[Paper Review] Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent

Tian Tong, Cong Ma+1 more|arXiv (Cornell University)|May 18, 2020

Sparse and Compressive Sensing Techniques39 citations

TL;DR

Introduces Scaled Gradient Descent (ScaledGD), a preconditioned factorized-gradient method that achieves linear convergence independent of the condition number for ill-conditioned low-rank matrix estimation tasks, with lightweight per-iteration cost.

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.

Motivation & Objective

• Motivate the need for faster, condition-number-insensitive convergence in low-rank matrix estimation under ill-conditioning.
• Propose ScaledGD as a diagonally scaled, adaptive preconditioned gradient method with low overhead.
• Show theoretical guarantees of linear convergence independent of the condition number across key problems (matrix sensing, robust PCA, matrix completion).
• Provide a unified analysis via a new distance metric that accounts for preconditioning.
• Demonstrate empirical acceleration over vanilla gradient descent across multiple tasks.

Proposed method

• Parametrize X as LR^T to obtain a rank-r factorization and optimize over L and R jointly.
• Apply a diagonal scaling to gradient steps, using (R^T R)^{-1} and (L^T L)^{-1} to form the scaled updates.
• Introduce a new distance metric that accounts for the preconditioners and the identifiability up to GL(r) actions.
• Instantiate ScaledGD for three tasks: matrix sensing, robust PCA, and matrix completion, with spectral initialization.
• Prove contraction of the ScaledGD iterates under the new distance metric, yielding linear convergence with rate independent of the condition number.
• Generalize the analysis to general loss functions that are restricted strongly convex and smooth over low-rank matrices.

Experimental results

Research questions

• RQ1 Can a first-order method with per-iteration costs similar to gradient descent achieve linear convergence that is independent of the condition number for ill-conditioned low-rank matrix estimation?
• RQ2 What preconditioning strategy yields fast, provable convergence across matrix sensing, robust PCA, and matrix completion?
• RQ3 How does the proposed distance metric capture the effect of preconditioning on convergence?
• RQ4 Does scalability hold under standard sampling/models (RIP, incoherence, etc.) while maintaining near-optimal sample complexity?

Key findings

• ScaledGD achieves linear convergence with a rate that does not depend on the condition number κ, under suitable initialization.
• For matrix sensing, ScaledGD converges in O(log(1/ε)) iterations with m = O(n r^2 κ^2) measurements and RIP with δ_{2r} ≤ 0.02/(√r κ).
• For robust PCA, ScaledGD with hard thresholding converges in O(log(1/ε)) iterations under α ≤ c/(μ r^{3/2} κ) corruption, assuming μ-incoherence.
• For matrix completion, ScaledGD attains ε-accuracy in O(log(1/ε)) iterations with sample complexity n1 n2 p ≳ (μ κ^2 ∨ log n) μ n r^2 κ^2.
• ScaledGD does not require regularizations to balance L and R or extra incoherence projections, unlike some prior methods.
• The framework extends to general loss functions that are restricted strongly convex and smooth over low-rank matrices.