Primal-Dual RL

We study representation learning in low-rank Constrained Markov Decision Processes (CMDPs), where both value and reward functions are approximated by a set of unknown representation vectors, and the transition dynamics admit a low-rank factorization. The objective is to maximize expected cumulative rewards while complying with constraints on the expected cumulative utility. To achieve this, we propose MFRLC (Model-Free Representation Learning for Low-Rank CMDP), a model-free algorithm that uses a primal-dual scheme to effectively balance reward regret and constraint violations. To the best of our knowledge, MFRLC is the first representation-learning method for low-rank CMDPs that combines the Least-Squares Value Iteration with Upper Confidence Bound (LSVI-UCB) and primal-dual techniques. MFRLC further enhances value function estimation with bonuses for function approximation and learns the mapping from representations to value functions directly, avoiding explicit estimation of the transition kernel. We prove that, with high probability, MFRLC achieves $\widetilde O(K^{3/4})$ reward regret and cumulative constraint violation over $K$ deployment rounds, independent of the size of the state space. To our knowledge, these are the first sublinear regret and violation guarantees for CMDPs with unknown features.

Primal-dual methods enforce safety constraints in reinforcement learning through Lagrangian relaxation. For a fixed policy, the exact bounded multiplier response exhibits bang-bang switching at the constraint boundary. Practical gradient updates can oscillate through coupling with policy optimization. We propose Structural DuaL Adaptation (SDLA), which replaces this bang-bang response with a unique continuous response through current-cost centering, trajectory weighting, and quadratic regularization. We characterize activation thresholds and response sensitivity. Ternary search computes multipliers without a dual gradient stepsize. For entropy-regularized tabular policies, we bound equilibrium constraint violations alongside reward gaps. We establish convergence of delayed updates under mild assumptions. Neural PPO/SAC experiments on OmniSafe and HASARD demonstrate competitive rewards with improved constraint tracking. On HASARD, PPO-SDLA achieves higher reported mean reward than the human baseline in six of eighteen settings while meeting the respective mean-cost budgets, including four with lower mean cost. To the best of our knowledge, this is the first report of higher mean reward at lower mean cost than the human baseline across multiple HASARD settings.

We study reinforcement learning in constrained partially observable Markov decision processes (CPOMDPs), where an agent maximizes reward subject to a cost constraint using only incomplete information about the underlying state. The lack of state observability may require optimal policies to depend on arbitrarily long observation-action histories, making planning and learning challenging. We leverage recent advances in finite-memory approximations of POMDPs and represent finite history windows as states of a constrained MDP. We then develop a primal--dual algorithm that provides the first non-asymptotic guarantees that jointly bound average reward suboptimality and constraint violation in the original CPOMDP. A key challenge is that this finite-window approximation affects both reward values and the set of feasible policies. Under a filter-stability assumption, we derive bounds relating optimality and feasibility in the finite-state model to those in the original CPOMDP, with approximation errors that decay geometrically with the window length. Experiments in partially observable gridworlds demonstrate the effectiveness of our approach.

We identify and study a structural mechanism for Markovian nonconvex ADMM in reinforcement learning. Using finite discounted MDPs as a canonical proving ground, we show that the discounted Bellman resolvent $(I-γP_π)^{-1}$ can provide the multiplier stability that classical nonconvex ADMM analyses often obtain from a designated smooth block. Starting from this mechanism, we establish convergence under controlled Markov sampling and then under stochastic observations using an empirical Bellman surrogate that jointly represents the random residual and its Jacobian. Markov mixing, initialization drift, observation noise, and decaying bias enter as one operator perturbation, avoiding unbiased product and double sampling requirements. When the perturbations are square summable, the true KKT residual converges almost surely to zero. Under a finite conditional fourth moment condition, a companion iterate satisfies $\mathbb{E}[\widetilde G_{K+1}] \le A/T+(B/T)\sum_{k<T}m_k^{-1},$ which becomes $O(T^{-1}+T/N)$ for total Markov sample budget $N$, giving $O(ε^{-1})$ iteration complexity and $O(ε^{-2})$ sample complexity for squared KKT accuracy $ε$. Beyond stationarity, discounted occupancy coverage yields $J^\star-J(π)=O(\sqrt G)$ for direct tabular policies, so covered exact KKT points are globally optimal, while a statewise quadratic Bellman-improvement condition sharpens the relation to $O(G)$. Finally, nonlinear policy, projected Bellman, and explicit occupancy formulations exhibit the same chain of operator invertibility, dual representation, and multiplier stability. This supports discounted operator invertibility as a reusable structural principle for primal-dual reinforcement learning.

