Theoretical limitations of multi-layer Transformer

Most of it is linear algebra and convex optimization. You can learn a lot of it with free resources from MIT, Stanford, Georgia Tech, or YouTube. If you want more of a school style learning environment you can enroll in the Georgia Tech OMSCS program and just take the classes related to the math etc that you are interested in. No reason you have to graduate and it is maybe $800 a course.

I'm not a math person either, but I'm familiar with some of the terminology. The two things I got out of this paper were:

1. Depth is more important than breadth for making transformers smarter. That is, for a given size of model it will be more powerful with more, smaller layers than it would be with less, bigger ones. Interestingly, Mistral just updated their small model yesterday with a big reduction in layers in order to improve performance. Among the many ways they say it more technically they do say directly "depth plays a more critical role than width in reasoning and composition tasks".

2. As I understand it, they are claiming to be able to prove mathematically that Chain of Thought such as seen in the new DeepSeek R1 and GPT o1/o3 models creates results that wouldn't be possible without it. The thought chains effectively act as additional layers, and per the previous point, the more layers the better. "From a theoretical view, CoT provides Transformer with extra computation space, and previous work ... proved that log-precision Transformer with CoT could simulate any polynomial-time algorithm. Therefore, by further assuming certain complexity conjecture ... their results imply that constant depth Transformer with CoT could simulate poly-time algorithm, while constant depth Transform ... itself can not solve P-complete task."

Would be fun to ask o3