# Corruption-Robust Linear Bandits: Minimax Optimality and Gap-Dependent Misspecification

arXiv: https://arxiv.org/abs/2410.07533 · HTML: https://mlsanity.com/paper/2410.07533/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 51 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 30 of 32)

- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Learning Near Optimal Policies with Low Inherent Bellman Error](https://mlsanity.com/paper/2003.00153/) (2020)
- [Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning](https://mlsanity.com/paper/1703.07710/) (2017)
- [Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning?](https://mlsanity.com/paper/1910.03016/) (2019)
- [Learning with Good Feature Representations in Bandits and in RL with a Generative Model](https://mlsanity.com/paper/1911.07676/) (2019)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [Adapting to Misspecification in Contextual Bandits](https://mlsanity.com/paper/2107.05745/) (2021)
- [Stochastic Linear Bandits Robust to Adversarial Attacks](https://mlsanity.com/paper/2007.03285/) (2020)
- [Beating Stochastic and Adversarial Semi-bandits Optimally and Simultaneously](https://mlsanity.com/paper/1901.08779/) (2019)
- [Bias no more: high-probability data-dependent regret bounds for adversarial bandits and MDPs](https://mlsanity.com/paper/2006.08040/) (2020)
- [Nearly Optimal Algorithms for Linear Contextual Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2205.06811/) (2022)
- [Achieving Near Instance-Optimality and Minimax-Optimality in Stochastic and Adversarial Linear Bandits Simultaneously](https://mlsanity.com/paper/2102.05858/) (2021)
- [An algorithm with nearly optimal pseudo-regret for both stochastic and adversarial bandits](https://mlsanity.com/paper/1605.08722/) (2016)
- [A Model Selection Approach for Corruption Robust Reinforcement Learning](https://mlsanity.com/paper/2110.03580/) (2021)
- [An Improved Parametrization and Analysis of the EXP3++ Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1702.06103/) (2017)
- [Efficient and Robust Algorithms for Adversarial Linear Contextual Bandits](https://mlsanity.com/paper/2002.00287/) (2020)
- [Corruption-Tolerant Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2003.01971/) (2020)
- [Corruption-Robust Algorithms with Uncertainty Weighting for Nonlinear Contextual Bandits and Markov Decision Processes](https://mlsanity.com/paper/2212.05949/) (2022)
- [A Blackbox Approach to Best of Both Worlds in Bandits and Beyond](https://mlsanity.com/paper/2302.09739/) (2023)
- [Stochastic Linear Optimization with Adversarial Corruption](https://mlsanity.com/paper/1909.02109/) (2019)
- [Bypassing the Simulator: Near-Optimal Adversarial Linear Contextual Bandits](https://mlsanity.com/paper/2309.00814/) (2023)
- [Best-of-Three-Worlds Linear Bandit Algorithm with Variance-Adaptive Regret Bounds](https://mlsanity.com/paper/2302.12370/) (2023)
- [No-Regret Online Reinforcement Learning with Adversarial Losses and Transitions](https://mlsanity.com/paper/2305.17380/) (2023)
- [Towards Optimal Regret in Adversarial Linear MDPs with Bandit Feedback](https://mlsanity.com/paper/2310.11550/) (2023)
- [Best-of-three-worlds Analysis for Linear Bandits with Follow-the-regularized-leader Algorithm](https://mlsanity.com/paper/2303.06825/) (2023)
- [A Robust Phased Elimination Algorithm for Corruption-Tolerant Gaussian Process Bandits](https://mlsanity.com/paper/2202.01850/) (2022)
- [On the Model-Misspecification in Reinforcement Learning](https://mlsanity.com/paper/2306.10694/) (2023)
- [The entropic barrier is n-self-concordant](https://mlsanity.com/paper/2112.10947/) (2021)

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

- [An Improved Algorithm for Adversarial Linear Contextual Bandits via Reduction](https://mlsanity.com/paper/2508.11931/) (2025)
- [A Jointly Efficient and Optimal Algorithm for Heteroskedastic Generalized Linear Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2602.10971/) (2026)
- [Batched Stochastic Linear Bandits with 1-Bit Communication Constraints](https://mlsanity.com/paper/2605.30976/) (2026)
- [Policy Regret for Embedding Model Routing: Contextual Bandits with Low-Rank Experts](https://mlsanity.com/paper/2606.14929/) (2026)
- [Robust Linear Dueling Bandits with Post-serving Context under Unknown Delays and Adversarial Corruptions](https://mlsanity.com/paper/2605.01752/) (2026)
- [Bandits with Multiple Optimal Arms: Minimax Regret and Non-Adaptivity](https://mlsanity.com/paper/2609.38659/) (2026)
- [Follow the Perturbed Leader for Linear Bandits: √T Regret over General Convex Sets](https://mlsanity.com/paper/2610.17818/) (2026)

## Abstract (arXiv)

In linear bandits, how can a learner effectively learn when facing corrupted rewards? While significant work has explored this question, a holistic understanding across different adversarial models and corruption measures is lacking, as is a full characterization of the minimax regret bounds. In this work, we compare two types of corruptions commonly considered: strong corruption, where the corruption level depends on the action chosen by the learner, and weak corruption, where the corruption level does not depend on the action chosen by the learner. We provide a unified framework to analyze these corruptions. For stochastic linear bandits, we fully characterize the gap between the minimax regret under strong and weak corruptions. We also initiate the study of corrupted adversarial linear bandits, obtaining upper and lower bounds with matching dependencies on the corruption level. Next, we reveal a connection between corruption-robust learning and learning with gap-dependent mis-specification, a setting first studied by Liu et al. (2023a), where the misspecification level of an action or policy is proportional to its suboptimality. We present a general reduction that enables any corruption-robust algorithm to handle gap-dependent misspecification. This allows us to recover the results of Liu et al. (2023a) in a black-box manner and significantly generalize them to settings like linear MDPs, yielding the first results for gap-dependent misspecification in reinforcement learning. However, this general reduction does not attain the optimal rate for gap-dependent misspecification. Motivated by this, we develop a specialized algorithm that achieves optimal bounds for gap-dependent misspecification in linear bandits, thus answering an open question posed by Liu et al. (2023a).