### 统计代写|机器学习代写machine learning代考|Regularized Risk Functionals

statistics-lab™ 为您的留学生涯保驾护航 在代写机器学习machine learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写机器学习machine learning代写方面经验极为丰富，各种代写机器学习machine learning相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|机器学习代写machine learning代考|Regularized Risk Functionals

One possible method of overcoming the lack of knowledge about $\mathbf{P}{\mathbf{Z}}$ is to replace it by its empirical estimate $\mathbf{v}{z}$. This principle, discussed in the previous section, justifies the perceptron learning algorithm. However, minimizing the empirical risk, as done by the perceptron learning algorithm, has several drawbacks:

1. Many examples are required to ensure a small generalization error $R\left[\mathcal{A}_{\mathrm{ERM}}, z\right]$ with high probability taken over the random choice of $z$.
2. There is no unique minimum, i.e., each weight vector $\mathbf{w} \in V(z)$ in version space parameterizes a classifier $f_{\mathrm{w}}$ that has $R_{\text {emp }}\left[f_{\mathrm{w}}, z\right]=0$.
3. Without any further assumptions on $\mathbf{P}_{\mathbf{Z}}$ the number of steps until convergence of the perceptron learning algorithm is not bounded.
• A training sample $z \in Z^{m}$ that is linearly separable in feature space is required.
The second point in particular shows that ERM learning makes the learning task an ill-posed one (see Appendix A.4): A slight variation $\tilde{z}$ in the training sample $z$ might lead to a large deviation between the expected risks of the classifiers learned using the ERM principle, $\left|R\left[\mathcal{A}{\text {ERM }}(z)\right]-R\left[\mathcal{A}{\text {ERM }}(\tilde{z})\right]\right|$. As will be seen in Part II of this book, a very influential factor in this deviation is the possibility of the hypothesis space $\mathcal{F}$ adopting different labelings $\boldsymbol{y}$ for randomly drawn objects $\boldsymbol{x}$. The more diverse the set of functions a hypothesis space contains, the more easily

it can produce a given labeling $y$ regardless of how bad the subsequent prediction might be on new, as yet unseen, data points $z=(x, y)$. This effect is also known as overfitting, i.e., the empirical risk as given by equation (2.11) is much smaller than the expected risk ( $2.8$ ) we originally aimed at minimizing.

One way to overcome this problem is the method of regularization. In our example this amounts to introducing a regularizer a-priori, that is, a functional $\Omega: \mathcal{F} \rightarrow \mathbb{R}^{+}$, and defining the solution to the learning problem to be
$$\mathcal{A}{\Omega}(z) \stackrel{\text { def }}{=} \underset{f \in \mathcal{F}}{\operatorname{argmin}} \underbrace{R{\mathrm{emp}}[f, z]+\lambda \Omega[f]}{R{\mathrm{reg}}[f, z]}$$

## 统计代写|机器学习代写machine learning代考|Kernels and Linear Classifiers

As we assume $\phi$ to be given we will call this the explicit way to non-linearize a linear classification model. We already mentioned in Section $2.2$ that the number of dimensions, $n$, of the feature space has a great impact on the generalization ability of empirical risk minimization algorithms. Thus, one conceivable criterion for defining features $\phi_{i}$ is to seek a small set of basis functions $\phi_{i}$ which allow perfect discrimination between the classes in $\mathcal{X}$. This task is called feature selection.

Let us return to the primal perceptron learning algorithm mentioned in the last subsection. As we start at $\mathbf{w}{0}=\mathbf{0}$ and add training examples only when a mistake is committed by the current hypothesis, it follows that the each solution has to admit a representation of the form, $$\mathbf{w}{t}=\sum_{i=1}^{m} \alpha_{i} \phi\left(x_{i}\right)=\sum_{i=1}^{m} \alpha_{i} \mathbf{x}{i}$$ Hence, instead of formulating the perceptron algorithm in terms of the $n$ variables $\left(w{1}, \ldots, w_{n}\right)^{\prime}=\mathbf{w}$ we could learn the $m$ variables $\left(\alpha_{1}, \ldots, \alpha_{m}\right)^{\prime}=\alpha$ which we call the dual space of variables. In the case of perceptron learning we start with $\alpha_{0}=\mathbf{0}$ and then employ the representation of equation (2.17) to update $\boldsymbol{\alpha}{t}$ whenever a mistake occurs. To this end, we need to evaluate $$y{j}\left\langle\mathbf{x}{j}, \mathbf{w}{t}\right\rangle=y_{j}\left\langle\mathbf{x}{j}, \sum{i=1}^{m} \alpha_{i} \mathbf{x}{i}\right\rangle=y{j} \sum_{i=1}^{m} \alpha_{i}\left\langle\mathbf{x}{j}, \mathbf{x}{i}\right\rangle$$
which requires only knowledge of the inner product function $\langle\cdot, \cdot\rangle$ between the mapped training objects $\mathbf{x}$. Further, for the classification of a novel test object $x$ it suffices to know the solution vector $\alpha_{t}$ as well as the inner product function, because
$$\left\langle\mathbf{x}, \mathbf{w}{t}\right\rangle=\left\langle\mathbf{x}, \sum{i=1}^{m} \alpha_{i} \mathbf{x}{i}\right\rangle=\sum{i=1}^{m} \alpha_{i}\left\langle\mathbf{x}, \mathbf{x}_{i}\right\rangle .$$

## 统计代写|机器学习代写machine learning代考|The Kernel Technique

The key idea of the kernel technique is to invert the chain of arguments, i.e., choose a kernel $k$ rather than a mapping before applying a learning algorithm. Of course, not any symmetric function $k$ can serve as a kernel. The necessary and sufficient conditions of $k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ to be a kernel are given by Mercer’s theorem. Before we rephrase the original theorem we give a more intuitive characterization of Mercer kernels.

