| Title: |
Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities |
| Authors: |
Caprio, Rocco; Kuntz, Juan; Power, Sam; Johansen, Adam |
| Source: |
Caprio, R, Kuntz, J, Power, S & Johansen, A 2025, 'Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities', Journal of Machine Learning Research, vol. 26, no. 103, pp. 1-38. < http://jmlr.org/papers/v26/24-0437.html > |
| Publication Year: |
2025 |
| Collection: |
University of Bristol: Bristol Reserach |
| Description: |
We derive 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. |
| Document Type: |
article in journal/newspaper |
| Language: |
English |
| Relation: |
info:eu-repo/semantics/altIdentifier/hdl/https://hdl.handle.net/1983/9bf15303-f913-4e06-8f1f-4aa69ef5c0b1 |
| Availability: |
https://hdl.handle.net/1983/9bf15303-f913-4e06-8f1f-4aa69ef5c0b1; https://research-information.bris.ac.uk/en/publications/9bf15303-f913-4e06-8f1f-4aa69ef5c0b1; http://jmlr.org/papers/v26/24-0437.html |
| Rights: |
info:eu-repo/semantics/openAccess ; http://creativecommons.org/licenses/by/4.0/ |
| Accession Number: |
edsbas.A1048584 |
| Database: |
BASE |