# On the Complexity of Best Arm Identification in Multi-Armed Bandit Models

arXiv: https://arxiv.org/abs/1407.4443 · HTML: https://mlsanity.com/paper/1407.4443/

## Key facts

- Cited by 286 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2014.

## Built on by (most cited first, 30 of 286)

- [Leveraging Offline Data in Online Reinforcement Learning](https://mlsanity.com/paper/2211.04974/) (2022)
- [Beyond No Regret: Instance-Dependent PAC Reinforcement Learning](https://mlsanity.com/paper/2108.02717/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [FOCUS: Efficient Keyframe Selection for Long Video Understanding](https://mlsanity.com/paper/2510.27280/) (2025)
- [Instance-Dependent Near-Optimal Policy Identification in Linear MDPs via Online Experiment Design](https://mlsanity.com/paper/2207.02575/) (2022)
- [A Closer Look at the Worst-case Behavior of Multi-armed Bandit Algorithms](https://mlsanity.com/paper/2106.02126/) (2021)
- [Navigating to the Best Policy in Markov Decision Processes](https://mlsanity.com/paper/2106.02847/) (2021)
- [Best Arm Identification with Safety Constraints](https://mlsanity.com/paper/2111.12151/) (2021)
- [Differentially Private Multi-Armed Bandits in the Shuffle Model](https://mlsanity.com/paper/2106.02900/) (2021)
- [Fixed-Budget Best-Arm Identification in Structured Bandits](https://mlsanity.com/paper/2106.04763/) (2021)
- [Efficient Pure Exploration for Combinatorial Bandits with Semi-Bandit Feedback](https://mlsanity.com/paper/2101.08534/) (2021)
- [Minimax Optimal Algorithms for Fixed-Budget Best Arm Identification](https://mlsanity.com/paper/2206.04646/) (2022)
- [On the Existence of a Complexity in Fixed Budget Bandit Identification](https://mlsanity.com/paper/2303.09468/) (2023)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [Loss-Aversively Fair Classification](https://mlsanity.com/paper/2105.04273/) (2021)
- [Open Problem: Optimal Best Arm Identification with Fixed Budget](https://mlsanity.com/paper/2303.00950/) (2023)
- [The Fragility of Optimized Bandit Algorithms](https://mlsanity.com/paper/2109.13595/) (2021)
- [A/B/n Testing with Control in the Presence of Subpopulations](https://mlsanity.com/paper/2110.15573/) (2021)
- [Nearly Optimal Algorithms for Level Set Estimation](https://mlsanity.com/paper/2111.01768/) (2021)
- [Near Instance-Optimal PAC Reinforcement Learning for Deterministic MDPs](https://mlsanity.com/paper/2203.09251/) (2022)
- [A Near-Optimal Algorithm for Safe Reinforcement Learning Under Instantaneous Hard Constraints](https://mlsanity.com/paper/2302.04375/) (2023)
- [More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity](https://mlsanity.com/paper/2306.12214/) (2023)
- [Almost Optimal Variance-Constrained Best Arm Identification](https://mlsanity.com/paper/2201.10142/) (2022)
- [Active Learning with Safety Constraints](https://mlsanity.com/paper/2206.11183/) (2022)
- [Instance-optimal PAC Algorithms for Contextual Bandits](https://mlsanity.com/paper/2207.02357/) (2022)
- [Task-Optimal Exploration in Linear Dynamical Systems](https://mlsanity.com/paper/2102.05214/) (2021)
- [A Fully Problem-Dependent Regret Lower Bound for Finite-Horizon MDPs](https://mlsanity.com/paper/2106.13013/) (2021)
- [Policy Choice and Best Arm Identification: Asymptotic Analysis of Exploration Sampling](https://mlsanity.com/paper/2109.08229/) (2021)
- [Optimal Clustering with Bandit Feedback](https://mlsanity.com/paper/2202.04294/) (2022)
- [Information-Directed Selection for Top-Two Algorithms](https://mlsanity.com/paper/2205.12086/) (2022)

## Abstract (arXiv)

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).