Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems

• URL: http://arxiv.org/abs/2112.14738v1
• Date: Wed, 29 Dec 2021 18:46:52 GMT
• Title: Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems
• Authors: Chris Junchi Li, Michael I. Jordan
• Abstract summary: Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
• Score: 98.34292831923335
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by the problem of online canonical correlation analysis, we propose the \emph{Stochastic Scaled-Gradient Descent} (SSGD) algorithm for minimizing the expectation of a stochastic function over a generic Riemannian manifold. SSGD generalizes the idea of projected stochastic gradient descent and allows the use of scaled stochastic gradients instead of stochastic gradients. In the special case of a spherical constraint, which arises in generalized eigenvector problems, we establish a nonasymptotic finite-sample bound of $\sqrt{1/T}$, and show that this rate is minimax optimal, up to a polylogarithmic factor of relevant parameters. On the asymptotic side, a novel trajectory-averaging argument allows us to achieve local asymptotic normality with a rate that matches that of Ruppert-Polyak-Juditsky averaging. We bring these ideas together in an application to online canonical correlation analysis, deriving, for the first time in the literature, an optimal one-time-scale algorithm with an explicit rate of local asymptotic convergence to normality. Numerical studies of canonical correlation analysis are also provided for synthetic data.

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