Boosting Gradient Ascent for Continuous DR-submodular Maximization

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 $\gamma$-weakly DR-submodular, we propose an auxiliary function $F$ whose stationary points can provide a better $(1-e^{-\gamma})$-approximation than the $(\gamma^2/(1+\gamma^2))$-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.

Communication-Efficient Decentralized Online Continuous DR-Submodular Maximization

Maximizing a monotone submodular function is a fundamental task in machine learning, economics, and statistics. In this paper, we present two communication-efficient decentralized online algorithms for the monotone continuous DR-submodular maximization problem, both of which reduce the number of per-function gradient evaluations and per-round communication complexity from $T^{3/2}$ to $1$. The first one, One-shot Decentralized Meta-Frank-Wolfe (Mono-DMFW), achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. As far as we know, this is the first one-shot and projection-free decentralized online algorithm for monotone continuous DR-submodular maximization. Next, inspired by the non-oblivious boosting function \citep{zhang2022boosting}, we propose the Decentralized Online Boosting Gradient Ascent (DOBGA) algorithm, which attains a $(1-1/e)$-regret of $O(\sqrt{T})$. To the best of our knowledge, this is the first result to obtain the optimal $O(\sqrt{T})$ against a $(1-1/e)$-approximation with only one gradient inquiry for each local objective function per step. Finally, various experimental results confirm the effectiveness of the proposed methods.

Online Learning for Non-monotone Submodular Maximization: From Full Information to Bandit Feedback

In this paper, we revisit the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, which finds wide real-world applications in the domain of machine learning, economics, and operations research. At first, we present the Meta-MFW algorithm achieving a $1/e$-regret of $O(\sqrt{T})$ at the cost of $T^{3/2}$ stochastic gradient evaluations per round. As far as we know, Meta-MFW is the first algorithm to obtain $1/e$-regret of $O(\sqrt{T})$ for the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set. Furthermore, in sharp contrast with ODC algorithm \citep{thang2021online}, Meta-MFW relies on the simple online linear oracle without discretization, lifting, or rounding operations. Considering the practical restrictions, we then propose the Mono-MFW algorithm, which reduces the per-function stochastic gradient evaluations from $T^{3/2}$ to 1 and achieves a $1/e$-regret bound of $O(T^{4/5})$. Next, we extend Mono-MFW to the bandit setting and propose the Bandit-MFW algorithm which attains a $1/e$-regret bound of $O(T^{8/9})$. To the best of our knowledge, Mono-MFW and Bandit-MFW are the first sublinear-regret algorithms to explore the one-shot and bandit setting for online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, respectively. Finally, we conduct numerical experiments on both synthetic and real-world datasets to verify the effectiveness of our methods.

MNL-Bandits under Inventory and Limited Switches Constraints

Optimizing the assortment of products to display to customers is a key to increasing revenue for both offline and online retailers. To trade-off between exploring customers' preference and exploiting customers' choices learned from data, in this paper, by adopting the Multi-Nomial Logit (MNL) choice model to capture customers' choices over products, we study the problem of optimizing assortments over a planning horizon $T$ for maximizing the profit of the retailer. To make the problem setting more practical, we consider both the inventory constraint and the limited switches constraint, where the retailer cannot use up the resource inventory before time $T$ and is forbidden to switch the assortment shown to customers too many times. Such a setting suits the case when an online retailer wants to dynamically optimize the assortment selection for a population of customers. We develop an efficient UCB-like algorithm to optimize the assortments while learning customers' choices from data. We prove that our algorithm can achieve a sub-linear regret bound $\tilde{O}\left(T^{1-\alpha/2}\right)$ if $O(T^\alpha)$ switches are allowed. %, and our regret bound is optimal with respect to $T$. Extensive numerical experiments show that our algorithm outperforms baselines and the gap between our algorithm's performance and the theoretical upper bound is small.

Continuous Submodular Maximization: Boosting via Non-oblivious Function

In this paper, we revisit the constrained and stochastic continuous submodular maximization in both offline and online settings. For each $\gamma$-weakly DR-submodular function $f$, we use the factor-revealing optimization equation to derive an optimal auxiliary function $F$, whose stationary points provide a $(1-e^{-\gamma})$-approximation to the global maximum value (denoted as $OPT$) of problem $\max_{\boldsymbol{x}\in\mathcal{C}}f(\boldsymbol{x})$. Naturally, the projected (mirror) gradient ascent relied on this non-oblivious function achieves $(1-e^{-\gamma}-\epsilon^{2})OPT-\epsilon$ after $O(1/\epsilon^{2})$ iterations, beating the traditional $(\frac{\gamma^{2}}{1+\gamma^{2}})$-approximation gradient ascent \citep{hassani2017gradient} for submodular maximization. Similarly, based on $F$, the classical Frank-Wolfe algorithm equipped with variance reduction technique \citep{mokhtari2018conditional} also returns a solution with objective value larger than $(1-e^{-\gamma}-\epsilon^{2})OPT-\epsilon$ after $O(1/\epsilon^{3})$ iterations. In the online setting, we first consider the adversarial delays for stochastic gradient feedback, under which we propose a boosting online gradient algorithm with the same non-oblivious search, achieving a regret of $\sqrt{D}$ (where $D$ is the sum of delays of gradient feedback) against a $(1-e^{-\gamma})$-approximation to the best feasible solution in hindsight. Finally, extensive numerical experiments demonstrate the efficiency of our boosting methods.

Online Allocation with Two-sided Resource Constraints

Motivated by many interesting real-world applications in logistics and online advertising, we consider an online allocation problem subject to lower and upper resource constraints, where the requests arrive sequentially, sampled i.i.d. from an unknown distribution, and we need to promptly make a decision given limited resources and lower bounds requirements. First, with knowledge of the measure of feasibility, i.e., $\alpha$, we propose a new algorithm that obtains $1-O(\frac{\epsilon}{\alpha-\epsilon})$ -competitive ratio for the offline problems that know the entire requests ahead of time. Inspired by the previous studies, this algorithm adopts an innovative technique to dynamically update a threshold price vector for making decisions. Moreover, an optimization method to estimate the optimal measure of feasibility is proposed with theoretical guarantee at the end of this paper. Based on this method, if we tolerate slight violation of the lower bounds constraints with parameter $\eta$, the proposed algorithm is naturally extended to the settings without strong feasible assumption, which cover the significantly unexplored infeasible scenarios.

Towards Better Laplacian Representation in Reinforcement Learning with Generalized Graph Drawing

The Laplacian representation recently gains increasing attention for reinforcement learning as it provides succinct and informative representation for states, by taking the eigenvectors of the Laplacian matrix of the state-transition graph as state embeddings. Such representation captures the geometry of the underlying state space and is beneficial to RL tasks such as option discovery and reward shaping. To approximate the Laplacian representation in large (or even continuous) state spaces, recent works propose to minimize a spectral graph drawing objective, which however has infinitely many global minimizers other than the eigenvectors. As a result, their learned Laplacian representation may differ from the ground truth. To solve this problem, we reformulate the graph drawing objective into a generalized form and derive a new learning objective, which is proved to have eigenvectors as its unique global minimizer. It enables learning high-quality Laplacian representations that faithfully approximate the ground truth. We validate this via comprehensive experiments on a set of gridworld and continuous control environments. Moreover, we show that our learned Laplacian representations lead to more exploratory options and better reward shaping.