机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Kernel AdaTron in Classification

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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

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Kernel AdaTron in Classification

The classic AdaTron algorithm as given in [12] is developed for a linear classifier. As mentioned previously, the KA is a variant of the classic AdaTron algorithm in the feature space of SVMs. The KA algorithm solves the maximization of the dual Lagrangian (3.2a) by implementing the gradient ascent algorithm. The update $\Delta \alpha_{i}$ of the dual variables $\alpha_{i}$ is given as:
$$\Delta \alpha_{i}=\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=\eta_{i}\left(1-y_{i} \sum_{j=1}^{n} \alpha_{j} y_{j} K\left(\mathbf{x}{i}, \mathbf{x}{j}\right)\right)=\eta_{i}\left(1-y_{i} d_{i}\right)$$
The update of the dual variables $\alpha_{i}$ is given as
$$\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n .$$ In other words, the dual variables $\alpha_{i}$ are clipped to zero if $\left(\alpha_{i}+\Delta \alpha_{i}\right)<0$. In the case of the soft nonlinear classifier $(C<\infty) \alpha_{i}$ are clipped between zero and $C,\left(0 \leq \alpha_{i} \leq C\right)$. The algorithm converges from any initial setting for the Lagrange multipliers $\alpha_{i}$.

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|SMO without Bias Term b in Classification

Recently [148] derived the update rule for multipliers $\alpha_{i}$ that includes a detailed analysis of the Karush-Kuhn-Tucker (KKT) conditions for checking the optimality of the solution. (As referred above, a fixed bias update was mentioned only in Platt’s papers). The no-bias SMO algorithm can be broken down into three different steps as follows:

1. The first step is to find the data points or the $\alpha_{i}$ variables to be optimized. This is done by checking the KKT complementarity conditions of the $\alpha_{i}$ variables. An $\alpha_{i}$ that violates the $\mathrm{KKT}$ condition will be referred to as a $\mathrm{KKT}$ violator. If there are no $\mathrm{KKT}$ violators in the entire data set, the optimal solution for (3.2) is found and the algorithm will stop. The $\alpha_{i}$ need to be updated if:
$\alpha_{i}0 \quad \wedge \quad y_{i} E_{i}>\tau$
where $E_{i}=d_{i}-y_{i}$ denotes the difference between the value of the decision function $d_{i}$ (i.e., it is a SVM output) at the point $\mathbf{x}{i}$ and the desired target (label) $y{i}$ and $\tau$ is the precision of the KKT conditions which should be fulfilled.
2. In the second step, the $\alpha_{i}$ variables that do not fulfill the $K K T$ conditions will be updated. The following update rule for $\alpha_{i}$ was proposed in [148]:
$$\Delta \alpha_{i}=-\frac{y_{i} E_{i}}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)}=-\frac{y_{i} d_{i}-1}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)}=\frac{1-y_{i} d_{i}}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)}$$
After an update, the same clipping operation as in (3.5) is performed
$$\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n$$
3. After the updating of an $\alpha_{i}$ variable, the $y_{j} E_{j}$ terms in the KKT conditions of all the $\alpha_{j}$ variables will be updated by the following rules:
$$y_{j} E_{j}=y_{j} E_{j}^{o l d}+\left(\alpha_{i}-\alpha_{i}^{\text {old }}\right) K\left(\mathbf{x}{i}, \mathbf{x}{j}\right) y_{j} \quad j=1, \ldots, n$$
The algorithm will return to Step 1 in order to find a new KKT violator for updating.

Note the equality of the updating term between KA (3.4) and (3.8) of SMO without the bias term when the learning rate in $(3.4)$ is chosen to be $\eta=$ $1 / K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)$. Because SMO without-bias-term algorithm also uses the same clipping operation in (3.9), both algorithms are strictly equal. This equality is not that obvious in the case of a ‘classic’ SMO algorithm with bias term due to the heuristics involved in the selection of active points which should ensure the largest increase of the dual Lagrangian $L_{d}$ during the iterative optimization steps.

