I must admit that at times, I’ve been frustrated by a lack of mathematical competency during my graduate studies in Reinforcement Learning. Other times though, I feel like my current level of mathematical competency is perfectly fine. I generally have enough intuition to understand the papers I’m reading. Part of me feels that when I allow myself to be frustrated, I’m allowing a certain type of author win – the type of researcher dressing up fairly underwhelming results in overly complex mathematics to make it look more impressive than it really is. Often times, complex math is needed … but other times it feels like I’m being shown dancing girls by a biz dev expert.
That said, the frustration is real.
I finished my undergraduate degree in 2002 (Currently 15 years ago) with a major in Computer Science. I took the required mathematics courses (2 calculus courses, a linear algebra course, statistics, and a numerical analysis course). I thrived in these courses (I’d have to look but I’d guess my average was around 95% where the course average was 60%) and generally enjoyed them. I’m not exactly sure why I didn’t pursue more. That said, the knowledge I gained from these courses is both distant and lacking to what would be required for easily understanding the papers I read. For example, I was reading a paper the other day talking about Fourier Transforms. A simple enough concept that I’ve studied before, but distant enough that I couldn’t quite remember the details I needed to parse the paper. The week before “LSI Systems.” In both cases I googled each term which unearthed 10 more terms and mathematical concepts I was unfamiliar with. Each term may require a weeks dedication, if not more to truly understand. It felt like pealing an onion.
I’ve thought about possible solutions. In my first year (last year) I audited a few classes – linear algebra and a statistics course. I should probably do more of that during my grad studies. But as always, 3 hours of lectures a week + time getting to and from adds up and is a pretty expensive investment. Not to mention the fact that to truly get the most out of these courses, one must not just attend classes, but do the assignments, prepare for exams etc. So this bottom up approach (read analysis, probability text books, MOOCs) is problematic because of it’s time cost when you have conferences to attend, pressure to publish, other classes to take for credit etc. Furthermore, the signal to noise ratio in this approach is pretty low. This statement isn’t mean to minimize the value in understanding the entire content of a mathematics textbook. But for the utility of applying it to research directly, a lot of pages simply aren’t relevant. However, the only way to unearth the valuable bits might be to read the entire content. In a perfect world, there’d be a system in place that would give me just enough content to understand the concept I wished. This system would understand my current level of knowledge. I could query “Langevin flow” and be returned a list of pointers to articles / moocs, etc. given my current level of knowledge. Google cant do this.
I’ve thought about a more top down approach – googling every term that’s unfamiliar – but as I stated before – that seems like exponential tree expansion.
Another more extreme solution would be spending a semester or two doing nothing but studying the appropriate math courses. This would be taking the actual courses (maybe for credit or at least doing assignments etc)? But obviously, as stated before, grad students have deadlines and should really publish / advance research … not just take background courses. This would have to be framed as a pseudo – sabbatical.
All this said, maybe it’s much ado about nothing. I have a general intuitive understanding of most concepts I’ve encountered. And for those that required a more intimate understanding, I’ve been able to pick this up. Perhaps I could even argue that a beginner understanding of mathematical concepts could come as an advantage to some researchers more mathematically minded – treating their mastery of equations and proofs as a hammer seeking a nail – solving problems that don’t really matter. My lack of mathematical know how hasn’t narrowed my research interests into purely application based areas (although I suppose that wouldn’t be a problem if it did. There’s nothing wrong with application research … as opposed to more theoretical).
None the less, the frustration is real. I desire to understand a bit more. And I don’t think I’m merely being tricked by dancing girls and flashy math equations. A better mathematical foundation would help. I’ll continue to use a more top down approach (researching terms and ideas on demand ad hoc) while sprinkling in a bit more bottom up. Listening to math lectures when I have the chance. Doing the latter is easier said than done though as “down time” is something that doesn’t occur that often. Perhaps I’ll dedicate myself to a MOOC to force the issue (the question of which content / MOOC not withstanding).
I’d love to hear other solutions to this frustration though ….