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Understanding the Under-Coverage Bias in Uncertainty Estimation
Yu Bai · Song Mei · Huan Wang · Caiming Xiong

Estimating the data uncertainty in regression tasks is often done by learning a quantile function or a prediction interval of the true label conditioned on the input. It is frequently observed that quantile regression---a vanilla algorithm for learning quantiles with asymptotic guarantees---tends to *under-cover* than the desired coverage level in reality. While various fixes have been proposed, a more fundamental understanding of why this under-coverage bias happens in the first place remains elusive.In this paper, we present a rigorous theoretical study on the coverage of uncertainty estimation algorithms in learning quantiles. We prove that quantile regression suffers from an inherent under-coverage bias, in a vanilla setting where we learn a realizable linear quantile function and there is more data than parameters. More quantitatively, for $\alpha>0.5$ and small $d/n$, the $\alpha$-quantile learned by quantile regression roughly achieves coverage $\alpha - (\alpha-1/2)\cdot d/n$ regardless of the noise distribution, where $d$ is the input dimension and $n$ is the number of training data. Our theory reveals that this under-coverage bias stems from a certain high-dimensional parameter estimation error that is not implied by existing theories on quantile regression. Experiments on simulated and real data verify our theory and further illustrate the effect of various factors such as sample size and model capacity on the under-coverage bias in more practical setups.

Author Information

Yu Bai (Salesforce Research)
Song Mei (University of California, Berkeley)
Huan Wang (Salesforce Research)

Huan Wang is an senior research scientist at Salesforce Research. His research interests include machine learning, big data analytics, computer vision and NLP. He used to be a research scientist at Microsoft AI Research, Yahoo’s New York Labs, and an adjunct professor at the engineering school of New York University. He graduated as a Ph.D in Computer Science at Yale University in 2013. Before that, he received an M.Phil. from The Chinese University of Hong Kong and a B.Eng. from Zhejiang University, both in information engineering.

Caiming Xiong (State Univerisity of New York at Buffalo)

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