Minimax Optimal Rate Research Articles

Understanding Implicit Regularization in Over-Parameterized Single Index Model

In this article, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding of the role played by implicit regularization without excess technicality, we assume that the distribution of the covariates is known a priori. For both the vector and matrix settings, we construct an over-parameterized least-squares loss function by employing the score function transform and a robust truncation step designed specifically for heavy-tailed data. We propose to estimate the true parameter by applying regularization-free gradient descent to the loss function. When the initialization is close to the origin and the stepsize is sufficiently small, we prove that the obtained solution achieves minimax optimal statistical rates of convergence in both the vector and matrix cases. In addition, our experimental results support our theoretical findings and also demonstrate that our methods empirically outperform classical methods with explicit regularization in terms of both l 2 -statistical rate and variable selection consistency. Supplementary materials for this article are available online.

An optimal statistical and computational framework for generalized tensor estimation

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of nonconvexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data.

Estimation of conditional mean operator under the bandable covariance structure

We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate the regression coefficient matrix, which is a function of the bandable covariance matrix. Although the tapering estimator of covariance has the minimax optimal convergence rate for the class of bandable covariances, we show that it is sub-optimal for the regression coefficient; that is, a minimax estimator for the class of bandable covariances may not be a minimax estimator for its functionals. We propose the blockwise tapering estimator of the regression coefficient, which has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We also propose a Bayesian procedure called the blockwise tapering post-processed posterior of the regression coefficient and show that the proposed Bayesian procedure has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We show that the proposed methods outperform the existing methods via numerical studies.

Minimax bounds for estimating multivariate Gaussian location mixtures

We prove minimax bounds for estimating Gaussian location mixtures on Rd under the squared L2 and the squared Hellinger loss functions. Under the squared L2 loss, we prove that the minimax optimal rate is upper and lower bounded by a constant multiple of n−1(logn)d∕2. Under the squared Hellinger loss, we consider two subclasses based on the behavior of the tails of the mixing measure. When the mixing measure has a sub-Gaussian tail, the minimax rate under the squared Hellinger loss is bounded from below by (logn)d∕n, which implies that the optimal minimax rate is between (logn)d∕n and the upper bound (logn)d+1∕n obtained by [11]. On the other hand, when the mixing measure is only assumed to have a bounded pth moment for a fixed p>0, the minimax rate under the squared Hellinger loss is bounded from below by n−p∕(p+d)(logn)−3d∕2. This rate is minimax optimal up to logarithmic factors.

On robust learning in the canonical change point problem under heavy tailed errors in finite and growing dimensions

This paper presents a number of new findings about the canonical change point estimation problem. The first part studies the estimation of a change point on the real line in a simple stump model using the robust Huber estimating function which interpolates between the ℓ1 (absolute deviation) and ℓ2 (least squares) based criteria. While the ℓ2 criterion has been studied extensively, its robust counterparts and in particular, the ℓ1 minimization problem have not. We derive the limit distribution of the estimated change point under the Huber estimating function and compare it to that under the ℓ2 criterion. Theoretical and empirical studies indicate that it is more profitable to use the Huber estimating function (and in particular, the ℓ1 criterion) under heavy tailed errors as it leads to smaller asymptotic confidence intervals at the usual levels compared to the ℓ2 criterion. We also compare the ℓ1 and ℓ2 approaches in a parallel setting, where one has m independent single change point problems and the goal is to control the maximal deviation of the estimated change points from the true values, and establish rigorously that the ℓ1 estimation criterion provides a superior rate of convergence to the ℓ2, and that this relative advantage is driven by the heaviness of the tail of the error distribution. Finally, we derive minimax optimal rates for the change plane estimation problem in growing dimensions and demonstrate that Huber estimation attains the optimal rate while the ℓ2 scheme produces a rate sub-optimal estimator for heavy tailed errors. In the process of deriving our results, we establish a number of properties about the minimizers of compound Binomial and compound Poisson processes which are of independent interest.

Minimax optimal conditional density estimation under total variation smoothness

This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in a multivariate setting. We first demonstrate that conditional density estimation is impossible if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$. This motivates us to consider a sub-class of absolutely continuous distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only be H\"older smooth in $x$, but also be total variation smooth in $z$. We propose a corresponding kernel-based estimator and prove that it achieves the minimax rate. We give some simple examples of densities satisfying our assumptions which imply that our results are not vacuous. Finally, we propose an estimator which achieves the minimax optimal rate adaptively, i.e., without the need to know the smoothness parameter values in advance. Crucially, both of our estimators (the adaptive and non-adaptive ones) impose no assumptions on the marginal density $p_Z$, and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.

