The paper introduces a versatile adaptive sampling procedure called Confident Sample (CS) that can efficiently determine whether the marginal gain of a submodular function is approximately above or below a given threshold, using as few noisy samples as possible.
The authors then leverage CS as a subroutine to develop efficient algorithms for various submodular maximization problems under the noisy setting:
For Monotone Submodular Maximization with Cardinality constraint (MSMC), the ConfThreshGreedy (CTG) algorithm is proposed, which achieves an approximation ratio arbitrarily close to 1-1/e with high probability.
For Unconstrained Submodular Maximization (USM), the Confident Double Greedy (CDG) algorithm is introduced, which achieves an approximation ratio arbitrarily close to 1/3 with high probability.
For Monotone Submodular Maximization with Matroid constraint (MSMM), the ConfContinuousThreshGreedy (CCTG) algorithm is proposed, which accesses the multilinear extension of the submodular function via noisy samples and achieves an approximation ratio arbitrarily close to 1-1/e with high probability.
The proposed algorithms demonstrate improved sample complexity compared to standard approaches that rely on fixed-precision approximations of the objective function.
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