Mathematical Aspects of Algorithmic Learning and Deep Neural Networks
October and November 2021 (six weeks total), Tuesdays and Thursdays. Each session will start at 16:00 CEST and end 18:30 CEST, with a 30 min break.
*** Oct 21 / Nov 16: 14:00 – 16:30
Online via ZOOM.
The main purpose of the course is to describe the nature of algorithmic learning (AL), of its most relevant modalities and most successful applications, together with a presentation of the main mathematical ingredients that provide the basis for both the definition and study of models and for the analysis of the algorithms. Along the way, open questions and problems will be highlighted.
The course is meant to enable students at the level of graduate or late undergraduate, with knowledge of basic mathematical topics (differential and integral calculus in several variables, probability theory, algebra and geometry), but with little or no knowledge of AL, to become reasonably at ease with its current trends, achievements, open problems, and publications.
There will be twelve 2-hour sessions planned for the first three weeks of October and November 2021, with one session on Tuesday and one on Thursday for each of the six weeks. Each session will start at 16:00 CEST and end 18:30 CEST, with a 30 min break after the first hour.
The following table (Summary) provides a short description of each session, the day in which it will be delivered, and the name of the speaker.
16:00: Course presentation by Lluís Alsedà, CRM Director, and Carme Cascante, BGSMath Director.
Informal outline and some background topics
The curse of dimensionality. NNs and approximation properties
Reproducing kernel Hilbert spaces
Gradient descent and stochastic approximation
Training dynamics: lazy regime and Neural Tangent Kernel
Training dynamics: active regime and mean-field description
Group theory and differential geometry basics. Noether’s theorem
|Beyond Barron spaces: geometric stability||JB|
Harmonic Analysis: Fourier, Wavelets, Graph spectral transforms
|The Scattering Transform||JB|
Beyond Euclidean Domains: the 5G
|Open Problems and closing remarks||JB|
Syllabus and Basic References for Each Session
Informal outline. General references. Optimization techniques, with emphasis on the convex case. Examples. A model for inductive learning. Remarks on computational resources.
References: , , , , .
2 Neural networks and their approximation properties. The curse of dimensionality
The curse of dimensionality in statistical learning. Lipschitz and Sobolev hypothesis classes. From low-dimensional to high-dimensional function approximation. Polynomial approximation theorems. Universal approximation theorems. Shallow neural networks.
References: , .
3 Reproducing Kernel Hilbert Spaces
Kernels. Positive definite kernels. The kernel trick. Properties of kernels. The reproducing Hilbert space associated to a kernel. Examples of kernels. The representer theorem. Learning with kernels.
References: , , , , , , , , , .
4 Gradient descent and stochastic approximation
Optimization by gradient descent. GD with momentum. Stochastic versus batch optimization methods. Stochastic gradient approximation. Stochastic variance reduced gradient. Algorithms.
References: , , .
5 Training dynamics: Lazy regime and Neural TangentKernel (NTK)
Kernel gradient. Neural tangent kernel. Lazy training. Convergence of the SGD.
References: , , .
6 Training dynamics: active regime and mean-field description
Systems of interaction particles and thermodynamic limits. Overparameterised limits of shallow neural networks: The Barron Space. Wasserstein gradient flows and measure transport dynamics. Global convergence properties. Open questions.
References: [21, 22, 23].
7 Group theory and differential geometry basics. Noether’s theorem.
Differential manifolds. Lie groups and Lie algebras. Lagrangian and Hamiltonian mechanics. Symmetries in physical systems and conserved quantities. Noether’s theorem.
References: , , , , , , , , .
8 Beyond Barron spaces: geometric stability
Curse of dimensionality for Barron Spaces. Geometric Machine Learning, geometric domains and geometric priors. Examples: Grids, Graphs, Gauges, Groups, Geodesics. Invariance, Equivariance and Scale separation.
9 Harmonic Analysis: Fourier, Wavelets, Graph spectral transforms.
Summary of Fourier analysis. Gabor basis. The wavelet transform. Multiresolution analysis. The fast wavelet transform. Spectral techniques on graphs.
References: , , , , .
10 The Scattering Transform
Instability of Fourier Invariants. Stability of Wavelet equivariants. Putting everything together: wavelet scattering transform. Main mathematical properties: energy conservation and deformation stability. Examples. Open problems.
References: [37, 38].
11 Beyond Euclidean Domains
The Geometric Deep Learning Blueprint. Application to geometric domains: grids, groups, graphs, gauges, geodesics. Graph Neural Networks, Convolutional Neural Networks. Open problems.
12 Closing remarks
In this last lecture, we will wrap the course by tying together the open problems seen throughout the lectures, and outline key open research directions. Role of Depth. Computational Lower bounds.
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Courant Institute and the Center for Data Science, New York University
Universitat Politècnica de Catalunya