Ridge regression is a well established regression estimator which can conveniently be adapted for classification problems. One compelling reason is probably the fact that ridge regression emits a closed-form solution thereby facilitating the training phase. However in the case of high-dimensional problems, the closed-form solution which involves inverting the regularised covariance matrix is rather expensive to compute. The high computational demand of such operation also renders difficulty in constructing ensemble of ridge regressions. In this paper, we consider learning an ensemble of ridge regressors where each regressor is trained in its own randomly projected subspace. Subspace regressors are later combined via adaptive boosting methodology. Experiments based on five high-dimensional classification problems demonstrated the effectiveness of the proposed method in terms of learning time and in some cases improved predictive performance can be observed.
Characterization of local minima draws much attention in theoretical studies of deep learning. In this study, we investigate the distribution of parameters in an over-parametrized finite neural network trained by ridge regularized empirical square risk minimization (RERM). We develop a new theory of ridgelet transform, a wavelet-like integral transform that provides a powerful and general framework for the theoretical study of neural networks involving not only the ReLU but general activation functions. We show that the distribution of the parameters converges to a spectrum of the ridgelet tra…
Many machine learning problems can be formulated as predicting labels for a pair of objects. Problems of that kind are often referred to as pairwise learning, dyadic prediction or network inference problems. During the last decade kernel methods have played a dominant role in pairwise learning. They still obtain a state-of-the-art predictive performance, but a theoretical analysis of their behavior has been underexplored in the machine learning literature. In this work we review and unify existing kernel-based algorithms that are commonly used in different pairwise learning settings, ranging f…
Conformal prediction is a method of producing prediction sets that can be applied on top of a wide range of prediction algorithms. The method has a guaranteed coverage probability under the standard IID assumption regardless of whether the assumptions (often considerably more restrictive) of the underlying algorithm are satisfied. However, for the method to be really useful it is desirable that in the case where the assumptions of the underlying algorithm are satisfied, the conformal predictor loses little in efficiency as compared with the underlying algorithm (whereas being a conformal predi…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
This paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call \emph{centered kernel ridge regression} (CKRR), also known in the literature as kernel ridge regression with offset. This modified technique is obtained by accounting for the bias in the regression problem resulting in the old kernel ridge regression but with \emph{centered} kernels. The analysis is carried out under the assumption that the data is drawn from a Gaussian distribution and heavily relies on tools from random matrix theory (RMT). Under the regime in which the data dimension a…
University of Genova（热那亚大学）；DeepMind Paris（DeepMind巴黎）；Istituto Italiano di Tecnologia（意大利理工学院）
We introduce ParK, a new large-scale solver for kernel ridge regression. Our approach combines partitioning with random projections and iterative optimization to reduce space and time complexity while provably maintaining the same statistical accuracy. In particular, constructing suitable partitions directly in the feature space rather than in the input space, we promote orthogonality between the local estimators, thus ensuring that key quantities such as local effective dimension and bias remain under control. We characterize the statistical-computational tradeoff of our model, and demonstrat…
Xi’an Jiaotong University（西安交通大学）；City University of Hong Kong（香港城市大学）
This paper focuses on generalization performance analysis for distributed algorithms in the framework of learning theory. Taking distributed kernel ridge regression (DKRR) for example, we succeed in deriving its optimal learning rates in expectation and providing theoretically optimal ranges of the number of local processors. Due to the gap between theory and experiments, we also deduce optimal learning rates for DKRR in probability to essentially reflect the generalization performance and limitations of DKRR. Furthermore, we propose a communication strategy to improve the learning performance…
Tsinghua University（清华大学）；The University of Arizona（亚利桑那大学）
The divide-and-conquer method has been widely used for estimating large-scale kernel ridge regression estimates. Unfortunately, when the response variable is highly skewed, the divide-and-conquer kernel ridge regression (dacKRR) may overlook the underrepresented region and result in unacceptable results. We combine a novel response-adaptive partition strategy with the oversampling technique synergistically to overcome the limitation. Through the proposed novel algorithm, we allocate some carefully identified informative observations to multiple nodes (local processors). Although the oversampli…
Kernel methods have had great success in Statistics and Machine Learning. Despite their growing popularity, however, less effort has been drawn towards developing kernel based classification methods on Riemannian manifolds due to difficulty in dealing with non-Euclidean geometry. In this paper, motivated by the extrinsic framework of manifold-valued data analysis, we propose a new positive definite kernel on planar Kendall shape space $Σ_2^k$, called extrinsic Veronese Whitney Gaussian kernel. We show that our approach can be extended to develop Gaussian kernels on any embedded manifold. Furth…
