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# 计算机代写|机器学习代写Machine Learning代考|Decision stumps

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## 计算机代写|机器学习代写Machine Learning代考|Decision stumps

As an example of a weak classifier, we consider “decision stumps,” which are a trivial special case of decision trees. A decision stump has the following form:
$$f(\mathbf{x})=s\left(x_k>c\right)$$
where the value in the parentheses is 1 if the $k$-th element of the vector $\mathbf{x}$ is greater than $c$, and -1 otherwise. The scalar $s$ is either -1 or 1 which allows one the classifier to respond with class 1 when $x_k \leq c$. Accordingly, there are three parameters to a decision stump:

$c \in \mathbb{R}$

$k \in{1, \ldots K}$, where $K$ is the dimension of $\mathbf{x}$, and

$s \in{-1,1}$
Because the number of possible parameter settings is relatively small, a decision stump is often trained by brute force: discretize the real numbers from the smallest to the largest value in the training set, enumerate all possible classifiers, and pick the one with the lowest training error. One can be more clever in the discretization: between each pair of data points, only one classifier must be tested (since any stump in this range will give the same value). More sophisticated methods, for example, based on binning the data, or building CDFs of the data, may also be possible.

## 计算机代写|机器学习代写Machine Learning代考|Why does it work?

There are many different ways to analyze AdaBoost; none of them alone gives a full picture of why AdaBoost works so well. AdaBoost was first invented based on optimization of certain bounds on training, and, since then, a number of new theoretical properties have been discovered.

Loss function view. Here we discuss the loss function interpretation of AdaBoost. As was shown (decades after AdaBoost was first invented), AdaBoost can be viewed as greedy optimization of a particular loss function. We define $f(\mathbf{x})=\frac{1}{2} \sum_m \alpha_m f_m(\mathbf{x})$, and rewrite the classifier as $g(\mathbf{x})=$ $\operatorname{sign}(f(\mathbf{x}))$ (the factor of $1 / 2$ has no effect on the classifier output). AdaBoost can then be viewed as optimizing the exponential loss:
$$L_{\exp }(\mathbf{x}, y)=e^{-y f(\mathbf{x})}$$
so that the full learning objective function is
$$E=\sum_i e^{-\frac{1}{2} y_i \sum_{m=1}^M \alpha_m f_m(\mathbf{x})}$$
which must be optimized with respect to the weights $\alpha$ and the parameters of the weak classifiers. The optimization process is greedy and sequential: we add one weak classifier at a time, choosing it and its $\alpha$ to be optimal with respect to $E$, and then never change it again. Note that the exponential loss is an upper-bound on the $0-1$ loss:
$$L_{\exp }(\mathbf{x}, y) \geq L_{0-1}(\mathbf{x}, y)$$
Hence, if exponential loss of zero is achieved, then the 0-1 loss is zero as well, and all training points are correctly classified.

## 计算机代写|机器学习代写MACHINE LEARNING代考|DECISION STUMPS

$$f(\mathbf{x})=s\left(x_k>c\right)$$

$$c \in \mathbb{R}$$
$k \in 1, \ldots K$ ， 在哪里 $K$ 是维度 $\mathbf{x}$ ，和
$s \in-1,1$

## 计算机代写|机器学习代写MACHINE LEARNING代考|WHY DOES IT WORK?

L_{\exp }(\mathbf{x}, y)=e^{-y f(\mathbf{x})}
$$这样完整的学习目标函数是$$
E=\sum_i e^{-\frac{1}{2} y_i \sum_{m=1}^M \alpha_m f_m(\mathbf{x})}
$$必须针对权重进行优化 \alpha 和弱分类器的参数。优化过程是贪婪的和顺序的：我们一次添加一个弱分类器，选择它和它的 \alpha 相对于最佳 E ，然后再也 不会更改它。请注意，指数损失是 0-1 损失:$$
L_{\exp }(\mathbf{x}, y) \geq L_{0-1}(\mathbf{x}, y)


## Matlab代写

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