We address the problem of algorithmic fairness: ensuring that sensitive information does not unfairly influence the outcome of a classifier. We present an approach based on empirical risk minimization, which incorporates a fairness constraint into the learning problem. It encourages the conditional risk of the learned classifier to be approximately constant with respect to the sensitive variable. We derive both risk and fairness bounds that support the statistical consistency of our methodology. We specify our approach to kernel methods and observe that the fairness requirement implies an orthogonality constraint which can be easily added to these methods. We further observe that for linear models the constraint translates into a simple data preprocessing step. Experiments indicate that the method is empirically effective and performs favorably against state-of-the-art approaches.

Regression with group-sparsity penalty plays a central role in high-dimensional prediction problems. However, most existing methods require the group structure to be known a priori. In practice, this may be a too strong assumption, potentially hampering the effectiveness of the regularization method. To circumvent this issue, we present a method to estimate the group structure by means of a continuous bilevel optimization problem where the data is split into training and validation sets. Our approach relies on an approximation scheme where the lower level problem is replaced by a smooth dual forward-backward algorithm with Bregman distances. We provide guarantees regarding the convergence of the approximate procedure to the exact problem and demonstrate the well behaviour of the proposed method on synthetic experiments. Finally, a preliminary application to genes expression data is tackled with the purpose of unveiling functional groups.

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.