Decentralized Projection-free Online Upper-Linearizable Optimization with Applications to DR-Submodular Optimization
Yiyang Lu, Mohammad Pedramfar, Vaneet Aggarwal
TL;DR
The paper develops a decentralized, projection-free framework for online optimization of upper-linearizable functions, unifying and extending DR-submodular and concave settings to general convex domains. It introduces the DROCULO algorithm, a block-structured method that uses a linearizable oracle and infeasible projection via an LOO, achieving a tunable regret-communication trade-off: for any θ in [0,1], E[R_α^i] = O(T^{1-θ/2}) with communication O(T^θ) and LOO calls O(T^{2θ}). The authors further adapt the framework to semi-bandit, zeroth-order, and bandit feedback across monotone and non-monotone up-concave subclasses (A1–A3), yielding new first-known guarantees in decentralized settings. These results show that projection-free decentralized optimization can handle broad function classes and heterogeneous feedback with competitive regret, highlighting practical utility in dynamic networks. The work also suggests avenues for empirical validation and potential extensions to more complex network dynamics and constraint sets.
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 $O(T^{1-θ/2})$ with communication complexity of $O(T^θ)$ and number of linear optimization oracle calls of $O(T^{2θ})$ for decentralized upper-linearizable function optimization, for any $0\le θ\le 1$. 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.