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

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