Markov chain stochastic DCA and applications in deep learning with PDEs regularization.

This paper addresses a large class of nonsmooth nonconvex stochastic DC (difference-of-convex functions) programs where endogenous uncertainty is involved and i.i.d. (independent and identically distributed) samples are not available. Instead, we assume that it is only possible to access Markov chains whose sequences of distributions converge to the target distributions. This setting is legitimate as Markovian noise arises in many contexts including Bayesian inference, reinforcement learning, and stochastic optimization in high-dimensional or combinatorial spaces. We then design a stochastic algorithm named Markov chain stochastic DCA (MCSDCA) based on DCA (DC algorithm) - a well-known method for nonconvex optimization. We establish the convergence analysis in both asymptotic and nonasymptotic senses. The MCSDCA is then applied to deep learning via PDEs (partial differential equations) regularization, where two realizations of MCSDCA are constructed, namely MCSDCA-odLD and MCSDCA-udLD, based on overdamped and underdamped Langevin dynamics, respectively. Numerical experiments on time series prediction and image classification problems with a variety of neural network topologies show the merits of the proposed methods.

Online Stochastic DCA With Applications to Principal Component Analysis.

Stochastic algorithms are well-known for their performance in the era of big data. In this article, we study nonsmooth stochastic Difference-of-Convex functions (DC) programs-the major class of nonconvex stochastic optimization, which have a variety of applications in divers domains, in particular, machine learning. We propose new online stochastic algorithms based on the state-of-the-art DC Algorithm (DCA)-a powerful approach in nonconvex programming framework, in the online context of streaming data continuously generated by some (unknown) source distribution. The new schemes use the stochastic approximations (SAs) principle: deterministic quantities of the standard DCA are replaced by their noisy estimators constructed using newly arriving samples. The convergence analysis of the proposed algorithms is studied intensively with the help of tools from modern convex analysis and martingale theory. Finally, we study several aspects of the proposed algorithms on an important problem in machine learning: the expected problem in principal component analysis (PCA).