On the power of standard information for tractability for $L_2$-approximation in the randomized setting

We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $\Lambda^{\rm std}$ consisting of function values or general linear information $\Lambda^{\rm all}$ consisting of arbitrary linear functionals. We use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using $\Lambda^{\rm std}$. We investigate the equivalences of various notions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and $\Lambda^{\rm all}$ for the normalized or absolute error criterion. We show that in the randomized setting for the normalized or absolute error criterion, the power of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ for all notions of exponential and algebraic tractability without any condition. Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak and H.Wo\'zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Z\"urich, 2012.