Multi-Agent Distributed Optimization With Feasible Set Privacy

We consider the problem of decentralized constrained optimization with multiple agents $E_1,\ldots,E_N$ who jointly wish to learn the optimal solution set while keeping their feasible sets $\mathcal{P}_1,\ldots,\mathcal{P}_N$ private from each other. We assume that the objective function $f$ is known to all agents and each feasible set is a collection of points from a universal alphabet $\mathcal{P}_{alph}$. A designated agent (leader) starts the communication with the remaining (non-leader) agents, and is the first to retrieve the solution set. The leader searches for the solution by sending queries to and receiving answers from the non-leaders, such that the information on the individual feasible sets revealed to the leader should be no more than nominal, i.e., what is revealed from learning the solution set alone. We develop achievable schemes for obtaining the solution set at nominal information leakage, and characterize their communication costs under two communication setups between agents. In this work, we focus on two kinds of network setups: i) ring, where each agent communicates with two adjacent agents, and ii) star, where only the leader communicates with the remaining agents. We show that, if the leader first learns the joint feasible set through an existing private set intersection (PSI) protocol and then deduces the solution set, the information leaked to the leader is greater than nominal. Moreover, we draw connection of our schemes to threshold PSI (ThPSI), which is a PSI-variant where the intersection is revealed only when its cardinality is larger than a threshold value. Finally, for various realizations of $f$ mapped uniformly at random to a fixed range of values, our schemes are more communication-efficient with a high probability compared to retrieving the entire feasible set through PSI.