Solving optimization problems with Blackwell approachability
This work provides a parameter-free solution for convex-concave saddle-point problems, benefiting researchers and practitioners in optimization and game theory by eliminating the need for step-size tuning.
The paper tackles the problem of solving convex-concave saddle-point problems by introducing SP-CBA+, a parameter-free algorithm based on the Conic Blackwell Algorithm+ (CBA+) regret minimizer, which achieves an O(1/√T) ergodic convergence rate and outperforms classical methods in simulations across various settings like matrix games and distributionally robust logistic regression.
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.