Minimax Linear Regression under the Quantile Risk Article Swipe

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Ayoub El Hanchi , Chris J. Maddison , Murat A. Erdogdu ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2406.12145

We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the known lower bounds for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the recently proposed min-max regression procedure, thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all $p$-th power error functions for $p \in (2, \infty)$. Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

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Minimax Quantile regression Linear regression Econometrics Quantile Mathematics Statistics Computer science Mathematical optimization

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2406.12145
PDF: https://arxiv.org/pdf/2406.12145
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4399836889

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DOI: https://doi.org/10.48550/arxiv.2406.12145

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Title: Minimax Linear Regression under the Quantile Risk

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Type: preprint

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Language: en

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Publication year: 2024

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Publication date: 2024-06-17

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Authors: Ayoub El Hanchi, Chris J. Maddison, Murat A. Erdogdu

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Landing page: https://arxiv.org/abs/2406.12145

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PDF URL: https://arxiv.org/pdf/2406.12145

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OA URL: https://arxiv.org/pdf/2406.12145

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Concepts: Minimax, Quantile regression, Linear regression, Econometrics, Quantile, Mathematics, Statistics, Computer science, Mathematical optimization

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Cited by: 0

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Related works (count): 10

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