Theory for the neural-network self-test estimator

Establish rigorous convergence guarantees and finite-sample error bounds for the neural-network estimator that minimizes the trajectory-free self-test loss for learning the interaction potential Φ and external potential V from unlabeled snapshots, accounting for the non-convex optimization landscape and the implicit regularization induced by stochastic gradient methods.

Background

The paper provides nonasymptotic error bounds for parametric (basis-function) estimators minimizing the quadratic self-test loss, showing O((Δt)α + M{-1/2}) rates. However, the practical and flexible neural-network version of the estimator introduces non-convex optimization and implicit regularization effects that are not covered by the current analysis.

A theoretical treatment would require establishing identifiability, coercivity, and statistical concentration in the neural-network function class, together with optimization stability and generalization guarantees that reflect the implicit bias of first-order methods.

References

The nonparametric neural network estimator is more challenging to analyze due to the non-convexity of the loss landscape and the implicit regularization effects of the optimization, which we leave for future work.

— Learning interacting particle systems from unlabeled data  (2604.02581 - Wei et al., 2 Apr 2026) in Section 4 (Error bounds for the parametric estimator), opening paragraph

Since the convergence rate of our proposed estimator for the original interacting particle system essentially matches the rate suggested by the toy model, despite the substantially greater complexity of the original setting, we further conjecture that our estimator is minimax optimal over the considered class of functions, up to a \log\log(N) factor.

— Nonparametric Estimation of the Diffusion-Interaction Function in Particle Systems  (2609.24374 - Amorino et al., 21 Sep 2026) in Section 4, subsection “Understanding the convergence rate”