# Beyond the Hazard Rate: More Perturbation Algorithms for Adversarial Multi-armed Bandits

## TL;DR

This paper extends the analysis of follow the perturbed leader algorithms for adversarial multi-armed bandits by introducing the generalized hazard rate, allowing for regret bounds with distributions like Gaussian and uniform.

## Contribution

It introduces the generalized hazard rate concept and provides regret bounds for FTPL algorithms without the bounded hazard rate assumption, including for Gaussian and uniform distributions.

## Key findings

- Gaussian distribution can achieve near-optimal regret.
- Regret bounds are established for distributions with unbounded support.
- Disproves the conjecture that Gaussian cannot be used for low-regret algorithms.

## Abstract

Recent work on follow the perturbed leader (FTPL) algorithms for the adversarial multi-armed bandit problem has highlighted the role of the hazard rate of the distribution generating the perturbations. Assuming that the hazard rate is bounded, it is possible to provide regret analyses for a variety of FTPL algorithms for the multi-armed bandit problem. This paper pushes the inquiry into regret bounds for FTPL algorithms beyond the bounded hazard rate condition. There are good reasons to do so: natural distributions such as the uniform and Gaussian violate the condition. We give regret bounds for both bounded support and unbounded support distributions without assuming the hazard rate condition. We also disprove a conjecture that the Gaussian distribution cannot lead to a low-regret algorithm. In fact, it turns out that it leads to near optimal regret, up to logarithmic factors. A key ingredient in our approach is the introduction of a new notion called the generalized hazard rate.