Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent (2005.08898v4)

Abstract: Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approaches can still be computationally expensive especially for ill-conditioned matrices: the convergence rate of gradient descent depends linearly on the condition number of the low-rank matrix, while the per-iteration cost of alternating minimization is often prohibitive for large matrices. The goal of this paper is to set forth a competitive algorithmic approach dubbed Scaled Gradient Descent (ScaledGD) which can be viewed as pre-conditioned or diagonally-scaled gradient descent, where the pre-conditioners are adaptive and iteration-varying with a minimal computational overhead. With tailored variants for low-rank matrix sensing, robust principal component analysis and matrix completion, we theoretically show that ScaledGD achieves the best of both worlds: it converges linearly at a rate independent of the condition number of the low-rank matrix similar as alternating minimization, while maintaining the low per-iteration cost of gradient descent. Our analysis is also applicable to general loss functions that are restricted strongly convex and smooth over low-rank matrices. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties over a wide range of low-rank matrix estimation tasks.

• The paper introduces Scaled Gradient Descent (ScaledGD), an algorithm that significantly accelerates ill-conditioned low-rank matrix estimation by achieving linear convergence independent of the matrix condition number.
• ScaledGD employs adaptive gradient scaling with minimal per-iteration overhead, demonstrating substantial computational savings over traditional methods, especially for matrices with high condition numbers.
• This research offers both theoretical insights into scaling strategies and practical benefits for applications involving large-scale data problems like image processing and machine learning.

Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent

The paper explores an algorithmic advancement in low-rank matrix estimation using scaled gradient descent (ScaledGD) and focuses on improving computational efficiency and convergence rates, particularly for ill-conditioned matrices. Low-rank matrix estimation is pivotal for several domains, including signal processing and machine learning, and traditionally involves factorizing a matrix into low-rank components and optimizing these using iterative methods.

Main Contribution and Methodology

A critical bottleneck in conventional methods, such as gradient descent and alternating minimization, is their sensitivity to the condition number of the matrix. For gradient descent, the convergence rate depends linearly on this condition number, while alternating minimization often incurs high iteration costs. The paper proposes ScaledGD, a preconditioned gradient descent variant that adaptively scales the gradients for each iteration with minimal computational overhead, aiming to balance linear convergence rates and low per-iteration costs effectively.

The paper outlines ScaledGD's theoretical underpinning, providing analytical evidence of its linear convergence rate independent of the matrix's condition number. The approach involves introducing a carefully designed metric for measuring iterative progress, accounting for preconditioners when evaluating distances between estimates and true matrix values. Among the contributions is the analysis of ScaledGD for matrix sensing, robust principal component analysis, and matrix completion, each with a tailored implementation and theoretical validation.

Numerical Results and Theoretical Insights

Numerical experiments conducted in the paper demonstrate that ScaledGD offers significant computational savings, particularly for matrices with high condition numbers, compared to traditional gradient descent. It effectively achieves the theoretical guarantees of fast convergence outlined in the paper. The key metrics such as per-iteration cost, sample complexity, and condition independence, validate the effectiveness of ScaledGD against traditional methods in the domain.

Implications and Future Directions

This innovation has both theoretical and practical implications. Theoretically, the paper enhances our understanding of scaling strategies in gradient-based methods, stimulating more interest in investigating new distance metrics that integrate scaling and regularization techniques. Practically, ScaledGD can provide computational efficiency in real-world applications like image processing and machine learning tasks involving large-scale data.

Future research could explore extending ScaledGD beyond low-rank matrices, applying the scaling framework to other domains such as tensor decomposition or exploring how such strategies can be robustified against noise and initialization sensitivity.

In conclusion, this paper contributes a significant advancement in nonconvex optimization strategies for low-rank matrix estimation, challenging traditional perceptions of gradient descent limitations, and laying ground for further exploration in scalable and efficient algorithmic solutions for high-dimensional data problems.