Title: The Multi-Block DC Function Class: Theory, Algorithms, and Applications
Primary area: optimization
Keywords: Multi-block DC, DC programming, non-convex optimization

Abstract: We present the Multi-Block DC (BDC) class, a broad class of structured nonconvex functions that admit a DC (“difference-of-convex”) decomposition across parameter blocks. This block structure not only subsumes the usual DC programming, it turns out to be provably more powerful. Specifically, we demonstrate how standard models (e.g., polynomials and tensor factorization) must have DC decompositions of exponential size, while their BDC formulation is polynomial. This separation in complexity also underscores another key aspect: unlike DC formulations, obtaining BDC formulations for problems is vastly easier and constructive. We illustrate this aspect by presenting explicit BDC formulations for modern tasks such as deep ReLU networks, a result with no known equivalent in the DC class. Moreover, we complement the theory by developing algorithms with non-asymptotic convergence theory, including both batch and stochastic settings, and illustrate their empirical performance through several experiments.

TL;DR: This work proposes and studies the broad class of multi-block DC functions with many applications.