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Abstract We consider the problem of estimating the factors of a rank-$$1$$ matrix with i.i.d. Gaussian, rank-$$1$$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this non-convex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic one-step recursion that is accurate even in high-dimensional problems. Notably, while the infinite-sample population update is uninformative and suggests exact recovery in a single step, the algorithm—and our deterministic one-step prediction—converges geometrically fast from a random initialization. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behaviour. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic one-step updates within fluctuations of the order $$n^{-1/2}$$ when each iteration is run with $$n$$ observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional $$M$$-estimation and provides an avenue for sharply analyzing complex iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.
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Hyperparameter tuning via trajectory predictions: stochastic prox-linear methods in matrix sensing
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 canonical 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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- PAR ID:
- 10636541
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematical Programming
- ISSN:
- 0025-5610
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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