Out-of-domain (OOD) generalization is a significant challenge for machine learning models. Many techniques have been proposed to overcome this challenge, often focused on learning models with certain invariance properties. In this work, we draw a link between OOD performance and model calibration, arguing that calibration across multiple domains can be viewed as a special case of an invariant representation leading to better OOD generalization. Specifically, we show that under certain conditions, models which achieve \emph{multi-domain calibration} are provably free of spurious correlations. This leads us to propose multi-domain calibration as a measurable and trainable surrogate for the OOD performance of a classifier. We therefore introduce methods that are easy to apply and allow practitioners to improve multi-domain calibration by training or modifying an existing model, leading to better performance on unseen domains. Using four datasets from the recently proposed WILDS OOD benchmark, as well as the Colored MNIST, we demonstrate that training or tuning models so they are calibrated across multiple domains leads to significantly improved performance on unseen test domains. We believe this intriguing connection between calibration and OOD generalization is promising from both a practical and theoretical point of view.

We investigate the use of a non-parametric independence measure, the Hilbert-Schmidt Independence Criterion (HSIC), as a loss-function for learning robust regression and classification models. This loss-function encourages learning models where the distribution of the residuals between the label and the model prediction is statistically independent of the distribution of the instances themselves. This loss-function was first proposed by \citet{mooij2009regression} in the context of learning causal graphs. We adapt it to the task of learning for unsupervised covariate shift: learning on a source domain without access to any instances or labels from the unknown target domain, but with the assumption that $p(y|x)$ (the conditional probability of labels given instances) remains the same in the target domain. We show that the proposed loss is expected to give rise to models that generalize well on a class of target domains characterised by the complexity of their description within a reproducing kernel Hilbert space. Experiments on unsupervised covariate shift tasks demonstrate that models learned with the proposed loss-function outperform models learned with standard loss functions, achieving state-of-the-art results on a challenging cell-microscopy unsupervised covariate shift task.

Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a multigrid solver is the prolongation matrix, which relates between different scales of the problem. This matrix is strongly problem-dependent, and its optimal construction is critical to the efficiency of the solver. In practice, however, devising multigrid algorithms for new problems often poses formidable challenges. In this paper we propose a framework for learning multigrid solvers. Our method learns a (single) mapping from discretized PDEs to prolongation operators for a broad class of 2D diffusion problems. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Our tests demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.