This paper studies kernel PCA in a decentralized setting, where data are distributively observed with full features in local nodes, and a fusion center is prohibited. Compared with linear PCA, the use of kernel brings challenges to the design of decentralized consensus optimization: the local projection directions are data-dependent. As a result, the consensus constraint in distributed linear PCA is no longer valid. To overcome this problem, we propose a projection consensus constraint and obtain an effective decentralized consensus framework, where local solutions are expected to be the projection of the global solution on the column space of the local dataset. We also derive a fully non-parametric, fast, and convergent algorithm based on the alternative direction method of multiplier, of which each iteration is analytic and communication-efficient. Experiments on a truly parallel architecture are conducted on real-world data, showing that the proposed decentralized algorithm is effective in utilizing information from other nodes and takes great advantages in running time over the central kernel PCA.

Deep neural networks (DNNs) trained with the logistic loss (also known as the cross entropy loss) have made impressive advancements in various binary classification tasks. Despite the considerable success in practice, generalization analysis for binary classification with deep neural networks and the logistic loss remains scarce. The unboundedness of the target function for the logistic loss in binary classification is the main obstacle to deriving satisfactory generalization bounds. In this paper, we aim to fill this gap by developing a novel theoretical analysis and using it to establish tight generalization bounds for training fully connected ReLU DNNs with logistic loss in binary classification. Our generalization analysis is based on an elegant oracle-type inequality which enables us to deal with the boundedness restriction of the target function. Using this oracle-type inequality, we establish generalization bounds for fully connected ReLU DNN classifiers $\hat{f}^{\text{FNN}}_n$ trained by empirical logistic risk minimization with respect to i.i.d. samples of size $n$, which lead to sharp rates of convergence as $n\to\infty$. In particular, we obtain optimal convergence rates for $\hat{f}^{\text{FNN}}_n$ (up to some logarithmic factor) only requiring the Hölder smoothness of the conditional class probability $\eta$ of data. Moreover, we consider a compositional assumption that requires $\eta$ to be the composition of several vector-valued multivariate functions of which each component function is either a maximum value function or a Hölder smooth function only depending on a small number of its input variables. Under this assumption, we can even derive optimal convergence rates for $\hat{f}^{\text{FNN}}_n$ (up to some logarithmic factor) which are independent of the input dimension of data. This result explains why in practice DNN classifiers can overcome the curse of dimensionality and perform well in high-dimensional classification problems. Furthermore, we establish dimension-free rates of convergence under other circumstances such as when the decision boundary is piecewise smooth and the input data are bounded away from it. Besides the novel oracle-type inequality, the sharp convergence rates presented in our paper also owe to a tight error bound for approximating the natural logarithm function near zero (where it is unbounded) by ReLU DNNs. In addition, we justify our claims for the optimality of rates by proving corresponding minimax lower bounds. All these results are new in the literature and will deepen our theoretical understanding of classification with deep neural networks.

Imposing suitably designed nonconvex regularization is effective to enhance sparsity, but the corresponding global search algorithm has not been well established. In this article, we propose a global search algorithm for the nonconvex two-level$\ell _{1}$penalty based on its piecewise linear property and apply it to machine learning tasks. With the search capability, the optimization performance of the proposed algorithm could be improved, resulting in better sparsity and accuracy than most state-of-the-art global and local algorithms. Besides, we also provide an approximation analysis to demonstrate the effectiveness of our global search algorithm in sparse quantile regression.

Asymmetric kernels naturally exist in real life, e.g., for conditional probability and directed graphs. However, most of the existing kernel-based learning methods require kernels to be symmetric, which prevents the use of asymmetric kernels. This paper addresses the asymmetric kernel-based learning in the framework of the least squares support vector machine named AsK-LS, resulting in the first classification method that can utilize asymmetric kernels directly. We will show that AsK-LS can learn with asymmetric features, namely source and target features, while the kernel trick remains applicable, i.e., the source and target features exist but are not necessarily known. Besides, the computational burden of AsK-LS is as cheap as dealing with symmetric kernels. Experimental results on various tasks, including Corel, PASCAL VOC, Satellite, directed graphs, and UCI database, all show that in the case asymmetric information is crucial, the proposed AsK-LS can learn with asymmetric kernels and performs much better than the existing kernel methods that rely on symmetrization to accommodate asymmetric kernels.

