Improved Bounds for Reward-Agnostic and Reward-Free Exploration

We study an unknown-transition finite-horizon Markov decision process (MDP) with a finite collection of known reward functions $\{r^1, r^2, \ldots, r^M\}$. The goal is to output an $ε$-optimal policy for every reward using online episodic interaction only. Performance is measured by the policy error $V_{0}^{*, m} - V_{0}^{\widehatπ^{m}, m}$ where $m\in [M]$ represents the reward function and $V_{0}^{*, m}=\mathbb{E}_{s_1\sim μ}[V_{1}^{*, m}(s_1)]$. Under this setting, we design a provably efficient algorithm to establish a minimax sample complexity bound of $O\left(\frac{SAH^3}{ε^2}\log M \mathrm{polylog}\left(\frac{SAH\log M}{\min\left\{ε, 1\right\}δ}\right)\right)$ episodes, with no additional burn-in cost. This matches the information-theoretic lower bound up to a factor of $\mathrm{polylog}(SAH\log M/(\min\left\{ε, 1\right\}δ))$. Our method combines three technical ingredients. First, we adapt MVP to reward-switching learning to construct optimistic value estimates. Second, we use fresh replay samples to conservatively evaluate the candidate policies. Third, gap-based multiplicative weights updates adjust the reward-sampling distribution using the differences between these estimates, converting weighted learning progress into simultaneous guarantees for all rewards.

Pure exploration in episodic Reinforcement Learning has primarily focused on Best Policy Identification (BPI), which seeks to identify a (near)-optimal policy with high confidence. Motivated by practical settings where a ``good enough'' policy suffices, we study an alternate objective of Good Policy Identification (GPI). For a given reward threshold $μ_0$, GPI only requires identifying a policy with expected reward in an episode at least $μ_0$ if such a policy exists (positive instance), or declaring None if no such policy exists (negative instance). We formalize GPI under the fixed-confidence setting. We require the output to be correct with probability $\geq 1-δ$, and seek to minimize the expected sample complexity, which is the expected number of episodes explored for the output. We propose a novel algorithm BEE-GPI, and derive theoretically-grounded upper bounds on its sample complexity for positive and negative instances. Notably, for positive instances, the coefficient of $\log 1/δ$ in our upper bound is $O(H^2/(V^* - μ_0)^2)$, where $H$ is the episode length and $V^*$ is the optimal expected reward in an episode. The coefficient does not depend on the action and state space sizes otherwise, in sharp contrast to the sample complexity in BPI. We further establish lower bound results to show the near-optimality of BEE-GPI and the necessity of the $1/(V^* -μ)^2$ term. Numerical experiments further validate the efficiency of our approach.

The exploration-exploitation dilemma in reinforcement learning (RL) is a fundamental challenge to efficient RL algorithms. Existing algorithms for finite state and action discounted RL problems address this by assuming sufficient exploration over both state and action spaces. However, this yields non-implementable algorithms and sub-optimal performance. To resolve these limitations, we introduce a new class of methods with auto-exploration, or methods that automatically explore both state and action spaces. Auto-exploration can be applied in both the tabular and linear function approximation setting. Under algorithm-independent assumptions on the existence of an exploring optimal policy, both settings attain $O(ε^{-2})$ sample complexity to solve to $ε$ error. These complexities are novel since they avoid algorithm-dependent parameters seen in prior works, which may be arbitrarily large. The methods are also simple to implement because they are parameter-free. We achieve these results by integrating auto-exploration into policy mirror descent to avoid the (unknown) stationary distribution seen in prior art. In the tabular setting, we introduce a dynamic exploration time with a data-driven stopping time, while for linear function approximation we propose a new sampling distribution based on the discounted visitation distribution that covers a more general class of Markov chains.