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.

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.

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.

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.

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.

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.

This paper is concerned with the problem of uncertainty estimation in regression problems. Uncertainty estimation is an increasingly important task in modern machine learning applications--Models should not only make high-accuracy predictions, but also have a sense of how much the true label may deviate from the prediction. This capability is crucial for deploying machine learning in the real world, in particular in risk-sensitive domains such as medical AI [15, 29], self-driving cars [47], and so on. A common approach for uncertainty estimation in regression is to learn a quantile function or a prediction interval of the true label conditioned on the input, which provides useful distributional information about the label. Such learned quantiles are typically evaluated by their coverage, i.e., probability that it covers the true label on a new test example. For example, a learned 90% upper quantile function should be an actual upper bound of the true label at least 90% of the time. Algorithms for learning quantiles date back to the classical quantile regression [35], which estimates the quantile function by solving an empirical risk minimization problem with a suitable loss function that depends on the desired quantile level α.