统计代写|机器学习代写machine learning代考|The Representer Theorem

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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 Representer Theorem

We have seen that kernels are a powerful tool that enrich the applicability of linear classifiers by a large extent. Nonetheless, apart from the solution of the perceptron learning algorithm it is not yet clear when this method can successfully be applied, i.e., for which learning algorithms $\mathcal{A}: \cup_{m=1}^{\infty} \mathcal{Z}^{m} \rightarrow \mathcal{F}$ the solution $\mathcal{A}(z)$ admits a representation of the form
$$(\mathcal{A}(z))(\cdot)=\sum_{i=1}^{m} \alpha_{i} k\left(x_{i}, \cdot\right)$$
Before identifying this class of learning algorithms we introduce a purely functional analytic point of view on kernels. We will show that each Mercer kernel automatically defines a reproducing kernel Hilbert space (RKHS) of functions as given by equation $(2.34)$. Finally, we identify the class of cost functions whose solution has the form $(2.34)$.

Suppose we are given a Mercer kernel $k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$. Then let $\mathcal{F}{0}$ be the linear space of real-valued functions on $\mathcal{X}$ generated by the functions ${k(x,-) \mid x \in \mathcal{X}}$. Consider any two functions $f(\cdot)=\sum{i=1}^{r} \alpha_{i} k\left(x_{i}, \cdot\right)$ and $g(\cdot)=\sum_{j=1}^{s} \beta_{j} k\left(\tilde{x}{j}, \cdot\right)$ in $\mathcal{F}{0}$ where $\alpha \in \mathbb{R}^{r}, \beta \in \mathbb{R}^{s}$ and $x_{i}, \tilde{x}{j} \in \mathcal{X}$. Define the inner product $\langle f, g\rangle$ between $f$ and $g$ in $\mathcal{F}{0}$ as
$\langle f, g\rangle \stackrel{\text { def }}{=} \sum_{i=1}^{r} \sum_{j=1}^{s} \alpha_{i} \beta_{j} k\left(x_{i}, \tilde{x}{j}\right)=\sum{j=1}^{s} \beta_{j} f\left(\tilde{x}{j}\right)=\sum{i=1}^{r} \alpha_{i} g\left(x_{i}\right)$
where the last equality follows from the symmetry of the kernel $k$. Note that this inner product $\langle\cdot, \cdot\rangle$ is independent of the representation of the function $f$ and $g$ because changing the representation of $f$, i.e., changing $r, \alpha$ and $\left{x_{1}, \ldots, x_{r}\right}$, would not change $\sum_{j=1}^{s} \beta_{j} f\left(\tilde{x}_{j}\right.$ ) (similarly for $g$ ). Moreover, we see that

1. $\langle f, g\rangle=\langle g, f\rangle$ for all functions $f, g \in \mathcal{F}_{0}$,
2. $\langle c f+d g, h\rangle=c\langle f, h\rangle+d\langle g, h\rangle$ for all functions $f, g, h \in \mathcal{F}_{0}$ and all $c, d \in \mathbb{R}$,
3. $\langle f, f\rangle=\sum_{i=1}^{r} \sum_{j=1}^{r} \alpha_{i} \alpha_{j} k\left(x_{i}, x_{j}\right) \geq 0$ for all functions $f \in \mathcal{F}_{0}$ because $k$ is a Mercer kernel.

It still remains to established that $\langle f, f\rangle=0$ implies that $f=0$. To show this we need first the following important reproducing property: For all functions $f \in \mathcal{F}{0}$ and all $x \in \mathcal{X}$ $\langle f, k(x, \cdot)\rangle=f(x) ，$ which follows directly from choosing $s=1, \beta{1}=1$ and $\tilde{x}{1}=x$ in (2.35) -hence $g(\cdot)=k(x, \cdot)$. Now using the Cauchy-Schwarz inequality (see Theorem A.106 and preceding comments) we know that $$0 \leq(f(x))^{2}=(\langle f, k(x, \cdot)\rangle)^{2} \leq\langle f, f\rangle \underbrace{\langle k(x, \cdot), k(x, \cdot)\rangle}{k(x, x)}$$

