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Laplacian regularized stratified models (LRSM) are models that utilize theexplicit or implicit network structure of the sub-problems as defined by thecategorical features called strata (e.g., age, region, time, forecast horizon,etc.), and draw upon data from neighboring strata to enhance the parameterlearning of each sub-problem. They have been widely applied in machine learningand signal processing problems, including but not limited to time seriesforecasting, representation learning, graph clustering, max-marginclassification, and general few-shot learning. Nevertheless, existing works onLRSM have either assumed a known graph or are restricted to specificapplications. In this paper, we start by showing the importance and sensitivityof graph weights in LRSM, and provably show that the sensitivity can bearbitrarily large when the parameter scales and sample sizes are heavilyimbalanced across nodes. We then propose a generic approach to jointly learnthe graph while fitting the model parameters by solving a single optimizationproblem. We interpret the proposed formulation from both a graph connectivityviewpoint and an end-to-end Bayesian perspective, and propose an efficientalgorithm to solve the problem. Convergence guarantees of the proposedoptimization algorithm is also provided despite the lack of global stronglysmoothness of the Laplacian regularization term typically required in theexisting literature, which may be of independent interest. Finally, weillustrate the efficiency of our approach compared to existing methods byvarious real-world numerical examples.

Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models