  Motivated by the desire to understand stochastic algorithms for nonconvex
optimization that are robust to their hyperparameter choices, we analyze a
mini-batched prox-linear iterative algorithm for the problem of recovering an
unknown rank-1 matrix from rank-1 Gaussian measurements corrupted by noise. We
derive a deterministic recursion that predicts the error of this method and
show, using a non-asymptotic framework, that this prediction is accurate for
any batch-size and a large range of step-sizes. In particular, our analysis
reveals that this method, though stochastic, converges linearly from a local
initialization with a fixed step-size to a statistical error floor. Our
analysis also exposes how the batch-size, step-size, and noise level affect the
(linear) convergence rate and the eventual statistical estimation error, and we
demonstrate how to use our deterministic predictions to perform hyperparameter
tuning (e.g. step-size and batch-size selection) without ever running the
method. On a technical level, our analysis is enabled in part by showing that
the fluctuations of the empirical iterates around our deterministic predictions
scale with the error of the previous iterate.
