Low-rank matrix approximation is an effective tool in alleviating the memory and computational burdens of kernel methods and sampling, as the mainstream of such algorithms, has drawn considerable attention in both theory and practice. This paper presents detailed studies on the Nyström sampling scheme and in particular, an error analysis that directly relates the Nyström approximation quality with the encoding powers of the landmark points in summarizing the data. The resultant error bound suggests a simple and efficient sampling scheme, the k-means clustering algorithm, for Nyström low-rank approximation. We compare it with state-of-the-art approaches that range from greedy schemes to probabilistic sampling. Our algorithm achieves significant performance gains in a number of supervised/unsupervised learning tasks including kernel PCA and least squares SVM.

Large-scale learning problems require a plethora of labels that can be efficiently collected from crowdsourcing services at low cost. However, labels annotated by crowdsourced workers are often noisy, which inevitably degrades the performance of large-scale optimizations including the prevalent stochastic gradient descent (SGD). Specifically, these noisy labels adversely affect updates of the primal variable in conventional SGD. To solve this challenge, we propose a robust SGD mechanism called progressive stochastic learning (POSTAL), which naturally integrates the learning regime of curriculum learning (CL) with the update process of vanilla SGD. Our inspiration comes from the progressive learning process of CL, namely learning from "easy" tasks to "complex" tasks. Through the robust learning process of CL, POSTAL aims to yield robust updates of the primal variable on an ordered label sequence, namely, from "reliable" labels to "noisy" labels. To realize POSTAL mechanism, we design a cluster of "screening losses," which sorts all labels from the reliable region to the noisy region. To sum up, POSTAL using screening losses ensures robust updates of the primal variable on reliable labels first, then on noisy labels incrementally until convergence. In theory, we derive the convergence rate of POSTAL realized by screening losses. Meanwhile, we provide the robustness analysis of representative screening losses. Experimental results on UCI1 simulated and Amazon Mechanical Turk crowdsourcing data sets show that the POSTAL using screening losses is more effective and robust than several existing baselines.1UCI is the abbreviation of University of California Irvine.

Deep learning with noisy labels is practically challenging, as the capacity of deep models is so high that they can totally memorize these noisy labels sooner or later during training. Nonetheless, recent studies on the memorization effects of deep neural networks show that they would first memorize training data of clean labels and then those of noisy labels. Therefore in this paper, we propose a new deep learning paradigm called "Co-teaching" for combating with noisy labels. Namely, we train two deep neural networks simultaneously, and let them teach each other given every mini-batch: firstly, each network feeds forward all data and selects some data of possibly clean labels; secondly, two networks communicate with each other what data in this mini-batch should be used for training; finally, each network back propagates the data selected by its peer network and updates itself. Empirical results on noisy versions of MNIST, CIFAR-10 and CIFAR-100 demonstrate that Co-teaching is much superior to the state-of-the-art methods in the robustness of trained deep models. * The first two authors (Bo Han and Quanming Yao) made equal contributions. The implementation is available at https://github.com/bhanML/Co-teaching.32nd Conference on Neural Information Processing Systems (NIPS 2018),

The aim of multi-output learning is to simultaneously predict multiple outputs given an input. It is an important learning problem for decision-making, since making decisions in the real world often involves multiple complex factors and criteria. In recent times, an increasing number of research studies have focused on ways to predict multiple outputs at once. Such efforts have transpired in different forms according to the particular multi-output learning problem under study. Classic cases of multi-output learning include multi-label learning, multidimensional learning, multi-target regression and others. From our survey of the topic, we were struck by a lack in studies that generalize the different forms of multi-output learning into a common framework. This paper fills that gap with a comprehensive review and analysis of the multi-output learning paradigm. In particular, we characterize the 4 Vs of multi-output learning, i.e., volume, velocity, variety, and veracity, and the ways in which the 4 Vs both benefit and bring challenges to multioutput learning by taking inspiration from big data. We analyze the life cycle of output labeling, present the main mathematical definitions of multi-output learning, and examine the field's key challenges and corresponding solutions as found in the literature. Several model evaluation metrics and popular data repositories are also discussed. Last but not least, we highlight some emerging challenges with multi-output learning from the perspective of the 4 Vs as potential research directions worthy of further studies.

Learning with noisy labels is imperative in the Big Data era since it reduces expensive labor on accurate annotations. Previous method, learning with noise transition, has enjoyed theoretical guarantees when it is applied to the scenario with the class-conditional noise. However, this approach critically depends on an accurate pre-estimated noise transition, which is usually impractical. Subsequent improvement adapts the preestimation in the form of a Softmax layer along with the training progress. However, the parameters in the Softmax layer are highly tweaked for the fragile performance and easily get stuck into undesired local minimums. To overcome this issue, we propose a Latent Class-Conditional Noise model (LCCN) that models the noise transition in a Bayesian form. By projecting the noise transition into a Dirichlet-distributed space, the learning is constrained on a simplex instead of some adhoc parametric space. Furthermore, we specially deduce a dynamic label regression method for LCCN to iteratively infer the latent true labels and jointly train the classifier and model the noise. Our approach theoretically safeguards the bounded update of the noise transition, which avoids arbitrarily tuning via a batch of samples. Extensive experiments have been conducted on controllable noise data with CIFAR10 and CIFAR-100 datasets, and the agnostic noise data with Clothing1M and WebVision17 datasets. Experimental results have demonstrated that the proposed model outperforms several state-of-the-art methods.