Gauss--Hermite Quadrature for Gaussian-Mixture Entropy with an Action-Space Hermite Surrogate

What the math gives to ML

The paper provides a deterministic way to evaluate the differential entropy of a Gaussian mixture by standardizing each component and applying tensor-product Gauss--Hermite quadrature, avoiding noisy Monte Carlo estimates. This is directly transferable to mixture-density networks, uncertainty-aware world models, and entropy-regularized policies, where mixture parameters depend on network outputs and entropy must be differentiated repeatedly. Its second contribution is an action-space Hermite polynomial surrogate: objective evaluations at a symmetric local stencil estimate curvature without requiring analytic or autodifferentiated derivatives, which is useful for expensive neural simulators or model-predictive policies. The strongest experiments are to replace Monte Carlo mixture-entropy losses with deterministic quadrature and to benchmark the Hermite stencil as a cheap optimizer for a learned continuous-action objective.