  Recent theoretical work studies sample-efficient reinforcement learning (RL)
extensively in two settings: learning interactively in the environment (online
RL), or learning from an offline dataset (offline RL). However, existing
algorithms and theories for learning near-optimal policies in these two
settings are rather different and disconnected. Towards bridging this gap, this
paper initiates the theoretical study of policy finetuning, that is, online RL
where the learner has additional access to a "reference policy" $\mu$ close to
the optimal policy $\pi_\star$ in a certain sense. We consider the policy
finetuning problem in episodic Markov Decision Processes (MDPs) with $S$
states, $A$ actions, and horizon length $H$. We first design a sharp offline
reduction algorithm -- which simply executes $\mu$ and runs offline policy
optimization on the collected dataset -- that finds an $\varepsilon$
near-optimal policy within $\widetilde{O}(H^3SC^\star/\varepsilon^2)$ episodes,
where $C^\star$ is the single-policy concentrability coefficient between $\mu$
and $\pi_\star$. This offline result is the first that matches the sample
complexity lower bound in this setting, and resolves a recent open question in
offline RL. We then establish an $\Omega(H^3S\min\{C^\star, A\}/\varepsilon^2)$
sample complexity lower bound for any policy finetuning algorithm, including
those that can adaptively explore the environment. This implies that -- perhaps
surprisingly -- the optimal policy finetuning algorithm is either offline
reduction or a purely online RL algorithm that does not use $\mu$. Finally, we
design a new hybrid offline/online algorithm for policy finetuning that
achieves better sample complexity than both vanilla offline reduction and
purely online RL algorithms, in a relaxed setting where $\mu$ only satisfies
concentrability partially up to a certain time step.
