Sharp predictions for mini-batched prox-linear iterations in rank one matrix sensing

We consider the problem of estimating the factors of a rank-1 matrix with i.i.d. Gaussian, rank-1 measurements that are corrupted by noise. We derive a deterministic, low dimensional recursion which predicts the error of a mini-batched prox-linear iteration for the nonconvex least squares loss associated to this statistical model. Our guarantees are fully non-asymptotic and are accurate for any batch size m ≥ 1 for a large range of step-sizes. Moreover, and in contrast with previous work, we show that the magnitude of the fluctuations of the empirical iterates around our deterministic predictions adapt to the problem error. Finally, we demonstrate how to use our deterministic predictions to perform hyperparameter tuning (e.g. step-size and minibatch size selection) without ever running the method.