# Statistical Properties of the log-cosh Loss Function Used in Machine Learning

arXiv: https://arxiv.org/abs/2208.04564 · HTML: https://mlsanity.com/paper/2208.04564/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 9 Aug 2022.

## Builds on (most cited first, 5 of 5)

- [A survey of loss functions for semantic segmentation](https://mlsanity.com/paper/2006.14822/) (2020)
- [Smoothed Quantile Regression with Large-Scale Inference](https://mlsanity.com/paper/2012.05187/) (2020)
- [Smoothing quantile regressions](https://mlsanity.com/paper/1905.08535/) (2019)
- [Classification of Breast Cancer Lesions in Ultrasound Images by using Attention Layer and loss Ensembles in Deep Convolutional Neural Networks](https://mlsanity.com/paper/2102.11519/) (2021)
- [Solution to the Non-Monotonicity and Crossing Problems in Quantile Regression](https://mlsanity.com/paper/2111.04805/) (2021)

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## Abstract (arXiv)

This paper analyzes a popular loss function used in machine learning called the log-cosh loss function. A number of papers have been published using this loss function but, to date, no statistical analysis has been presented in the literature. In this paper, we present the distribution function from which the log-cosh loss arises. We compare it to a similar distribution, called the Cauchy distribution, and carry out various statistical procedures that characterize its properties. In particular, we examine its associated pdf, cdf, likelihood function and Fisher information. Side-by-side we consider the Cauchy and Cosh distributions as well as the MLE of the location parameter with asymptotic bias, asymptotic variance, and confidence intervals. We also provide a comparison of robust estimators from several other loss functions, including the Huber loss function and the rank dispersion function. Further, we examine the use of the log-cosh function for quantile regression. In particular, we identify a quantile distribution function from which a maximum likelihood estimator for quantile regression can be derived. Finally, we compare a quantile M-estimator based on log-cosh with robust monotonicity against another approach to quantile regression based on convolutional smoothing.