Smooth ECE: Principled Reliability Diagrams via Kernel Smoothing (2309.12236v1)

Abstract: Calibration measures and reliability diagrams are two fundamental tools for measuring and interpreting the calibration of probabilistic predictors. Calibration measures quantify the degree of miscalibration, and reliability diagrams visualize the structure of this miscalibration. However, the most common constructions of reliability diagrams and calibration measures -- binning and ECE -- both suffer from well-known flaws (e.g. discontinuity). We show that a simple modification fixes both constructions: first smooth the observations using an RBF kernel, then compute the Expected Calibration Error (ECE) of this smoothed function. We prove that with a careful choice of bandwidth, this method yields a calibration measure that is well-behaved in the sense of (B{\l}asiok, Gopalan, Hu, and Nakkiran 2023a) -- a consistent calibration measure. We call this measure the SmoothECE. Moreover, the reliability diagram obtained from this smoothed function visually encodes the SmoothECE, just as binned reliability diagrams encode the BinnedECE. We also provide a Python package with simple, hyperparameter-free methods for measuring and plotting calibration: pip install relplot\.

• The paper's main contribution is the SmoothECE measure that leverages RBF kernel smoothing to address discontinuities in traditional calibration methods.
• It employs a reflected Gaussian kernel and rigorous theoretical analysis to ensure a consistent and well-behaved calibration across prediction intervals.
• Experimental results on real-world datasets demonstrate improved interpretability and reduced estimation errors compared to conventional reliability diagrams.

A Comprehensive Overview of the "Smooth ECE: Principled Reliability Diagrams via Kernel Smoothing" Paper

The paper "Smooth ECE: Principled Reliability Diagrams via Kernel Smoothing" by Jarosław Błasiok and Preetum Nakkiran addresses the shortcomings of traditional methods for measuring the calibration of probabilistic predictors, specifically the Expected Calibration Error (ECE) and reliability diagrams. The authors introduce a novel approach called SmoothECE, which leverages kernel smoothing to provide a well-behaved calibration measure and a visually meaningful reliability diagram.

Key Contributions and Methodology

The primary contribution of this paper lies in the development of the SmoothECE measure, which utilizes radial basis function (RBF) kernel smoothing to overcome the inherent limitations of conventional binning methods in calibration evaluation. The authors identify significant flaws in current practices, such as the discontinuous nature of ECE and sensitivity to binning choices, which they aim to address through their novel methodology.

SmoothECE is conceptualized by first applying RBF kernel smoothing to probabilistic predictions before calculating the ECE of the smoothed predictions. This approach not only ensures continuity but also allows for the derivation of a consistent calibration measure, as defined by theoretical frameworks in existing literature. The authors meticulously prove that with proper bandwidth selection, the resulting calibration measure is well-defined and consistent.

Numerical and Theoretical Insights

The paper's theoretical foundation is robust, with the authors defining "consistency" in calibration measures through the lens of Wasserstein distance between distributions. The consistency is evidenced through mathematical proof and empirical demonstrations. Furthermore, the authors propose the use of a reflected Gaussian kernel to handle boundary issues in smoothing, ensuring that the calibration function remains well-behaved across the entire prediction interval.

Numerical results are demonstrated through several experiments on real-world datasets, ranging from image classification tasks to meteorological predictions. These experiments compare the proposed smooth reliability diagrams against traditional binned diagrams, illustrating the advantages in terms of interpretability and reduced estimation errors.

Implications and Future Directions

The introduction of SmoothECE has both practical and theoretical implications. Practically, it provides a more reliable and visually intuitive method for analyzing the calibration of machine learning models, potentially impacting how probabilistic predictions are evaluated across various applications. Theoretically, the work enriches the literature on calibration error measurement by proposing an innovative integration of regression techniques into calibration assessments.

Looking toward future developments, SmoothECE's framework can be extended to other forms of calibration assessments, including multi-class and multi-label settings, where calibration metrics are less standardized. Additionally, the kernel smoothing approach might be combined with other statistical techniques to better accommodate domain-specific nuances in predictions.

In conclusion, Błasiok and Nakkiran's work on SmoothECE offers a significant step forward in the calibration evaluation of probabilistic predictors, addressing long-standing issues associated with legacy methodologies while setting the stage for further refinement and adoption in diverse fields.