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21 results

Machine Learning 1. Lecture 3. Neural Networks: Gradient Optimization Methods

2026LectureКонстантин Вячеславович Воронцов

The neuron as a linear model; implementing Boolean functions and the exclusive-OR problem; two-layer networks as universal approximators, Cybenko's theorem; the multilayer network and the backpropagation method; the adaptive gradient methods RMSProp, AdaDelta, Adam, Nadam; gradient explosion and gradient clipping; Dropout, ReLU activation functions, batch normalization, initialization and network pruning (Optimal Brain Damage).
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130

Machine Learning 2. Lecture 1. Deep Neural Networks

2026LectureКонстантин Вячеславович Воронцов

Reminder: the neuron model, the multilayer network, stochastic gradient, momentum, vanishing and exploding gradients, ReLU, dropout, batch normalisation; depth matters more than width; feature generation and end-to-end learning; convolutional neural networks: convolution and pooling layers, the standard CNN scheme, AlexNet and ImageNet, residual networks ResNet and Highway Networks; convolutional networks for speech, texts and games; recurrent neural networks, training by BPTT, long short-term memory networks LSTM, the peephole variant, GRU.
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390

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

Algebra 1. Lecture 2

2026LectureАртём Авилов

The order of an element and of a group, the order of a permutation as the lcm of its cycle lengths; generators and cyclic groups; homomorphisms and isomorphisms of groups, the classification of cyclic groups; examples — the sign of a permutation, inversion in an abelian group, the exponential, the determinant, S3S_3 and D3D_3; subgroups, kernel and image, the injectivity criterion; left and right cosets and the partition into them; normal subgroups and conjugation, the index, Lagrange's theorem, subgroups of index 2; the quotient group; the homomorphism theorem with examples; conjugate elements, the centre, a group action on a set.

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150

Probability Theory. Lecture 1

2026LectureСтанислав Шапошников

The finite probability space and five properties of probability, a description of all probability measures on a finite set, equally likely outcomes and the choice of a model, classical paradoxes (two dice, the division of the stakes, Bertrand's paradox), conditional probability and the product rule, the law of total probability as a way to define a measure, Bayes' formula, independence of events.
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170

Topology 2. Lecture 3

2026LectureФёдор Вылегжанин

Singular homology — standard simplices and the inclusions of facets, singular simplices and chains, the differential; zeroth homology and path components, the homology of a point, reduced homology; chain maps and maps of homology, functoriality, the chain map induced by a continuous map; chain homotopies and the prism; the homotopy invariance theorem with a full proof via the prism operator; subcomplexes, quotient complexes and the homology of a pair.
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550

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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150

Machine Learning 1. Lecture 6. Linear Regression Models

2026LectureКонстантин Вячеславович Воронцов

Multiple linear regression and least squares, the pseudoinverse and the geometric meaning of least squares; the solution through the singular value decomposition; multicollinearity and the condition number; ridge regression, choosing the regularization parameter and the effective dimension; LASSO and non-smooth regularizers for feature selection; principal component analysis as a low-rank matrix decomposition; the spectral method of least squares.
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130

Machine Learning 1. Lecture 7. Generalizations of Linear Models

2026LectureКонстантин Вячеславович Воронцов

Nonlinear regression: the Newton–Raphson and Gauss–Newton methods as a series of linear regressions; logistic regression and iteratively reweighted least squares (IRLS); the generalized additive model and backfitting; the probabilistic meaning of least squares; the generalized linear model (GLM), the exponential family of distributions and the link function; logistic regression as a special case of GLM; the median, quantile regression and robust loss functions.
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130

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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160