Chu and Raginsky Bound Soft Maxima of Gaussian Processes

Bounding the extremes of a Gaussian process is a central problem in probability theory, machine learning, and statistical physics. A 56-page preprint by Yifeng Chu and Maxim Raginsky on arXiv, submitted June 21, 2026, addresses a smoothed version of that problem, deriving upper and lower bounds on the soft maxima of centered Gaussian processes with finite or countable index sets.

In the paper's framing, soft maxima are computed in terms of expected values of random Gibbs averages at inverse temperature β > 0, and they reduce to expected suprema in the zero-temperature limit as β approaches infinity. That setup lets the authors treat the hard supremum as a limiting special case and study how the bounds behave as the smoothing parameter varies.

The key structural result is that the bounds retain the same multiscale structure as expressions for the expected supremum derived using the method of generic chaining, a classical and sharp framework in Gaussian process theory. The difference in the smoothed setting is a truncation term governed by the inverse temperature β. That correspondence matters because it means the analytic machinery of generic chaining extends naturally into the smoothed regime, with a quantifiable error tied to how closely β approximates a hard maximum.

The authors draw on ideas from statistical physics and information theory, and rely in part on the tensorization technique introduced recently by Liu (2025). As a worked example, they apply the framework to the Sherrington-Kirkpatrick model and obtain a Parisi formula in the finite system size setting. The Parisi formula is typically established in the thermodynamic limit, so a finite-size version carries independent theoretical interest for researchers at the intersection of probability and statistical physics.

The honest caveat is that the paper applies to Gaussian processes with finite or countable index sets, and what the abstract does not address is how the bounds extend to the continuous-index processes common in Bayesian optimization and regression. Whether the gap between upper and lower bounds is tight enough to be useful for those settings, and at which values of β, is the open question left for follow-on work.