Submodular Optimization under Noise

We consider the problem of maximizing a monotone submodular function under noise. There has been a great deal of work on optimization of submodular functions under various constraints, resulting in algorithms that provide desirable approximation guarantees. In many applications, however, 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 a cardinality constraint when given access to a noisy oracle of the function. We show that:

• For a cardinality constraint $k\ge 2$, there is an approximation algorithm whose approximation ratio is arbitrarily close to $1-1\mathrm{/}e$;
• For $k=1$ there is an algorithm whose approximation ratio is arbitrarily close to $1\mathrm{/}2$. No randomized algorithm can obtain an approximation ratio better than $1\mathrm{/}2+o(1)$; -If the noise is adversarial, no non-trivial approximation guarantee can be obtained.