Juan Kuntz

Juan Kuntz, Jen Ning Lim, Adam M. Johansen

(Neal and Hinton, 1998) recast maximum likelihood estimation of any given latent variable model as the minimization of a free energy functional $F$, and the EM algorithm as coordinate descent applied to $F$. Here, we explore alternative ways to optimize the functional. In particular, we identify various gradient flows associated with $F$ and show that their limits coincide with $F$'s stationary points. By discretizing the flows, we obtain practical particle-based algorithms for maximum likelihood estimation in broad classes of latent variable models. The novel algorithms scale to high-dimensional settings and perform well in numerical experiments.

Rocco Caprio, Juan Kuntz, Samuel Power et al.

We prove non-asymptotic error bounds for particle gradient descent (PGD, Kuntz et al., 2023), a recently introduced algorithm for maximum likelihood estimation of large latent variable models obtained by discretizing a gradient flow of the free energy. We begin by showing that the flow converges exponentially fast to the free energy's minimizers for models satisfying a condition that generalizes both the log-Sobolev and the Polyak--Łojasiewicz inequalities (LSI and PŁI, respectively). We achieve this by extending a result well-known in the optimal transport literature (that the LSI implies the Talagrand inequality) and its counterpart in the optimization literature (that the PŁI implies the so-called quadratic growth condition), and applying the extension to our new setting. We also generalize the Bakry--Émery Theorem and show that the LSI/PŁI extension holds for models with strongly concave log-likelihoods. For such models, we further control PGD's discretization error and obtain the non-asymptotic error bounds. While we are motivated by the study of PGD, we believe that the inequalities and results we extend may be of independent interest.

Jen Ning Lim, Juan Kuntz, Samuel Power et al.

Maximum likelihood estimation (MLE) of latent variable models is often recast as the minimization of a free energy functional over an extended space of parameters and probability distributions. This perspective was recently combined with insights from optimal transport to obtain novel particle-based algorithms for fitting latent variable models to data. Drawing inspiration from prior works which interpret `momentum-enriched' optimization algorithms as discretizations of ordinary differential equations, we propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. The result is a dynamical system that blends elements of Nesterov's Accelerated Gradient method, the underdamped Langevin diffusion, and particle methods. Under suitable assumptions, we prove that the continuous-time system minimizes the functional. By discretizing the system, we obtain a practical algorithm for MLE in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.

Tim Y. J. Wang, Juan Kuntz, O. Deniz Akyildiz

We introduce a novel particle-based algorithm for end-to-end training of latent diffusion models. We reformulate the training task as minimizing a free energy functional and obtain a gradient flow that does so. By approximating the latter with a system of interacting particles, we obtain the algorithm, which we underpin theoretically by providing error guarantees. The novel algorithm compares favorably in experiments with previous particle-based methods and variational inference analogues.