Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator.

Key to multitask learning is exploiting the relationships between different tasks in order to improve prediction performance. Most previous methods have focused on the case where tasks relations can be modeled as linear operators and regularization approaches can be used successfully. However, in practice assuming the tasks to be linearly related is often restrictive, and allowing for nonlinear structures is a challenge. In this paper, we tackle this issue by casting the problem within the framework of structured prediction. Our main contribution is a novel algorithm for learning multiple tasks which are related by a system of nonlinear equations that their joint outputs need to satisfy. We show that our algorithm can be efficiently implemented and study its generalization properties, proving universal consistency and learning rates. Our theoretical analysis highlights the benefits of non-linear multitask learning over learning the tasks independently. Encouraging experimental results show the benefits of the proposed method in practice.

We show that, in this setting, the L TL problem can be reformulated as a Least Squares (LS) problem and we exploit a novel meta-algorithm to efficiently solve it.

Applications of optimal transport have recently gained remarkable attention as a result of the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation to the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn approximation, proving that it enjoys the same smoothness of its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula is used to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.

Key to multitask learning is exploiting the relationships between different tasks in order to improve prediction performance. Most previous methods have focused on the case where tasks relations can be modeled as linear operators and regularization approaches can be used successfully. However, in practice assuming the tasks to be linearly related is often restrictive, and allowing for nonlinear structures is a challenge. In this paper, we tackle this issue by casting the problem within the framework of structured prediction. Our main contribution is a novel algorithm for learning multiple tasks which are related by a system of nonlinear equations that their joint outputs need to satisfy. We show that our algorithm can be efficiently implemented and study its generalization properties, proving universal consistency and learning rates. Our theoretical analysis highlights the benefits of non-linear multitask learning over learning the tasks independently. Encouraging experimental results show the benefits of the proposed method in practice.