ATCS – Selected Topics in Learning, Prediction, and Optimization (with applications in Finance)
2024 Spring
Lecturer: Jian Li ( lapordge at gmail dot com)
TA: Zeren Tan (tanzr20@mails.tsinghua.edu.cn)
time: every Monday 9:50am-12:15am
Room: 四教4201
We intend to cover a subset of the following topics (tentative):
(1) I assume you already know all basics (convex optimization and machine learning, stochastic gradient descent, gradient boosting, deep learning basics, CNN, RNN, please see my undergrad course). If you don't know much machine learning (e.g., you do not know how to derive the dual of SVM yet), please do NOT take this course. I will recall some concepts briefly when necessary.(1) statistical learning theory (2) theory of deep learning (3) I will talk about some (new) topics in ML: diffusion, LLM, robustness, explainable AI, fairness, calibration
I won't stickly follow the above order....I may skip something mentioned above and cover something not mentioned above...It is a graduate course.
I will be talking about several applications of ML and optimization in Finance (trading, pricing derivatives etc), and of course in typical CS areas like vision, nlp, social networks as well...
I will teach about 2/3-3/4 of the classes. For the rest, I will choose some topics and students need to do class presentation.
Tentative topics for class presentation: generative models (GAN), adversarial learning and robustness, unsupervised learning (co-training, pseudolabeling, contrastive learning), meta-learning, AutoML, various financial applications.
Basic machine learning knowledge is a must. Andrew Ng's undergrad lecture notes
The course may use various math tools from convex optimization, spectral theory, matrix pertubation, probability, high dimensional geometry, functional analysis, fourier analysis, real algebra geometry, stochastic differential geometry, information theory and so on. Only standard CS undergrad math and machine learning knowledge are required, otherwise the course will be self-contained. But certain math maturity is required.
Some knowledge about convex optimization may be useful. See this course (by S. Boyd) and a previous course by myself. But it will be fine if you didn't take those courses.
The course is a blending of theory and practice. We will cover both the underlying mathematics as well as interesting heuristics.
Grading:
Schedule:
| Feb 26 | Introduction of the course Gaussian Process Basics of Brownian Motion Stochastic differential equation (SDE) Diffusion process |
optional reading:
Stochastic Calculus, Filtering, and Stochastic Control (an excellent introductory book for SDE) |
scribed notes |
| Mar 4 | Ito Integral, Ito Process, Ito's Lemma, Feymann-Kac, Fokker-Planck, Intro to generative diffusion process |
optional reading:
Stochastic Calculus, Filtering, and Stochastic Control (an excellent introductory book for SDE), Score-Based Generative Modeling through Stochastic Differential Equations |
scribed notes |
| Mar 11 | Diffusion Process Score-based Generative Diffusion Models SMLD,DDPM, Probability Flow ODE Variational Perspective of Diffusion process DDIM |
Score-Based Generative Modeling through Stochastic Differential
Equations Denoising Diffusion Probabilistic Models Denoising Diffusion Implicit Models |
scribed notes |
| Mar 18 | DPM-Solver VQ-GAN Latent Diffusion Models (Stable Diffusion), ControlNet Consistency Models Latent Consistency Models (LCM), LCM-Lora |
DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling
in Around 10 Steps
High-Resolution Image Synthesis with Latent Diffusion Models Adding Conditional Control to Text-to-Image Diffusion Models Consistency Models Latent Consistency Models Synthesizing High-Resolution Images with Few-step Inference LoRA: Low-Rank Adaptation of Large Language Models |
scribed notes |
| Mar 25 | Rectified Flow DiT (diffusion transformer) ViT (vision transformer) Flow Matching Stable Diffusion 3 SORA Discussion |
Flow Straight and Fast: Learning to Generate and Transfer Data with
Rectified Flow Flow Matching for Generative Modeling Scalable diffusion models with transformers https://stability.ai/news/stable-diffusion-3-research-paper https://openai.com/sora https://github.com/hpcaitech/Open-Sora |
scribed notes |
| Apr 1 | Quick review of classical
statistical learning theory, Symmetrization, Chaining, Covering number, VC-dimension |
We follow the exposition from the book [Book] Probability in High Dimension | scribed notes |
| Apr 8 | Pseudo-dimension, Fat-shattering
dimension, Margin Theory, Intro to deep learning theory |
Foundation of Machine
Learning. Sec 4.4 Margin theory. understand deep learning requires rethinking generalization Spectrally-normalized margin bounds for neural networks Stronger Generalization Bounds for Deep Nets via a Compression Approach uniform convergence may be unable to explain generalization in deep learning |
scribed notes |
