Solving optimization problems with Blackwell approachability

TL;DR

This paper introduces a new parameter-free regret minimizer based on Blackwell approachability, applicable to various convex sets, and demonstrates its effectiveness in solving convex-concave saddle-point problems with state-of-the-art results.

Contribution

The paper presents the Conic Blackwell Algorithm$^+$ (CBA$^+$), a novel parameter- and scale-free regret minimizer, and the SP-CBA$^+$ algorithm for saddle-point problems, both with theoretical guarantees and practical efficiency.

Findings

CBA$^+$ achieves $O(\sqrt{T})$ regret for convex sets.

SP-CBA$^+$ attains $O(1/\sqrt{T})$ ergodic convergence rate.

SP-CBA$^+$ outperforms classical methods in various saddle-point applications.

Abstract

We introduce the Conic Blackwell Algorithm$^+$ (CBA$^+$) regret minimizer, a new parameter- and scale-free regret minimizer for general convex sets. CBA$^+$ is based on Blackwell approachability and attains $O(\sqrt{T})$ regret. We show how to efficiently instantiate CBA$^+$ for many decision sets of interest, including the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the simplex. Based on CBA$^+$, we introduce SP-CBA$^+$, a new parameter-free algorithm for solving convex-concave saddle-point problems, which achieves a $O(1/\sqrt{T})$ ergodic rate of convergence. In our simulations, we demonstrate the wide applicability of SP-CBA$^+$ on several standard saddle-point problems, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, SP-CBA$^+$ achieves state-of-the-art numerical performance, and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters.