Covariate shift occurs prevalently in practice, where the input distributions of the source and target data are substantially different. Despite its practical importance in various learning problems, most of the existing methods only focus on some specific learning tasks and are not well validated theoretically and numerically. To tackle this problem, we propose a unified analysis of general nonparametric methods in a reproducing kernel Hilbert space (RKHS) under covariate shift. Our theoretical results are established for a general loss belonging to a rich loss function family, which includes many commonly used methods as special cases, such as mean regression, quantile regression, likelihood-based classification, and margin-based classification. Two types of covariate shift problems are the focus of this paper and the sharp convergence rates are established for a general loss function to provide a unified theoretical analysis, which concurs with the optimal results in literature where the squared loss is used. Extensive numerical studies on synthetic and real examples confirm our theoretical findings and further illustrate the effectiveness of our proposed method.

We propose a new initialization scheme for the k-median problem in the general metric space (e.g., discrete space induced by graphs), based on the construction of metric embedding tree structure of the data. We propose a novel and efficient search algorithm which finds initial centers that can be used subsequently for the local search algorithm. The so-called HST initialization method can produce initial centers achieving lower error than those from another popular method k-median++, also with higher efficiency when k is not too small. Our HST initialization are then extended to the setting of differential privacy (DP) to generate private initial centers. We show that the error of applying DP local search followed by our private HST initialization improves prior results on the approximation error, and approaches the lower bound within a small factor. Experiments demonstrate the effectiveness of our proposed methods.

Clustering is a fundamental task in data science with wide-ranging applications. In k-medoids clustering, cluster centers must be actual datapoints and arbitrary distance metrics may be used; these features allow for greater interpretability of the cluster centers and the clustering of exotic objects in k-medoids clustering, respectively.

Clustering is a fundamental task in data science with wide-ranging applications. In k-medoids clustering, cluster centers must be actual datapoints and arbitrary distance metrics may be used; these features allow for greater interpretability of the cluster centers and the clustering of exotic objects in k-medoids clustering, respectively.

B.1 Complexity Analysis Denote the number of nodes and edges in the graph as N and E, the number of latent factors as K, the number of operation choices as |O|, the dimensionality of hidden representations as d. The time complexity of the disentangled super-network is O(K|E|d+K|V|d2), where the computation for each factor is fully parallelizable and amenable to GPU acceleration, and K is usually a small constant. The time complexity of the self-supervised training and contrastive search modules is both O(K2d2). As architectures under different factors share the parameters, the number of learnable parameters is the same as classical graph super-network, i.e., O(|O|d2). Therefore, the complexity of our method is comparable to classical GNAS methods.

Preserving training dynamics across batch sizes is an important tool for practical machine learning as it enables the trade-off between batch size and wall-clock time. This trade-off is typically enabled by a scaling rule, for example, in stochastic gradient descent, one should scale the learning rate linearly with the batch size. Another important machine learning tool is the model EMA, a functional copy of a target model, whose parameters move towards those of its target model according to an Exponential Moving Average (EMA) at a rate parameterized by a momentum hyperparameter. This model EMA can improve the robustness and generalization of supervised learning, stabilize pseudo-labeling, and provide a learning signal for Self-Supervised Learning (SSL). Prior works have not considered the optimization of the model EMA when performing scaling, leading to different training dynamics across batch sizes and lower model performance. In this work, we provide a scaling rule for optimization in the presence of a model EMA and demonstrate the rule's validity across a range of architectures, optimizers, and data modalities. We also show the rule's validity where the model EMA contributes to the optimization of the target model, enabling us to train EMA-based pseudo-labeling and SSL methods at small and large batch sizes. For SSL, we enable training of BYOL up to batch size 24,576 without sacrificing performance, a 6 wall-clock time reduction under idealized hardware settings.