  We consider the order of convergence for linear and nonlinear Monte Carlo
approximation of compact embeddings from Sobolev spaces of dominating mixed
smoothness defined on the torus $\mathbb{T}^d$ into the space
$L_{\infty}(\mathbb{T}^d)$ via methods that use arbitrary linear information.
These cases are interesting because we can gain a speedup of up to $1/2$ in the
main rate compared to the worst case approximation. In doing so we determine
the rate for some cases that have been left open by Fang and Duan.
