Global Optimality for Constrained Exploration via Penalty Regularization
Authors: Florian Wolf, Ilyas Fatkhullin, Niao He
Organizations: The Computing & Mathematical Sciences Department, California Institute of Technology, Pasadena, CA. · Department of Computer Science, ETH Zurich, Switzerland. · 3ETH AI Center, ETH Zurich, Switzerland.
Abstract
Efficient exploration is a central problem in reinforcement learning and is often formalized as maximizing the entropy of the state-action occupancy measure. While unconstrained maximum-entropy exploration is relatively well understood, real-world exploration is often constrained by safety, resource, or imitation requirements. This constrained setting is particularly challenging because entropy maximization lacks additive structure, rendering Bellman-equation-based methods inapplicable. Moreover, scalable approaches require policy parameterization, inducing non-convexity in both the objective and the constraints. To our knowledge, the only prior model-free policy-gradient approach for this setting under general policy parameterization is due to Ying et al. (2025). Unfortunately, their guarantees are limited to weak regret and ergodic averages, which do not imply that the final output is a single deployable policy that is near-optimal and nearly feasible. In this work we take a different approach to this problem, and propose Policy Gradient Penalty (PGP) method, a single-loop policy-space method that enforces general convex occupancy-measure constraints via quadratic-penalty regularization. PGP constructs pseudo-rewards that yield gradient estimates of the penalized objective, subsequently exploiting the classical Policy Gradient Theorem. We further establish the regularity of the penalized objective, providing the smoothness properties needed to justify the convergence of PGP. Leveraging hidden convexity and strong duality, we then establish global last-iterate convergence guarantees, attaining an ε-optimal constrained entropy value with ε bounded constraint violation despite policy-induced non-convexity. We validate PGP through ablations on a grid-world benchmark and further demonstrate scalability on two challenging continuous-control tasks.
Safe reinforcement learning maximizes reward subject to safety constraints. For Constrained Markov Decision Processes, the linear-programming view over occupancy measures implies that whenever the constraint is active at optimality, the optimal policy lies exactly on the constraint boundary, yet standard gradient-based methods do not exploit this structure and often settle in the feasible interior. We introduce Boundary-Seeking Policy Gradient (BSPG), a first-order method whose update combines a tangential component that improves reward while preserving cost to first order with a signed, residual-driven normal component that regulates the policy toward the active boundary from either side; the combined direction admits an algebraic Lagrangian form with an induced coefficient and no learned dual variable. Under exact gradients and stated regularity conditions, the constraint residual converges to zero from either side with a finite-horizon O(1/T) bound, the tangential component is a reward-ascent direction on the boundary, and any convergent parameter sequence is stationary on the active constraint set, satisfying the KKT conditions when the limit is also a local maximizer over the feasible set. This complements existing analyses, which certify feasibility but do not characterize the constraint value at convergence. On a standard Safety-Gymnasium navigation task, BSPG attains higher reward while tracking the boundary more tightly than the compared baselines.
Wasserstein policy gradient (WPG) is a policy optimization method for reinforcement learning (RL) that exploits the optimal-transport geometry of action distributions. For the entropy-regularized RL objective, WPG evolves each state-conditional policy by transporting it along the action gradient of the soft Q-function together with a Langevin-type diffusion. Despite its appeal for continuous-control problems, its global convergence properties remain poorly understood. Standard Langevin analyses do not directly apply, because the RL objective depends on the policy through the Bellman recursion rather than through a static convex functional, and the Langevin drift is determined by the soft Q-function, whose regularity must be controlled along the policy iterates. In this paper, we develop a global convergence theory for WPG by exploiting the Bellman structure of entropy-regularized RL. We show that the role usually played by convexity can be replaced by a Bellman-based argument: the soft Bellman residual admits a statewise KL representation with respect to a Gibbs policy; Bellman contraction relates this residual to the global optimality gap; and a Bellman resolvent identity connects value improvement to relative Fisher information. Combined with a uniform log-Sobolev inequality (LSI) for the evolving Gibbs family, these ingredients yield a distributional Polyak--Łojasiewicz condition. We further establish the regularity and uniform bounds needed to control the discretization error, thereby obtaining geometric contraction up to a discretization bias. Conceptually, our analysis shows that although entropy-regularized RL is not convex in the usual flat sense, the Bellman recursion induces a favorable Polyak--Lojasiewicz-type (PL) geometry that supports global convergence of WPG.
In reinforcement learning (RL), agents benefit from exploration only because they repeatedly encounter similar states: trying different actions can improve performance or reduce uncertainty; without such retries, a greedy policy is optimal. We formalize this intuition with ReMax, an objective that evaluates a policy by the expected maximum return over M samples, where M is a positive integer, while accounting for return uncertainty. Optimizing this objective induces stochastic exploration as an emergent property, without explicit bonus terms. For efficient policy optimization, we derive a new policy-gradient formulation for ReMax and introduce ReMax PPO (RePPO), a PPO variant that optimizes ReMax while generalizing the discrete retry count M to a continuous parameter m>0, enabling fine-grained control of exploration. Empirically, RePPO promotes exploration, without any explicit exploration bonuses, on the MinAtar and Craftax benchmarks.
Soichiro Nishimori, Paavo Parmas, Sotetsu Koyamada +4