In blocking $K$-armed bandits, playing an arm renders it unavailable for a fixed number of future rounds. While this model is relatively well understood in the stochastic regime, much less is known when rewards are generated adversarially. Via a novel reduction, we first show that computing the total reward of the best dynamic policy is NP-hard, even when the blocking time $d > 1$ is identical across arms. We therefore turn to tractable comparators and study the class of $d$-periodic policies, proving that the optimal periodic policy is efficiently computable and always obtains at least a $\frac{1}{K}$ fraction of the dynamic optimum. We also show that this $\frac{1}{K}$ factor is information-theoretically tight: no algorithm can achieve sublinear $\alpha$-regret with respect to the offline optimal dynamic policy for any $\alpha > \frac{1}{K}$. Our main result shows that $T^{2/3}$ is the minimax rate for the regret (against periodic policies) for adversarial blocking bandits with identical blocking times, and that this rate is achievable by an efficient algorithm. Our main technical contribution is the lower bound, which establishes that blocking bandits are at least as hard as bandits with switching costs. The matching upper bound instead follows from a reduction to combinatorial semi-bandits over bipartite matchings. Finally, we show that $\sqrt{T}$ regret rates are efficiently achievable in the full information setting, and more generally via $\alpha$-regret with $\alpha = \frac{1}{2}$.

We derive a new analysis of Follow The Regularized Leader (FTRL) for online learning with delayed bandit feedback. By separating the cost of delayed feedback from that of bandit feedback, our analysis allows us to obtain new results in four important settings. We derive the first optimal (up to logarithmic factors) regret bounds for combinatorial semi-bandits with delay and adversarial Markov Decision Processes with delay (both known and unknown transition functions). Furthermore, we use our analysis to develop an efficient algorithm for linear bandits with delay achieving near-optimal regret bounds. In order to derive these results we show that FTRL remains stable across multiple rounds under mild assumptions on the regularizer.

We study a contextual version of online combinatorial optimisation with full and semi-bandit feedback. In this sequential decision-making problem, an online learner has to select an action from a combinatorial decision space after seeing a vector-valued context in each round. As a result of its action, the learner incurs a loss that is a bilinear function of the context vector and the vector representation of the chosen action. We consider two natural versions of the problem: semi-bandit where the losses are revealed for each component appearing in the learner’s combinatorial action, and full-bandit where only the total loss is observed. We design computationally efficient algorithms based on a new loss estimator that takes advantage of the special structure of the problem, and show regret bounds order $\sqrt{T}$ with respect to the time horizon. The bounds demonstrate polynomial scaling with the relevant problem parameters which is shown to be nearly optimal. The theoretical results are complemented by a set of experiments on simulated data.

We derive a new analysis of Follow The Regularized Leader (FTRL) for online learning with delayed bandit feedback. By separating the cost of delayed feedback from that of bandit feedback, our analysis allows us to obtain new results in three important settings. On the one hand, we derive the first optimal (up to logarithmic factors) regret bounds for combinatorial semi-bandits with delay and adversarial Markov decision processes with delay (and known transition functions). On the other hand, we use our analysis to derive an efficient algorithm for linear bandits with delay achieving near-optimal regret bounds. Our novel regret decomposition shows that FTRL remains stable across multiple rounds under mild assumptions on the Hessian of the regularizer.