Particles Don’t Care About Z: Towards Scaling Entropy Estimation of Unnormalized Densities

Computing the differential entropy of distributions known only up to a normalization constant is a fundamental challenge with broad theoretical and practical significance. While variational inference is highly scalable for density approximation from samples, its application to unnormalized densities remains under-explored due to the difficulty of constructing variational distributions that exploit the structure of the unnormalized density and are simultaneously expressive, tractable, efficiently samplable. Recently, Messaoud et al. (ICLR 24) introduced _P-SVGD_, a Stein variational inference method for this setting. We show, however, that \textit{P-SVGD} fails to scale to high dimensions due to incorrect invertibility assumptions, omission of a critical trace-of-Hessian term, and unstable divergence-control heuristics. We propose _MET-SVGD_, a principled extension of _P-SVGD_ that provides a general framework for stable _SVGD_ hyperparameter selection with global invertibility and convergence guarantees. Empirically, _MET-SVGD_ achieves up to $12\times$ and $16\times$ lower entropy estimation error than _P-SVGD_ and existing _SVGD_ baselines, respectively. On CIFAR-10 energy-based image generation, it improves FID by $80.4\%$ and yields $64\times$ higher training stability. In maximum-entropy reinforcement learning, it achieves up to $16\%$ higher returns than _P-SVGD_. Code is available at https://shorturl.at/fTG0G.

Many modern AI systems rely on estimating uncertainty and randomness in complex probability distributions, including applications in image generation, reinforcement learning, and scientific modeling. In many real-world problems, however, these distributions are only known approximately and cannot be computed exactly, making entropy estimation particularly difficult. Existing scalable methods often become unstable or inaccurate as the dimensionality of the problem grows. In this work, we identify several fundamental issues in a recent entropy estimation method called P-SVGD that prevent it from scaling reliably to high-dimensional problems. We then introduce MET-SVGD, a more stable and principled approach that fixes these issues while preserving computational efficiency. Our method provides stronger theoretical guarantees and significantly improves robustness in practice. Across a range of experiments, MET-SVGD substantially improves entropy estimation accuracy, produces more stable image generation models, and improves performance in reinforcement learning tasks. These results suggest that the proposed approach can make scalable probabilistic modeling more reliable for future AI systems.