Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures

Zhiyuan Yu, Jingbo Liu
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:4519-4527, 2026.

Abstract

Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical prior at all noise levels ($\Delta>0$), extending the result of Qiu and Sen (2022) in the high-noise regime. Our argument constructs a ridge regression functional that dominates the TAP free energy, yielding the first rigorous analysis of the global optimizer of the non-concave TAP functional for a planted inference model at an arbitrary noise level. This verifies that TAP, rather than mean-field, is the correct variational description in this setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-yu26d, title = { Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures }, author = {Yu, Zhiyuan and Liu, Jingbo}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {4519--4527}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/yu26d/yu26d.pdf}, url = {https://proceedings.mlr.press/v300/yu26d.html}, abstract = { Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical prior at all noise levels ($\Delta>0$), extending the result of Qiu and Sen (2022) in the high-noise regime. Our argument constructs a ridge regression functional that dominates the TAP free energy, yielding the first rigorous analysis of the global optimizer of the non-concave TAP functional for a planted inference model at an arbitrary noise level. This verifies that TAP, rather than mean-field, is the correct variational description in this setting. } }
Endnote
%0 Conference Paper %T Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures %A Zhiyuan Yu %A Jingbo Liu %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-yu26d %I PMLR %P 4519--4527 %U https://proceedings.mlr.press/v300/yu26d.html %V 300 %X Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical prior at all noise levels ($\Delta>0$), extending the result of Qiu and Sen (2022) in the high-noise regime. Our argument constructs a ridge regression functional that dominates the TAP free energy, yielding the first rigorous analysis of the global optimizer of the non-concave TAP functional for a planted inference model at an arbitrary noise level. This verifies that TAP, rather than mean-field, is the correct variational description in this setting.
APA
Yu, Z. & Liu, J.. (2026). Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:4519-4527 Available from https://proceedings.mlr.press/v300/yu26d.html.

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