# Thompson Sampling for Adversarial Bit Prediction

## TL;DR

This paper analyzes the performance of Thompson sampling in adversarial bit prediction, identifying sequences with minimal and maximal regret, and extending results to weighted error models.

## Contribution

It characterizes adversarial sequences with extreme regret bounds and extends analysis to weighted false positive and false negative errors.

## Key findings

- Sequences with alternating bits have maximal regret of O(√T).
- Sequences of all ones or zeros have minimal regret of O(1).
- Results extend to models with weighted false positives and negatives.

## Abstract

We study the Thompson sampling algorithm in an adversarial setting, specifically, for adversarial bit prediction. We characterize the bit sequences with the smallest and largest expected regret. Among sequences of length $T$ with $k < \frac{T}{2}$ zeros, the sequences of largest regret consist of alternating zeros and ones followed by the remaining ones, and the sequence of smallest regret consists of ones followed by zeros. We also bound the regret of those sequences, the worse case sequences have regret $O(\sqrt{T})$ and the best case sequence have regret $O(1)$. We extend our results to a model where false positive and false negative errors have different weights. We characterize the sequences with largest expected regret in this generalized setting, and derive their regret bounds. We also show that there are sequences with $O(1)$ regret.