  Gaussian comparison theorems are useful tools in probability theory; they are
essential ingredients in the classical proofs of many results in empirical
processes and extreme value theory. More recently, they have been used
extensively in the analysis of non-smooth optimization problems that arise in
the recovery of structured signals from noisy linear observations. We refer to
such problems as Primary Optimization (PO) problems. A prominent role in the
study of the (PO) problems is played by Gordon's Gaussian min-max theorem (GMT)
which provides probabilistic lower bounds on the optimal cost via a simpler
Auxiliary Optimization (AO) problem. Motivated by resent work of M. Stojnic, we
show that under appropriate convexity assumptions the (AO) problem allows one
to tightly bound both the optimal cost, as well as the norm of the solution of
the (PO). As an application, we use our result to develop a general framework
to tightly characterize the performance (e.g. squared-error) of a wide class of
convex optimization algorithms used in the context of noisy signal recovery.
