Optimistic algorithms have been extensively studied for regret minimization in episodic tabular Markov Decision Processes (MDPs), both from a minimax and an instance-dependent view. However, for the PAC RL problem, where the goal is to identify a near-optimal policy with high probability, little is known about their instance-dependent sample complexity. A negative result of Wagenmaker et al. (2022) suggests that optimistic sampling rules cannot be used to attain the (still elusive) optimal instance-dependent sample complexity. On the positive side, we provide the first instance-dependent bound for an optimistic algorithm for PAC RL, BPI-UCRL, for which only minimax guarantees were available (Kaufmann et al., 2021). While our bound features some minimal visitation probabilities, it also features a refined notion of sub-optimality gap compared to the value gaps that appear in prior work. Moreover, in MDPs with deterministic transitions, we show that BPI-UCRL is actually near instance-optimal (up to a factor of the horizon). On the technical side, our analysis is very simple thanks to a new “target trick” of independent interest. We complement these findings with a novel hardness result explaining why the instance-dependent complexity of PAC RL cannot be easily related to that of regret minimization, unlike in the minimax regime.

Collecting and leveraging data with good coverage properties plays a crucial role in different aspects of reinforcement learning (RL), including reward-free exploration and offline learning. However, the notion of “good coverage” really depends on the application at hand, as data suitable for one context may not be so for another. In this paper, we formalize the problem of \emph{active coverage} in episodic Markov decision processes (MDPs), where the goal is to interact with the environment so as to fulfill given sampling requirements. This framework is sufficiently flexible to specify any desired coverage property, making it applicable to any problem that involves online exploration. Our main contribution is an \emph{instance-dependent} lower bound on the sample complexity of active coverage and a simple game-theoretic algorithm, CovGame, that nearly matches it. We then show that CovGame can be used as a building block to solve different PAC RL tasks. In particular, we obtain a simple algorithm for PAC reward-free exploration with an instance-dependent sample complexity that, in certain MDPs which are “easy to explore”, is lower than the minimax one. By further coupling this exploration algorithm with a new technique to do implicit eliminations in policy space, we obtain a computationally-efficient algorithm for best-policy identification whose instance-dependent sample complexity scales with gaps between policy values.

Best Arm Identification (BAI) problems are progressively used for data-sensitive applications, such as designing adaptive clinical trials, tuning hyper-parameters, and conducting user studies to name a few. Motivated by the data privacy concerns invoked by these applications, we study the problem of BAI with fixed confidence under $\epsilon$-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive a lower bound on the sample complexity of any $\delta$-correct BAI algorithm satisfying $\epsilon$-global DP. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget $\epsilon$. In the high-privacy regime (small $\epsilon$), the hardness depends on a coupled effect of privacy and a novel information-theoretic quantity, called the Total Variation Characteristic Time. In the low-privacy regime (large $\epsilon$), the sample complexity lower bound reduces to the classical non-private lower bound. Second, we propose AdaP-TT, an $\epsilon$-global DP variant of the Top Two algorithm. AdaP-TT runs in *arm-dependent adaptive episodes* and adds *Laplace noise* to ensure a good privacy-utility trade-off. We derive an asymptotic upper bound on the sample complexity of AdaP-TT that matches with the lower bound up to multiplicative constants in the high-privacy regime. Finally, we provide an experimental analysis of AdaP-TT that validates our theoretical results.

In probably approximately correct (PAC) reinforcement learning (RL), an agent is required to identify an $\epsilon$-optimal policy with probability $1-\delta$. While minimax optimal algorithms exist for this problem, its instance-dependent complexity remains elusive in episodic Markov decision processes (MDPs). In this paper, we propose the first nearly matching (up to a horizon squared factor and logarithmic terms) upper and lower bounds on the sample complexity of PAC RL in deterministic episodic MDPs with finite state and action spaces. In particular, our bounds feature a new notion of sub-optimality gap for state-action pairs that we call the deterministic return gap. While our instance-dependent lower bound is written as a linear program, our algorithms are very simple and do not require solving such an optimization problem during learning. Their design and analyses employ novel ideas, including graph-theoretical concepts (minimum flows) and a new maximum-coverage exploration strategy.

We investigate the problem of best-policy identification in discounted Markov Decision Processes (MDPs) when the learner has access to a generative model. The objective is to devise a learning algorithm returning the best policy as early as possible. We first derive a problem-specific lower bound of the sample complexity satisfied by any learning algorithm. This lower bound corresponds to an optimal sample allocation that solves a non-convex program, and hence, is hard to exploit in the design of efficient algorithms. We then provide a simple and tight upper bound of the sample complexity lower bound, whose corresponding nearly-optimal sample allocation becomes explicit. The upper bound depends on specific functionals of the MDP such as the sub-optimality gaps and the variance of the next-state value function, and thus really captures the hardness of the MDP. Finally, we devise KLB-TS (KL Ball Track-and-Stop), an algorithm tracking this nearly-optimal allocation, and provide asymptotic guarantees for its sample complexity (both almost surely and in expectation). The advantages of KLB-TS against state-of-the-art algorithms are discussed and illustrated numerically.

We investigate the classical active pure exploration problem in Markov Decision Processes, where the agent sequentially selects actions and, from the resulting system trajectory, aims at identifying the best policy as fast as possible. We propose a problem-dependent lower bound on the average number of steps required before a correct answer can be given with probability at least $1-\delta$. We further provide the first algorithm with an instance-specific sample complexity in this setting. This algorithm addresses the general case of communicating MDPs; we also propose a variant with a reduced exploration rate (and hence faster convergence) under an additional ergodicity assumption. This work extends previous results relative to the \emph{generative setting}~\cite{pmlr-v139-marjani21a}, where the agent could at each step query the random outcome of any (state, action) pair. In contrast, we show here how to deal with the \emph{navigation constraints}, induced by the \emph{online setting}. Our analysis relies on an ergodic theorem for non-homogeneous Markov chains which we consider of wide interest in the analysis of Markov Decision Processes.