Online Two-Stage Submodular Maximization

TL;DR

The paper tackles Online Two-Stage Submodular Maximization (O2SSM), where online revelation of submodular second-stage objectives complicates first-stage restriction design. It introduces RAOCO-, an online algorithm that leverages Online Convex Optimization on a concave relaxation of the first-stage objective and Randomized Pipage Rounding to produce integral restricted ground sets, achieving sublinear regret that scales with the second-stage matroid structure. For general matroids, the algorithm attains a $(1-1/e)^2$-regret bound, while for uniform matroids of rank $k$ it improves to $(1-1/e)(1-e^{-k}k^k/k!)$, matching state-of-the-art offline guarantees in certain regimes. The approach combines a fractional relaxation with a rounding scheme that preserves expected value, underpinned by Contention Resolution Schemes adapted to dependent rounding, and is supported by experiments on real and synthetic datasets showing robust performance gains over one-stage online baselines.

Abstract

Given a collection of monotone submodular functions, the goal of Two-Stage Submodular Maximization (2SSM) [Balkanski et al., 2016] is to restrict the ground set so an objective selected u.a.r. from the collection attains a high maximal value, on average, when optimized over the restricted ground set. We introduce the Online Two-Stage Submodular Maximization (O2SSM) problem, in which the submodular objectives are revealed in an online fashion. We study this problem for weighted threshold potential functions, a large and important subclass of monotone submodular functions that includes influence maximization, data summarization, and facility location, to name a few. We design an algorithm that achieves sublinear $(1 - 1/e)^2$-regret under general matroid constraints and $(1 - 1/e)(1-e^{-k}k^k/k!)$-regret in the case of uniform matroids of rank $k$; the latter also yields a state-of-the-art bound for the (offline) 2SSM problem. We empirically validate the performance of our online algorithm with experiments on real datasets.

Figures (5)

• Figure 1: Cumulative average reward $C_t$ of RAOCO variants and baselines for each dataset. We report the average value of $C_t$ over 5 runs. The shaded regions correspond to one standard deviation above and below the average. Note that OFLN-RGR coincides with OPT in all datasets.
• Figure 2: $k$ vs $C_T$ tradeoff for RAOCOOGA-OGA and One-stage-OGA, where $T$ is defined for each dataset at Table \ref{['tab:datasets']}. We report the average value of $C_T$ over $5$ runs. The shaded regions correspond to one standard deviation above and below the average.
• Figure 3: $\ell$ vs $C_T$ tradeoff for RAOCOOGA-OGA and One-stage-OGA, where $T$ is defined for each dataset at Table \ref{['tab:datasets']}. We report the average value of $C_T$ over $5$ runs. The shaded regions correspond to one standard deviation above and below the average.
• Figure 4: Cummulative reward of RAOCOOGA-OGA, Random and OPT on HotpotQA
• Figure 5: Comparison of approximation guarantees as a function of $k$ for the (offline) 2SSM. The guarantee of both our algorithm and the one by balkanski2016learning converge to the optimal value $1 - 1/e$ as $k \to \infty$, but our algorithm achieves strictly better guarantees even for finite $k\in \mathbb N$. Notably, our guarantee for cardinality constraints also outperforms the state-of-the-art bound for matroid constraints from krause2017 for all $k \geq 2$. For general matroids, our guarantee for 2SSM ($(1-1/e)^2\approx0.40$) is slightly worse than the one by krause2017 ($1/2(1-1/e^2)\approx 0.43$).

Theorems & Definitions (44)

• Theorem 3.1
• Theorem 5.1
• Lemma 5.1
• Lemma 5.1
• Lemma 5.1
• proof : Proof of Theorem \ref{['thm:alg-c-regret']}
• Definition B.1: Contention Resolution Scheme (CRS)
• Definition B.2: $\gamma$-selectable CRS
• Definition B.3: Submodular dominance
• Lemma B.4: chekuri_vondrak_crs