# Achieving Near Instance-Optimality and Minimax-Optimality in Stochastic and Adversarial Linear Bandits Simultaneously

arXiv: https://arxiv.org/abs/2102.05858 · HTML: https://mlsanity.com/paper/2102.05858/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 41 references that are in the map.
- First posted on arXiv 11 Feb 2021.

## Builds on (most cited first, 21 of 21)

- [Taming the Monster: A Fast and Simple Algorithm for Contextual Bandits](https://mlsanity.com/paper/1402.0555/) (2014)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [Stochastic bandits robust to adversarial corruptions](https://mlsanity.com/paper/1803.09353/) (2018)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [Better Algorithms for Stochastic Bandits with Adversarial Corruptions](https://mlsanity.com/paper/1902.08647/) (2019)
- [Corruption-robust exploration in episodic reinforcement learning](https://mlsanity.com/paper/1911.08689/) (2019)
- [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)
- [The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits](https://mlsanity.com/paper/1610.04491/) (2016)
- [Simultaneously Learning Stochastic and Adversarial Episodic MDPs with Known Transition](https://mlsanity.com/paper/2006.05606/) (2020)
- [An algorithm with nearly optimal pseudo-regret for both stochastic and adversarial bandits](https://mlsanity.com/paper/1605.08722/) (2016)
- [An Improved Parametrization and Analysis of the EXP3++ Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1702.06103/) (2017)
- [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)
- [An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits](https://mlsanity.com/paper/2010.12247/) (2020)
- [Stochastic Linear Optimization with Adversarial Corruption](https://mlsanity.com/paper/1909.02109/) (2019)
- [Refined Lower Bounds for Adversarial Bandits](https://mlsanity.com/paper/1605.07416/) (2016)
- [Taking a hint: How to leverage loss predictors in contextual bandits?](https://mlsanity.com/paper/2003.01922/) (2020)
- [Regret Lower Bound and Optimal Algorithm in Finite Stochastic Partial Monitoring](https://mlsanity.com/paper/1509.09011/) (2015)
- [Crush Optimism with Pessimism: Structured Bandits Beyond Asymptotic Optimality](https://mlsanity.com/paper/2006.08754/) (2020)

## Built on by (most cited first, 30 of 48)

- [High-Dimensional Experimental Design and Kernel Bandits](https://mlsanity.com/paper/2105.05806/) (2021)
- [Nearly Optimal Algorithms for Linear Contextual Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2205.06811/) (2022)
- [A Model Selection Approach for Corruption Robust Reinforcement Learning](https://mlsanity.com/paper/2110.03580/) (2021)
- [The best of both worlds: stochastic and adversarial episodic MDPs with unknown transition](https://mlsanity.com/paper/2106.04117/) (2021)
- [First-Order Regret in Reinforcement Learning with Linear Function Approximation: A Robust Estimation Approach](https://mlsanity.com/paper/2112.03432/) (2021)
- [Corruption-Robust Algorithms with Uncertainty Weighting for Nonlinear Contextual Bandits and Markov Decision Processes](https://mlsanity.com/paper/2212.05949/) (2022)
- [Corruption-Robust Offline Reinforcement Learning with General Function Approximation](https://mlsanity.com/paper/2310.14550/) (2023)
- [A Blackbox Approach to Best of Both Worlds in Bandits and Beyond](https://mlsanity.com/paper/2302.09739/) (2023)
- [Nearly Optimal Best-of-Both-Worlds Algorithms for Online Learning with Feedback Graphs](https://mlsanity.com/paper/2206.00873/) (2022)
- [On Optimal Robustness to Adversarial Corruption in Online Decision Problems](https://mlsanity.com/paper/2109.10963/) (2021)
- [Linear Contextual Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2110.12615/) (2021)
- [Improved Corruption Robust Algorithms for Episodic Reinforcement Learning](https://mlsanity.com/paper/2102.06875/) (2021)
- [Best-of-Both-Worlds Algorithms for Partial Monitoring](https://mlsanity.com/paper/2207.14550/) (2022)
- [Contexts can be Cheap: Solving Stochastic Contextual Bandits with Linear Bandit Algorithms](https://mlsanity.com/paper/2211.05632/) (2022)
- [Improved Best-of-Both-Worlds Guarantees for Multi-Armed Bandits: FTRL with General Regularizers and Multiple Optimal Arms](https://mlsanity.com/paper/2302.13534/) (2023)
- [An Optimization-based Algorithm for Non-stationary Kernel Bandits without Prior Knowledge](https://mlsanity.com/paper/2205.14775/) (2022)
- [A Near-Optimal Best-of-Both-Worlds Algorithm for Online Learning with Feedback Graphs](https://mlsanity.com/paper/2206.00557/) (2022)
- [Robust Lipschitz Bandits to Adversarial Corruptions](https://mlsanity.com/paper/2305.18543/) (2023)
- [Best-of-Three-Worlds Linear Bandit Algorithm with Variance-Adaptive Regret Bounds](https://mlsanity.com/paper/2302.12370/) (2023)
- [Adversarially Robust Multi-Armed Bandit Algorithm with Variance-Dependent Regret Bounds](https://mlsanity.com/paper/2206.06810/) (2022)
- [Best-of-three-worlds Analysis for Linear Bandits with Follow-the-regularized-leader Algorithm](https://mlsanity.com/paper/2303.06825/) (2023)
- [Towards Robust Model-Based Reinforcement Learning Against Adversarial Corruption](https://mlsanity.com/paper/2402.08991/) (2024)
- [Best of Both Worlds Model Selection](https://mlsanity.com/paper/2206.14912/) (2022)
- [Best-of-Both-Worlds Algorithms for Linear Contextual Bandits](https://mlsanity.com/paper/2312.15433/) (2023)
- [Follow-the-Perturbed-Leader with Fr'echet-type Tail Distributions: Optimality in Adversarial Bandits and Best-of-Both-Worlds](https://mlsanity.com/paper/2403.05134/) (2024)
- [Nearly Optimal Algorithms for Contextual Dueling Bandits from Adversarial Feedback](https://mlsanity.com/paper/2404.10776/) (2024)
- [Accelerated Rates between Stochastic and Adversarial Online Convex Optimization](https://mlsanity.com/paper/2303.03272/) (2023)
- [Corruption-Robust Linear Bandits: Minimax Optimality and Gap-Dependent Misspecification](https://mlsanity.com/paper/2410.07533/) (2024)
- [Simultaneously Learning Stochastic and Adversarial Bandits with General Graph Feedback](https://mlsanity.com/paper/2206.07908/) (2022)
- [Cooperative Stochastic Multi-agent Multi-armed Bandits Robust to Adversarial Corruptions](https://mlsanity.com/paper/2106.04207/) (2021)

## Abstract (arXiv)

In this work, we develop linear bandit algorithms that automatically adapt to different environments. By plugging a novel loss estimator into the optimization problem that characterizes the instance-optimal strategy, our first algorithm not only achieves nearly instance-optimal regret in stochastic environments, but also works in corrupted environments with additional regret being the amount of corruption, while the state-of-the-art (Li et al., 2019) achieves neither instance-optimality nor the optimal dependence on the corruption amount. Moreover, by equipping this algorithm with an adversarial component and carefully-designed testings, our second algorithm additionally enjoys minimax-optimal regret in completely adversarial environments, which is the first of this kind to our knowledge. Finally, all our guarantees hold with high probability, while existing instance-optimal guarantees only hold in expectation.