Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise

Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise

The paper "Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise" by Thomas Pouplin, Alan Jeffares, Nabeel Seedat, and Mihaela van der Schaar explores methodologies for constructing valid prediction intervals in regression scenarios characterized by asymmetric noise distributions. The authors propose an innovative approach called Relaxed Quantile Regression (RQR) which addresses specific limitations of traditional quantile regression (QR) methods, particularly their inability to effectively deal with non-symmetric noise distributions.

Introduction

The construction of valid prediction intervals, rather than point estimates, is critical for robust uncertainty quantification in regression models. This is particularly vital in high-stakes applications such as medical decision-making, autonomous driving, and energy forecasting, where the cost of prediction errors can be substantial. Traditional QR, while popular due to its computational efficiency and simplicity, assumes that the specific quantiles to be estimated are known beforehand and typically focuses on symmetric intervals around the median. This assumption is suboptimal for real-world distributions that are often skewed, leading to either symmetric yet inefficient intervals or the necessity to estimate an excessive number of quantiles.

Contributions

The paper's main contributions can be summarized as follows:

1. Identification of Quantile Regression Limitations: The authors highlight the inefficiencies of traditional QR in dealing with non-symmetric noise distributions.
2. Proposal of Relaxed Quantile Regression (RQR): A novel approach which does not necessitate the pre-specification of quantiles, thus allowing the construction of more efficient prediction intervals.
3. Regularization Terms for Desired Properties: Introduction of regularization techniques to reward narrower intervals (RQR-W) and improved conditional coverage (RQR-O).
4. Theoretical Analysis and Empirical Validation: Rigorous proofs of coverage properties and extensive empirical validation across benchmark datasets.

Methodology

Relaxed Quantile Regression (RQR)

The RQR framework eliminates the need to pre-specify quantiles and directly learns interval bounds by optimizing an objective function that maintains desired coverage. Specifically, for a given coverage level $\alpha$, the model minimizes: [ \mathcal{L}^{{\text{RQR}{\alpha}((\mu1,\mu_2),} \mathbf{x}, y)} = \begin{cases} \alpha \kappa & \text{if } \kappa \geq 0, \ (\alpha-1) \kappa & \text{if } \kappa < 0, \ \end{cases} ] where $\kappa = (y - \mu1)(y - \mu2)$.

This formulation ensures that the intervals maintain the desired average coverage while allowing for non-symmetry. The permutation invariant nature of the loss function further avoids the crossing bounds issue inherent in traditional QR.

Regularization

To further refine the interval properties:

1. RQR-W: This variant minimizes interval width by adding a regularization term proportional to the squared interval width: [ \mathcal{L}^{{\text{RQR-W}\alpha((\mu1,\mu_2),} \mathbf{x}, y)} = \mathcal{L}^{{\text{RQR}{\alpha}((\mu1,\mu_2),} \mathbf{x}, y)} + \lambda \frac{(\mu2 - \mu1)^2}{2}. ]
2. RQR-O: This variant improves conditional coverage using a regularization term that promotes the independence of interval width from instances of (mis)coverage: [ \mathcal{R}(\cdot) = \left|\frac{\text{Cov}(\mathbf{w}, \mathbf{m})}{\text{Var}(\mathbf{w})\text{Var}(\mathbf{m})}\right|, ] where $\mathbf{w}$ and $\mathbf{m}$ denote vectors of interval widths and coverage indicators, respectively.

Theoretical and Empirical Results

The theoretical analysis confirms that RQR and its variants achieve the target coverage in expectation. Importantly, RQR-W was shown to provide narrower intervals without sacrificing coverage, while RQR-O improved conditional coverage metrics, demonstrating the method's flexibility in addressing various practical needs.

Empirically, the methods were tested on several benchmark datasets, showing superior performance over traditional quantile regression approaches. RQR consistently maintained the target coverage while often achieving narrower intervals. Additionally, the regularized RQR variants outperformed existing methods, such as QR and Simultaneous Quantile Regression (SQR), in terms of both interval width and conditional coverage.

Implications and Future Directions

The introduction of RQR has significant implications for practical applications of regression models. By moving beyond the constraints of symmetrical intervals and allowing for regularization that can optimize for specific needs, RQR offers a more adaptable and efficient framework for uncertainty quantification.

Future research could further explore the integration of RQR with conformal prediction methods, potentially enhancing finite sample validity guarantees. Additionally, developing new regularization terms tailored to other domain-specific requirements could extend the applicability and robustness of RQR in diverse real-world scenarios.