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From complex to simple : hierarchical free-energy landscape renormalized in deep neural networks

by Hajime Yoshino

Submission summary

As Contributors: Hajime Yoshino
Arxiv Link: https://arxiv.org/abs/1910.09918v1
Date submitted: 2019-10-23
Submitted by: Yoshino, Hajime
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Statistical and Soft Matter Physics
Approach: Theoretical

Abstract

We develop a statistical mechanical approach based on the replica method to study the solution space of deep neural networks. Specifically we analyze the configuration space of the synaptic weights in a simple feed-forward perceptron network within a Gaussian approximation for two scenarios : a setting with random inputs/outputs and a teacher-student setting. By increasing the strength of constraints, i. e. increasing the number of imposed patterns, successive 2nd order glass transition (random inputs/outputs) or 2nd order crystalline transition (teacher-student setting) take place place layer-by-layer starting next to the inputs/outputs boundaries going deeper into the bulk. For deep enough network the central part of the network remains in the liquid phase. We argue that in systems of finite width, weak bias field remain in the central part and plays the role of a symmetry breaking field which connects the opposite sides of the system. In the setting with random inputs/outputs, the successive glass transitions bring about a hierarchical free-energy landscape with ultra-metricity, which evolves in space: it is most complex close to the boundaries but becomes renormalized into progressively simpler one in deeper layers. These observations provide clues to understand why deep neural networks operate efficiently. Finally we present results of a set of numerical simulations to examine the theoretical predictions.

Current status:
Editor-in-charge assigned


Submission & Refereeing History

Submission 1910.09918v1 on 23 October 2019

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