Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation

TL;DR

This paper introduces the Interacting Particle Langevin Algorithm (IPLA), a novel diffusion-based method for maximum marginal likelihood estimation that offers theoretical guarantees and practical advantages over traditional algorithms like EM.

Contribution

The paper develops a new particle system formulation for MMLE, proves its ergodicity and error bounds, and introduces IPLA with nonasymptotic guarantees and practical stochastic gradient extensions.

Findings

IPLA converges with explicit nonasymptotic error bounds.

The system's stationary measure is a Gibbs distribution with inverse temperature.

Numerical experiments validate the algorithm's effectiveness in logistic regression.

Abstract

We develop a class of interacting particle systems for implementing a maximum marginal likelihood estimation (MMLE) procedure to estimate the parameters of a latent variable model. We achieve this by formulating a continuous-time interacting particle system which can be seen as a Langevin diffusion over an extended state space of parameters and latent variables. In particular, we prove that the parameter marginal of the stationary measure of this diffusion has the form of a Gibbs measure where number of particles acts as the inverse temperature parameter in classical settings for global optimisation. Using a particular rescaling, we then prove geometric ergodicity of this system and bound the discretisation error in a manner that is uniform in time and does not increase with the number of particles. The discretisation results in an algorithm, termed Interacting Particle Langevin Algorithm (IPLA) which can be used for MMLE. We further prove nonasymptotic bounds for the optimisation error of our estimator in terms of key parameters of the problem, and also extend this result to the case of stochastic gradients covering practical scenarios. We provide numerical experiments to illustrate the empirical behaviour of our algorithm in the context of logistic regression with verifiable assumptions. Our setting provides a straightforward way to implement a diffusion-based optimisation routine compared to more classical approaches such as the Expectation Maximisation (EM) algorithm, and allows for especially explicit nonasymptotic bounds.