Windows on Theory

I have posted about this on https://windowsontheory.org/2018/05/02/short-non-review-of-caplans-case-against-education/

I am of course biased as a faculty (though I think his policy prescriptions would negatively effect public schools more than private universities such as Harvard), but I think Caplan's analysis is extremely shallow.

First, note that if he's right, higher education should be an incredible drag on the economy - we're taking 4 highly productive years out of the workforce. Such an extreme conclusion shouldn't be that hard to test without resorting to anecdotes and contorted reading of data. Educational policies and subsidies vary so greatly between different countries and even states, that if it was such a colossal waste we should be seeing it in the higher GDP or productivity of the less educated countries and localities.
(The article is in general very US-centric for an issue that is not specific to the US at all.)

When a locality has less access to education, there would be naturally less of a "credential arms race" and so more people that have just as much base talent but did not go to college. Why aren't employers flocking to those places and hiring high school graduates who would be so much cheaper?

I know one theory Caplan has is that completing college certifies "conformity" and "tolerance for boredom": is spending 4 years and tens or hundreds or thousands of dollars the only way to test for these properties?

Also (at least from my experience in the…

The short version is, if this is correct (which is exceedingly unlikely) then either quantum mechanics is wrong or whether P=NP has just become irrelevant. This is because, as Scott Aaronson is often forced to tirelessly point out, an inability to build a Quantum Computer must also show Quantum mechanics to somehow be in error. If instead, we can build a Quantum computer and this result is true then the universe has suddenly become an incredibly more interesting place (for one example, building this would also solve AI).

The result also has consequences for quantum computation specifically, in that it also takes care of the negative sign problem, which arises when simulating certain kinds of important quantum many body systems.

And last but certainly not least, all those press releases breathlessly proclaiming Quantum computers work by trying a gazillion possibilities at once (with no worries about the plausibility of actually reading out the correct answer with a non-negligible probability) weren't so far off after all.

All in all, either there is a mistake hidden somewhere in the preprint or this is the greatest scientific achievement since Evolution invented Human Level intelligence.

See here for intuition on why NP-complete problems if at all solvable without brute force, should also be efficiently solvable: http://windowsontheory.org/2013/05/06/reasons-to-care-in-honor-of-scott-aaronson/

I think what they are getting at is the following, which isn't a proof or anything (if you are looking for rules to follow, the much simpler "your model isn't differentially private" is probably even more accurate):

Let X be a set of iid samples drawn from a distribution D, let M be a model trained with eps-DP on the set X, and let x' be a fresh sample from D.

Let's think about the random variable M(x'), which represents your test loss or whatever. Differential privacy says that the density of this random variable is pointwise within a factor of exp(eps) of M'(x') where M' is the random model resulting from training on X \cup { x' }. But M'(x') is distributed per a random x', random X, and M' trained on both just like your training loss or whatever.

You can push linearity of expectations through all of this to show that your expected loss on test and training should vary by at most a factor of exp(eps), but I'm not aware of a pointwise test for a specific model (and it may not make sense, as DP isn't a property of specific models, but rather distributions of them).

Edit: you can read a bit more about a special case of this at https://windowsontheory.org/2014/02/04/differential-privacy-for-measure-concentration/

Quick note that the article is from 2015, and the MSS result dates back to 2013.

There's a blog post by Nikhil here that explains the proof of the discrepancy result that implies Kadison-Singer: https://windowsontheory.org/2013/07/11/discrepancy-graphs-and-the-kadison-singer-conjecture-2/

The main result is a theorem that replaces a Chernoff bound (saying here that a random partition has discrepancy logarithmic in the number of vectors with high probability) with a bound that says achieves a better bound on the discrepancy, independent of the number of vectors, at the cost of replacing the "with high probability" with "there exists".

The proof uses some really beautiful techniques from the geometry of polynomial roots to prove this and is a pretty fun read: https://arxiv.org/abs/1306.3969

Some related links:

An introduction to the area from before this new breakthrough -- https://www.ams.org/journals/notices/201910/rnoti-p1618.pdf

Gil Kalai on the mathematical implications -- https://gilkalai.wordpress.com/2020/01/17/amazing-zhengfeng-ji-anand-natarajan-thomas-vidick-john-wright-and-henry-yuen-proved-that-mip-re-and-thus-disproved-connes-1976-embedding-conjecture-and-provided-a-negative-answer-to-tsirelso/

Lance Fortnow -- https://blog.computationalcomplexity.org/2020/01/quantum-provers-to-infinity-and-beyond.html

KW Regen -- https://rjlipton.wordpress.com/2020/01/15/halting-is-poly-time-quantum-provable/

Boaz Barak -- https://windowsontheory.org/2020/01/14/mipre-connes-embedding-conjecture-disproved/

Scott Aaronson -- https://www.scottaaronson.com/blog/?p=4512

I think we should look at the correct operation T on the set of all computer scientists. If T is an operation that restricts the set to computer scientists in the field of theoretical computer science (TCS), then the opposite seems to hold as mentioned in this post, which talks about TCS math being more rigorous than applied math: https://windowsontheory.org/2014/10/12/applied-mathematicians-vs-theoretical-computer-scientists/

If, loosely speaking, T maps you to 'average computer scientist', then the situation is different. And so on.

So depending on the type of T, subset or groups of computer scientists or metrics you are looking at, Lamport's observations hold water.