Inertial Block Proximal Methods for Non-Convex Non-Smooth Optimization
This work addresses optimization challenges in machine learning and data analysis, particularly for non-negative factorizations, but is incremental as it builds on existing accelerated first-order methods.
The paper tackles non-convex non-smooth optimization problems by proposing inertial block coordinate descent methods, which offer advantages like dual extrapolation points and random block updates, and demonstrates competitive performance in non-negative matrix factorization and tensor factorization tasks.
We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the gradients and to add the inertial force (we will empirically show that it is more efficient than using a single extrapolation point), (2) they allow to randomly picking the block of variables to update, and (3) they do not require a restarting step. We prove the subsequential convergence of the generated sequence under mild assumptions, prove the global convergence under some additional assumptions, and provide convergence rates. We deploy the proposed methods to solve non-negative matrix factorization (NMF) and show that they compete favorably with the state-of-the-art NMF algorithms. Additional experiments on non-negative approximate canonical polyadic decomposition, also known as non-negative tensor factorization, are also provided.