Estimation error analysis for variational committor methods with arbitrary parameterizations

Analyze the role of estimation error in committor estimation via the variational approach that minimizes V[v] = E_π[(v(X_τ) − v(X_0))^2] subject to boundary conditions v|_A = 0 and v|_B = 1, for arbitrary parameterizations of v (including nonlinear models such as neural networks). Establish how estimation error arises under finite sampling from the equilibrium measure and quantify its impact on the accuracy of variationally computed committors across system settings.

Background

The paper discusses a variational principle for computing committors by minimizing V[v] = E_π[(v(X_τ) − v(X_0))^2] with constraints v|_A=0 and v|_B=1, noting that this approach can be applied to arbitrary parameterizations such as neural networks and can reduce approximation error. However, because the expectation is taken over the equilibrium measure, the authors argue that estimation error may be large, particularly for systems with high-energy barriers where important transition states are undersampled.

They further show that when the committor is parameterized as a sum of indicator functions, minimizing the functional is equivalent to constructing an MSM and computing its committor from equilibrium samples, thereby directly connecting the variational approach to MSM estimation error. Based on this, the authors caution that while the variational approach may offer strong approximation properties, its estimation error characteristics remain to be analyzed in detail for general parameterizations, motivating this open problem.