Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. Consider the points $X_1, X_2,..., X_n$ are vectors drawn i.i.d. from a distribution with mean zero and covariance $Σ$, where $Σ$ is unknown. Let $A_n = X_nX_n^T$, then $E[A_n] = Σ$. This paper consider the problem of finding the least eigenvalue and eigenvector of matrix $Σ$. A classical such estimator are due to Krasulina\cite{krasulina_method_1969}. We are going to state the convergence proof of Krasulina for the least eigenvalue and corresponding eigenvector, and then find their convergence rate.
University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Google Brain（谷歌大脑）
We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly Q…
We present Matrix Krasulina, an algorithm for online k-PCA, by generalizing the classic Krasulina's method (Krasulina, 1969) from vector to matrix case. We show, both theoretically and empirically, that the algorithm naturally adapts to data low-rankness and converges exponentially fast to the ground-truth principal subspace. Notably, our result suggests that despite various recent efforts to accelerate the convergence of stochastic-gradient based methods by adding a O(n)-time variance reduction step, for the k-PCA problem, a truly online SGD variant suffices to achieve exponential convergence…
University of Michigan（密歇根大学）；Rutgers University–New Brunswick（罗格斯大学新伯朗士威分校）
This paper considers the problem of estimating the principal eigenvector of a covariance matrix from independent and identically distributed data samples in streaming settings. The streaming rate of data in many contemporary applications can be high enough that a single processor cannot finish an iteration of existing methods for eigenvector estimation before a new sample arrives. This paper formulates and analyzes a distributed variant of the classical Krasulina's method (D-Krasulina) that can keep up with the high streaming rate of data by distributing the computational load across multiple …
We consider a situation in which we see samples in $\mathbb{R}^d$ drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both.
KSG mutual information estimator, which is based on the distances of each sample to its k-th nearest neighbor, is widely used to estimate mutual information between two continuous random variables. Existing work has analyzed the convergence rate of this estimator for random variables whose densities are bounded away from zero in its support. In practice, however, KSG estimator also performs well for a much broader class of distributions, including not only those with bounded support and densities bounded away from zero, but also those with bounded support but densities approaching zero, and th…
Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor classification has not fully reflected these subtle properties. We analyze the behavior of these estimators in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. As a by-product, we are able to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previo…
We derive the $l_{\infty}$ convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two different assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small.
In this paper, we investigate the statistical convergence rate of a Bayesian low-rank tensor estimator. Our problem setting is the regression problem where a tensor structure underlying the data is estimated. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a near optimal rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adapt…
Zou [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and the initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle l…
University of Electronic Science and Technology of China（电子科技大学）；Princeton University（普林斯顿大学）
This paper considers the problem of estimation of the Fisher information for location from a random sample of size $n$. First, an estimator proposed by Bhattacharya is revisited and improved convergence rates are derived. Second, a new estimator, termed a clipped estimator, is proposed. Superior upper bounds on the rates of convergence can be shown for the new estimator compared to the Bhattacharya estimator, albeit with different regularity conditions. Third, both of the estimators are evaluated for the practically relevant case of a random variable contaminated by Gaussian noise. Moreover, u…
In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal estimator is the square-root Lasso, formulated along with its derivatives as a "non-smooth + non-smooth" optimization problem. Modern techniques to solve these include smoothing the datafitting term, to benefit from fast efficient proximal algorithms. In this work we show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators. Thanks to our theoretic…
Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable densities. Kernel estimators based on Stein's methods or score matching have shown promise, however their theoretical properties and relationships have not been fully-understood. We provide a unifying view of these estimators under the framework of regularized nonparametric regression. It allows us to analyse existing estimators and construct new ones with desira…
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）
Differential privacy is a cryptographically-motivated definition of privacy which has gained significant attention over the past few years. Differentially private solutions enforce privacy by adding random noise to a function computed over the data, and the challenge in designing such algorithms is to control the added noise in order to optimize the privacy-accuracy-sample size tradeoff. This work studies differentially-private statistical estimation, and shows upper and lower bounds on the convergence rates of differentially private approximations to statistical estimators. Our results reveal…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
Stochastic gradient descent (SGD) is the workhorse of modern machine learning. Sometimes, there are many different potential gradient estimators that can be used. When so, choosing the one with the best tradeoff between cost and variance is important. This paper analyzes the convergence rates of SGD as a function of time, rather than iterations. This results in a simple rule to select the estimator that leads to the best optimization convergence guarantee. This choice is the same for different variants of SGD, and with different assumptions about the objective (e.g. convexity or smoothness). I…
Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained $\ell_1$ minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically. A major step in establishing the minimax rate of c…
We consider the problem of robustifying high-dimensional structured estimation. Robust techniques are key in real-world applications which often involve outliers and data corruption. We focus on trimmed versions of structurally regularized M-estimators in the high-dimensional setting, including the popular Least Trimmed Squares estimator, as well as analogous estimators for generalized linear models and graphical models, using possibly non-convex loss functions. We present a general analysis of their statistical convergence rates and consistency, and then take a closer look at the trimmed vers…
University of Michigan（密歇根大学）；Yale University（耶鲁大学）
Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze-Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman-Rafsky (FR) estimator of…
Crowdsourcing has become a primary means for label collection in many real-world machine learning applications. A classical method for inferring the true labels from the noisy labels provided by crowdsourcing workers is Dawid-Skene estimator. In this paper, we prove convergence rates of a projected EM algorithm for the Dawid-Skene estimator. The revealed exponent in the rate of convergence is shown to be optimal via a lower bound argument. Our work resolves the long standing issue of whether Dawid-Skene estimator has sound theoretical guarantees besides its good performance observed in practic…
This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the reproducing kernel Hilbert space to the conditional expectation and this implies local and uniform convergence of this function estimator. By preselecting the kernel, the problem does not suffer from the curse of dimensionality. The paper analyzes the statistical prop…
We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression and its plug-in property for differential operators. For a general class of kernels, we establish convergence rates of the posterior measure of the regression function and its derivatives, which are both minimax optimal up to a logarithmic factor for functions in certain classes. Our calculation shows that the rate-optimal estimation of the regression function and its derivatives share the same choice of hyperparameter, indicating that the Bayes procedure remarkably adapts to …
Optimal transport maps between two probability distributions $μ$ and $ν$ on $\mathbb{R}^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to $μ$ and $ν$. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projectio…
Several new estimation methods have been recently proposed for the linear regression model with observation error in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar δ$, and (2) zero mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar δ$, while the second assumption has been applied when an estimator for the second moment of the o…
We consider nonparametric estimation of $L_2$, Renyi-$α$ and Tsallis-$α$ divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plug-in estimator. We show that these estimators achieve the parametric convergence rate of $n^{-1/2}$ when the densities' smoothness, $s$, are both at least $d/4$ where $d$ is the dimension. We also derive minimax lower bounds for this problem which con…
This paper presents a class of new algorithms for distributed statistical estimation that exploit divide-and-conquer approach. We show that one of the key benefits of the divide-and-conquer strategy is robustness, an important characteristic for large distributed systems. We establish connections between performance of these distributed algorithms and the rates of convergence in normal approximation, and prove non-asymptotic deviations guarantees, as well as limit theorems, for the resulting estimators. Our techniques are illustrated through several examples: in particular, we obtain new resul…
Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistical implications of nesting MC estimators, including cases of multiple levels of nesting, and establish the conditions under which they converge. We derive corresponding rates of convergence and provide empirical evidence that these rates are observed in practice. We further establ…
We give a comprehensive theoretical characterization of a nonparametric estimator for the $L_2^2$ divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is $\sqrt{n}$-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.
