### 统计代写|机器学习代写machine learning代考|Kernel Classifiers from a Machine Learning Perspective

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|The Basic Setting

This chapter presents the machine learning approach to learning kernel classifiers. After a short introduction to the problem of learning a linear classifier, it shows how learning can be viewed as an optimization task. As an example, the classical perceptron algorithm is presented. This algorithm is an implementation of a more general principle known as empirical risk minimization. The chapter also presents a descendant of this principle, known as regularized (structural) risk minimization. Both these principles can be applied in the primal or dual space of variables. It is shown that the latter is computationally less demanding if the method is extended to nonlinear classifiers in input space. Here, the kernel technique is the essential method used to invoke the nonlinearity in input space. The chapter presents several families of kernels that allow linear classification methods to be applicable even if no vectorial representation is given, e.g., strings. Following this, the support vector method for classification learning is introduced. This method elegantly combines the kernel technique and the principle of structural risk minimization. The chapter finishes with a presentation of a more recent kernel algorithm called adaptive margin machines. In contrast to the support vector method, the latter aims at minimizing a leave-one-out error bound rather than a structural risk.

The task of classification learning is the problem of finding a good strategy to assign classes to objects based on past observations of object-class pairs. We shall only assume that all objects $x$ are contained in the set $\mathcal{X}$, often referred to as the input space. Let $\mathcal{Y}$ be a finite set of classes called the output space. If not otherwise stated, we will only consider the two-element output space ${-1,+1}$, in which case

the learning problem is called a binary classification learning task. Suppose we are given a sample of $m$ training objects,
$$\boldsymbol{x}=\left(x_{1}, \ldots, x_{m}\right) \in \mathcal{X}^{m},$$
together with a sample of corresponding classes,
$$\boldsymbol{y}=\left(y_{1}, \ldots, y_{m}\right) \in \mathcal{Y}^{m}$$
We will often consider the labeled training sample, ${ }^{1}$
$$z=(\boldsymbol{x}, \boldsymbol{y})=\left(\left(x_{1}, y_{1}\right), \ldots,\left(x_{m}, y_{m}\right)\right) \in(\mathcal{X} \times \mathcal{Y})^{m}=\mathcal{Z}^{m}$$
and assume that $z$ is a sample drawn identically and independently distributed (iid) according to some unknown probability measure $\mathbf{P}_{Z}$.

## 统计代写|机器学习代写machine learning代考|Learning by Risk Minimization

Apart from algorithmical problems, as soon as we have a fixed object space $\mathcal{X}$, a fixed set (or space) $\mathcal{F}$ of hypotheses and a fixed loss function $l$, learning reduces to a pure optimization task on the functional $R[f]$.

Definition $2.9$ (Learning algorithm) Given an object space $\mathcal{X}$, an output space $\mathcal{Y}$ and a fixed set $\mathcal{F} \subseteq \mathbb{R}^{\mathcal{X}}$ of functions mapping $\mathcal{X}$ to $\mathbb{R}$, a learning algorithm $\mathcal{A}$

for the hypothesis space $\mathcal{F}$ is a mapping ${ }^{6}$
$\mathcal{A}: \bigcup_{m=1}^{\infty}(\mathcal{X} \times \mathcal{Y})^{m} \rightarrow \mathcal{F} .$
The biggest difficulty so far is that we have no knowledge of the function to be optimized, i.e., we are only given an iid sample $z$ instead of the full measure $\mathbf{P}_{\mathrm{Z}}$. Thus, it is impossible to solve the learning problem exactly. Nevertheless, for any learning method we shall require its performance to improve with increasing training sample size, i.e., the probability of drawing a training sample $z$ such that the generalization error is large will decrease with increasing $m$. Here, the generalization error is defined as follows.

Definition $2.10$ (Generalization error) Given a learning algorithm $\mathcal{A}$ and a loss $l: \mathbb{R} \times \mathcal{Y} \rightarrow \mathbb{R}$ the generalization error of $\mathcal{A}$ is defined as
$$R[\mathcal{A}, z] \stackrel{\text { def }}{=} R[\mathcal{A}(z)]-\inf _{f \in \mathcal{F}} R[f]$$
In other words, the generalization error measures the deviation of the expected risk of the function learned from the minimum expected risk.