| Apr 15 | Algorithmic Stability Generalization of SGD (convex setting) Generalization of SGLD (nonconvex) Uniform convergence in deep learning Generalization measure |
uniform convergence may be unable to explain generalization in deep
learning Train faster, generalize better: Stability of stochastic gradient descent On generalization error bounds of noisy gradient methods for non-convex learning fantastic generalization measures and where to find them |
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| Apr 22 | PAC-Bayesian framework A nonvacuous generalization bound based on PAC-bayesian Generalization of SGLD (with l2 regularization) Quick intro of convergence of Markov process (Poincare inequality, Log-sobolev inequality Kaiyue presented his work on SAM (this paper and this paper) |
Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. generalization bounds of sgld for non-convex learning: two theoretical viewpoints generalization bounds for gradient methods via discrete and continuous prior Sharpness-Aware Minimization (SAM) for Efficiently Improving Generalization Logarithmic Sobolev Inequalities Essentials |
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| Apr 29 | Adverserial Robustness: Shortcut in Learning Adverserial examples Attack: FGSM, PGD Defense: Data Augmentation, Adverserial training (AT), Diffusion based Certified robustness Randomized Smoothing (Neyman-Pearson Lemma) Robustness of MLLM Dimpled manifold model |
Shortcut learning in deep neural networks Explaining and harnessing adversarial examples Certified adversarial robustness via randomized smoothing (CERTIFIED!!) ADVERSARIAL ROBUSTNESS FOR FREE! Adversarial purification with score-based generative models Adversarial Robustness BenchmarkHow Robust is Google's Bard to Adversarial Image Attacks? The Dimpled Manifold Model of Adversarial Examples in Machine Learning |
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| Mar 6 | Adverserial Robustness: Robust features and nonrobust features Dimpled manifold model Implicit Bias in Deep Learning: Margin Maximization, Simplicity Bias: Simple classification boundaries, Low rank solutions, Low frequency solutions, Early phase of GD: like a linear model, Feature Averaging (lead to nonrobust solutions), Sharpness Minimization |
Adversarial examples are not bugs, they are features The Dimpled Manifold Model of Adversarial Examples in Machine Learning Gradient Descent Maximizes the Margin of Homogeneous Neural Networks Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias The Pitfalls of Simplicity Bias in Neural Networks On the Spectral Bias of Neural Networks Fourier Analysis Sheds Light on Deep Neural Networks The Surprising Simplicity of the Early-Time Learning Dynamics of Neural Networks |
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| Mar 13 | Reproducing Kernel Hilbert Space RKHS Max Mean Discrepancy (MMD) |
A Primer on Reproducing Kernel Hilbert Spaces A Kernel Two-Sample Test |
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| Mar 20 | Guest lecture Dingli Yu (Tensor program) Kaifeng Lyu (Grokking) |
Tensor Programs VI: Feature Learning in
Infinite-Depth Neural Networks
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| Mar 27 | Gaussian process and Kernel
Regression Random Fourier Features Kernel Generalization Bounds Neural Tanget Kernel |
A Primer on Reproducing Kernel Hilbert Spaces Universality, characteristic kernels and RKHS embedding of measures Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences Rademacher and Gaussian complexities: Risk bounds and structural results Learning Kernels Using Local Rademacher Complexity Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks |
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| Jun 3 | Self-Contrastive learning: Word2Vec, Deepwalk, relation between self-contrastive learning and spectral clustering |
References:
[Book] Introduction to online convex optimization
[Book] Learning, Prediction and Games
[Book] Options, Futures and Other Derivatives
[Book] Advances in Financial Machine Learning
[Book] Convex Optimization
[Book] Foundation of Machine Learning
[Book] Understanding Machine Learning: From Theory to Algorithms
Python is the default programming language we will use in the course.
If you haven't use it before, don't worry. It is very easy to learn (if you know any other programming language), and is a very efficient language, especially
for prototyping things related scientific computing and numerical optimization. Python codes are usually much shorter than C/C++ code (the lauguage has done a lot for you). It is also more flexible and generally faster than matlab.
A standard combination for this class is Python+numpy (a numeric lib for python)+scipy (a scientific computing lib for python)+matplotlab (for generating nice plots)
Another somewhat easier way is to install Anaconda (it is a free Python distribution with most popular packages).