Talagrand Meets Talagrand: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes with Finite Index Sets

Yifeng Chu, Maxim Raginsky
Proceedings of The 37th International Conference on Algorithmic Learning Theory, PMLR 313:1-17, 2026.

Abstract

Analysis of extremal behavior of stochastic processes is a key ingredient in a wide variety of applications, including probability, statistical physics, theoretical computer science, and learning theory. In this paper, we consider centered Gaussian processes on finite index sets and investigate expected values of their smoothed, or “soft,” maxima. We obtain upper and lower bounds for these expected values using a combination of ideas from statistical physics (the Gibbs variational principle for the equilibrium free energy and replica-symmetric representations of Gibbs averages) and from probability theory (Sudakov minoration). These bounds are parametrized by an inverse temperature $\beta > 0$ and reduce to the usual Gaussian maximal inequalities in the zero-temperature limit $\beta \to \infty$. We provide an illustration of our methods in the context of the Random Energy Model, one of the simplest models of physical systems with random disorder.

Cite this Paper

BibTeX
@InProceedings{pmlr-v313-chu26a, title = {Talagrand Meets Talagrand: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes with Finite Index Sets}, author = {Chu, Yifeng and Raginsky, Maxim}, booktitle = {Proceedings of The 37th International Conference on Algorithmic Learning Theory}, pages = {1--17}, year = {2026}, editor = {Telgarsky, Matus and Ullman, Jonathan}, volume = {313}, series = {Proceedings of Machine Learning Research}, month = {23--26 Feb}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v313/main/assets/chu26a/chu26a.pdf}, url = {https://proceedings.mlr.press/v313/chu26a.html}, abstract = {Analysis of extremal behavior of stochastic processes is a key ingredient in a wide variety of applications, including probability, statistical physics, theoretical computer science, and learning theory. In this paper, we consider centered Gaussian processes on finite index sets and investigate expected values of their smoothed, or “soft,” maxima. We obtain upper and lower bounds for these expected values using a combination of ideas from statistical physics (the Gibbs variational principle for the equilibrium free energy and replica-symmetric representations of Gibbs averages) and from probability theory (Sudakov minoration). These bounds are parametrized by an inverse temperature $\beta > 0$ and reduce to the usual Gaussian maximal inequalities in the zero-temperature limit $\beta \to \infty$. We provide an illustration of our methods in the context of the Random Energy Model, one of the simplest models of physical systems with random disorder.} }
Endnote
%0 Conference Paper %T Talagrand Meets Talagrand: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes with Finite Index Sets %A Yifeng Chu %A Maxim Raginsky %B Proceedings of The 37th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Matus Telgarsky %E Jonathan Ullman %F pmlr-v313-chu26a %I PMLR %P 1--17 %U https://proceedings.mlr.press/v313/chu26a.html %V 313 %X Analysis of extremal behavior of stochastic processes is a key ingredient in a wide variety of applications, including probability, statistical physics, theoretical computer science, and learning theory. In this paper, we consider centered Gaussian processes on finite index sets and investigate expected values of their smoothed, or “soft,” maxima. We obtain upper and lower bounds for these expected values using a combination of ideas from statistical physics (the Gibbs variational principle for the equilibrium free energy and replica-symmetric representations of Gibbs averages) and from probability theory (Sudakov minoration). These bounds are parametrized by an inverse temperature $\beta > 0$ and reduce to the usual Gaussian maximal inequalities in the zero-temperature limit $\beta \to \infty$. We provide an illustration of our methods in the context of the Random Energy Model, one of the simplest models of physical systems with random disorder.
APA
Chu, Y. & Raginsky, M.. (2026). Talagrand Meets Talagrand: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes with Finite Index Sets. Proceedings of The 37th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 313:1-17 Available from https://proceedings.mlr.press/v313/chu26a.html.

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