We present a new finite-time analysis of the estimation error of the Ordinary Least Squares (OLS) estimator for stable linear time-invariant systems. We characterize the number of observed samples (the length of the observed trajectory) sufficient for the OLS estimator to be $(\varepsilon,δ)$-PAC, i.e., to yield an estimation error less than $\varepsilon$ with probability at least $1-δ$. We show that this number matches existing sample complexity lower bounds [1,2] up to universal multiplicative factors (independent of ($\varepsilon,δ)$ and of the system). This paper hence establishes the optimality of the OLS estimator for stable systems, a result conjectured in [1]. Our analysis of the performance of the OLS estimator is simpler, sharper, and easier to interpret than existing analyses. It relies on new concentration results for the covariates matrix.
We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size $k$ and desired accuracy $\varepsilon$: $\textbf{Distribution estimation}$ Under $\ell_1$ distance, PML yields optimal $Θ(k/(\varepsilon^2\log k))$ sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual dis…
University of Notre Dame（圣母大学）；Seoul National University（首尔国立大学）
Recent theoretical studies proved that deep neural network (DNN) estimators obtained by minimizing empirical risk with a certain sparsity constraint can attain optimal convergence rates for regression and classification problems. However, the sparsity constraint requires to know certain properties of the true model, which are not available in practice. Moreover, computation is difficult due to the discrete nature of the sparsity constraint. In this paper, we propose a novel penalized estimation method for sparse DNNs, which resolves the aforementioned problems existing in the sparsity constrai…
Sequential change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. When the post-change parameters are unknown, we consider a set of detection procedures based on sequential likelihood ratios with non-anticipating estimators constructed using online convex optimization algorithms such as online mirror descent, which provides a more versatile approach to tackle complex situations where recursive maximum likelihood estimators cannot be found. When the underlying distributions belong to a exponential family and the estimator…
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Indiana University–Purdue University Indianapolis（印第安纳大学与普渡大学印第安纳波利斯联合分校）
Estimator algorithms in learning automata are useful tools for adaptive, real-time optimization in computer science and engineering applications. This paper investigates theoretical convergence properties for a special case of estimator algorithms: the pursuit learning algorithm. In this note, we identify and fill a gap in existing proofs of probabilistic convergence for pursuit learning. It is tradition to take the pursuit learning tuning parameter to be fixed in practical applications, but our proof sheds light on the importance of a vanishing sequence of tuning parameters in a theoretical c…
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…
Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are statistically inefficient as estimators of the true parameter value. To address these two issues, we propose a new iterative procedure termed averaged implicit SGD (AI-SGD). For statistical efficiency, AI-SGD employs averaging of the iterates, which achieves the optimal Cramér-Rao bound under strong convexity, i.e., it is an optimal unbiased estimator of the true pa…
Princeton University（普林斯顿大学）；University of Liège（列日大学）；Université Paul Sabatier（保罗·萨巴捷大学）
We consider an original problem that arises from the issue of security analysis of a power system and that we name optimal discovery with probabilistic expert advice. We address it with an algorithm based on the optimistic paradigm and on the Good-Turing missing mass estimator. We prove two different regret bounds on the performance of this algorithm under weak assumptions on the probabilistic experts. Under more restrictive hypotheses, we also prove a macroscopic optimality result, comparing the algorithm both with an oracle strategy and with uniform sampling. Finally, we provide numerical ex…
The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk…
The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory and is in common use for analyzing the far tail of observed phenomena. Curiously, important asymptotic properties of likelihood-based estimation under this standard model have yet to be established. In this paper, we formally prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighborhood of the shape parameter.
