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I need your expertise in solving the following problem:

Let $n \in \mathbb{N}$ such that $n > 1$, $\left\lbrace X_i \right\rbrace_{i=1}^n$ be a set of sub-Gaussian variables with mean $0$ and unit variance and $\left\lbrace Y_i \right\rbrace_{i=1}^n$ be a set of random variables where for each $i \in [n]$, $Y_i \in \left\lbrace-1,1 \right\rbrace$. ($Y_i$ is dependent on $X_i$ for each $i \in [n]$)

We want to have bounds on the expectation of the dot product of $X$'s and $Y'$s i.e. $$ E\big[ \sum\limits_{i=1}^n x_iy_i \big]$$

How can we establish bounds on this?

Please advise.

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  • $\begingroup$ Any more info on how $Y_i$ depends on $X_i$? $\endgroup$ – Clement C. Jun 15 '17 at 13:20
  • $\begingroup$ think of $X_i$ is as a data point and $Y_i$ it's label. $\endgroup$ – user3492773 Jun 15 '17 at 13:22
  • $\begingroup$ But then, in terms of what do you expect to bound this expectation? For instance, if $Y_i = \operatorname{sign}(X_i)$ for all $i$, then you get $\sum_{i=1}^n\mathbb{E}[\lvert X_i\rvert]$ (which is as bad as it gets); while if $Y_i$ is independent of $X_i$ and uniform (labels corresponding to a random function, I guess), then you'll get $0$. $\endgroup$ – Clement C. Jun 15 '17 at 13:25
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    $\begingroup$ $E(\sum\limits_{i=1}^{n}X_iY_i) = \sum\limits_{i=1}^{n}E(X_iY_i) = \sum\limits_{i=1}^{n}E(E(X_iY_i|Y_i)) = \sum\limits_{i=1}^{n}E(Y_iE_{X_i|Y_i}(X_i))$. So one must know the distributions $f(X_i|Y_i)$ and $f(Y_i)$ or f(X_i,Y_i) to solve this. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Jun 15 '17 at 16:56

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