In this paper, we study the finite-sample statistical rates of distributional temporal difference (TD) learning with linear function approximation. The aim of distributional TD learning is to estimate the return distribution of a discounted Markov decision process for a given policy π. Previous works on statistical analysis of distributional TD learning mainly focus on the tabular case. In contrast, we first consider the linear function approximation setting and derive sharp finite-sample rates. Our theoretical results demonstrate that the sample complexity of linear distributional TD learning matches that of classic linear TD learning. This implies that, with linear function approximation, learning the full distribution of the return from streaming data is no more difficult than learning its expectation (value function). To derive tight sample complexity bounds, we conduct a fine-grained analysis of the linear-categorical Bellman equation and employ the exponential stability arguments for products of random matrices. Our results provide new insights into the statistical efficiency of distributional reinforcement learning algorithms.

This work presents the first finite-time analysis of average-reward Q-learning with an asynchronous implementation. A key feature of the algorithm we study is the use of adaptive stepsizes that act as local clocks for each state-action pair. We show that the mean-square error of this Q-learning algorithm, measured in the span seminorm, converges at a rate of O(1/k). To establish this result, we demonstrate that adaptive stepsizes are necessary: without them, the algorithm fails to converge to the correct target. Moreover, adaptive stepsizes can be viewed as a form of implicit importance sampling that counteracts the effect of asynchronous updates. Technically, the use of adaptive stepsizes causes each Q-learning update to depend on the full sample history, introducing strong correlations and making the algorithm a non-Markovian stochastic approximation (SA) scheme. Our approach to overcoming this challenge involves (1) a time-inhomogeneous Markovian reformulation of non-Markovian SA, and (2) a combination of almost-sure time-varying bounds, conditioning arguments, and Markov chain concentration inequalities to break the strong correlations between the adaptive stepsizes and the iterates.

We study value-iteration (VI) algorithms for solving general (a.k.a.

The fine-tuning of pre-trained large language models (LLMs) using reinforcement learning (RL) is generally formulated as direct policy optimization. This approach was naturally favored as it efficiently improves a pretrained LLM, seen as an initial policy. Another RL paradigm, Q-learning methods, has received far less attention in the LLM community while demonstrating major success in various non-LLMRL tasks. In particular, Q-learning effectiveness comes from its sample efficiency and ability to learn offline, which is particularly valuable given the high computational cost of sampling with LLM. However, naively applying a Q-learning-style update to the model's logits is ineffective due to the specificity of LLMs. Our core contribution is to derive theoretically grounded loss functions from Bellman equations to adapt Q-learning methods to LLMs. To do so, we carefully adapt insights from the RL literature to account for LLM-specific characteristics, ensuring that the logits become reliable Q-value estimates. We then use this loss to build a practical algorithm, ShiQfor Shifted-Q, that supports off-policy, token-wise learning while remaining simple to implement. Finally, we evaluate ShiQ on both synthetic data and real-world benchmarks, e.g., UltraFeedback, BFCL-V3, demonstrating its effectiveness in both single-turn and multi-turn LLM settings.

Expert demonstrations, such as those from car drivers, help navigate environments with unknown rewards, but are often collected in controlled settings, such as closed-course test tracks, while learned control policies must be deployed in new environments, such as city streets. We can imitate experts to perform well in the same source environment where demonstrations are observed, and we may even use inverse reinforcement learning (IRL) to improve on simple behavior cloning (Ng and Russell, 2000; Abbeel and Ng, 2004; Ziebart et al., 2008; Fu et al., 2018; Geng et al., 2020). But the target environment may have a different transition law, discount factor, or soft-control regularization. For this, IRL is crucial: we can learn a reward from demonstrations in the source environment and transfer it to the target environment, learning a policy that optimizes the same reward function in a new setting (Fu et al., 2018; Schlaginhaufen and Kamgarpour, 2024). In this paper, we characterize how well this transfer can be done and which approaches are preferable. In particular, we show the value in a coupled approach that takes the target environment into account even when learning from the source. In ordinary offline control, the Bellman equation uses a known reward, so the main statistical error comes from target transitions.

We propose a new algorithm for model-based distributional reinforcement learning (RL), and prove that it is minimax-optimal for approximating return distributions in the generative model regime (up to logarithmic factors), the first result of this kind for any distributional RL algorithm. Our analysis also provides new theoretical perspectives on categorical approaches to distributional RL, as well as introducing a new distributional Bellman equation, the stochastic categorical CDF Bellman equation, which we expect to be of independent interest. Finally, we provide an experimental study comparing a variety of model-based distributional RL algorithms, with several key takeaways for practitioners.

We conduct a novel theoretical analysis to demonstrate CODA's