Provable Bregman-divergence based Methods for Nonconvex and Non-Lipschitz Problems
This solves an open problem in optimization theory by providing convergence guarantees for alternating minimization methods in non-Lipschitz settings, which is incremental but important for theoretical foundations.
The authors tackled optimization problems that lack Lipschitz smoothness, which is common in machine learning, by developing Bregman-divergence based algorithms that provably converge to second-order stationary points for relatively smooth problems, including nonconvex and non-Lipschitz cases.
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed optimization methods cover both the proximal alternating minimization and the proximal alternating linearized minimization when we specialize the Bregman divergence to the Euclidian distance. Therefore, this work not only develops guaranteed optimization methods for non-Lipschitz smooth problems but also solves an open problem of showing the second-order convergence guarantees for these alternating minimization methods.