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, for models
satisfying a condition generalizing both the log-Sobolev and the
Polyak--{\L}ojasiewicz inequalities (LSI and P{\L}I, respectively), the flow
converges exponentially fast to the set of minimizers of the free energy. 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 P{\L}I implies the so-called quadratic
growth condition), and applying it to our new setting. We also generalize the
Bakry--\'Emery Theorem and show that the LSI/P{\L}I generalization holds for
models with strongly concave log-likelihoods. For such models, we further
control PGD's discretization error, obtaining 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.