### 统计代写|机器学习代写machine learning代考| Adaptive Margin Machines

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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|Leave-One-Out Machines

Theorem $2.37$ suggests an algorithm which directly minimizes the expression in the bound. The difficulty is that the resulting objective function will contain the step function $\mathbf{I}{t \geq 0}$. The idea we exploit is similar to the idea of soft margins in SVMs, where the step function is upper bounded by a piecewise linear function, also known as the hinge loss (see Figure 2.7). Hence, introducing slack variables, gives the following optimization problem: $$\begin{array}{ll} \text { minimize } & \sum{i=1}^{m} \xi_{i} \ \text { subject to } \quad & y_{i} \sum_{\substack{j=1 \ j \neq i}}^{m} \alpha_{j} y_{j} k\left(x_{i}, x_{j}\right) \geq 1-\xi_{i} \quad i=1, \ldots, m, \ & \boldsymbol{\alpha} \geq \mathbf{0}, \boldsymbol{\xi} \geq \mathbf{0} \end{array}$$

For further classification of new test objects we use the decision rule given in equation (2.54). Let us study the resulting method which we call a leave-one-out machine (LOOM).

First, the technique appears to have no free regularization parameter. This should be compared with support vector machines, which control the amount of regularization through the free parameter $\lambda$. For SVMs, in the case of $\lambda \rightarrow 0$ one obtains a hard margin classifier with no training errors. In the case of linearly inseparable datasets in feature space (through noise, outliers or class overlap) one must admit some training errors (by constructing soft margins). To find the best choice of training error/margin tradeoff one must choose the appropriate value of $\lambda$. In leave-one-out machines a soft margin is automatically constructed. This happens because the algorithm does not attempt to minimize the number of training errors-it minimizes the number of training points that are classified incorrectly even when they are removed from the linear combination which forms the decision rule. However, if one can classify a training point correctly when it is removed from the linear combination, then it will always be classified correctly when it is placed back into the rule. This can be seen as $\alpha_{i} y_{i} k\left(x_{i}, x_{i}\right)$ always has the same sign as $y_{i}$; any training point is pushed further from the decision boundary by its own component of the linear combination. Note also that summing for all $j \neq i$ in the constraint $(2.56)$ is equivalent to setting the diagonal of the Gram matrix $\mathbf{G}$ to zero and instead summing for all $j$. Thus, the regularization employed by leave-one-out machines disregards the values $k\left(x_{i}, x_{i}\right)$ for all $i$.

Second, as for support vector machines, the solutions $\hat{\alpha} \in \mathbb{R}^{m}$ can be sparse in terms of the expansion vector; that is, only some of the coefficients $\hat{\alpha}_{i}$ are nonzero. As the coefficient of a training point does not contribute to its leave-one-out error in constraint (2.56), the algorithm does not assign a non-zero value to the coefficient of a training point in order to correctly classify it. A training point has to be classified correctly by the training points of the same label that are close to it, but the point itself makes no contribution to its own classification in training.

## 统计代写|机器学习代写machine learning代考|Pitfalls of Minimizing a Leave-One-Out Bound

The core idea of the presented algorithm is to directly minimize the leave-one-out bound. Thus, it seems that we are able to control the generalization ability of an algorithm disregarding quantities like the margin. This is not true in general ${ }^{18}$ and

in particular the presented algorithm is not able to achieve this goal. There are some pitfalls associated with minimizing a leave-one-out bound:

1. In order to get a bound on the leave-one-out error we must specify the algorithm $\mathcal{A}$ beforehand. This is often done by specifying the form of the objective function which is to be maximized (or minimized) during learning. In our particular case we see that Theorem $2.37$ only considers algorithms defined by the maximization of $W$ ( $\alpha$ ) with the “box” constraint $0 \leq \boldsymbol{\alpha} \leq \mathbf{u}$. By changing the learning algorithm to minimize the bound itself we may well develop an optimization algorithm which is no longer compatible with the assumptions of the theorem. This is true in particular for leave-one-out machines which are no longer in the class of algorithms considered by Theorem $2.37$-whose bound they are aimed at minimizing. Further, instead of minimizing the bound directly we are using the hinge loss as anper bound on the Heaviside step function.
2. The leave-one-out bound does not provide any guarantee about the generalization error $R[\mathcal{A}, z]$ (see Definition 2.10). Nonetheless, if the leave-one-out error is small then we know that, for most training samples $z \in \mathcal{Z}^{m}$, the resulting classifier has to have an expected risk close to that given by the bound. This is due to Hoeffding’s bound which says that for bounded loss (the expected risk of a hypothesis $f$ is bounded to the interval $[0,1])$ the expected risk $R[\mathcal{A}(z)]$ of the learned classifier $\mathcal{A}(z)$ is close to the expectation of the expected risk (bounded by the leave-one-out bound) with high probability over the random choice of the training sample. ${ }^{19}$ Note, however, that the leave-one-out estimate does not provide any information about the variance of the expected risk. Such information would allow the application of tighter bounds, for example, Chebyshev’s bound.
3. The original motivation behind the use of the leave-one-out error was to measure the goodness of the hypothesis space $\mathcal{F}$ and of the learning algorithm $\mathcal{A}$ for the learning problem given by the unknown probability measure $\mathbf{P}{\mathbf{Z}}$. Commonly, the leave-one-out error is used to select among different models $\mathcal{F}{1}, \mathcal{F}_{2}, \ldots$ for a given learning algorithm $\mathcal{A}$. In this sense, minimizing the leave-one-out error is more a model selection strategy than a learning paradigm within a fixed model.

