Boosting Gradient Ascent for Continuous DR-submodular Maximization
Qixin Zhang, Zongqi Wan, Zengde Deng, Zaiyi Chen, Xiaoming Sun, Jialin Zhang, Yu Yang
TL;DR
The paper addresses the suboptimal approximation of projected gradient ascent for continuous DR-submodular maximization by introducing a non-oblivious auxiliary function F that reweights gradient information. By solving a factor-revealing optimization, the authors derive an optimal weight function, yielding stationary points of F with a (1−e^{−γ})-approximation for monotone γ-weak DR-submodular objectives and a (1−∥x̲∥∞)/4-approximation for non-monotone objectives, thereby boosting PGA in offline, online, bandit, and minimax contexts. They develop unbiased gradient estimators for F, extend the boosting framework to online delayed and bandit feedback, and prove regret guarantees that match or improve existing bounds, along with empirical demonstrations on four problem families. The approach provides a scalable, theory-backed method to achieve tight approximation ratios in diverse settings, offering practical impact for large-scale continuous submodular optimization. Overall, the work unifies and advances boosting techniques for continuous submodular maximization, with broad applicability and provable performance improvements.
Abstract
Projected Gradient Ascent (PGA) is the most commonly used optimization scheme in machine learning and operations research areas. Nevertheless, numerous studies and examples have shown that the PGA methods may fail to achieve the tight approximation ratio for continuous DR-submodular maximization problems. To address this challenge, we present a boosting technique in this paper, which can efficiently improve the approximation guarantee of the standard PGA to \emph{optimal} with only small modifications on the objective function. The fundamental idea of our boosting technique is to exploit non-oblivious search to derive a novel auxiliary function $F$, whose stationary points are excellent approximations to the global maximum of the original DR-submodular objective $f$. Specifically, when $f$ is monotone and $γ$-weakly DR-submodular, we propose an auxiliary function $F$ whose stationary points can provide a better $(1-e^{-γ})$-approximation than the $(γ^2/(1+γ^2))$-approximation guaranteed by the stationary points of $f$ itself. Similarly, for the non-monotone case, we devise another auxiliary function $F$ whose stationary points can achieve an optimal $\frac{1-\min_{\boldsymbol{x}\in\mathcal{C}}\|\boldsymbol{x}\|_{\infty}}{4}$-approximation guarantee where $\mathcal{C}$ is a convex constraint set. In contrast, the stationary points of the original non-monotone DR-submodular function can be arbitrarily bad~\citep{chen2023continuous}. Furthermore, we demonstrate the scalability of our boosting technique on four problems. In all of these four problems, our resulting variants of boosting PGA algorithm beat the previous standard PGA in several aspects such as approximation ratio and efficiency. Finally, we corroborate our theoretical findings with numerical experiments, which demonstrate the effectiveness of our boosting PGA methods.