In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions $f_1$, such that no other estimator can attain a better rate for both $f_0$ and $f_1$. We…
An important question in constructing Cross Validation (CV) estimators of the generalization error is whether rules can be established that allow "optimal" selection of the size of the training set, for fixed sample size $n$. We define the {\it resampling effectiveness} of random CV estimators of the generalization error as the ratio of the limiting value of the variance of the CV estimator over the estimated from the data variance. The variance and the covariance of different average test set errors are independent of their indices, thus, the resampling effectiveness depends on the correlatio…
Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asy…
Unmeasured confounding is a threat to causal inference and gives rise to biased estimates. In this article, we consider the problem of individualized decision-making under partial identification. Firstly, we argue that when faced with unmeasured confounding, one should pursue individualized decision-making using partial identification in a comprehensive manner. We establish a formal link between individualized decision-making under partial identification and classical decision theory by considering a lower bound perspective of value/utility function. Secondly, building on this unified framewor…
Variational methods are extremely popular in the analysis of network data. Statistical guarantees obtained for these methods typically provide asymptotic normality for the problem of estimation of global model parameters under the stochastic block model. In the present work, we consider the case of networks with missing links that is important in application and show that the variational approximation to the maximum likelihood estimator converges at the minimax rate. This provides the first minimax optimal and tractable estimator for the problem of parameter estimation for the stochastic block…
We consider the problem of optimality in manifold reconstruction. A random sample $\mathbb{X}_n = \left\{X_1,\ldots,X_n\right\}\subset \mathbb{R}^D$ composed of points close to a $d$-dimensional submanifold $M$, with or without outliers drawn in the ambient space, is observed. Based on the Tangential Delaunay Complex, we construct an estimator $\hat{M}$ that is ambient isotopic and Hausdorff-close to $M$ with high probability. The estimator $\hat{M}$ is built from existing algorithms. In a model with additive noise of small amplitude, we show that this estimator is asymptotically minimax optim…
It is a standard criterium in statistics to define an optimal estimator the one with the minimum variance. Thus, the optimality is proved with inequality among variances of competing estimators. The inequalities, demonstrated here, disfavor the standard least squares estimators. Inequalities among estimators are connected to names of Cramer, Rao and Frechet. The standard demonstrations of these inequalities require very special analytical properties of the probability functions, globally indicated as regular models. These limiting conditions are too restrictive to handle realistic problems in …
We consider estimation of the common probability density $f$ of i.i.d. random variables $X_i$ that are observed with an additive i.i.d. noise. We assume that the unknown density $f$ belongs to a class $\mathcal{A}$ of densities whose characteristic function is described by the exponent $\exp(-α|u|^r)$ as $|u|\to \infty$, where $α>0$, $r>0$. The noise density is supposed to be known and such that its characteristic function decays as $\exp(-β|u|^s)$, as $|u| \to \infty$, where $β>0$, $s>0$. Assuming that $r<s$, we suggest a kernel type estimator that is optimal in sharp asymptotical minimax sen…
In this article we study the existence and strong consistency of GEE estimators, when the generalized estimating functions are martingales with random coefficients. Furthermore, we characterize estimating functions which are asymptotically optimal.
Accurate approximation of the sampling distribution of nonparametric kernel density estimators is crucial for many statistical inference problems. Since these estimators have complex asymptotic distributions, bootstrap methods are often used for this purpose. With i.i.d. observations, a large literature exists concerning optimal bootstrap methods which achieve the fastest possible convergence rate of the bootstrap estimator of the sampling distribution of the kernel density estimator. With dependent data, such an optimality theory is an important open problem. We establish a general theory of …
We consider the problem of estimating the mean of a random vector based on i.i.d. observations and adversarial contamination. We introduce a multivariate extension of the trimmed-mean estimator and show its optimal performance under minimal conditions.
We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We show that the novel estimator achieves minimax rates in supremum norm and the low-dimensional components of the estimator have a Gaussian limiting distribution. These results hold uniformly over the class of precision matrices with row sparsity of small order $\sqrt{n}/\log p$ and spectrum uniformly bounded, under a sub-Gaussian tail assumption on the margins…
To fast approximate maximum likelihood estimators with massive data, this paper studies the Optimal Subsampling Method under the A-optimality Criterion (OSMAC) for generalized linear models. The consistency and asymptotic normality of the estimator from a general subsampling algorithm are established, and optimal subsampling probabilities under the A- and L-optimality criteria are derived. Furthermore, using Frobenius norm matrix concentration inequalities, finite sample properties of the subsample estimator based on optimal subsampling probabilities are also derived. Since the optimal subsamp…
Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the …
Recently, S. Arlot and R. Genuer have shown that a model of random forests outperforms its single-tree counterpart in the estimation of $α-$Hölder functions, $α\leq2$. This backs up the idea that ensembles of tree estimators are smoother estimators than single trees. On the other hand, most positive optimality results on Bayesian tree-based methods assume that $α\leq1$. Naturally, one wonders whether Bayesian counterparts of forest estimators are optimal on smoother classes, just like it has been observed for frequentist estimators for $α\leq 2$. We dwell on the problem of density estimation a…
We address the problem of density estimation with $\mathbb{L}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].
