  In this paper, we study a family of non-convex and possibly non-smooth
inf-projection minimization problems, where the target objective function is
equal to minimization of a joint function over another variable. This problem
include difference of convex (DC) functions and a family of bi-convex functions
as special cases. We develop stochastic algorithms and establish their
first-order convergence for finding a (nearly) stationary solution of the
target non-convex function under different conditions of the component
functions. To the best of our knowledge, this is the first work that
comprehensively studies stochastic optimization of non-convex inf-projection
minimization problems with provable convergence guarantee. Our algorithms
enable efficient stochastic optimization of a family of non-decomposable DC
functions and a family of bi-convex functions. To demonstrate the power of the
proposed algorithms we consider an important application in variance-based
regularization. Experiments verify the effectiveness of our inf-projection
based formulation and the proposed stochastic algorithm in comparison with
previous stochastic algorithms based on the min-max formulation for achieving
the same effect.
