Exploring the MIT Mathematics and EECS Curriculum Using Large Language Models

We curate a comprehensive dataset of 4,550 questions and solutions from problem sets, midterm exams, and final exams across all MIT Mathematics and Electrical Engineering and Computer Science (EECS) courses required for obtaining a degree. We evaluate the ability of large language models to fulfill the graduation requirements for any MIT major in Mathematics and EECS. Our results demonstrate that GPT-3.5 successfully solves a third of the entire MIT curriculum, while GPT-4, with prompt engineering, achieves a perfect solve rate on a test set excluding questions based on images. We fine-tune an open-source large language model on this dataset. We employ GPT-4 to automatically grade model responses, providing a detailed performance breakdown by course, question, and answer type. By embedding questions in a low-dimensional space, we explore the relationships between questions, topics, and classes and discover which questions and classes are required for solving other questions and classes through few-shot learning. Our analysis offers valuable insights into course prerequisites and curriculum design, highlighting language models' potential for learning and improving Mathematics and EECS education.

Dissecting Hessian: Understanding Common Structure of Hessian in Neural Networks

Hessian captures important properties of the deep neural network loss landscape. We observe that eigenvectors and eigenspaces of the layer-wise Hessian for neural network objective have several interesting structures -- top eigenspaces for different models have high overlap, and top eigenvectors form low rank matrices when they are reshaped into the same shape as the corresponding weight matrix. These structures, as well as the low rank structure of the Hessian observed in previous studies, can be explained by approximating the Hessian using Kronecker factorization. Our new understanding can also explain why some of these structures become weaker when the network is trained with batch normalization. Finally, we show that the Kronecker factorization can be combined with PAC-Bayes techniques to get better explicit generalization bounds.