in my department [we are an outlier probably] post doc wages are actually pretty comparable to industry i.e. >= 100k. 40k to 70k seems low to me.
I would expect the range to be more like 60-100k.
I haven't been on the real job market yet-but the key to getting a TT or postdoc position seems to be a) collaborators who want to hire you b) people who have heard you give a talk in person and are familiar with your research.
The other thing to note in ML is that it seems like a few people go to industry research labs for a few years i.e MSR/FAiR/google brain and then come back to the academy since there are industry roles that involve research and publication. for instance moritz hardt.
my personal plan for the first 3 years of grad school is to work really hard and try to keep both academia and industry open and after year 3 evaluate the number of publications I have and my current skill set to see if I can make it in the academy or shift more towards industry.
I think the biggest factor I would comment on is look very closely about what jobs the graduated students from the department you matriculate at AND more importantly the professor you want to work with go on to do post Phd. There are a lot of naysayers in this thread about the risks of an academic career and I share those concerns but I felt a lot more comfortable taking the plunge after I looked at the career record of the graduated students of my advisor. They were all either tenure track or had good industry positions.
edit: if your advisor has collaborators in industry groups I think it is pretty straight forward to get an industry gig.
I haven't really come across algebra in machine learning other than people applying it to deep learning.
i.e. https://arxiv.org/pdf/1802.03690.pdf
ie. https://icml.cc/Conferences/2018/Schedule?showEvent=2048
I don't personally find papers like this valuable but idk I have never really enjoyed abstract algebra.
For areas of mathematics to do theory in ML (and to do ML more generally!)
-probability/concentration/hoeffding bounds [the PAC model] [Key]
-linear algebra [key]
-optimization [key]
for books
-understanding machine learning by shai ben david
This book is nice since it really balances theory with a more practical understanding.
-An Introduction to Computational Learning Theory by kearns is a classic [low priority]. this is fun since the proofs are simple and deep but is very very far away from practical algorithms.
-convex optimization by boyd
Course Notes:
[I think a good alternative to blogs is stalking course notes for other schools-they are very often public.]
- http://ttic.uchicago.edu/~avrim/MLT18/index.html
good learning theory course by avrim blum who is a big deal in learning theory and theory.
- tim roughgardens notes are a blessing for algorithms and theory [seriously he should have a patreon or something]
https://theory.stanford.edu/~tim/notes.html
Blogs:
this is ben recht's blog and is filled with ML wisdom.
-https://blogs.princeton.edu/imabandit/ not quite learning theory but a lot of ML adjacent stuff
I don't read many blogs as I should tbh so other people can give better advice
VIDEOS https://www.youtube.com/channel/UCW1C2xOfXsIzPgjXyuhkw9g
This is the simons institute youtube channel. probably the best single location for recordings of TALKS in computer science-good amount of ML talks.
One exception I can think of is what I call "closet programmers," which are folks that work in various areas which rely on software such as experimental physics, astronomy, molecular biology. and end up mostly doing programming because they love it. We have a bunch of engineers like that and they are all excellent :-)
Also those closet programmers are always really fun to talk to since their problems and culture are breath of fresh air.