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Machine Learning 1. Bayesian Classification Theory

2026LectureКонстантин Вячеславович Воронцов

The probabilistic formulation of classification: prior and posterior class probabilities, class likelihood functions, the maximum a posteriori principle; the optimal Bayes classifier and losses from errors; empirical estimation of prior probabilities and class densities; the naive Bayes classifier, features with exponential-family densities and the linear naive Bayes classifier; a survey of Bayesian classifiers: the Parzen window method, the quadratic discriminant, Fisher's linear discriminant, Gaussian mixtures with diagonal covariance matrices, and the radial basis function network.
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Machine Learning 1. Lecture 14. Anomaly Detection and Robust Methods

2026LectureКонстантин Вячеславович Воронцов

Anomaly and novelty detection; heuristic anomaly scores: local density, isolation forest, a mixture with a background component, robust principal component analysis; outlier removal in nonparametric regression, the LOWESS algorithm; anomaly as the value of the loss function; the theory of robust learning: slowly growing functions, iterative reweighting of objects with IRS and IRLS, robust loss functions, the median, the quantile and the censored mean, robust aggregation functions and their smoothing, the IR-ERM algorithm; one-class SVM, learning from positive and unlabeled objects (PU-learning), problems with new and unknown classes.
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480

Machine Learning 1. Lecture 5. The Support Vector Machine

2026LectureКонстантин Вячеславович Воронцов

The optimal separating hyperplane and the width of the separating strip; the hinge loss and L2 regularization; the Karush–Kuhn–Tucker conditions, the dual problem and the types of objects; kernels, rectifying spaces and the constructive building of kernels; SVM as a two-layer neural network; choosing the constant C; SVM regression; L1 regularization, Elastic Net SVM, SFM and RVM.
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Machine Learning 1. Lecture 9. Logical Classification Methods

2026LectureКонстантин Вячеславович Воронцов

The interpretability of models; decision trees and the greedy ID3 algorithm; the axioms of impurity, the entropy and Gini criteria; handling missing values; the merits and drawbacks of the greedy strategy; tree pruning (post-pruning); CART regression trees and Minimal Cost-Complexity Pruning; rule induction: logical regularities, families of rules (conjunctions, syndromes, half-planes, balls), search metaheuristics, a tree as a covering set of conjunctions; the Pareto front in (p,n) space, logical and statistical regularities, Fisher's exact test, informativeness criteria; building a classifier from regularities.
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Machine Learning 1. Lecture 11. Advanced Ensemble Methods

2026LectureКонстантин Вячеславович Воронцов

Bias–variance analysis: the decomposition of the error into noise, bias and variance, the bias and variance of simple voting, why complex ensembles do not overfit; gradient boosting for an arbitrary loss function as a parametric approximation of a gradient step; stochastic gradient boosting; special cases: regression, AdaBoost, AnyBoost, variants of loss functions; XGBoost: the tree as a linear model, L0 and L2 regularisation, analytical leaf values and the splitting criterion; CatBoost: leakage in the gradients, ordered boosting on subsamples of doubling length, unbiased target statistics for categorical features.
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Machine Learning 1. Lecture 12. Density Estimation and Bayesian Classification

2026LectureКонстантин Вячеславович Воронцов

Estimating a probability density from a sample: the maximum likelihood principle, the multivariate Gaussian density and its geometric meaning, multicollinearity and regularisation of the covariance matrix; separating a mixture of distributions, the EM algorithm as a method of simple iterations, the Gaussian mixture (GMM); the nonparametric Parzen–Rosenblatt estimate, the choice of kernel and window width; Bayesian classification theory: the generative and discriminative approaches, the optimal Bayes classifier, the Parzen window method, the quadratic discriminant, Fisher's linear discriminant, the radial basis function network and its training by the EM algorithm.
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Machine Learning 1. Lecture 4. Metric Methods of Classification and Regression

2026LectureКонстантин Вячеславович Воронцов

The continuity and compactness hypotheses; the Minkowski metric and feature-free distances; the generalized metric classifier, the k nearest neighbours method, the Parzen window and the method of potential functions as a linear classifier; complete cross-validation, the compactness profile and the exact CCV formula for the nearest neighbour method; prototype selection; Nadaraya–Watson kernel smoothing, the choice of kernel and bandwidth.
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Machine Learning 2. Lecture 2. Neural Networks for Unsupervised Learning

