Source-linked AI summary
Differentially Private Distributed Optimization
Zhenqi Huang, Sayan Mitra, Nitin Vaidya
TL;DR
The paper studies how agents can jointly optimize a summed objective without revealing their individual cost functions through communication. It proposes noisy iterative distributed algorithms and analyzes their convergence, privacy, and accuracy. The reported accuracy scales as O(1/ε^2), while a lower bound relating achievable privacy and accuracy remains open.
Problem
The problem is to minimize a global sum of private individual cost functions while preventing an adversary observing all exchanged messages from inferring those functions.
Method
The paper proposes synchronous iterative distributed algorithms in which agents broadcast Laplace-noised estimates and update them using received neighbor information and private costs.
Results
The proposed algorithms guarantee ε-differential privacy and convergence under the stated conditions, with observed accuracy of order O(1/ε^2).
Takeaways & Limitations
Privacy and optimization accuracy can be analyzed together through the algorithm’s privacy, step-size, step-size decay, and noise decay parameters.
Takeaways & Limitations
A lower bound on the maximum differential privacy achievable at a given accuracy remains open for this class of iterative algorithms.
Abstract
from arXiv · showhide
In distributed optimization and iterative consensus literature, a standard problem is for $N$ agents to minimize a function $f$ over a subset of Euclidean space, where the cost function is expressed as a sum $\sum f_i$. In this paper, we study the private distributed optimization (PDOP) problem with the additional requirement that the cost function of the individual agents should remain differentially private. The adversary attempts to infer information about the private cost functions from the messages that the agents exchange. Achieving differential privacy requires that any change of an individual's cost function only results in unsubstantial changes in the statistics of the messages. We propose a class of iterative algorithms for solving PDOP, which achieves differential privacy and convergence to the optimal value. Our analysis reveals the dependence of the achieved accuracy and the privacy levels on the the parameters of the algorithm. We observe that to achieve $ε$-differential privacy the accuracy of the algorithm has the order of $O(\frac{1}{ε^2})$.
1 Introduction
The paper formulates private distributed optimization as minimizing a sum of agents’ private cost functions while preventing an adversary from inferring them from exchanged messages. It proposes iterative distributed algorithms using noisy estimates and analyzes privacy, convergence, and accuracy.
- PDOP asks N agents to minimize a global cost function formed by summing their individual cost functions.
- Distributed agents exchange estimates over potentially changing networks while keeping individual cost functions private from an adversary.
- Centralized optimization requires a trusted leader, route maintenance, and exposes systems to leader failure and topology-related privacy concerns.
- The paper adopts ε-differential privacy so complete communication observations reveal no significant information about any agent’s cost function.
- The proposed algorithms add Laplace noise to estimates, with successive noise decaying to support convergence toward the optimum.
- O(1/ε^2) accuracy is observed for achieving ε-differential privacy.
2 Preliminaries
The preliminaries introduce Laplace noise, vector and norm notation, Euclidean projection, convexity, strong convexity, and a sequence-limit condition used in the analysis.
- Laplace-distributed random real numbers are used by the paper’s algorithms.
- The Laplace distribution described has mean zero and variance 2c^2.
- The preliminaries define vector components, transposes, standard Lp norms, and the relationship between L1 and L2 norms.
- Euclidean projection selects a closest point in a set and does not increase distances between projected points.
- Strongly convex functions on compact domains have a unique minimum.
- A convergent scalar sequence with limit zero is introduced as a condition in a preliminary proposition.
3 The Private Distributed Optimization Problem
The paper defines PDOP through agents, private strongly convex costs, a summed global objective, and a time-varying communication graph. Its iterative template alternates noisy reporting, aggregation, and cost-reducing updates, with convergence and privacy as goals.
- The PDOP model: A PDOP consists of N agents, a compact convex domain, individual strongly convex differentiable costs, a summed global cost, and a communication-matrix sequence.
- The PDOP model: Each individual cost function is known only to its associated agent, while all agents jointly minimize the global objective.
