Over-the-Air Gaussian Process Regression Based on Product of Experts Koya Sato

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Over-the-Air Gaussian Process Regression

Based on Product of Experts

Koya Sato

Artiﬁcial Intelligence eXploration Research Center,

The University of Electro-Communications, 1-5-1, Chofugaoka, Chofu-shi, Tokyo, Japan

E-mail: k sato@ieee.org

Abstract—This paper proposes a distributed Gaussian process

regression (GPR) with over-the-air computation, termed AirComp

GPR, for communication- and computation-efﬁcient data analysis

over wireless networks. GPR is a non-parametric regression

method that can model the target ﬂexibly. However, its com-

putational complexity and communication efﬁciency tend to be

signiﬁcant as the number of data increases. AirComp GPR

focuses on that product-of-experts-based GPR approximates the

exact GPR by a sum of values reported from distributed nodes.

We introduce AirComp for the training and prediction steps to

allow the nodes to transmit their local computation results simul-

taneously; the communication strategies are presented, including

distributed training based on perfect and statistical channel state

information cases. Applying to a radio map construction task, we

demonstrate that AirComp GPR speeds up the computation time

while maintaining the communication cost in training constant

regardless of the numbers of data and nodes.

Index Terms—Over-the-air computation, distributed machine

learning, Gaussian processes, radio map construction

I. INTRODUCTION

Gaussian process regression (GPR) is a non-parametric

approach to regression tasks, which realizes ﬂexible modeling

of a dataset without specifying low-level assumptions [1], [2].

Assuming GP for the target data, we can obtain both the mean

and variance of the regression results. There has been a wide

range of applications for GPR such as environmental monitor-

ing based on spatial statistics [3], [4], experimental design [5]

and motion trajectory analysis [6]; in wireless communication

systems, recent results have shown its advances in coverage

analysis and communication design, with the term of radio

map [7]–[9]. GPR will play an important role in the next

Internet of Things (IoT) era.

However, GPR has some critical drawbacks regarding com-

munication and computational costs in such applications. Let

us consider a situation where multiple nodes are distributed on

a network to monitor an environmental state and connected to

a server wirelessly, as envisioned in [10]. When the server

performs GPR to analyze the sensing results, the nodes need

to upload their sensing data to the server. The exact GPR

requires inverse matrices in training and prediction steps. This

leads the complexity of O(N3)for Ntraining data; further,

for nin input dimension data, the nodes upload (nin + 1)N

This work was supported in part by JST, ACT-X, JPMJAX21AA and JST

SICORP, JPMJSC20C1.

variables to the server. The ﬁrst problem can be improved by

distributed GPR based on the product of experts [2], [11].

This method approximates GPR by the sum of computation

results at nodes to reduce the computational complexity from

O(N3)at the server to O((N/M)3)at Mdistributed nodes;

however, the communication slots still depend on M.

In this paper, toward a communication- and computation-

efﬁcient IoT monitoring system, we propose a distributed GPR

scheme with over-the-air computation, termed AirComp GPR.

Over-the-air computation is a technique for communication-

efﬁcient distributed computation over shared channels based

on nomographic functions [12], [13]. Each node transmits

its message with an analog modulation function. Then, the

receiver obtains the target computation result from the super-

imposed signal based on a decoding function. Since multiple

nodes transmit their analog-modulated signals simultaneously,

we can realize a low-latency computation over networks. We

focus on that both training/regression results in the distributed

GPR are based on the sum of computation results reported

from the nodes. The proposed method aggregates the local

computation results based on the over-the-air computation; as

a result, the communication cost does not depend on the data

size and the number of nodes.

Major contributions of this paper are listed as follows.

•We propose AirComp-aided distributed GPR for com-

munication/computation efﬁcient regression over wireless

networks. It is shown that the computational complexity

can be reduced from O(N3)at BS to O((N/M)3)at

Mdistributed nodes, and its communication cost at the

training step can be constant regardless of Mand N.

•Two schemes are introduced for the training step: per-

fect channel state information (CSI)-based and statistical

CSI-based schemes. The ﬁrst approach can perform the

distributed GPR with a limited accuracy degradation from

full GPR; further, the latter enables no requirements for

the uplink instantaneous channel estimations.

•Performance of AirComp GPR is analyzed in the radio

map construction task. We demonstrate that an accurate

radio map can be constructed efﬁciently.

Notations: throughout this paper, the transpose, determinant,

and inverse operators are denoted by (·)T,det(·)and (·)−1,

while the expectation and the variance are expressed by E[·]

arXiv:2210.02204v2 [eess.SP] 6 Oct 2022

...

Result of local training or regression

Fig. 1. Signal transmission model.

and Var[·], respectively. Further, |·|and || · || are deﬁned

as operators to obtain the absolute and Euclidean distance,

respectively.

