FEDERATED DROPOUT LEARNING FOR HYBRID BEAMFORMING WITH SPATIAL PATH INDEX MODULATION IN MULTI-USER MMWAVE-MIMO SYSTEMS
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FEDERATED DROPOUT LEARNING FOR HYBRID BEAMFORMING WITH SPATIAL PATH INDEX
MODULATION IN MULTI-USER MMWAVE-MIMO SYSTEMS
Ahmet M. Elbir† , Sinem Coleri† and Kumar Vijay Mishra+
†
Department of Electrical and Electronics Engineering, Koç University, Istanbul, Turkey
+
United States CCDC Army Research Laboratory, Adelphi, MD 20783 USA
arXiv:2102.07450v1 [eess.SP] 15 Feb 2021
ABSTRACT from reduced array gain because it always uses a subarray. For a
Millimeter wave multiple-input multiple-output (mmWave-MIMO) single-user scenario, [8] proposed beamspace-based approaches for
systems with small number of radio-frequency (RF) chains have spatial path index modulation (SPIM), which modulates the indices
limited multiplexing gain. Spatial path index modulation (SPIM) of the spatial paths to create different spatial patterns. The use of
is helpful in improving this gain by utilizing additional signal bits beamspace model is further exploited in [9] by employing lens arrays
modulated by the indices of spatial paths. In this paper, we introduce at both transmitter and receiver to improve the bit-error-rate (BER).
model-based and model-free frameworks for beamformer design in Apart from BER, spectral efficiency is utilized as a performance
multi-user SPIM-MIMO systems. We first design the beamformers metric in [7] for SPIM-based transmitter design. Here, theoretical
via model-based manifold optimization algorithm. Then, we leverage conditions for SPIM-MIMO to outperform mmWave-MIMO are intro-
federated learning (FL) with dropout learning (DL) to train a learning duced. The SPIM structure in [7] considers only analog beamformer
model on the local dataset of users, who estimate the beamformers design, for which the same baseband beamformers are used even if
by feeding the model with their channel data. The DL randomly the structure of the analog beamformer is changed due to the selec-
selects different set of model parameters during training, thereby tion of different spatial patterns. Analog-only beamformer design
further reducing the transmission overhead compared to conventional is also considered in [11] for uplink multi-user scenario with code-
FL. Numerical experiments show that the proposed framework ex- book design. A joint design for analog and baseband precoders for
hibits higher spectral efficiency than the state-of-the-art SPIM-MIMO SPIM is performed in [12] by implementing zero-forcing baseband
methods and mmWave-MIMO, which relies on the strongest propa- precoding and selecting the steering vectors as analog beamformer
gation path. Furthermore, the proposed FL approach provides at least candidates. Similar to [12], [7, 8] also design the analog precoders
10 times lower transmission overhead than the centralized learning with a predefined codebook of steering vectors, which entails a beam
techniques. training task prior to the precoder design. Most of the aforementioned
works investigate the single-user scenario. Their extension to the
Index Terms— Dropout learning, federated learning, manifold
multi-user case remains a challenge. Although [11] considered the
optimization, massive MIMO, spatial modulation.
uplink multi-user SPIM architecture, it included the codebook of
analog-only beamformers at the user end.
1. INTRODUCTION
In this paper, we design both analog and digital beamformers for
The millimeter wave multiple-input multiple-output (mmWave- a downlink multi-user scenario using model-based and model-free
MIMO) communications systems substantially improve the through- techniques. We leverage the optimality of the manifold optimization
put in the fifth generation (5G) networks [1, 2]. As an emerging 5G (MO) [13, 14] for the model-based approach. Then, taking advantage
technology, index modulation (IM) is attractive primarily because it of the model-free structure of learning-based methods [15–17] to
offers both improved energy efficiency and spectral efficiency over improve robustness and computational efficiency, we train a global
conventional modulations. The IM encodes additional information model through federated learning (FL). All users contribute to the
in the indices of the transmission media such as subcarriers [3, 4], learning process by computing the model updates with respect to their
antennas [5, 6], and spatial paths [7–9]. In this paper, we focus local datasets. The model updates are then collected at the base station
on spatial modulation (SM) in the context of mmWave-MIMO (BS) for model aggregation and then sent back to the users for the
systems [10]. next communication round and the global model is iteratively updated.
