Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities
<|reference_start|>Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities: 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.<|reference_end|>
@article{caprio2024error,
    title={Error bounds for particle gradient descent, and extensions of the
  log-Sobolev and Talagrand inequalities},
    author={Rocco Caprio, Juan Kuntz, Samuel Power and Adam M. Johansen},
    journal={arXiv preprint arXiv:2403.02004},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.02004},
    primaryClass={cs.LG math.FA math.OC stat.CO stat.ML}
}