# Optimizing Over-the-Air Computation in IRS-Aided C-RAN Systems

Over-the-air computation (AirComp) is an efficient solution to enable federated learning on wireless channels. AirComp assumes that the wireless channels from different devices can be controlled, e.g., via transmitter-side phase compensation, in order to ensure coherent on-air combining. Intelligent reflecting surfaces (IRSs) can provide an alternative, or additional, means of controlling channel propagation conditions. This work studies the advantages of deploying IRSs for AirComp systems in a large-scale cloud radio access network (C-RAN). In this system, worker devices upload locally updated models to a parameter server (PS) through distributed access points (APs) that communicate with the PS on finite-capacity fronthaul links. The problem of jointly optimizing the IRSs' reflecting phases and a linear detector at the PS is tackled with the goal of minimizing the mean squared error (MSE) of a parameter estimated at the PS. Numerical results validate the advantages of deploying IRSs with optimized phases for AirComp in C-RAN systems.

## Authors

• 6 publications
• 13 publications
• 127 publications
• 54 publications
11/10/2020

### Federated Learning via Intelligent Reflecting Surface

Over-the-air computation (AirComp) based federated learning (FL) is capa...
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### Channel Estimation for IRS-aided Multiuser Communications with Reduced Error Propagation

Intelligent reflecting surface (IRS) has emerged as a promising paradigm...
01/12/2022

### Joint Transmit and Reflective Beamformer Design for Secure Estimation in IRS-Aided WSNs

Wireless sensor networks (WSNs) are vulnerable to eavesdropping as the s...
07/29/2020

### Optimized Amplify-and-Forward Relaying for Hierarchical Over-the-Air Computation

Over-the-air computation (AirComp) is an emerging wireless technique wit...
05/19/2020

### Over-the-Air Computation Systems: Optimal Design with Sum-Power Constraint

Over-the-air computation (AirComp), which leverages the superposition pr...
04/29/2019

### Over-the-Air Computation via Intelligent Reflecting Surfaces

Over-the-air computation (AirComp) becomes a promising approach for fast...
06/10/2019

### Intelligent Reflecting Surface vs. Decode-and-Forward: How Large Surfaces Are Needed to Beat Relaying?

The rate and energy efficiency of wireless channels can be improved by d...
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## I Introduction

footnotetext: This work was supported by Basic Science Research Program through the National Research Foundation of Korea grants funded by the Ministry of Education [NRF-2018R1D1A1B07040322, NRF-2019R1A6A1A09031717]. The work of O. Simeone was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 725731). The work of S. Shamai was supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694630).

Federated learning is an emerging distributed learning paradigm in which mobile devices collaboratively train a machine learning model while preserving the privacy of local data sets

[1]. In the presence of latency and bandwidth constraints, the implementation of federated learning on wireless systems is challenging if many workers, or devices, are involved. A potential solution to this problem is over-the-air computation (AirComp), which leverages the superposition property of the multiple access channel (MAC) from worker devices to a parameter server (PS) to allow for simultaneous transmissions from multiple devices [2, 3, 4]. It was reported in [3] that AirComp outperforms a conventional multiple access technique in terms of test accuracy, and that the gain is particularly significant at low transmit power and large number of workers.

AirComp assumes that the wireless channels from different devices can be controlled, e.g., via transmitter-side phase compensation, in order to ensure coherent on-air combining [5]. To alleviate this problem, the work [6] considered a deployment of intelligent reflecting surfaces (IRSs). IRSs, also referred to as reconfigurable intelligent surfaces, can be controlled through integrated electronics in order to shape their response to impinging electromagnetic waves [7]. This enables the modification of the propagation channel between nearby transceivers. As a result, IRSs are considered as a cost-effective solution to improve spectral and energy efficiency of wireless systems [8, 9, 10, 11]. As examples of recent works on IRSs, references [9] and [10] addressed the joint design of downlink beamforming and IRSs’ phases for interference management in multi-user [9] and multi-cell systems [10]. Reference [8] analyzed the number of reflecting elements of IRSs needed to beat conventional wireless relaying techniques (see also [12]). Finally, an information-theoretic study was provided in [13].

## Ii System Model

As illustrated in Fig. 1, we consider an over-the-air computation task performed on a C-RAN system. In the system, single-antenna worker devices send locally updated models to a PS through single-antenna APs. Each AP is connected to the PS via a fronthaul link, which we model as a digital link of capacity bit/sample [14]. We define the sets and for the workers’ and APs’ indices, respectively.

### Ii-a Over-the-Air Computation Model

We focus on the transmission at a specific time slot where each worker sends a scalar parameter , and the PS estimates a function of the transmitted parameters . The parameter

can be an element of the gradient vector

[3] or the local model [4] updated at worker using its local dataset. The PS typically estimates the weighted sum , with , where denotes the number of training samples at device [4]. To simplify the discussion, we assume for all , and that the target parameter denoted by is given by the sum

 ¯θ=f(θ)=∑k∈NWθk. (1)

We also assume that the parameters are independent, and we define the power of parameter as . Thus, the target parameter has power .

