On the symmetrized arithmetic-geometric mean inequality for opertors

We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and R'{e} $$\mathrm{\parallel }\frac{(n-d)!}{n!}\sum _{j_1,\mathrm{.}\mathrm{.}\mathrm{.},j_{d}different}{A}_{j_1}^{\ast }{A}_{j_2}^{\ast }\mathrm{.}\mathrm{.}\mathrm{.}{A}_{j_d}^{\ast }{A}_{j_d}\mathrm{.}\mathrm{.}\mathrm{.}{A}_{j_2}{A}_{j_1}\mathrm{\parallel }\le C(d,n)\mathrm{\parallel }\frac{1}{n}\sum _{j=1}^n{A}_j^{\ast }{A}_j{\mathrm{\parallel }}^d\mathrm{.}$$ Complementing the results from Recht and R'{e}, we find upper bounds for C(d,n) under additional assumptions. Moreover, using free probability, we show that $C(d,n)>1$, thereby disproving the most optimistic conjecture from Recht and R'{e}.We also prove a deviation result for the symmetrized-AGM inequality which shows that the symmetric inequality almost holds for many classes of random matrices. Finally we apply our results to the incremental gradient method(IGM).