TY - JOUR T1 - Embedding Principle: A Hierarchical Structure of Loss Landscape of Deep Neural Networks AU - Zhang , Yaoyu AU - Li , Yuqing AU - Zhang , Zhongwang AU - Luo , Tao AU - John Xu , Zhi-Qin JO - Journal of Machine Learning VL - 1 SP - 60 EP - 113 PY - 2022 DA - 2022/03 SN - 1 DO - http://doi.org/10.4208/jml.220108 UR - https://global-sci.org/intro/article_detail/jml/20372.html KW - Neural network, Loss landscape, Critical point, Embedding principle. AB -

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 irreversibility 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.