Gaussian-Mixture-Model Q-Functions for Policy Iteration in Reinforcement Learning

Unlike their conventional use as estimators of probability density functions in reinforcement learning (RL), this paper introduces a novel function-approximation role for Gaussian mixture models (GMMs) as direct surrogates for Q-function losses. The proposed estimators, termed GMM-QFs, possess substantial representational capacity, as they are shown to be universal approximators over a broad class of functions. They are further embedded within Bellman residuals, where their learnable parameters--a fixed number of mixing weights, together with Gaussian mean vectors and covariance matrices--are inferred from data via optimization on a product Riemannian manifold. This geometric perspective on the parameter space naturally introduces Riemannian optimization into the policy-evaluation step of standard policy-iteration (PI) frameworks. Moreover, rigorous theoretical analysis establishes performance bounds on Q-function estimation error under the proposed PI scheme. Supporting numerical tests show that GMM-QFs deliver competitive performance and, in some cases, outperform state-of-the-art approaches across a range of benchmark RL tasks, all while maintaining a significantly smaller computational footprint than deep-learning methods.