The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space

Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of gλ random variable and its distribution function, expected value, and variance are then presented. Markov inequality, Chebyshev＇s inequality and the Khinchine＇s Law of Large Numbers on Sugeno measure space are also proven. Furthermore, the concepts of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugeno measure space are proposed. According to these properties and concepts, the key theorem of learning theory, the bounds on the rate of convergence of learning process and the relations between these bounds and capacity of the set of functions on Sugeno measure space are given.

On the rate of convergence of two generalized Bernstein type operators

In this paper,we introduce the Bézier variant of two new families of generalized Bernstein type operators.We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity.By means of construction of suitable functions and the method of Bojanic and Cheng,we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO－PARAMETER MARTINGALE DIFFERENCES

In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.

The Convergence of 1-Periodic Branched Continued Fraction of the Special Form in Parabolic Regions

Branched continued fractions are one of the multidimensional generalization of the continued fractions. Branched continued fractions with not equivalent variables are an analog of the regular C-fractions for multiple power series. We consider 1-periodic branched continued fraction of the special form which is an analog fraction with not equivalent variables if the values of that variables are fixed. We establish an analog of the parabola theorem for that fraction and estimate truncation error bounds for that fractions at some restrictions. We also propose to use weight coefficients for obtaining different parabolic regions for the same fraction without any additional restriction for first element.

A New Type of Solution Method for the Generalized Linear Complementarity Problem over a Polyhedral Cone

This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.

The k Nearest Neighbors Estimator of the M-Regression in Functional Statistics

It is well known that the nonparametric estimation of the regression function is highly sensitive to the presence of even a small proportion of outliers in the data.To solve the problem of typical observations when the covariates of the nonparametric component are functional,the robust estimates for the regression parameter and regression operator are introduced.The main propose of the paper is to consider data-driven methods of selecting the number of neighbors in order to make the proposed processes fully automatic.We use thek Nearest Neighbors procedure(kNN)to construct the kernel estimator of the proposed robust model.Under some regularity conditions,we state consistency results for kNN functional estimators,which are uniform in the number of neighbors(UINN).Furthermore,a simulation study and an empirical application to a real data analysis of octane gasoline predictions are carried out to illustrate the higher predictive performances and the usefulness of the kNN approach.

ON CONVERGENCE OF GENERAL GAMMA TYPE OPERATORS

The present paper deals with the new type of Gamma operators, here we estimate the rate of pointwise convergence of these new Gamma type operators Mn,k for functions of bounded variation, by using some techniques of probability theory.

THE RATE OF UNIFORM CONVERGENCE OF THE SURVIVAL FUNCTION ESTIMATOR FOR TRUNCATED AND CENSORED DATA

For left truncated and right censored data, based on a strong representation of the product-limit estimator of the survival function, we derive the sufficient and necessary condition for the rate of strong uniform convergence of the product-limit estimator over the whole line.

ON THE BEST UNIFORM CONVERGENCE RATE OF EB ESTIMATORS

ＯＮＴＨＥＢＥＳＴＵＮＩＦＯＲＭＣＯＮＶＥＲＧＥＮＣＥＲＡＴＥＯＦＥＢＥＳＴＩＭＡＴＯＲＳＬＩＡＮＧＨｕａ（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ）ＹＵＡＮＹａｈｕａ（...

Rates of convergence of powered order statistics from general error distribution

Let{Xn:n≥1}be a sequence of independent random variables with common general error distribution GED(v)with shape parameter v>0,and let Mn,r denote the r-th largest order statistics of X1,X2,...,Xn.With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics|Mn,r|p are established.An alternative method is presented to estimate the probability of the r-th extremes.Numerical analyses are provided to support the main results.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model , where y i ’s are responses, are known and nonrandom design points, is a compact set in the real line , β = (β 1, ··· , β p )' is an unknown parameter vector, g(·) is an unknown function and {ε i } is a linear process, i.e., , where e j are i.i.d. random variables with zero mean and variance . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coeffcients of the process. Moreover, our result can be used to construct the asymptotically effcient estimators for parameters in the ARMA error process.

A General Law of Moment Convergence Rates for Uniform Empirical Process

Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = sup0≤t≤ 1 │Fn（t）│. In this paper, the exact convergence rates of a general law of weighted infinite series of E{││Fn││ -εg^s（n）}＋ are obtained.

