Topics course Mathematics of Deep Learning, NYU, Spring 19. CSCI-GA 3033.

• Mondays from 7.10pm-9pm. CIWW 102

• Tutoring Session with Parallel Curricula (optional): Fridays 11am-12:15pm CIWW 101.

• Piazza: sign-up here

• Office Hours: Tuesdays 4:30pm-6:00pm, office 612, 60 5th ave.

Lecture Instructor: Joan Bruna ([email protected])

Tutor (Parallel Curricula): Luca Venturi ([email protected])

Tutor (Parallel Curricula): Aaron Zweig ([email protected])

This Graduate-level topics course aims at offering a glimpse into the emerging mathematical questions around Deep Learning. In particular, we will focus on the different geometrical aspects surounding these models, from input geometric stability priors to the geometry of optimization, generalisation and learning. We will cover both the background and the current open problems.

Besides the lectures, we will also run a parallel curricula (optional), following the Depth First Learning methodology. We will start with an inverse curriculum on the Neural ODE paper by Chen et al.

• Introduction: the Curse of Dimensionality

• Part I: Geometry of Data

  • Euclidean Geometry: transportation metrics, CNNs , scattering.
  • Non-Euclidean Geometry: Graph Neural Networks.
  • Unsupervised Learning under Geometric Priors (Implicit vs explicit models, microcanonical, transportation metrics).
  • Applications and Open Problems: adversarial examples, graph inference, inverse problems.

• Part II: Geometry of Optimization and Generalization

  • Stochastic Optimization (Robbins & Munro, Convergence of SGD)
  • Stochastic Differential Equations (Fokker-Plank, Gradient Flow, Langevin Dynamics, links with SGD; open problems)
  • Dynamics of Neural Network Optimization (Mean Field Models using Optimal Transport, Kernel Methods)
  • Landscape of Deep Learning Optimization (Tensor/Matrix factorization, Deep Nets; open problems).
  • Generalization in Deep Learning.

• Part III (time permitting): Open qustions on Reinforcement Learning

Multivariate Calculus, Linear Algebra, Probability and Statistics at solid undergraduate level.

Notions of Harmonic Analysis, Differential Geometry and Stochastic Calculus are nice-to-have, but not essential.

The course will be graded with a final project -- consisting in an in-depth survey of a topic related to the syllabus, plus a participation grade. The detailed abstract of the project will be graded at the mid-term.

Final Project is due May 1st by email to the instructors

• Class 1: Basics of RL and Q learning
  • Required Reading:
    • Sutton and Barto (Ch 3, Ch 4, Ch 5, Ch 6.5)
      • The standard introduction to RL. Focus in Chapter 3 on getting used to the notation we’ll use throughout the module, and an introduction to the Bellman operator and fixed point equations. In Chapter 4 the most important idea is value iteration (and exercise 4.10 will ask you to show why iterating the Q function is basically the same algorithm).
      • Chapter 5 considers using full rollouts to estimate our value / Q function, rather than the DP updates. Focus on the difference between on-policy and off-policy, which will be relevant to the final algorithm.
      • Including 6.5 is an introduction to Q-learning in practice, updating one state-action pair at a time (without worrying about function approximation yet).
    • Contraction Mapping Theorem (3.1)
      • We’ll need the notion of contractions repeatedly throughout the module. Their essential property is a unique fixed point, and you should have a clear understanding of the constructive proof of this fixed point (don’t worry about the ODE applications).
  • Questions:
    • Exercise 3.14, Exercise 4.10 in S & B
    • Prove the Bellman operator contracts Q functions with regard to the infinity norm
    • What is a sanity-check lower bound on complexity for Q learning? Why might this be infeasible for RL problems in the wild?

• Class 6: Neural ODEs

  • Motivation: Let’s read the paper!
  • Required Reading:
    • Neural Ordinary Differential Equations
    • A blog post on NeuralODEs
  • Optional Reading:
    • A follow-up paper by the authors on scalable continuous normalizing flows: Free-form Continuous Dynamics for Scalable Reversible Generative Models

• Class 5: The adjoint method (and auto-diff)

  • Motivation: The adjoint method is a numerical method for efficiently computing the gradient of a function in numerical optimization problems. Understanding this method is essential to understand how to train ‘continuous depth’ nets. We also review the basics of Automatic Differentiation, which will help us understand the efficiency of the algorithm proposed in the NeuralODE paper.
  • Required Reading:
    • Section 8.7 from Computational Science and Engineering (CSE)
    • Sections 2,3 from Automatic Differentiation in Machine Learning: a Survey
  • Optional Reading:
    • Prof. Steven G. Johnson's notes on adjoint method
  • Questions:
    • Exercises 1,2,3 from Section 8.7 of CSE
    • Consider the problem of optimizing a real-valued function g over the solution of the ODE y' = Ay , y(0) = y_0 at time T>0: min_{y0, A} g(y(T)). What is the solution of the adjoint equation?
    • How do you get eq. (14) in Section 8.7 of CSE?

