Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization

Abstract

This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm. Funding: This work was supported by Norway Grants 2014-2021 [Grant ELO-Hyp 24/2020]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii [Grants PN-III-P4-PCE-2021-0720, L2O-MOC, nr 70/2022]; and the ITN-ETN project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement [Grant 861137].