Block Majorization Minimization with Extrapolation and Application to $β$-NMF
This work addresses optimization challenges in machine learning for researchers and practitioners, but it appears incremental as it builds on existing block majorization minimization methods.
The authors tackled the problem of solving multi-convex optimization problems, particularly nonnegative matrix factorization with β-divergences, by proposing a Block Majorization Minimization method with Extrapolation (BMMe) and an adaptive update rule, resulting in empirically significant acceleration.
We propose a Block Majorization Minimization method with Extrapolation (BMMe) for solving a class of multi-convex optimization problems. The extrapolation parameters of BMMe are updated using a novel adaptive update rule. By showing that block majorization minimization can be reformulated as a block mirror descent method, with the Bregman divergence adaptively updated at each iteration, we establish subsequential convergence for BMMe. We use this method to design efficient algorithms to tackle nonnegative matrix factorization problems with the $β$-divergences ($β$-NMF) for $β\in [1,2]$. These algorithms, which are multiplicative updates with extrapolation, benefit from our novel results that offer convergence guarantees. We also empirically illustrate the significant acceleration of BMMe for $β$-NMF through extensive experiments.