Asymptotics of Multi-Scale McKean--Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach

TL;DR

This paper analyzes the small-noise asymptotics of multi-scale McKean--Vlasov diffusions with super-linear kernels, introducing a lifted semigroup approach and establishing large deviation principles with explicit convergence rates.

Contribution

It develops a novel lifted semigroup method and generalized Khasminskii scheme to analyze complex multi-scale McKean--Vlasov dynamics with super-linear interactions.

Findings

Established the functional law of large numbers for the system.

Proved a large deviation principle with explicit convergence rates.

Applied results to multi-scale models in machine learning and optimization.

Abstract

In this work, we establish the small-noise asymptotic behaviour (namely, the functional law of large numbers and the large deviation principle) for multi-scale McKean--Vlasov diffusions with super-linear kernels. In this setting, the interaction depends on the laws of both the slow component and the fast oscillating process. Consequently, the frozen (parameterized) system exhibits McKean--Vlasov dynamics, forming a nonlinear Markov process and thereby rendering the analysis more complex compared to existing works. We develop a lifted semigroup argument and employ a generalized Khasminskii time discretization scheme to derive the small-noise limit of the slow variable, providing explicit convergence rates. Furthermore, we introduce the notion of a lifted viable pair and utilize a generalized functional occupation measure approach to establish the Laplace principle, which is equivalent to the large deviation principle. The main results of this work find broad applications in multi-scale models arising in fields such as machine learning and optimization theory. In particular, our results can be employed to analyze the dynamics of multi-scale consensus-based methods for multilevel optimization, where the coefficients typically satisfy local Lipschitz continuity on the interaction kernels.