Grid entropy in a "choose the best of D samples" model

TL;DR

This paper investigates the limiting behavior of empirical measures in a model where samples are chosen from a distribution using strategies, revealing that the set of limit points aligns with measures of finite grid entropy and characterizing its structure.

Contribution

It characterizes the set of limit points of empirical measures in a sample selection model, linking it to grid entropy and identifying extremal strategies.

Findings

Limit points coincide with measures of finite grid entropy.

Extreme points are given by a natural greedy strategy with zero grid entropy.

The set of limit points is the convex hull of measures with a Beta distribution density.

Abstract

Grid entropy is a deterministic quantity inherent to lattice models which captures the entropy of empirical measures along paths that converge weakly to a given target measure. In this paper, we study the limiting behaviour of empirical measures in a model consisting of repeatedly taking $D$ samples from a distribution and picking out one according to an omniscient "strategy." We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all past and future choices, and furthermore, that this set coincides with the set of measures with finite grid entropy. This set is convex and weakly compact; we characterize its extreme points as those given by a natural "greedy" deterministic strategy and we compute the grid entropy of said extreme points to be 0. This yields a description of the set of limit points of empirical measures as the closed convex hull of measures given by a density which is $D \cdot Beta(1,D)$ distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.