2026LectureКонстантин Вячеславович Воронцов

Kohonen networks: clustering as a two-layer neural network, the hard (WTA) and soft (WTM) competition rules, self-organising Kohonen maps and their interpretation; autoencoders: the formulation of the problem, the linear autoencoder and the principal component method, sparse, denoising, contractive, relational and variational autoencoders, reparametrisation, autoencoders for supervised learning, multilayer autoencoders; joint training of several models: pre-training and transfer learning, multi-task learning and few-shot learning, self-supervised learning, distillation and surrogate modelling, learning using privileged information (LUPI), generative adversarial networks (GAN).
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Machine Learning 2. Lecture 6. Learning to Rank

2026LectureКонстантин Вячеславович Воронцов

The learning-to-rank problem: recovering a partial order relation, the pointwise, pairwise and listwise approaches, the linear ranking model, applications; ordinal regression and the ordinal SVM (SVOR); the pairwise criterion, Ranking SVM, RankNet, the transition to the listwise method LambdaRank; ranking search results: query–document objects, assessor ratings, text, link and click features, TF-IDF and its probabilistic meaning, Okapi BM25, PageRank; quality criteria: precision and recall, MAP, the share of defective pairs and its link with AUC, DCG and NDCG, the pFound user behaviour model; the DSSM neural model: a siamese network, word hashing by letter trigrams, training on clicks with negative sampling.
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Geometry. Lecture 2

2026LectureФёдор Попеленский

Finite geometries — the Platonic solids. A reminder — right actions, orbits, stabilisers, fundamental domains (corrected definition) and the example of Z+Z\mathbb{Z} + \mathbb{Z} on the plane; the formula ∣G∣=∣Orb⁡(x)∣⋅∣Stab⁡(x)∣|G| = |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)| and the 48 symmetries of the cube; regular polyhedra — Sosinsky's definition and the definition via flags, a stellated counterexample; the five Platonic solids; regular polytopes in R4\mathbb{R}^4 (six) and in Rn\mathbb{R}^n for n≥5n \ge 5 (the cube, the simplex, the hyperoctahedron) with the coordinates of their vertices; why there are no more than five — face angles and the Schläfli symbol; uniqueness of the gluing and the rigidity theorem; Euler's formula; the finite subgroups of SO(3)SO(3) — Zn\mathbb{Z}_n, DnD_n and the rotation groups of the tetrahedron, cube and icosahedron; the duality cube–octahedron, icosahedron–dodecahedron; which figures on the sphere realise Zn\mathbb{Z}_n, DnD_n and Z2\mathbb{Z}_2.

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Артём Авилов
@artem-avilov

Артём Алексеевич Авилов — доцент факультета математики НИУ ВШЭ и научный сотрудник Лаборатории алгебраической геометрии и её приложений (с 2017 года). Работает в Высшей школе экономики с 2012 года. Окончил механико-математический факультет МГУ имени М. В. Ломоносова (2012). Кандидат физико-математических наук (2016), диссертация «Автоморфизмы алгебраических многообразий и минимальные модели». В НИУ ВШЭ читает курсы по алгебраической геометрии — в 2026/27 учебном году «Алгебраические поверхности» и «Введение в теорию схем»; в Независимом московском университете осенью 2026 года читает курс «Алгебра-1».

