Differentiable Greedy Submodular Maximization: Guarantees, Gradient Estimators, and Applications
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We consider making outputs of the greedy algorithm for monotone submodular function maximization differentiable w.r.t. parameters of objective functions; this is motivated by many applications, e.g., sensitivity analysis and end-to-end learning. Our contribution is a theoretically guaranteed and widely applicable smoothing framework based on randomization. We prove that our smoothed greedy algorithm almost recovers original approximation guarantees in expectation for the cases of cardinality and κ-extensible system constrains. We also show how to efficiently compute unbiased gradient estimators of any expected output-dependent quantities by sampling outputs. We demonstrate the utility and effectiveness of our framework by applying it to various situations including the aforementioned ones.
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