Yifeng Chu, Maxim Raginsky (2025), arXiv preprint (v2 says “accepted to ALT 2026”) — ARXIV-2502.06709
Key takeaways
- For a centered Gaussian process on a finite index set, the expected softmax-weighted average g(β) = E[Σ X_t e^{βX_t}/Z] scales like σ·√(E D(ν_β‖ν_0)) (upper bound always, Sudakov-type lower bound at low temperature, matching for i.i.d.). The quenched free energy scales likewise with the Rényi-½ divergence. Both reduce to the classical √log|T| maximal inequalities as β→∞.
Seeded from the abstract and a skim, before the reading in NOTE-046. What follows is what the work says about itself.
Extremal behaviour of stochastic processes matters across probability, statistical physics, theoretical computer science and learning theory. The paper studies centered Gaussian processes on finite index sets and the expected values of their smoothed (“soft”) maxima. Upper and lower bounds are obtained by combining statistical-physics tools (the Gibbs variational principle and replica-symmetric representations of Gibbs averages) with Sudakov minoration from probability. The bounds are indexed by an inverse temperature β and recover the usual Gaussian maximal inequalities as β→∞. The Random Energy Model serves as an illustration.
Standing in the record
Filed from the survey of 2026-09-25 of work the anthology set aside as out of scope (tier C): 440 seconds of active reading over 2 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-046 is the close reading of 2026-09-25, and it placed the work: Active — a short, fully proved and correct finite-temperature refinement of the Gaussian maximal inequality and Sudakov minoration. It is worth reading for the replica-symmetric identity (Lemma 2) and the Gibbs-variational upper bound. It carries no ML claim of its own.
When seeded, before the reading in NOTE-046, the entry gave a priority for a deeper reading: medium — Owner time is modest (t=440, n=2). The skim captures the results, but whether it is relevant to the anthology depends on connecting it to softmax/attention analysis, which needs a targeted read of §3.
What the seeding asked a deeper reading to check (NOTE-046 is that reading):
- g(β) is exactly the softmax-weighted expectation used in attention and in Boltzmann/Gibbs policies with Gaussian logits. The √(KL from uniform) scaling is a candidate tool for bounding expected attention output or entropy-regularised selection. This is the “softmax analysis” relevance flagged in the triage. It is not ML work itself, so filing would need a concrete ML-facing claim to hang it on.
- Check the constants and the threshold β* in Theorem 1’s lower bound (how it depends on covariance), and whether the i.i.d. matching extends to weakly correlated logits.
Access when seeded: arXiv abs page (v1 10 Feb 2025, v2 15 Jan 2026, “17 pages; accepted to ALT 2026”) and full PDF of v2 read directly: abstract, §1 with its informal summary of results, §1.1 related work, section structure, and the §5 REM application. I did not find the ALT 2026 proceedings version and did not look for it. The survey’s title carries the suffix “(boundary)”. That is the survey’s marker, not part of the title, and the full title is “…with Finite Index Sets”.