# Batch-Size Independent Regret Bounds for the Combinatorial Multi-Armed Bandit Problem

arXiv: https://arxiv.org/abs/1905.03125 · HTML: https://mlsanity.com/paper/1905.03125/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2019.

## Built on by (most cited first, 14 of 14)

- [Cost-Effective Online Multi-LLM Selection with Versatile Reward Models](https://mlsanity.com/paper/2405.16587/) (2024)
- [Contextual Combinatorial Bandits with Probabilistically Triggered Arms](https://mlsanity.com/paper/2303.17110/) (2023)
- [Batch-Size Independent Regret Bounds for Combinatorial Semi-Bandits with Probabilistically Triggered Arms or Independent Arms](https://mlsanity.com/paper/2208.14837/) (2022)
- [Adversarial Combinatorial Bandits with General Non-linear Reward Functions](https://mlsanity.com/paper/2101.01301/) (2021)
- [Confidence-Budget Matching for Sequential Budgeted Learning](https://mlsanity.com/paper/2102.03400/) (2021)
- [Top-k eXtreme Contextual Bandits with Arm Hierarchy](https://mlsanity.com/paper/2102.07800/) (2021)
- [Non-stationary Bandits and Meta-Learning with a Small Set of Optimal Arms](https://mlsanity.com/paper/2202.13001/) (2022)
- [Effective Off-Policy Evaluation and Learning in Contextual Combinatorial Bandits](https://mlsanity.com/paper/2408.11202/) (2024)
- [Combinatorial Multivariant Multi-Armed Bandits with Applications to Episodic Reinforcement Learning and Beyond](https://mlsanity.com/paper/2406.01386/) (2024)
- [Offline Learning for Combinatorial Multi-armed Bandits](https://mlsanity.com/paper/2501.19300/) (2025)
- [Risk-Aware Algorithms for Combinatorial Semi-Bandits](https://mlsanity.com/paper/2112.01141/) (2021)
- [Stable Matching with Ties: Approximation Ratios and Learning](https://mlsanity.com/paper/2411.03270/) (2024)
- [Online Mixture of Experts: No-Regret Learning for Optimal Collective Decision-Making](https://mlsanity.com/paper/2510.21788/) (2025)
- [Covariance-adapting algorithm for semi-bandits with application to sparse rewards](https://mlsanity.com/paper/2604.13738/) (2026)

## Abstract (arXiv)

We consider the combinatorial multi-armed bandit (CMAB) problem, where the reward function is nonlinear. In this setting, the agent chooses a batch of arms on each round and receives feedback from each arm of the batch. The reward that the agent aims to maximize is a function of the selected arms and their expectations. In many applications, the reward function is highly nonlinear, and the performance of existing algorithms relies on a global Lipschitz constant to encapsulate the function's nonlinearity. This may lead to loose regret bounds, since by itself, a large gradient does not necessarily cause a large regret, but only in regions where the uncertainty in the reward's parameters is high. To overcome this problem, we introduce a new smoothness criterion, which we term Gini-weighted smoothness, that takes into account both the nonlinearity of the reward and concentration properties of the arms. We show that a linear dependence of the regret in the batch size in existing algorithms can be replaced by this smoothness parameter. This, in turn, leads to much tighter regret bounds when the smoothness parameter is batch-size independent. For example, in the probabilistic maximum coverage (PMC) problem, that has many applications, including influence maximization, diverse recommendations and more, we achieve dramatic improvements in the upper bounds. We also prove matching lower bounds for the PMC problem and show that our algorithm is tight, up to a logarithmic factor in the problem's parameters.