Conic Blackwell Algorithm: Parameter-Free Convex-Concave Saddle-Point Solving
This provides a scalable, parameter-free solution for optimization problems in game theory and robust decision-making, though it is incremental as it builds on ideas from existing regret minimization methods.
The paper tackles convex-concave saddle-point problems by developing a parameter-free algorithm, the Conic Blackwell Algorithm⁺ (CBA⁺), which achieves O(1/√T) average regret and outperforms state-of-the-art methods in experiments on matrix games and distributionally robust optimization.
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.