A fundamental notion of distance between train and test distributions from the field of domain adaptation is discrepancy distance. While in general hard to compute, here we provide the first set of provably efficient algorithms for testing discrepancy distance, where discrepancy is computed with respect to a fixed output classifier. These results imply a broad set of new, efficient learning algorithms in the recently introduced model of Testable Learning with Distribution Shift (TDS learning) due to Klivans et al. (2023).Our approach generalizes and improves all prior work on TDS learning: (1) we obtain learners that succeed simultaneously for large classes of test distributions, (2) achieve near-optimal error rates, and (3) give exponential improvements for constant depth circuits. Our methods further extend to semi-parametric settings and imply the first positive results for low-dimensional convex sets. Additionally, we separate learning and testing phases and obtain algorithms that run in fully polynomial time at test time.

We study the problem of learning under arbitrary distribution shift, where the learner is trained on a labeled set from one distribution but evaluated on a different, potentially adversarially generated test distribution. We focus on two frameworks: [GKKM'20], allowing abstention on adversarially generated parts of the test distribution, and [KSV'23], permitting abstention on the entire test distribution if distribution shift is detected. All prior known algorithms either rely on learning primitives that are computationally hard even for simple function classes, or end up abstaining entirely even in the presence of a tiny amount of distribution shift. We address both these challenges for natural function classes, including intersections of halfspaces and decision trees, and standard training distributions, including Gaussians. For PQ learning, we give efficient learning algorithms, while for TDS learning, our algorithms can tolerate moderate amounts of distribution shift. At the core of our approach is an improved analysis of spectral outlier-removal techniques from learning with nasty noise. Our analysis can (1) handle arbitrarily large fraction of outliers, which is crucial for handling arbitrary distribution shifts, and (2) obtain stronger bounds on polynomial moments of the distribution after outlier removal, yielding new insights into polynomial regression under distribution shifts. Lastly, our techniques lead to novel results for tolerant [RV'23], and learning with nasty noise.

There are increasing cases where the class labels of test samples are unavailable, creating a significant need and challenge in measuring the discrepancy between training and test distributions. This distribution discrepancy complicates the assessment of whether the hypothesis selected by an algorithm on training samples remains applicable to test samples. We present a novel approach called Importance Divergence (I-Div) to address the challenge of test label unavailability, enabling distribution discrepancy evaluation using only training samples. I-Div transfers the sampling patterns from the test distribution to the training distribution by estimating density and likelihood ratios. Specifically, the density ratio, informed by the selected hypothesis, is obtained by minimizing the Kullback-Leibler divergence between the actual and estimated input distributions. Simultaneously, the likelihood ratio is adjusted according to the density ratio by reducing the generalization error of the distribution discrepancy as transformed through the two ratios. Experimentally, I-Div accurately quantifies the distribution discrepancy, as evidenced by a wide range of complex data scenarios and tasks.

Maximizing the area under the receiver operating characteristic curve (AUC) is a popular approach to imbalanced binary classification problems. Existing AUC maximization methods usually assume that training and test distributions are identical. However, this assumption is often violated in practice due to {\it a positive distribution shift}, where the negative-conditional density does not change but the positive-conditional density can vary. This shift often occurs in imbalanced classification since positive data are often more diverse and time-varying than negative data. To deal with this shift, we theoretically show that the AUC on the test distribution can be expressed by using the positive and marginal training densities and the marginal test density. Based on this result, we can maximize the AUC on the test distribution by using positive and unlabeled data in the training distribution and unlabeled data in the test distribution. The proposed method requires only positive labels in the training distribution as supervision. Moreover, the derived AUC has a simple form and thus is easy to implement. The effectiveness of the proposed method is shown with four real-world datasets.

Uncertainty quantification is essential for the reliable deployment of machine learning models to high-stakes application domains. Uncertainty quantification is all the more challenging when training distribution and test distribution are different, even if the distribution shifts are mild. Despite the ubiquity of distribution shifts in real-world applications, existing uncertainty quantification approaches mainly study the in-distribution setting where the train and test distributions are the same. In this paper, we develop a systematic calibration model to handle distribution shifts by leveraging data from multiple domains. Our proposed method---multi-domain temperature scaling---uses the heterogeneity in the domains to improve calibration robustness under distribution shift. Through experiments on three benchmark data sets, we find our proposed method outperforms existing methods as measured on both in-distribution and out-of-distribution test sets.

Adapting to dynamic data distributions is a practical yet challenging task.

Uncertainty quantification is a key component for safety-critical systems such as healthcare and robotics, since it enables agents to account for risk when making decisions.

We organize the supplementary materials as follows: Appendix A: the proofs for Theorem 1. Appendix B: the pseudo-code of the proposed method. Appendix E: more ablation studies on expert learning and the proposed inverse softmax loss. We first recall several key notations and define some new notations. As shown in Eq. (4), the optimization objective of our test-time self-supervised aggregation method Meanwhile, the mutual information between predictions ˆ Y and labels Y can be represented by: I ( ˆ Y; Y) = H ( ˆ Y) H( ˆ Y |Y). In this appendix, we provide more details on experimental settings.

Consider building a model for diagnosing a specific heart disease, and suppose that most participants of the study are middle to high-aged men.