Distributed submodular maximization: trading performance for privacy

This paper considers a multi-agent submodular set function maximization problem subject to partition matroid in which the utility is shared, but the agents’ policy choices are constrained locally. The paper’s main contribution is a distributed algorithm that enables each agent to find a suboptimal policy locally with a guaranteed level of privacy. The submodular set function maximization problems are NP-hard. For agents communicating over a connected graph, this paper proposes a polynomial-time distributed algorithm to obtain a guaranteed near optimal solution. The proposed algorithm is based on a distributed randomized gradient ascent scheme on the multilinear extension of the submodular set function in the continuous domain. Our next contribution is the design of a distributed rounding algorithm that does not need any inter-agent communication. We base our algorithm’s privacy preservation characteristic on our proposed stochastic rounding method and tie the level of privacy to the variable γ ∈ [0, 1]. That is, the policy choice of an agent can be determined with the probability of at most γ. We show that our distributed algorithm results in a strategy set that when the team’s objective function is evaluated in the worst case, the objective function value is in 1 − (1/e)
^h(γ)
− O(1/T ) of the optimal solution, highlighting the interplay between level of optimality gap and guaranteed level of privacy where T is the number of communication rounds between the agents.