  In this paper we study the problem of recovering a structured but unknown
parameter ${\bf{\theta}}^*$ from $n$ nonlinear observations of the form
$y_i=f(\langle {\bf{x}}_i,{\bf{\theta}}^*\rangle)$ for $i=1,2,\ldots,n$. We
develop a framework for characterizing time-data tradeoffs for a variety of
parameter estimation algorithms when the nonlinear function $f$ is unknown.
This framework includes many popular heuristics such as projected/proximal
gradient descent and stochastic schemes. For example, we show that a projected
gradient descent scheme converges at a linear rate to a reliable solution with
a near minimal number of samples. We provide a sharp characterization of the
convergence rate of such algorithms as a function of sample size, amount of
a-prior knowledge available about the parameter and a measure of the
nonlinearity of the function $f$. These results provide a precise understanding
of the various tradeoffs involved between statistical and computational
resources as well as a-prior side information available for such nonlinear
parameter estimation problems.
