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

1. You know what deep learning is, and that deep models are trained with variants of SGD;
2. You are curious to understand what influences training speed and generalization in vision and language models.
3. 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