Online learning of Cauchy loss with mini-batch gradient

Online kernel regression methods often degrade in the presence of heavy-tailed noise and outliers. In this paper, we propose OMBC, an Online Mini-Batch Cauchy kernel regression algorithm that employs the Cauchy loss to achieve both efficient and robust model updates in streaming settings. We prove that, under standard source conditions, OMBC attains an O(T−1/2) convergence rate and we characterize how the mini-batch size and the Cauchy scale parameter affect the bias–variance trade-off. Extensive experiments on synthetic and real-world datasets demonstrate that OMBC outperforms state-of-the-art online learning techniques in high-noise scenarios. Moreover, in a DepMap multi-omics tumor cell line gene–protein expression prediction task, OMBC achieves substantial performance gains. These findings indicate that OMBC provides an effective and scalable solution for online learning under heavy-tailed perturbations.