A fundamental question in the theory of reinforcement learning is: suppose the optimal Q-function lies in the linear span of a given ddimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolves this question in the negative, providing an exponential (in d) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that such a lower can be circumvented with an even stronger assumption that there is a constant gap between the optimal Q-value of the best action and that of the second-best action (for all states); indeed, the construction in Weisz et al. (2020) relies on having an exponentially small gap. This work resolves this subsequent question, showing that an exponential sample complexity lower bound still holds even if a constant gap is assumed. Perhaps surprisingly, this result implies an exponential separation between the online RL setting and the generative model setting, where sample-efficient RL is in fact possible in the latter setting with a constant gap. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption.

While much progress has been made in understanding the minimax sample complexity of reinforcement learning (RL)--the complexity of learning on the "worstcase" instance--such measures of complexity often do not capture the true difficulty of learning. In practice, on an "easy" instance, we might hope to achieve a complexity far better than that achievable on the worst-case instance. In this work we seek to understand the "instance-dependent" complexity of learning near-optimal policies (PACRL) in the setting of RL with linear function approximation. We propose an algorithm, PEDEL, which achieves a fine-grained instance-dependent measure of complexity, the first of its kind in the RL with function approximation setting, thereby capturing the difficulty of learning on each particular problem instance. Through an explicit example, we show that PEDEL yields provable gains over low-regret, minimax-optimal algorithms and that such algorithms are unable to hit the instance-optimal rate. Our approach relies on a novel online experiment design-based procedure which focuses the exploration budget on the "directions" most relevant to learning a near-optimal policy, and may be of independent interest.

Learning near-optimal behaviour from an expert's demonstrations typically relies on the assumption that the learner knows the features that the true reward function depends on. In this paper, we study the problem of learning from demonstrations in the setting where this is not the case, i.e., where there is a mismatch between the worldviews of the learner and the expert. We introduce a natural quantity, the teaching risk, which measures the potential suboptimality of policies that look optimal to the learner in this setting. We show that bounds on the teaching risk guarantee that the learner is able to find a near-optimal policy using standard algorithms based on inverse reinforcement learning. Based on these findings, we suggest a teaching scheme in which the expert can decrease the teaching risk by updating the learner's worldview, and thus ultimately enable her to find a near-optimal policy.

In this paper we consider the dynamic assortment selection problem under an uncapacitated multinomial-logit (MNL) model. By carefully analyzing a revenue potential function, we show that a trisection based algorithm achieves an item-independent regret bound of Op? T log log T q, which matches information theoretical lower bounds up to iterated logarithmic terms. Our proof technique draws tools from the unimodal/convex bandit literature as well as adaptive confidence parameters in minimax multi-armed bandit problems.

We then give a practical approximation using neural networks anddemonstrate itsperformance andsampleefficiencyinpractice.