Embedding Principle: a hierarchical structure of loss landscape of deep neural networks

TL;DR

This paper proves a hierarchical structure of the loss landscape in deep neural networks, showing how critical points of narrower networks embed into wider ones and explaining the prevalence of strict saddle points in wide NNs.

Contribution

It introduces the Embedding Principle, revealing a hierarchical structure of loss landscapes and critical points across different network widths, with implications for optimization.

Findings

Loss landscape of an NN contains all critical points of narrower NNs.

Number of negative/zero/positive eigenvalues can increase but not decrease with width.

Strict saddle points are common in wide NNs, aiding optimization.

Abstract

We prove a general Embedding Principle of loss landscape of deep neural networks (NNs) that unravels a hierarchical structure of the loss landscape of NNs, i.e., loss landscape of an NN contains all critical points of all the narrower NNs. This result is obtained by constructing a class of critical embeddings which map any critical point of a narrower NN to a critical point of the target NN with the same output function. By discovering a wide class of general compatible critical embeddings, we provide a gross estimate of the dimension of critical submanifolds embedded from critical points of narrower NNs. We further prove an irreversiblility property of any critical embedding that the number of negative/zero/positive eigenvalues of the Hessian matrix of a critical point may increase but never decrease as an NN becomes wider through the embedding. Using a special realization of general compatible critical embedding, we prove a stringent necessary condition for being a "truly-bad" critical point that never becomes a strict-saddle point through any critical embedding. This result implies the commonplace of strict-saddle points in wide NNs, which may be an important reason underlying the easy optimization of wide NNs widely observed in practice.