Example 2.16 (Mercer’s theorem) Suppose our input space $\mathcal{X}$ has a finite number of elements, i.e., $\mathcal{X}=\left{x_{1}, \ldots, x_{r}\right}$. Then, the $r \times r$ kernel matrix $\mathbf{K}$ with $\mathbf{K}{i j}=k\left(x{i}, x_{j}\right)$ is by definition a symmetric matrix. Consider the eigenvalue decomposition of $\mathbf{K}=\mathbf{U} \mathbf{\Lambda} \mathbf{U}^{\prime}$, where $\mathbf{U}=\left(\mathbf{u}{1}^{\prime} ; \ldots ; \mathbf{u}{r}^{\prime}\right)$ is an $r \times n$ matrix such that $\mathbf{U}^{\prime} \mathbf{U}=\mathbf{I}{n}, \boldsymbol{\Lambda}=\operatorname{diag}\left(\lambda{1}, \ldots, \lambda_{n}\right), \lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{n}>0$ and $n \leq r$ being known as the rank of the matrix $\mathbf{K}$ (see also Theorem A.83 and Definition A.62).

Now the mapping $\phi: \mathcal{X} \rightarrow \mathcal{K} \subseteq \ell_{2}^{n}$,
$\phi\left(x_{i}\right)=\boldsymbol{\Lambda}^{\frac{1}{2}} \mathbf{u}{i}$, leads to a Gram matrix $\mathbf{G}$ given by $$\mathbf{G}{i j}=\left\langle\phi\left(x_{i}\right), \phi\left(x_{j}\right)\right\rangle_{\kappa}=\left(\boldsymbol{\Lambda}^{\frac{1}{2}} \mathbf{u}{i}\right)^{\prime}\left(\boldsymbol{\Lambda}^{\frac{1}{2}} \mathbf{u}{j}\right)=\mathbf{u}{i}^{\prime} \boldsymbol{\Lambda} \mathbf{u}{j}=\mathbf{K}{i j} .$$ We have constructed a feature space $\mathcal{K}$ and a mapping $\mathbf{\Lambda}$ into it purely from the kernel $k$. Note that $\lambda{n}>0$ is equivalent to assuming that $\mathbf{K}$ is positive semidefinite denoted by $\mathbf{K} \geq 0$ (see Definition A.40). In order to show that $\mathbf{K} \geq 0$ is also necessary for $k$ to be a kernel, we assume that $\lambda_{n}<0$. Then, the squared length of the nth mapped object $x_{n}$ is
$$\left|\boldsymbol{\phi}\left(x_{n}\right)\right|^{2}=\mathbf{u}{n}^{\prime} \boldsymbol{\Lambda} \mathbf{u}{n}=\lambda_{n}<0,$$
which contradicts the geometry in an inner product space.
Mercer’s theorem is an extension of this property, mainly achieved by studying the eigenvalue problem for integral equations of the form
$$\int_{\mathcal{X}} k(x, \tilde{x}) f(\tilde{x}) d \tilde{x}=\lambda f(x),$$
where $k$ is a bounded, symmetric and positive semidefinite function.

## 统计代写|机器学习代写machine learning代考|Regularized Risk Functionals

1. 需要许多示例来确保小的泛化错误R[一种和R米,和]以高概率接管随机选择和.
2. 没有唯一的最小值，即每个权重向量在∈在(和)在版本空间中参数化分类器F在有R雇员 [F在,和]=0.
3. 没有任何进一步的假设磷从直到感知器学习算法收敛的步数没有限制。
• 训练样本和∈从米需要在特征空间中线性可分。
第二点特别表明 ERM 学习使学习任务成为一个不适定的任务（见附录 A.4）：略有不同和~在训练样本中和可能导致使用 ERM 原理学习的分类器的预期风险之间存在较大偏差，|R[一种风险管理 (和)]−R[一种风险管理 (和~)]|. 正如本书第二部分中将看到的，这种偏差的一个非常有影响的因素是假设空间的可能性F采用不同的标签是对于随机绘制的对象X. 假设空间包含的函数集越多样化，就越容易

## 统计代写|机器学习代写machine learning代考|Kernels and Linear Classifiers

⟨X,在吨⟩=⟨X,∑一世=1米一种一世X一世⟩=∑一世=1米一种一世⟨X,X一世⟩.

## 统计代写|机器学习代写machine learning代考|The Kernel Technique

φ(X一世)=Λ12在一世, 导致一个 Gram 矩阵G由G一世j=⟨φ(X一世),φ(Xj)⟩ķ=(Λ12在一世)′(Λ12在j)=在一世′Λ在j=ķ一世j.我们构建了一个特征空间ķ和一个映射Λ纯粹从内核进入它ķ. 注意λn>0相当于假设ķ是半正定的，表示为ķ≥0（见定义 A.40）。为了表明ķ≥0也是必要的ķ作为一个内核，我们假设λn<0. 然后，第 n 个映射对象的平方长度Xn是
|φ(Xn)|2=在n′Λ在n=λn<0,

Mercer 定理是这一性质的扩展，主要通过研究形式为的积分方程的特征值问题来实现
∫Xķ(X,X~)F(X~)dX~=λF(X),

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。