There is a stage in a machine learning education where the practical questions run out. You can train a network, you can tune it, you know which architectures suit which problems, and then you start noticing that the standard explanations do not hold up. A model with far more parameters than data points should overfit catastrophically and does not. Regularisation is described as preventing overfitting, but stochastic gradient descent appears to regularise on its own without anybody asking it to.
Classical statistical learning theory predicts that the bias-variance trade-off should punish exactly the models that work best in practice. At that point you either stop asking or you go and find a course like 9.520. MIT 9.520 is the statistical learning theory course run out of the Center for Brains, Minds and Machines, taught by Tomaso Poggio with Lorenzo Rosasco, Alexander Rakhlin and colleagues. Poggio has been working on the mathematics of learning since long before it was a career, and the course reflects that lineage: it treats learning as a problem in approximation theory and regularisation rather than as a set of engineering techniques that happen to work.
The structure divides into three parts. The algorithmic foundations cover regularisation methods, kernel methods, stochastic gradient descent and the phenomenon of implicit regularisation, where an optimisation procedure imposes a bias toward certain solutions without any explicit penalty term. The statistical learning theory portion covers generalisation, uniform convergence and the complexity measures used to bound it. The deep learning theory portion, which is what makes this course distinctive, separates the question of why deep networks work into three genuinely distinct questions: approximation, meaning what functions a deep network can represent and why depth helps; optimisation, meaning why gradient descent on a non-convex landscape finds good solutions at all; and generalisation, meaning why the solutions it finds work on data it has not seen.
Treating those as three separate problems is the single most clarifying idea in the course. Most informal explanations of deep learning blur them together, which is why most informal explanations are unsatisfying. The mathematical demand is real and I would not soften it. This expects comfort with functional analysis, reproducing kernel Hilbert spaces, measure-theoretic probability and graduate linear algebra.
Someone whose mathematical background is an undergraduate engineering degree and some self-study will find the first few lectures manageable and then hit a wall. That is not a criticism, it is a statement about who the course is for, and the honest advice is to build the prerequisites first rather than attempting it and bouncing. The dating is the main practical caveat. The public course site is most recently maintained around the Fall 2019 offering, with lecture videos from Fall 2018 available publicly.
Learning theory moves more slowly than applied machine learning, so most of the material has aged well, and the treatment of regularisation, kernels and generalisation bounds is as valid now as it was then. The deep learning theory section is the part where time has passed most visibly. Work on scaling behaviour, on what emerges at larger model sizes, and on the theory of attention and transformers has developed considerably since 2019, and none of it is here. Treat the course as the foundation and the recent literature as the continuation.
The practical value is indirect and worth being honest about. This will not help you ship anything. It will make you a substantially better reader of the literature, because you will be able to tell when a paper's explanation of why a method works is actually an explanation rather than a story told after the fact. That skill compounds.
A field producing this volume of published claims needs people who can distinguish evidence from narrative, and the theory is what equips you to do it. Four point three. Genuinely difficult, taught by the people who did the work, addressing the questions that practical courses leave hanging, slightly dated in its most modern section and still the best public treatment of why machine learning works.