Mathematics for Machine Learning: Multivariate Calculus

Imperial College London via Coursera

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Overview

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...

Syllabus

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. 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.

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.

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.

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.

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 well 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 well 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. Well 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.

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 well first apply to fitting a straight line - linear regression. Then well look at how to optimise our fitting function using chi-squared in the general case using the gradient descent method. Finally, well look at how to do this easily in Python in just a few lines of code, which will wrap up the course.