Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908623426289664 |
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| author | Du, Kai Ren, Zhenjie Suciu, Florin Wang, Songbo |
| author_facet | Du, Kai Ren, Zhenjie Suciu, Florin Wang, Songbo |
| contents | For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11428 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method Du, Kai Ren, Zhenjie Suciu, Florin Wang, Songbo Probability For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron. |
| title | Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method |
| topic | Probability |
| url | https://arxiv.org/abs/2311.11428 |