Accelerated Flow for Probability Distributions
This work addresses the problem of accelerating convergence in probability distribution optimization, which is incremental as it adapts existing vector-space methods to the distributional setting.
The paper extends accelerated gradient methods from vector spaces to probability distributions by formulating a mean-field optimal control problem, deriving Hamilton's equations for optimal gradient flow that achieves accelerated density transport, and provides a quantitative convergence rate estimate for displacement convex objectives.
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 (wibisono, et. al. 2016) 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.