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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/multivariate-calculus-machine-learning
课程评论：没有评论
课程名称：机器学习的数学：多变量微积分 课程概述：本课程简要介绍了构建许多常见机器学习技术所需的多变量微积分。我们从基本概念开始，复习“升/跑”公式，并将其转化为函数梯度的正式定义。接着，构建一套工具，使微积分的计算更为简便和快速。我们学习如何计算指向多维曲面上坡的向量，并通过互动游戏实践这一概念。课程还探讨了如何利用微积分构建函数的近似值，并量化这些近似值的准确性。此外，我们还讨论微积分在神经网络训练中的应用，最后展示其在线性回归模型中的应用。课程旨在提供直观的微积分理解及必要的语言，使学员在遇到困难时能够自主查找相关概念，希望学员在不过多深入细节的情况下，能够自信地进入更集中于机器学习的后续课程。 课程大纲： 1. 什么是微积分？ 理解微积分是理解机器学习的关键。微积分是一套分析函数及其输入之间关系的工具。课程从基础开始，回顾函数的定义以及图形绘制中的斜率的概念，随后推导出导数的严谨定义，并通过实例发展出几条节省时间的导数规则。 2. 多变量微积分 在前一模块的基础上扩展微积分工具，处理多变量系统，分析多个输入的函数对结果的影响，并介绍存储多变量微积分分析结果的线性代数结构。 3. 多变量链式法则及其应用 学习多变量链式法则在神经网络中的应用，神经网络受到生物大脑的启发，由一系列互联的神经元构成。每个神经元的行为受一组控制参数影响，这些参数需要优化以更好地拟合数据，链式法则用于计算这些参数的影响。 4. 泰勒级数与线性化 泰勒级数是一种将函数重新表示为多项式级数的方法，它解释了使用简单线性近似复杂函数的原理。课程推导了一元泰勒级数的正式表达，并讨论其在机器学习中的重要后果，随后探讨多变量情况下的雅可比矩阵和海森矩阵。 5. 优化简介 利用多变量微积分寻找函数的极小值和极大值以优化参数。首先处理一维情况，使用梯度估算函数零点，再扩展到多维，通过求解雅可比矩阵的梯度向量来找到极值。 6. 回归 为了优化拟合函数的参数，需要定义拟合的好坏。我们首先将卡方应用于直线拟合——线性回归，接着利用梯度下降法在一般情况下优化拟合函数，最后学习如何用简洁的Python代码实现这一过程，课程至此结束。
Name:What is calculus?
Description:Understanding calculus is central to understanding machine learning! You can think of calculus as simply a set of tools for analysing the relationship between functions and their inputs. Often, in machine learning, we are trying to find the inputs which enable a function to best match the data. We start this module from the basics, by recalling what a function is and where we might encounter one. Following this, we talk about the how, when sketching a function on a graph, the slope describes the rate of change of the output with respect to an input. Using this visual intuition we next derive a robust mathematical definition of a derivative, which we then use to differentiate some interesting functions. Finally, by studying a few examples, we develop four handy time saving rules that enable us to speed up differentiation for many common scenarios.
Name:Multivariate calculus
Description:Building on the foundations of the previous module, we now generalise our calculus tools to handle multivariable systems. This means we can take a function with multiple inputs and determine the influence of each of them separately. It would not be unusual for a machine learning method to require the analysis of a function with thousands of inputs, so we will also introduce the linear algebra structures necessary for storing the results of our multivariate calculus analysis in an orderly fashion.
Name:Multivariate chain rule and its applications
Description:Having seen that multivariate calculus is really no more complicated than the univariate case, we now focus on applications of the chain rule. Neural networks are one of the most popular and successful conceptual structures in machine learning. They are build up from a connected web of neurons and inspired by the structure of biological brains. The behaviour of each neuron is influenced by a set of control parameters, each of which needs to be optimised to best fit the data. The multivariate chain rule can be used to calculate the influence of each parameter of the networks, allow them to be updated during training.
Name:Taylor series and linearisation
Description:The Taylor series is a method for re-expressing functions as polynomial series. This approach is the rational behind the use of simple linear approximations to complicated functions. In this module, we will derive the formal expression for the univariate Taylor series and discuss some important consequences of this result relevant to machine learning. Finally, we will discuss the multivariate case and see how the Jacobian and the Hessian come in to play.
Name:Intro to optimisation
Description:If we want to find the minimum and maximum points of a function then we can use multivariate calculus to do this, say to optimise the parameters (the space) of a function to fit some data. First we’ll do this in one dimension and use the gradient to give us estimates of where the zero points of that function are, and then iterate in the Newton-Raphson method. Then we’ll extend the idea to multiple dimensions by finding the gradient vector, Grad, which is the vector of the Jacobian. This will then let us find our way to the minima and maxima in what is called the gradient descent method. We’ll then take a moment to use Grad to find the minima and maxima along a constraint in the space, which is the Lagrange multipliers method.
Name:Regression
Description:In order to optimise the fitting parameters of a fitting function to the best fit for some data, we need a way to define how good our fit is. This goodness of fit is called chi-squared, which we’ll first apply to fitting a straight line - linear regression. Then we’ll look at how to optimise our fitting function using chi-squared in the general case using the gradient descent method. Finally, we’ll look at how to do this easily in Python in just a few lines of code, which will wrap up the course.
This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.
