another learner to quickly adapt to their new environment. Such group-adaptive learning has numerous applications, from predicting financial time-series, through content recommendation systems, to visual understanding for adaptive autonomous agents. Mathematics Reproducing Hilbert space of vector valued functions (RKHSvv) The MTL problem can be cast within the context of RKHSvv (a complete inner product space of vector-valued functions equipped with a reproducing kernel). In particular, recent focus has been on cases where task structure can be identified via a separable kernel, described below. The presentation here derives from Ciliberto et al., 2015. RKHSvv concepts Suppose the training data set is , with , , where indexes task, and . Let . In this setting there is a consistent input and output space and the same loss function for each task: . This results in the regularized machine learning problem: where is a vector valued reproducing kernel Hilbert space with functions having components . The reproducing kernel for the space of functions is a symmetric matrix-valued function , such that and the following reproducing property holds: The reproducing kernel gives rise to a representer theorem showing that any solution to equation has the form: Separable kernels The form of the kernel induces both the representation of the feature space and structures the output across tasks. A natural simplification is to choose a separable kernel, which factors into separate kernels on the input space and on the tasks . In this case the kernel relating scalar components and is given by . For vector valued functions we can write , where is a scalar reproducing kernel, and is a symmetric positive semi-definite matrix. Henceforth denote . This factorization property, separability, implies the input feature space representation does not vary by task. That is, there is no interaction between the input kernel and the task kernel. The structure on tasks is represented solely by . Methods for non-separable kernels is an current field of research. For the separable case, the representation theorem is reduced to . The model output on the training data is then , where is the empirical kernel matrix with entries , and is the matrix of rows . With the separable kernel, equation can be rewritten as where is a (weighted) average of applied entry-wise to and . (The weight is zero if is a missing observation). Note the second term in can be derived as follows: Known task