Replicable Bandits with UCB based Exploration

Abstract: We study replicable algorithms for stochastic multi-armed bandits (MAB) and linear bandits with UCB (Upper Confidence Bound) based exploration. A bandit algorithm is $ρ$-replicable if two executions using shared internal randomness but independent reward realizations, produce the same action sequence with probability at least $1-ρ$. Prior work is primarily elimination-based and, in linear bandits with infinitely many actions, relies on discretization, leading to suboptimal dependence on the dimension $d$ and $ρ$. We develop optimistic alternatives for both settings. For stochastic multi-armed bandits, we propose RepUCB, a replicable batched UCB algorithm and show that it attains a regret $O!\left(\frac{K^2\log^2 T}{ρ^2}\sum_{a:Δ_a>0}\left(Δ_a+\frac{\log(KT\log T)}{Δ_a}\right)\right)$. For stochastic linear bandits, we first introduce RepRidge, a replicable ridge regression estimator that satisfies both a confidence guarantee and a $ρ$-replicability guarantee. Beyond its role in our bandit algorithm, this estimator and its guarantees may also be of independent interest in other statistical estimation settings. We then use RepRidge to design RepLinUCB, a replicable optimistic algorithm for stochastic linear bandits, and show that its regret is bounded by $\widetilde{O}!\big(\big(d+\frac{d^3}ρ\big)\sqrt{T}\big)$. This improves the best prior regret guarantee by a factor of $O(d/ρ)$, showing that our optimistic algorithm can substantially reduce the price of replicability.