# Optimal Batched Linear Bandits

arXiv: https://arxiv.org/abs/2406.04137 · HTML: https://mlsanity.com/paper/2406.04137/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 6 Jun 2024.

## Builds on (most cited first, 18 of 18)

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Optimal Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/1602.04589/) (2016)
- [Batched Multi-armed Bandits Problem](https://mlsanity.com/paper/1904.01763/) (2019)
- [Optimal Best-arm Identification in Linear Bandits](https://mlsanity.com/paper/2006.16073/) (2020)
- [Sequential Batch Learning in Finite-Action Linear Contextual Bandits](https://mlsanity.com/paper/2004.06321/) (2020)
- [Linear Bandits with Limited Adaptivity and Learning Distributional Optimal Design](https://mlsanity.com/paper/2007.01980/) (2020)
- [The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits](https://mlsanity.com/paper/1610.04491/) (2016)
- [Regret Bounds for Batched Bandits](https://mlsanity.com/paper/1910.04959/) (2019)
- [Adaptive Exploration in Linear Contextual Bandit](https://mlsanity.com/paper/1910.06996/) (2019)
- [Minimal Exploration in Structured Stochastic Bandits](https://mlsanity.com/paper/1711.00400/) (2017)
- [Asymptotically Optimal Information-Directed Sampling](https://mlsanity.com/paper/2011.05944/) (2020)
- [An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits](https://mlsanity.com/paper/2010.12247/) (2020)
- [Double Explore-then-Commit: Asymptotic Optimality and Beyond](https://mlsanity.com/paper/2002.09174/) (2020)
- [Langevin Monte Carlo for Contextual Bandits](https://mlsanity.com/paper/2206.11254/) (2022)
- [Contexts can be Cheap: Solving Stochastic Contextual Bandits with Linear Bandit Algorithms](https://mlsanity.com/paper/2211.05632/) (2022)
- [Instance-Optimality in Interactive Decision Making: Toward a Non-Asymptotic Theory](https://mlsanity.com/paper/2304.12466/) (2023)
- [Almost Optimal Batch-Regret Tradeoff for Batch Linear Contextual Bandits](https://mlsanity.com/paper/2110.08057/) (2021)
- [Optimal Batched Best Arm Identification](https://mlsanity.com/paper/2310.14129/) (2023)

## Built on by (most cited first, 7 of 7)

- [Optimal and Practical Batched Linear Bandit Algorithm](https://mlsanity.com/paper/2507.08438/) (2025)
- [Practical and Optimal Algorithm for Linear Contextual Bandits with Rare Parameter Updates](https://mlsanity.com/paper/2606.00984/) (2026)
- [Comparative Performance of Collaborative Bandit Algorithms: Effect of Sparsity and Exploration Intensity](https://mlsanity.com/paper/2410.12086/) (2024)
- [Few Batches or Little Memory, But Not Both: Simultaneous Space and Adaptivity Constraints in Stochastic Bandits](https://mlsanity.com/paper/2603.13742/) (2026)
- [Batched Stochastic Linear Bandits with 1-Bit Communication Constraints](https://mlsanity.com/paper/2605.30976/) (2026)
- [Stochastic Matching Bandits with Rare Optimization Updates](https://mlsanity.com/paper/2509.04194/) (2025)
- [Oracle-Efficient Combinatorial Semi-Bandits](https://mlsanity.com/paper/2510.21431/) (2025)

## Abstract (arXiv)

We introduce the E4 algorithm for the batched linear bandit problem, incorporating an Explore-Estimate-Eliminate-Exploit framework. With a proper choice of exploration rate, we prove E4 achieves the finite-time minimax optimal regret with only O(loglog T) batches, and the asymptotically optimal regret with only 3 batches as Trightarrow∞, where T is the time horizon. We further prove a lower bound on the batch complexity of linear contextual bandits showing that any asymptotically optimal algorithm must require at least 3 batches in expectation as Trightarrow∞, which indicates E4 achieves the asymptotic optimality in regret and batch complexity simultaneously. To the best of our knowledge, E4 is the first algorithm for linear bandits that simultaneously achieves the minimax and asymptotic optimality in regret with the corresponding optimal batch complexities. In addition, we show that with another choice of exploration rate E4 achieves an instance-dependent regret bound requiring at most O(log T) batches, and maintains the minimax optimality and asymptotic optimality. We conduct thorough experiments to evaluate our algorithm on randomly generated instances and the challenging textitEnd of Optimism instances citeplattimore2017end which were shown to be hard to learn for optimism based algorithms. Empirical results show that E4 consistently outperforms baseline algorithms with respect to regret minimization, batch complexity, and computational efficiency.