Global Convergence of Non-Convex Gradient Descent for Computing Matrix Squareroot

TL;DR

This paper proves the first global convergence and good rate of gradient descent for computing the square root of a positive definite matrix, a fundamental problem in numerical linear algebra with broad applications.

Contribution

It establishes the first global convergence result with a near-optimal rate for non-convex gradient descent in matrix square root computation, including robustness to errors.

Findings

Gradient descent finds an ε-approximate square root in O(α log(‖M-U₀²‖_F/ε)) iterations.

The convergence rate is robust to iteration errors.

The proof technique may extend to other non-convex optimization problems.

Abstract

While there has been a significant amount of work studying gradient descent techniques for non-convex optimization problems over the last few years, all existing results establish either local convergence with good rates or global convergence with highly suboptimal rates, for many problems of interest. In this paper, we take the first step in getting the best of both worlds -- establishing global convergence and obtaining a good rate of convergence for the problem of computing squareroot of a positive definite (PD) matrix, which is a widely studied problem in numerical linear algebra with applications in machine learning and statistics among others. Given a PD matrix $M$ and a PD starting point $U_0$, we show that gradient descent with appropriately chosen step-size finds an $\epsilon$-accurate squareroot of $M$ in $O(\alpha \log (\|M-U_0^2\|_F /\epsilon))$ iterations, where $\alpha = (\max\{\|U_0\|_2^2,\|M\|_2\} / \min \{\sigma_{\min}^2(U_0),\sigma_{\min}(M) \} )^{3/2}$. Our result is the first to establish global convergence for this problem and that it is robust to errors in each iteration. A key contribution of our work is the general proof technique which we believe should further excite research in understanding deterministic and stochastic variants of simple non-convex gradient descent algorithms with good global convergence rates for other problems in machine learning and numerical linear algebra.