### Ii-B Channel Model

To assist edge communication from the workers to the APs, we assume the presence of IRSs [6] in the network. Each IRS has reflecting elements, whose reflecting phases are dynamically adjusted to adapt to the instantaneous channel state information (CSI). We define the set for the IRSs’ indices.

Under a flat-fading channel model, the received signal of AP can be written as

 yi=∑k∈NWhi,kxk+zi, (2)

where is the signal transmitted by worker ; denotes the channel coefficient from worker to AP ; and represents the additive noise. The signal satisfies the transmit power constraint .

Due to the presence of IRSs, the channel coefficient is modelled as [8, 9, 10]

 (3)

where denotes the small-scale fading channel from worker to AP ; represents the small-scale fading channel vector from IRS to AP ; is the small-scale fading channel vector from worker to IRS ; denotes the path-loss of the direct link from worker to AP ; is the path-loss of the composite link from worker to AP through IRS ; and is a diagonal matrix that represents the reflecting operation of IRS , which is defined as

 Θj=diag({ejϕj,m}nIm=1), (4)

where denotes the reflecting phase of the th element of IRS .

We model the path-loss between worker and AP as , where is the Euclidean distance in meter between the two input vectors, and denote the position vectors of worker and AP , respectively, is the path-loss exponent, and denotes the path-loss at the reference distance of m. For the path-loss of the composite channel from worker to AP through IRS , we adopt the sum-distance model [7] which models as

 ρr,i,j,k=c0(D(pW,k,pI,j)+D(pI,j,pA,i))−η, (5)

where denotes the position vector of IRS .

## Iii Over-the-Air Computation in IRS-aided C-RAN

In this section, we illustrate the operations at the worker devices, the APs, and the PS in the IRS-aided C-RAN system described in Sec. II.

### Iii-a Transmission at Worker Devices

Without claim of optimality (see [15]), we assume that each worker uses the maximum transmit power , so that the transmit signal is given as

 xk=αkθk, (6)

with the coefficient . We note that this does not require CSI at worker devices.

### Iii-B Quantization at APs

AP sends a quantized version of the received signal to the PS through a fronthaul link of capacity bit/sample. Under the assumptions that the updated model vectors have a sufficiently large dimension, the quantized signal denoted by can be modelled as [14, 16]

 ^yi=yi+qi, (7)

where models the quantization distortion as being independent of and distributed as . According to standard rate-distortion theoretic results [17], the quantization noise power satisfies the condition

 I(yi;^yi)=log2(1+σ2y,iωi)≤C, (8)

where

denotes the variance of the received signal

given as

 σ2y,i=∑k∈NW|hi,k|2P+σ2z. (9)

The minimum distortion power that satisfies the condition (8) is given as

 ωi=σ2y,i/(2C−1). (10)

Note that the optimal distortion level (10) is a function of the reflecting phases , since affects the channel coefficients as seen in (3).

### Iii-C Estimation at PS

Based on the received quantized signals , the PS estimates the target parameter in (1). To elaborate, let us define a vector which stacks the quantized signals. Then, the vector can be expressed as

 ^y=∑k∈NWαkhkθk+z+q, (11)

where we have defined the vectors , and with .

The channel vector from worker to all the APs can be written as a function of the IRSs’ phases as

 hk=hd,k+∑j∈NIRr,j,kGjdiag(hr,j,k)vj, (12)

where the matrices , , and the vectors , are defined as , , , and , respectively. Note that the optimization of the phases of IRS is equivalent to that of the vector as long as the conditions

 |vj(m)|2=1 (13)

are satisfied for all , where denotes the th element of . From the vector , each phase can be obtained as .

We assume that the PS performs a linear estimation of the target parameter from . Accordingly, an estimate of is given as

 ^¯θ=fH^y, (14)

with a linear detection vector .

For given phases , i.e., , and linear detection vector , the MSE between the estimate and the target parameter is evaluated as

 e(v,f) =E[|^¯θ−¯θ|2] (15) =∑k∈NW|αkfHhk−1|2σ2θ,k+fH(σ2zI+Ω)f.

## Iv Optimization

We tackle the problem of jointly optimizing the IRSs’ reflecting phases and the linear detection vector of the PS with the goal of minimizing the MSE in (15) while satisfying the unit modulus constraints (13). The problem can be stated as

 minimizev,f e(v,f) (16a) s.t. |vj(m)|2=1,j∈NI,m∈{1,…,nI}. (16b)

Since it is difficult to jointly optimize the variables and , we propose an iterative algorithm that alternately optimizes one variable while fixing other.

If we fix the IRSs’ phases in problem (16), finding the optimal detector becomes an unconstrained quadratic optimization problem, whose closed-form solution is given as

 (17)

To tackle the problem of optimizing the IRSs’ phases for fixed , we remove the terms that are not dependent on the IRSs’ phases from the cost function. Stating the obtained problem with respect to a stacked vector with yields

 minimize¯v ⎛⎜⎝∑k∈NW(|aHk¯v|2+2R{b∗kaHk¯v})σ2θ,k+∑i∈NA,k∈NW(|cHi,k¯v|2+2R{d∗i,kcHi,k¯v})⎞⎟⎠ (18a) s.t. |¯v(m)|2=1,m∈{1,…,¯nI}, (18b)

where we have defined the notations , , , , , and with and being the th element of and the

th column of an identity matrix of size

, respectively.

