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

Yifeng Chu, Maxim Raginsky

TL;DR

This paper considers centered Gaussian processes on finite index sets and investigates expected values of their smoothed, or ``soft,''maxima, values, and obtains upper and lower bounds for these expected values using a combination of ideas from statistical physics and probability theory.

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 $β> 0$ and reduce to the usual Gaussian maximal inequalities in the zero-temperature limit $β\to \infty$. 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.

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

TL;DR

This paper considers centered Gaussian processes on finite index sets and investigates expected values of their smoothed, or ``soft,''maxima, values, and obtains upper and lower bounds for these expected values using a combination of ideas from statistical physics and probability theory.

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 and reduce to the usual Gaussian maximal inequalities in the zero-temperature limit . 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.

Paper Structure

This paper contains 15 sections, 10 theorems, 68 equations.

Table of Contents

  1. Introduction and informal summary of results
  2. Related work
  3. Organization of the paper
  4. Preliminaries and auxiliary results
  5. Some notions from information theory
  6. A replica-symmetric representation of the quenched Gibbs average
  7. Upper and lower bounds on the quenched Gibbs average
  8. Upper bound via the Gibbs variational principle
  9. A low-temperature lower bound via Sudakov minoration
  10. A lower bound for i.i.d. processes
  11. Upper and lower bounds on the quenched free energy
  12. Upper bound
  13. A lower bound for i.i.d. processes
  14. A Sudakov-type lower bound without independence
  15. Application: the quenched pressure in the random energy model

Key Result

Lemma 1

The function $\beta \mapsto \|\nu_\beta\|^2_2 = \sum_{t \in T}\nu_\beta^2(t)$ is nondecreasing.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 11 more