One of the most significant developments in artificial intelligence is *machine learning* – where rather than teaching a machine explicitly how to do a complex task (in the sense of a traditional computer program), instead the machine learns directly from the experience of repeatedly doing the task itself. The machine is given a lot of flexibility in how it approaches this well-defined task, a way to measure how successful it is, and an algorithm for how to try to improve.

It might be easiest to think about this in the context of playing games: rather than giving a computer explicit instructions such as "in this situation, do this move”" you let the computer play lots of games and nudge it to choose moves that ended with a victory more often than ones that ended in defeat.

Advances in engineering and computer science are key to progress in this area. But the real nuts and bolts of machine learning is done with mathematics. When we say a machine learning algorithm is designed to solve a particular task, what we mean is that the algorithm is designed to construct a mathematical function that will take some inputs (eg, the current state of the game, described mathematically) and reliably give the desired output (eg, the best next move in the game).

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Initially the machine learning algorithm doesn’t know the best mathematical function to reliably do this. Instead the algorithm is designed to start with some initial function and adjusts this function in response to feedback from repeatedly doing the task. The feedback might be some sort of measure of how well the computer is doing the task (say, if it won or lost the game in the example above, and how quickly that happened) – an approach called *reinforcement learning*. Or the algorithm repeatedly performs the task using some problem-specific data obtained in advance (called *training data*) as a reference and making appropriate adjustments, hopefully resulting in a mathematical function that can reliably do the required task for new, unseen data.

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You can read more in our introduction Maths in a minute: Machine learning and neural networks and find more details in Chris Budd’s article, What is machine learning?
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### About this article

This article was written by Kweku Abraham, Chris Budd, Marianne Freiberger, Yury Korolev and Rachel Thomas.

Kweku Abraham is a postdoctoral researcher in statistics at the University of Cambridge, working on the mathematics of deep learning.

Chris Budd is based at the University of Bath, where he is Professor of Applied Mathematics. He is also Professor of Maths at the Royal Institution and Gresham Professor of Geometry. He has also been the Education Officer of the London Mathematical Society, and Vice-President of the Institute of Mathematics and its Applications.

Yury Korolev is a Lecturer in Mathematics and Data Science and an EPSRC Postdoctoral Fellow at the Department of Mathematical Sciences at the University of Bath, and a Quondam Fellow of Hughes Hall, University of Cambridge.

Marianne Freiberger and Rachel Thomas are Editors of *Plus*.

*This article was produced as part of our collaboration with the Mathematics for Deep Learning (Maths4DL) research programme.* Maths4DL brings together researchers from the universities of Bath and Cambridge, and University College London and aims to combine theory, modelling, data and computation to unlock the next generation of deep learning. You can see more content produced with Maths4DL here.