What question did this study set out to answer?

The aim is to establish a theoretical framework for distance-based loss functions in binary classification using the Hassanat distance.

February 13, 2026Open Access

HasLoss: a novel Hassanat distance-based loss functions for binary classification

Key Points

The aim is to establish a theoretical framework for distance-based loss functions in binary classification using the Hassanat distance.
Proved bounded gradients using gradient analysis and finite Lipschitz constants.
Formulated six variants with different error sensitivities.
Validated theoretical properties empirically on synthetic and real-world datasets.
Compared performance against established loss functions like BCE and Focal Loss.
Achieved competitive performance across several evaluation metrics.
Demonstrated improved calibration, convergence speed, precision, recall, F1-score, and AUC.
Showed robustness to outliers and noise.
Identified some variants with larger practical effect sizes compared to BCE.

Abstract

Introduction Loss functions play a critical role in machine learning, particularly in training neural networks for classification tasks. In this work, we establish a theoretical framework for distance-based loss functions by adapting the Hassanat distance for binary classification. Methods Through gradient analysis, we prove that Hassanat losses exhibit bounded gradients with finite Lipschitz constants, providing convergence guarantees and robustness to outliers. We formulate six variants with different error sensitivities and validate these theoretical properties empirically. Their effectiveness is evaluated on synthetic datasets and nine real-world datasets, ranging from a few hundred to nearly 48,000 samples, under controlled experimental conditions. A comprehensive comparison is conducted against widely used loss functions, including Binary Cross-Entropy (BCE), Focal Loss, Mean Squared Error (MSE), and L1 Loss. Results Results show that the proposed Hassanat-based losses achieve competitive performance across evaluation metrics, with comparable or slightly improved results in calibration, convergence speed (in terms of epochs), precision, recall, F1-score, and AUC on several datasets, while exhibiting notable robustness to outliers and noise. The estimated Floating Point Operations (FLOPs) shows that the wall-clock time difference is due to implementation gap, not algorithmic. Importantly, Cohen's d effect size and confidence interval analyses shows that some of the proposed variants introduce a larger practical effect size than popular loss functions such as BCE. Discussion This work establishes both theoretical foundations and empirical validation for distance-based loss functions. The bounded gradient framework with finite Lipschitz constants provides principled optimization guarantees while explaining observed robustness and convergence behavior. This foundation enables systematic development of robust loss functions tailored to specific application requirements.

Bookmark

View Full Paper