Generalized Majorization-Minimization

School of Engineering, Brown University Providence, RI 02912, USA Abstract Non-convex optimization is ubiquitous in machine learning. The bound at each iteration is required to touch the objective function at the optimizer of the previous bound. We show that this touching constraint is unnecessary and overly restrictive. We generalize MM by relaxing this constraint, and propose a new optimization framework, named Generalized Majorization-Minimization (G-MM) that is more flexible compared to MM. For instance, it can incorporate application-specific biases into the optimization procedure without changing the objective function. We derive G-MM algorithms for several latent variable models and show empirically that they consistently outperform their MM counterparts in optimizing non-convex objectives. In particular, G-MM algorithms appear to be less sensitive to initialization. Keywords: majorization-minimization, non-convex optimization, latent variable models, expectation maximization 1. Introduction Non-convex optimization is ubiquitous in machine learning. Majorization-Minimization (MM) (Hunter et al., 2000) is an optimization framework for designing well-behaved optimization algorithms for non-convex functions. MM algorithms work by iteratively optimizing a sequence of easy-to-optimize surrogate functions that bound the objective. Two of the most successful instances of MM algorithms are Expectation-Maximization (EM) (Dempster et al., 1977) and the Concave-Convex Proce-1 arXiv:1506.07613v2 However, both have a number of drawbacks in practice, such as sensitivity to initialization and lack of uncertainty modeling for latent variables. This has been noted in works such as (Neal and Hinton, 1998; Felzenszwalb et al., 2010; Parizi et al., 2012; Kumar et al., 2012; Ping et al., 2014). We propose a new procedure, Generalized Majorization-Minimization (G-MM), for optimizing non-convex objective functions. Our approach is inspired by MM, but we generalize the bound construction process.