Overview

Generative AI (GenAI) is reshaping the future of humanity. From ChatGPT revolutionizing work and productivity to self-driving cars navigating complex environments, GenAI is driving a transformative revolution across every facet of society. At the core, GenAI constructs probabilistic density functions to model language, images, proteins, molecules, and beyond.

This course not only teaches you how to build DeepSeek-R1 and Stable Diffusion from scratch, but also takes you on a deep dive into the mathematical foundations that power GenAI:

  • 🎲 Probability & Statistics: To model uncertainty with functions.
  • 🤖 Non-Convex Optimization (Deep Learning) – To train neural networks to approximate intricate functions.
  • 📡 Information Theory – To enforce low-dimensional structures within the function.
  • 🌀 Dynamical Systems – To model the evolution of the function over time and space.
  • 🌪️ Stochastic Processes – To model the evolution of the function with controlled randomness.

This ambitious course culminates in a final project where you design a startup mockup with GenAI tools, aiming to create impact for the common good at the intersection of technology and business.

Prerequisites:
  • MATH: Linear Algebra (MATH 2210, MATH 2211), Multivariate Calculus (MATH 2202, MATH 2203)
  • CSCI: Randomness (CSCI 2244), one 3000-level AI course

Schedule

Theme Date Topic Materials Assignments Final Project
Tue, Aug 26 Lec 1. Introduction
What: demos, formulation
Why: opportunities, risks
How: syllabus, logistics
The Social Dilemma lab1
Language Generation
(Week 1-7) 		Symbolic Representation
Thu, Aug 28 Lec 2. Logic and Grammars
Logic: propositional logic, first-order logic
Grammar: lexicon, syntax, Eliza
Eliza hw1 (sol)
Statistical Representation
Tue, Sep 2 Lec 3. N-Gram Language Models
Probability: language modeling
Statistics: MLE, MAP
N-Gram: smoothing, perplexity
Sasha Rush: LLM In 5 Formulas lab2 survey
Neural Representation
Thu, Sep 4 Lec 4. Linear Models
Task: regression, classification
Data: (X, Y-continuous/discrete)
Model: linear layer, sigmoid/softmax
Loss: MSE/cross entropy
Optimization: analytical/SGD
3Blue1Brown: Essence of Linear Algebra, Essence of Calculus hw2 (sol)
Tue, Sep 9 Lec 5. Multi-Layer Perceptrons
Model: Activation, Dropout layer
Optimization: Backpropagation
lab3
Thu, Sep 11 Lec 6. Word Embeddings
Counting: Word-context matrix
Learning: NLM, Word2Vec (Skip-gram, CBOW)
Vector Properties: Analogies
Papers: NLM, Word2Vec hw3 (sol)
Tue, Sep 16 Lec 7. Recurrent Neural Networks
RNN: Hidden state & sequential modeling
LSTM: Gates & cell state for long memory
NLP applications: encoder, decoder
Papers: RNN, LSTM lab4
Thu, Sep 18 Lec 8. Attention Module
Machine Translation: Seq2Seq
Attention Mechanism: Query, Key, Value
Papers: Seq2Seq, Attention hw4 (sol) team info due
Tue, Sep 23 Lec 9. Transformer Model
Encoder: MHA, FFN,LayerNorm, Residual Connection
Decoder: Masked Attn, Cross Attn
Embedding: BPE, Sinuositional Encoding
Attention is All You Need!
Paper: Transformer
Code: Annotated Transformer
Thu, Sep 25 Lec 10. GPT 1-3
GPT-1: Finetuning
GPT-2: Zero-shot
GPT-3: Few-shot (In-context Learning)
 Scaling is all you need!
Papers: GPT-1, GPT-2, GPT-3 		lab5 problem statement due (Fri)
Tue, Sep 30 Lec 11. InstructGPT
Supervised Fine-Tuning
Reinforcement Learning from Human Feedback
Alignment is all you need!
Paper: InstructGPT
Thu, Oct 2 Lec 12. Reinforcement Learning Basics
RL Framework: state, action, reward
Value-based RL: Bellman Equation (Q-learning)
Policy-based RL: ε-greedy
hw5
Tue, Oct 7 Lec 13. ChatGPT
RLHF: PPO, DPO
DeepSeekR1: GRPO
Thu, Oct 9 Mid-term Exam topics, practice (sol)
Tue, Oct 14 No Class (Happy Fall Break)
Image Generation
(Week 8-12)
Thu, Oct 16 Lec 14. Image Basics and Filtering
Digital Representation: 2D arrays, grayscale, RGB channels
Convolution Operation: Mathematical definition & properties
Classic Filters: Gaussian blur, Laplacian edge detection
Probability and Statistics
Tue, Oct 21 Lec 15. Statistical Image Modeling
MRFs & CRFs: Spatial dependencies & energy functions
Patch-Based Synthesis: Efros & Leung, PatchMatch
Early Deep Models: RBMs, Deep Belief Networks
Deep Learning
Thu, Oct 23 Lec 16. CNN Models for Image Generation
CNN Architecture: Convolutional, pooling, activation layers
Encoder-Decoder: U-Net, skip connections, transposed convs
Advanced CNNs: ResNet, DenseNet, attention mechanisms
Game Theory
Thu, Oct 30 Lec 17. Generative Adversarial Networks (GANs)
GAN Framework: Generator vs. discriminator game
Architecture Evolution: DCGAN, Progressive Growing, StyleGAN
Challenges: Mode collapse, training instability, solutions
Information Theory
Tue, Oct 28 Lec 16. Variational Autoencoders (VAEs)
VAE Architecture: Encoder q(z|x), decoder p(x|z), prior p(z)
ELBO Objective: Reconstruction + KL divergence
Training: Reparameterization trick, KL vanishing
Dynamical Systems
Tue, Nov 4 Lec 18. Neural ODEs
Autoregressive Principle: Pixel-by-pixel generation
PixelRNN/PixelCNN: Sequential modeling with LSTM/conv
Image Transformers: Self-attention over image patches
Thu, Nov 6 Lec 19. Normalizing Flows
Change of Variables: Exact likelihood computation
Flow Architectures: RealNVP, Glow, continuous flows
Properties: Invertibility, differentiability, composition
Stochastic Process
Tue, Nov 11 Lec 20. Stochastic Differential Equations
Forward Process: Gradual noise addition to images
Reverse Process: Learning to denoise step by step
DDPM: Denoising diffusion probabilistic models
Thu, Nov 13 Lec 21. Diffusion Models
Text-to-Image: DALL·E 2, Midjourney, Imagen
Advanced Guidance: Classifier & classifier-free guidance
Latent Diffusion: Stable Diffusion, faster training
Guest Lectures
(Week 13-14)
Tue, Nov 18 Lec 22. Image Style Transfer
Siyu Huang (Clemson University)
Thu, Nov 20 Lec 23. Molecule Generation for Drug Discovery
Wengong Jin (Northeastern University)
Tue, Nov 25 Lec 24. Biomedical 3D Generation
Jiancheng Yang (Aalto University)
Thu, Nov 27 No Class (Happy Thanksgiving)
Finals
(Week 15-16) 		Tue, Dec 2 Project Presentations I
Thu, Dec 4 Project Presentations II
Thu, Dec. 11 report/code due


