Unified Projection-Free Algorithms for Adversarial DR-Submodular Optimization

TL;DR

This work tackles online adversarial optimization of continuous DR-submodular functions under projection-free constraints. It introduces a unified Frank-Wolfe-based framework that covers monotone and non-monotone objectives, full-information, semi-bandit, and bandit feedback, across downward-closed and general convex feasible sets. The approach hinges on four technical ideas—offline bounds, meta-actions, random permutations, and smoothing with a shrunk constraint set—that yield sublinear $α$-regret in many settings, including several first-of-its-kind projected-free results for monotone DR-submodular maximization. The framework also provides explicit regret and query-complexity trade-offs, remains projection-free (solving only linear subproblems), and extends to noisy feedback without variance reduction techniques. Empirical evaluations confirm favorable regret rates and computational efficiency, highlighting the method’s practical applicability to large-scale online DR-submodular problems.

Abstract

This paper introduces unified projection-free Frank-Wolfe type algorithms for adversarial continuous DR-submodular optimization, spanning scenarios such as full information and (semi-)bandit feedback, monotone and non-monotone functions, different constraints, and types of stochastic queries. For every problem considered in the non-monotone setting, the proposed algorithms are either the first with proven sub-linear $α$-regret bounds or have better $α$-regret bounds than the state of the art, where $α$ is a corresponding approximation bound in the offline setting. In the monotone setting, the proposed approach gives state-of-the-art sub-linear $α$-regret bounds among projection-free algorithms in 7 of the 8 considered cases while matching the result of the remaining case. Additionally, this paper addresses semi-bandit and bandit feedback for adversarial DR-submodular optimization, advancing the understanding of this optimization area.

Figures (4)

• Figure 1: $\frac{1}{e}$-regret bounds for non-monotone functions with a downward-closed feasible region and full-information (noisy) gradient feedback, as a function of query-complexity. ODC is from thang21_onlin_non_monot_dr_maxim. Meta and Mono are from zhang23_onlin_learn_non_submod_maxim. Better performance corresponds to the bottom left corner. Our algorithm's regret bounds dominate the state of the art.
• Figure 2: Empirical regret plots for the experiments. The top row depicts time-averaged regret for each round $t$ for a horizon of $T=500$. The bottom row depicts cumulative regret for multiple horizons with logarithmic scaling. Grey lines in the bottom row represent $y = aT^{1/2}$ curves for different $a$ for visual reference. Colors correspond to regret bounds (i.e. black for $\tilde{O}(T^{1/2})$). Our methods (solid lines) use significantly fewer queries and less computation than baselines (dashed and dotted lines) with similar regret bounds and achieve better regret than baselines using similar numbers of queries and computation.
• Figure : Generalized Meta-Frank-Wolfe
• Figure : Generalized (Semi-)Bandit-Frank-Wolfe

Theorems & Definitions (19)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Definition 1: Smoothing Trick
• Theorem 1
• Theorem 2
• Lemma 1: Lemma 11 of pedramfar23_unified_approac_maxim_contin_dr_funct
• Lemma 2: Lemma 13 in pedramfar23_unified_approac_maxim_contin_dr_funct
• Lemma 3: Lemma 14 in pedramfar23_unified_approac_maxim_contin_dr_funct