Submodular Optimization under Noise

We consider the problem of maximizing monotone submodular functions under noise, which to the best of our knowledge has not been studied in the past. There has been a great deal of work on optimization of submodular functions under various constraints, with many algorithms that provide desirable approximation guarantees. However, in many applications we do not have access to the submodular function we aim to optimize, but rather to some erroneous or noisy version of it. This raises the question of whether provable guarantees are obtainable in presence of error and noise. We provide initial answers, by focusing on the question of maximizing a monotone submodular function under cardinality constraints when given access to a noisy oracle of the function. We show that: For a cardinality constraint $k \geq 2$, there is an approximation algorithm whose approximation ratio is arbitrarily close to $1-1/e$; For $k=1$ there is an approximation algorithm whose approximation ratio is arbitrarily close to $1/2$ in expectation. No randomized algorithm can obtain an approximation ratio in expectation better than $1/2+O(1/\sqrt n)$ and $(2k - 1)/2k + O(1/\sqrt{n})$ for general $k$; If the noise is adversarial, no non-trivial approximation guarantee can be obtained.