### 机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Regression by Support Vector 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

## 机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Regression by Support Vector Machines

In the regression, we estimate the functional dependence of the dependent (output) variable $y \in \Re$ on an $m$-dimensional input variable $\mathbf{x}$. Thus, unlike in pattern recognition problems (where the desired outputs $y_{i}$ are discrete values e.g., Boolean) we deal with real valued functions and we model an $\Re^{m}$ to $\Re^{1}$ mapping here. Same as in the case of classification, this will be achieved by training the SVM model on a training data set first. Interestingly and importantly, a learning stage will end in the same shape of a dual Lagrangian as in classification, only difference being in a dimensionalities of the Hessian matrix and corresponding vectors which are of a double size now e.g., $\mathbf{H}$ is a $(2 n, 2 n)$ matrix. Initially developed for solving classification problems, SV techniques can be successfully applied in regression, i.e., for a functional approximation problems $[45,142]$. The general regression learning problem is set as follows – the learning machine is given $n$ training data from which it attempts to learn the input-output relationship (dependency, mapping or function) $f(\mathbf{x})$. A training data set $\mathcal{X}=[\mathbf{x}(i), y(i)] \in \Re^{m} \times \Re, i=1, \ldots, n$ consists of $n$ pairs $\left(\mathbf{x}{1}, y{1}\right),\left(\mathbf{x}{2}, y{2}\right), \ldots,\left(\mathbf{x}{n}, y{n}\right)$, where the inputs $\mathbf{x}$ are $m$-dimensional vectors $\mathbf{x} \in \Re^{m}$ and system responses $y \in \Re$, are continuous values. We introduce all the relevant and necessary concepts of SVM’s regression in a gentle way starting again with a linear regression hyperplane $f(\mathbf{x}, \mathbf{w})$ given as
$$f(\mathbf{x}, \mathbf{w})=\mathbf{w}^{T} \mathbf{x}+b$$
In the case of SVM’s regression, we measure the error of approximation instead of the margin used in classification. The most important difference in respect to classic regression is that we use a novel loss (error) functions here. This is the Vapnik’s linear loss function with e-insensitivity zone defined as
$$E(\mathbf{x}, y, f)=|y-f(\mathbf{x}, \mathbf{w})|_{e}= \begin{cases}0 & \text { if }|y-f(\mathbf{x}, \mathbf{w})| \leq \varepsilon \ |y-f(\mathbf{x}, \mathbf{w})|-\varepsilon & \text { otherwise }\end{cases}$$

or as,
$$E(\mathbf{x}, y, f)=\max (0,|y-f(\mathbf{x}, \mathbf{w})|-\varepsilon)$$
Thus, the loss is equal to zero if the difference between the predicted $f\left(\mathbf{x}{i}, \mathbf{w}\right)$ and the measured value $y{i}$ is less than $\varepsilon$. In contrast, if the difference is larger than $\varepsilon$, this difference is used as the error. Vapnik’s $\varepsilon$-insensitivity loss function (2.40) defines an $\varepsilon$ tube as shown in Fig. 2.18. If the predicted value is within the tube, the loss (error or cost) is zero. For all other predicted points outside the tube, the loss equals the magnitude of the difference between the predicted value and the radius $\varepsilon$ of the tube.

## 机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Implementation Issues

In both the classification and the regression the learning problem boils down to solving the QP problem subject to the so-called ‘box-constraints’ and to the equality constraint in the case that a model with a bias term $b$ is used. The SV training works almost perfectly for not too large data basis. However, when the number of data points is large (say $n>2,000$ ) the QP problem becomes extremely difficult to solve with standard QP solvers and methods. For example, a classification training set of 50,000 examples amounts to a Hessian matrix $\mathbf{H}$ with $2.5 * 10^{9}$ (2.5 billion) elements. Using an 8 -byte floating-point representation we need 20,000 Megabytes $=20$ Gigabytes of memory [109]. This cannot be easily fit into memory of present standard computers, and this is the single basic disadvantage of the SVM method. There are three approaches that resolve the QP for large data sets. Vapnik in [144] proposed the chunking method that is the decomposition approach. Another decomposition approach is suggested in [109]. The sequential minimal optimization [115] algorithm is of different character and it seems to be an ‘error back propagation’ for an SVM learning. A systematic exposition of these various techniques is not given here, as all three would require a lot of space. However, the interested reader can find a description and discussion about the algorithms mentioned above in next chapter and $[84,150]$. The Vogt and Kecman’s chapter $[150]$ discusses the application of an active set algorithm in solving small to medium sized QP problems. For such data sets and when the high precision is required the active set approach in solving QP problems seems to be superior to other approaches (notably to the interior point methods and to the sequential minimal optimization (SMO) algorithm). Next chapter introduces the efficient iterative single data algorithm (ISDA) for solving huge data sets (say more than 100,000 or 500,000 or over 1 million training data pairs).

## 机器学习代写|监督学习代考Supervised and Unsupervised learning代写|Iterative Single Data Algorithm

One of the mainstream research fields in learning from empirical data by support vector machines (SVMs), and solving both the classification and the regression problems is an implementation of the iterative (incremental) learning schemes when the training data set is huge. The challenge of applying SVMs on huge data sets comes from the fact that the amount of computer memory required for solving the quadratic programming (QP) problem presented in the previous chapter increases drastically with the size of the training data set $n$. Depending on the memory requirement, all the solvers of SVMs can be classified into one of the three basic types as shown in Fig. 3.1 [150]. Direct methods (such as interior point methods) can efficiently obtain solution in machine precision, but they require at least $\mathcal{O}\left(n^{2}\right)$ of memory to store the Hessian matrix of the QP problem. As a result, they are often used to solve small-sized problems which require high precision. At the other end of the spectrum are the working-set (decomposition) algorithms whose memory requirements are only $\mathcal{O}\left(n+q^{2}\right)$ where $q$ is the size of the working-set (for the ISDAs developed in this book, $q$ is equal to 1). The reason for the low memory footprint is due to the fact that the solution is obtained iteratively instead of directly as in most of the QP solvers. They are the only possible algorithms for solving large-scale learning problems, but they are not suitable for obtaining high precision solutions because of the iterative nature of the algorithm. The relative size of the learning problem depends on the computer being used. As a result, a learning problem will be regarded as a “large” or “huge” problem in this book if the Hessian matrix of its unbounded SVs $\left(\mathbf{H}{S{f}} S_{f}\right.$ where $S_{f}$ denotes the set of free SVs) cannot be stored in the computer memory. Between the two ends of the spectrum are the active-set algorithms $[150]$ and their memory requirements are $\mathcal{O}\left(N_{F S V}^{2}\right)$, i.e. they depend on the number of unbounded support vectors of the problem. The main focus of this book is to develop efficient algorithms that can solve large-scale QP problems for SVMs in practice. Although many applications in engineering also require the solving of large-scale QP problems (and there are many solvers available), the QP problems induced by SVMs are different from these applications. In the case of SVMs, the Hessian matrix of (2.38a) is extremely dense, whereas in most of the engineering applications, the optimization problems have relatively sparse Hessian matrices. This is why many of the existing QP solvers are not suitable for SVMs and new approaches need to be invented and developed. Among several candidates that avoid the use of standard QP solvers, the two learning approaches which recently have drawn the attention are the Iterative Single Data Algorithm (ISDA), and the Sequential Minimal Optimization (SMO) $[69,78,115,148]$.

F(X,在)=在吨X+b

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