Active Exploration via Autoregressive Generation of Missing Data

Decision-making from offline datasets typically warm-starts a policy or score model from fixed offline data and then refines it with limited online interaction. Offline data reduces uncertainty, but it does not remove the need for exploration; it changes what remains to be explored. We formalise this residual uncertainty by the conditional mutual information $I(χ;τ_{1:T}\mid\mathcal{D}_N)$ between a learning target $χ$ and the online trajectories after conditioning on the offline dataset. This view leads naturally to information-directed sampling (IDS), a family parameterised by $η\ge 0$ that selects actions by trading off instantaneous regret against information gain. We prove a generic offline-to-online Bayesian regret bound for IDS through a ratio certificate: any information-ratio bound satisfied by a reference Thompson-sampling policy over the same randomised policy class is inherited by IDS. In a known-dynamics Bayesian linear-reward model, the conditional mutual information has a log-determinant form, and vanilla IDS ($η=0$) satisfies $\widetilde O\!\left(Hd\min\left\{\sqrt T,\,T\sqrt{C^\dagger_{β,\mathrm{IDS}_0}(N,T)/N}\right\}\right),$ where the coverage coefficient is tied to the visitation distribution induced by vanilla IDS itself. We also identify a warm-start regime with a dominated but informative probe in which vanilla IDS selects the probe while Thompson sampling never does, giving a constant-factor Bayesian regret separation. Controlled bandit experiments and D4RL offline-to-online RL experiments validate this mechanism: IDS is most beneficial when offline data is informative but leaves biased or low-probability residual uncertainty that targeted online actions can resolve, a regime shared by offline RL, offline black-box optimization, and Bayesian optimization.

Autoregressive generation lies at the heart of the mechanism of large language models. It can be viewed as the repeated application of a next-token generator: starting from an input string (prompt), the generator is applied for $M$ steps, and the last generated token is taken as the final output. [Joshi et al., 2025] proposed a PAC model for studying the learnability of the input-output maps arising from this process. We develop an online analogue of this framework, focusing on the mistake bound of learning the final output induced by an unknown next-token generator. We distinguish between two forms of feedback. In the End-to-End model, after each round the learner observes only the final token produced after $M$ autoregressive steps. In the Chain-of-Thought model, the learner is additionally shown the entire $M$-step trajectory. Our goal is to understand how the optimal mistake bound depends on the generation horizon $M$, and to what extent observing intermediate tokens can reduce this dependence. Our main results show that the online theory of autoregressive learning exhibits a qualitative picture analogous to the statistical one found by [Hanneke et al., 2026], but with a different scale of dependence on the generation horizon. In the End-to-End model, we prove a taxonomy of possible mistake-bound growth rates in the generation horizon $M$: essentially any rate between constant and logarithmic can arise. We further show that this logarithmic ceiling is unavoidable. In the Chain-of-Thought model, we show that access to the full generated trajectory eliminates the dependence on $M$ altogether. We also analyze autoregressive linear threshold classes, and prove optimal mistake bounds, as well as a new lower bound for the statistical setting. Along the way, our results resolve several questions left open by [Joshi et al., 2025].

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.