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We develop a class of interacting particle systems for implementing a maximummarginal likelihood estimation (MMLE) procedure to estimate the parameters of alatent variable model. We achieve this by formulating a continuous-timeinteracting particle system which can be seen as a Langevin diffusion over anextended state space of parameters and latent variables. In particular, weprove that the parameter marginal of the stationary measure of this diffusionhas the form of a Gibbs measure where number of particles acts as the inversetemperature parameter in classical settings for global optimisation. Using aparticular rescaling, we then prove geometric ergodicity of this system andbound the discretisation error in a manner that is uniform in time and does notincrease with the number of particles. The discretisation results in analgorithm, termed Interacting Particle Langevin Algorithm (IPLA) which can beused for MMLE. We further prove nonasymptotic bounds for the optimisation errorof our estimator in terms of key parameters of the problem, and also extendthis result to the case of stochastic gradients covering practical scenarios.We provide numerical experiments to illustrate the empirical behaviour of ouralgorithm in the context of logistic regression with verifiable assumptions.Our setting provides a straightforward way to implement a diffusion-basedoptimisation routine compared to more classical approaches such as theExpectation Maximisation (EM) algorithm, and allows for especially explicitnonasymptotic bounds.

Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation