A Seagull Loss Function With Application to Recommender Systems

Outliers are intrinsic to recommender systems (RSs) due to user uncertainty and bring large deviations to the total loss. Classical loss functions such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> do not consider this issue, thus biasing the model toward abnormal ratings. Advanced ones such as Logcosh, Huber, and Wing losses try to control model biasing; however, they do not consider the rating ranges of recommender systems. In this article, we propose a Seagull loss combining quadratic and softsign functions to handle this issue. The quadratic function aims to preserve or appropriately increase losses caused by small deviations, while softsign function aims to suppress losses caused by large deviations. It is distinguished from three types of popular losses. Compared with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula>, it controls the upper bound, therefore suppressing the influence of outliers. Compared with Logcosh and Huber, it not only adjusts the loss of small deviations but also suppresses the loss of large deviations. Compared with Wing loss, it considers the characteristics of data in RSs and therefore is more suitable to the matrix factorization model. Experiments are undertaken on six real-world recommendation datasets in comparison with six popular loss functions. Results show that Seagull loss function has superiority to all losses above on mean absolute error (MAE) and root mean square error (RMSE) and slight inferiority to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> and Wing on hit ratio (HR), mean average precision (MAP), and normalized discounted cumulative gain (NDCG).