Tobias Wegel

Modern machine learning methods often have to rely on high-dimensional data that is expensive to label, while unlabeled data is abundant. When the data exhibits low-dimensional structure such as sparsity, conventional regularization techniques are known to improve generalization for a single objective (e.g., prediction risk). However, it is largely unexplored how to leverage this structure in the context of multi-objective learning (MOL) with multiple competing objectives. In this work, we discuss how the application of vanilla regularization approaches can fail, and propose the first MOL estimator that provably yields improved performance in the presence of sparsity and unlabeled data. We demonstrate its effectiveness experimentally for multi-distribution learning and fairness-risk trade-offs.

In multi-objective learning (MOL), several possibly competing prediction tasks must be solved jointly by a single model. Achieving good trade-offs may require a model class G with larger capacity than what is necessary for solving the individual tasks. This, in turn, increases the statistical cost, as reflected in known MOL bounds that depend on the complexity of G. We show that this cost is unavoidable for some losses, even in an idealized semi-supervised setting, where the learner has access to the Bayes-optimal solutions for the individual tasks as well as the marginal distributions over the covariates. On the other hand, for objectives defined with Bregman losses, we prove that the complexity of G may come into play only in terms of unlabeled data. Concretely, we establish sample complexity upper bounds, showing precisely when and how unlabeled data can significantly alleviate the need for labeled data. These rates are achieved by a simple, semi-supervised algorithm via pseudo-labeling.