$\gamma$-weakly $\theta$-up-concavity: A Unified Framework for Non-Convex Optimization Beyond DR-Submodular and OSS Functions

TL;DR

This paper introduces a new unifying framework called $b3$-weakly $b8$-up-concavity that generalizes many non-convex functions, enabling unified approximation guarantees for optimization problems in machine learning and combinatorics.

Contribution

It proposes a novel first-order condition that encompasses various function classes and provides a unified approach to approximation guarantees in both offline and online optimization.

Findings

Generalizes DR-submodular and OSS functions under a single framework.

Establishes upper-linearizability with explicit approximation bounds.

Improves approximation coefficients for OSS optimization over matroid constraints.

Abstract

Optimizing non-convex functions is a fundamental challenge across machine learning and combinatorial optimization. We introduce and study $\gamma$-weakly $\theta$-up-concavity, a novel first-order condition that characterizes a broad class of such functions. This condition provides a powerful unifying framework, strictly generalizing both DR-submodular and One-Sided Smooth (OSS) functions while capturing broader forms of scale-dependent curvature, including accumulating-then-diminishing returns and flat-start behavior. Our central theoretical contribution demonstrates that $\gamma$-weakly $\theta$-up-concave functions are upper-linearizable: for any feasible point, we can construct a linear surrogate whose gains provably approximate the original non-linear objective. A key technical contribution is a nonuniform upper-linearization argument yielding approximation coefficients that depend explicitly on the curvature parameters and the geometry of the feasible region. This linearizability yields immediate and unified approximation guarantees for a wide range of problems. Specifically, we obtain unified approximation guarantees for offline optimization as well as static and dynamic regret bounds in online settings via standard reductions to linear optimization. Moreover, our framework recovers the optimal approximation coefficient for DR-submodular maximization and improves existing approximation coefficients for OSS optimization, particularly over matroid constraints.