Coordinated Science Laboratory, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校协调科学实验室）；ECE Department, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电子与计算机工程系）
We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node's state variable is a convex comb…
This paper addresses the following simple question about sparsity. For the estimation of an $n$-dimensional mean vector $\boldsymbolθ$ in the Gaussian sequence model, is it possible to find an adaptive optimal threshold estimator in a full range of sparsity levels where nonadaptive optimality can be achieved by threshold estimators? We provide an explicit affirmative answer as follows. Under the squared loss, adaptive minimaxity in strong and weak $\ell_p$ balls with $0\le p<2$ is achieved by a class of smooth threshold estimators with the threshold level of the Benjamini-Hochberg FDR rule or …
We consider a unified framework of sequential change-point detection and hypothesis testing modeled by means of hidden Markov chains. One observes a sequence of random variables whose distributions are functionals of a hidden Markov chain. The objective is to detect quickly the event that the hidden Markov chain leaves a certain set of states, and to identify accurately the class of states into which it is absorbed. We propose computationally tractable sequential detection and identification strategies and obtain sufficient conditions for the asymptotic optimality in two Bayesian formulations.…
We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the exact minimax prediction and $\ell_2$ estimation rate $(s/n)\log (p/s)$ in high-dimensional linear regression on the class of $s$-sparse target vectors in $\mathbb R^p$. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error. The minimax optimal bo…
In this brief note we present two new parameter identifiers whose estimates converge in finite time under weak interval excitation assumptions. The main novelty is that, in contrast with other finite-convergence time (FCT) estimators, our schemes preserve the FCT property when the parameters change. The previous versions of our FCT estimators can track the parameter variations only asymptotically. Continuous-time and discrete-time versions of the new estimators are presented
We consider estimation of a multivariate normal mean vector under sum of squared error loss. We propose a new class of smooth estimators parameterized by αdominating the James-Stein estimator. The estimator for α=1 corresponds to the generalized Bayes estimator with respect to the harmonic prior. When αgoes to infinity, the estimator converges to the James-Stein positive-part estimator. Thus the class of our estimators is a bridge between the admissible estimator (α=1) and the inadmissible estimator (α=\infty). Although the estimators have quasi-admissibility which is a weaker optimality than …
In large-scale modern data analysis, first-order optimization methods are usually favored to obtain sparse estimators in high dimensions. This paper performs theoretical analysis of a class of iterative thresholding based estimators defined in this way. Oracle inequalities are built to show the nearly minimax rate optimality of such estimators under a new type of regularity conditions. Moreover, the sequence of iterates is found to be able to approach the statistical truth within the best statistical accuracy geometrically fast. Our results also reveal different benefits brought by convex and …
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Thus, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yi…
In this paper, we study two problems: (1) estimation of a $d$-dimensional log-concave distribution and (2) bounded multivariate convex regression with random design with an underlying log-concave density or a compactly supported distribution with a continuous density. First, we show that for all $d \ge 4$ the maximum likelihood estimators of both problems achieve an optimal risk of $Θ_d(n^{-2/(d+1)})$ (up to a logarithmic factor) in terms of squared Hellinger distance and $L_2$ squared distance, respectively. Previously, the optimality of both these estimators was known only for $d\le 3$. We a…
The difference equations $ξ_{k}=af(ξ_{k-1})+ε_{k}$, where $(ε_k)$ is a square integrable difference martingale, and the differential equation ${\rm d}ξ=-af(ξ){\rm d}t+{\rm d}η$, where $η$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated.
We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficien…
Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.
In the realm of the model selection context, Akaike's and Schwarz's information criteria, AIC and BIC, have been applied successfully for decades for model order identification. The Efficient Determination Criterion (EDC) is a generalization of these criteria, proposed originally to define a strongly consistent class of estimators for the dependency order of a multiple Markov chain. In this work, the EDC is generalized to partially nested models, which encompass many other order identification problems. Based on some assumptions, a class of strongly consistent estimators is established in this…
Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.
We consider the problem of the estimation of the invariant distribution function of an ergodic diffusion process when the drift coefficient is unknown. The empirical distribution function is a natural estimator which is unbiased, uniformly consistent and efficient in different metrics. Here we study the properties of optimality for an other kind of estimator recently proposed. We consider a class of unbiased estimators and we show that they are also efficient in the sense that their asymptotic risk, defined as the integrated mean square error, attains an asymptotic minimax lower bound.
In the multiple testing context, a challenging problem is the estimation of the proportion $π_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real data, or may be at least relaxed. Under independence, we propose an estimator $\hatπ_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $π_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $…