II. SYSTEM MODEL

A. Task Deﬁnition

We consider a situation where Msensing nodes are con-

nected to a base station (BS) over wireless networks. The i-th

node has a dataset,

Di={(xi,k, yi,k )|k= 1,2,··· , Ni},(1)

where Niis the number of data, xi,k is the input vector

(e.g., sensing location) and yi,k =f(xi,k) + is its output

value generated from N(f(xi,k), σ2

)(∼ N(0, σ2

)is the

independently and identically distributed (i.i.d.) noise). When

local datasets are non-overlapped each other, the full dataset

over the network can be expressed as

[

i=1 Di,(2)

where the number of full data can be deﬁned as N=

i=1 Ni. Further, it is assumed that all data in Dfollows a

Gaussian process: i.e., f∼GP (µ(x), k(x,x0)), where µ(x)

is the expectation value at x, and k(x,x0)is the covariance

between µ(x)and µ(x0). The task in this context is to estimate

ffor test inputs X∗= [x∗,1,x∗,2,··· ,x∗,ntest ]from Di

distributedly.

Possible applications of the above task include environmen-

tal monitoring [14] and radio map construction [7].

B. Signal Model

AirComp GPR can be divided into training and regression

steps. We herein deﬁne the signal model for these steps. Fig. 1

summarizes the signal transmission model, where all nodes

simultaneously transmit their messages to BS through a shared

wireless channel. The i-th node ﬁrst encodes its message si

so that BS can extract the sum of si; we denote this process

as xi= Enc(si). When all nodes are time synchronized, the

received signal at BS can be given by

i=1 pγihixi+z,(3)

where √γi∈Ris the average channel gain and hi∼

CN(0,1) is the i.i.d. instantaneous channel gain assuming ﬂat

over one transmission. Further, xiis the transmitted vector

constrained by the maximum transmission power Pmax as

||xi||2≤Pmax, and zis the additive white Gaussian noise

(AWGN) vector following CN(0, σ2

z), where σ2

zis the noise

ﬂoor. Then, BS extract the sum of siusing a decoding

operation, deﬁned by Dec(y).

Note that BS has to share a message that contains a

few hyper-parameters in the training step to the nodes. To

enable the channel estimation at the nodes, BS broadcasts

it with digital encoding with sufﬁcient transmission power;

we assume that the nodes can decode it correctly. Further,

assuming channel reciprocity, the nodes can estimate the

instantaneous channel state information (CSI) √γihiowing

to the broadcasted downlink signals. In contrast, we consider

two situations for BS; (a) global CSI and (b) statistical CSI

(i.e., only γiis available). This condition affects the AirComp

in the training step (see IV-A).

III. GAUSSIAN PROCESS REGRESSION

Before explaining the proposed method, this section intro-

duces full GPR, and its distributed method based on products

of experts [2]. Note that, for simplicity, this section assumes

ntest = 1 and denotes the test input as x∗.

A. Full GPR

Consider a situation where BS has the full dataset Dand

performs the exact GPR. From the full dataset D, we deﬁne

y={yi,k | ∀(i, k)}and X={xi,k | ∀(i, k)}. GPR ﬁrst

needs to tune hyper-parameters θ={ψ, σ}, where ψis the

hyper-parameter vector for a kernel function k. Finding θcan

be realized by maximizing the log-marginal likelihood,

log p(y|X,θ) = −1

2(y−m)TK+σ2

I−1(y−m)

−1

2log det K+σ2

I−N

2log 2π, (4)

where Iis the N×Nidentity matrix and K∈RN×Nis the

kernel matrix, where its element is Kij =k(xi,xj)(xiis the

i-th element in X). Further, mis a vector with Nelements,

where its i-th element m(xi)is the prior mean at xi1.

Based on a vector θ, the full GPR predicts the distribution

of the output at the test input x∗as the Gaussian distribution

with mean (E[f(x∗)] = µ(x∗)) and variance (Var[f(x∗)] =

σ2(x∗)) given by the following equations, respectively:

µ(x∗) = m(x∗) + kT

∗K+σ2

I−1(y−m)(5)

σ2(x∗) = k∗∗ −kT

∗K+σ2

I−1k∗,(6)

1For example, vector mis given from m(x1) = m(x2) = ··· =

m(xN) = 1

NPN

i=1 yi, where yiis the i-th element in y.

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Over-the-AirGaussianProcessRegressionBasedonProductofExpertsKoyaSatoArticialIntelligenceeXplorationResearchCenter,TheUniversityofElectro-Communications,1-5-1,Chofugaoka,Chofu-shi,Tokyo,JapanE-mail:ksato@ieee.orgAbstractThispaperproposesadistributedGaussianprocessregression(GPR)withover-the-aircomp...

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Over-the-Air Gaussian Process Regression Based on Product of Experts Koya Sato

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