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.

翻译：我们证明了粒子梯度下降（PGD）(Kuntz et al., 2023)的非渐近误差界，该算法是最近提出的一种用于大型潜变量模型最大似然估计的方法，通过对自由能的梯度流进行离散化得到。首先，我们证明，对于满足同时推广对数Sobolev不等式（LSI）和Polyak-Łojasiewicz不等式（PŁI）条件的模型，该流指数收敛到自由能最小化集合。我们通过推广最优输运文献中已知的结果（即LSI蕴含Talagrand不等式）及其在优化文献中的对应结论（即PŁI蕴含所谓的二次增长条件），并将其应用于新设定来实现这一点。我们还推广了Bakry-Émery定理，并证明对于具有强凸对数似然的模型，LSI/PŁI的推广成立。对于此类模型，我们进一步控制了PGD的离散化误差，得到了非渐近误差界。尽管我们的研究动机源于PGD，但我们相信所推广的不等式和结果可能具有独立的研究价值。