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Online Two-Stage Submodular Maximization
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Introduces online two-stage submodular maximization (O2SSM) for weighted threshold potential objectives and gives RAOCO, an OCO-plus-pipage-rounding algorithm with sublinear alpha-regret under matroid and uniform matroid constraints.
Problem
Online version of two-stage submodular maximization, where monotone submodular objectives are revealed sequentially and a restricted ground set must be chosen online without oracle access to the (NP-hard, non-submodular) first-stage objective.
Approach
Relax the first-stage reward to a concave function over fractional ground sets for WTP objectives, run standard OCO (FTRL/OGA) on it, and round the fractional ground set with randomized pipage rounding whose output is analyzed via a contention-resolution argument exploiting submodular dominance.
Key results
- RAOCO achieves sublinear (1-1/e)^2-regret under general matroid constraints and (1-1/e)(1-e^{-k}k^k/k!)-regret under uniform matroids, the latter also improving the offline 2SSM approximation for WTP objectives.
- On the Coverage dataset, RAOCO variants achieve almost double the average reward of 1S-OGA.
- Two-stage RAOCO variants outperform or match the one-stage baseline 1S-OGA on all datasets, with largest gaps on Influence, Images, and Coverage.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use two-stage RAOCO (OGA, FTRL-L2, or FTRL-H) instead of one-stage OGA (1S-OGA) | O2SSM · All datasets except HotpotQA | cumulative average reward | Mixed |
| Use two-stage RAOCO (OGA, FTRL-L2, or FTRL-H) instead of one-stage OGA (1S-OGA) | O2SSM · Coverage | cumulative average reward | Helped |
Limitations
- Analysis restricted to WTP objectives
- Assumes full-information feedback
- Regret bounds are alpha-regret relative to an approximate offline solution rather than the optimum
Experimental setup
Datasets: Wikipedia Images MovieRec Influence HotpotQA TeamFormation Coverage
Benchmarks: Wikipedia Images MovieRec Influence HotpotQA TeamFormation Coverage
Metrics: cumulative average reward C_t alpha-regret
Baselines: Random One-stage OGA (1S-OGA) Continuous-Optimization (offline) Replacement-Greedy (offline) OPT in hindsight
Builds on: Two-stage submodular maximization (Balkanski et al. 2016) Weighted threshold potential functions Randomized pipage rounding Online gradient ascent Follow the regularized leader
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