Recently there has been significant theoretical progress on understanding the convergence and
generalization of gradient-based methods on nonconvex losses with overparameterized models.
Nevertheless, many aspects of optimization and generalization and in particular the critical
role of small random initialization are not fully understood. In this paper, we take a step
towards demystifying this role by proving that small random initialization followed by a few
iterations of gradient descent behaves akin to popular spectral methods. We also show that this
implicit spectral bias from small random initialization, which is provably more prominent for
overparameterized models, also puts the gradient descent iterations on a particular trajectory
towards solutions that are not only globally optimal but also generalize well. Concretely, we
focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural
nonconvex formulation. In this setting, we show that the trajectory of the gradient descent
iterations from small random initialization can be approximately decomposed into three phases:
(I) a spectral or alignment phase where we show that that the iterates have an implicit spectral
bias akin to spectral initialization allowing us to show that at the end of this phase the column
space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle
avoidance/refinement phase where we show that the trajectory of the gradient iterates moves
away from certain degenerate saddle points, and (III) a local refinement phase where we show
that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix.
Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization
schemes that may have implications for computational problems beyond low-rank reconstruction
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This content will become publicly available on July 1, 2024
Implicit Balancing and Regularization: Generalization and Convergence Guarantees for Overparameterized Asymmetric Matrix Sensing
Recently, there has been significant progress in understanding the convergence and generalization properties of gradient-based methods for training overparameterized learning models. However, many aspects including the role of small random initialization and how the various parameters of the model are coupled during gradient-based updates to facilitate good generalization, remain largely mysterious. A series of recent papers have begun to study this role for non-convex formulations of symmetric Positive Semi-Definite (PSD) matrix sensing problems which involve reconstructing a low-rank PSD matrix from a few linear measurements. The underlying symmetry/PSDness is crucial to existing convergence and generalization guarantees for this problem. In this paper, we study a general overparameterized low-rank matrix sensing problem where one wishes to reconstruct an asymmetric rectangular low-rank matrix from a few linear measurements. We prove that an overparameterized model trained via factorized gradient descent converges to the low-rank matrix generating the measurements. We show that in this setting, factorized gradient descent enjoys two implicit properties: (1) coupling of the trajectory of gradient descent where the factors are coupled in various ways throughout the gradient update trajectory and (2) an algorithmic regularization property where the iterates show a propensity towards low-rank models despite the overparameterized nature of the factorized model. These two implicit properties in turn allow us to show that the gradient descent trajectory from small random initialization moves towards solutions that are both globally optimal and generalize well.
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- NSF-PAR ID:
- 10483605
- Publisher / Repository:
- Proceedings of Thirty Sixth Conference on Learning Theory
- Date Published:
- Format(s):
- Medium: X
- Location:
- Banglore, India
- Sponsoring Org:
- National Science Foundation
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Funding: This work was supported by grants from the National Science Foundation, Office of Naval Research, Air Force Office of Scientific Research, and Army Research Office.
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