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

Mixture policies theoretically offer greater flexibility than unimodal policies in continuous action reinforcement learning, but the practical benefits of this complexity remain elusive. Mixture policies are notably absent from most state-of-the-art algorithms, raising a fundamental question: Is the added representational overhead useful? We show that increased flexibility can theoretically enhance solution quality and entropy robustness. Yet standard algorithms like SAC do not leverage these advantages. A core issue is the lack of a low-variance reparameterization trick for mixtures, a luxury Gaussian policies enjoy. We propose a marginalized reparameterization (MRP) estimator to address this, proving it offers lower variance than the standard likelihood-ratio (LR) approach. Our experiments across Gym MuJoCo, DeepMind Control Suite, and MetaWorld show that MRP mixture policies significantly outperform their LR ones, and reach parity (sometimes better) with Gaussian counterparts. In addition, we do find several cases where MRP mixture policies exhibit clear empirical advantages. In this paper, we provide a clearer understanding of the trade-offs involved, elevating MRP mixture policies from theoretical curiosity to a practical tool.

The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the maximum mean discrepancy (MMD), that is, the worst-case integration error over a reproducing kernel Hilbert space (RKHS). These MMD minimization procedures come with strong experimental support, but comparatively less theoretical footing. In particular, apart from recent progress in distribution compression, little has been proved in favor of an improvement of MMD minimization over classical Monte Carlo quadrature when the RKHS is infinite-dimensional. In this paper, we study a joint probability distribution over quadrature nodes, a tailored Gibbs distribution, whose support intuitively tends to concentrate around MMD minimizers as a temperature parameter is decreased. Our main contribution is to prove that drawing integration nodes from our distribution does outperform i.i.d Monte Carlo. While our bounds on the worst-case integration error feature the same rate as i.i.d. Monte Carlo, we do obtain a tighter concentration inequality as the temperature parameter decreases. This means smaller confidence intervals as the number of quadrature nodes increases. While arguably a first step, our results demonstrate that the mathematical toolbox developed around Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. There remains the issue of sampling from our Gibbs distribution. In our numerical experiments, we demonstrate that a simple MCMC chain already yields approximate samples that lead to improved confidence intervals around the target integrals, as supported by our theoretical results.