According to the below inequality : $$E\big[\sqrt{\sum_{m=1}^{M}z_{m}^{i}}\big]\geq{E\big[\sqrt{M}\sqrt[2M]{\prod_{m=1}^{M}z_m^i\big]}}={E\big[\sqrt{M}\prod_{m=1}^{M}z_m^i\big]}$$ we scaled the left side to the right side, where $\eta$ is constant vector, and $z_m$ is random, binary, unit-basis vector.

But we found the scale level is too large. We wanna remove the sqrt of the left side and have a moderate scale, are there any approach?