Staff & Office Hours


Instructor
Omer Yurekli
TA
Zimeng Yang
TA
Name Office hours
Donglai Wei (Tu/W) 3-4 pm @ Rm 528F, 245 Beacon ST
Omer Yurekli (M) 10-11 am, (F) 12-1 pm @ Rm 122, 245 Beacon ST
Zimeng Yang (W) 2-4 pm @ Rm 122, 245 Beacon ST
  • Email/Slack Donglai for any other things to help you succeed in the course.


Course information

1. Get help (besides office hours)

  • Slack: For labs/psets/final projects, we will create dedicated channels for you to ask public questions. If you cannot make your post public (e.g., due to revealing problem set solutions), please create a room with instructors and TAs separately, or come to office hours. Please note, however, that the course staff cannot provide help debugging code, and there is no guarantee that they'll be able to answer last-minute assignment questions before the deadline. We also appreciate it when you respond to questions from other students! If you have an important question that you would prefer to discuss over email, you may email the course staff, or the instructor directly.
  • Support: The university counseling services center provides a variety of programs and activities.
  • Accommodations for students with disabilities: If you are a student with a documented disability seeking reasonable accommodations in this course, please contact Kathy Duggan, (617) 552-8093, dugganka@bc.edu, at the Connors Family Learning Center regarding learning disabilities and ADHD, or Rory Stein, (617) 552-3470, steinr@bc.edu, in the Disability Services Office regarding all other types of disabilities, including temporary disabilities. Advance notice and appropriate documentation are required for accommodations.

2. Assignments/Grading

  • Submission: Submit the required files to Canvas.
  • 10% - Attendance: Each week has a coding exercise in Colab to help you gain the hands-on understanding about the material.
  • 15% - Labs (weekly, Colab): Hands‑on GenAI implementations.
  • 15% - HWs (weekly, hand-written): Mathematical exercises (derivations/proofs) to solidify theory.
  • 25% - Midterm (closed‑book, handwritten): Focused on mathematical concepts from HWs; one double-page notes sheet allowed.
  • 35% - Final project (startup mockup with GenAI): Proposal (5%), Milestone demo (10%), Final demo (10%), Written report (5%), Individual reflection/peer eval (5%). (More info)

3. Academic policy

  • Late policy: You'll have 10 late days each (counting weekends) for labs and HWs respectively over the course of the semester. Each time you use one, you may submit an assignment one day late without penalty. You are allowed to use multiple late days on a single assignment. For example, you can use all of your days at once to turn in one assignment a week late. You do not need to notify us when you use a late day; we'll deduct it automatically. If you run out of late days and still submit late, your assignment will be penalized at a rate of 10% per day. We will not provide additional late time, except under exceptional circumstances, and for these we'll require documentation (e.g., a doctor's note). Please note that the late days are provided to help you deal with minor setbacks, such as routine illness or injury, paper deadlines, interviews, and computer problems; these do not generally qualify for an additional extension.
  • Academic integrity: While you are encouraged to discuss homework assignments with GenAI or other students, your programming work must be completed individually. Thus it is acceptable to learn from GenAI or another student the general idea for writing program code to perform a particular task, or the technique for solving a mathematical problem, but unacceptable for two students to prepare their assignments together and submit what are essentially two copies of identical work. If you have any uncertainty about the application of this policy, please check with me. Failure to comply with these guidelines will be considered a violation of the University policies on academic integrity. Please make sure that you are familiar with these policies.

4. Additional resource (Free student accounts!)

Acknowledgements: This course draws heavily from ...

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