Parametrized Stochastic Multi-armed Bandits with Binary Rewards

TL;DR

This paper introduces a Two-Phase Algorithm for stochastic multi-armed bandits with binary rewards, achieving sub-linear regret bounds that are independent of the number of arms, applicable to both finite and infinite arm scenarios.

Contribution

The paper proposes a novel algorithm with regret bounds that do not depend on the total number of arms, suitable for large or infinite arm sets with correlated Bernoulli rewards.

Findings

Regret bounds are sub-linear and uniform in time.

Finite arm regret scales as O(n (T)) for f in (\u221e(\log(T)))

Infinite arm regret is O(((T)))

Abstract

In this paper, we consider the problem of multi-armed bandits with a large, possibly infinite number of correlated arms. We assume that the arms have Bernoulli distributed rewards, independent across time, where the probabilities of success are parametrized by known attribute vectors for each arm, as well as an unknown preference vector, each of dimension $n$. For this model, we seek an algorithm with a total regret that is sub-linear in time and independent of the number of arms. We present such an algorithm, which we call the Two-Phase Algorithm, and analyze its performance. We show upper bounds on the total regret which applies uniformly in time, for both the finite and infinite arm cases. The asymptotics of the finite arm bound show that for any $f \in \omega(\log(T))$, the total regret can be made to be $O(n \cdot f(T))$. In the infinite arm case, the total regret is $O(\sqrt{n^3 T})$.