Near-Optimal Regret for Efficient Stochastic Combinatorial Semi-Bandits

TL;DR

This paper introduces CMOSS, a computationally efficient algorithm for stochastic combinatorial semi-bandits that achieves near-optimal regret bounds without the logarithmic dependence on time, outperforming existing methods.

Contribution

The paper proposes CMOSS, a novel algorithm that attains instance-independent regret bounds matching lower bounds and reduces computational complexity in stochastic semi-bandit problems.

Findings

CMOSS achieves regret bounds of $O( ( ext{log }k)\sqrt{kmT})$ and $O((m-k)\sqrt{ ext{log }k ext{log }(m-k)T})$.

CMOSS eliminates the $ ext{log }T$ dependence present in previous algorithms.

Experimental results show CMOSS outperforms benchmark algorithms in regret and runtime.

Abstract

The combinatorial multi-armed bandit (CMAB) is a cornerstone of sequential decision-making framework, dominated by two algorithmic families: UCB-based and adversarial methods such as follow the regularized leader (FTRL) and online mirror descent (OMD). However, prominent UCB-based approaches like CUCB suffer from additional regret factor $\log T$ that is detrimental over long horizons, while adversarial methods such as EXP3.M and HYBRID impose significant computational overhead. To resolve this trade-off, we introduce the Combinatorial Minimax Optimal Strategy in the Stochastic setting (CMOSS). CMOSS is a computationally efficient algorithm that achieves an instance-independent regret of $O\big( (\log k)\sqrt{kmT}\big )$ when $k\leq \frac{m}{2}$ and $O\big((m-k)\sqrt{\log k\log(m-k)T}\big )$ when $k>\frac{m}{2}$ under semi-bandit feedback, where $m$ is the number of arms and $k$ is the maximum cardinality of a feasible action. Crucially, this result eliminates the dependency on $\log T$ and matches the established lower bounds of $\Omega\big(\sqrt{kmT}\big)$ when $k\leq \frac{m}{2}$ and $\Omega\big((m-k)\sqrt{\log (\frac{m}{m-k}) T}\big)$ when $k>\frac{m}{2}$ up to logarithmic terms of $k$ and $m$. We then extend our analysis to show that CMOSS is also applicable to cascading feedback. Experiments on synthetic and real-world datasets validate that CMOSS consistently outperforms benchmark algorithms in both regret and runtime efficiency.