Константин Вячеславович Воронцов
@konstantin_vorontsov

Константин Вячеславович Воронцов — профессор РАН, доктор физико-математических наук. Заведующий кафедрой «Математические методы прогнозирования» ВМК МГУ, где с 2021 года — профессор; заведующий лабораторией «Машинное обучение и семантический анализ» Института искусственного интеллекта МГУ (с 2022 года); заведующий кафедрой «Машинное обучение и цифровая гуманитаристика» МФТИ (с 2021 года); заведующий кафедрой «Интеллектуальные системы» МФТИ (с 2022 года), на которой работает с 2004 года; главный научный сотрудник отдела «Интеллектуальные системы» ВЦ ФИЦ ИУ РАН, где работает с 2001 года. Окончил МФТИ (1994) и аспирантуру ВЦ РАН (1997). Кандидатская диссертация «Локальные базисы в алгебраическом подходе к проблеме распознавания» (1999, специальность 01.01.09); докторская диссертация «Комбинаторная теория надёжности обучения по прецедентам» (2010, специальность 05.13.17). Автор комбинаторной теории надёжности обучения по прецедентам, оценивающей вероятность переобучения с учётом эффектов расслоения и сходства в семействах алгоритмов; основное направление его исследований — теория и методы аддитивной регуляризации тематических моделей (ARTM). С 2014 года — научный руководитель разработки библиотеки тематического моделирования с открытым кодом BigARTM, с 2019 года — поисково-рекомендательной системы SciSearch. Один из идеологов и администраторов ресурса MachineLearning.ru (с 2008 года). Медаль РАН с премией для молодых учёных (1999, совместно с Е. В. Нарыжным); почётное звание профессора РАН (2015); юбилейная медаль РАН «300 лет Российской академии наук» (2024); премия Yandex ML Prize в номинации «Зал славы» (2025). Автор курсов «Вероятностные тематические модели», «Математические методы анализа текстов», «Прикладной статистический анализ данных» и спецкурса по теории надёжности обучения по прецедентам; в 2008–2020 годах преподавал в Школе анализа данных Яндекса. С 2004 года читает годовой курс «Машинное обучение» в МФТИ и ВМК МГУ.

Topology 2. Lecture 2

2026LectureФёдор Вылегжанин

Where simplicial homology shows up in real life — topological data analysis. The union of balls around a point cloud, persistence modules and their decomposition into a sum of interval modules over a field (the structure theorem and the algorithm), the barcode; the Vietoris–Rips and Čech complexes; the stability theorem for barcodes, the Hausdorff, Gromov–Hausdorff and Wasserstein distances; open covers and their nerves, Leray covers and the nerve theorem, the category of covers, paracompactness, Lebesgue covering dimension, Čech homology and cohomology.
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520

Algebra 1. Lecture 1

2026LectureАртём Авилов

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group SnS_n — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.

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310

Geometry. Lecture 1

2026LectureФёдор Попеленский

What geometries there are and what they have in common — Klein's Erlangen programme; the school 3–4–5 triangle and the ellipsoid; the cube and the brick as finite geometries; metric spaces; groups, quantifiers in the axioms; the group of bijections and the order of composition; left and right actions of a group on a set; the metrics d1d_1, d2d_2, d∞d_\infty on Rn\mathbb{R}^n; isometries, subgroups; the order of an element; orbits, stabilisers, transitive actions; the list of geometries in the course — the Euclidean plane, the regular triangle and S3S_3, the cube and its 48 symmetries, the Poincaré model of the Lobachevsky plane; fundamental domains; subgeometries — the square in the cube.

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Machine Learning 1. Generalization Theory

2026LectureКонстантин Вячеславович Воронцов

The problem of estimating generalization ability; the error matrix and the probability of overfitting; the case of a single algorithm and the hypergeometric distribution; the principle of uniform convergence of frequencies and Boole's inequality; the Vapnik–Chervonenkis bound, the growth function and the capacity of a family of algorithms; inverting the bound and the complexity penalty; structural risk minimization; the splitting and similarity effects; the Occam's razor bound and its link to regularization.
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140

Machine Learning. Additional Lecture 1. The Theory of Overfitting

2026LectureКонстантин Вячеславович Воронцов

The combinatorial theory of overfitting: the probability of overfitting and why the Vapnik–Chervonenkis bounds are loose; experiments with overfitting — a monotone chain of algorithms, the effect of splitting and connectivity of the family on the probability of overfitting and on the complete cross-validation estimate, overfitting of chains, the exact formula for the probability of overfitting when choosing between two algorithms; the splitting–connectivity graph of a family of algorithms, examples for linear classifiers, a monotone chain and a two-dimensional net; generating and forbidden sets, upper connectivity and inferiority of an algorithm; the splitting–connectivity upper bound, the idea of its proof and its properties.
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Topology 2. Lecture 1

2026LectureФёдор Вылегжанин

Why homology is needed — the fundamental group as a functor, homotopy groups, bordism and its combinatorial approximation; chain complexes and their homology; abstract simplicial complexes, geometric realisation and triangulations; the complex of simplicial chains, the proof of d2=0d^2 = 0, the simplicial homology of a segment, of two points and of the boundary of a triangle; reduced homology.

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