The large amount of online data and vast array of computing resources enable current researchers in both industry and academia to employ the power of deep learning with neural networks. While deep models trained with massive amounts of data demonstrate promising generalisation ability on unseen data from relevant domains, the computational cost of finetuning gradually becomes a bottleneck in transfering the learning to new domains. We address this issue by leveraging the low-rank property of learnt feature vectors produced from deep neural networks (DNNs) with the closed-form solution provided…
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated int…
Nara Institute of Science and Technology（奈良先端科学技术大学院大学）；The Institute of Statistical Mathematics（统计数学研究所）；National Institute of Genetics（国立遗传学研究所）
This paper discusses the effect of hubness in zero-shot learning, when ridge regression is used to find a mapping between the example space to the label space. Contrary to the existing approach, which attempts to find a mapping from the example space to the label space, we show that mapping labels into the example space is desirable to suppress the emergence of hubs in the subsequent nearest neighbor search step. Assuming a simple data model, we prove that the proposed approach indeed reduces hubness. This was verified empirically on the tasks of bilingual lexicon extraction and image labeling…
Boosting is a learning scheme that combines weak prediction rules to produce a strong composite estimator, with the underlying intuition that one can obtain accurate prediction rules by combining "rough" ones. Although boosting is proved to be consistent and overfitting-resistant, its numerical convergence rate is relatively slow. The aim of this paper is to develop a new boosting strategy, called the re-scale boosting (RBoosting), to accelerate the numerical convergence rate and, consequently, improve the learning performance of boosting. Our studies show that RBoosting possesses the almost o…
Indian Institute of Technology Indore（印度理工学院印多尔分校）；Nanyang Technological University（南洋理工大学）；Aarhus University（奥胡斯大学）
In this paper, a multi-layer architecture (in a hierarchical fashion) by stacking various Kernel Ridge Regression (KRR) based Auto-Encoder for one-class classification is proposed and is referred as MKOC. MKOC has many layers of Auto-Encoders to project the input features into new feature space and the last layer was regression based one class classifier. The Auto-Encoders use an unsupervised approach of learning and the final layer uses semi-supervised (trained by only positive samples) approach of learning. The proposed MKOC is experimentally evaluated on 15 publicly available benchmark data…
Computer Learning Research Centre（计算机学习研究中心）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）
This paper derives an identity connecting the square loss of ridge regression in on-line mode with the loss of the retrospectively best regressor. Some corollaries about the properties of the cumulative loss of on-line ridge regression are also obtained.
In this paper, we investigate a divide and conquer approach to Kernel Ridge Regression (KRR). Given n samples, the division step involves separating the points based on some underlying disjoint partition of the input space (possibly via clustering), and then computing a KRR estimate for each partition. The conquering step is simple: for each partition, we only consider its own local estimate for prediction. We establish conditions under which we can give generalization bounds for this estimator, as well as achieve optimal minimax rates. We also show that the approximation error component of th…
The University of Washington eScience Institute（华盛顿大学电子科学研究所）；University of Washington（华盛顿大学）；Center for Magnetic Resonance Research（磁共振研究中心）；University of Minnesota（明尼苏达大学）
Ridge regression (RR) is a regularization technique that penalizes the L2-norm of the coefficients in linear regression. One of the challenges of using RR is the need to set a hyperparameter ($α$) that controls the amount of regularization. Cross-validation is typically used to select the best $α$ from a set of candidates. However, efficient and appropriate selection of $α$ can be challenging, particularly where large amounts of data are analyzed. Because the selected $α$ depends on the scale of the data and predictors, it is not straightforwardly interpretable. Here, we propose to reparameter…
Lucerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）
Boosting algorithms are frequently used in applied data science and in research. To date, the distinction between boosting with either gradient descent or second-order Newton updates is often not made in both applied and methodological research, and it is thus implicitly assumed that the difference is irrelevant. The goal of this article is to clarify this situation. In particular, we present gradient and Newton boosting, as well as a hybrid variant of the two, in a unified framework. We compare these boosting algorithms with trees as base learners using various datasets and loss functions. Ou…
We develop a novel preconditioning method for ridge regression, based on recent linear sketching methods. By equipping Stochastic Variance Reduced Gradient (SVRG) with this preconditioning process, we obtain a significant speed-up relative to fast stochastic methods such as SVRG, SDCA and SAG.