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Kernel AdaTron in Regression

The first extension of the Kernel AdaTron algorithm for regression is presented in [147] as the following gradient ascent update rules for $\alpha_{i}$ and $\alpha_{i}^{}$, \begin{aligned} \Delta \alpha_{i} &=\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=\eta_{i}\left(y_{i}-\varepsilon-\sum_{j=1}^{n}\left(\alpha_{j}-\alpha_{j}^{}\right) K\left(\mathbf{x}{j}, \mathbf{x}{i}\right)\right)=\eta_{i}\left(y_{i}-\varepsilon-f_{i}\right) \ &=-\eta_{i}\left(E_{i}+\varepsilon\right) \ \Delta \alpha_{i}^{} &=\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}^{}}=\eta_{i}\left(-y_{i}-\varepsilon+\sum_{j=1}^{n}\left(\alpha_{j}-\alpha_{j}^{}\right) K\left(\mathbf{x}{j}, \mathbf{x}{i}\right)\right)=\eta_{i}\left(-y_{i}-\varepsilon+f_{i}\right) \ &=\eta_{i}\left(E_{i}-\varepsilon\right) \end{aligned}
where $E_{i}$ is an error value given as a difference between the output of the SVM $f_{i}$ and desired value $y_{i}$. The calculation of the gradient above does not take into account the geometric reality that no training data can be on both sides of the tube. In other words, it does not use the fact that either $\alpha_{i}$ or $\alpha_{i}^{}$ or both will be nonzero, i.e. that $\alpha_{i} \alpha_{i}^{}=0$ must be fulfilled in each iteration step. Below the gradients of the dual Lagrangian $L_{d}$ accounting for geometry will be derived following [85]. This new formulation of the KA algorithm strictly equals the SMO method given below in Sect. 3.2.4 and it is given as \begin{aligned} \frac{\partial L_{d}}{\partial \alpha_{i}}=&-K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}-\sum_{j=1, j \neq i}^{n}\left(\alpha_{j}-\alpha_{j}^{}\right) K\left(\mathbf{x}{j}, \mathbf{x}{i}\right)+y_{i}-\varepsilon+K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{} \ &-K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{} \ =&-K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}-\left(\alpha_{i}-\alpha_{i}^{}\right) K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)-\sum_{j=1, j \neq i}^{n}\left(\alpha_{j}-\alpha_{j}^{}\right) K\left(\mathbf{x}{j}, \mathbf{x}{i}\right) \ &+y_{i}-\varepsilon \ =&-K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}+y_{i}-\varepsilon-f_{i}=-\left(K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}+E_{i}+\varepsilon\right) . \end{aligned} For the $\alpha^{}$ multipliers, the value of the gradient is
$$\frac{\partial L_{d}}{\partial \alpha_{i}^{*}}=-K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}+E_{i}-\varepsilon$$
The update value for $\alpha_{i}$ is now

$$\begin{gathered} \Delta \alpha_{i}=\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=-\eta_{i}\left(K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}+E_{i}+\varepsilon\right) \ \alpha_{i} \leftarrow \alpha_{i}+\Delta \alpha_{i}=\alpha_{i}+\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=\alpha_{i}-\eta_{i}\left(K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}+E_{i}+\varepsilon\right) \end{gathered}$$
For the learning rate $\eta=1 / K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)$ the gradient ascent learning $\mathrm{KA}$ is defined as,
$$\alpha_{i} \leftarrow \alpha_{i}-\alpha_{i}^{}-\frac{E_{i}+\varepsilon}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)}$$
Similarly, the update rule for $\alpha_{i}^{}$ is
$$\alpha_{i}^{} \leftarrow \alpha_{i}^{}-\alpha_{i}+\frac{E_{i}-\varepsilon}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)}$$
Same as in the classification, $\alpha_{i}$ and $\alpha_{i}^{}$ are clipped between zero and $C$, \begin{aligned} &\alpha_{i} \leftarrow \min \left(\max \left(0, \alpha_{i}+\Delta \alpha_{i}\right), C\right) \quad i=1, \ldots, n \ &\alpha_{i}^{} \leftarrow \min \left(\max \left(0, \alpha_{i}^{} \Delta \alpha_{i}^{}\right), C\right) \quad i=1, \ldots, n \end{aligned}

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Kernel AdaTron in Classification

Δ一种一世=这一世∂大号d∂一种一世=这一世(1−是一世∑j=1n一种j是jķ(X一世,Xj))=这一世(1−是一世d一世)

\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n .\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n .换句话说，对偶变量一种一世被剪裁为零，如果(一种一世+Δ一种一世)<0. 在软非线性分类器的情况下(C<∞)一种一世被夹在零和C,(0≤一种一世≤C). 该算法从拉格朗日乘数的任何初始设置收敛一种一世.

