Adapting to Unknown Sparsity by controlling the False Discovery Rate

We attempt to recover an $n$-dimensional vector observed in white noise, where $n$ is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the $\ell_p$ norm for $p$ small. We obtain a procedure which is asymptotically minimax for $\ell^r$ loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the {\it False Discovery Rate} (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter $q_n$ also plays a determining role in asymptotic minimaxity. If $q = \lim q_n \in [0,1/2]$ and also $q_n > \gamma/\log(n)$ we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, $q = \lim q_n \in (1/2,1]$, forces the risk to exceed the minimax risk by a factor growing with $q$. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form $2 \cdot \log({potential model size} / {actual model size})$. We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.