As a sidebar, it has always seemed to me that there is a giant gulf between truly beginner-friendly math books, which are aimed at children, and introductory math books aimed at adults. The latter almost always read like foreign language textbooks where you must first know the language before you can start, while the former are too elementary. I'd love to find "College level Math for English Majors" or something of the like, if anyone knows of such a book :)
The route for someone with no (or very little) math background is a lot longer, and I don't think this book is trying to provide it. I think that one either has to 1) bite the bullet and learn a year or two of undergrad math first, which provides the necessary foundation for this stuff, then learn this for real; or 2) be content with understanding and using machine learning at a hand-wavy level (which I am not denigrating). It might be nice to have a "hand-wavy machine learning" book around (there are certainly enough blog posts of that sort), but this isn't trying to be it.
The course requires a 1st year linear algebra course and a 2nd year statistics course, which might explain why this book doesn't cover some of the basic concepts
The NO BULLSHIT guide to MATH & PHYSICS is a 3-in-1 combo book that includes a high school math review, mechanics, and calculus. Preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...
The NO BULLSHIT guide to LINEAR ALGEBRA covers all the standard LA material, and also discusses lots of interesting applications (graphs, graphics, crypto, quantum computing).
I put a lot of effort into making them with an "easy ramp up" so anyone with basic math background can pick up. If you don't remember you high school math in details, you can still handle. You can buy print versions on amazons or ebook on gumroad. See website for links: https://minireference.com/
Here’s some off the top of my head. I really think these will help you build a good foundation for mathematical thinking.
Concrete Mathematics by Graham, Knuth, Patashnik
How to Prove It by Velleman
Polya’s How to Solve It
E.T. Jaynes’ Probability Theory
Conceptual Mathematics by Lawvere and Schanuel
Intro to Linear Algebra:
Strang does a GREAT job explaining the intuition behind linear algebra. The book is targeted at first-year undergrads and an adult with just high-school math could work through it.
Bonus: if you're interested in machine learning, you must learn linear algebra.
More advanced is this Strang masterpiece: Intro to Applied Mathematics.
It covers a lot of the applied math that we focused on pre-AI and pre-cloud computing. Diff eqs, diffusion equations, Fourier analysis, numerical methods, phase plane analysis, optimization, complex anlaysis.
All good books, all pitched at a level that is very ill suited for what is being asked for.
To the GP, "math for English majors" is a common enough course schema that I think looking at a few syllabi might score you something.
Otherwise I basically agree with your comment. I just take issue with calling Concrete Mathematics a graduate textbook, because I hear people say that as though it’s not an appropriate recommendation for learning. That gives me the impression they’ve not actually opened up a graduate textbook in math or computer science. Concrete Mathematics might not be year one material, but you can do it after a calculus course and maybe an algorithms course. Contrast this with an actual graduate course, like convex analysis and optimization. Textbooks at that level would definitely not be accessible for most undergrads.
I would certainly recommend giving it a shot for anyone interested, as it’s a lovely book full of fun problems. As you say, it’s accessible to well prepared undergraduates.
Took me 3 yrs of evenings and weeekendw to get through this :\.
If you look in Goodfellow et al's Deep Learning book, Murphy's Machine Learning text and others mentioned here (Learning from Data, Shalev-Shwartz/Ben-David) the prereq's are always some variation of above and I think you could do a lot of the above at U.S. community colleges, at least the CC's around me.
Frankly, you'd have to do a bunch of self study beyond CC and there's no shortcut/royal road. So the key is self study, that's a discipline anyone that wants to do Data Science/machine learning for real needs
Machine learning lends itself to easily learning additional pieces of math once you have a nice foundation, and it is nice enough that the foundation is pretty small---vector calc, (mostly) real analysis, linear algebra (it helps if you know infinite but orthogonal eigenfunctions), little bit of physics knowledge (statistical models and hamiltonians), and a little bit of differential geometry.
Unfortunately, that foundation is slowly getting bigger. Read some papers in topological data analysis recently. Category theory is crawling in!
Murphy’s Machine Learning: A Probabilistic Perspective is just over 1000 pages long. It’s an exceptionally good book for the mathematical theory behind machine learning. What would this book look like under your proposal? If we add in Axler’s Linear Algebra Done Right and Chung’s A First Course in Probability Theory, we’re adding another 350 pages or so each. Both of those books have prerequisites, and Murphy’s book isn’t the top either.
In my opinion your proposal is unrealistic, yes. I don’t mean for that to sound harsh, I just don’t see a way to achieve what you’re asking for without making a monolithic tome larger than Knuth’s The Art of Computer Programming, and potentially even more diverse in scope.
An online discussion forum isn’t typically going to cut it, IMO. People won’t be at the same level, won’t be working on the same topics, won’t have the same goals and interests, the same level of commitment, coordinated schedules, etc. This has been tried many times, but I have not heard many successful reports.
Online discussion forums are great for handling independent limited-scope questions–answers (including for current university students), and okay for coordinating research efforts among people who are already experts, but in general are not very effective for getting a group of people to diligently work through years of prerequisite material in an organized fashion.
You shouldn’t spend your life in a university, but if you have a goal of learning advanced mathematics, you might benefit from spending a few years there. YMMV.
As an alternative, if you can find an expert tutor/coach to meet with one-on-one on a regular basis, that is even better than a formal course. But that is expensive and hard to scale.
> it’s not easy or typically very efficient
It is also very possible for folks to think they have it right, and go off in entirely wrong directions. In my experience it is likely, even in quite good and hard-working students.
I teach an inquiry-based proofs class and so I sit through the students's discussion. Even on material that would be considered trivial in many of the books mentioned here, there are people saying aloud things in that discussion that are just simply wrong. It is not that these folks are dummies, it is that this stuff is hard.
