Solving optimization problems with Blackwell approachability

• URL: http://arxiv.org/abs/2202.12277v1
• Date: Thu, 24 Feb 2022 18:19:21 GMT
• Title: Solving optimization problems with Blackwell approachability
• Authors: Julien Grand-Cl\'ement and Christian Kroer
• Abstract summary: Conic Blackwell Algorithm$+$ (CBA$+$) regret minimizer. CBA$+$ is based on Blackwell approachability and attains $O(sqrtT)$ regret. Based on CBA$+$, we introduce SP-CBA$+$, a new parameter-free algorithm for solving convex-concave saddle-point problems.
• Score: 31.29341288414507
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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