Online ICA: Understanding Global Dynamics of Nonconvex Optimization via Diffusion Processes
arXiv:1808.09642
Abstract
Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain less well understood in theory. In this paper, we propose a new analytic paradigm based on diffusion processes to characterize the global dynamics of nonconvex statistical optimization. As a concrete example, we study stochastic gradient descent (SGD) for the tensor decomposition formulation of independent component analysis. In particular, we cast different phases of SGD into diffusion processes, i.e., solutions to stochastic differential equations. Initialized from an unstable equilibrium, the global dynamics of SGD transit over three consecutive phases: (i) an unstable Ornstein-Uhlenbeck process slowly departing from the initialization, (ii) the solution to an ordinary differential equation, which quickly evolves towards the desirable local minimum, and (iii) a stable Ornstein-Uhlenbeck process oscillating around the desirable local minimum. Our proof techniques are based upon Stroock and Varadhan's weak convergence of Markov chains to diffusion processes, which are of independent interest.
Appeared in NIPS 2016
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- Asymptotic Analysis via Stochastic Differential Equations of Gradient Descent Algorithms in Statistical and Computational Paradigms
- Misspecified Nonconvex Statistical Optimization for Phase Retrieval
- Differential Equations for Modeling Asynchronous Algorithms
- Dropping Convexity for More Efficient and Scalable Online Multiview Learning
- Stochastic Approximation for Online Tensorial Independent Component Analysis