Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution (1711.10467v3)

Abstract: Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e. phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control --- measured entrywise and by the spectral norm --- which might be of independent interest.

• The paper demonstrates that gradient descent inherently enforces incoherence constraints, leading to linear convergence in nonconvex statistical estimation.
• It employs a leave-one-out perturbation technique to control statistical dependencies and achieve near-optimal sample complexity in phase retrieval and related tasks.
• Theoretical insights are validated by numerical experiments that confirm faster convergence and enhanced computational efficiency compared to traditional methods.

Implicit Regularization in Nonconvex Statistical Estimation: Insights from Gradient Descent

This paper scrutinizes the phenomenon of implicit regularization in nonconvex statistical estimation tasks, specifically using gradient descent without explicit regularization mechanisms. The investigation focuses on several key applications: phase retrieval, matrix completion, and blind deconvolution. The authors spearhead an analytical framework revealing how gradient descent, despite the absence of explicit regularization, implicitly enforces favorable geometrical conditions that foster efficient convergence. This implicit regularization significantly enhances the computational efficiency of gradient descent beyond traditional expectations.

Key Findings

• Implicit Regularization Phenomenon: In nonconvex optimization, the need for explicit regularization, such as trimming or projection, is typically stressed to ensure controllable convergence. This paper challenges that norm by establishing that even vanilla gradient descent can inherently enforce incoherence constraints—termed implicit regularization. This arises naturally under specific statistical models, such as Gaussian designs and Bernoulli sampling, which are common in practical scenarios.
• Performance in Specific Tasks: The paper presents rigorous analysis demonstrating near-optimal sample complexity and computational efficiency in a variety of tasks. For phase retrieval, matrix completion, and blind deconvolution, gradient descent achieves substantial improvement in both iteration complexity and convergence speed. For instance, in phase retrieval, the step size becomes significantly more aggressive, reducing iteration complexity remarkably as opposed to prior conservative estimates.
• Theoretical Implications: The paper provides conditions under which the Hessian matrix satisfies restricted strong convexity and smoothness, promoting geometric conditions favorable for linear convergence. The authors utilize a leave-one-out perturbation technique to navigate the statistical dependencies inherent in the sampled datasets, proving that the iterates remain within a region that exhibits these beneficial geometric properties throughout the algorithm's execution.
• Numerical Verification: The empirical results bolster theoretical findings, showcasing linear convergence across all tasks tackled, effectively validating the implicit regularization hypothesis. The experiments delineate a stark improvement in convergence speed and demonstrate the retention of essential incoherence measures across iterations.

Methodological Insights

The approach hinges on a delicate leave-one-out technique, a probabilistic method that involves analyzing a variant of the optimization sequence with one sample omitted. This analytical trick mitigates dependencies between iterates and samples, facilitating control over the maximum allowable deviation of iterates and ensuring the satisfaction of incoherence constraints. This method is pivotal for linking statistical modeling with nonconvex optimization theory, affirming generalized utility in broader contexts beyond the three cases detailed.

Implications and Future Perspectives

• Algorithm Design: This work hints at the potential for designing gradient-based algorithms that harness statistical properties naturally without recourse to extensive tuning or regularization. Such methods are beneficial in modern machine learning, especially where model sizes and data scales reach formidable proportions.
• Generalized Application: While focusing on three canonical problems, the underlying principles of this investigation have broader implications, suggesting possible extensions to other sophisticated machine learning models, such as neural networks, where understanding of optimization dynamics is crucial.
• Continuous Research: Future research can explore understanding when and how implicit regularization manifests in other complex systems beyond the statistical regimes considered here. Exploring this phenomenon in stochastic settings or under adversarial conditions could yield insights relevant to robust machine learning practices.

This work contributes to both theoretical understanding and practical efficacy of gradient descent in nonconvex optimization, inviting further explorations into the nuances of implicit regularization across broader statistical and computational realms.