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Fix $m$ arbitrary values $x_1, x_2, ..., x_m$ in $[0,1]$, and an integer $n$. Obtain $n$-set $S$ by drawing $n \le m$ times randomly without replacement from $\{1,2,..,m\}$. Define r.v. $X = \sum_{i \in S} x_i$ and $\mu=E[X]$.

Is it the case that

$$\Pr[X \ge \mu + \epsilon\mu] ~=~ \exp(- \Omega(\epsilon^2 \mu))~?~~~~~(1)$$

Following the hint here, we can use McDiarmid's inequality to prove the weaker bound

$$\Pr[X \ge \mu + \epsilon n] ~=~ \exp(-\Omega(\epsilon^2 n)),~~~~~(2)$$

but this bound depends on $n$ and is weaker when $\mu \ll n$.

A related but different question was asked here.


[EDIT: My original question asked first if bound (1) holds in all applications of of McDiarmid's inequality where $X=f(y_1,..,y_n)$ is a function of $n$ independent random variables, and $f$ changes by at most 1 when any $x_i$ is changed. For the record, the answer to that seems to be NO. For a counter-example, take $$\textstyle X = f(x_1, x_2, \ldots, x_n) = \big|k \,-\, \big((\sum_{i=1}^n x_i) \bmod 2k\big)\big| $$ where each $x_i$ is i.i.d. uniformly over $[0,1]$ and, say, $k=\lceil \sqrt n\rceil$. Then $X$ is roughly uniformly distributed over $[0,k]$, so $\mu \approx k/2$, and $\Pr[X \ge \mu + \mu/2] = \Omega(1) \ne \exp(-\Omega(\mu))$.]

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  • $\begingroup$ It seems to me that multiplicative or additive are the same, of the form $\mathbb{P}(X \geqslant \mu + A)$ with $ A = \varepsilon n $ in one case and $ A = \varepsilon \mu $ in the other case (the wikipedia link states that this is for all $ A > 0$). If you want to state the MacDiarmid inequality in a multiplicative form, replace the $A$ accordingly in the second case, i.e. take $ \mu t = \varepsilon k $ and the bound in $ \exp(-\varepsilon^2 k/3) $ becomes $ \exp( - ( \mu t / k )^2 k/3) = \exp( - t^2 \mu^2/(3 k) ) $ (hence $c = 3k/\mu $). Of course, $c$ depends of $ \mu $ here... $\endgroup$ – Synia Sep 13 '17 at 18:53
  • $\begingroup$ @Synia, In the form stated on Wikipedia, the bound is $$\Pr[X \ge \mu + A] \le \exp(-A^2/n),$$ where $n$ is the number of random variables. So, (in the general case) if you substitute $A=\epsilon\mu$, the bound is $\exp(-\epsilon^2 \mu^2/n)$, not $\exp(-\epsilon^2 \mu)$. (And note that, likewise, in the application above we want $c$ to be a constant independent of $k$ and $\mu$, e.g. $c=3$.) Also, if you look at the proof of McDiarmid, the Doob martingale that it uses can either increase or decrease in each step, which forces the use of an additive error bound. $\endgroup$ – Neal Sep 13 '17 at 21:07
  • $\begingroup$ Yes, I was suspecting something like that, namely that $c $ be independent of $n $ (no need to edit, I guess). Stated like this, the question becomes of course much more interesting. There may be some inequalities available in the literature in the case of a sum that "beat" the classical MacDiarmid inequality (I am thinking of Chatterjee's method with exchangeable pairs). I will try to look it up. $\endgroup$ – Synia Sep 13 '17 at 21:44
  • $\begingroup$ I realized that the answer to the first general question I asked is NO. I've edited the post to reflect this. $\endgroup$ – Neal Sep 13 '17 at 22:14
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    $\begingroup$ Looks likely that the answer to the current question is yes, from Theorem 10 of this paper and Proposition 5 of this one. $\endgroup$ – Neal Sep 14 '17 at 2:37

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