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

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.

Solution to the Non-Monotonicity and Crossing Problems in Quantile Regression

This paper proposes a new method to address the long-standing problem of lack of monotonicity in estimation of the conditional and structural quantile function, also known as quantile crossing problem. Quantile regression is a very powerful tool in data science in general and econometrics in particular. Unfortunately, the crossing problem has been confounding researchers and practitioners alike for over 4 decades. Numerous attempts have been made to find a simple and general solution. This paper describes a unique and elegant solution to the problem based on a flexible check function that is easy to understand and implement in R and Python, while greatly reducing or even eliminating the crossing problem entirely. It will be very important in all areas where quantile regression is routinely used and may also find application in robust regression, especially in the context of machine learning. From this perspective, we also utilize the flexible check function to provide insights into the root causes of the crossing problem.