The performance of supervised machine learning models is directly related to the quality of the training dataset. In particular, the presence of significantly many outliers in the training data can lead to low accuracy because popular loss function like the Mean Squared Error (MSE) assign very high importance to large errors. The Mean Absolute Error (MAE) loss assigns equal importance to all errors and is most robust to outliers, but suffers from being non-differentiable at the origin. MAE also has large derivative values close to its minimum leading to instability and oscillations during training. Thus differentiable approximations to MAE namely Huber and Log-Cosh losses were introduced for robust regression tasks. This paper introduces two new infinitely differentiable loss functions that more closely approximate the MAE loss and provide improved performance on regression tasks with significantly many outliers in the training dataset. A comparison of the performance of regression models with different loss functions on a wide variety of benchmarks and datasets is presented to demonstrate the superior performance of the Square Root Loss (SRL) and Smooth Mean Absolute Error (SMAE) losses proposed in this paper. The SRL loss is shown to be strictly convex and the SMAE loss is shown to be strictly quasi-convex. Given the fundamental importance of linear regression, two new robust linear regression models are presented.

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