Regularized Multi-Task Learning for Multi-Dimensional Log-Density Gradient Estimation

Multi-task learning is a paradigm of machine learning for solving multiple related learning tasks simultaneously with the expectation that information brought by other related tasks can be mutually exploited to improve the accuracy [Caruana, 1997]. Multi-task learning is particularly useful when one has many related learning tasks to solve but only few training samples are available for each task, which is often the case in many real-world problems such as therapy screening [Bickel et al., 2008] and face verification [Wang et al., 2009]. Multi-task learning has been gathering a great deal of attention, and extensive studies have been conducted both theoretically and experimentally [Thrun, 1996, Evgeniou and Pontil, 2004, Ando and Zhang, 2005, Zhang, 2013, Baxter, 2000]. Thrun [1996] proposed the lifelong learning framework, which transfers the knowledge obtained from the tasks experienced in the past to a newly given task, and it was demonstrated to improve the performance of image recognition. Baxter Baxter [2000] defined a multi-task learning framework called inductive bias learning, and derived a generalization error bound. The semi-supervised multi-task learning method proposed by Ando and Zhang [2005] generates many auxiliary learning 2 tasks from unlabeled data and seeks a good feature mapping for the target learning task.