Least squares variational inference
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918146047213568 |
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| author | Fay, Yvann Le Chopin, Nicolas Barthelmé, Simon |
| author_facet | Fay, Yvann Le Chopin, Nicolas Barthelmé, Simon |
| contents | Variational inference consists in finding the best approximation of a target distribution within a certain family, where `best' means (typically) smallest Kullback-Leiber divergence. We show that, when the approximation family is exponential, the best approximation is the solution of a fixed-point equation. We introduce LSVI (Least-Squares Variational Inference), a Monte Carlo variant of the corresponding fixed-point recursion, where each iteration boils down to ordinary least squares regression and does not require computing gradients. We show that LSVI is equivalent to stochastic mirror descent; we use this insight to derive convergence guarantees. We introduce various ideas to improve LSVI further when the approximation family is Gaussian, leading to a $O(d^3)$ complexity in the dimension $d$ of the target in the full-covariance case, and a $O(d)$ complexity in the mean-field case. We show that LSVI outperforms state-of-the-art methods in a range of examples, while remaining gradient-free, that is, it does not require computing gradients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18475 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least squares variational inference Fay, Yvann Le Chopin, Nicolas Barthelmé, Simon Computation Variational inference consists in finding the best approximation of a target distribution within a certain family, where `best' means (typically) smallest Kullback-Leiber divergence. We show that, when the approximation family is exponential, the best approximation is the solution of a fixed-point equation. We introduce LSVI (Least-Squares Variational Inference), a Monte Carlo variant of the corresponding fixed-point recursion, where each iteration boils down to ordinary least squares regression and does not require computing gradients. We show that LSVI is equivalent to stochastic mirror descent; we use this insight to derive convergence guarantees. We introduce various ideas to improve LSVI further when the approximation family is Gaussian, leading to a $O(d^3)$ complexity in the dimension $d$ of the target in the full-covariance case, and a $O(d)$ complexity in the mean-field case. We show that LSVI outperforms state-of-the-art methods in a range of examples, while remaining gradient-free, that is, it does not require computing gradients. |
| title | Least squares variational inference |
| topic | Computation |
| url | https://arxiv.org/abs/2502.18475 |