Decentralized optimization is an emerging paradigm in distributed learning in which agents achieve network-wide solutions by peer-to-peer communication without the central server. Since communication tends to be slower than computation, when each agent communicates with only a few neighboring agents per iteration, they can complete iterations faster than with more agents or a central server. However, the total number of iterations to reach a network-wide solution is affected by the speed at which the information of the agents is ``mixed'' by communication. We found that popular communication topologies either have large degrees (such as stars and complete graphs) or are ineffective at mixing information (such as rings and grids). To address this problem, we propose a new family of topologies, EquiTopo, which has an (almost) constant degree and network-size-independent consensus rate which is used to measure the mixing efficiency.In the proposed family, EquiStatic has a degree of $\Theta(\ln(n))$, where $n$ is the network size, and a series of time-varying one-peer topologies, EquiDyn, has a constant degree of 1. We generate EquiDyn through a certain random sampling procedure. Both of them achieve $n$-independent consensus rate. We apply them to decentralized SGD and decentralized gradient tracking and obtain faster communication and better convergence, both theoretically and empirically. Our code is implemented through BlueFog and available at https://github.com/kexinjinnn/EquiTopo.

This paper generalizes regularized regression problems in a hyper-reproducing kernel Hilbert space (hyper-RKHS), illustrates its utility for kernel learning and out-of-sample extensions, and proves asymptotic convergence results for the introduced regression models in an approximation theory view. Algorithmically, we consider two regularized regression models with bivariate forms in this space, including kernel ridge regression (KRR) and support vector regression (SVR) endowed with hyper-RKHS, and further combine divide-and-conquer with Nyström approximation for scalability in large sample cases. This framework is general: the underlying kernel is learned from a broad class, and can be positive definite or not, which adapts to various requirements in kernel learning. Theoretically, we study the convergence behavior of regularized regression algorithms in hyper-RKHS and derive the learning rates, which goes beyond the classical analysis on RKHS due to the non-trivial independence of pairwise samples and the characterisation of hyper-RKHS. Experimentally, results on several benchmarks suggest that the employed framework is able to learn a general kernel function form an arbitrary similarity matrix, and thus achieves a satisfactory performance on classification tasks.

This article considers the power of deep neural networks (deep nets) in realizing data features. Based on refined covering number estimates, we find that, to realize data features such as the locality, rotation invariance, and manifold structure, deep nets essentially improve the performances of shallow neural networks (shallow nets) without requiring additional capacity costs. Conversely, to realize some data features, such as the smoothness, we show that deep nets perform similar as shallow nets, provided the depth is not extremely large. Both sides show the advantages and limitations of deep nets in realizing data features and demonstrate that deep nets are not always better than shallow nets.

Random Fourier features (RFFs) have been successfully employed to kernel approximation in large-scale situations. The rationale behind RFF relies on Bochner's theorem, but the condition is too strict and excludes many widely used kernels, e.g., dot-product kernels (violates the shift-invariant condition) and indefinite kernels [violates the positive definite (PD) condition]. In this article, we present a unified RFF framework for indefinite kernel approximation in the reproducing kernel Kreĭn spaces (RKKSs). Besides, our model is also suited to approximate a dot-product kernel on the unit sphere, as it can be transformed into a shift-invariant but indefinite kernel. By the Kolmogorov decomposition scheme, an indefinite kernel in RKKS can be decomposed into the difference of two unknown PD kernels. The spectral distribution of each underlying PD kernel can be formulated as a nonparametric Bayesian Gaussian mixtures model. Based on this, we propose a double-infinite Gaussian mixture model in RFF by placing the Dirichlet process prior. It takes full advantage of high flexibility on the number of components and has the capability of approximating indefinite kernels on a wide scale. In model inference, we develop a non-conjugate variational algorithm with a sub-sampling scheme for the posterior inference. It allows for the non-conjugate case in our model and is quite efficient due to the sub-sampling strategy. Experimental results on several large classification data sets demonstrate the effectiveness of our nonparametric Bayesian model for indefinite kernel approximation when compared to other representative random feature-based methods.