- The PDOP model: The rendezvous example models each agent’s travel cost as squared distance to a chosen assembly point without sharing addresses.
- Assumptions: The model assumes compactness, convexity, uniformly bounded gradients, and strong convexity of individual costs.
- Communication model: Communication is synchronous but may vary over time, with weighted graph matrices assumed doubly stochastic and strongly connected.
- Iterative algorithm: Each round reports a value, exchanges neighbor messages, aggregates them, and updates the agent’s estimate using its private cost.
- Privacy and convergence: The observable execution is the sequence of broadcast messages, and solving PDOP requires common convergence near the optimum while preserving differential privacy.
4 An Algorithm for PDOP instantiating the Subroutines
The paper instantiates PDOP with synchronous iterative updates that combine noisy broadcasts, neighbor averaging, projected local gradient steps, and geometrically decaying noise and step sizes. The resulting algorithm is ε-differentially private, convergent, and accuracy depends on tunable parameters, with accuracy worsening as privacy becomes stricter.
- Algorithm Description: The instantiated algorithm uses privacy parameter ε, initial step size c, step-size decay q, and noise-decay p, with p ∈(q, 1).The noise and step size are chosen as geometrically decaying sequences based on these parameters.
- Algorithm Description: The algorithm uses three subroutines: agents add Laplace noise to local estimates, average neighbors’ broadcasts, then update by a projected local gradient step.The broadcast subroutine reports xi + wi; the mixing subroutine uses a doubly stochastic weighted average; the update subroutine projects onto X.
- Algorithm Description: The noise distribution converges to a Dirac distribution while the step size converges to 0, and any c > 0, q ∈(0, 1), p ∈(q, 1) yields privacy and convergence guarantees.The paper identifies c, q, and p as the tunable parameters after fixing the privacy parameter.
- Differential Privacy: Sensitivity measures how messages change when one agent’s cost function changes, and its bound determines the Laplace noise required for differential privacy.The paper defines sensitivity using adjacent PDOPs and an L1 norm, then uses a sufficient cumulative-noise condition for privacy.
- Differential Privacy: The proposed algorithm guarantees ε-differential privacy for any c > 0, q ∈(0, 1), and p ∈(q, 1).The guarantee follows from the Laplace-noise privacy lemma and the algorithm’s sensitivity bound.
- Convergence: Theorem 8 establishes convergence, requiring expected disagreement between agents’ local values to approach 0 so that agents eventually agree on a common value.The convergence bound contains decaying terms whose limits depend on the noise magnitude and step size.
- Accuracy and Parameter Choice: As ε approaches 0, representing complete privacy, the accuracy becomes arbitrarily bad.The paper reports that the accuracy level has inverse-square dependence on ε.
- Accuracy and Parameter Choice: The accuracy level d is not convex in c, q, and p, so globally optimal parameter choices lack a clean closed-form expression.With any two parameters fixed, the remaining parameter has a local optimum.
5 Conclusion
The paper formulates PDOP for agents that minimize a sum of individual cost functions while keeping those functions differentially private. It proposes a structurally simple iterative algorithm and characterizes the privacy–accuracy tradeoff, while leaving a lower-bound question open.
- Problem: The PDOP problem requires agents to minimize a global cost formed from individual cost functions while keeping those functions differentially private.Agents exchange estimates, but an adversary with access to all communication should not deduce the individual cost functions.
- Algorithm: The proposed algorithm adds Laplace noise, averages noisy neighbor estimates, and moves along each agent’s own cost-function gradient.The noise decays across rounds, and communication neighbors may change between rounds.
- Algorithm: The algorithm remains applicable when the communication topology and neighbors change from one round to another.The conclusion states that this structurally simple iterative method solves PDOP under changing communication topology.
- Open question: A lower bound on the maximum differential privacy achievable at a given accuracy remains an open problem, even for iterative algorithms.This is identified as an unresolved question in the paper’s conclusion.