# Locally minimax optimal confidence sets for the best model

arXiv: https://arxiv.org/abs/2503.21639 · HTML: https://mlsanity.com/paper/2503.21639/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 17 of 17)

- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [The Projected Covariance Measure for assumption-lean variable significance testing](https://mlsanity.com/paper/2211.02039/) (2022)
- [Dimension-agnostic inference using cross U-statistics](https://mlsanity.com/paper/2011.05068/) (2020)
- [Rank Verification for Exponential Families](https://mlsanity.com/paper/1610.03944/) (2016)
- [A Permutation-Free Kernel Independence Test](https://mlsanity.com/paper/2212.09108/) (2022)
- [An Efficient Doubly-Robust Test for the Kernel Treatment Effect](https://mlsanity.com/paper/2304.13237/) (2023)
- [Winners with Confidence: Discrete Argmin Inference with an Application to Model Selection](https://mlsanity.com/paper/2408.02060/) (2024)
- [Multiple-Splitting Projection Test for High-Dimensional Mean Vectors](https://mlsanity.com/paper/2110.15480/) (2021)
- [Rank-transformed subsampling: inference for multiple data splitting and exchangeable p-values](https://mlsanity.com/paper/2301.02739/) (2023)
- [Sequential model confidence sets](https://mlsanity.com/paper/2404.18678/) (2024)
- [Selective inference is easier with p-values](https://mlsanity.com/paper/2411.13764/) (2024)
- [Powerful rank verification for multivariate Gaussian data with any covariance structure](https://mlsanity.com/paper/2503.01065/) (2025)
- [Dimension-agnostic Change Point Detection](https://mlsanity.com/paper/2303.10808/) (2023)
- [Gaussian Rank Verification](https://mlsanity.com/paper/2501.14142/) (2025)

## Built on by (most cited first, 3 of 3)

- [CITE: Anytime-Valid Statistical Inference in LLM Self-Consistency](https://mlsanity.com/paper/2605.05873/) (2026)
- [Reliable Selection of Heterogeneous Treatment Effect Estimators](https://mlsanity.com/paper/2511.18464/) (2025)
- [Sequential Confidence Sets for Coverage-Constrained Conformal Model Selection](https://mlsanity.com/paper/2609.28522/) (2026)

## Abstract (arXiv)

This paper tackles a fundamental inference problem: given n observations from a distribution P over ℝd with unknown mean boldsymbolμ, we must form a confidence set for the index (or indices) corresponding to the smallest component of boldsymbolμ. By duality, we reduce this to testing, for each r in 1,ldots,d, whether μr is the smallest. Based on the sample splitting and self-normalization approach of Kim and Ramdas (2024), we propose "dimension-agnostic" tests that maintain validity regardless of how d scales with n, and regardless of arbitrary ties in boldsymbolμ. Notably, our validity holds under mild moment conditions, requiring little more than finiteness of a second moment, and permitting possibly strong dependence between coordinates. In addition, we establish the local minimax separation rate for this problem, which adapts to the cardinality of a confusion set, and show that the proposed tests attain this rate. Furthermore, we develop robust variants that continue to achieve the same minimax rate under heavy-tailed distributions with only finite second moments. While these results highlight the theoretical strength of our method, a practical concern is that sample splitting can reduce finite-sample power. We show that this drawback can be substantially alleviated by the multi-split aggregation method of Guo and Shah (2025). Finally, empirical results on simulated and real data illustrate the strong performance of our approach in terms of type I error control and power compared to existing methods.