Winners with Confidence: Discrete Argmin Inference with an Application to Model Selection

TL;DR

This paper introduces a new asymptotic test for identifying the minimum index in noisy data, applicable in high-dimensional and tied scenarios, with practical tuning and demonstrated effectiveness.

Contribution

It develops a novel asymptotic normal test statistic for discrete argmin inference that handles high-dimensional data and ties, integrating cross-validation and differential privacy techniques.

Findings

Effective in high-dimensional settings

Handles ties in population means

Achieves favorable bias-variance trade-off

Abstract

We study the problem of finding the index of the minimum value of a vector from noisy observations. This problem is relevant in population/policy comparison, discrete maximum likelihood, and model selection. We develop an asymptotically normal test statistic, even in high-dimensional settings and with potentially many ties in the population mean vector, by integrating concepts and tools from cross-validation and differential privacy. The key technical ingredient is a central limit theorem for globally dependent data. We also propose practical ways to select the tuning parameter that adapts to the signal landscape. Numerical experiments and data examples demonstrate the ability of the proposed method to achieve a favorable bias-variance trade-off in practical scenarios.