# Best-of-Three-Worlds Linear Bandit Algorithm with Variance-Adaptive Regret Bounds

## TL;DR

This paper introduces a hierarchical adaptive linear bandit algorithm that achieves optimal regret bounds across adversarial and stochastic environments, with variance-adaptive performance and data-dependent regret guarantees.

## Contribution

It presents a novel hierarchical algorithm that attains best-of-three-worlds regret bounds and incorporates new techniques for high-level and low-level adaptability in linear bandits.

## Key findings

- Achieves ${O}(\sqrt{T \log T})$ regret in adversarial settings.
- Attains $O(rac{\log T}{\Delta_{ ext{min}}} + \sqrt{rac{C \log T}{\Delta_{ ext{min}}})$ regret in stochastic environments with corruption.
- Provides variance-adaptive regret bounds of $O(rac{\sigma^2 \log T}{\Delta_{ ext{min}}})$.

## Abstract

This paper proposes a linear bandit algorithm that is adaptive to environments at two different levels of hierarchy. At the higher level, the proposed algorithm adapts to a variety of types of environments. More precisely, it achieves best-of-three-worlds regret bounds, i.e., of ${O}(\sqrt{T \log T})$ for adversarial environments and of $O(\frac{\log T}{\Delta_{\min}} + \sqrt{\frac{C \log T}{\Delta_{\min}}})$ for stochastic environments with adversarial corruptions, where $T$, $\Delta_{\min}$, and $C$ denote, respectively, the time horizon, the minimum sub-optimality gap, and the total amount of the corruption. Note that polynomial factors in the dimensionality are omitted here. At the lower level, in each of the adversarial and stochastic regimes, the proposed algorithm adapts to certain environmental characteristics, thereby performing better. The proposed algorithm has data-dependent regret bounds that depend on all of the cumulative loss for the optimal action, the total quadratic variation, and the path-length of the loss vector sequence. In addition, for stochastic environments, the proposed algorithm has a variance-adaptive regret bound of $O(\frac{\sigma^2 \log T}{\Delta_{\min}})$ as well, where $\sigma^2$ denotes the maximum variance of the feedback loss. The proposed algorithm is based on the SCRiBLe algorithm. By incorporating into this a new technique we call scaled-up sampling, we obtain high-level adaptability, and by incorporating the technique of optimistic online learning, we obtain low-level adaptability.