Adaptive transfer learning

In transfer learning, we wish to make inference about a target population when we have access to data both from the distribution itself, and from a different but related source distribution. We introduce a flexible framework for transfer learning in the context of binary classification, allowing for covariate-dependent relationships between the source and target distributions that are not required to preserve the Bayes decision boundary. Our main contributions are to derive the minimax optimal rates of convergence (up to poly-logarithmic factors) in this problem, and show that the optimal rate can be achieved by an algorithm that adapts to key aspects of the unknown transfer relationship, as well as the smoothness and tail parameters of our distributional classes. This optimal rate turns out to have several regimes, depending on the interplay between the relative sample sizes and the strength of the transfer relationship, and our algorithm achieves optimality by careful, decision tree-based calibration of local nearest-neighbour procedures.

Over-parametrized deep neural networks minimizing the empirical risk do not generalize well

Recently it was shown in several papers that backpropagation is able to find the global minimum of the empirical risk on the training data using over-parametrized deep neural networks. In this paper, a similar result is shown for deep neural networks with the sigmoidal squasher activation function in a regression setting, and a lower bound is presented which proves that these networks do not generalize well on a new data in the sense that networks which minimize the empirical risk do not achieve the optimal minimax rate of convergence for estimation of smooth regression functions.

Framelet block thresholding estimator for sparse functional data

Nonparametric estimation of mean and covariance functions based on discretely observed data is important in functional data analysis. In this paper, we propose a framelet block thresholding method for the case of sparsely observed functional data. The procedure is easily implemented and the resultant estimators are represented as explicit B-spline expressions. For sparsely observed functional data, we establish, under some mild conditions but without knowing the smoothness parameter, convergence rates of mean integrated squared errors for mean and covariance estimators respectively. In particular, the mean estimator attains minimax optimal rate. The simulated and real data examples are provided to offer empirical support of the theoretical properties. Compared to the existing methods, the proposed method outperforms in adapting automatically to local variations.

Optimal minimax rates against nonsmooth alternatives

Summary This study investigates optimal minimax rates for specification testing when the alternative hypothesis is built on a set of nonsmooth functions. The set consists of bounded functions that are not necessarily differentiable with no smoothness constraints imposed on their derivatives. In the instrumental variable regression set up with an unknown error variance structure, we find that the optimal minimax rate is $n^{-1/4}$, where n is the sample size. The rate is achieved by a simple test based on the difference between nonparametric and parametric variance estimators. Simulation studies illustrate that the test has reasonable power against various nonsmooth alternatives. The empirical application to Engel curves specification emphasizes the good applicability of the test.

Adaptive transfer learning

In transfer learning, we wish to make inference about a target population when we have access to data both from the distribution itself, and from a different but related source distribution. We introduce a flexible framework for transfer learning in the context of binary classification, allowing for covariate-dependent relationships between the source and target distributions that are not required to preserve the Bayes decision boundary. Our main contributions are to derive the minimax optimal rates of convergence (up to poly-logarithmic factors) in this problem, and show that the optimal rate can be achieved by an algorithm that adapts to key aspects of the unknown transfer relationship, as well as the smoothness and tail parameters of our distributional classes. This optimal rate turns out to have several regimes, depending on the interplay between the relative sample sizes and the strength of the transfer relationship, and our algorithm achieves optimality by careful, decision tree-based calibration of local nearest-neighbour procedures.

Fast and Separable Estimation in High-Dimensional Tensor Gaussian Graphical Models

In the tensor data analysis, the Kronecker covariance structure plays a vital role in unsupervised learning and regression. Under the Kronecker covariance model assumption, the covariance of an M-way tensor is parameterized as the Kronecker product of M individual covariance matrices. With normally distributed tensors, the key to high-dimensional tensor graphical models becomes the sparse estimation of the M inverse covariance matrices. Unable to maximize the tensor normal likelihood analytically, existing approaches often require cyclic updates of the M sparse matrices. For the high-dimensional tensor graphical models, each update step solves a regularized inverse covariance estimation problem that is computationally nontrivial. This computational challenge motivates our study of whether a noncyclic approach can be as good as the cyclic algorithms in theory and practice. To handle the potentially very high-dimensional and high-order tensors, we propose a separable and parallel estimation scheme. We show that the new estimator achieves the same minimax optimal convergence rate as the cyclic estimation approaches. Numerically, the new estimator is much faster and often more accurate than the cyclic approach. Moreover, another advantage of the separable estimation scheme is its flexibility in modeling, where we can easily incorporate user-specified or specially structured covariances on any modes of the tensor. We demonstrate the efficiency of the proposed method through both simulations and a neuroimaging application. Supplementary materials for this article are available online.

