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Abstract Large datasets make it possible to build predictive models that can capture heterogenous relationships between the response variable and features. The mixture of high-dimensional linear experts model posits that observations come from a mixture of high-dimensional linear regression models, where the mixture weights are themselves feature-dependent. In this article, we show how to construct valid prediction sets for an ℓ1-penalized mixture of experts model in the high-dimensional setting. We make use of a debiasing procedure to account for the bias induced by the penalization and propose a novel strategy for combining intervals to form a prediction set with coverage guarantees in the mixture setting. Synthetic examples and an application to the prediction of critical temperatures of superconducting materials show our method to have reliable practical performance. 

Abstract Performance of classifiers is often measured in terms of average accuracy on test data. Despite being a standard measure, average accuracy fails in characterising the fit of the model to the underlying conditional law of labels given the features vector (Y∣X), e.g. due to model misspecification, over fitting, and high-dimensionality. In this paper, we consider the fundamental problem of assessing the goodness-of-fit for a general binary classifier. Our framework does not make any parametric assumption on the conditional law Y∣X and treats that as a black-box oracle model which can be accessed only through queries. We formulate the goodness-of-fit assessment problem as a tolerance hypothesis testing of the form H0:E[Df(Bern(η(X))‖Bern(η^(X)))]≤τ where Df represents an f-divergence function, and η(x), η^(x), respectively, denote the true and an estimate likelihood for a feature vector x admitting a positive label. We propose a novel test, called Goodness-of-fit with Randomisation and Scoring Procedure (GRASP) for testing H0, which works in finite sample settings, no matter the features (distribution-free). We also propose model-X GRASP designed for model-X settings where the joint distribution of the features vector is known. Model-X GRASP uses this distributional information to achieve better power. We evaluate the performance of our tests through extensive numerical experiments. 

Low-dimensional structure of data can solve the adversarial robustness-accuracy conflict for machine learning systems. Modern machine learning systems have demonstrated breakthrough performance in a multitude of applications. However, they are known to be highly vulnerable to small perturbations to the input data, known as adversarial attacks. There are many well-documented examples of such behavior, for example small perturbations of an image, which is imperceptible to a human, can significantly degrade performance of modern classifiers. Adversarial training has been put forward as a way to improve robustness of learning algorithms to adversarial attacks. However, this benefit often comes at the cost of decreasing accuracy on natural unperturbed inputs, pointing to a potential conflict between adversarial robustness and standard accuracy. In “Adversarial robustness for latent models: Revisiting the robust-standard accuracies tradeoff,” Adel Javanmard and Mohammad Mehrabi develop a theory to show that when the data enjoys low-dimensional structure, then it is possible to train models that are nearly optimal with respect to both, the standard and robust accuracies.