organize into functional structures, revealing conditions for achieving zero training error.

Contribution

It proposes a statistical-mechanical model of DNNs using adaptive symmetries, extending physics concepts to analyze complex neural systems and their optimization processes.

Findings

Large DNNs have extensive adaptive symmetries reservoir.

When information capacity exceeds dataset complexity, DNNs self-organize with zero training error.

The model generalizes statistical physics to describe high-dimensional regularities in DNNs.

Abstract

An antithetical concept, adaptive symmetry, to conservative symmetry in physics is proposed to understand the deep neural networks (DNNs). It characterizes the invariance of variance, where a biotic system explores different pathways of evolution with equal probability in absence of feedback signals, and complex functional structure emerges from quantitative accumulation of adaptive-symmetries breaking in response to feedback signals. Theoretically and experimentally, we characterize the optimization process of a DNN system as an extended adaptive-symmetry-breaking process. One particular finding is that a hierarchically large DNN would have a large reservoir of adaptive symmetries, and when the information capacity of the reservoir exceeds the complexity of the dataset, the system could absorb all perturbations of the examples and self-organize into a functional structure of zero training errors measured by a certain surrogate risk. More specifically, this process is characterized by a statistical-mechanical model that could be appreciated as a generalization of statistics physics to the DNN organized complex system, and characterizes regularities in higher dimensionality. The model consists of three constitutes that could be appreciated as the counterparts of Boltzmann distribution, Ising model, and conservative symmetry, respectively: (1) a stochastic definition/interpretation of DNNs that is a multilayer probabilistic graphical model, (2) a formalism of circuits that perform biological computation, (3) a circuit symmetry from which self-similarity between the microscopic and the macroscopic adaptability manifests. The model is analyzed with a method referred as the statistical assembly method that analyzes the coarse-grained behaviors (over a symmetry group) of the heterogeneous hierarchical many-body interaction in DNNs.