The study of collaborative multi-agent bandits has attracted significant attention recently. In light of this, we initiate the study of a new collaborative setting, consisting of $N$ agents such that each agent is learning one of $M$ stochastic multi-armed bandits to minimize their group cumulative regret. We develop decentralized algorithms which facilitate collaboration between the agents under two scenarios. We characterize the performance of these algorithms by deriving the per agent cumulative regret and group regret upper bounds. We also prove lower bounds for the group regret in this setting, which demonstrates the near-optimal behavior of the proposed algorithms.

For the misspecified linear Markov decision process (MLMDP) model of Jin et al. [2020], we propose an algorithm with three desirable properties. (P1) Its regret after K episodes scales as Kmax{\ensuremath{\varepsilon}mis,\ensuremath{\varepsilon}tol}, where \ensuremath{\varepsilon}mis is the degree of misspecification and \ensuremath{\varepsilon}tol is a user-specified error tolerance. (P2) Its space and per-episode time complexities remain bounded as $K\rightarrow\infty$. (P3) It does not require \ensuremath{\varepsilon}mis as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of \ensuremath{\varepsilon}tol, we also improve existing regret bounds (up to log factors) while achieving either (P2) or (P3) (existing algorithms satisfy neither). At a high level, our algorithm generalizes (to MLMDPs) and refines the Sup-Lin-UCB algorithm, which Takemura et al. [2021] recently showed satisfies (P3) in the contextual bandit setting. We also provide an intuitive interpretation of their result, which informs the design of our algorithm.

We propose an algorithm that uses linear function approximation (LFA) for stochastic shortest path (SSP). Under minimal assumptions, it obtains sublinear regret, is computationally efficient, and uses stationary policies. To our knowledge, this is the first such algorithm in the LFA literature (for SSP or other formulations). Our algorithm is a special case of a more general one, which achieves regret square root in the number of episodes given access to a computation oracle.

Cascading bandits is a natural and popular model that frames the task of learning to rank from Bernoulli click feedback in a bandit setting. For the case of unstructured rewards, we prove matching upper and lower bounds for the problem-independent (i.e., gap-free) regret, both of which strictly improve the best known. A key observation is that the hard instances of this problem are those with small mean rewards, i.e., the small click-through rates that are most relevant in practice. Based on this, and the fact that small mean implies small variance for Bernoullis, our key technical result shows that variance-aware confidence sets derived from the Bernstein and Chernoff bounds lead to optimal algorithms (up to log terms), whereas Hoeffding-based algorithms suffer order-wise suboptimal regret. This sharply contrasts with the standard (non-cascading) bandit setting, where the variance-aware algorithms only improve constants. In light of this and as an additional contribution, we propose a variance-aware algorithm for the structured case of linear rewards and show its regret strictly improves the state-of-the-art.

Recent works have shown that agents facing independent instances of a stochastic K-armed bandit can collaborate to decrease regret. However, these works assume that each agent always recommends their individual best-arm estimates to other agents, which is unrealistic in envisioned applications (machine faults in distributed computing or spam in social recommendation systems). Hence, we generalize the setting to include n honest and m malicious agents who recommend best-arm estimates and arbitrary arms, respectively. We first show that even with a single malicious agent, existing collaboration-based algorithms fail to improve regret guarantees over a single-agent baseline. We propose a scheme where honest agents learn who is malicious and dynamically reduce communication with (i.e., "block") them. We show that collaboration indeed decreases regret for this algorithm, assuming m is small compared to K but without assumptions on malicious agents' behavior, thus ensuring that our algorithm is robust against any malicious recommendation strategy.