统计代写|机器学习代写machine learning代考|Support Vector Classification Learning

The methods presented in the last two sections, namely the idea of regularization, and the kernel technique, are elegantly combined in a learning algorithm known as support vector learning (SV learning). ${ }^{16}$ In the study of SV learning the notion of margins is of particular importance. We shall see that the support vector machine (SVM) is an implementation of a more general regularization principle known as the large margin principle. The greatest drawback of SVMs, that is, the need for zero training error, is resolved by the introduction of soft margins. We will demonstrate how both large margin and soft margin algorithms can be viewed in the geometrical picture given in Figure $2.1$ on page 23 . Finally, we discuss several extensions of the classical SVM algorithm achieved by reparameterization.Let us begin by defining what we mean by the margin of a classifier. In Figure $2.6$ a training sample $z$ in $\mathbb{R}^{2}$ together with a classifier (illustrated by the incurred decision surface) is shown. The classifier $f_{\mathrm{w}}$ in Figure $2.6$ (a) has a “dead zone” (gray area) separating the two sets of points which is larger than the classifier $f_{\bar{w}}$ chosen in Figure $2.6$ (b). In both pictures the “dead zone” is the tube around the (linear) decision surface which does not contains any training example $\left(x_{i}, y_{i}\right) \in z$. To measure the extent of such a tube we can use the norm of the weight vector w parameterizing the classifier $f_{\mathrm{w}}$. In fact, the size of this tube must be inversely proportional to the minimum real-valued output $y_{i}\left\langle\mathbf{x}_{i}, \mathbf{w}\right\rangle$ of a classifier $\mathbf{w}$ on a given training sample $z$. This quantity is also known as the functional margin on the training sample $z$ and needs to be normalized to be useful for comparison across different weight vectors w not necessarily of unit length. More precisely, when normalizing the real-valued outputs by the norm of the weight vector $w$ (which is equivalent to considering the real-valued outputs of normalized weight vectors $\mathbf{w} /|\mathbf{w}|$ only) we obtain a confidence measure comparable across different hyperplanes. The following definition introduces the different notions of margins more formally.

统计代写|机器学习代写machine learning代考|Soft Margins—Learning with Training Error

The algorithm presented in the last subsection is clearly restricted to training samples which are linearly separable. One way to deal with this insufficiency is to use “powerful” kernels (like an RBF kernel with very small $\sigma$ ) which makes each training sample separable in feature space. Although this would not cause any computational difficulties, the “large expressive” power of the classifiers in

feature space may lead to overfitting, that is, a large discrepancy between empirical risk (which was previously zero) and true risk of a classifier. Moreover, the above algorithm is “nonrobust” in the sense that one outlier (a training point $\left(x_{i}, y_{i}\right) \in z$ whose removal would lead to a large increase in margin) can cause the learning algorithm to converge very slowly or, even worse, make it impossible to apply at all (if $\gamma_{i}(\mathbf{w})<0$ for all $\mathbf{w} \in \mathcal{W}$ ).

In order to overcome this insufficiency we introduce a heuristic which has become known as the soft margin SVM. The idea exploited is to upper bound the zero-one loss $l_{0-1}$ as given in equation (2.9) by a linear or quadratic function (see Figure 2.7),
\begin{aligned} &l_{0-1}(f(x), y)=I_{-y f(x)>0} \leq \max {1-y f(x), 0}=l_{\text {lin }}(f(x), y) \ &l_{0-1}(f(x), y)=I_{-y f(x)>0} \leq \max {1-y f(x), 0}^{2}=l_{\text {quad }}(f(x), y) \end{aligned}
It is worth mentioning that, due to the cut off at a real-valued output of one (on the correct side of the decision surface), the norm $|f|$ can still serve as a regularizer. Viewed this way, the idea is in the spirit of the second parameterization of the optimization problem of large margins (see equation $(2.40)$ ).

统计代写|机器学习代写machine learning代考|The Representer Theorem

(一种(和))(⋅)=∑一世=1米一种一世ķ(X一世,⋅)

⟨F,G⟩= 定义 ∑一世=1r∑j=1s一种一世bjķ(X一世,X~j)=∑j=1sbjF(X~j)=∑一世=1r一种一世G(X一世)

1. ⟨F,G⟩=⟨G,F⟩适用于所有功能F,G∈F0,
2. ⟨CF+dG,H⟩=C⟨F,H⟩+d⟨G,H⟩适用于所有功能F,G,H∈F0和所有C,d∈R,
3. ⟨F,F⟩=∑一世=1r∑j=1r一种一世一种jķ(X一世,Xj)≥0适用于所有功能F∈F0因为ķ是美世内核。

统计代写|机器学习代写machine learning代考|Soft Margins—Learning with Training Error

l0−1(F(X),是)=一世−是F(X)>0≤最大限度1−是F(X),0=llin (F(X),是) l0−1(F(X),是)=一世−是F(X)>0≤最大限度1−是F(X),02=l四边形 (F(X),是)

有限元方法代写

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MATLAB代写

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