My experience applying for a phd and talking to professors is it has gotten a lot more competitive. When I applied a year ago to 8 schools my recommenders thought I'd picked enough schools and had a strong enough background for top schools (2 first author papers, one at a workshop in a top conference). That led to 2 interviews, but at the end still 8 rejections. The acceptance rate for cs overall in stanford/similar schools is getting to around 3 percent. I was applying in ML which is extra popular and remember hearing that Montreal had 1000 cs graduate applicants this past year, with 500 for ML and 900 some ML overlap. Given they only want 15-20ish students a year, that's a fairly poor acceptance rate. My own undergrad was a STEM school and the number of CS majors has somewhere around tripled in the past 5ish years even though it started already high around 20 percent (yeah a majority is CS at this point). I know my school talked about how there's plenty of universities across the country that have had CS enrollment grow a ton and are having problems supporting all of them.

There are a few factors I can think of. I'm currently a postdoc and completed my PhD in 2016, so I was applying in 2011/2012 ish. This is a long answer, but it's an interesting question.

First, this is not indicative of other fields. I did my PhD in space science, I now work out of an astrophysics office. Astronomy is a competitive field by academic standards, certainly beyond PhD, but even at places like Oxford and Cambridge, the acceptance rate is probably something like 1:2 (really not bad) if you have a good undergrad result. Postdocs at top places like Harvard-Smithsonian can be extremely competitive though. I asked around my colleagues here and they didn't seem to think that PhD applications are significantly up recently.

Computer vision has always been quite a competitive field. Again, I think because it's a field where a lot of cutting edge stuff is happening actively and academically these researchers are famous. In itself that's quite weird when you think about it. This may be bias because I'm familiar with the field. However I was at NeurIPS last year and there was a perpetual queue of people waiting to get a photo with Andrew Ng at the closing party. I've never seen anything like that at another conference. It's actually quite annoying because it makes it very difficult to speak to top people in the field. Compared to say ecology where I can just email people and I'm highly likely to get a positive response.

Some places are just competitive full stop. Max Planck Institute (MPI), ETH Zurich, Oxbridge, etc. These places will always have a lot of international interest and everyone knows the big names in the US. My theory is that this is in part due to media representation. Movies teach us that getting into Harvard/Yale/Princeton/MIT/Stanford is a big deal. Can you name the top 2 or 3 universities in France beyond the Ecole Polytechnique, for example?

Thirdly, ML has skewed this significantly, especially since it's actively a good field to get into and a lot of the big names are still in the business of supervising people. Skeptically, I think many people applying are in it for the money and the prestige, not due to a deep interest in the theory. Bear in mind there are tons of fields where you can do applied ML - where there's more low hanging fruit, easy high impact papers, opportunities to work on novel architectures and datasets, etc. I would really recommend this route, it worked well for me.

The matching process varies with institution. In the UK it's common for a department to get funding for X students. Those students are doled out between supervisors - there may be many more potential projects than there is actually funding. This is the case for research council funding. However there are also funded PhDs which are usually some deliverable on a project that has a well-defined goal. When I started, I knew what my project would be at a high level. Lots of people in my cohort picked an advisor after they had been accepted. This is quite normal, most physics departments allow you to put in a general application.

So you do get problems when students say, I want to come to your department and work for X professor. That isn't necessarily something that can be guaranteed. The department might hire 5 really good applicants who all want the same advisor. However again, this is quite rare outside CS I think.

Finally I think people don't really understand what they're taking on when they want to work with a hotshot advisor. They need to ask questions like does this person actually have time for me? Working in a top lab is undoubtedly good, but it can be very high pressure and some people will suffer under those conditions.

> From my experience during my short PhD attempt, the amount of time spent studying (classes, reading papers, etc) versus simply working for an advisor was smaller than I expected, and I don't know if that's because of a real change in the PhD process or simply my own naïveté.

My advisor was more of a PM as he was (is) out of the active research game; he sat on tons of high profile committees at this point. So I worked largely independently and then got feedback at regular meetings. That was fine for me, but other people who wanted more active mentorship struggled I think. I did find it hard that nobody in my group had any significant overlap with my project. But it varies a lot. Some people get given a very tightly defined project due to funding, some work closely with their supervisor or act as a cog in a larger group.