Global Convergence of NonConvex Gradient Descent for Computing Matrix Squareroot
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Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:479488, 2017.
Abstract
While there has been a significant amount of work studying gradient descent techniques for nonconvex 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 semidefinite (PSD) matrix, which is a widely studied problem in numerical linear algebra with applications in machine learning and statistics among others. Given a PSD matrix M and a PSD starting point $U_0$, we show that gradient descent with appropriately chosen stepsize finds an epsilonaccurate squareroot of M in $O(α\log(MU_0_F^2 / ε))$ iterations, where $α= (\max{U_0^2, M} / \min{\sigma_min^2(U_0), \sigma_min(M)} )^3/2$. 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 nonconvex gradient descent algorithms with good global convergence rates for other problems in machine learning and numerical linear algebra.
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