Abstract
We study an interacting particle system in Rd motivated by Stein variational gradient descent [Q. Liu and D. Wang, Proceedings of NIPS, 2016], a deterministic algorithm for approximating a given probability density with unknown normalization based on particles. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a nonlocal and nonlinear PDE. We also prove the global existence, uniqueness, and regularity of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in a long time limit.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 648-671 |
| Number of pages | 24 |
| Journal | SIAM Journal on Mathematical Analysis |
| Volume | 51 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2019 |
| Externally published | Yes |
Bibliographical note
Funding Information:\ast Received by the editors May 16, 2018; accepted for publication (in revised form) December 10, 2018; published electronically March 5, 2019. http://www.siam.org/journals/sima/51-2/M118761.html Funding: The work of the first author was supported in part by the National Science Foundation through grant DMS-1454939. The work of the third author was supported in part by the National Science Foundation through grant DMS-1351653. \dagger Department of Mathematics, Duke University, Durham NC 27708 ([email protected], [email protected], [email protected]).
Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics
Keywords
- Interacting particle system
- Mean field limit
- Sampling
- Stein variational gradient descent