In order to generalize leave-one-out machines we see that the $m$ constraints in equation (2.56) can be rewritten as
\begin{aligned} y_{i} \sum_{\substack{j=1 \ j \neq i}}^{m} \alpha_{j} y_{j} k\left(x_{i}, x_{j}\right)+\alpha_{i} k\left(x_{i}, x_{i}\right) & \geq 1-\xi_{i}+\alpha_{i} k\left(x_{i}, x_{i}\right) \quad i=1, \ldots, m, \ y_{i} f\left(x_{i}\right) & \geq 1-\xi_{i}+\alpha_{i} k\left(x_{i}, x_{i}\right) \quad i=1, \ldots, m . \end{aligned}
Now, it is easy to see that a training point $\left(x_{i}, y_{i}\right) \in z$ is linearly penalized for failing to obtain a functional margin of $\bar{\gamma}{i}(\mathbf{w}) \geq 1+\alpha{i} k\left(x_{i}, x_{i}\right)$. In other words, the larger the contribution the training point makes to the decision rule (the larger the value of $\alpha_{i}$ ), the larger its functional margin must be. Thus, the algorithm controls the margin for each training point adaptively. From this formulation one can generalize the algorithm to control regularization through the margin loss. To make the margin at each training point a controlling variable we propose the following learning algorithm:
$$\begin{array}{ll} \text { minimize } & \sum_{i=1}^{m} \xi_{i} \ \text { subject to } \quad & y_{i} \sum_{j=1}^{m} \alpha_{j} y_{j} k\left(x_{i}, x_{j}\right) \geq 1-\xi_{i}+\lambda \alpha_{i} k\left(x_{i}, x_{i}\right), \quad i=1, \ldots, m . \ & \boldsymbol{\alpha} \geq \mathbf{0}, \xi \geq \mathbf{0} \end{array}$$ This algorithm-which we call adaptive margin machines-can also be viewed in the following way: If an object $x_{0} \in \boldsymbol{x}$ is an outlier (the kernel values w.r.t. points in its class are small and w.r.t. points in the other class are large), $\alpha_{o}$ in equation (2.58) must be large in order to classify $x_{o}$ correctly. Whilst support vector machines use the same functional margin of one for such an outlier, they attempt to classify $x_{o}$ correctly. In adaptive margin machines the functional margin is automatically increased to $1+\lambda \alpha_{o} k\left(x_{o}, x_{o}\right)$ for $x_{o}$ and thus less effort is made to change the decision function because each increase in $\alpha_{o}$ would lead to an even larger increase in $\xi_{o}$ and can therefore not be optimal.

## 统计代写|机器学习代写machine learning代考|Pitfalls of Minimizing a Leave-One-Out Bound

1. 为了得到留一错误的界限，我们必须指定算法一种预先。这通常通过指定在学习期间要最大化（或最小化）的目标函数的形式来完成。在我们的特殊情况下，我们看到定理2.37只考虑由最大化定义的算法在 ( 一种) 带有“盒子”约束0≤一种≤在. 通过改变学习算法以最小化界限本身，我们可以很好地开发一种不再与定理假设兼容的优化算法。对于不再属于 Theorem 所考虑的算法类别的留一法机器尤其如此2.37- 他们旨在最小化谁的界限。此外，我们不是直接最小化边界，而是使用铰链损失作为 Heaviside 阶跃函数的边界。
2. 留一法不能保证泛化误差R[一种,和]（见定义 2.10）。尽管如此，如果留一法误差很小，那么我们知道，对于大多数训练样本和∈从米，由此产生的分类器必须具有接近给定边界的预期风险。这是由于 Hoeffding 的界限，它表示对于有限损失（假设的预期风险F有界于区间[0,1])预期风险R[一种(和)]学习分类器的一种(和)在训练样本的随机选择上，以高概率接近预期风险的期望（以留一法为界）。19但是请注意，留一法估计不提供有关预期风险方差的任何信息。此类信息将允许应用更严格的界限，例如切比雪夫界限。
3. 使用留一法误差的最初动机是衡量假设空间的优劣F和学习算法一种对于由未知概率测度给出的学习问题磷从. 通常，留一法误差用于在不同模型之间进行选择F1,F2,…对于给定的学习算法一种. 从这个意义上说，最小化留一法错误更像是一种模型选择策略，而不是固定模型中的学习范式。

最小化 ∑一世=1米X一世  受制于 是一世∑j=1米一种j是jķ(X一世,Xj)≥1−X一世+λ一种一世ķ(X一世,X一世),一世=1,…,米. 一种≥0,X≥0这种算法——我们称之为自适应边距机——也可以用以下方式查看：如果一个对象X0∈X是一个异常值（其类中的内核值 wrt 点很小，而其他类中的 wrt 点很大），一种这等式 (2.58) 中的值必须很大才能分类X这正确。虽然支持向量机对这样的异常值使用相同的功能余量，但它们试图对X这正确。在自适应余量机器中，功能余量自动增加到1+λ一种这ķ(X这,X这)为了X这因此改变决策函数的努力会更少，因为每次增加一种这将导致更大的增长X这因此不可能是最优的。

## 有限元方法代写

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## MATLAB代写

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