Tighter upper bound for $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma]$

Question

Following this question I was thinking about ways to improve the upper bound and came up with the following argument. We want to find an upper bound for

\begin{equation} \mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma] \end{equation}

where $W$ is a symmetric matrix with independent entries $W_{ij} \sim \mathcal{N}(0,1)$, except for the symmetry condition. This is a slightly different version of the problem mentioned in the link but the argument is analogous. I came across the following result for gaussian processes

Let $(X_1, \ldots, X_n)$ and $(Y_1, \ldots, Y_n)$ be gaussian random vectors with $\mathbb{E}(X_i) = \mathbb{E}(Y_i)$ for each $i$. For $1 \leq i,j \leq n$, let $\gamma_{ij}^{X} = \mathbb{E}(X_i - X_j)^2$ and $\gamma_{ij}^{Y} = \mathbb{E}(Y_i - Y_j)^2$, and let $\gamma = \max_{1 \leq i,j \leq n} | \gamma_{ij}^{X} - \gamma_{ij}^{Y}|$. Then

\begin{equation} |\mathbb{E}(\max_{1 \leq i \leq n} X_i) - \mathbb{E}(\max_{1 \leq i \leq n} Y_i) | \leq \sqrt{\gamma \log n}. \end{equation}

I was thinking about applying this result with the random vectors $Y,X \in \mathbb{R}^{2^n}$ s.t. $Y_i = 0$ and $X_i = 2 \sum \limits_{s<t} A_{ij} \sigma_{s}^{i} \sigma_{t}^{i}$ for $1 \leq i \leq 2^{n} $, where $\sigma^{i}$ is the $i$-th hypercube vertex for some ordering of the vertices.

Question: Is the following Argument sound? Did I make a mistake or overlook something?

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It is clear that $Y$ is a gaussian random vector. I think $X$ is also a gaussian random vector because for any real numbers $\alpha_1, \ldots, \alpha_n$ we have

\begin{equation} \alpha_1X_1 + \ldots + \alpha_n X_n = \sum \limits_{s<t} A_{st}(\sum \limits_{i=1}^{n} \alpha_i \sigma_{s}^{i} \sigma_{t}^{i}) \end{equation}

Which is a gaussian random variable. Furthermore we have

\begin{equation} \mathbb{E}(\sum \limits_{s<t} A_{st}\sigma_{s}^{i} \sigma_{t}^{i}) = \sum \limits_{s<t} \mathbb{E}(A_{st})\sigma_{s}^{i} \sigma_{t}^{i} = 0 \end{equation}

and

\begin{equation} \gamma_{ij}^{X} = \sum \limits_{s<t}(\sigma_{s}^i \sigma_{t}^i - \sigma_{s}^{j} \sigma_{t}^{j})^2 \leq 2n^2 \end{equation}

Together with $\gamma_{ij}^Y = 0$ we have

\begin{equation} \mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma] \leq \sqrt{2n^2 \log 2^n} \end{equation}

The right hand side can be simplified to $\sqrt{2\log(2)} n^{3/2}$. If I understood correctly the author of the cited paper uses $\log$ to denote the natural logarithm. This would lead us to an upper bound of the order $\sim 1.177 n^{3/2}$, which is not too far away from the actual value for large $n$ which is $\sqrt{2} \cdot 0.7633 n^{3/2} \sim 1.079 n^{3/2}$.

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Thank you very much for your help.

Answer 1

$\newcommand\si{\sigma}$ $\newcommand\Si{\Sigma}$ $\newcommand\R{\mathbb R}$ Let $\Si:=\{\pm 1\}^n$. The map $$\R^{n\times n}\ni w\mapsto f(w):=(w_\si)_{\si\in\Si}\in\R^\Si, $$ where $w_\si:=\si^T w\si$, is linear. Therefore and because $W$ is zero-mean Gaussian, we see that $$(W_\si)_{\si\in\Si}:=f(W):=f\circ W$$ is zero-mean Gaussian, with $W_\si:=\si^T W\si=\sum_{i,j}\si_i\si_j W_{ij}$; everywhere here, the summation indices run over the set $\{1,\dots,n\}$. Also, for all $\rho$ and $\si$ in $\Si$ $$EW_\rho W_\si=\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l EW_{ij}W_{kl} \\ =\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{\{i,j\}=\{k,l\}} \\ =\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{i=j=k=l}\\ +\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{i=k\ne j=l} \\ +\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{i=l\ne j=k} \\ =2\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{i=k,j=l} \\ -\sum_{i,j,k,l}\rho_i\rho_j\si_k\si_l 1_{i=j=k=l} \\ =2\sum_{i,j}\rho_i\rho_j\si_i\si_j -\sum_i\rho_i\rho_i\si_i\si_i \\ =2(\rho\cdot\si)^2-n, $$ where $\rho\cdot\si:=\sum_i\rho_i\si_i$; in particular, $EW_\si^2=EW_\rho^2=2n^2-n$. So, $$E(W_\si-W_\rho)^2=EW_\si^2+EW_\rho^2-2EW_\rho W_\si =4n^2-4(\rho\cdot\si)^2\le4n^2. $$ So, the bound you are getting is actually $2\sqrt{\log2\,}\, n^{3/2}$, $\sqrt2$ times as large as you suggested.