The Multi-Block DC Function Class: Theory, Algorithms, and Applications

TL;DR

This paper introduces the Multi-Block DC class, a structured nonconvex function class with efficient decomposition and algorithms, enabling new applications like deep ReLU networks.

Contribution

It defines the Multi-Block DC class, proves its superiority over traditional DC formulations in complexity and ease of construction, and develops algorithms with convergence guarantees.

Findings

Multi-Block DC formulations are polynomial in size for models with exponential DC decompositions.

Explicit BDC formulations are provided for deep ReLU networks.

Algorithms with non-asymptotic convergence are developed for both batch and stochastic settings.

Abstract

We present the Multi-Block DC (BDC) class, a rich class of structured nonconvex functions that admit a DC ("difference-of-convex") decomposition across parameter blocks. This multi-block class not only subsumes the usual DC programming, but also 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 demonstrate the broad applicability of our method through several applications.