  This paper studies adaptive sensing for estimating the nonzero amplitudes of
a sparse signal with the aim of providing analytical guarantees on the
performance gain due to adaptive resource allocation. We consider a previously
proposed optimal two-stage policy for allocating sensing resources. For
positive powers q, we derive tight upper bounds on the mean qth-power error
resulting from the optimal two-stage policy and corresponding lower bounds on
the improvement over non-adaptive uniform sensing. It is shown that the
adaptation gain is related to the detectability of nonzero signal components as
characterized by Chernoff coefficients, thus quantifying analytically the
dependence on the sparsity level of the signal, the signal-to-noise ratio, and
the sensing resource budget. For fixed sparsity levels and increasing
signal-to-noise ratio or sensing budget, we obtain the rate of convergence to
oracle performance and the rate at which the fraction of resources spent on the
first exploratory stage decreases to zero. For a vanishing fraction of nonzero
components, the gain increases without bound as a function of signal-to-noise
ratio and sensing budget. Numerical simulations demonstrate that the bounds on
adaptation gain are quite tight in non-asymptotic regimes as well.
