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 等，2023）的非渐近误差界。该算法通过离散化自由能的梯度流，最近被提出用于大规模潜变量模型的极大似然估计。我们首先证明，对于满足同时推广对数索博列夫不等式（LSI）和Polyak--Łojasiewicz不等式（PŁI）条件的模型，该梯度流以指数速度收敛到自由能的最小值集。为此，我们推广了最优传输文献中已知的结果（LSI蕴含塔拉格朗德不等式）及其在优化文献中的对应结论（PŁI蕴含所谓的二次增长条件），并将其应用于新场景。我们还推广了Bakry--Émery定理，表明对于具有强凹对数似然的模型，LSI/PŁI的推广成立。针对此类模型，我们进一步控制了PGD的离散化误差，从而获得非渐近误差界。尽管我们的研究动机源于对PGD的分析，但所推广的不等式与结论可能具有独立的研究价值。