Rotting Infinitely Many-armed Bandits

TL;DR

This paper studies the infinitely many-armed bandit problem with rotting rewards, establishing tight regret bounds and proposing algorithms that adapt to unknown rotting rates, advancing understanding of non-stationary bandit challenges.

Contribution

It provides tight regret bounds for rotting rewards in infinite-armed bandits and introduces algorithms that adapt to unknown rotting rates.

Findings

Matching lower and upper regret bounds up to poly-log factors.

Algorithms with known and unknown rotting rates achieve near-optimal regret.

Adaptive algorithms effectively handle non-stationary reward decay.

Abstract

We consider the infinitely many-armed bandit problem with rotting rewards, where the mean reward of an arm decreases at each pull of the arm according to an arbitrary trend with maximum rotting rate $\varrho=o(1)$. We show that this learning problem has an $\Omega(\max\{\varrho^{1/3}T,\sqrt{T}\})$ worst-case regret lower bound where $T$ is the horizon time. We show that a matching upper bound $\tilde{O}(\max\{\varrho^{1/3}T,\sqrt{T}\})$, up to a poly-logarithmic factor, can be achieved by an algorithm that uses a UCB index for each arm and a threshold value to decide whether to continue pulling an arm or remove the arm from further consideration, when the algorithm knows the value of the maximum rotting rate $\varrho$. We also show that an $\tilde{O}(\max\{\varrho^{1/3}T,T^{3/4}\})$ regret upper bound can be achieved by an algorithm that does not know the value of $\varrho$, by using an adaptive UCB index along with an adaptive threshold value.