(Winter 2019) IFT 6085: Theoretical principles for deep learning

This is an old version of the class. For the latest iteration of the class check here.

Class discussion group: please sign up to receive announcements and participate in discussion.


Research in deep learning produces state-of-the-art results on a number of machine learning tasks. Most of those advances are driven by intuition and massive exploration through trial and error. As a result, theory is currently lagging behind practice. The ML community does not fully understand why the best methods work.

A symptom of this lack of understanding is that deep learning methods largely lack guarantees and interpretability, two necessary properties for mission-critical applications. More importantly, a solid theoretical foundation can aid the design of a new generation of efficient methods—sans the need for blind trial-and-error-based exploration.

In this class we will go over a number of recent publications that attempt to shed light onto these questions. Before discussing the new results in each paper we will first introduce the necessary fundamental tools from optimization, statistics, information theory and statistical mechanics. The purpose of this class is to get students engaged with new research in the area. To that end, the majority of credit will be given for a class project report and presentation on a relevant topic.

Prerequisites: This is meant to be an advanced graduate class for students who want to engage in theory-driven deep learning research. We will introduce some theoretical tools necessary, but start with the assumption that students are comfortable with:

If you have not taken classes on all of the above topics, this class is not designed for you. You should take another class instead.


Lecturer: Ioannis Mitliagkas,

Class info

Winter 2019 semester:

Room: The new Mila auditorium, 6650 St. Urbain.

Office hours: 10:45am-11:45am on Thursday right after class.


We will use the class discussion group for all questions. If you have something personal/sensitive to discuss, feel free to email me. Starting your email subject with ‘IFT6085:’ will ensure that your email is not miscategorized.


Class project: 40% Paper presentation: 25% Surprise quizzes, midterm: 20% Scribing: 10% Class participation: 5%

Use this Latex template for scribing.

Tentative topics–to be updated as we go along

Extensive class bibliography


NOTE: For the first half of the class we will be closely following the previous iteration of the class.

January 9th Class introduction [slides, quiz]

Crash course in optimization

January 10th Basics of convex analysis and gradient descent [new scribed notes]


Convex analysis basics from ‘Convex Optimization’ by Boyd, Vandenberge ([5] under references):

Convergence proofs: from Chapter 3 of [1] (‘Convex Optimization…’ by S.Bubeck under References)

January 16th-17th The different rates of gradient descent: from Lipschitz to strongly convex [new scribed notes]


Convergence proofs from Chapter 3 of [1] (‘Convex Optimization…’ by S.Bubeck under References)

January 23rd Black box models and lower bounds [new scribed notes]

Reading: [1, Theorem 3.15], [6]

January 24th Accelerated methods: Polyak’s momentum (the heavy ball method) [new scribed notes] [slides]

Reading: [6], [7, pages 67-76], [8], [9]

January 30th Nesterov’s Accelerated Gradient, Stochastic gradient descent [new scribed notes]

Reading: Section 6 until 6.2 of [1], Section 14.3 of [4]

Crash course in statistical learning theory

January 31st Elements of statistical learning theory [new scribed notes]

Reading: Sections 2 (if you need the intro), 3, 4 and 6 of [4].

February 6th PAC-Bayes bounds [new scribed notes]

Reading: [12]

Reading (harder): Section 6 of [2]

February 7th Stability and generalization [new scribed notes]

Reading: [13,14]

Seminar part of class

February 13th: Guest lecturer, Guillaume Rabusseau Expressivity and universal approximation theorems [new scribed notes]

February 20th Applications of stability [new scribed notes]

Reading: [14]

February 21st Generative models [new scribed notes]

Reading: [16,17]

February 27th Wasserstein GANs [new scribed notes]

Reading: [18,19]

February 28th The Numerics of GANs [new scribed notes] [slides]

March 6th BREAK No class

March 7th BREAK No class

March 13th Midterm exam

March 14th: Guest lecturer, Guillaume Lajoie Intro to dynamical systems with application to neural networks [new scribed notes]

Reading: [31, 32, 33]

March 20th, morning Student paper presentations, A

Evaluation form

March 20th, afternoon Student paper presentations, B

Evaluation form

March 21st Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning [new scribed notes]

Reading: [23]

March 27th, morning Basic results on reinforcement learning [new scribed notes]

Reading: [34], Chapters 3 and 4

March 27th, afternoon Student paper presentations, D

Evaluation form

March 28th, morning Student paper presentations, C

Evaluation form

April 3rd Distributional reinforcement learning [new scribed notes]

Bonus lectures

This section contains useful material (older lectures) that have been replaced by new material, but are still very interesting and useful. Students will not be examined on this material.

