### cs代写|机器学习代写machine learning代考|Support vector machines

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• Foundations of Data Science 数据科学基础

## cs代写|机器学习代写machine learning代考|Soft margin classifier

Thus far we have only discussed the linear separable case, but how about the case when there are overlapping classes? It is possible to extend the optimization problem by allowing some data points to be in the margin while penalizing these points somewhat. We therefore include some slag variables $\xi_{i}$ that reduce the effective margin for each data point, but we add a penalty term to the optimization that penalizes if the sum of these slag variables are large,
$$\min {\mathbf{w}, b} \frac{1}{2}|\mathbf{w}|^{2}+C \sum{i} \xi_{i}$$
subject to the constraints
\begin{aligned} y^{(i)}\left(\mathbf{w}^{T} \mathbf{x}+b\right) & \geq 1-\xi_{i} \ \xi_{i} & \geq 0 \end{aligned}
The constant $C$ is a free parameter in this algorithm. Making this constant large means allowing fewer points to be in the margin. This parameter must be tuned and it is advisable at least to try to vary this parameter in order to verify that the results do not dramatically depend on an initial choice.

## cs代写|机器学习代写machine learning代考|Non-linear support vector machines

We have treated the case of overlapping classes while assuming that the best we can do is a linear separation. However, what if the underlying problem is separable with a function that might be more complex? An example is shown in Fig. 3.10. Nonlinear separation and regression models are of course much more common in machine learning, and we will now look into the non-linear generalization of the SVM.

Let us illustrate the basic idea with an example in two-dimensions. A linear function with two attributes that span the 2-dimensional feature space is given by
$$y=w_{0}+w_{1} x_{1}+w_{2} x_{2}=\mathbf{w}^{T} \mathbf{x},$$
with
$$\mathbf{x}=\left(\begin{array}{c} 1 \ x_{1} \ x_{2} \end{array}\right)$$
and weight vector
$$\mathbf{w}^{T}=\left(w_{0}, w_{1}, w_{2}\right) .$$
Let us say that we cannot separate the data with this linear function but that we could separate it with a polynomial that include second-order terms like
$$y=\tilde{w}{0}+\tilde{w}{1} x_{1}+\tilde{w}{2} x{2}+\tilde{w}{3} x{1} x_{2}+\tilde{w}{4} x{1}^{2}+\tilde{w}{5} x{2}^{2}=\tilde{\mathbf{w}} \phi(\mathbf{x}) .$$
We can view the second equation as a linear separation on a feature vector
$$\mathbf{x} \rightarrow \phi(\mathbf{x})=\left(\begin{array}{c} 1 \ x_{1} \ x_{2} \ x_{1} x_{2} \ x_{1}^{2} \ x_{2}^{2} \end{array}\right) .$$
This can be seen as mapping the attribute space $\left(1, x_{1}, x_{2}\right)$ to a higher-dimensional space with the mapping function $\phi(\mathbf{x})$. We call this mapping a feature map. The separating hyperplane is then linear in this higher-dimensional space. Thus, we can use the above linear maximum margin classification method in non-linear cases if we replace all occurrences of the attribute vector $x$ with the mapped feature vector $\phi(\mathbf{x})$.
There are only three problems remaining. One is that we don’t know what the mapping function should be. The somewhat ad-hoc solution to this problem will be that we try out some functions and see which one works best. We will discuss this further later in this chapter. The second problem is that we have the problem of overfitting

as we might use too many feature dimensions and corresponding free parameters $w_{i}$. In the next section, we provide a glimpse of an argument why SVMs might address this problem. The third problem is that with an increased number of dimensions the evaluation of the equations becomes more computational intensive. However, there is a useful trick to alleviate the last problem in the case when the calculations always contain only dot products between feature vectors. An example of this is the solution of the minimization problem of the dual problem in the earlier discussions of the linear SVM. The function to be minimized in this formulation, Egn $3.26$ with the feature maps, only depends on the dot products between a vector $\mathbf{x}^{(i)}$ of one example and another example $\mathbf{x}^{(j)}$. Also, when predicting the class for a new input vector $\mathbf{x}$ from Egn $3.24$ when adding the feature maps, we only need the resulting values for the dot products $\phi\left(\mathbf{x}^{(i)}\right)^{T} \phi(\mathbf{x})$. We now discuss that such dot products can sometimes be represented with functions called kernel functions,
$$K(\mathbf{x}, \mathbf{z})=\phi(\mathbf{x})^{T} \phi(\mathbf{z})$$
Instead of actually specifying a feature map, which is often a guess to start with, we could actually specify a kernel function. For example, let us consider a quadratic kernel function between two vectors $\mathbf{x}$ and $\mathbf{z}$,
$$K(\mathbf{x}, \mathbf{z})=\left(\mathbf{x}^{T} \mathbf{z}+1\right)^{2}$$

## cs代写|机器学习代写machine learning代考|Statistical learning theory and VC dimension

SVMs are good and practical classification algorithms for several reasons. In particular, they are formulated as a convex optimization problem that has many good theoretical properties and that can be solved with quadratic programming. They are formulated to

take advantage of the kernel trick, they have a compact representation of the decision hyperplane with support vectors, and turn out to be fairly robust with respect to the hyper parameters. However, in order to act as a good learner, they need to moderate the overfitting problem discussed earlier. A great theoretical contributions of Vapnik and colleagues was the embedding of supervised learning into statistical learning theory and to derive some bounds that make statements on the average ability to learn form data. We briefly outline here the ideas and state some of the results without too much details, and we discuss this issue here entirely in the context of binary classification. However, similar observations can be made in the case of multiclass classification and regression. This section uses language from probability theory that we only introduce in more detail later. Therefore, this section might be best viewed at a later stage. Again, the main reason in placing this section is to outline the deeper reasoning for specific models.

As can’t be stressed enough, our objective in supervised machine learning is to find a good model which minimizes the generalization error. To state this differently by using nomenclature common in these discussions, we call the error function here the risk function $R$; in particular, the expected risk. In the case of binary classification, this is the probability of missclassification,
$$R(h)=P(h(x) \neq y)$$
Of course, we generally do not know this density function. We assume here that the samples are iid (independent and identical distributed) data, and we can then estimate what is called the empirical risk with the help of the test data,
$$\hat{R}(h)=\frac{1}{m} \sum_{i=1}^{m} \mathbb{1}\left(h\left(\mathbf{x}^{(i)} ; \theta\right)=y^{(i)}\right)$$

## cs代写|机器学习代写machine learning代考|Non-linear support vector machines

X=(1 X1 X2)

X→φ(X)=(1 X1 X2 X1X2 X12 X22).

ķ(X,和)=φ(X)吨φ(和)

ķ(X,和)=(X吨和+1)2

## cs代写|机器学习代写machine learning代考|Statistical learning theory and VC dimension

R(H)=磷(H(X)≠是)

R^(H)=1米∑一世=1米1(H(X(一世);θ)=是(一世))

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