I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound: $$ \phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d} \mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(1+\gamma\langle X^{(j)},Z\rangle)]^2}{\prod_{i=1}^d \cosh(\gamma\sum _{j=1}^nX^{(j)}_i)}\right]\tag{1}$$ where $\gamma\in(0,1]$, $d\gg 1$, $X=(X^{(j)})_{1\leq j\leq n}$ is a collection of i.i.d. standard $d$-dimensional Gaussian r.v.'s, and $Z$ is uniform on $\{-1,1\}^d$ (Rademacher) independent of the $X^{(j)}$'s.

Conjecture. Assuming $\gamma^2 d \leq 1$, as long as $n\gamma^4 d \ll 1$ we have $$\phi(\gamma,d,n)\ll 1\tag{2}$$

I haven't been able to show it, even for $n=2$; there seems to be a delicate balancing act for things to exactly cancel the $e^{\frac{1}{2}n\gamma^2 d}$ factor...

Update: For (1) in the case $n=2$, my numerical experiments (a bit noisier than I had hoped) seem to be consistent with the conjecture:

  • $\gamma$ fixed, varying $d$

  • $d=5$ fixed, varying $\gamma$ (fit made with SciPy, in Python) enter image description here

Update 2: For $d=1$ and $n\in\{1,2,3\}$, Mathematica could compute explicitly the expectation (though already it takes some time for $n=3$. The behavior on these few points is clearly linear wrt $n$: enter image description here enter image description here

  • $\begingroup$ Is $Y$ in $E_{XY}$ a misprint or some other r.v. $Y$ is, indeed, present? $\endgroup$ – fedja Feb 24 '20 at 2:41
  • $\begingroup$ @fedja Oh, good catch, that was a remain of what I wrote in an earlier version for $n=2$ (I had $X,Y$ instead of $X_1,X_2$). Edited. $\endgroup$ – Clement C. Feb 24 '20 at 2:44
  • $\begingroup$ One more stupid question: when $n=d=1$ we have $E_X(\frac {E_Z\dots}{\dots})=E_X(\frac {1+\gamma^2X^2}{\dots})\ge \frac{[E_X\sqrt{1+\gamma^2X^2}]^2}{E_X\dots}=e^{-\gamma^2/2}[E_X\sqrt{1+\gamma^2X^2}]^2$, so the whole expression is at least $[E_X\sqrt{1+\gamma^2X^2}]^2-1\sim \gamma^2$. How does this agree with your conjectured $\gamma^4$ decay as $\gamma\to 0$? Am I missing or misunderstanding something? $\endgroup$ – fedja Feb 24 '20 at 22:53
  • $\begingroup$ @fedja (from my phone) I am not sure how you get this denominator for $n=1$. Since the square is outside of the $\mathbb{E}_Z$, the denominator should be 1 (as $\langle X,Z\rangle$ has expectation zero), shouldn't it? $\endgroup$ – Clement C. Feb 25 '20 at 1:55
  • $\begingroup$ You mean "the numerator"? I see. $E_Z[\rm{big\ formula}]^2$ is ambiguous: I interpreted it as the expectation of the square rather than the square of the expectation. So it should be the square of the expectation in $Z$, right? $\endgroup$ – fedja Feb 25 '20 at 2:12

Something looks fishy at least with the first conjecture. Perhaps I'm again misinterpreting something, but the argument is as follows:

Consider $n=2$. Write $X^{(1)}=\frac{X+Y}{\sqrt 2}, X^{(2)}=\frac{X-Y}{\sqrt 2}$. Then $X,Y$ are independent standard Gaussians in $\mathbb R^d$. Also $$ E_Z[(1+\gamma\langle X^{(1)},Z\rangle)(1+\gamma\langle X^{(2)},Z\rangle)]=1+\gamma^2\langle X^{(1)},X^{(2)}\rangle \\ =1+\frac{\gamma^2}2(\|X\|^2-\|Y\|^2) $$ Now, since $\|Y\|^2$ is a sum of squares of $d$ independent Gaussians, it deviates from any fixed number by about $\sqrt d$ with constant probability, so the expectation $E_Y[E_Z[(1+\gamma\langle X^{(1)},Z\rangle)(1+\gamma\langle X^{(2)},Z\rangle)]^2]$ is at least $c\gamma^4d$ regardless of $X$ and the whole expression you are interested in is at least $$ -1+c\gamma^4d\left(E[\cosh(\gamma \sqrt 2 W)]E[\frac 1{\cosh(\gamma \sqrt 2 W)}]\right)^d $$ but for fixed $\gamma$ and $d\to+\infty$, this is, clearly, exponential in $d$. What am I missing this time?

  • $\begingroup$ I have to read carefully what you wrote (what is $W$ in the last line?),but I may have been making a hidden assumption there. Namely (as used in the last part of my answer, dealing with the Gaussian variant) I am implicitly assuming $\gamma^2 d$ is bounded (as I get $e^{\gamma^2 d}\cdot \gamma^4 d$ as an upper bound), which is clearly not an assumption stated in my original post, but is something I'll have in my end application. I'll check if, under that assumption, what you wrote still provides a counterexample. $\endgroup$ – Clement C. Feb 27 '20 at 1:23
  • $\begingroup$ Yes, under that extra assumption $\gamma2 d \lesssim 1$, the last part indeed stays bounded, so things are consistent. I apologize for wasting your time by not stating that upfront. $\endgroup$ – Clement C. Feb 27 '20 at 1:31
  • $\begingroup$ @ClementC. $W$ is the standard Gaussian. Sorry for not stating that. Yeah, if $\gamma^2d$ is bounded, then the exponential growth gets tamed. OK, let me try to work it out under that assumption but expect more stupid questions :-) $\endgroup$ – fedja Feb 27 '20 at 1:46
  • $\begingroup$ @ClementC. One comes right away: is $n$ bounded too? (if so, what's the point of the $n$ factor in the proposed equivalence; if not, then I'm afraid that the exponential factor may still prevail as $n\to\infty$) $\endgroup$ – fedja Feb 27 '20 at 1:50
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    $\begingroup$ @ClementC. So what is the actual game? Just to show that $\psi(\gamma,d,n)$ is small if $n\gamma^4 d$ is small or it is subtler than that? $\endgroup$ – fedja Feb 27 '20 at 2:35

