Overview

Machine learning is often introduced as a collection of algorithms and software tools. This course takes a different approach. It studies the mathematical structures that make learning from data possible: probability, optimisation, linear algebra, approximation and statistical decision-making.

The course develops these foundations through a small number of recurring questions. What does it mean for a model to learn from a finite sample? Why can a simpler model outperform a more flexible one? How do assumptions about data become assumptions about prediction? When does optimisation produce a useful statistical solution rather than merely a numerical one?

The emphasis is on reasoning rather than software proficiency. Computational examples are used to connect formal results with model behaviour, but the purpose is to understand why established methods work, when they fail, and what their limitations imply for applied research.

Learning objectives

By the end of the course, learners should be able to:

• formulate supervised-learning problems using probabilistic and decision-theoretic language;
• explain the relationship between empirical loss, expected risk and generalisation;
• derive and interpret ordinary least squares, regularised regression and selected classification methods;
• analyse the role of convexity, gradients and numerical optimisation in model estimation;
• distinguish model complexity from computational complexity and evaluate the consequences of each;
• use simulation and numerical experiments to test mathematical claims about learning systems.

Intended participants and prerequisites

Prior study of machine learning is not required. The course is unsuitable, however, for learners seeking a non-technical introduction or a software-only tutorial.

Course structure

The course is organised into six units. Each unit combines formal exposition, worked examples and a short set of analytical or computational exercises.

Unit	Subject	Principal questions
1	Learning as statistical decision-making	How are data, models, loss functions and decisions connected? What exactly is being estimated?
2	Linear prediction and projection	Why is least squares a projection problem? Which conclusions depend on distributional assumptions?
3	Generalisation and model complexity	Why does performance on observed data differ from performance on new data? How should complexity be controlled?
4	Regularisation and constrained estimation	How do penalties encode structure? Why can biased estimates produce better predictions?
5	Classification and probabilistic boundaries	How do regression, likelihood and geometric separation lead to different classification rules?
6	Optimisation, computation and model failure	When does a numerical solution correspond to a meaningful learning solution? How can simulation expose failure?

Teaching and assessment

Teaching is delivered through concise lectures, technical notes, worked derivations and computational demonstrations. Learners should reproduce the central derivations independently and use numerical experiments to examine cases in which theoretical assumptions are weakened.

Component	Purpose	Weight
Analytical exercises	Derivation and interpretation of core results	40%
Computational investigation	Simulation-based examination of model behaviour	30%
Final assessment	Integrated reasoning across the six course units	30%

A certificate of completion is issued only when the required assessments have been completed at the stated standard. Participation alone does not constitute completion.

Faculty

Materials and technical requirements

Computing environment

Learners need a current web browser and access to a Python environment capable of running NumPy, pandas, Matplotlib and scikit-learn. Equivalent numerical software may be used when an exercise does not depend on a specific Python implementation.

Mathematical notation

Course notes use standard notation from linear algebra, probability and multivariable calculus. A notation guide is provided, but it is not a substitute for the stated prerequisites.

Course materials

Required readings and datasets are identified within each unit. Where external material is prescribed, the course page states whether it is freely available or must be obtained separately.