Nonparametric Variance-Penalized Actor-Critic: Statistical Inference for Risk-Sensitive Reinforcement Learning

Abstract

Variance penalization is a principled approach to risk-sensitive reinforcement learning (RL) that explicitly trades expected return for policy stability. Existing methods require a dedicated second critic to estimate return variance online, adding architectural complexity and compounding estimation error during learning. We propose a nonparametric variance-penalized actor-critic (VPAC) framework that replaces the variance critic with statistically grounded online estimators based on bootstrapping and random scaling, techniques drawn from the statistical inference literature for stochastic approximation. These estimators require no auxiliary network, maintain a single-critic architecture, and produce variance penalties that are bounded by construction, enabling clean convergence analysis. We establish almost-sure convergence for both a variance-penalized Q-learning algorithm and a two-timescale actor-critic variant via the ordinary differential equation (ODE) method, requiring only that variance estimates remain bounded rather than consistent. Empirically, we evaluate across discrete and continuous stochastic environments, demonstrating that the proposed methods match or exceed the variance reduction achieved by the existing dual-critic VPAC baseline while eliminating the overhead of a second critic. We further validate on a high-temperature superconductor (HTS) manufacturing case study, where VPAC-RS (Random Scaling) achieves a 74% reduction in steady-state critical current variability and a 63% reduction in episode return standard deviation, translating directly to improved yield consistency. Our results establish nonparametric statistical inference as a practical and theoretically sound alternative to auxiliary critics for risk-sensitive RL.