Hyperparameter tuning via trajectory predictions: Stochastic prox-linear methods in matrix sensing

• URL: http://arxiv.org/abs/2402.01599v1
• Date: Fri, 2 Feb 2024 17:55:12 GMT
• Title: Hyperparameter tuning via trajectory predictions: Stochastic prox-linear methods in matrix sensing
• Authors: Mengqi Lou and Kabir Aladin Verchand and Ashwin Pananjady
• Abstract summary: We analyze a problem recovering an unknown rank-1 matrix from rank-1 Gaussian measurements corrupted by noise. We derive a deterministic recursion that predicts an accurate error of this method. We show, using a non-asymotic framework, that this is for any batch-size and a large range of step-sizes.
• Score: 6.446062819763263
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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