The goal of the course is to equip students with the mathematical foundations necessary to understand and rigorously analyse high-dimensional data and machine learning concepts and algorithms, in particular those that can be used in quantitative finance. Specifically, and after a general brief introduction to the concept of machine learning, this course will provide students with the theoretical framework and the tools to:

o Visualise and cluster high dimensional data.
o Understand the approximation abilities (of an arbitrary input-output map) of a given machine learning algorithm,
o Build classification and regression maps using machine learning techniques,
o Use available data to train a machine learning algorithm in order to increase its performance,
o Exploit any moderately accurate machine learning algorithm to build one with an arbitrary high accuracy (boosting).
The theoretical aims of the course will be illustrated with practical examples from quantitative finance throughout.

(LO1) Recommend appropriate unsupervised learning techniques for gaining insight on high-dimensional data.

(LO2) Explain how supervised linear clustering algorithms (SVM) are deployed and trained.

(LO3) Solve basic convex constrained optimisation problems.

(LO4) Apply kernel methods to solve supervised non-linear classification problems.

(LO5) Describe and show the convergence of gradient descent methods.

(LO6) Explain how feed-forward neural networks are deployed and trained.

(LO7) Establish the universality of feed-forward neural networks.

(S1) To be able to pursue further study in data science and machine learning and more advanced types of neural networks.

(S2) To be able to approach methodically machine learning problems and understand the mathematics of learning systems.

• Introduction: what is machine learning? Applications and problems, classes of learning problems, machine learning setup and scenarios.
• Visualisation of high-dimensional data: Principal Component Analysis (PCA), introduction to t-distributed Stochastic Neighbour Embedding (t-SNE).
• k-means clustering algorithm
• Support Vector Machines (SVM): Data classification problem, margin of a hyperplane classifier, convex optimisation and Lagrangian duality, Hard and Soft SVM.
• Kernel methods: non-linear classification, definition and properties of positive definite symmetric (PDS) kernels, the Reproducing Kernel Hilbert Space (RKHS) theorem and Representer theorem.
• Optimisation techniques: logistic regression, maximum likelihood principle, gradient descent, stochastic gradient descent.
• Neural Networks: definition of a neural network, backpropagation, universal approximation theorems.
• Boosting: the boosting learning paradigm, the AdaBoost algorithm.