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

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