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

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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|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),

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