Entropy Regularization Improves Policy Robustness in Continuous-Time Reinforcement Learning

Regularization-based methods have become a standard approach for training Deep Reinforcement Learning policies against adversarial input perturbations. In this paper, we unify these methods by deriving new upper bounds on the performance gap between the nominal and worst-case policies. Each upper bound is expressed as an existing regularization objective plus a KL-divergence penalty between the nominal and worst-case policies, which further explains why adding a KL penalty improves robustness in practice. Building on these bounds, we formulate robust training as a constrained optimization problem, showing that existing methods correspond to the special case of a fixed Lagrange multiplier. We instead update the multiplier jointly with the policy to automatically tune the regularization weight. Finally, we conduct extensive adversarial evaluations across several continuous control tasks to validate our theoretical analysis.

The framework of robust Markov decision processes (RMDPs) allows the design of reinforcement learning agents that satisfy performance guarantees under worst-case transition dynamics. Traditional RMDPs consider discrete-time dynamics and recently, sample-efficient policy gradient algorithms have been considered in this context. This paper investigates policy gradient algorithms within a continuous-time RMDP framework. Policy gradients and adversarial gradients are derived using pathwise and adjoint-based formulas for stochastic and ordinary differential equations. We propose double-loop optimisers to obtain linear convergence in the oracle-based setting and an $\tilde{\mathcal{O}}(\frac{1}{ε^2})$ sample complexity in the sample-based setting in an analysis which also derives novel tools for the framework of undiscounted total cost MDPs. Additionally, we propose mean-field optimisers as distributional optimisers with an $\tilde{\mathcal{O}}(\frac{1}{K})$ oracle-based convergence rate and an $\tilde{\mathcal{O}}(\frac{N^2}ε)$ sample complexity under $N$-particle approximation. The effectiveness of continuous-time policy gradient algorithms is confirmed for both optimisers on continuous-time RMDPs with neural ordinary differential equation dynamics.

Reinforcement learning (RL) achieves strong performance in sequential decision-making but remains brittle under dynamic uncertainty and distributional shifts. Robust Adversarial Reinforcement Learning (RARL) improves robustness via worst-case perturbations, but existing approaches frequently suffer from unstable optimization and degraded value estimation. In particular, overly aggressive adversaries can drive the agent toward uninformative failure states, while adversarial perturbations amplify disagreement between double critics and introduce biased value targets. We propose a unified framework, RACER (Risk-sensitive robust Adversarial critic ConsistEncy-regularized Reinforcement learning), that revisits adversarial RL from a risk-sensitive perspective. First, we introduce a state-dependent adversarial objective that adaptively regulates perturbation strength, suppressing harmful disturbances while preserving informative exploration. Second, we propose critic consistency regularization to reduce disagreement between Q-value estimators and stabilize learning. Comprehensive experiments on challenging continuous control benchmarks demonstrate that RACER consistently improves performance, robustness, and training stability over strong robust RL baselines.