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

arXiv: https://arxiv.org/abs/2606.26009 · HTML: https://mlsanity.com/paper/2606.26009/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Sub-sampled Cubic Regularization for Non-convex Optimization](https://mlsanity.com/paper/1705.05933/) (2017)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [Sampling-free Inference for Ab-Initio Potential Energy Surface Networks](https://mlsanity.com/paper/2205.14962/) (2022)
- [Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation](https://mlsanity.com/paper/2112.03491/) (2021)
- [Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions](https://mlsanity.com/paper/2307.09337/) (2023)
- [Accurate Ab-initio Neural-network Solutions to Large-Scale Electronic Structure Problems](https://mlsanity.com/paper/2504.06087/) (2025)
- [Towards a Foundation Model for Neural Network Wavefunctions](https://mlsanity.com/paper/2303.09949/) (2023)
- [Transferable Neural Wavefunctions for Solids](https://mlsanity.com/paper/2405.07599/) (2024)
- [Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators](https://mlsanity.com/paper/2303.10599/) (2023)
- [Functional Neural Wavefunction Optimization](https://mlsanity.com/paper/2507.10835/) (2025)
- [Convergence of variational Monte Carlo simulation and scale-invariant pre-training](https://mlsanity.com/paper/2303.11602/) (2023)
- [O(N2) Universal Antisymmetry in Fermionic Neural Networks](https://mlsanity.com/paper/2205.13205/) (2022)
- [Robust and Fast Training via Per-Sample Clipping](https://mlsanity.com/paper/2605.02701/) (2026)

## Abstract (arXiv)

Variational Monte Carlo (VMC) is a central algorithm in electronic structure theory and has gained renewed importance through modern neural-network ans"atze such as FermiNet. At its core, VMC seeks ground states by minimizing the Rayleigh quotient by stochastic optimization. In this work, we show that the resulting stochastic optimization problem is intrinsically governed by the nodal geometry of the underlying wave function. More precisely, we establish that properties of the nodal set determine the integrability of the local energy and gradient estimators that drive VMC. For broad and practically relevant ansatz classes, including Slater-Jastrow wave functions with variable-exponent Slater-type orbitals, we prove that these estimators are generically heavy-tailed and fail to admit higher moments. At the same time, for general analytic ans"atze, we prove weak moment bounds for the relevant estimators and identify precise low-moment regimes, showing how generic and degenerate nodal structures lead to different integrability thresholds. Building on this analysis, we introduce a new robust variant of VMC unicodex2013 coined PS-Clip-VMC unicodex2013 which is based on clipping both the local energies and the per-sample gradients. We prove that PS-Clip-VMC converges both in expectation and with high probability in the weak moment regime of VMC. The robustness of our method is confirmed experimentally by training FermiNet on atoms with up to 18 electrons.