Decentralized Projection-free Online Upper-Linearizable Optimization with Applications to DR-Submodular Optimization

TL;DR

This paper presents a new decentralized projection-free optimization framework for upper-linearizable functions, achieving sublinear regret and communication complexity, applicable to various feedback settings and general convex constraints.

Contribution

It introduces a novel decentralized projection-free method for upper-linearizable functions, extending to non-monotone cases and various feedback types, with theoretical regret and complexity guarantees.

Findings

Achieves regret of O(T^{1-θ/2}) with communication O(T^θ) and oracle calls O(T^{2θ})

First to handle monotone and non-monotone up-concave optimization with convex constraints

Extends results to zeroth order, semi-bandit, and bandit feedback settings

Abstract

We introduce a novel framework for decentralized projection-free optimization, extending projection-free methods to a broader class of upper-linearizable functions. Our approach leverages decentralized optimization techniques with the flexibility of upper-linearizable function frameworks, effectively generalizing traditional DR-submodular function optimization. We obtain the regret of $O(T^{1-\theta/2})$ with communication complexity of $O(T^{\theta})$ and number of linear optimization oracle calls of $O(T^{2\theta})$ for decentralized upper-linearizable function optimization, for any $0\le \theta \le 1$. This approach allows for the first results for monotone up-concave optimization with general convex constraints and non-monotone up-concave optimization with general convex constraints. Further, the above results for first order feedback are extended to zeroth order, semi-bandit, and bandit feedback.