I'm not a people person but one thing I've seen from this class is that a lot of learning is about community. It makes a big difference.
The problem is, that simply isn't possible / practical for many people. Assuming we're talking about people who are career professionals who are already past their first (and maybe only) stint at a university, most people just don't have the free time (and other resources) to sit in classes at a university. Especially since most universities still cater almost exclusively to "traditional" students who take classes in person, during the day.
If universities made more night classes available, or more online/hybrid classes, I could buy this. But for a lot of us, the best we can do is some combination of "watch Youtube videos, do MOOC's, self-study, do ungraded problems, and ask for help on Internet forums".
That said, not all problems you do on your own have to be exactly "ungraded". For many mathematical subjects you can find books of exercises that include the answers so you can "grade" yourself (assuming you can resist the temptation to cheat and look at the answers first), or you can find previous years problem sets and exams on many course websites.
Yeah, learning the stuff outside of a university setting is probably harder in some sense, but it's not impossible. And if that avenue is the only one available to somebody, then it is - by definition - the best avenue available to them.
Personally I am not an academic, I never went to grad school, and I do lots of self-studying of various technical topics, up to hopefully some reasonable level of understanding.
If you want to self-study, all the material you could ever hope to need is available in a mulitude of books targeted to various levels of prior experience and with many different styles of exposition. If you want to study with others, this is precisely what college is for. People who don’t want to study in a college setting basically don’t want to study in a group setting in general, because the college setting actually is the most efficient group setting.
I think most people underestimate the sheer vastness of the material you have to learn and the requisite time you need to learn it. Reading isn’t enough, you have to do it - over and over. It’s frankly a slog, and there’s no more efficient way than being immersed in coursework.
The entire benefit of a course structure is the instructor. The community forum helps, but really it’s an extension of the instructor. A qualified instuctor can quickly resolve a complex question that might take you hours or days to (maybe incorrectly) answer yourself. If you’re capable of learning genuinely unfamiliar math without an instructor, that means you have sufficient mathematical maturity that you don’t need a group forum. At that point it’s actually more efficient to self-study.
I get the sense people believe mathematicians are pretentious when they say things like this, but I think that’s uncharitable. They typically give advice like this because they have a significant amount of personal experience and know very well what tends to work (and how efficiently it works).
Either work for it for years or don’t. Or wait for brain implants or something.
Just labeling things I had never seen before, like indicator functions, was extremely valuable.
Especially for this kind of book that is introducing mathematics to people from a broad background - I think it's important to understand how much of an impediment not knowing notation is by sight. Trying to Google or search for notation is a nightmare.
Get both unless you're only getting one, in which case, get Shai Shalev-Schwartz's.
I simply clarified that the question was about computational learning theory, a subfield largely started by Leslie Valiant in the form of PAC (Probably Approximately Correct) learning. The difference in emphasis between the machine learning conferences I mentioned helps point out how practical machine learning (like ICML, matching PRML/ML/ESL) and feature extraction/representation learning (like ICLR, perhaps matching portions of both ICML and ICLR), while important, are not what the previous poster was asking about.
This covers calculus, linear algebra, probability, statistics, convex optimization and a math for ML course thrown in for the HN audience:
(The first two are "MOOCs" recorded in the 1970s! probably the first ever recorded MOOC, even before the internet, and the lecturer is absolute gold)
Calculus Revisited: Single Variable Calculus | MIT https://ocw.mit.edu/resources/res-18-006-calculus-revisited-....
Calculus Revisited: Multivariable Calculus | MIT https://ocw.mit.edu/resources/res-18-007-calculus-revisited-....
Complex Variables, Differential Equations, and Linear Algebra | MIT https://ocw.mit.edu/resources/res-18-008-calculus-revisited-....
Linear Algebra | MIT - https://www.youtube.com/watch?v=ZK3O402wf1c&list=PLE7DDD9101....
Introduction to Linear Dynamical Systems |Stanford https://see.stanford.edu/Course/EE263
Probability | Harvard https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6....
Intermediate Statistics | CMU https://www.youtube.com/playlist?list=PLcW8xNfZoh7eI7KSWneVW....
Convex Optimization I | Stanford https://see.stanford.edu/Course/EE364A
Math Background for ML | CMU https://www.youtube.com/playlist?list=PL7y-1rk2cCsA339crwXMW....
I you aren't willing to invest the energy and effort to do that, all the video watching won't do anything for you. It will get you 5% of the way at best. The true learning comes from staring at a problem for hours, trying 100 dead ends, and then finally having an insight two days later while taking a shower that suddenly make that intractable problem seem trivial.
These things aren't mutually exclusive. I don't know about anybody else, but I'd rather watch a video of a human explaining the subject, then sit down with a textbook and start working through problems.
> you can't sit thru lots of pages of mathematics text
Even if you watch those videos, you still need to sit through lots of pages of mathematics if you want to master the subject.
Calculus Revisited: Single Variable Calculus | MIT https://ocw.mit.edu/resources/res-18-006-calculus-revisited-...
Calculus Revisited: Multivariable Calculus | MIT https://ocw.mit.edu/resources/res-18-007-calculus-revisited-...
Complex Variables, Differential Equations, and Linear Algebra | MIT https://ocw.mit.edu/resources/res-18-008-calculus-revisited-...
Linear Algebra | MIT - https://www.youtube.com/watch?v=ZK3O402wf1c&list=PLE7DDD9101...
Probability | Harvard https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6...
Intermediate Statistics | CMU https://www.youtube.com/playlist?list=PLcW8xNfZoh7eI7KSWneVW...
Math Background for ML | CMU https://www.youtube.com/playlist?list=PL7y-1rk2cCsA339crwXMW...