Non-stationary Bandits and Meta-Learning with a Small Set of Optimal Arms

We study a sequential decision problem where the learner faces a sequence of $K$-armed stochastic bandit tasks. The tasks may be designed by an adversary, but the adversary is constrained to choose the optimal arm of each task in a smaller (but unknown) subset of $M$ arms. The task boundaries might be known (the bandit meta-learning setting), or unknown (the non-stationary bandit setting), and the number of tasks $N$ as well as the total number of rounds $T$ are known ($N$ could be unknown in the meta-learning setting). We design an algorithm based on a reduction to bandit submodular maximization, and show that its regret in both settings is smaller than the simple baseline of $\tilde{O}(\sqrt{KNT})$ that can be obtained by using standard algorithms designed for non-stationary bandit problems. For the bandit meta-learning problem with fixed task length $\tau$, we show that the regret of the algorithm is bounded as $\tilde{O}(N\sqrt{M \tau}+N^{2/3})$. Under additional assumptions on the identifiability of the optimal arms in each task, we show a bandit meta-learning algorithm with an improved $\tilde{O}(N\sqrt{M \tau}+N^{1/2})$ regret.