# Contextual Linear Bandits under Noisy Features: Towards Bayesian Oracles

## TL;DR

This paper investigates contextual linear bandit problems with noisy and incomplete features, proposing a Bayesian-based algorithm that achieves near-optimal regret bounds despite feature uncertainty.

## Contribution

It introduces a Bayesian analysis for noisy feature bandits and develops an algorithm that approximates the Bayesian oracle, achieving $ ilde{O}(d ext{ } ext{sqrt}(T))$ regret.

## Key findings

- Bayesian analysis reveals significant deviations from classical assumptions due to noise.
- The proposed algorithm attains near-optimal regret bounds in noisy feature settings.
- Experimental results on synthetic and real datasets validate the approach.

## Abstract

We study contextual linear bandit problems under feature uncertainty, where the features are noisy and have missing entries. To address the challenges posed by this noise, we analyze Bayesian oracles given the observed noisy features. Our Bayesian analysis reveals that the optimal hypothesis can significantly deviate from the underlying realizability function, depending on the noise characteristics. These deviations are highly non-intuitive and do not occur in classical noiseless setups. This implies that classical approaches cannot guarantee a non-trivial regret bound. Therefore, we propose an algorithm that aims to approximate the Bayesian oracle based on the observed information under this model, achieving $\tilde{O}(d\sqrt{T})$ regret bound when there is a large number of arms. We demonstrate the proposed algorithm using synthetic and real-world datasets.