I have a lot of books on measure theoretic probability theory, functional analysis, graduate level topology, convex optimization, stochastic calculus, numerical analysis using a functional analysis framework, statistical theory, Bayesian Analysis, decision theory, elements of statistical learning, abstract algebra, advanced linear algebra, PDEs and information theory
I want to learn as much of these topics as possible over the next 12 years while machine learning advances, but I wonder how many of these advanced topics are actually used in Machine Learning today?
Particuarly, I want to study deep and reinforcement learning that does NOT need training data sets except for mathematical structures. I.e., Imagine an algorithm that knows which approach to take and learns to classify objects by topology, then further classifies them by geometry, and ever further by color, periodicity, symmetry etc.
For example, even if you trained the algorithm on recognizing squares, it would be able to distinguish a circle as being similar but fundamentally different. Maybe it could also learn to put pictures of limbs together and generalize them into the shape of a person while figuring out how they interact.
Best Answer
I'd recommend getting an overview of the math that's currently used in deep learning architectures that are used for supervised settings (this does mean looking into approaches that involve "training sets"), before you dive deeper into other math.
http://www.deeplearningbook.org/ has a very good overview of the math you'd need to understand what's going on in neural nets/deep nets. Once you're comfortable with the current approaches, you'd be able to understand the research in the field, and the directions it's heading in. (from ICML, NIPS papers, for instance)
At that point, you will likely find open problems that seem to interest you, and you can begin to actively work on them. It's often useful to have a problem you want to solve in mind, and then explore all the work that's been done on that problem (prior approaches, the math involved, etc) - Sometimes, you'll find that there are some problems that interest you deeply, but the current approaches to solve them are unsatisfying - this is really the point when you might have to invent(discover?) the math needed to solve it, or "borrow" the math from a different field. The main benefit you'll have if you work on problems that are similar to what other researchers are interested in is that there's a community that's publishing work at a breakneck pace, and you'll be able to quickly get feedback on approaches that have been tried and haven't quite worked just yet.
I'm not quite discounting the value of learning math by itself, but just saying that if you learn the math in the light of a problem (or ten), you'll learn how to apply the existing math well (+ how to do a good literature search), and you'll also learn to figure out when new math is required.