A Measure-Theoretic Finite-Sample Theory for Adaptive-Data Fitted Q-Iteration

Fitted $Q$-iteration (FQI) is a standard regression-based method for optimal control in offline reinforcement learning, but its stability under function approximation often relies on Bellman completeness, which requires Bellman images of the fitted class to remain in the class. We study Kullback--Leibler (KL)-regularized, or soft, FQI relative to a fixed reference policy without this assumption. Our key insight is that soft control locally inherits the contraction of policy evaluation in a discounted-occupancy norm. At the soft-optimal fixed point, the linearization of the soft Bellman operator is exactly the Bellman operator for the soft-optimal policy, which contracts in its discounted-occupancy norm; projection in the same norm preserves this contraction. Standard soft FQI instead projects under the offline state-action distribution and need not preserve this property. Motivated by this observation, we propose \emph{occupancy-reweighted soft FQI}, which retains standard Bellman targets and least-squares updates while reweighting regressions by discounted-occupancy ratios induced by the current soft policy. Under $Q$-function realizability and local regularity, we establish local contraction and finite-sample convergence with estimated ratios, without Bellman completeness. We then use temperature annealing to convert the local result into global convergence from arbitrary initialization: sufficiently high temperature provides a globally contractive starting regime, while gradual cooling connects successive local contraction regions to any prescribed positive target temperature. Under an action-gap margin condition, switching at a fixed positive temperature to hard FQI with refreshed occupancy weights also yields population and finite-sample convergence to the unregularized optimum.

Distributionally robust reinforcement learning seeks policies that remain effective when the deployment environment differs from the one that generated the training data. We study model-free robust Q-learning with $χ^2$ uncertainty sets and linear function approximation, using data from a single trajectory of an unknown nominal MDP. Evaluating the $χ^2$ robust Bellman target introduces the square root of a conditional second moment, which cannot be estimated unbiasedly from one transition, while the projected robust Bellman operator need not be contractive. We address these obstacles through a variational reformulation of the robust Bellman target and a blockwise frozen-target scheme, and establish a finite-time error bound relative to the optimal robust Q-function for every $γ\in(0,1)$. A neural-network experiment illustrates how the variational target can be used in a continuous-state nonlinear-control task.

We present a novel theoretical framework, Q-MMR, for off-policy evaluation in finite-horizon MDPs. Q-MMR learns a set of scalar weights, one for each data point, such that the reweighted rewards approximate the expected return under the target policy. The weights are learned inductively in a top-down manner via a moment matching objective against a value-function discriminator class. Notably, and perhaps surprisingly, a data-dependent finite-sample guarantee for general function approximation can be established under only the realizability of $Q^π$, with a dimension-free bound -- that is, the error does not depend on the statistical complexity of the function class. We also establish connections to several existing methods, such as importance sampling and linear FQE. Further theoretical analyses shed new light on the nature of coverage, a concept of fundamental importance to offline RL.