Proximal Point Method for Online Saddle Point Problem

TL;DR

This paper introduces three proximal point algorithms for online saddle point problems, providing theoretical guarantees and experimental validation for their effectiveness in nonstationary environments.

Contribution

It proposes three variants of the proximal point method tailored for online saddle point problems, with theoretical bounds and practical performance analysis.

Findings

Algorithms achieve near-optimal duality gap bounds.

Maintains nearly constant performance in stationary environments.

Experimental results confirm effectiveness of the proposed methods.

Abstract

This paper focuses on the online saddle point problem, which involves a sequence of two-player time-varying convex-concave games. Considering the nonstationarity of the environment, we adopt the duality gap and the dynamic Nash equilibrium regret as performance metrics for algorithm design. We present three variants of the proximal point method: the Online Proximal Point Method (OPPM), the Optimistic OPPM (OptOPPM), and the OptOPPM with multiple predictors. Each algorithm guarantees upper bounds for both the duality gap and dynamic Nash equilibrium regret, achieving near-optimality when measured against the duality gap. Specifically, in certain benign environments, such as sequences of stationary payoff functions, these algorithms maintain a nearly constant metric bound. Experimental results further validate the effectiveness of these algorithms. Lastly, this paper discusses potential reliability concerns associated with using dynamic Nash equilibrium regret as a performance metric. The technical appendix and code can be found at https://github.com/qingxin6174/PPM-for-OSP.