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.

Understanding the loss landscape of a deep neural network is obviously important in analyzing the training trajectory and the generalization performance. This work proposes a new approaching for studying this problem by examining the (embedding) relation between the loss landscapes of neural networks of different widths. The paper proves a general Embedding Principle, namely the loss landscape of a neural network "contains" all critical points of the landscape for narrower neural networks. The paper also demonstrates that (i) critical points embedded from narrower neural networks form submanifolds; (ii) a local minimum is more likely to become a strict saddle point in the landscape of wider neural networks but not vice versa.