[Paper Review] Sharpness of improved Fr\'echet-Hoeffding bounds: an optimal transport approach

Daniel Bartl, Michael Kupper+2 more|arXiv (Cornell University)|Sep 1, 2017

Markov Chains and Monte Carlo Methods18 references3 citations

TL;DR

This paper provides an optimal transport-based derivation of the improved Fréchet--Hoeffding upper bound for distributions with given marginals and prescribed values on a subset of R^d. By establishing strong duality for a constrained transport problem and identifying the improved bound as the dual optimizer of a relaxed problem, the authors prove its sharpness for the relaxed Fréchet class.

ABSTRACT

The improved Fr\'echet--Hoeffding bounds are ad-hoc estimates on the Fr\'echet class of probability distributions with given marginals and prescribed values on a subset of R d . Using an optimal transport approach, we first provide an alternative derivation of the improved upper bound. To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr\'echet class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem. The proof of strong duality is based on a general representation result for increasing convex functionals and the explicit computation of the conjugates. We show further that the improved upper Fr\'echet--Hoeffding bound is not the dual optimizer of this transport problem. In turn we prove that the improved upper bound is the dual optimizer of a relaxed version of the initial transport problem, thus proving sharpness of the improved upper Fr\'echet--Hoeffding bound for the relaxed Fr\'echet class. This is achieved by direct construction of the dual optimizers for a certain class of objective functions.

Motivation & Objective

• To re-derive the improved Fréchet--Hoeffding upper bound using optimal transport theory.
• To establish a dual representation for a constrained transport problem over the Fréchet class with fixed values on a subset of R^d.
• To prove that the improved upper bound is the dual optimizer of a relaxed version of the original transport problem.
• To demonstrate the sharpness of the improved bound within the relaxed Fréchet class.

Proposed method

• Formulate the problem as a constrained optimal transport problem over the Fréchet class with fixed values on a subset of R^d.
• Establish a dual representation of the transport problem using increasing convex functionals and conjugate duality.
• Apply a general representation result for increasing convex functionals to prove strong duality.
• Explicitly compute conjugates to verify admissibility of the improved upper Fréchet--Hoeffding bound in the dual problem.
• Construct dual optimizers for a specific class of objective functions to validate the optimality of the bound.
• Show that the improved bound is not the dual optimizer of the original problem but of a relaxed version, proving its sharpness in the relaxed setting.

Experimental results

Research questions

• RQ1 Can the improved Fréchet--Hoeffding upper bound be derived via an optimal transport framework?
• RQ2 Is there a dual representation for the constrained transport problem over the Fréchet class with fixed values on a subset of R^d?
• RQ3 Does strong duality hold for this constrained transport problem, and what is the role of conjugate duality in the proof?
• RQ4 Why is the improved upper bound not the dual optimizer of the original problem, and what is its role in a relaxed version?
• RQ5 Is the improved upper Fréchet--Hoeffding bound sharp for the relaxed Fréchet class?

Key findings

• The improved upper Fréchet--Hoeffding bound is shown to be admissible in the dual optimization problem, confirming its validity as a bound.
• Strong duality is established for the constrained transport problem using a general representation result for increasing convex functionals.
• The improved upper bound is not the dual optimizer of the original problem, indicating a structural distinction between the original and relaxed formulations.
• The improved upper bound is proven to be the dual optimizer of a relaxed version of the original transport problem.
• Sharpness of the improved bound is rigorously established for the relaxed Fréchet class through direct construction of dual optimizers.
• The proof relies on explicit computation of conjugates and demonstrates the optimality of the bound in the relaxed setting.