Polyak–Łojasiewicz condition for NN‑VMC with FiRE

Establish whether the Polyak–Łojasiewicz (PL) condition holds for the energy optimization landscape of neural-network variational Monte Carlo using the finite-range embeddings (FiRE) wave function, to justify and theoretically explain the observed O(1/t) convergence rate of SGD-type optimization methods.

Background

The authors empirically observe approximately t{-1} convergence in optimization across multiple systems, which is faster than rates established in recent theoretical analyses of VMC under certain assumptions.

They note that optimal O(1/t) rates can be derived for SGD-type methods under a Polyak–Łojasiewicz condition, but explicitly state it is unclear whether this condition holds in their setting, leaving a theoretical gap between empirical observations and existing convergence guarantees.

References

While this rate would match our empirical findings, it is unclear if a Polyak-Lojasiewicz condition holds in our setting.

— Accurate Ab-initio Neural-network Solutions to Large-Scale Electronic Structure Problems (2504.06087 - Scherbela et al., 8 Apr 2025) in Appendix, Subsection Theoretical VMC convergence rates