  High-dimensional statistical inference deals with models in which the the
number of parameters p is comparable to or larger than the sample size n. Since
it is usually impossible to obtain consistent procedures unless
$p/n\rightarrow0$, a line of recent work has studied models with various types
of low-dimensional structure, including sparse vectors, sparse and structured
matrices, low-rank matrices and combinations thereof. In such settings, a
general approach to estimation is to solve a regularized optimization problem,
which combines a loss function measuring how well the model fits the data with
some regularization function that encourages the assumed structure. This paper
provides a unified framework for establishing consistency and convergence rates
for such regularized M-estimators under high-dimensional scaling. We state one
main theorem and show how it can be used to re-derive some existing results,
and also to obtain a number of new results on consistency and convergence
rates, in both $\ell_2$-error and related norms. Our analysis also identifies
two key properties of loss and regularization functions, referred to as
restricted strong convexity and decomposability, that ensure corresponding
regularized M-estimators have fast convergence rates and which are optimal in
many well-studied cases.
