My background is in CS and I’m currently working as an ML engineer, but I’m very eager to go back for a PhD soon.
In the mean time, I’ve noticed my maths isn’t as strong as I’d like it to be. That aside, I’ve also taken a big interest into calculus.
Are there any online courses (free or non-free) that would allow me to learn mathematics either through self study, or as a MOOC? I’ll gladly take undegrad level courses. I just want something structured.
I’ve found that even trying to really study through books things like number theory, proofs, and integration takes huuuuge time investments. I feel like this will be less so with a structured course and teacher.
This channel (by the sounds of it) isn't exactly what your looking for, but I feel like it's appropriate for the topic and might act as a supplement to whatever course(s) you end up taking, depending on where you are having difficulties.
He has an amazing ability to break math down into intuitive visuals which highlight the underlying nature of math, rather than being hyper focussed on a specific problem. For myself, the videos have allowed me to pick up other/adjacent math concepts significantly easier because I've developed a signficantly better understanding of what's happening "under the hood", so to speak.
Might be worth a quick look at least!
Grant Sanderson has made a lot of great contributions to Khan Academy as well, particularly in the Linear Algebra and Multivariable Calculus sections.
> I’ve found that even trying to really study through books things like number theory, proofs, and integration takes huuuuge time investments.
Yep, that's how to do it if you want to do it well. Practice. There's a reason why these degrees take so long. I would advise against just about any online course where you only watch videos and get the feeling that you've learned something while in reality you retain close to nothing.
The University of Illinois at Urbana-Champaign offers a wide variety of mathematics courses online for self-paced study. The courses range from 100 - 400 level. 100 - 300 level courses confer undergraduate course credit at UIUC, whereas 400 level courses are eligible for graduate credit. (Note that you are not enrolled as a degree-seeking student in a mathematics program, you're simply taking courses online).
You have 16 weeks from the date of registration to complete each course. The courses themselves have weekly homework, two midterms and a final. You must find an eligible proctor to complete the midterms and final in person, but otherwise you need not go anywhere. Each course costs about $1500 - 2000. The lectures for each unit are video recordings of lectures in the corresponding course onsite at UIUC. There is also a certificate you can earn, but it's primarily focused on completion of lower-level courses. Your homeworks are graded (with feedback) by a lecturer or math TA at the university. There are also remote office hours available.
UIUC is probably the highest caliber mathematics program which offers something like this. It's generally difficult to find a top ~20 math university willing to support online, self-paced study for credit. The ones which do offer such a system are usually very expensive or predominantly focus on lower-level courses like a simple calculus sequence.
Look at UIUC and Open University, MIT's OCW, community/junior colleges can give you a big boost, there's probably a lot of other advice at /r/math, /r/LearnMath and math.stackexchange.
Some older threads:
I assume you're looking to do an ML PhD, so you probably don't need to learn up on number theory. Mathematics is a huge topic, so you need to condense what you're going to study down to a couple of courses. Also, ML doesn't use that much actual maths, so you don't need to know proofs, real analysis, algebra, differential equations, etc. Though doing a a decent proofs course is always helpful everywhere.
I would start with linear algebra. This post  links to a great course. Then you might want to do a course on multivariate calculus and a course on optimization but it's not really required. Second, I would do this  course on machine learning.
After these two courses, which would probably take you 3 to 4 months if you have a job, then you'll have a better idea of what math courses to take.
Final note, when I say do a course, I mean watch a lecture, take notes like you would do in a real class, go to the course website, go through the lecture notes, do the homework set associated with the lecture, and then go to the next lecture. Just watching the lecture, especially for a math lecture, especially of you're not familiar with the subject, is not as helpful.
University of London does post grad maths diplomas . They are solid and somewhat structured for self study. They are also cheap .
Aside from that find yourself a tutor over Skype and crack on.
Just be sure you know what you’re getting yourself into. Unless you are one of the tiny number of people that have a genuine interest in maths, be prepared to slog through things you find senseless or boring in order to get to the prize. Further, and I think this is very true: there are no shortcuts. If you can implement something like mastery learning you will be all the better for it and if you don’t you may just regret it.
Finally, think really carefully about how you’re going to retain the maths you learn because if you don’t you can be sure that a month after your exam you won’t remember a scooby.