In mmWave-MIMO, hybrid analog-digital beamformers are em- Once trained, the model parameters are shared with each user, which
ployed, where the number of radio-frequency (RF) chains is much can estimate the beamformers by simply feeding the model with its
smaller than the antennas. While this saves cost and power, its mul- downlink channel matrix. As a result, a non-linear data mapping is
tiplexing gain is limited [1]. The SM techniques have been shown constructed between the channel data (input) and the beamformers
to be helpful in addressing this problem [7–9]. In [5], an antenna (output), wherein a convolutional neural network (CNN) with dropout
grouping (AG) approach is proposed for point-to-point communica- learning (DL) is designed [18]. The DL allows randomly selecting a
tion, wherein some antenna elements are (de)activated to provide fraction (up to ~50%) of the model parameters, thus further reducing
SM in terms of active/passive antenna indices. This approach suffers the communication cost during FL-based training.
Unlike the conventional centralized learning (CL) methods [19–
S. Coleri acknowledges the support of the Scientific and Technological Research
Council of Turkey (TUBITAK) EU CHIST-ERA grant 119E350. A. M. Elbir acknowl- 21], where the BS collects all of the training datasets from users,
edges the support of TUBITAK. our proposed FL-based approach is advantageous because of lesstransmission overhead; it is further reduced by employing DL to send
approximately half of the model parameters to the users. We validate
this through extensive numerical experiments and demonstrate that
the proposed model-based and model-free approaches have supe-
rior spectral efficiency than the state-of-the-art model-based SPIM
techniques [7] as well as outperforming the conventional mmWave-
MIMO [22]. Apart from maintaining satisfactory prediction perfor-
mance, the model-free FL offers a communication-efficient train-
ing, which requires approximately 10 times lower communication
exchange for model parameter transmission than the conventional
CL-based techniques.
Fig. 1. The SPIM-MIMO architecture processes the incoming data
2. SYSTEM MODEL streams {su }u∈U and employs spatial path index information s0
in a switching network, which connects NRF = U RF chains to
Consider a multi-user MIMO scenario with SPIM (SPIM-MIMO), M̄ = U M taps on the analog beamformers to exploit one of the M
where the BS has NT antennas to communicate with U users, each of spatial paths per user.
which has NR antennas, via a single data stream. Then, the vector of
all data symbols are given by s = [s1 , . . . , sU ]T ∈ CU . Additionally, (u)
the spatial path index information represented by s0 is fed to the and [aT (ϕ)]n = √1 exp{−jπ (n − 1) sin(ϕ)}, respectively.
NT
switching network (Fig. 1) to randomly assign the outputs of NRF = √ √
Σu = diag{ γu,1 , . . . , γu,M } is an M × M diagonal matrix
U ≤ M̄ RF chains to the M̄ taps of the analog beamformer. Thus, the including the scattering path gains γu,m [7].
BS can process at most M̄ spatial paths, for which M̄ = U M ≤ NT , (i)
The received signal yu is then processed by analog combiner
where M denotes the number of available spatial paths for each user. (u,i) NR
wRF ∈ C as
Compared to the conventional mmWave-MIMO, SPIM-MIMO has
(u,i)H (i) (i)
the advantage of transmitting additional data streams by exploiting yeu(i) = wRF Hu FRF FBB s + n
eu , (4)
the spatial pattern of the mmWave channel with limited RF chains, (u,i)H
i.e., NRF ≤ M̄ [7]. If M = 1, i.e., NRF = M̄ , then SPIM-MIMO where n eu = wRF nu . Similar to the analog precoders, the
(u,i)
reduces to conventional mmWave-MIMO because there is only one analog combiner wRF also has constant-modulus elements, i.e.,
(u,i) √
choice of transmission [9]. |[wRF ]n | = 1/ NR , n = 1, . . . , NR .
