  This paper studies a novel algorithm for nonconvex composite minimization
which can be interpreted in terms of dual space nonlinear preconditioning for
the classical proximal gradient method. The proposed scheme can be applied to
additive composite minimization problems whose smooth part exhibits an
anisotropic descent inequality relative to a reference function. It is proved
that the anisotropic descent property is closed under pointwise average if the
dual Bregman distance is jointly convex and, more specifically, closed under
pointwise conic combinations for the KL-divergence. We analyze the method's
asymptotic convergence and prove its linear convergence under an anisotropic
proximal gradient dominance condition. Applications are discussed including
exponentially regularized LPs and logistic regression with nonsmooth
regularization. In numerical experiments we show significant improvements of
the proposed method over its Euclidean counterparts.
