LOCAL GRAPH-HOMOMORPHIC PROCESSING FOR PRIV ATIZED DISTRIBUTED SYSTEMS Elsa Rizk Stefan Vlaski Ali H. Sayed

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LOCAL GRAPH-HOMOMORPHIC PROCESSING FOR PRIVATIZED DISTRIBUTED

SYSTEMS

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

ABSTRACT

We study the generation of dependent random numbers in a dis-

tributed fashion in order to enable privatized distributed learning by

networked agents. We propose a method that we refer to as local

graph-homomorphic processing; it relies on the construction of par-

ticular noises over the edges to ensure a certain level of differen-

tial privacy. We show that the added noise does not affect the per-

formance of the learned model. This is a signiﬁcant improvement

to previous works on differential privacy for distributed algorithms,

where the noise was added in a less structured manner without re-

specting the graph topology and has often led to performance dete-

rioration. We illustrate the theoretical results by considering a linear

regression problem over a network of agents.

Index Terms—distributed systems, distributed learning, differ-

ential privacy, random number generator

1. INTRODUCTION AND RELATED MATERIAL

Distributed systems consist of a network of agents that collaborate

to achieve a common goal. Some examples include distributed com-

puting [1] when components of a software are shared over a network,

or distributed machine learning [2] where the goal is to ﬁt a global

model to the data dispersed at different computing locations. Dur-

ing collaboration in such systems, communication between neigh-

bours is necessary. However, the shared information might be sensi-

tive, such as in distributed systems handling health or ﬁnancial data.

Thus, there is a need to privatize communication channels. One way

to achieve secure communications is through cryptographic meth-

ods [3–6], while another is by adding random noise to make the

communication differentially private [7–11].

In the standard implementations, agents add independent noise

to their shared messages. This property degrades the performance of

the learned model since the noises propagate over the graph through

cooperation, as already shown in Theorem 1 of [12]. In order to

endow agents with enhanced privacy with minimal effect on perfor-

mance, it is necessary for the additional noise sources to be mindful

of the graph topology [11]. However, this information is not avail-

able globally and, therefore, one needs to devise a scheme to gen-

erate graph-dependent random noise sources in a distributed man-

ner and without assuming any global information about the graph

structure. Motivated by this observation, we develop in this work a

scheme that constructs privacy perturbations in a manner that their

negative effect on performance is canceled out. One solution was

suggested in [3] for the case of federated learning. Pairs of agent

collaborate to add noise that cancels out at the server. However, the

suggested method generates pseudo-random numbers, which is less

School of Engineering, ´

Ecole Polytechnique F´

ed´

erale de Lausanne (e-

mail:{elsa.rizk, ali.sayed}@epﬂ.ch).

Department of Electrical and Electronic Engineering, Imperial College

London (e-mail: s.vlaski@imperial.ac.uk).

secure than true random numbers [13] and without any guarantees

of differential privacy.

The objective of this work is therefore to generate dependent

random numbers in a distributed manner across a graph. The prob-

lem is challenging for at least two reasons. Firstly, generating ran-

dom numbers is usually difﬁcult without enforcing beforehand some

distribution for the random process. In practice, random number

generators exploit a variety of entropy sources in a computer such

as mouse movements, available memory, or temperature [14]. Sec-

ondly, it is not evident how agents should exploit independent en-

tropy sources to generate dependent random numbers. Most avail-

able solutions [15–21] rely on a central orchestrator or consider a

fully connected network. A truly distributed method does not appear

to exist.

2. LOCAL GRAPH-HOMOMORPHIC PROCESS

2.1. Problem Setup

We consider a network of Kagents connected by some graph topol-

ogy (Fig. 1). We let amk >0denote the weight attributed to the

message sent by neighbour mto agent kand let A= [amk]denote

the corresponding combination matrix. We assume Ais symmetric

and doubly-stochastic, i.e.:

1TA=1TA1=1.(1)

We further denote the neighbourhood of agent kby Nk; it consists

of all agents connected to kby an edge.

Fig. 1: Illustration of a network of agents.

We consider problems where agents aggregate the received mes-

sages from their neighbours. In other words, if we let ψmk,i denote

arXiv:2210.15414v1 [cs.CR] 26 Oct 2022

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LOCALGRAPH-HOMOMORPHICPROCESSINGFORPRIVATIZEDDISTRIBUTEDSYSTEMSElsaRizk,StefanVlaski,AliH.SayedABSTRACTWestudythegenerationofdependentrandomnumbersinadis-tributedfashioninordertoenableprivatizeddistributedlearningbynetworkedagents.Weproposeamethodthatwerefertoaslocalgraph-homomorphicprocessing;itrel...

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LOCAL GRAPH-HOMOMORPHIC PROCESSING FOR PRIV ATIZED DISTRIBUTED SYSTEMS Elsa Rizk Stefan Vlaski Ali H. Sayed

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