Do i.i.d Sums Concentrate Any Faster Than Martingales?

Suppose $ X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $ \|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae).

The simplest concentration inequality I know only applies in the case $ d=1$ and only when $ X_1,X_2, \ldots, X_N$ are i.i.d. The Hoeffding Lemma gives for each $ \epsilon >0$ the bound

$ $ P(|X_1 + \ldots + X_N| \ge \epsilon) \le \exp\left (-\frac{2\epsilon^2}{N} \right).\tag{1}$ $

On the other end of the spectrum are results that work under the weaker assumption that $ X_1,X_2, \ldots, X_N$ is a martingale, and work for any $ d \in \mathbb N$ , or indeed for infinite dimensional Banach spaces provided some variant of the parallelogram is satisfied. For example Theorem 3.5 of this paper of Pinelis leads to the following variant of the Azuma-Hoeffding inequality.

$ $ P(\|X_1 + \ldots + X_n\|_2 \ge \epsilon \text{ for some }n\le N) \le \exp\left (-\frac{\epsilon^2}{2N} \right).\tag{2}$ $

The exponent is the same as the scalar Azuma Hoeffding. Notice the $ 2$ is now downstairs rather than upstairs like before.

If we are only dealing with i.i.d scalars and only interested in the final element, we should use $ (1)$ because it gives a better bound. If we are dealing with either vectors, martingales, or want a uniform inequality we better use $ (2)$ instead.

My problem is between the two extremes. I am dealing with a sequence of i.i.d vectors in $ \mathbb R^d$ and I am interested in a uniform bound. I wonder does there exist an appropriate middle-ground between these two results? Perhaps combining the $ -2\epsilon^2/N$ of the first with the uniform nature of the second, at the expense of only applying to i.i.d sequences as opposed to martingales.

Is an encrypted private key which never leaves my home directory more secure than an unencrypted one?

On a Linux system I’m running an utility like this:

$   /usr/bin/myapp myprivatekey Enter passphrase for the private key:... ...application runs and uses the private key 

My understanding is that if I have a private key encrypted with a passphrase it is more secure than an unencrypted one because the private key cannot be accessed even if the user account is compromised. So if the private key is loaded by a process running as a different user and the passphrase is typed manually by the user then one cannot intercept the above passphrase. Please note that the /usr/bin/myapp can only be written by root.

On the other hand a colleague argues that, if the user account is compromised then the private key is compromised too even if it’s protected by a passphrase, because if the account is compromised then the password typed by the user can be intercepted and one cannot be protect himself in such a situation.

Which one is correct? Is it possible to setup a system such that the private key is protected in the above situation?


Is scrolling better than clicking to reveal more content?

Which is faster? Scrolling down or clicking a button to reveal more content? Currently, our site requires clicking of page numbers to reveal more content. We have 9 channels per page, which fits on one screen in most common resolutions.

Personally, I believe having 2 or 3 pages worth of content and scrolling down to reveal those pages would be easier. We should have 20+ channels per page. I just flick my mouse wheel or hit the down key. I don’t have to aim at a tiny button (which takes longer as per Fitts’s Law).

Received a different US visa category than I applied for

I am from germany and an employee of a large US-based IT company. I recently applied for a B1/B2 and I went through the whole process. I then went to the consulate last week and they accepted my application and told me that they would ship my passport to me as I ordered. I just received my passport via the mail but when I checked everything, I noticed that instead of B1/B2 the visa only states B1. Is this just a human error or is there more to it? And most importantly, is there anything I can/should do about this to change it to a B1/B2 and how long would this take?

“A good programmer can be as 10X times more productive than a mediocre one” [closed]

I had read an interview with a great programmer (it is not in English) and in it he said that “a great programmer can be as 10 times as good as a mediocre one” giving reason for why good programmers are very well paid and why programming companies give many facilities for their employees. The idea was that there is a very large demand for good programmers, because of the above reason and that’s why companies pay very much to bring them.

Do you agree with this statement? Do you know any objective facts that could support it?

Edit: The question has nothing to do with experience; if you talk about one great programmer with 1 year experience then s/he should be 10 times more productive than a mediocre programmer with 1 year experience. I agree that from certain experience years onwards, things start to dissipate but that’s not the purpose of the question.

Why is the triangulated category of motives easier than the abelian one?

There are several expository articles with the title “You could have invented [insert something mysterious here]” (a notable one being about spectral sequences, possibly it even started this genre). This question is somewhat similar in spirit to them.

Here it is stated that “Deligne first suggested that it might be easier to define the derived category $ DM(S,\mathbb{Q})$ of the hypothetical abelian category of mixed motives.” First time I heard about this, it seemed a little bit counterintuitive to me. We were doing some abelian stuff, why should passing to the derived category make anything any simpler? Of course, it is easy now to point to the success of Voevodsky and others and say that it was totally obvious.

The question is assuming you never heard about Voevodsky, Morel, etc., you are in the 1960’s, how could you arrive at the idea that the triangulated category is easier to construct than the abelian category of mixed motives?