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# Non-convex Conditional Gradient Sliding

*Proceedings of the 35th International Conference on Machine Learning*, PMLR 80:4208-4217, 2018.

#### Abstract

We investigate a projection free optimization method, namely non-convex conditional gradient sliding (NCGS) for non-convex optimization problems on the batch, stochastic and finite-sum settings. Conditional gradient sliding (CGS) method, by integrating Nesterov’s accelerated gradient method with Frank-Wolfe (FW) method in a smart way, outperforms FW for convex optimization, by reducing the amount of gradient computations. However, the study of CGS in the non-convex setting is limited. In this paper, we propose the non-convex conditional gradient sliding (NCGS) methods and analyze their convergence properties. We also leverage the idea of variance reduction from the recent progress in convex optimization to obtain a new algorithm termed

*variance reduced NCGS*(NCGS-VR), and obtain faster convergence rate than the batch NCGS in the finite-sum setting. We show that NCGS algorithms outperform their Frank-Wolfe counterparts both in theory and in practice, for all three settings, namely the batch, stochastic and finite-sum setting. This significantly improves our understanding of optimizing non-convex functions with complicated feasible sets (where projection is prohibitively expensive).