Conic Blackwell Algorithm: Parameter-Free Convex-Concave Saddle-Point Solving

TL;DR

This paper introduces the Conic Blackwell Algorithm$^+$, a parameter-free, scale-free method for convex-concave saddle-point problems that achieves optimal regret bounds and outperforms existing algorithms in practical experiments.

Contribution

The paper presents a new parameter-free, scale-free algorithm CBA$^+$ for saddle-point problems, extending CFR$^+$ ideas to new decision sets with strong empirical performance.

Findings

Achieves $O(1/ oot{T}{}$ regret bound.

Outperforms state-of-the-art methods on synthetic and real data.

Effective for various decision sets including simplexes and norm balls.

Abstract

We develop new parameter-free and scale-free algorithms for solving convex-concave saddle-point problems. Our results are based on a new simple regret minimizer, the Conic Blackwell Algorithm$^+$ (CBA$^+$), which attains $O(1/\sqrt{T})$ average regret. Intuitively, our approach generalizes to other decision sets of interest ideas from the Counterfactual Regret minimization (CFR$^+$) algorithm, which has very strong practical performance for solving sequential games on simplexes. We show how to implement CBA$^+$ for the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the simplex, and we present numerical experiments for solving matrix games and distributionally robust optimization problems. Our empirical results show that CBA$^+$ is a simple algorithm that outperforms state-of-the-art methods on synthetic data and real data instances, without the need for any choice of step sizes or other algorithmic parameters.