# Stochastic Approximation

Convergence analyses of synchronous Q-learning usually keep only the Gaussian limit, and central-limit normalization removes the smaller-scale displacement produced by the initial nonlinear evolution. This analysis splits the last-iterate error into an optimal-policy temporal-difference component and a nonnegative displacement caused by maximization, proves the latter converges almost surely to a random limit at the n^-1/4 scale, and uses an Edgeworth expansion to give corrections to the Gaussian approximation. In a two-state example, the first-order corrected density tracks the empirical distribution better than the central-limit Gaussian.

Convergence analyses of synchronous Q-learning usually keep only the Gaussian limit, and central-limit normalization removes the smaller-scale displacement produced by the initial nonlinear evolution. This analysis splits the last-iterate error into an optimal-policy temporal-difference component and a nonnegative displacement caused by maximization, proves the latter converges almost surely to a random limit at the n^-1/4 scale, and uses an Edgeworth expansion to give corrections to the Gaussian approximation. In a two-state example, the first-order corrected density tracks the empirical distribution better than the central-limit Gaussian.

Standard cryo-EM pipelines first reconstruct a 3D density from 2D projections and only then fit an atomic model, so atomic-coordinate priors never reach the most under-determined stage. EB-FM trains a flow-matching prior inside a Monte Carlo stochastic-approximation expectation-maximization loop, repeatedly inverting the nonlinear imaging forward model during training, so no clean structures are needed for pretraining. On simulated cryo-EM data with known poses it recovers conformational distributions directly in atomic-coordinate space for four proteins, though predicted motion amplitudes stay below the molecular-dynamics ground truth.

Standard cryo-EM pipelines first reconstruct a 3D density from 2D projections and only then fit an atomic model, so atomic-coordinate priors never reach the most under-determined stage. EB-FM trains a flow-matching prior inside a Monte Carlo stochastic-approximation expectation-maximization loop, repeatedly inverting the nonlinear imaging forward model during training, so no clean structures are needed for pretraining. On simulated cryo-EM data with known poses it recovers conformational distributions directly in atomic-coordinate space for four proteins, though predicted motion amplitudes stay below the molecular-dynamics ground truth.