High-dimensional index volatility models via Stein’s identity

We study the estimation of the parametric components of single and multiple index volatility models. Using the first- and second-order Stein’s identities, we develop methods that are applicable for the estimation of the variance index in the high-dimensional setting requiring finite moment condition, which allows for heavy-tailed data. Our approach complements the existing literature in the low-dimensional setting, while relaxing the conditions on estimation, and provides a novel approach in the high-dimensional setting. We prove that the statistical rate of convergence of our variance index estimators consists of a parametric rate and a nonparametric rate, where the latter appears from the estimation of the mean link function. However, under standard assumptions, the parametric rate dominates the rate of convergence and our results match the minimax optimal rate for the mean index estimation. Simulation results illustrate finite sample properties of our methodology and back our theoretical conclusions.

High-dimensional nonparametric density estimation via symmetry and shape constraints

We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on Rp and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are K-homothetic (i.e., scalar multiples of a convex body K⊆Rp). When K is known, we prove that the K-homothetic log-concave maximum likelihood estimator based on n independent observations from such a density achieves the minimax optimal rate of convergence with respect to, for example, squared Hellinger loss, of order n−4/5, independent of p. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst case and adaptive risk bounds in cases where K is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when n and p are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data.

De-Biased Sparse PCA: Inference for Eigenstructure of Large Covariance Matrices

Sparse principal component analysis has become one of the most widely used techniques for dimensionality reduction in high-dimensional datasets. While many methods are available for point estimation of eigenstructure in high-dimensional settings, in this paper we propose methodology for uncertainty quantification, such as construction of confidence intervals and tests for the principal eigenvector and the corresponding largest eigenvalue. We base our methodology on an M-estimator with Lasso penalty which achieves minimax optimal rates and is used to construct a de-biased sparse PCA estimator. The novel estimator has a Gaussian limiting distribution and can be used for hypothesis testing or support recovery of the first eigenvector. The empirical performance of the new estimator is demonstrated on synthetic data and we also show that the estimator compares favourably with the classical PCA in moderately high-dimensional regimes.

Multiscale Quantile Segmentation

We introduce a new methodology for analyzing serial data by quantile regression assuming that the underlying quantile function consists of constant segments. The procedure does not rely on any distributional assumption besides serial independence. It is based on a multiscale statistic, which allows to control the (finite sample) probability for selecting the correct number of segments S at a given error level, which serves as a tuning parameter. For a proper choice of this parameter, this probability tends exponentially fast to one, as sample size increases. We further show that the location and size of segments are estimated at minimax optimal rate (compared to a Gaussian setting) up to a log-factor. Thereby, our approach leads to (asymptotically) uniform confidence bands for the entire quantile regression function in a fully nonparametric setup. The procedure is efficiently implemented using dynamic programming techniques with double heap structures, and software is provided. Simulations and data examples from genetic sequencing and ion channel recordings confirm the robustness of the proposed procedure, which at the same time reliably detects changes in quantiles from arbitrary distributions with precise statistical guarantees. Supplementary materials for this article are available online.

Statistically optimal and computationally efficient low rank tensor completion from noisy entries

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell _{p}$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.

Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs

We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on n independent samples from them. Curiously, we show that estimating the IPM itself between probability measures is not significantly easier than estimating the probability measures under the IPM. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a log log(n)/ log(n) factor.

How Well Generative Adversarial Networks Learn Distributions

This paper studies the rates of convergence for learning distributions implicitly with the adversarial framework and Generative Adversarial Networks (GAN), which subsume Wasserstein, Sobolev, MMD GAN, and Generalized/Simulated Method of Moments (GMM/SMM) as special cases. We study a wide range of parametric and nonparametric target distributions, under a host of objective evaluation metrics. We investigate how to obtain a good statistical guarantee for GANs through the lens of regularization. On the nonparametric end, we derive the optimal minimax rates for distribution estimation under the adversarial framework. On the parametric end, we establish a theory for general neural network classes (including deep leaky ReLU networks), that characterizes the interplay on the choice of generator and discriminator pair. We discover and isolate a new notion of regularization, called the generator-discriminator-pair regularization, that sheds light on the advantage of GANs compared to classical parametric and nonparametric approaches for explicit distribution estimation. We develop novel oracle inequalities as the main technical tools for analyzing GANs, which is of independent interest.

Minimax Rates in Network Analysis: Graphon Estimation, Community Detection and Hypothesis Testing

This paper surveys some recent developments in fundamental limits and optimal algorithms for network analysis. We focus on minimax optimal rates in three fundamental problems of network analysis: graphon estimation, community detection and hypothesis testing. For each problem, we review state-of-the-art results in the literature followed by general principles behind the optimal procedures that lead to minimax estimation and testing. This allows us to connect problems in network analysis to other statistical inference problems from a general perspective.