(i) (i)
Assume FRF ∈ CNT ×U and FBB ∈ CU ×U be the analog and (i) (i)
Our goal is to design the beamformers, FRF , FBB and WRF =
(i)
baseband beamformers corresponding to the i-th spatial pattern, re- (1,i) (U,i)
[wRF , . . . , wRF ] by exploiting SPIM. The downlink channel Hu
spectively, for i = 1, . . . , M U , i.e., selecting one of the M paths for is available for u ∈ U = {1, . . . , U } and used to design the beam-
each user. The signal vector transmitted by the NT antennas is formers with FL-based training, in which a learning model is trained
(i) (i)
x(i) = FRF FBB s. (1) to provide a mapping from the channel matrix to the beamformers.
Note that (1) includes the design of both analog and baseband beam-
formers for each spatial pattern whereas the method in [7] designs 3. BEAMFORMING VIA MODEL-BASED APPROACH
only analog beamformers and uses a fixed baseband beamformer.
(i)
The RF precoders FRF , which are constructed by phase shifters, We construct the analog beamformers via simultaneously incorporat-
(i)
have constant-modulus elements, i.e., |[FRF ]m,n | = √1 . In ad- ing all of the M spatial paths, in which the analog beamformers corre-
NT (u) (u,1) (u,M )
(i) (i) sponding to all spatial paths FRF = [f RF , . . . , f RF ] ∈ CNT ×M
dition, we have the power constraint kFRF FBB k2F
= NRF that is (u) (u,1) (u,M )
(i) per user and WRF = [wRF , . . . , wRF ] ∈ CNR ×M are designed
enforced by the normalization of FBB . Finally, the NR × 1 received
for u ∈ U. Then, we design the baseband precoders after taking into
signal by the u-th user becomes
account the interference among the users. Given Hu 1 , the analog
(i) (i)
yu(i) = Hu FRF FBB s + nu , (2) (u)
beamformer FRF ∈ CNT ×M is designed by minimizing the distance
NR ×NT (u) (u)
where Hu ∈ C represents the mmWave channel matrix be- between the beamformer FRF f BB ∈ CNT and the optimal digital
tween the BS and the u-th user and nu ∼ CN (0, σn2 INR ) is tem- opt NT
precoder fu ∈ C , available from singular value decomposition
porarily and spatially white zero-mean Gaussian noise with variance (SVD) of Hu [23]. Thus, the maximizing the spectral efficiency [20]
σn2 . The mmWave channel can be modeled as the contribution of M is equivalent to solve
clustered paths from each user [22, 23]. Thus, Hu can be given by (u) (u)
minimize kfuopt − FRF f BB k2F
(u) (u)H (u) (u)
Hu = AR Σu AT , (3) FRF ,f BB
(u) (u) (u) (u) 1 (u) (u)
where the steering matrices = [aR (φ1 ), . . . , aR (φM )] ∈
AR subject to |[FRF ]m,n | = √ , kFRF f BB k2F = M, (5)
C NR ×M (u) (u) (u)
and AT = [aT (ϕm ), . . . , aT (ϕM )] ∈ CNT ×M cor- NT
(u)
respond to the angle-of-arrival/angle-of-departure (AoA/AoD) an- which is solved for u ∈ U to obtain the analog precoders {FRF }u∈U .
gles φm and ϕm , for m = 1, . . . , M , respectively. For a uni- Similarly, the following optimization yields analog combiners
(u) (u)
form linear array (ULA), the n-th element of aR (φ) and aT (ϕ)
1
(u) 1 The estimate of Hu is obtained via both learning- [17, 24, 25] and non-learning-
can be defined as [aR (φ)]n = √ exp{−jπ (n − 1) sin(φ)} based [22, 26] approaches. We assume Hu is obtained prior to the beamformer design.
