Efficient Matroid Bandit Linear Optimization Leveraging Unimodality

TL;DR

This work tackles the matroid-constrained combinatorial semi-bandit problem, where standard algorithms incur prohibitive time due to repeated greedy calls. The authors introduce MAUB, a unimodal-bandit algorithm that leverages the matroid’s local exchange structure to maintain a leader and explore only within a small neighborhood, reducing membership-oracle queries to O(log log T) with negligible impact on regret. They prove a regret bound matching optimal orders and derive a time complexity that scales favorably with the matroid size and oracle costs. Empirical results across uniform, linear, graphic, and transversal matroids show MAUB achieves regret comparable to state-of-the-art methods while substantially reducing oracle calls and computation time. The approach suggests a broader potential for unimodal structures to improve efficiency in combinatorial bandit problems.

Abstract

We study the combinatorial semi-bandit problem under matroid constraints. The regret achieved by recent approaches is optimal, in the sense that it matches the lower bound. Yet, time complexity remains an issue for large matroids or for matroids with costly membership oracles (e.g. online recommendation that ensures diversity). This paper sheds a new light on the matroid semi-bandit problem by exploiting its underlying unimodal structure. We demonstrate that, with negligible loss in regret, the number of iterations involving the membership oracle can be limited to \mathcal{O}(\log \log T)$. This results in an overall improved time complexity of the learning process. Experiments conducted on various matroid benchmarks show (i) no loss in regret compared to state-of-the-art approaches; and (ii) reduced time complexity and number of calls to the membership oracle.

Figures (2)

• Figure 1: Regret vs iterations for uniform, linear, graphic and transversal matroids (the smaller, the better). U($D$,$|E|$) is the uniform matroid of rank $D$ on $|E|$ elements. $K_N$ it the graphic matroid associated to the complete graph of size $N$. Our algorithm MAUB consistently matches or outperforms OMM.
• Figure 2: Proof diagram of finite-time analysis of MAUB. We remind the reader that whenever unimodality is satisfied, $B_2$ is an arm in $\mathcal{N}_{B, \boldsymbol{\mu}}$ such that $\boldsymbol{\mu}_{B_2}>\boldsymbol{\mu}_{B}$.

Theorems & Definitions (20)

• Definition 3.1: Matroid, groundset, independent subsets, rank, bases
• Lemma 3.2: basis exchange property sch-cwi-03, tal-aamas-16
• Theorem 4.1: Unimodality
• proof
• Definition 4.2: Unimodal Bandit Statistics
• Lemma 5.1
• Lemma 5.1
• Theorem 5.2
• Theorem 5.3
• Corollary 5.4