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

• URL: http://arxiv.org/abs/2111.15527v1
• Date: Tue, 30 Nov 2021 16:15:50 GMT
• Title: Embedding Principle: a hierarchical structure of loss landscape of deep neural networks
• Authors: Yaoyu Zhang, Yuqing Li, Zhongwang Zhang, Tao Luo, Zhi-Qin John Xu
• Abstract summary: We prove a general Embedding Principle of loss landscape of deep neural networks (NNs) We provide a gross estimate of the dimension of critical submanifolds embedded from critical points of narrower NNs.
• Score: 3.0871079010101963
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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