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# Online ICA: Understanding Global Dynamics of Nonconvex Optimization via Diffusion Processes

## Author(s): Li, CJ; Wang, Z; Liu, H

To refer to this page use: http://arks.princeton.edu/ark:/88435/pr1g862
 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. Publication Date: 2016 Citation: Li, Chris Junchi, Zhaoran Wang, and Han Liu. "Online ica: Understanding global dynamics of nonconvex optimization via diffusion processes." In Advances in Neural Information Processing Systems 29 (2016), pp. 4967-4975. ISSN: 1049-5258 Pages: 4967 - 4975 Type of Material: Conference Article Journal/Proceeding Title: Advances in Neural Information Processing Systems Version: Author's manuscript

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