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# 计算机代写|机器学习代写Machine Learning代考|COMP7703 Deep Learning

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

Another example of a hypothesis space uses a signal-flow representation of a hypothesis map $h: \mathbb{R}^n \rightarrow \mathbb{R}$. This signal-flow representation is referred to as artificial neural network. Figure $3.8$ depicts an example for a artificial neural network that is used to represent a (parameterized) hypothesis $h^{(\mathbf{w})}: \mathbb{R}^n \rightarrow \mathbb{R}$. A feature vector $\mathbf{x} \in \mathbb{R}^n$ is fed into the input units, each of which reads in one single feature $x_j \in \mathbb{R}$. The features $x_j$ are then multiplied with the weights $w_{j, j^{\prime}}$ associated with the link between the $j$ th input node (“neuron”) with the $j^{\prime}$ th node in the middle (hidden) layer. The output of the $j^{\prime}$-th node in the hidden layer is given by $s_{j^{\prime}}=g\left(\sum_{j=1}^n w_{j, j^{\prime}} x_j\right)$ with some (typically non-linear) activation function $f: \mathbb{R} \rightarrow \mathbb{R}$. The input argument to the activation function is the weighted combination $\sum_{j=1}^n w_{j, j^{\prime}} s_{j^{\prime}}$ of the outputs $s_j$ of the nodes in a previous layer. For the artificial neural network depicted in Fig. 3.11, the output of neuron $s_1$ is $f(z)$ with $z=w_{1,1} x_1+w_{1,2} x_2$.

Two popular choices for the activation function used within artificial neural networks are the sigmoid function $f(z)=\frac{1}{1+\exp (-z)}$ or the deep net $f(z)=\max {0, z}$. Artificial neural networks using many, say 10 , hidden layers, is often referred to as a deep net. ML methods using hypothesis spaces obtained from deep nets are known as deep learning methods [7].

Remarkably, using some simple non-linear activation function $f(z)$ as the building block for artificial neural networks allows us to represent an extremely large class of predictor maps $h^{(\mathbf{w})}: \mathbb{R}^n \rightarrow \mathbb{R}$. The hypothesis space generated by a given artificial neural network structure, i.e., the set of all predictor maps which can be implemented by a given artificial neural network and suitable weights $\mathbf{w}$, tends to be much larger than the hypothesis space (2.4) of linear predictors using weight vectors $\mathbf{w}$ of the same length [7, Chap. 6.4.1.]. It can be shown that an artificial neural network with only one single (but arbitrarily large) hidden layer can approximate any given map $h: \mathcal{X} \rightarrow \mathcal{Y}=\mathbb{R}$ to any desired accuracy [8]. However, a key insight which underlies many deep learning methods is that using several layers with few neurons, instead of one single layer containing many neurons, is computationally favourable [9].

## 计算机代写|机器学习代写Machine Learning代考|Maximum Likelihood

For many applications it is useful to model the observed datapoints $\mathbf{z}^{(i)}$, with $i=$ $1, \ldots, m$, as i.i.d. realizations of a random variable $\mathbf{z}$ with probability distribution $p(\mathbf{z} ; \mathbf{w})$. This probability distribution is parameterized in the sense of depending on a weight vector $\mathbf{w} \in \mathbb{R}^n$. A principled approach to estimating the vector $\mathbf{w}$ based on a set of i.i.d. realizations $\mathbf{z}^{(1)}, \ldots, \mathbf{z}^{(m)} \sim p(\mathbf{z} ; \mathbf{w})$ is maximum likelihood estimation [10].

Maximum likelihood estimation can be interpreted as an ML problem with a hypothesis space parameterized by the weight vector $\mathbf{w}$, i.e., each element $h^{(\mathbf{w})}$ of the hypothesis space $\mathcal{H}$ corresponds to one particular choice for the weight vector $\mathbf{w}$, and the loss function
$$L\left(\mathbf{z}, h^{(\mathbf{w})}\right):=-\log p(\mathbf{z} ; \mathbf{w}) .$$
A widely used choice for the probability distribution $p(\mathbf{z} ; \mathbf{w})$ is a multivariate normal (Gaussian) distribution with mean $\boldsymbol{\mu}$ and covariance matrix $\Sigma$, both of which constitute the weight vector $\mathbf{w}=(\boldsymbol{\mu}, \Sigma$ ) (we have to reshape the matrix $\Sigma$ suitably into a vector form). Given the i.i.d. realizations $\mathbf{z}^{(1)}, \ldots, \mathbf{z}^{(m)} \sim p(\mathbf{z} ; \mathbf{w})$, the maximum likelihood estimates $\hat{\boldsymbol{\mu}}, \widehat{\Sigma}$ of the mean vector and the covariance matrix are obtained via
$$\hat{\boldsymbol{\mu}}, \widehat{\Sigma}=\underset{\boldsymbol{\mu} \in \mathbb{R}^n, \Sigma \in \mathbb{S}{+}^n}{\operatorname{argmin}}(1 / m) \sum{i=1}^m-\log p\left(\mathbf{z}^{(i)} ;(\boldsymbol{\mu}, \Sigma)\right) .$$

## 计算机代写|机器学习代写Machine Learning代考|Maximum Likelihood

$$L\left(\mathbf{z}, h^{(\mathbf{w})}\right):=-\log p(\mathbf{z} ; \mathbf{w}) .$$

$$\hat{\boldsymbol{\mu}}, \widehat{\Sigma}=\underset{\boldsymbol{\mu} \in \mathbb{R}^n, \Sigma \in \mathrm{S}_{+}{ }^n}{\operatorname{argmin}}(1 / m) \sum i=1^m-\log p\left(\mathbf{z}^{(i)} ;(\boldsymbol{\mu}, \Sigma)\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。