Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise

TL;DR

This work targets uncertainty quantification in regression by constructing valid prediction intervals under asymmetric noise. It introduces Relaxed Quantile Regression (RQR), a direct interval-learning objective that omits the need to prespecify quantiles and still achieves the desired coverage level $\alpha$, while allowing regularizers to trade off interval width or conditional coverage. The authors prove that the RQR objective yields interval coverage in expectation with bounded variance and show how variants like RQR-W and RQR-O bias the solution toward narrower intervals or improved conditional coverage, respectively. Empirically, RQR and its regularized forms outperform traditional quantile-based methods on benchmark datasets, particularly under skewed noise, demonstrating the practical value of flexible, single-model interval prediction. The approach offers a versatile framework for uncertainty quantification with direct control over interval properties relevant to real-world decision-making.

Abstract

Constructing valid prediction intervals rather than point estimates is a well-established approach for uncertainty quantification in the regression setting. Models equipped with this capacity output an interval of values in which the ground truth target will fall with some prespecified probability. This is an essential requirement in many real-world applications where simple point predictions' inability to convey the magnitude and frequency of errors renders them insufficient for high-stakes decisions. Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the (non-parametric) distribution of outputs. This method is simple, computationally inexpensive, interpretable, assumption-free, and effective. However, it does require that the specific quantiles being learned are chosen a priori. This results in (a) intervals that are arbitrarily symmetric around the median which is sub-optimal for realistic skewed distributions, or (b) learning an excessive number of intervals. In this work, we propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint whilst maintaining its strengths. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities (e.g. mean width) whilst retaining the essential coverage guarantees of quantile regression.

Figures (6)

• Figure 1: Symmetric quantiles. We compare two pairs of intervals on an identical (non-symmetric) log-normal probability distribution where in both cases a fixed level of coverage $\alpha$ is obtained. In the upper figure, the intervals are selected to be symmetric in terms of probability mass around the median $q = 0.5$ (i.e. the two lined regions and contain equal probability mass) as in the case of quantile regression. In the lower figure, we remove this constraint and obtain a much narrower interval with identical coverage.
• Figure 2: Gradient analysis. The gradients with respect to the RQR objective (upper) and RQR-W objective (lower) for a single $x$ over a range of potential values of $y$. The current predicted bounds $(\mu_1, \mu_2) = (0, 1)$ are updated using these gradients.
• Figure 3: Applying RQR-W & RQR-O in practice. Resulting intervals on robotics distance estimation task. Upper: The RQR-W objective achieves generally narrower intervals across the 1639 test examples. Lower: The RQR-O objective achieves more consistent coverage across different interval widths.
• Figure 4: Variance of the estimated bounds. Estimating the minimal width interval rather than symmetric quantiles is also likely to result in lower variance estimates of the two bounds due to being estimated in more dense regions of the distribution. This is illustrated on the log-normal distribution example from \ref{['fig:skewillustration']}. We take samples of various sizes from this distribution and estimate both the quantiles and the minimum width bounds. We find that the variance of these estimates is significantly larger when estimating the former.
• Figure 5: A closed form solution for the Gumbel distribution. Comparison of the minimum reached by the one-dimensional losses $\mathcal{L}^\text{RQR}(\mu_l)$ and $\mathcal{L}^\text{RQR-W}(\mu_l)$. Note that the dashed lines visually map each loss functions minimum loss to their corresponding interval width.

Theorems & Definitions (15)

• Theorem 3.1: RQR In-sample Coverage
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• Theorem 2.1: RQR In-sample Coverage
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