The most well known learning principle is the empirical risk minimization (ERM) principle. Here, we replace $\mathbf{P}{z}$ by $\mathbf{v}{z}$, which contains all knowledge that can be drawn from the training sample $z$. As a consequence the expected risk becomes an empirically computable quantity known as the empirical risk.

## 统计代写|机器学习代写machine learning代考|The Perceptron Algorithm

The first iterative procedure for learning linear classifiers presented is the perceptron learning algorithm proposed by F. Rosenblatt. The learning algorithm is given on page 321 and operates as follows:

1. At the start the weight vector $\mathbf{w}$ is set to $\mathbf{0}$.
2. For each training example $\left(x_{i}, y_{i}\right)$ it is checked whether the current hypothesis correctly classifies or not. This can be achieved by evaluating the sign of $y_{i}\left\langle\mathbf{x}{i}, \mathbf{w}\right\rangle$. If the $i$ th training sample is not correctly classified then the misclassified pattern $\mathbf{x}{i}$ is added to or subtracted from the current weight vector depending on the correct class $y_{i}$. In summary, the weight vector $\mathbf{w}$ is updated to $\mathbf{w}+y_{i} \mathbf{x}_{i}$.
3. If no mistakes occur during an iteration through the training sample $z$ the algorithm stops and outputs $\mathbf{w}$.

The optimization algorithm is a mistake-driven procedure, and it assumes the existence of a version space $V(z) \subseteq \mathcal{W}$, i.e., it assumes that there exists at least one classifier $f$ such that $R_{\text {emp }}[f, z]=0$.Since our classifiers are linear in feature space, such training samples are called linearly separable. In order that the perceptron learning algorithm works for any training sample it must be ensured that the unknown probability measure $\mathbf{P}{\mathrm{Z}}$ satisfies $R\left[f^{}\right]=0$. Viewed differently, this means that $\mathbf{P}{Y \mid X=x}(y)=\mathbf{I}{y=h^{}(x)}, h^{} \in \mathcal{H}$, where $h^{}$ is sometimes known as the teacher perceptron. It should be noticed that the number of parameters learned by the perceptron algorithm is $n$, i.e., the dimensionality of the feature space $\mathcal{K}$. We shall call this space of parameters the primal space, and the corresponding algorithm the primal perceptron learning algorithm. As depicted in Figure 2.2, perceptron learning is best viewed as starting from an arbitrary $^{7}$ point $\mathbf{w}{0}$ on the hypersphere $\mathcal{W}$, and each time we observe a misclassification with a training example $\left(x_{i}, y_{i}\right)$, we update $\mathbf{w}{I}$ toward the misclassified training object $y{i} \mathbf{x}_{i}$ (see also Figure $2.1$ (left)). Thus, geometrically, the perceptron learning algorithm performs a walk through the primal parameter space with each step made in the direction of decreasing training error. Note, however, that in the formulation of the algorithm given on page 321 we do not normalize the weight vector $\mathbf{w}$ after each update.

X=(X1,…,X米)∈X米,

## 统计代写|机器学习代写machine learning代考|Learning by Risk Minimization

R[一种,和]= 定义 R[一种(和)]−信息F∈FR[F]

## 统计代写|机器学习代写machine learning代考|The Perceptron Algorithm

1. 开始时的权重向量在设定为0.
2. 对于每个训练示例(X一世,是一世)检查当前假设是否正确分类。这可以通过评估是一世⟨X一世,在⟩. 如果一世训练样本未正确分类，则错误分类模式X一世取决于正确的类被添加到当前权重向量或从当前权重向量中减去是一世. 总之，权重向量在更新为在+是一世X一世.
3. 如果在训练样本的迭代过程中没有出现错误和算法停止并输出在.

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## MATLAB代写

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