Kernel method-based one-class classifier is mainly used for outlier or novelty detection. In this letter, kernel ridge regression (KRR) based one-class classifier (KOC) has been extended for learning using privileged information (LUPI). LUPI-based KOC method is referred to as KOC+. This privileged information is available as a feature with the dataset but only for training (not for testing). KOC+ utilizes the privileged information differently compared to normal feature information by using a so-called correction function. Privileged information helps KOC+ in achieving better generalization pe…
Despite its impressive theory \& practical performance, Frequent Directions (\acrshort{fd}) has not been widely adopted for large-scale regression tasks. Prior work has shown randomized sketches (i) perform worse in estimating the covariance matrix of the data than \acrshort{fd}; (ii) incur high error when estimating the bias and/or variance on sketched ridge regression. We give the first constant factor relative error bounds on the bias \& variance for sketched ridge regression using \acrshort{fd}. We complement these statistical results by showing that \acrshort{fd} can be used in the optimi…
General predictive models do not provide a measure of confidence in predictions without Bayesian assumptions. A way to circumvent potential restrictions is to use conformal methods for constructing non-parametric confidence regions, that offer guarantees regarding validity. In this paper we provide a detailed description of a computationally efficient conformal procedure for Kernel Ridge Regression (KRR), and conduct a comparative numerical study to see how well conformal regions perform against the Bayesian confidence sets. The results suggest that conformalized KRR can yield predictive confi…
In this paper, we propose a random projection approach to estimate variance in kernel ridge regression. Our approach leads to a consistent estimator of the true variance, while being computationally more efficient. Our variance estimator is optimal for a large family of kernels, including cubic splines and Gaussian kernels. Simulation analysis is conducted to support our theory.
CREST（经济与统计研究中心）；ENSAE（法国国立统计与经济管理学院）；Istituto Italiano di Tecnologia（意大利理工学院）；MIT（麻省理工学院）
In this note, we provide an elementary analysis of the prediction error of ridge regression with random design. The proof is short and self-contained. In particular, it bypasses the use of Rudelson's deviation inequality for covariance matrices, through a combination of exchangeability arguments, matrix perturbation and operator convexity.
Technion — Israel Institute of Technology（以色列理工学院）
We consider the most common variants of linear regression, including Ridge, Lasso and Support-vector regression, in a setting where the learner is allowed to observe only a fixed number of attributes of each example at training time. We present simple and efficient algorithms for these problems: for Lasso and Ridge regression they need the same total number of attributes (up to constants) as do full-information algorithms, for reaching a certain accuracy. For Support-vector regression, we require exponentially less attributes compared to the state of the art. By that, we resolve an open proble…
We propose an $L_{2}$-based penalization algorithm for functional linear regression models, where the coefficient function is shrunk towards a data-driven shape template $γ$, which is constrained to belong to a class of piecewise functions by restricting its basis expansion. In particular, we focus on the case where $γ$ can be expressed as a sum of $q$ rectangles that are adaptively positioned with respect to the regression error. As the problem of finding the optimal knot placement of a piecewise function is nonconvex, the proposed parametrization allows to reduce the number of variables in t…
Random binning features, introduced in the seminal paper of Rahimi and Recht (2007), are an efficient method for approximating a kernel matrix using locality sensitive hashing. Random binning features provide a very simple and efficient way of approximating the Laplace kernel but unfortunately do not apply to many important classes of kernels, notably ones that generate smooth Gaussian processes, such as the Gaussian kernel and Matern kernel. In this paper, we introduce a simple weighted version of random binning features and show that the corresponding kernel function generates Gaussian proce…
We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online ridge regression and the forward algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without…
We provide a deterministic space-efficient algorithm for estimating ridge regression. For $n$ data points with $d$ features and a large enough regularization parameter, we provide a solution within $\varepsilon$ L$_2$ error using only $O(d/\varepsilon)$ space. This is the first $o(d^2)$ space deterministic streaming algorithm with guaranteed solution error and risk bound for this classic problem. The algorithm sketches the covariance matrix by variants of Frequent Directions, which implies it can operate in insertion-only streams and a variety of distributed data settings. In comparisons to ra…
