  Motivated by recent increased interest in optimization algorithms for
non-convex optimization in application to training deep neural networks and
other optimization problems in data analysis, we give an overview of recent
theoretical results on global performance guarantees of optimization algorithms
for non-convex optimization. We start with classical arguments showing that
general non-convex problems could not be solved efficiently in a reasonable
time. Then we give a list of problems that can be solved efficiently to find
the global minimizer by exploiting the structure of the problem as much as it
is possible. Another way to deal with non-convexity is to relax the goal from
finding the global minimum to finding a stationary point or a local minimum.
For this setting, we first present known results for the convergence rates of
deterministic first-order methods, which are then followed by a general
theoretical analysis of optimal stochastic and randomized gradient schemes, and
an overview of the stochastic first-order methods. After that, we discuss quite
general classes of non-convex problems, such as minimization of
$\alpha$-weakly-quasi-convex functions and functions that satisfy
Polyak--Lojasiewicz condition, which still allow obtaining theoretical
convergence guarantees of first-order methods. Then we consider higher-order
and zeroth-order/derivative-free methods and their convergence rates for
non-convex optimization problems.
