What Doubling Tricks Can and Can't Do for Multi-Armed Bandits

TL;DR

This paper investigates the effectiveness of doubling tricks in converting non-anytime algorithms into anytime algorithms for multi-armed bandits, analyzing their ability to preserve different regret bounds.

Contribution

It provides a theoretical analysis of geometric and exponential doubling tricks, detailing which regret bounds they can conserve in multi-armed bandit problems.

Findings

Geometric doubling tricks conserve $O(\sqrt{T})$ regret bounds.

Geometric doubling tricks cannot conserve $O(\log T)$ regret bounds.

Exponential doubling tricks better conserve $O(\log T)$ bounds.

Abstract

An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of an algorithm is measured by its regret, and we study two families of sequences of growing horizons (geometric and exponential) to generalize previously known results that certain doubling tricks can be used to conserve certain regret bounds. In a broad setting, we prove that a geometric doubling trick can be used to conserve (minimax) bounds in $R\_T = O(\sqrt{T})$ but cannot conserve (distribution-dependent) bounds in $R\_T = O(\log T)$. We give insights as to why exponential doubling tricks may be better, as they conserve bounds in $R\_T = O(\log T)$, and are close to conserving bounds in $R\_T = O(\sqrt{T})$.