• Class 4: Normalizing Flow

  • Motivation: In this class we take a little detour through the topic of Normalizing Flows. This is used for density estimation and generative modeling, and it is another model which can be seen a time-discretization of its continuous-time counterpart.
  • Required Reading:
    • Density Estimation by Dual Ascent of the Log-likelihood, minus Section 3 (DE)
    • A family of non-parametric density estimation algorithms
    • A post on Normalizing flow
  • Optional Reading:
    • Variational Inference with Normalizing Flows
    • High-Dimensional Probability Estimation with Deep Density Models
  • Questions:
    • In DE, what is the difference between t and t, i.e. what do they represent?
    • In DE, why does eq. (4.2) imply convergence t as t ?
    • What is the computational complexity of evaluating a determinant of a N x N matrix, and why is that relevant in this context?

• Class 3: ResNets

  • Motivation: The introduction of Residual Networks (ResNets) made possible to train very deep networks. In this section we study some residual architectures variants and their properties. We then look into how ResNets approximates ODEs and how this interpretation can motivate neural net architectures and new training approaches.
  • Required Reading:
    • ResNets: ResNets and An Overview of ResNet and its Variants
    • ResNets and ODEs:
      • Sections 1-3 of Multi-level Residual Networks from Dynamical Systems View
      • Reversible Architectures for Arbitrarily Deep Residual Neural Networks
  • Optional Reading:
    • The original ResNets paper: Deep Residual Learning for Image Recognition
    • Another blog post on ResNets: Understanding and Implementing Architectures of ResNet and ResNeXt for state-of-the-art Image Classification
    • Invertible ResNets: The Reversible Residual Network: Backpropagation Without Storing Activations
    • Stable Architectures for Deep Neural Networks
  • Questions:
    • Can you think of any other neural network architectures which can be seen as discretizations of some ODE?
    • Do you understand why adding ‘residual layers’ should not degrade the network performance?
    • How do the authors of (Multi-level […]) explain the phenomena of still having almost as good performances in residual networks when removing a layer?
    • Implement your favourite variant ResNet variant

• Class 2: Numerical solution of ODEs II

  • Motivation: In the previous class we introduced some simple schemes to numerically solve ODEs. In this class we go through some more involved schemes and their convergence analysis.
  • Required Reading:
    • Runge-Kutta methods: Section 11.8 from NM or Sections 12.{5,12} from NA
    • Multi-step methods: Sections 12.6-9 from NA or Section 11.5-6 from NM
    • System of ODEs: Sections 11.9-10 from NM or Sections 12.10-11 from NA
  • Optional Reading:
    • Prof. Trefethen's class ODEs and Nonlinear Dynamics 4.1
    • Predictor-corrector methods: Section 11.7 from NM
    • Richardson extrapolation: Section 16.4 from Numerical Recipes
    • Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations
  • Questions:
    • From NA, Section 12: Exercises 12.11, 12.12, 12.19

• Class 1: Numerical solution of ODEs I

  • Motivation: ODEs are used to mathematically model a number of natural processes and phenomena. The study of their numerical simulations is one of the main topics in numerical analysis and of fundamental importance in applied sciences.
  • Required Reading:
    • Sections 12.1-4 from An Introduction to Numerical Analysis (NA) or Sections 11.1-3 from Numerical Mathematics (NM)
  • Optional Reading:
    • Section 12.5 from NM
    • Prof. Trefethen's class ODEs and Nonlinear Dynamics 4.2
  • Questions:
    • From NM, Section 11.12: Exercise 1
    • From NA, Section 12: Exercises 12.3,12.4, 12.7
    • Consider the following method for solving y' = f(y): y_{n+1} = y_n + h*(theta*f(y_n) + (1-theta)*f(y_{n+1}))
      Assuming sufficient smoothness of y and f, for what value of 0 <= theta <= 1 is the truncation error the smallest? What does this mean about the accuracy of the method?
    • Notebook