Lecture: Nonconvex Optimization for Deep Learning
University of Tübingen, Wintersemester 2024/2025
Dr. Antonio Orvieto, ELLIS Principal Investigator at MPI.
TLDR;
Take this module if
- You know what deep learning is, and that deep models are trained with variants of SGD;
- You are curious to understand what influences training speed and generalization in vision and language models.
- You like maths and want to know more about deep learning theory.
Time/Date (to be confirmed)
TTR2 on Tuesday, 10-12 (lecture) + 12-14 (tutorial).
Basic Information
Note: This lecture does not overlap with "Convex and Nonconvex Optimization." While students are encouraged to take "Convex and Nonconvex Optimization" to solidify their understanding of SGD and basic optimization concepts (duality, interior point methods, constraints), we will only discuss optimization in the context of training deep neural networks and often drift into discussions regarding model design and initialization.
Successful training of deep learning models requires non-trivial optimization techniques. This course gives a formal introduction to the field of nonconvex optimization by discussing training of large deep models. We will start with a recap of essential optimization concepts and then proceed to convergence analysis of SGD in the general nonconvex smooth setting. Here, we will explain why a standard nonconvex optimization analysis cannot fully explain the training of neural networks. After discussing the properties of stationary points (e.g., saddle points and local minima), we will study the geometry of neural network landscapes; in particular, we will discuss the existence of "bad" local minima.
Next, to gain some insight into the training dynamics of SGD in deep networks, we will explore specific and insightful nonconvex toy problems, such as deep chains and matrix factorization/decomposition/sensing. These are to be considered warm-ups (primitives) for deep learning problems. We will then examine training of standard deep neural networks and discuss the impact of initialization and (over)parametrization on optimization speed and generalization. We will also touch on the benefits of normalization and skip connections.
Finally, we will analyze adaptive methods like Adam and discuss their theoretical guarantees and performance on language models. If time permits, we will touch on advanced topics such as label noise, sharpness-aware minimization, neural tangent kernel (NTK), and maximal update parametrization (muP).
Here are a few crucial papers discussed in the lecture (math will be greatly simplified): https://arxiv.org/abs/1605.07110, https://arxiv.org/pdf/1802.06509, https://arxiv.org/abs/1812.0795, https://proceedings.mlr.press/v9/glorot10a/glorot10a.pdf,
https://arxiv.org/abs/1502.01852, https://arxiv.org/pdf/2402.16788v1.
Requirements
The course requires some deep learning familiarity and basic knowledge of gradient-based optimization. Students who have already attended "Convex and Nonconvex Optimization" or any machine learning lecture that discusses gradient descent will have no problem following the lecture. In general, the semester requires good mathematical skills, roughly at the level of the lecture "Mathematics for Machine Learning." In particular, multivariate calculus and linear algebra are needed.
Textbook
Lecture notes will be provided. A good reference that takes a similar approach is "Optimization for deep learning: theory and algorithms" by Ruoyu Sun, available at https://arxiv.org/pdf/1912.08957.
Exercises
Exercise sheets will be provided every week and discussed during the next session. Exercises are not graded, and no project has been planned.
Exam
Written exam.
Schedule (work in progress)
|
Date |
Lecture |
Tutorial |
Session 1 |
Oct 15 |
Impact of Optimization Theory on DL
Recap of Basic Neural Networks
|
no tutorial |
Session 2 |
Oct 22 |
SGD, Adam, and other optimizers
Quadratic Models of NN minima |
Loss Landscape Visualization |
Session 3 |
Oct 29 |
Local Minima and Saddle Points |
SGD on Quadratic Models |
Session 4 |
Nov 5 |
Initialization of Deep Networks |
Pytorch tutorial |
Session 5 |
Nov 12 |
Upper/Lower bounds for SGD |
SGD vs. Adam on Deep MLPs |
Session 6 |
Nov 19 |
Overparametrization |
Tightness of Bounds |
Session 7 |
Nov 26 |
Reparametrization tricks (muP, NTP) |
Lazy Training |
Session 8 |
Dec 3 |
Implicit Bias of SGD |
Learning rate transfer |
NeurIPS - Canceled |
Dec 10 |
|
|
Session 9 |
Dec 17 |
Matrix Sensing and Factorization |
Q&A before break |
Break - Canceled |
Dec 24 |
|
|
Break - Canceled |
Dec 31 |
|
|
Session 10 |
Jan 7 |
Batch/Layer Normalization |
SAM performance |
Session 11 |
Jan 14 |
Adaptive Methods Theory |
Stabilization |
Session 12 |
Jan 21 |
Transformers and Rank Collapse |
Monotonic stepsize decrease |
Session 13 |
Jan 28 |
Training Language Models |
Effects of Rank Collapse |
Session 14 |
Feb 4 |
Exam tips |
Exam tips |