In supervised batch learning, the predictive normalized maximum likelihood (pNML) has been proposed as the min-max regret solution for the distribution-free setting, where no distributional assumptions are made on the data. However, the pNML is not defined for a large capacity hypothesis class as over-parameterized linear regression. For a large class, a common approach is to use regularization or a model prior. In the context of online prediction where the min-max solution is the Normalized Maximum Likelihood (NML), it has been suggested to use NML with ``luckiness'': A prior-like function is…
We introduce single-set spectral sparsification as a deterministic sampling based feature selection technique for regularized least squares classification, which is the classification analogue to ridge regression. The method is unsupervised and gives worst-case guarantees of the generalization power of the classification function after feature selection with respect to the classification function obtained using all features. We also introduce leverage-score sampling as an unsupervised randomized feature selection method for ridge regression. We provide risk bounds for both single-set spectral …
We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimal…
We consider sketched approximate matrix multiplication and ridge regression in the novel setting of localized sketching, where at any given point, only part of the data matrix is available. This corresponds to a block diagonal structure on the sketching matrix. We show that, under mild conditions, block diagonal sketching matrices require only O(stable rank / ε^2) and $O( stat. dim. ε)$ total sample complexity for matrix multiplication and ridge regression, respectively. This matches the state-of-the-art bounds that are obtained using global sketching matrices. The localized nature of sketchin…
Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees. Cross-validation (CV) is widely used for hyperparameter tuning in these models, but do practical optimization methods minimize the true out-of-sample loss? A recent line of research promises to show that the optimum of the CV loss matches the optimum of the out-of-sample loss (possibly after simple corrections). It remains to show how tractable it is to minimize the CV loss. In the present paper, we show that, in the case of ridge regression, th…
University of California, Irvine（加州大学欧文分校）；Technical University of Berlin（柏林工业大学）；Max Planck Institute of Microstructure Physics（马克斯·普朗克微结构物理研究所）；University of Basel（巴塞尔大学）；Korea University（高丽大学）
Accurate approximations to density functionals have recently been obtained via machine learning (ML). By applying ML to a simple function of one variable without any random sampling, we extract the qualitative dependence of errors on hyperparameters. We find universal features of the behavior in extreme limits, including both very small and very large length scales, and the noise-free limit. We show how such features arise in ML models of density functionals.
Amazon（亚马逊）；University of Oxford（牛津大学）；University of Pennsylvania（宾夕法尼亚大学）；Oxford University（牛津大学）
Methods for learning from data depend on various types of tuning parameters, such as penalization strength or step size. Since performance can depend strongly on these parameters, it is important to compare classes of estimators-by considering prescribed finite sets of tuning parameters-not just particularly tuned methods. In this work, we investigate classes of methods via the relative performance of the best method in the class. We consider the central problem of linear regression-with a random isotropic ground truth-and investigate the estimation performance of two fundamental methods, grad…
Lucerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）
We introduce a novel boosting algorithm called `KTBoost' which combines kernel boosting and tree boosting. In each boosting iteration, the algorithm adds either a regression tree or reproducing kernel Hilbert space (RKHS) regression function to the ensemble of base learners. Intuitively, the idea is that discontinuous trees and continuous RKHS regression functions complement each other, and that this combination allows for better learning of functions that have parts with varying degrees of regularity such as discontinuities and smooth parts. We empirically show that KTBoost significantly outp…
University of Chicago, Booth School of Business（芝加哥大学布斯商学院）；Massachusetts Institute of Technology（麻省理工学院）
In the absence of explicit regularization, Kernel "Ridgeless" Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition t…
Columbia University（哥伦比亚大学）；Microsoft Research（微软研究院）；Rutgers University（罗格斯大学）
This work gives a simultaneous analysis of both the ordinary least squares estimator and the ridge regression estimator in the random design setting under mild assumptions on the covariate/response distributions. In particular, the analysis provides sharp results on the ``out-of-sample'' prediction error, as opposed to the ``in-sample'' (fixed design) error. The analysis also reveals the effect of errors in the estimated covariance structure, as well as the effect of modeling errors, neither of which effects are present in the fixed design setting. The proofs of the main results are based on a…
