My Ph.D. is in pure mathematics, and I admit I don't know much (i.e. anything) about theoretical CS. However, I have started exploring non-academic options for my career and in introducing myself to machine learning, stumbled across statements such as "No one understands why neural networks work well," which I found interesting.
My question, essentially, is what kinds of answers do researchers want? Here's what I've found in my brief search on the topic:
- The algorithms implementing simple neural networks are pretty straightforward.
- The process of SGD is well-understood mathematically, as is the statistical theory.
- The universal approximation theorem is powerful and proven.
- There's a nice recent paper https://arxiv.org/abs/1608.08225 which essentially gives the answer that universal approximation is much more than we actually need in practice because we can make strong simplifying assumptions about the functions we are trying to model with the neural network.
In the aforementioned paper, they state (paraphrasing) "GOFAI algorithms are fully understood analytically, but many ANN algorithms are only heuristically understood." Convergence theorems for the implemented algorithms are an example of analytic understanding that it seems we DO have about neural networks, so a statement at this level of generality doesn't tell me much about what's known vs. unknown or what would be considered "an answer."
The authors do suggest in the conclusion that questions such as effective bounds on the size of the neural network needed to approximate a given polynomial are open and interesting. What are other examples of mathematically specific analytical questions that would need to be answered to say that we "understand" neural networks? Are there questions that may be answered in more pure mathematical language?
(I am specifically thinking of methods in representation theory due to the use of physics in this paper --- and, selfishly, because it is my field of study. However, I can also imagine areas such as combinatorics/graph theory, algebraic geometry, and topology providing viable tools.)