From Linear to Linearizable Optimization: A Novel Framework with Applications to Stationary and Non-stationary DR-submodular Optimization

TL;DR

This paper introduces a new class of functions called upper-linearizable/quadratizable, along with a meta-algorithm that transforms existing algorithms for linear/quadratic maximization into ones suitable for these functions, advancing optimization techniques for DR-submodular problems.

Contribution

The paper develops a unified framework for optimizing upper-linearizable/quadratizable functions, extending to various feedback models and improving regret guarantees in DR-submodular maximization.

Findings

New class of upper-linearizable/quadratizable functions introduced.

Meta-algorithm for converting linear/quadratic maximization algorithms.

Improved dynamic and adaptive regret guarantees for DR-submodular maximization.

Abstract

This paper introduces the notion of upper-linearizable/quadratizable functions, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different convex sets. A general meta-algorithm is devised to convert algorithms for linear/quadratic maximization into ones that optimize upper-linearizable/quadratizable functions, offering a unified approach to tackling concave and DR-submodular optimization problems. The paper extends these results to multiple feedback settings, facilitating conversions between semi-bandit/first-order feedback and bandit/zeroth-order feedback, as well as between first/zeroth-order feedback and semi-bandit/bandit feedback. Leveraging this framework, new algorithms are derived using existing results as base algorithms for convex optimization, improving upon state-of-the-art results in various cases. Dynamic and adaptive regret guarantees are obtained for DR-submodular maximization, marking the first algorithms to achieve such guarantees in these settings. Notably, the paper achieves these advancements with fewer assumptions compared to existing state-of-the-art results, underscoring its broad applicability and theoretical contributions to non-convex optimization.