[Paper Review] Dominating Points of Gaussian Extremes

Harsha Honnappa, Raghu Pasupathy+1 more|arXiv (Cornell University)|Oct 29, 2018

Probability and Risk ModelsDecision Sciences23 references3 citations

TL;DR

This paper establishes a restricted large deviation principle (rLDP) for the componentwise maximum of i.i.d. zero-mean Gaussian random vectors on closed convex sets in ℝᵈ. It identifies a unique dominating point 𝑥∗ on the boundary of the set 𝒞 that minimizes a convex quadratic program, with the decay rate of the tail probability being a simple quadratic function of 𝑥∗, enabling efficient rare-event simulation via importance sampling.

ABSTRACT

We quantify the large deviations of Gaussian extreme value statistics on closed convex sets in d-dimensional Euclidean space. The asymptotics imply that the extreme value distribution exhibits a rate function that is a simple quadratic function of a unique "dominating point" located on the boundary of the convex set. Furthermore, the dominating point is identified as the optimizer of a certain convex quadratic programming problem, indicating a "collusion" between the dependence structure of the Gaussian random vectors and the geometry of the convex set in determining the asymptotics. We specialize our main result to polyhedral sets which appear frequently in other contexts involving logarithmic asymptotics. We also extend the main result to characterize the large deviations of Gaussian-mixture extreme value statistics on general convex sets. Our results have implications to contexts arising in rare-event probability estimation and stochastic optimization, since the nature of the dominating point and the rate function suggest importance sampling measures.

Motivation & Objective

• To characterize the asymptotic decay rate of the probability that the scaled componentwise maximum of i.i.d. zero-mean Gaussian vectors lies in an atypical convex set 𝒞.
• To identify a unique dominating point 𝑥∗ on the boundary of 𝒞 that governs the large deviations behavior.
• To establish a restricted large deviation principle (rLDP) with a quadratic rate function derived from a convex quadratic programming problem.
• To extend the rLDP to Gaussian-mixture extreme value statistics on general convex sets.
• To provide a foundation for designing efficient importance sampling measures in rare-event simulation and stochastic optimization.

Proposed method

• Derives the rLDP for the scaled maximum 𝐀ₙ⁻¹𝑀ₙ by analyzing the logarithmic asymptotics of 𝔼[𝕀(𝐀ₙ⁻¹𝑀ₙ ∈ 𝒞)] as 𝑛 → ∞.
• Identifies the dominating point 𝑥∗ as the minimizer of a convex quadratic program involving the inverse covariance matrix and the geometry of the convex set 𝒞.
• Uses the principle of the largest term to analyze mixture models, focusing on the dominant component in the tail probability.
• Applies large deviation theory and the logarithmic asymptotics of Gaussian tail probabilities to derive the rate function.
• Establishes the rLDP for Gaussian mixtures by leveraging the rLDP for individual components and combining them via the principle of the largest term.
• Introduces a speed function 𝑣(‖𝐀ₙ‖) = ‖𝐀ₙ‖² and shows the limit of the normalized log-probability converges to −𝐽(𝑥∗), where 𝐽(𝑥∗) is a quadratic function of the dominating point.

Experimental results

Research questions

• RQ1 What is the asymptotic decay rate of the probability that the scaled componentwise maximum of i.i.d. zero-mean Gaussian vectors lies in a closed convex set 𝒞 with non-empty interior?
• RQ2 How is the dominating point 𝑥∗ on the boundary of 𝒞 characterized in terms of the dependence structure of the Gaussian vectors and the geometry of 𝒞?
• RQ3 Can the rLDP be extended to Gaussian-mixture extreme value statistics on general convex sets?
• RQ4 How does the dominating point 𝑥∗ inform the design of efficient importance sampling measures for rare-event estimation?
• RQ5 What is the role of the convex quadratic programming problem in determining the rate function for the large deviations of Gaussian extremes?

Key findings

• The asymptotic decay rate of 𝔼[𝕀(𝐀ₙ⁻¹𝑀ₙ ∈ 𝒞)] is governed by a quadratic rate function 𝐽(𝑥∗) = ½⟨𝑥∗, 𝐀⁻¹Σ⁻¹𝐀𝑥∗⟩, where 𝑥∗ is the unique dominating point on the boundary of 𝒞.
• The dominating point 𝑥∗ is the minimizer of the convex quadratic program min_{𝑥∈∂𝒞} ½⟨(𝑥−0), 𝐀⁻¹Σ⁻¹𝐀(𝑥−0)⟩, reflecting the interplay between Gaussian dependence and convex set geometry.
• For polyhedral convex sets, the rLDP reduces to a finite-dimensional optimization problem, enabling explicit computation of the rate function.
• The rLDP for Gaussian-mixture extremes is derived via the principle of the largest term, with the rate function being the minimum over components of the infimum of the quadratic form ⟨(𝑥−𝜇ⱼ), Σⱼ⁻¹(𝑥−𝜇ⱼ)⟩.
• The rate function for the Gaussian-mixture case is 𝐽(𝒞) = ½ − ½ ∧_{𝑗=1}^{𝐾} inf_{𝑥∈𝒞} ⟨(𝑥−𝜇ⱼ), 𝐀⁻¹Σⱼ⁻¹𝐀(𝑥−𝜇ⱼ)⟩, with speed 𝑣(𝑎) = 𝑎².
• The results provide a theoretical basis for constructing optimal importance sampling measures in rare-event simulation, where the mode of the IS distribution is centered at the dominating point 𝑥∗.