No-Regret M${}^{\natural}$-Concave Function Maximization: Stochastic Bandit Algorithms and Hardness of Adversarial Full-Information Setting

TL;DR

This paper develops algorithms for maximizing M${}^{ atural}$-concave functions in stochastic bandit settings with sublinear regret, and proves hardness results for adversarial full-information scenarios, highlighting the problem's complexity.

Contribution

It introduces new stochastic bandit algorithms with regret guarantees for M${}^{ atural}$-concave functions and establishes computational hardness in adversarial settings.

Findings

Achieved $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms in stochastic bandit setting.

Proved polynomial-time algorithms cannot attain sublinear regret in adversarial full-information setting.

Demonstrated robustness of greedy algorithms to local errors in M${}^{ atural}$-concave maximization.

Abstract

M${}^{\natural}$-concave functions, a.k.a. gross substitute valuation functions, play a fundamental role in many fields, including discrete mathematics and economics. In practice, perfect knowledge of M${}^{\natural}$-concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M${}^{\natural}$-concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). For the stochastic bandit setting, we present $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms under $T$ times access to unbiased noisy value oracles of M${}^{\natural}$-concave functions. A key to proving these results is the robustness of the greedy algorithm to local errors in M${}^{\natural}$-concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve $O(T^{1-c})$ regret for any constant $c > 0$. Our proof is based on a reduction from the matroid intersection problem for three matroids, which would be a novel approach to establishing the hardness in online learning.