  We derive non-asymptotic bounds for the minimax risk of variable selection
under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^d$ for
classes of $s$-sparse vectors separated from 0 by a constant $a > 0$. In some
cases, we get exact expressions for the nonasymptotic minimax risk as a
function of $d, s, a$ and find explicitly the minimax selectors. These results
are extended to dependent or non-Gaussian observations and to the problem of
crowdsourcing. Analogous conclusions are obtained for the probability of wrong
recovery of the sparsity pattern. As corollaries, we derive necessary and
sufficient conditions for such asymptotic properties as almost full recovery
and exact recovery. Moreover, we propose data-driven selectors that provide
almost full and exact recovery adaptively to the parameters of the classes.