NR(u) (l) (l)
WRF ∈ CNR ×M for all possible paths: model training, i.e., minimizeθ D1u Dl=1
P u
L(f (Xu |θt−1 ), Yu ),
(u) (u) (u) with the use of the local gradient gu (θt ), where θt denotes the
minimize kwMMSE − WRF wBB k2F
(u)
WRF ,wBB
(u) model parameters at the t-th iteration and L(·) is the loss function.
Then, the u-th user transmits gu (θt ) to the BS. Once the gradient
(u) 1
subject to |[WRF ]n,m | = √ , (6) data from all users are collected, the BS finally Pincorporates gu (θt )
NR for u ∈ U to update θt as θt+1 = θt − ηt U1 U
u=1 gu (θt ), where
(u)H H −1 optH H (l) (l)
where wMMSE = fuopt HHu Hu fuopt +σn2 gu (θt ) = D1u Dl=1
P u
fu Hu is the NR ×1 ∇θ L(f (Xu |θt ), Yu )) for learning rate ηt .
optimum combiner using minimum-mean-squared-error (MMSE) After model aggregation, the BS returns the updated model parame-
estimation, which is used to obtain unconstrained combiner [23]. ters θt+1 to the users, which will be used for the computation of the
(u) (u)H (u) (u)H (u) gradients in the next iteration.
wBB = (WRF Λyu WRF )−1 (WRF Λyu wMMSE ) ∈ CM is used
(u) The proposed network architecture is a CNN comprised of 10
to compute all analog combiners. Once WRF
is found, the receiver
(u) layers. The first layer is the input layer, which accepts the input data
only uses a single column of WRF as a combiner vector. The co- of size NR × NT × 3. The {2, 4, 6}-th layers are the convolutional
variance matrix of the received signal in (2), for which the ana- layers with NSF = 128 filters, each of which employs a 3 × 3 kernel
(u) (u)
log and baseband precoders are replaced with FRF and FBB , is for 2-D spatial feature extraction. The {3, 5, 7}-th layers are the
(u) (u) (u)H (u)H
Λyu = Hu FRF FBB FBB FRF HHu +σn2 INR ∈ CNR ×NR . normalization layers. The eighth layer is a fully connected layer
The optimization problems in (5) and (6) are effectively solved via with NFCL = 1024 units, whose main purpose is to provide feature
alternating minimization techniques, such as manifold optimization mapping. The ninth layer is a dropout layer with κ = 1/2 probability.
(MO) or “Manopt” algorithm [13, 14, 24]. This is optimal in the The dropout layer applies an NFCL × 1 mask on the weights of the
sense that it achieves the minimum Euclidean distance between the fully connected layer, whose elements are uniform randomly selected
unconstrained and hybrid beamformers. from {0, 1}. As a result, at each iteration, DL randomly selects
(u) (u) different set of weights in the fully connected layer, thereby reducing
To exploit SPIM, only one column of FRF and WRF is selected the size of θt and gu (θt ), thereby, reducing model transmission
(u,i) (u) (u,i)
for the i-th spatial pattern as fRF = FRF b(u,i) and wRF = overhead. Finally, the last layer is output regression layer, yielding
(u)
WRF b(u,i) . Denote B(u) = {b(u,i1 ) , . . . , b(u,iM ) } to be the set of the output channel estimate of size (2NT U + NR ) × 1. Once the
selected paths for all possible path configurations of the u-th user training is completed, each user feeds the model with Hu and obtains
(u,i)
for im = 1, . . . , M . Then, the entries of b(u,i) ∈ RM are all zeros its beamformer wRF and F(i) , which is fed back to the BS.
except the im -th element, which is unity and denotes selection of the We further examine the transmission overhead which can be
im -th spatial path for the u-th user. defined as the size of the transmitted data during model training.
