关于此课程： 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.
此课程适用人群： This class is for people who would like to learn more about machine learning techniques, but don’t currently have the fundamental mathematics in place to go into much detail. This course will include some exercises that require you to work with code. If you've not had much experience with code before DON'T PANIC, we will give you lots of guidance as you go.
制作方： Imperial College London
教学方： Samuel J. Cooper, Lecturer
Dyson School of Design Engineering
教学方： David Dye, Professor of Metallurgy
Department of Materials
教学方： A. Freddie Page, Strategic Teaching Fellow
Dyson School of Design Engineering
| 基本信息 | 课程 2（共 3 门，Mathematics for Machine Learning Specialization ） |
| 级别 | Beginner |
| 承诺学习时间 | 6 weeks of study, 2-5 hours/week |
| 语言 | English |
| 如何通过 | 通过所有计分作业以完成课程。 |
| 用户评分 |
授课大纲
第 1 周
What is calculus?
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. Typically, 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 off 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.
10 视频, 4 阅读材料, 5 练习测试
- Leyendo: About Imperial College & the team
- Leyendo: How to be successful in this course
- Leyendo: Grading Policy
- Leyendo: Additional Readings & Helpful References
- Cuadro de aviso de la discusión: Nice to meet you!
- Complemento no calificado: Survey
- 视频: Welcome to Module 1!
- 视频: Functions
- Cuestionario de práctica: Matching functions visually
- 视频: Rise Over Run
- Cuestionario de práctica: Matching the graph of a function to the graph of its derivative
- 视频: Definition of a derivative
- 视频: Differentiation examples & special cases
- Cuestionario de práctica: Let's differentiate some functions
- 视频: Product rule
- Cuestionario de práctica: Practicing the product rule
- 视频: Chain rule
- Cuestionario de práctica: Practicing the chain rule
- 视频: Taming a beast
- 视频: See you next module!
已评分: Unleashing the toolbox
第 2 周
Multivariate calculus
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.
9 视频, 4 练习测试
- 视频: Variables, constants & context
- 视频: Differentiate with respect to anything
- Cuestionario de práctica: Practicing partial differentiation
- 视频: The Jacobian
- Cuestionario de práctica: Calculating the Jacobian
- 视频: Jacobian applied
- Cuestionario de práctica: Bigger Jacobians!
- 视频: The Sandpit
- Libreta: The Sandpit
- Libreta: The Sandpit - Part 2
- 视频: The Hessian
- Cuestionario de práctica: Calculating Hessians
- 视频: Reality is hard
- 视频: See you next module!
已评分: Assessment: Jacobians and Hessians
第 3 周
Multivariate chain rule and its applications
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.
6 视频, 3 练习测试
- 视频: Multivariate chain rule
- 视频: More multivariate chain rule
- Cuestionario de práctica: Multivariate chain rule exercise
- 视频: Simple neural networks
- Cuestionario de práctica: Simple Artificial Neural Networks
- 视频: More simple neural networks
- Cuestionario de práctica: Training Neural Networks
- Libreta: Backpropagation
- Cuadro de aviso de la discusión: I ❤️ backpropagation
- 视频: See you next module!
已评分: Backpropagation
第 4 周
Taylor series and linearisation
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.
9 视频, 4 练习测试
- 视频: Building approximate functions
- 视频: Power series
- Cuestionario de práctica: Matching functions and approximations
- Complemento no calificado: Visualising Taylor Series
- 视频: Power series derivation
- 视频: Power series details
- Cuestionario de práctica: Applying the Taylor series
- 视频: Examples
- 视频: Linearisation
- Cuestionario de práctica: Taylor series - Special cases
- 视频: Multivariate Taylor
- Cuestionario de práctica: 2D Taylor series
- 视频: See you next module!
已评分: Taylor Series Assessment
第 5 周
Intro to optimisation
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.
4 视频, 2 练习测试
- Cuestionario de práctica: Newton-Raphson in one dimension
- 视频: Gradient Descent
- Libreta: Gradient descent in a sandpit
- Cuadro de aviso de la discusión: Steepest strategies
- 视频: Constrained optimisation
- Cuestionario de práctica: Lagrange multipliers
- 视频: See you next module!
已评分: Checking Newton-Raphson
已评分: Optimisation scenarios
第 6 周
Regression
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.
4 视频, 1 阅读材料, 2 练习测试
- Cuestionario de práctica: Linear regression
- 视频: General non linear least squares
- Cuestionario de práctica: Fitting a non-linear function
- 视频: Doing least squares regression analysis in practice
- Libreta: Fitting the distribution of heights data
- 视频: Wrap up of this course
- Leyendo: Did you like the course? Let us know!
- Complemento no calificado: Post-Course Survey
已评分: Fitting the distribution of height data
常见问题解答
运作方式
Trabajo del curso
Cada curso es como un libro de texto interactivo, con videos pregrabados, cuestionarios y proyectos.
Ayuda de tus compañeros
Conéctate con miles de estudiantes y debate ideas y materiales del curso, y obtén ayuda para dominar los conceptos.
Certificados
Obtén reconocimiento oficial por tu trabajo y comparte tu éxito con amigos, compañeros y empleadores.
制作方
Imperial College London
Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.
Imperial students benefit from a world-leading, inclusive educational experience, rooted in the College’s world-leading research. Our online courses are designed to promote interactivity, learning and the development of core skills, through the use of cutting-edge digital technology.
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评分和审阅
The course is still a bit young, some errors appear here and there sometimes, and some parts of it are a bit steep.
Otherwise, this is a good course, focused on derivatives.
Not enough detail in the explanations and little to no instructor participation in the forums.
Excellent course! It helps understand to take the sandpit as an example for learning Jacobian, Hessian and steepest algorithm stuff. More than boring math formulas.
SJ
Amazing