March 21st, 2018 Variance reduction techniques for stochastic optimization [scribed notes]

Reading: [22], Section 5.3 of [21]

March 29th, 2018 PacGAN: The power of two samples in generative adversarial networks

Reading: [24]

April 4th, 2018 Some results on non-convex optimization

Reading: [25,26]

April 5th, 2018 Escaping saddle points [scribed notes] slides by Yang Yuan

Reading: [27, 28]

April 11th, 2018 The theory of spin glasses

Guest lecture by Alex Fribergh, UdeM Math.

Reading: [29]

April 12th, 2018 The Loss Surfaces of Multilayer Networks

Reading: [30]


  1. Convex Optimization: Algorithms and Complexity, Sebastien Bubeck.
  2. Theory of classification: a survey of some recent advances Stephane Boucheron, Olivier Bousquet and Gabor Lugosi
  3. iPython notebook demonstrating basic ideas of gradient descent and stochastic gradient descent, simple and complex models as well as generalization.
  4. Understanding Machine Learning: From Theory to Algorithms, by Shai Shalev-Shwartz and Shai Ben-David.
  5. Convex Optimization, Stephen Boyd and Lieven Vandenberghe.
  6. Nesterov’s Accelerated Gradient Descent for Smooth and Strongly Convex Optimization, blog post by Sebastien Bubeck.
  7. Introductory lectures on convex optimization, Yurii Nesterov.
  8. Why momentum really works, blog post by Gabriel Goh (this blog post uses a slightly different parametrization of the momentum algorithm. The version we discuss in class, only applies the learning rate on the gradient.)
  9. YellowFin and the Art of Momentum Tuning, preprint J. Zhang, I. Mitliagkas.
  10. Large-scale Machine Learning and Optimization (class), Dimitris Papailiopoulos, University of Wisconsin.
  11. Advanced Machine Learning Systems (class), Chris De Sa, Cornell University.
  12. A PAC-Bayesian Tutorial with A Dropout Bound, David McAllester.
  13. Stability and generalization, O. Bousquet, A. Elisseeff.
  14. Train faster, generalize better: Stability of stochastic gradient descent, M. Hardt, B. Recht, Y. Singer.
  15. Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data, Gintare Karolina Dziugaite, Daniel M. Roy
  16. Lecture notes on generative learning algorithms, Andrew Ng
  17. Generative Adversarial Nets, Ian Goodfellow et al.
  18. Wasserstein GAN, Martin Arjovsky, Soumith Chintala, Léon Bottou
  19. Read-through: Wasserstein GAN, Alex Irpan
  20. The Numerics of GANs, Lars Mescheder, Sebastian Nowozin, Andreas Geiger
  21. Optimization Methods for Large-Scale Machine Learning, Léon Bottou, Frank E. Curtis, Jorge Nocedal
  22. Accelerating Stochastic Gradient Descent using Predictive Variance Reduction, Rie Johnson, Tong Zhang
  23. Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning, Ali Rahimi, Ben Recht
  24. PacGAN: The power of two samples in generative adversarial networks, Zinan Lin, Ashish Khetan, Giulia Fanti, Sewoong Oh
  25. Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming, Saeed Ghadimi, Guanghui Lan
  26. Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition, Hamed Karimi, Julie Nutini, Mark Schmidt
  27. Escaping From Saddle Points — Online Stochastic Gradient for Tensor Decomposition, Rong Ge, Furong Huang, Chi Jin, Yang Yuan
  28. How to Escape Saddle Points Efficiently, Chi Jin, Rong Ge, Praneeth Netrapalli, Sham M. Kakade, Michael I. Jordan
  30. The Loss Surfaces of Multilayer Networks, Anna Choromanska, Mikael Henaff, Michael Mathieu, Gérard Ben Arous, Yann LeCun
  31. Learning Long-Term Dependencies with Gradient Descent is Difficult, Y Bengio, P Simard, P Frasconi
  32. On the difficulty of training Recurrent Neural Networks, Razvan Pascanu, Tomas Mikolov, Yoshua Bengio
  33. Opening the Black Box: Low-Dimensional Dynamics in High-Dimensional Recurrent Neural Networks, David Sussillo and Omri Barak
  34. Reinforcement Learning: An Introduction, Richard S. Sutton and Andrew G. Barto

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