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Multi-Agent Distributed Optimization With Feasible Set Privacy

Shreya Meel, Sennur Ulukus

TL;DR

This work introduces distributed optimization with feasible set privacy (DOFSP), where multiple agents seek the optimal intersection of private feasible sets without leaking sensitive constraints. It builds a privacy-preserving framework based on private set operations (PSI, CarPSI, FindPSI, ThPSI) and non-colluding databases, and analyzes two network topologies (ring and star) to derive achievable schemes with explicit upload/download costs. For the two-agent case, the authors present a concrete scheme that sequentially tests equi-cost subsets via CarPSI and FindPSI, guaranteeing nominal information leakage and comparing favorably to naive PSI-first approaches; for $N>2$, they extend the primitives to ring and star settings, providing cost expressions and privacy proofs. The results demonstrate that DOFSP can substantially reduce communication overhead while preserving information-theoretic privacy of private feasible sets, with implications for privacy-sensitive distributed systems such as resource allocation and cyber-physical networks. The framework also highlights connections to threshold PSI and suggests directions for resilience against collusion and more general topologies.

Abstract

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.

Multi-Agent Distributed Optimization With Feasible Set Privacy

TL;DR

This work introduces distributed optimization with feasible set privacy (DOFSP), where multiple agents seek the optimal intersection of private feasible sets without leaking sensitive constraints. It builds a privacy-preserving framework based on private set operations (PSI, CarPSI, FindPSI, ThPSI) and non-colluding databases, and analyzes two network topologies (ring and star) to derive achievable schemes with explicit upload/download costs. For the two-agent case, the authors present a concrete scheme that sequentially tests equi-cost subsets via CarPSI and FindPSI, guaranteeing nominal information leakage and comparing favorably to naive PSI-first approaches; for , they extend the primitives to ring and star settings, providing cost expressions and privacy proofs. The results demonstrate that DOFSP can substantially reduce communication overhead while preserving information-theoretic privacy of private feasible sets, with implications for privacy-sensitive distributed systems such as resource allocation and cyber-physical networks. The framework also highlights connections to threshold PSI and suggests directions for resilience against collusion and more general topologies.

Abstract

We consider the problem of decentralized constrained optimization with multiple agents who jointly wish to learn the optimal solution set while keeping their feasible sets private from each other. We assume that the objective function is known to all agents and each feasible set is a collection of points from a universal alphabet . 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 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.

Paper Structure

This paper contains 35 sections, 3 theorems, 86 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. General Problem Setting
  4. Ring Setup
  5. Star Setup
  6. Main Results
  7. The Case of $N=2$ Entities
  8. Proposed Scheme
  9. CarPSI($X_{\mathcal{J}_r}, N_2$)
  10. FindPSI($\mathcal{J}_r,M_{i_r},N_2$)
  11. Main Algorithm
  12. Proof of Privacy
  13. Communication Cost
  14. Common Randomness Cost
  15. Comparison with the Naive Scheme
  16. ...and 20 more sections

Key Result

Theorem 1

For a $N=2$ agent DOFSP problem, given the feasible sets $\mathcal{P}_1$ and $\mathcal{P}_2$, the worst-case download cost $D$ depends on the position $R$ of the best function value in $X_1$ and is given by, where $X_{\mathcal{J}_r}=\sum_{j\in\mathcal{J}_r }{e}_j$ and ${e}_j$ is the $K$-length unit column vector with $1$ at the $j$th position. The upload cost $U$ for the above scheme is $KD$ symb

Figures (4)

  • Figure 1: Feasible set for a distributed optimization problem with four agents each with a linear inequality constraint.
  • Figure 2: Two types of communication setups for $N=4$ entities. Leader entity $E_1$ has a single database, each non-leader entity, $E_2$, $E_3$, $E_4$, has $N_i=2$ databases, $i=2, 3, 4$.
  • Figure 3: $P_{eq}$ against $\tau$ for different values of $M$ with $N=2$.
  • Figure 4: $P_{eq}$ against $\tau$ for various values of $M$ with $N>2$.

Theorems & Definitions (14)

  • Definition 1: Nominal information leakage
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 2
  • Remark 4
  • ...and 4 more