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Accelerated Flow for Probability Distributions
Amirhossein Taghvaei · Prashant Mehta

Thu Jun 13 10:15 AM -- 10:20 AM (PDT) @ Room 101
This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (Wibisono2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton's equations for the optimal gradient flow. The Hamilton's equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton's equations as a system of $N$ interacting particles. The continuous limit of the Nesterov's algorithm is shown to be a special case with $N=1$. The algorithm is illustrated with numerical examples and the performance is compared with the MCMC and Hamiltonian MCMC algorithms.

Author Information

Amir Taghvaei (University of Illinois at Urbana-Champaign)

Amirhossein Taghvaei was born and raised in southern beach of Caspian sea, Mazandaran, Iran. After obtaining two B.S degrees in Mechanical engineering and Physics from Sharif Univ. of Technology, Tehran, Iran, he joined University of Illinois at Urbana-Champaign as a PhD student in Mechanical Science and Engineering department. While pursuing PhD, he obtained his Masters degree in Mathematics, in May 2017. He is now in the last stage of his PhD program looking for opportunities to continue an academic path. He is a member of the Decision and Control group in Coordinated Science Laboratory advised by Prof. Prashant Mehta. His research interest lies mostly in the intersection of control theory and machine learning.

Prashant Mehta (University of Illinois at Urbana-CHampaign)

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