Bellman Equation

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.

Fitted $Q$-evaluation (FQE) is a standard regression-based method for off-policy evaluation, but under distribution shift, value-function realizability alone does not ensure convergence, and existing analyses often require Bellman completeness. We trace this instability to a geometric mismatch: standard FQE projects Bellman targets in the norm induced by the offline distribution, which need not preserve Bellman contraction. We therefore study \emph{occupancy-weighted FQE}, which changes only the regression weights. Weighting by a target-policy discounted occupancy ratio aligns the projection norm with the target-policy dynamics and restores contraction of the population projected Bellman operator. We derive finite-sample guarantees with estimated occupancy ratios and function-class misspecification, separating finite-iteration, statistical, approximation, and ratio-estimation errors. Exact occupancy weighting removes the need for Bellman completeness; with estimated weights, approximate completeness and value-function realizability reduce sensitivity to ratio-estimation error, with exact realizability yielding higher-order dependence. Combining occupancy-weighted FQE with fitted occupancy-ratio evaluation gives an end-to-end guarantee governed by the complexities and direct approximation errors of the value-function and occupancy-ratio classes. Under coverage, joint realizability of these two classes suffices for consistent estimation without Bellman or critic-side completeness. Controlled experiments illustrate the projection-norm mechanism and the finite-sample tradeoff between contraction and coverage.

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 propose and study distributionally robust Markov games (DR-MGs) with the average-reward criterion as a crucial framework for multi-agent decision-making under model mismatches and over extended horizons. Under a standard irreducible assumption, we first derive a correspondence between the optimal policies and the solutions of the robust Bellman equation, based on which we further show the existence of a stationary Nash Equilibrium (NE) of the game. We further study DR-MGs under a more general weakly communicating setting. We construct a set-valued map based on the constant-gain optimal robust Bellman operator and show that its value is a subset of the best-response policies. We further prove that this map admits a fixed point, which implies the existence of NE. We then design two algorithms, Robust Nash-Iteration and robust TD Descent, with provably convergent guarantees. Finally, we show that the NE under average-reward can be approximated by the ones for the discounted DR-MGs as the discount factor approaches one. Our studies provide a comprehensive theoretical and algorithmic foundation for decision-making in complex, uncertain, and long-running multi-player environments.

We study model-free methods for distributionally robust infinite-horizon average-reward Markov decision processes (MDPs). We present non-asymptotic convergence analyses of Q-learning and actor-critic algorithms for robust average-reward MDPs under contamination, total-variation distance, and Wasserstein uncertainty sets. A key ingredient of our analysis is showing that the optimal robust Bellman operator is a strict contraction with respect to a carefully designed semi-norm. This property enables a stochastic approximation update that learns the optimal robust $Q$-function with $\tilde{\mathcal{O}}(ε^{-2})$ dependence on the target accuracy. We also establish robust TD convergence bounds whose constants are uniform over all stationary policies, yielding an efficient data-driven routine for robust critic estimation. Building on this, we introduce an actor-critic algorithm that learns an $ε$-optimal robust policy with $\tilde{\mathcal{O}}(ε^{-2})$ dependence on the target accuracy. We provide numerical simulations to illustrate the qualitative behavior of the proposed algorithms. Our results contribute to the theoretical foundations of robust planning under model misspecification and to model-free approaches for building robust long-run policies directly from simulation data.

Regularization is a cornerstone of modern reinforcement learning. Regularized policy iteration (RPI) provides a fundamental scheme for solving regularized Markov decision processes (RMDPs), and the widely used soft actor-critic algorithm arises as a special case when the regularizer is Shannon entropy. Despite its empirical success, the theoretical underpinnings of RPI remain unclear. In this paper, we address this gap by proving that RPI is formally equivalent to the standard Newton-Raphson method applied to the Bellman equation smoothed by strongly convex regularizers. This equivalence enables a unified convergence analysis of existing methods and supports the development of accelerated algorithms. We show that RPI enjoys local quadratic convergence; notably, for Shannon entropy, the guarantee is dimension-free. We further study RPI with inexact policy evaluation, establishing its equivalence to an inexact Newton method in which each Newton step is solved via truncated iterations, and derive an asymptotic linear convergence rate of $γ^{M}$, where $M$ denotes the number of operator steps used in policy evaluation. Finally, motivated by higher-order Newton schemes, we propose a new algorithm for RMDPs that achieves third-order local convergence. Numerical experiments corroborate our theory and demonstrate the practical advantages of the proposed algorithm. Overall, our results advance the theoretical understanding of regularization in reinforcement learning and suggest new directions for efficient algorithm design.

We present an optimization-based framework for robust permissive synthesis for Interval Markov Decision Processes (IMDPs). While robust IMDP controller synthesis typically yields a single policy and most permissive-synthesis methods assume exact transition models, we synthesize multi-strategies that retain multiple actions while guaranteeing satisfaction of probabilistic reachability or expected-reward specifications under all admissible transition probabilities. We formulate the problem as a mixed-integer linear program (MILP) that maximizes the number of enabled state--action pairs subject to robust Bellman constraints. We develop two encodings: a direct vertex-enumeration formulation and a dualization-based formulation that avoids explicit enumeration of uncertainty-polytope vertices and has size linear in the number of successor transitions. Experiments on four benchmark domains show that both encodings achieve the same optimal permissiveness and scale to IMDPs with hundreds of thousands of states. Compared with standard robust single-policy synthesis, the resulting multi-strategies retain substantially more action choices.