In order to mitigate interference among the users, the baseband Let TFL and TCL denote thePtransmission overhead of FL and CL,
beamformer needs to be updated by computing the effective channel respectively. Define D = u∈U Du so that TCL = 3NT NR +
(1,i)
heff 2NT U + NR D, which includes the number of symbols in the uplink
(i)
as Heff =
.
. ∈C
U ×U (u,i) (u,i) (i)
, where heff = wRF Hu FRF ∈ transmission of the training dataset D from the users to the BS. In
.
(U,i) contrast, the transmission overhead of FL includes the transmission
heff
(i) (1,i) (U,i) of gu (θt ) and θt in uplink and downlink communication for t =
C1×U where FRF = [fRF , . . . , fRF ]. Finally, the baseband pre- 1, . . . , T , respectively. Finally, TFL is given by TFL = 2P T U. We
(i) (i) (i)−1
coder FBB is obtained as FBB = Heff and it is normalized as can see that the dominant terms are D and P , which are the number of
(i) (i) (i)
[FBB ]u,: = [FBB ]u,: /k[FBB ]u,: k2 . training data pairs and the number of CNN parameters, respectively.
While D can be adjusted according to the amount of available data
at the users, P is usually unchanged during model training. Here,
4. BEAMFORMING VIA FEDERATED LEARNING P = NCL (CNSF Wx Wy ) + κNSF Wx Wy NFCL , where NCL = 3
is the number of convolutional layers and C = 3 is the number of
The learning model accepts Hu as input and yields F(i) = spatial “channels”. Wx = Wy = 3 are the 2-D kernel sizes. As a
(i) (i) (u,i)
FRF FBB ∈ CNT ×U and wRF at the output. Define Du be result, we have P = 600, 192 whereas P = 1, 190, 016 if dropout
the local dataset of the u-th user, in which the l-th element is layer is removed.
(l) (l) (l) (l)
Dl = (Xu , Yu ), where Xu and Yu are the input and out-
put for l = 1, . . . , Du , and Du = |Du | is the size of the local
5. NUMERICAL SIMULATIONS
dataset. The input Xu ∈ RNR ×NT ×3 can be constructed by
“three-channel” data, whose the first and second “channel” can be We compared the performance of FL-based SPIM-MIMO with
designed as the element-wise real and imaginary part of Hu as mmWave-MIMO and the state-of-the-art model-based SPIM-MIMO
[Xu ]1 = Re{Hu } and [Xu ]2 = Im{Hu }, respectively. Also, the Wang et al. [7] in terms of spectral efficiency averaged over 1000
third channel can be constructed as [Xu ]3 = ∠{Hu }, which is Monte Carlo trials. The local dataset of each user includes N = 200
demonstrated to improve the feature extraction performance [19, different channel realizations for U = 8 users. The number of anten-
27]. Then, the output Yu ∈ R(2NT U +NR )×1 is constructed as nas at the BS and the users are NT = 128 and NR = 9, respectively.
(u,i)T
Yu = vec{vec{Re{F(i) }, Im{F(i) }}T , ∠wRF }T . We select the number of available spatial paths for each user as
In FL, the training dataset D is partitioned into small portions, i.e., M = 2. The location of each user is selected as φu,m ∈ Φu and
Du , u ∈ U , which are available at the users and not transmitted to the ϕu,m ∈ Ψ̄u , for m = 1, . . . , M , where Φu andSΨ̄u are the equally-
BS. Let θ ∈ RP denote the learnable parameters of size P , then FL
S
divided subregions of the angular domain Θ = u∈U Φu = u Ψ̄u ,
solves the following problem for the t-th communication round of the Θ ∈ [30◦ , 150◦ ] as in [16]. During training, each channel realization15 12
11
10
10
9
8
11.5 7
5 11
10.5 6
10
9.5 5
14 16 18 20 22
0 4
-20 -15 -10 -5 0 5 10 15 20 25 30 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Fig. 2. Spectral efficiency for mmWave-MIMO and SPIM-MIMO. Fig. 3. Spectral efficiency versus γ1 when SNR = 20 dB.
