Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification (2011.09588v4)

Abstract: Among the many ways of quantifying uncertainty in a regression setting, specifying the full quantile function is attractive, as quantiles are amenable to interpretation and evaluation. A model that predicts the true conditional quantiles for each input, at all quantile levels, presents a correct and efficient representation of the underlying uncertainty. To achieve this, many current quantile-based methods focus on optimizing the so-called pinball loss. However, this loss restricts the scope of applicable regression models, limits the ability to target many desirable properties (e.g. calibration, sharpness, centered intervals), and may produce poor conditional quantiles. In this work, we develop new quantile methods that address these shortcomings. In particular, we propose methods that can apply to any class of regression model, allow for selecting a trade-off between calibration and sharpness, optimize for calibration of centered intervals, and produce more accurate conditional quantiles. We provide a thorough experimental evaluation of our methods, which includes a high dimensional uncertainty quantification task in nuclear fusion.

• The paper introduces alternative quantile regression methods that decouple density estimation from regression to significantly enhance calibration in predictive models.
• It presents a combined calibration loss and interval score optimization to balance calibration and sharpness, yielding improved performance in high-dimensional tasks.
• The research proposes a group batching technique that improves subgroup calibration, demonstrating robust uncertainty quantification on UCI and nuclear fusion datasets.

Quantile Methods for Calibrated Uncertainty Quantification Without Pinball Loss

The paper, "Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification," introduces novel methodologies in uncertainty quantification (UQ) for regression models. Models that accurately predict the conditional quantiles across various levels provide a comprehensive representation of predictive uncertainty. This research critiques the limitations of pinball loss, the predominant method in quantile-based UQ approaches, and proposes alternative techniques to overcome these constraints.

Limitations of Pinball Loss

Pinball loss, although prevalent, imposes certain limitations on regression models that potentially hinder UQ quality. It can skew towards sharpness—narrow predictive intervals—at the expense of calibration, leading to sharp yet miscalibrated predictions. The paper highlights how highly expressive models using this loss can inadvertently ignore calibration while optimizing sharpness. The inefficacy of using regularization to address these issues in pinball loss is also discussed.

Proposed Methodologies

Model-Agnostic Quantile Regression (MAQR)

A flexible alternative introduced is the Model-Agnostic Quantile Regression (MAQR). It decouples the task into density estimation and regression, which can then be used with any regression model. This method, leveraging conditional density estimation, alleviates model-type restrictions and enhances calibration, making it suitable for varied regression models without additional overhead.

Combined Calibration Loss

The research proposes a combined calibration loss that explicitly balances calibration and sharpness. The novel loss function allows for a tunable trade-off, encouraging users to focus on calibration first before optimizing sharpness. The calibration objective ensures predicted quantiles align with expected probabilities, while sharpness is separately penalized, providing clarity and flexibility in model training.

Interval Score Optimization

Centered intervals are often desirable in practical applications. The paper advocates for the use of interval scores, a proper scoring rule, to simultaneously optimize centered prediction intervals. The approach significantly bolsters PI calibration and demonstrates improved performance in high-dimensional tasks compared to traditional methods.

Group Batching Technique

The authors introduce a novel batching approach to training quantile models, known as group batching. By deliberately forming training batches to enhance adversarial group calibration, this method significantly improves calibration metrics beyond average calibration, pushing towards the goal of individual calibration through better subgroup calibration.

Experimental Validation and Implications

Experimental results on UCI datasets and a high-dimensional nuclear fusion task demonstrate that the proposed methods consistently outperform pinball loss in terms of calibration, sharpness (when needed), and robustness across different model architectures. The implications of these methods extend to applications where calibrated uncertainty is crucial, such as safety-critical systems in autonomous vehicles and robust decision-making in robotics.

In summary, the research shifts focus from traditional pinball loss optimization to innovative quantile methods that prioritize accurate uncertainty representation. These advances hold promise for future developments in AI, where calibrated predictions are essential for decision-making processes.

Following this research, further exploration could focus on quantile methods' application in other domains, such as probabilistic graphical models or hybrid architectures, and investigation into alternative loss functions that could offer even more robust calibration in dynamic settings. These contributions could significantly enhance the theoretical framework and practical application of uncertainty quantification in AI and machine learning.