Stochastic Approximation Algorithms for Systems of Interacting Particles

Mohammad Reza Karimi Jaghargh, Ya-Ping Hsieh, Andreas Krause

Keywords: Stochastic Approximation, Mean-Field Dynamics, Dynamical Systems, Neural Networks, Sampling

Abstract:

Interacting particle systems have proven highly successful in various machine
learning tasks, including approximate Bayesian inference and neural network optimization. However, the analysis of these
systems often relies on the simplifying assumption of the \emph{mean-field} limit, where particle
numbers approach infinity and infinitesimal step sizes are used. In practice, discrete time steps,
finite particle numbers, and complex integration schemes are employed, creating a theoretical gap
between continuous-time and discrete-time processes. In this paper, we present a novel framework
that establishes a precise connection between these discrete-time schemes and their corresponding
mean-field limits in terms of convergence properties and asymptotic behavior. By adopting a dynamical system perspective, our framework seamlessly integrates various numerical schemes that are typically analyzed independently.
For example, our framework provides a unified treatment of optimizing an infinite-width two-layer neural network and sampling via Stein Variational Gradient descent, which were previously studied in isolation.

Submission Number: 11856