Linear Speedup in Saddle-Point Escape for Decentralized Non-Convex Optimization

TL;DR

This paper demonstrates that in decentralized non-convex optimization, the time to escape saddle points scales linearly with the number of agents, under certain spectral conditions of the combination policy.

Contribution

It provides a detailed analysis of second-order convergence guarantees in decentralized non-convex optimization, highlighting the impact of spectral properties and asymmetric weights on linear speedup.

Findings

Linear speedup in saddle-point escape time with symmetric combination policies

Potential for further speedup using asymmetric combination weights

Results apply to pursuit of second-order stationary points in non-convex settings

Abstract

Under appropriate cooperation protocols and parameter choices, fully decentralized solutions for stochastic optimization have been shown to match the performance of centralized solutions and result in linear speedup (in the number of agents) relative to non-cooperative approaches in the strongly-convex setting. More recently, these results have been extended to the pursuit of first-order stationary points in non-convex environments. In this work, we examine in detail the dependence of second-order convergence guarantees on the spectral properties of the combination policy for non-convex multi agent optimization. We establish linear speedup in saddle-point escape time in the number of agents for symmetric combination policies and study the potential for further improvement by employing asymmetric combination weights. The results imply that a linear speedup can be expected in the pursuit of second-order stationary points, which exclude local maxima as well as strict saddle-points and correspond to local or even global minima in many important learning settings.