Relaxing Probabilistic Latent Variable Models' Specification via Infinite-Horizon Optimal Control

Abstract

In this paper, we address the issue of model specification in probabilistic latent variable models (PLVMs) using an infinite-horizon optimal control approach. Traditional PLVMs rely on joint distributions to model complex data, but introducing latent variables results in an ill-posed parameter learning problem. To address this issue, regularization terms are typically introduced, leading to the development of the expectation-maximization (EM) algorithm, where the latent variable distribution is restricted to a predefined normalized distribution family to facilitate the expectation step. To overcome this limitation, we propose representing the latent variable distribution as a finite set of instances perturbed via an ordinary differential equation with a control policy. This approach ensures that the instances asymptotically converge to the true latent variable distribution as time approaches infinity. By doing so, we reformulate the distribution inference problem as an optimal control policy determination problem, relaxing the model specification to an infinite-horizon path space. Building on this formulation, we derive the corresponding optimal control policy using the Pontryagin's maximum principle and provide a closed-form expression for its implementation using the reproducing kernel Hilbert space. After that, we develop a novel, convergence-guaranteed EM algorithm for PLVMs based on this infinite-horizon-optimal-control-based inference strategy. Finally, extensive experiments are conducted to validate the effectiveness and superiority of the proposed approach.