### Session

## Oral Session 2: Reinforcement Learning

Moderator: Michal Valko

**An Exponential Lower Bound for Linearly Realizable MDP with Constant Suboptimality Gap**

Yuanhao Wang · Ruosong Wang · Sham Kakade

A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional 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 \emph{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.

**On the Expressivity of Markov Reward**

David Abel · Will Dabney · Anna Harutyunyan · Mark Ho · Michael Littman · Doina Precup · Satinder Singh

Reward is the driving force for reinforcement-learning agents. This paper is dedicated to understanding the expressivity of reward as a way to capture tasks that we would want an agent to perform. We frame this study around three new abstract notions of “task” that might be desirable: (1) a set of acceptable behaviors, (2) a partial ordering over behaviors, or (3) a partial ordering over trajectories. Our main results prove that while reward can express many of these tasks, there exist instances of each task type that no Markov reward function can capture. We then provide a set of polynomial-time algorithms that construct a Markov reward function that allows an agent to optimize tasks of each of these three types, and correctly determine when no such reward function exists. We conclude with an empirical study that corroborates and illustrates our theoretical findings.

**The best of both worlds: stochastic and adversarial episodic MDPs with unknown transition**

Tiancheng Jin · Longbo Huang · Haipeng Luo

We consider the best-of-both-worlds problem for learning an episodic Markov Decision Process through $T$ episodes, with the goal of achieving $\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the losses are adversarial and simultaneously $\mathcal{O}(\log T)$ regret when the losses are (almost) stochastic. Recent work by [Jin and Luo, 2020] achieves this goal when the fixed transition is known, and leaves the case of unknown transition as a major open question. In this work, we resolve this open problem by using the same Follow-the-Regularized-Leader (FTRL) framework together with a set of new techniques. Specifically, we first propose a loss-shifting trick in the FTRL analysis, which greatly simplifies the approach of [Jin and Luo, 2020] and already improves their results for the known transition case. Then, we extend this idea to the unknown transition case and develop a novel analysis which upper bounds the transition estimation error by the regret itself in the stochastic setting, a key property to ensure $\mathcal{O}(\log T)$ regret.