Is Variational Monte Carlo Robust? Sharp Moment Thresholds and Heavy-tailed Stochastic Optimization

Variational Monte Carlo (VMC) is a cornerstone of modern electronic structure theory, enabling the approximation of ground-state energies for complex many-particle systems. Despite its success with neural-network ansätze like FermiNet, the mathematical foundations of VMC optimization remain fragile. This paper demonstrates that the stochastic optimization problem at the heart of VMC is inherently governed by the nodal geometry of the wave function. Specifically, the authors prove that for standard Slater-Jastrow ansätze, the local energy and gradient estimators are generically heavy-tailed, meaning they lack the higher moments required for standard convergence guarantees.

The authors establish a direct link between the nodal set—the region where the wave function vanishes—and the integrability of the estimators used to drive the optimization. By analyzing the behavior of these estimators near nodal points, the study identifies precise thresholds for moment existence. While generic nodal structures allow for a limited number of moments, degenerate structures can cause these thresholds to drop, leading to severe instability in the stochastic gradient descent process. This explains why standard VMC training often encounters catastrophic spikes in energy or gradient variance.

To address these instabilities, the authors propose PS-Clip-VMC, a robust optimization algorithm that applies per-sample clipping to both the local energy and the gradient random variables. Unlike traditional methods that may only clip the local energy, this dual-clipping approach is shown to be necessary for convergence in the heavy-tailed regime. The authors provide rigorous proofs that PS-Clip-VMC converges both in expectation and with high probability. Numerical experiments on atoms with up to 18 electrons confirm that this method significantly improves training stability and performance compared to standard approaches, often achieving lower energies with fewer samples.