A Tight Lower Bound for Non-stochastic Multi-armed Bandits with Expert Advice

TL;DR

This paper establishes the exact minimax optimal expected regret for non-stochastic multi-armed bandits with expert advice, matching previous upper bounds and clarifying the fundamental limits of the problem.

Contribution

It provides a tight lower bound that confirms the minimax optimal regret rate, resolving a key theoretical question in the field.

Findings

Minimax optimal expected regret is Θ(√(T K log(N/K)))

Lower bound matches the previously known upper bound

Clarifies the fundamental limits of non-stochastic bandit with expert advice

Abstract

We determine the minimax optimal expected regret in the classic non-stochastic multi-armed bandit with expert advice problem, by proving a lower bound that matches the upper bound of Kale (2014). The two bounds determine the minimax optimal expected regret to be $\Theta\left( \sqrt{T K \log (N/K) } \right)$, where $K$ is the number of arms, $N$ is the number of experts, and $T$ is the time horizon.