structure Task structure representations There are three largely equivalent ways to represent task structure: through a regularizer; through an output metric, and through an output mapping. Task structure examples Via the regularizer formulation, one can represent a variety of task structures easily. Letting (where is the TxT identity matrix, and is the TxT matrix of ones) is equivalent to letting control the variance of tasks from their mean . For example, blood levels of some biomarker may be taken on patients at time points during the course of a day and interest may lie in regularizing the variance of the predictions across patients. Letting , where is equivalent to letting control the variance measured with respect to a group mean: . (Here the cardinality of group r, and is the indicator function). For example, people in different political parties (groups) might be regularized together with respect to predicting the favorability rating of a politician. Note that this penalty reduces to the first when all tasks are in the same group. Letting , where is the Laplacian for the graph with adjacency matrix M giving pairwise similarities of tasks. This is equivalent to giving a larger penalty to the distance separating tasks t and s when they are more similar (according to the weight ,) i.e. regularizes . All of the above choices of A also induce the additional regularization term which penalizes complexity in f more broadly. Learning tasks together with their structure Learning problem can be generalized to admit learning task matrix A as follows: Choice of must be designed to learn matrices A of a given type. See "Special cases" below. Optimization of Restricting to the case of convex losses and coercive penalties Ciliberto et al. have shown that although is not convex jointly in C and A, a related problem is jointly convex. Specifically on the convex set , the equivalent problem is convex with the same minimum value. And if is a minimizer for then is a minimizer for . may be solved by a barrier method on a closed set by introducing the following perturbation: The perturbation via the barrier forces the objective functions to be equal to on the boundary of . can be solved with a block coordinate descent method, alternating in C and A. This results in a sequence of minimizers in that converges to the solution in as , and hence