## Abstract

Gradient-descent (GD) based algorithms are an indispensable tool for optimizing modern machine learning models. The paper considers distributed stochastic GD (D-SGD)—a network-based variant of GD. Distributed algorithms play an important role in large-scale machine learning problems as well as the Internet of Things (IoT) and related applications. The paper considers two main issues. First, we study convergence of D-SGD to critical points when the loss function is nonconvex and nonsmooth. We consider a broad range of nonsmooth loss functions including those of practical interest in modern deep learning. It is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. It is well known that classical GD avoids saddle points; however, analogous results have been absent for distributed variants of GD. For this problem, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation.

Original language | English (US) |
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Article number | 328 |

Journal | Journal of Machine Learning Research |

Volume | 23 |

State | Published - Oct 1 2022 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Software
- Artificial Intelligence
- Control and Systems Engineering
- Statistics and Probability

## Keywords

- Nonconvex optimization
- distributed optimization
- gradient descent
- saddle point
- stochastic optimization