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We study approximation of multivariate functions from a separable Hilbertspace 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 squaresregression algorithm to obtain the upper estimates of the minimal randomizederror using $\Lambda^{\rm std}$. We investigate the equivalences of variousnotions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and$\Lambda^{\rm all}$ for the normalized or absolute error criterion. We showthat in the randomized setting for the normalized or absolute error criterion,the power of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ forall notions of exponential and algebraic tractability without any condition.Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak andH.Wo\'zniakowski in the book: Tractability of Multivariate Problems, VolumeIII: Standard Information for Operators, EMS Tracts in Mathematics, Z\"urich,2012.

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