is corrupted by synthetic noise on the input data for three SNRTRAIN
levels, i.e., SNRTRAIN = {20, 25, 30} dB, for G = 200 realizations FL completes data transmission
108
in order to provide robust performance against noisy input [25, 27]. after 952,000 blocks
As a result, the number of input-output pairs in the whole training FL completes data transmission
dataset is D = 3U N G = 3 × 8 × 200 × 200 = 960, 000. after 480,000 blocks
106
The proposed CNN model is realized and trained in MATLAB on
a PC with a 2304-core GPU. For CL, we use the stochastic gradient
104
descent (SGD) algorithm with momentum of 0.9 and the mini-batch
size MB = 128, and update the network parameters with learning 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
rate 0.001. For FL, we train the CNN for T = 50 iterations/rounds. 106
Once the training is completed, the labels of the validation data (i.e.,
20% of the whole dataset) are used in prediction stage. Fig. 4. Transmission overhead comparison of CL with FL, including
It was shown in [7] that SPIM-MIMO outperforms mmWave- and excluding DL.
MIMO for M = 2 with γ1 ≤ 4γ2 , where γm = γu,m for u ∈ U
and m = 1, 2. Figure 2 shows the spectral efficiency with respect
(3 · 128 · 9 + 2 · 128 · 8 + 9) · 960, 000 ≈ 5.3 × 109 , respectively. This
to SNR when the spatial path gains for all users are selected as
clearly shows the effectiveness of FL over CL, i.e., approximately
γ1 = γ2 = 0.5. Note that both SPIM-MIMO and mmWave-MIMO
10 times lower transmission overhead. In Fig. 4, we visualize the
use the same number of RF chains while SPIM-MIMO exploits the
number of transmitted symbols with respect to transmission blocks,
spatial distribution of the paths. In contrast, mmWave-MIMO designs
each of which carries 1000 symbols. We see that FL completes model
the precoders in accordance to the largest path gains, i.e., γ1 , in our
training quicker than CL after approximately 480, 000 and 952, 000
case. We observe that Wang et al. provides less spectral efficiency
transmission blocks with and without DL, respectively.
than the proposed model-based approach because it employs a single
baseband beamformer for all spatial patterns whereas the proposed
model-based approach updates the baseband beamformer FBB in
(i) 6. SUMMARY
accordance to the different spatial patterns as well as suppressing
the interference among the users. The proposed FL approach has We presented both model-based and model-free frameworks for beam-
slight performance loss than the model-based method due to the loss former design in multi-user SPIM-MIMO systems. Whereas the for-
during model training. It is worth noting that the performance of mer leverages MO for beamformer design, the latter employs FL to
FL is upper bounded by the model-based technique since FL cannot efficiently train the learning model. Our experiments showed that
perform better than its labels. the proposed approach has superior performance than the state-of-
the-art SPIM techniques as well as outperforming the conventional
In Fig. 3, we compare SPIM-MIMO and mmWave-MIMO with
mmWave-MIMO systems in terms of spectral efficiency. Further-
respect to γ1 when γ2 = 1 − γ1 . We observe that both techniques
more, the proposed FL approach exhibits a more communication-
meet when γ1 = 4γ2 for γ1 = 0.8. This clearly shows that the usage
efficient learning method than conventional CL for model training.
of SPIM is appropriate if the path gain are close. The SPIM-MIMO
We demonstrated that FL with (without) DL enjoys approximately 10
performance degrades as long as the difference between the path
(5) times lower transmission overhead during model training lower
gains are large. As a result, mmWave-MIMO becomes favorable.
transmission overhead than CL.
We note from both Fig. 2 and Fig. 3 that our proposed FL approach
closely follows the model-based technique.
Next, we present the effectiveness of FL-based model training
by comparison to the CL-based training. According to the analysis
in Sec. 4, the transmission overhead of FL and CL are 2P T U =
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