In safety-critical applications, deployment uses a single policy, whose performance and constraint satisfaction should hold directly rather than only for an average or mixture of training policies. This motivates last-iterate guarantees in constrained reinforcement learning. Recent progress has established such guarantees in exact-gradient or tabular online settings, yet scalable results for structured large-state problems remain open. We develop a general, statistically efficient framework for last-iterate convergence in structured Constrained MDPs (CMDPs). Our analysis separates contraction of the regularised primal-dual dynamics from actor approximation and statistical errors in policy evaluation under online exploration. This enables model-free on- and off-policy learning with structured function approximation: optimistic policy evaluation avoids explicit transition-model construction, while a compact parametric actor avoids maintaining mixtures or histories of past policies. We instantiate the framework for linear CMDPs and general function approximation, obtaining representation-dependent complexity and improved target-accuracy dependence over prior optimistic regularised primal-dual analyses. We further validate the stabilising effect predicted by our theory on a synthetic linear CMDP: the regularised method exhibits stable last-iterate behaviour, whereas its unregularised counterpart shows larger oscillations.

Safe reinforcement learning (Safe RL) aims to maximize expected return while satisfying safety constraints, typically modeled as Constrained Markov Decision Processes (CMDPs). While primal-dual methods scale well to deep RL, they often suffer from delayed constraint correction, leading to oscillatory behavior and prolonged safety violations. In this paper, we propose Constraint-Sensitive Policy Optimization (CSPO), a first-order primal-dual method that incorporates local constraint sensitivity into policy updates. CSPO augments the primal objective with a constraint-sensitive correction derived from the shortest signed distance to the safety boundary, enabling smarter recovery steps back to safety, compensating for delayed Lagrange multiplier updates, reducing oscillations near the boundary, and preserving the KKT solutions of the original constrained problem. Experiments on navigation and locomotion benchmarks demonstrate that CSPO achieves faster safety recovery and high reward preservation, resulting in higher constrained returns compared to state-of-the-art primal-dual and penalty-based methods

We study infinite-horizon average-reward constrained Markov decision processes (CMDPs) under the weakly communicating assumption. Our contributions are twofold. First, we establish strong duality for weakly communicating average-reward CMDPs over stationary policies with finite state and action spaces. Despite the absence of a linear programming formulation and the resulting nonconvexity under the weakly communicating setting, we show that strong duality still holds by carefully exploiting the geometric structure of the occupation measure set. Second, building on this result, we propose a primal--dual clipped value iteration algorithm for learning weakly communicating average-reward linear CMDPs. Our algorithm achieves regret and constraint violation bounds of $\widetilde{\mathcal{O}}(T^{2/3})$, improving upon the best known bounds, where $T$ denotes the number of interactions. Our approach extends clipped value iteration to the constrained setting and adapts it to a finite-horizon approximation, which stabilizes the dual variable and is crucial for achieving improved regret bounds. To analyze this, we develop a novel approach based on strong duality that enables the decomposition of the composite Lagrangian regret into separate bounds on regret and constraint violation.

We study safe reinforcement learning in finite-horizon linear mixture constrained Markov decision processes (CMDPs) with adversarial rewards under full-information feedback and an unknown transition kernel. We propose a primal-dual policy optimization algorithm that achieves regret and constraint violation bounds of $\widetilde{O}(\sqrt{d^2 H^3 K})$ under mild conditions, where $d$ is the feature dimension, $H$ is the horizon, and $K$ is the number of episodes. To the best of our knowledge, this is the first provably efficient algorithm for linear mixture CMDPs with adversarial rewards. In particular, our regret bound is near-optimal, matching the known minimax lower bound up to logarithmic factors. The key idea is to introduce a regularized dual update that enables a drift-based analysis. This step is essential, as strong duality-based analysis cannot be directly applied when reward functions change across episodes. In addition, we extend weighted ridge regression-based parameter estimation to the constrained setting, allowing us to construct tighter confidence intervals that are crucial for deriving the near-optimal regret bound.