The problem (18) is non-convex due to the unit modulus constraints (18b). To handle this issue, we adopt the matrix lifting approach proposed in [6]. Accordingly, we tackle the problem (18) with respect to a matrix defined as

 V=[¯v1][¯vH1]=[¯v¯vH¯v¯vH1]. (19)

The matrix is subject to the constraints , , and for all . From , the IRSs’ phase vector can be recovered as the first elements of the last column of .

We tackle (18) with respect to by using the following equalities:

 |aHk¯v|2+2R{b∗kaHk¯v} =[¯vH1][akaHkbkakb∗kaHk0][¯v1] =tr([akaHkbkakb∗kaHk0]V), (20) |cHi,k¯v|2+2R{d∗i,kcHi,k¯v} =[¯vH1][ci,kcHi,kdi,kci,kd∗i,kcHi,k0][[]c¯v1] =tr([ci,kcHi,kdi,kci,kd∗i,kcHi,k0]V). (21)

Specifically, by substituting (20) and (21) into problem (18), we obtain the problem

 minimizeV⪰0 tr(MV) (22a) s.t. V(m,m)=1,m∈{1,…,¯nI+1}, (22b) rank(V)≤1, (22c)

with the matrix defined as

 M (23)

To address the non-convexity of constraint (22c), we note that (22c) is equivalent to the constraint [6]

 tr(V)−σ1(V)=0, (24)

where

denotes the largest singular value of the input matrix. Function

is convex in [18]. Furthermore, for , the left-hand side (LHS) of (24) is 0 when and it becomes larger than 0 otherwise.

Based on this observation, as in [6], we tackle the problem

 minimizeV⪰0 tr(MV)+γ(tr(V)−σ1(V)) (25a) s.t. V(m,m)=1,m∈{1,…,¯nI+1}, (25b)

with a fixed weight . In problem (25), we have removed the rank constraint (22c) and instead added a penalty term to the cost function that increases if (22c) is not satisfied.

The problem (25) is a difference-of-convex (DC) problem whose locally optimal solution can be efficiently found via the concave convex procedure (CCP) approach [19]. CCP solves a sequence of convex problems obtained by linearizing the terms that induce non-convexity. In the DC problem (25), the only term that induces non-convexity is in the penalty term. Linearizing at a reference point yields the upper bound [6]

 −γ⋅σ1(V)≤−γ⋅tr(Vu1(V′)u1(V′)H), (26)

where

returns the eigenvector of the input matrix corresponding to the largest eigenvalue. The condition (

26) is satisfied with equality when . The CCP based algorithm for optimizing is summarized in Algorithm 1.

Overall, the proposed algorithm that alternately optimizes the IRSs’ phases and the linear detector is detailed in Algorithm 2. In the algorithm, we initialize and in Steps 1-2, and update for fixed in Steps 3-4. In Step 4, is modified only when it does not satisfy the modulus constraints (18b). In Step 5, is updated for fixed , and we check the convergence in Step 6.

## V Numerical Results

In simulation, we assume that the positions of workers, APs and

IRSs are uniformly distributed in a circular area of radius 100 m. We set the variance of local parameters to

for and assume dB, in the path-loss models and for the penalty coefficient in (25a). For all links, we consider independent Rayleigh fading channels that are distributed as , and . We compare the performance of the proposed optimized scheme with two baseline schemes, one without IRSs and one with IRSs whose reflecting phases are randomly chosen. In all figures, we plot the normalized MSE, which is defined as the MSE normalized by so that it lies in the range .

In Fig. 2, we plot the average normalized MSE versus the fronthaul capacity for an IRS-aided C-RAN system with , , , and dB. The figure shows that the proposed optimized scheme outperforms both baseline schemes without IRS and with random phases, and that the gain increases with the fronthaul capacity . This is because, when is small, the impact of carefully designing the IRSs’ phases becomes minor due to the impact of the quantization noise signals

. Also, the gain increases with the signal-to-noise ratio (SNR)

of the uplink channel, and this trend coincides with the observation reported in [10, Sec. IV].

Fig. 3 plots the average normalized MSE versus the number of APs for an IRS-aided C-RAN system with , , , and dB. When there are only a few APs, deploying IRSs provides relevant gains only when the reflecting phases are optimized according to Algorithm 2. However, the impact of optimizing the reflecting phases becomes minor for sufficiently large .

## Vi Concluding Remarks

We have studied the impacts of deploying IRSs on AirComp in a C-RAN system. To this end, we have tackled the joint optimization of the IRSs’ reflecting phases and the linear detector at the PS with the goal of minimizing the MSE of the parameter estimated at the PS. Numerical results were provided that investigate the effects of various parameters on the performance gain of the proposed optimization scheme compared to baseline schemes. Among open problems, we mention the design of channel estimation process, the investigation of the effect of imperfect CSI, and the design of AirComp jointly with information transfer.