No-Regret Dynamics in the Fenchel Game: A Unified Framework for Algorithmic Convex Optimization

TL;DR

This paper introduces a unified framework for convex optimization using no-regret game dynamics, connecting classical algorithms and deriving new methods through a game-theoretic perspective.

Contribution

It develops a general game-based framework that unifies many classical convex optimization algorithms and introduces novel methods by leveraging no-regret strategies.

Findings

Many classical methods are special cases of the framework

Convergence rates follow straightforwardly from regret bounds

New first-order methods are derived for specific convex problems

Abstract

We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization -- including average-iterate gradient descent, the Frank-Wolfe algorithm, Nesterov's acceleration methods, and the accelerated proximal method -- can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.