Partial progress: here is a proof for the analogous "Gaussian case" (i.e., $Z\sim N(\mathbf{0}_d, \mathbf{I}_d)$) and $n=2$. I was hoping to get some inspiration for it to handle the "Rademacher" $Z$ case I am interested in, though it doesn't seem to be helpful after all...

The goal is to bound an analogous quantity as (1), which for $n=2$ is $$ \psi(\gamma,d,2) = \frac{e^{-\gamma^2 d}}{(1-2\gamma^2)^{d/2}}-2 + e^{\gamma^2d}\mathbb{E}_{XY}\left[ {\frac{\mathbb{E}_{Z}[ (1+\gamma\langle X,{Z}\rangle)(1+\gamma\langle Y,{Z}\rangle) ]^2}{ e^{\frac{\gamma^2}{2}\lVert X+Y \rVert_2^2} } }\right] $$ (the denominator changes because it comes from $\mathbb{E}_Z[e^{\gamma \langle\sum_{j=1}^n X^{(j)},Z\rangle}]$, and that expression changes between the Gaussian and Rademacher cases. Similarly for why the additive "-1" is now a less nice expression.)

We can expand and compute the numerator (before squaring) as \begin{align*} \mathbb{E}_{Z}\left[(1+\gamma\langle X, Z\rangle)(1+\gamma\langle Y, Z\rangle) \right] = \mathbb{E}_{Z}\left[1+\gamma\langle{X+Y,Z}\rangle + \gamma^2\langle X, Z\rangle\langle Y, Z\rangle \right] = 1+ \gamma^2\langle X, Y\rangle \end{align*} since $Z$ is Gaussian. From there, observing that $V := \frac{\lVert X+Y\rVert_2^2}{2}$ is a $\chi^2(d)$ r.v., we get \begin{align*} e^{-\gamma^2d }&(1+\psi(d ,\gamma,2))\\ &= \mathbb{E}_{XY}[{ e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} (1+ \gamma^2\langle X, Y\rangle)^2 }]\\ &= \mathbb{E}_{V}[{ e^{-\gamma^2 V} }] + 2\gamma^2 \mathbb{E}_{XY}[{\langle X, Y\rangle e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] + \gamma^4\mathbb{E}_{XY}[{\langle X, Y\rangle^2e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] \\ &\leq \mathbb{E}_{V}[{ e^{-\gamma^2 V} }] + 2\gamma^2 \mathbb{E}_{XY}[{\langle X, Y\rangle e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] + \gamma^4\mathbb{E}_{XY}[{\langle X, Y\rangle^2}] \\ &= \frac{1}{(1+2\gamma^2)^{d /2}} + \gamma^2 \mathbb{E}_{XY}[{(\lVert X+Y\rVert_2^2 - \lVert X\rVert_2^2 - \lVert Y\rVert_2^2) e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] + d \gamma^4\,. \end{align*} The second term can be computed explicitly. Indeed, we have \begin{align*} \mathbb{E}_{XY}[{\lVert X+Y\rVert_2^2e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] &= 2 \mathbb{E}_{V}[{V e^{-\gamma^2V} }] = \frac{2d }{(1+2\gamma^2)^{1+d /2}} \\ \mathbb{E}_{XY}[{\lVert X\rVert_2^2e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] &= \mathbb{E}_{X}[{\lVert X\rVert_2^2 \mathbb{E}_{Y}[{ e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] }] = \mathbb{E}_{X}[{\lVert X\rVert_2^2 \prod_{i=1}^d \mathbb{E}_{Y}{ e^{-\frac{\gamma^2}{2}(X_i+Y_i)^2} } }]\\ &= {(1+\gamma^2)^{-d/2}}\mathbb{E}_{X}[{\lVert X\rVert_2^2e^{-\frac{\gamma^2}{2(1+\gamma^2)}\lVert X\rVert_2^2} }] = \frac{d (1+\gamma^2)}{(1+2\gamma^2)^{1+d /2}} \end{align*} from which the second term equals \begin{align*} \mathbb{E}_{XY}[{(\lVert X+Y\rVert_2^2 - \lVert X\rVert_2^2 - \lVert Y\rVert_2^2) e^{-\frac{\gamma^2}{2}\lVert X+Y\rVert_2^2} }] &= - \frac{2d \gamma^2}{(1+2\gamma^2)^{1+d /2}}\,. \end{align*} Overall, we get \begin{align*} e^{-\gamma^2d }\left(2-\frac{e^{-\gamma^2 d}}{(1-2\gamma^2)^{d/2}}+\psi(d ,\gamma,2)\right) &\leq \frac{1}{(1+2\gamma^2)^{d /2}}\left(1- \frac{2d \gamma^4}{1+2\gamma^2}\right) + d \gamma^4 \leq \frac{1}{(1+2\gamma^2)^{d /2}} + d \gamma^4\,, \end{align*} from which \begin{align*} \psi(d ,\gamma,2) = O( d \gamma^4 ) \end{align*} as long as $\gamma^2d = O(1)$.


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