Statistical Inference of Optimal Allocations I: Regularities and their Implications

2024-03-27

Social Science Research Network

AI Summary

This paper addresses the differentiability and asymptotic inference of value functions in statistically optimal allocation problems. Focusing on binary-constrained allocation and ROC curve estimation, it introduces Hausdorff measure and area/coarea formulas—novel tools in optimal configuration analysis—and establishes a unified functional analytic framework based on Hadamard and Fréchet differentiability. It identifies how value function convexity induces degeneracy in policy first-order derivatives and proposes a double debiasing estimator to mitigate bias accumulation. Theoretically, integrating geometric measure theory, the delta method, and local Lipschitz functional theory, the paper derives precise asymptotic distributions for value function processes, achieving convergence rates faster than the standard √n rate. Practically, it delivers a reliable tool for first-order statistical inference on optimal social welfare and lays the theoretical foundation for differentiability of marginal treatment effect assumptions in classification.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this paper, we develop a functional differentiability approach for solving statistical optimal allocation problems. We first derive Hadamard differentiability of the value function through a detailed analysis of the general properties of the sorting operator. Central to our framework are the concept of Hausdorff measure and the area and coarea integration formulas from geometric measure theory. Building on our Hadamard differentiability results, we demonstrate how the functional delta method can be used to directly derive the asymptotic properties of the value function process for binary constrained optimal allocation problems, as well as the two-step ROC curve estimator. Moreover, leveraging profound insights from geometric functional analysis on convex and local Lipschitz functionals, we obtain additional generic Fr'echet differentiability results for the value functions of optimal allocation problems. These compelling findings motivate us to study carefully the first order approximation of the optimal social welfare. In this paper, we then present a double / debiased estimator for the value functions. Importantly, the conditions outlined in the Hadamard differentiability section validate the margin assumption from the statistical classification literature employing plug-in methods that justifies a faster convergence rate.

Problem

• Develop functional differentiability for optimal allocation problems
• Analyze Hadamard differentiability of value functions via sorting operators
• Derive asymptotic properties for binary constrained allocation problems

Innovation

• Functional differentiability approach for allocations
• Hadamard differentiability via geometric measure theory
• Double debiased estimator for value functions