On the modes of convergence of Stochastic Optimistic Mirror Descent (OMD) for saddle point problems
This work addresses theoretical gaps in optimization algorithms for machine learning, but it is incremental as it primarily refines existing proofs.
The paper corrects convergence claims for Optimistic Mirror Descent (OMD) in saddle point problems, showing that monotone convergence occurs only after many iterations and proving high-probability convergence for stochastic cases, unlike previous almost-sure guarantees.
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.