In this paper, we consider the $\ell_q-$regularized kernel regression with $0 < q \leq 1$. In form, the algorithm minimizes a least-square loss functional adding a coefficient-based $\ell_q-$penalty term over a linear span of features generated by a kernel function. We study the asymptotic behavior of the algorithm under the framework of learning theory. The contribution of this paper is two-fold. First, we derive a tight bound on the $\ell_2-$empirical covering numbers of the related function space involved in the error analysis. Based on this result, we obtain the convergence rates for the $\ell_1-$regularized kernel regression which is the best so far. Second, for the case $0 < q < 1$, we show that the regularization parameter plays a role as a trade-off between sparsity and convergence rates. Under some mild conditions, the fraction of non-zero coefficients in a local minimizer of the algorithm will tend to $0$ at a polynomial decay rate when the sample size $m$ becomes large. As the concerned algorithm is non-convex, we also discuss how to generate a minimizing sequence iteratively, which can help us to search a local minimizer around any initial point.

Distributed learning is an effective way to analyze big data. In distributed regression, a typical approach is to divide the big data into multiple blocks, apply a base regression algorithm on each of them, and then simply average the output functions learnt from these blocks. Since the average process will decrease the variance, not the bias, bias correction is expected to improve the learning performance if the base regression algorithm is a biased one. Regularization kernel network is an effective and widely used method for nonlinear regression analysis. In this paper we will investigate a bias corrected version of regularization kernel network. We derive the error bounds when it is applied to a single data set and when it is applied as a base algorithm in distributed regression. We show that, under certain appropriate conditions, the optimal learning rates can be reached in both situations.

Applying the pinball loss in a support vector machine (SVM) classifier results in pin-SVM. The pinball loss is characterized by a parameter τ . Its value is related to the quantile level and different τ values are suitable for different problems. In this paper, we establish an algorithm to find the entire solution path for pin-SVM with different τ values. This algorithm is based on the fact that the optimal solution to pin-SVM is continuous and piecewise linear with respect to τ . We also show that the nonnegativity constraint on τ is not necessary, i.e., τ can be extended to negative values. First, in some applications, a negative τ leads to better accuracy. Second, τ = -1 corresponds to a simple solution that links SVM and the classical kernel rule. The solution for τ = -1 can be obtained directly and then be used as a starting point of the solution path. The proposed method efficiently traverses τ values through the solution path, and then achieves good performance by a suitable τ . In particular, τ = 0 corresponds to C-SVM, meaning that the traversal algorithm can output a result at least as good as C-SVM with respect to validation error.

Traditionally, the hinge loss is used to construct support vector machine (SVM) classifiers. The hinge loss is related to the shortest distance between sets and the corresponding classifier is hence sensitive to noise and unstable for re-sampling. In contrast, the pinball loss is related to the quantile distance and the result is less sensitive. The pinball loss has been deeply studied and widely applied in regression but it has not been used for classification. In this paper, we propose a SVM classifier with the pinball loss, called pin-SVM, and investigate its properties, including noise insensitivity, robustness, and misclassification error. Besides, insensitive zone is applied to the pin-SVM for a sparse model. Compared to the SVM with the hinge loss, the proposed pin-SVM has the same computational complexity and enjoys noise insensitivity and re-sampling stability.