Multi-task copula by sparse graph regression

This paper proposes multi-task copula (MTC) that can handle a much wider class of tasks than mean regression with Gaussian noise in most former multi-task learning (MTL). While former MTL emphasizes shared structure among models, MTC aims at joint prediction to exploit inter-output correlation. Given input, the outputs of MTC are allowed to follow arbitrary joint continuous distribution. MTC captures the joint likelihood of multi-output by learning the marginal of each output firstly and then a sparse and smooth output dependency graph function. While the former can be achieved by classical MTL, learning graphs dynamically varying with input is quite a challenge. We address this issue by developing sparse graph regression (SpaGraphR), a non-parametric estimator incorporating kernel smoothing, maximum likelihood, and sparse graph structure to gain fast learning algorithm. It starts from a few seed graphs on a few input points, and then updates the graphs on other input points by a fast operator via coarse-to-fine propagation. Due to the power of copula in modeling semi-parametric distributions, SpaGraphR can model a rich class of dynamic non-Gaussian correlations. We show that MTC can address more flexible and difficult tasks that do not fit the assumptions of former MTL nicely, and can fully exploit their relatedness. Experiments on robotic control and stock price prediction justify its appealing performance in challenging MTL problems.