You should probably look up the undergrad math courses of the grad school you want to go to, they will have lectures notes and textbook/chapter reading recommendations so structured and you aren't going through enormous reference books. Search the textbook name on youtube, often there will be some lectures for it sometimes even by the author.
There's an ML math prep book https://mml-book.com/ which is basically a crash course, and a series of lectures here for a background in math for an intro machine learning course at CMU https://www.youtube.com/user/professorgeoff though note they aren't as long as they seem, as these were live lectures so they start late, have breaks between exercises, etc. If you've read the Elements of Statistical Learning 2e you likely know much of this already.
Personally my recommendation is go through a Wilberger course, this set of undergrad lectures is for the Stillwell book 'Mathematics and it's History' https://www.youtube.com/playlist?list=PL55C7C83781CF4316 it will intuitively cover differential geometry, topology, group theory, polynomials etc, to the depth of Stillwell's book and if you see something that interests you or that you forgot, then you can pursue it taking formal courses. I'd recommend his Linear Algebra course too on the same youtube channel he uses clear definitions for everything so when you get to abstract 3D vector spaces it makes sense.
Anybody with a complete shit background in math like I used to have try the Wildberger foundations playlists on the same channel watching how he writes proofs, then pick up some large book written by Knuth and start attempting the exercises as a weekly hobby, which will now be possible to do. This is also how you retain these skills by using them on a regular basis, at least for me anyway.
> I feel like this will be less so with a structured course and teacher.
There is no royal road to geometry. Try this: https://aimath.org/textbooks/approved-textbooks/
Not sure what you mean by online. Do you mean distance courses?
I am based in the UK, and we have the Open University. I am currently taking a Mathematics and Physics degree with them.
The content is not directly equivalent to an undergraduate degree (probably around 40% of a normal undergraduate here in the UK). But they have lectures, assignments and examinations and you get a proper degree certificate at the end.
I have friend who is Spanish and he has been taking a distance Mathematics course with a Spanish University. That again seemed to be an excellent course (he is heavily into Category Theory).
I think your quote of 40% isn't right. From elsewhere on the net:
"Former students of the OU appying for a Masters at Cambridge have a 23% chance of getting of an offer. That's a bit below average but shows it is more than possible."
The level 1 courses are less complex than first-year regular uni because they'll take anyone and they have to get them up to par. But it quickly scales up.
In my experience Level 1 and 2 courses are not a match for normal Undergrad courses, but certainly the Level 3 courses are.
At level 2 maths I covered all the core subjects (i.e. differential equations, multi-variable calculus, fourier analysis, etc.) But each of these was just a chapter in the course (approx. 2 weeks of study). Whereas in normal undergrad they would be perhaps 8-10 weeks worth of lectures each.
But level 3 were certainly really great. The Quantum Mechanics covered everything up to perturbation theory, and the Cosmology module was very comprehensive and the Pure Maths courses covered Group Theory up to Sylow Theorems and Metric Spaces.
So, certainly if someone is considering a Maths course I would recommend the OU. But the cost is way too high for what you get.
> I’ve found that even trying to really study through books things like number theory, proofs, and integration takes huuuuge time investments
Well yeah, it's like three years of full-time study to get to an undergraduate level. Doing that part time is going to take you a decade or so. You need to put the time in. I'm not sure there's a shortcut.
There's no 'Royal road', as they say.
The Open University in the UK is still a fairly solid bet, despite not quite being what it used to be.
The range of choice in modules was radically cut and the cost of courses was increased considerably.
You used to be able to get a very interesting degree in maths with chosen specialisms in various pure and applied areas for about £3600. Might have been a bit more for people from outside the UK, but not massively.
There's little choice of courses now.
I did that; finished around 2015, I think. The fees went up while I was there, but I have a half-memory that it went up in steps, or maybe if you were already on it the hike wasn't so extreme.
In retrospect, paying somewhere around 3500 GBP for a masters (even if it was basically "here's the textbook, see you in nine months for the exam!") that I could do in my evenings and weekends was a bargain. I should have done more of them.
There are quite a few posts about this on HN, including this one 11 days ago. https://news.ycombinator.com/item?id=19811715
The reviews on Amazon were not particularly kind - regardless, good reading if you're interested in purchasing this book: https://www.amazon.com/gp/product/1727125452/ref=as_li_qf_as...