Accelerated Flow for Probability Distributions

Abstract

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 [23] 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.

Information Newton's flow: second-order optimization method in probability space

“…Remark 9. We can also approximate ∇ log ρ k via a kernel function by the blob method (Carrillo et al, 2019) and the diffusion map (Taghvaei and Mehta, 2019).…”

“…E.g., Bernton (2018); Wibisono (2019) apply the operator splitting technique to improve the unadjusted Langevin algorithm. Liu et al (2018); Taghvaei and Mehta (2019); Wang and Li (2019) study Nesterov's accelerated gradient methods in probability space.…”

Fokker-Planck particle systems for Bayesian inference: Computational approaches

“…See also [34] and [8] for earlier work on deterministic numerical methods for approximating diffusion processes. Furthermore, a diffusion map approach has been suggested in [38] that approximates the ∇ x ln π t term in the Fokker-Planck equation (2.17) without an explicit kernel density estimate for π t . While computationally attractive, it is unclear whether such an approximation leads to a particle system fitting the gradient flow structure (2.8).…”

Accelerated Information Gradient flow

“…Liu et al (2018Liu et al ( , 2019 propose an acceleration framework of ParVI methods based on manifold optimization. Taghvaei and Mehta (2019) introduce the accelerated flow from an optimal control perspective. On the other hand, Cheng et al (2017); Ma et al (2019) explore and analyzed the acceleration on MCMC, based on the underdamped Langevin dynamics.…”

“…Besides, we handle two difficulties in numerical implementations of AIG flows. On the one hand, as pointed out by Taghvaei and Mehta (2019); Liu et al (2019), the logarithm of density term (Wasserstein gradient of KL divergence) is hard to approximate in particle formulations. We propose a novel kernel selection method, whose bandwidth is learned by sampling from Brownian motions.…”