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|SMO without Bias Term b in Classification

1. 第一步是找到数据点或一种一世待优化的变量。这是通过检查 KKT 互补条件来完成的一种一世变量。一个一种一世这违反了ķķ吨条件将被称为ķķ吨违反者。如果没有ķķ吨整个数据集中的违规者，找到（3.2）的最优解，算法将停止。这一种一世在以下情况下需要更新：
一种一世0∧是一世和一世>τ
在哪里和一世=d一世−是一世表示决策函数的值之间的差异d一世（即，它是一个 SVM 输出）在该点X一世和所需的目标（标签）是一世和τ是应该满足的 KKT 条件的精度。
2. 在第二步中，一种一世不满足的变量ķķ吨条件将被更新。以下更新规则为一种一世在[148]中提出：
Δ一种一世=−是一世和一世ķ(X一世,X一世)=−是一世d一世−1ķ(X一世,X一世)=1−是一世d一世ķ(X一世,X一世)
更新后，执行与（3.5）相同的裁剪操作
\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n\alpha_{i} \leftarrow \min \left{\max \left{\alpha_{i}+\Delta \alpha_{i}, 0\right}, C\right} \quad i=1, \ldots, n
3. 更新后一种一世变数是j和j所有 KKT 条件中的条款一种j变量将按以下规则更新：
$$y_{j} E_{j}=y_{j} E_{j}^{old}+\left(\alpha_{i}-\alpha_{i}^{\ text {old }}\right) K\left(\mathbf{x} {i}, \mathbf{x} {j}\right) y_{j} \quad j=1, \ldots, n$$
算法将返回第 1 步，以查找新的 KKT 违规者进行更新。

机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Kernel AdaTron in Regression

∂大号d∂一种一世∗=−ķ(X一世,X一世)一种一世+和一世−e

$$\begin{聚集} \Delta \alpha_{i}=\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=-\eta_{i}\left(K \left(\mathbf{x} {i}, \mathbf{x} {i}\right) \alpha_{i}^{}+E_{i}+\varepsilon\right) \ \alpha_{i} \leftarrow \alpha_{i}+\Delta \alpha_{i}=\alpha_{i}+\eta_{i} \frac{\partial L_{d}}{\partial \alpha_{i}}=\alpha_{i} -\eta_{i}\left(K\left(\mathbf{x}{i}, \mathbf{x}{i}\right) \alpha_{i}^{}+E_{i}+\varepsilon\对） \结束{聚集} F这r吨H和l和一种rn一世nGr一种吨和这=1/ķ(X一世,X一世)吨H和Gr一种d一世和n吨一种sC和n吨l和一种rn一世nGķ一种一世sd和F一世n和d一种s, \alpha_{i} \leftarrow \alpha_{i}-\alpha_{i}^{}-\frac{E_{i}+\varepsilon}{K\left(\mathbf{x}{i}, \mathbf{ x}{i}\右）} 小号一世米一世l一种rl是,吨H和在pd一种吨和r在l和F这r一种一世一世s \alpha_{i}^{} \leftarrow \alpha_{i}^{}-\alpha_{i}+\frac{E_{i}-\varepsilon}{K\left(\mathbf{x}{i}, \mathbf{x}{i}\right)} 小号一种米和一种s一世n吨H和Cl一种ss一世F一世C一种吨一世这n,一种一世一种nd一种一世一种r和Cl一世pp和db和吨在和和n和和r这一种ndC,一种一世←分钟(最大限度(0,一种一世+Δ一种一世),C)一世=1,…,n 一种一世←分钟(最大限度(0,一种一世Δ一种一世),C)一世=1,…,n$$

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