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Additive models form a widely popular class of regression models which represent the relation between covariates and response variables as the sum of low-dimensional transfer functions. Besides flexibility and accuracy, a key benefit of these models is their interpretability: the transfer functions provide visual means for inspecting the models and identifying domain-specific relations between inputs and outputs. However, in large-scale problems involving the prediction of many related tasks, learning independently additive models results in a loss of model interpretability, and can cause overfitting when training data is scarce. We introduce a novel multi-task learning approach which provides a corpus of accurate and interpretable additive models for a large number of related forecasting tasks. Our key idea is to share transfer functions across models in order to reduce the model complexity and ease the exploration of the corpus. We establish a connection with sparse dictionary learning and propose a new efficient fitting algorithm which alternates between sparse coding and transfer function updates. The former step is solved via an extension of Orthogonal Matching Pursuit, whose properties are analyzed using a novel recovery condition which extends existing results in the literature. The latter step is addressed using a traditional dictionary update rule. Experiments on real-world data demonstrate that our approach compares favorably to baseline methods while yielding an interpretable corpus of models, revealing structure among the individual tasks and being more robust when training data is scarce. Our framework therefore extends the well-known benefits of additive models to common regression settings possibly involving thousands of tasks.