Old Dog Learns New Tricks: Randomized UCB for Bandit Problems

We propose $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$, a bandit strategy that builds on theoretically derived confidence intervals similar to upper confidence bound (UCB) algorithms, but akin to Thompson sampling (TS), it uses randomization to trade off exploration and exploitation. In the $K$-armed bandit setting, we show that there are infinitely many variants of $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$, all of which achieve the minimax-optimal $\tilde{O}(\sqrt{KT})$ regret after $T$ rounds. Moreover, for a specific multi-armed bandit setting, we show that both UCB and TS can be recovered as special cases of $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$. For structured bandits, where each arm is associated with a $d$-dimensional feature vector and rewards are distributed according to a linear or generalized linear model, we prove that $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$ achieves the minimax-optimal $\tilde{O}(d\sqrt{T})$ regret even in the case of infinitely many arms. Through experiments in both the multi-armed and structured bandit settings, we demonstrate that $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$ matches or outperforms TS and other randomized exploration strategies. Our theoretical and empirical results together imply that $\mathtt{R}\mathtt{a}\mathtt{n}\mathtt{d}\mathtt{U}\mathtt{C}\mathtt{B}$ achieves the best of both worlds.