# Lenient Regret for Multi-Armed Bandits

arXiv: https://arxiv.org/abs/2008.03959 · HTML: https://mlsanity.com/paper/2008.03959/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2020.

## Built on by (most cited first, 8 of 8)

- [Lenient Regret and Good-Action Identification in Gaussian Process Bandits](https://mlsanity.com/paper/2102.05793/) (2021)
- [Gap-Dependent Unsupervised Exploration for Reinforcement Learning](https://mlsanity.com/paper/2108.05439/) (2021)
- [Robust Bayesian Satisficing](https://mlsanity.com/paper/2308.08291/) (2023)
- [On Regret Bounds of Thompson Sampling for Bayesian Optimization](https://mlsanity.com/paper/2603.09276/) (2026)
- [Query-Reward Tradeoffs in Multi-Armed Bandits](https://mlsanity.com/paper/2110.05724/) (2021)
- [High-dimensional Nonparametric Contextual Bandit Problem](https://mlsanity.com/paper/2505.14102/) (2025)
- [Robust Satisficing Gaussian Process Bandits Under Adversarial Attacks](https://mlsanity.com/paper/2506.01625/) (2025)
- [Adaptive Experiments Under Data Sparse Settings: Applications for Educational Platforms](https://mlsanity.com/paper/2501.03999/) (2025)

## Abstract (arXiv)

We consider the Multi-Armed Bandit (MAB) problem, where an agent sequentially chooses actions and observes rewards for the actions it took. While the majority of algorithms try to minimize the regret, i.e., the cumulative difference between the reward of the best action and the agent's action, this criterion might lead to undesirable results. For example, in large problems, or when the interaction with the environment is brief, finding an optimal arm is infeasible, and regret-minimizing algorithms tend to over-explore. To overcome this issue, algorithms for such settings should instead focus on playing near-optimal arms. To this end, we suggest a new, more lenient, regret criterion that ignores suboptimality gaps smaller than some ε. We then present a variant of the Thompson Sampling (TS) algorithm, called ε-TS, and prove its asymptotic optimality in terms of the lenient regret. Importantly, we show that when the mean of the optimal arm is high enough, the lenient regret of ε-TS is bounded by a constant. Finally, we show that ε-TS can be applied to improve the performance when the agent knows a lower bound of the suboptimality gaps.