On the modes of convergence of Stochastic Optimistic Mirror Descent (OMD) for saddle point problems

TL;DR

This paper investigates the convergence properties of Optimistic Mirror Descent (OMD) for saddle point problems, clarifying previous claims and providing rigorous proofs for convergence under certain conditions.

Contribution

It corrects and clarifies the convergence results of OMD for saddle point problems, filling in missing proof details and establishing high probability convergence guarantees.

Findings

Convergence of OMD with exact gradients occurs only after many iterations.

Monotone convergence of OMD with exact gradients is not guaranteed for all coherent problems.

High probability convergence is established for stochastic gradients in strictly coherent saddle point problems.

Abstract

In this article, we study the convergence of Mirror Descent (MD) and Optimistic Mirror Descent (OMD) for saddle point problems satisfying the notion of coherence as proposed in Mertikopoulos et al. We prove convergence of OMD with exact gradients for coherent saddle point problems, and show that monotone convergence only occurs after some sufficiently large number of iterations. This is in contrast to the claim in Mertikopoulos et al. of monotone convergence of OMD with exact gradients for coherent saddle point problems. Besides highlighting this important subtlety, we note that the almost sure convergence guarantees of MD and OMD with stochastic gradients for strictly coherent saddle point problems that are claimed in Mertikopoulos et al. are not fully justified by their proof. As such, we fill out the missing details in the proof and as a result have only been able to prove convergence with high probability. We would like to note that our analysis relies heavily on the core ideas and proof techniques introduced in Zhou et al. and Mertikopoulos et al., and we only aim to re-state and correct the results in light of what we were able to prove rigorously while filling in the much needed missing details in their proofs.