  Approximate Message Passing (AMP) algorithms are a class of iterative
procedures for computationally-efficient estimation in high-dimensional
inference and estimation tasks. Due to the presence of an 'Onsager' correction
term in its iterates, for $N \times M$ design matrices $\mathbf{A}$ with i.i.d.
Gaussian entries, the asymptotic distribution of the estimate at any iteration
of the algorithm can be exactly characterized in the large system limit as $M/N
\rightarrow \delta \in (0, \infty)$ via a scalar recursion referred to as state
evolution. In this paper, we show that appropriate functionals of the iterates,
in fact, concentrate around their limiting values predicted by these asymptotic
distributions with rates exponentially fast in $N$ for a large class of
AMP-style algorithms, including those that are used when high-dimensional
generalized linear regression models are assumed to be the data-generating
process, like the generalized AMP algorithm, or those that are used when the
measurement matrix is assumed to be right rotationally invariant instead of
i.i.d. Gaussian, like vector AMP and generalized vector AMP. In practice, these
more general AMP algorithms have many applications, for example in in
communications or imaging, and this work provides the first study of finite
sample behavior of such algorithms.
