# Best-of-Both-Worlds Linear Contextual Bandits

arXiv: https://arxiv.org/abs/2312.16489 · HTML: https://mlsanity.com/paper/2312.16489/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 45 references that are in the map.
- First posted on arXiv 27 Dec 2023.

## Builds on (most cited first, 27 of 27)

- [Competitive caching with machine learned advice](https://mlsanity.com/paper/1802.05399/) (2018)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (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)
- [Nearly Optimal Algorithms for Linear Contextual Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2205.06811/) (2022)
- [The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits](https://mlsanity.com/paper/1610.04491/) (2016)
- [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)
- [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)
- [Robust Stochastic Linear Contextual Bandits Under Adversarial Attacks](https://mlsanity.com/paper/2106.02978/) (2021)
- [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)
- [Linear Contextual Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2110.12615/) (2021)
- [BISTRO: An Efficient Relaxation-Based Method for Contextual Bandits](https://mlsanity.com/paper/1602.02196/) (2016)
- [Best-of-Both-Worlds Algorithms for Partial Monitoring](https://mlsanity.com/paper/2207.14550/) (2022)
- [First- and Second-Order Bounds for Adversarial Linear Contextual Bandits](https://mlsanity.com/paper/2305.00832/) (2023)
- [Bypassing the Simulator: Near-Optimal Adversarial Linear Contextual Bandits](https://mlsanity.com/paper/2309.00814/) (2023)
- [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)
- [Best of Both Worlds Policy Optimization](https://mlsanity.com/paper/2302.09408/) (2023)
- [Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions](https://mlsanity.com/paper/2103.12487/) (2021)
- [Stability-penalty-adaptive follow-the-regularized-leader: Sparsity, game-dependency, and best-of-both-worlds](https://mlsanity.com/paper/2305.17301/) (2023)
- [Improved Regret Bounds for Oracle-Based Adversarial Contextual Bandits](https://mlsanity.com/paper/1606.00313/) (2016)
- [Best-of-three-worlds Analysis for Linear Bandits with Follow-the-regularized-leader Algorithm](https://mlsanity.com/paper/2303.06825/) (2023)
- [Best-of-Both-Worlds Algorithms for Linear Contextual Bandits](https://mlsanity.com/paper/2312.15433/) (2023)
- [Regret Lower Bound and Optimal Algorithm for High-Dimensional Contextual Linear Bandit](https://mlsanity.com/paper/2109.11612/) (2021)
- [Online Learning with Low Rank Experts](https://mlsanity.com/paper/1603.06352/) (2016)

## Built on by (most cited first, 2 of 2)

- [Last-Iterate Analyses of FTRL with the 1/2-Tsallis Entropy in Stochastic Bandits](https://mlsanity.com/paper/2510.22819/) (2025)
- [Best-of-Both Worlds for linear contextual bandits with paid observations](https://mlsanity.com/paper/2510.07424/) (2025)

## Abstract (arXiv)

This study investigates the problem of K-armed linear contextual bandits, an instance of the multi-armed bandit problem, under an adversarial corruption. At each round, a decision-maker observes an independent and identically distributed context and then selects an arm based on the context and past observations. After selecting an arm, the decision-maker incurs a loss corresponding to the selected arm. The decision-maker aims to minimize the cumulative loss over the trial. The goal of this study is to develop a strategy that is effective in both stochastic and adversarial environments, with theoretical guarantees. We first formulate the problem by introducing a novel setting of bandits with adversarial corruption, referred to as the contextual adversarial regime with a self-bounding constraint. We assume linear models for the relationship between the loss and the context. Then, we propose a strategy that extends the RealLinExp3 by Neu & Olkhovskaya (2020) and the Follow-The-Regularized-Leader (FTRL). The regret of our proposed algorithm is shown to be upper-bounded by Oleft(minleftfrac(log(T))3Delta* + √fracC(log(T))3Delta*,  √T(log(T))2rightright), where T ∈mathbbN is the number of rounds, Delta* > 0 is the constant minimum gap between the best and suboptimal arms for any context, and C∈[0, T]  is an adversarial corruption parameter. This regret upper bound implies Oleft(frac(log(T))3Delta*right) in a stochastic environment and by Oleft( √T(log(T))2right) in an adversarial environment. We refer to our strategy as the Best-of-Both-Worlds (BoBW) RealFTRL, due to its theoretical guarantees in both stochastic and adversarial regimes.