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arXiv · 2302.04753

Efficient displacement convex optimization with particle gradient descent

Hadi Daneshmand·Jason D. Lee·Chi Jin
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Abstract

Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient descent with a finite number of particles and establishes its theoretical guarantees to optimize functions that are \emph{displacement convex} in measures. Concretely, for Lipschitz displacement convex functions defined on probability over $\mathbb{R}^d$, we prove that $O(1/\epsilon^2)$ particles and $O(d/\epsilon^4)$ computations are sufficient to find the $\epsilon$-optimal solutions. We further provide improved complexity bounds for optimizing smooth displacement convex functions. We demonstrate the application of our results for function approximation with specific neural architectures with two-dimensional inputs.

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Hadi Daneshmand, Jason D. Lee, Chi Jin. 2023-02-09. Efficient displacement convex optimization with particle gradient descent. https://arxiv.org/abs/2302.04753

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