On the Convergence Rate of a Class of Proximal-Based Decomposition Methods for Monotone Variational Inequalities

A unified efficient algorithm framework of proximal-based decomposition methods has been proposed for monotone variational inequalities in 2012,while only global convergence is proved at the same time.In this paper,we give a unified proof on theO(1/t)iteration complexity,together with the linear convergence rate for this kind of proximal-based decomposition methods.Besides theε-optimal iteration complexity result defined by variational inequality,the non-ergodic relative error of adjacent iteration points is also proved to decrease in the same order.Further,the linear convergence rate of this algorithm framework can be constructed based on some special variational inequality properties,without necessary strong monotone conditions.

A Framework of Convergence Analysis of Mini-batch Stochastic Projected Gradient Methods

In this paper,we establish a unified framework to study the almost sure global convergence and the expected convergencerates of a class ofmini-batch stochastic(projected)gradient(SG)methods,including two popular types of SG:stepsize diminished SG and batch size increased SG.We also show that the standard variance uniformly bounded assumption,which is frequently used in the literature to investigate the convergence of SG,is actually not required when the gradient of the objective function is Lipschitz continuous.Finally,we show that our framework can also be used for analyzing the convergence of a mini-batch stochastic extragradient method for stochastic variational inequality.

一种基于条件梯度的加速分布式在线学习算法

由于容易实施,基于投影梯度的分布式在线优化模型逐渐成为一种主流的在线学习方法.然而,在处理大数据应用时,投影步骤成为该方法的计算瓶颈.近年来,研究者提出了面向凸代价函数的分布式在线条件梯度算法,其悔界为O(T^(3/4)),其中T是一个时间范围.该算法存在两方面的问题,一是其悔界劣于公认的悔界O(/T);二是没有分析非凸代价函数的收敛性能,而实际应用中代价函数大部分是非凸函数.因此,提出一种基于条件梯度的加速分布式在线学习算法,使用Frank-Wolfe步骤替代投影步骤,避免昂贵的投影计算.文中证明当局部代价函数为凸函数时,所提算法达到公认的悔界O(/T);当局部代价函数为潜在非凸函数时,所提算法以速率O(/T)收敛到平稳点.最后,仿真实验验证了所提算法的性能与理论证明的结论.

Ultimate generalization to monotonicity for uniform convergence of trigonometric series

Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.

CONVERGENCE RATES OF LAW OF ITERATED LOGARITHM FOR B-VALUED RANDOM VARIABLES

Let {X, X_n; n≥1} be i.i.d.r.v.'s taking values in a separable Banach space (B,||·||)such that EX=0 and Ef^2(X)<+∞, ?∈6B~*, and S_n=X_1+…+X_n for n≥1. The purposeof this paper is to study the rates of convergence to zero of P(inf||Sn/(2nloglogn)^(1/2)-x||≥ε) and P(sup inf||S_k/(2kloglogk)^(1/2)-x||≥ε) (?ε>0) under precisely necessary and sufficientconditions. We also give new necessary and sufficient conditions for X to satisfy the boundand compact law of the iterated logarithm, respectively. Our results improve some resultsof Darling and Robbins (1967) as well as Davis (1968) even in the case B=R.

CONVERGENCE PROPERTIES OF MULTI-DIMENSIONAL NEAREST NEIGHBOR DENSITY ESTIMATES FOR THE CASE OF GENERAL KERNEL

In this paper we give weaker conditions to ensure the strong uniform consis-tency of multi-dimensional nearest neighbor (N.N.) estimates with non-uniform kernel andobtain the convergence rates of these estimates on an arbitrary bounded set. The ratescan not be improved in some sense. Obviously, the problem of strong convergence rates ata given point is its special case. The range of applications of estimates is extended.

Double sampling derivatives and truncation error estimates

This paper investigates double sampling series derivatives for bivariate functions defined on R2 that are in the Bernstein space. For this sampling series, we estimate some of the pointwise and uniform bounds when the function satisfies some decay conditions. The truncated series of this formula allow us to approximate any order of partial derivatives for function from Bernstein space using only a finite number of samples from the function itself. This sampling formula will be useful in the approximation theory and its applications, especially after having the truncation error well-established. Examples with tables and figures are given at the end of the paper to illustrate the advantages of this formula.