# Stein Variational Gradient Descent-based Detection for Random Access with
Preambles in MTC

###### Abstract

Traditional preamble detection algorithms have low accuracy in the grant-based
random access scheme in massive machine-type communication (mMTC). We present
a novel preamble detection algorithm based on Stein variational gradient
descent (SVGD) at the second step of the random access procedure. It
efficiently leverages deterministic updates of particles for continuous
inference. To further enhance the performance of the SVGD detector, especially
in a dense user scenario, we propose a normalized SVGD detector with momentum.
It utilizes the momentum and a bias correction term to reduce the preamble
estimation errors during the gradient descent process. Simulation results show
that the proposed algorithm performs better than Markov Chain Monte Carlo-
based approaches in terms of detection accuracy.

Index Terms—  Preamble detection, Stein variational gradient descent, grant-
based random access, massive machine-type communication (mMTC).

##  1 Introduction

Under the background of massive machine-type communication (mMTC) [1, 2], the
future wireless communication system needs to hold a large number of devices.
However, the preamble resources are limited. With the number of devices
increasing, the preamble collision problem [3, 4, 5] becomes more serious,
which increases the difficulty of the communication systems design. To reduce
power consumption and achieve massive connection in the future Internet of
Things (IoT) scenarios [1], communication systems need a random access scheme
that can alleviate the preamble collision problem. At present, such random
access schemes are mainly divided into two categories: grant-based random
access (GBRA) and grant-free random access (GFRA) [6, 7, 8].

In this paper, we use the GBRA scheme. GBRA [9, 10, 11, 12] includes a four-
step handshake random access protocol: (1) Each active user randomly selects a
preamble from the preamble pool. (2) The base station (BS) delivers random
access responses to users. (3) Users return MSG3 to the BS. (4) The BS
performs random access conflict resolution. In the dense user scenario, a
large number of users access the BS at the same time. Therefore, preamble
collision can not be avoided [3], which significantly increases uplink
transmission signaling overhead. In addition, it also increases the access
delay and reduces the throughput of the system [13]. In the GBRA, colliding
users can only be recognized by the base station in the final step. However,
the base station will still allocate time-frequency resources to the colliding
users in the second step, which causes a waste of resources. Hence, efficient
preamble detection schemes are necessary for random access to detect preamble
collisions in advance.

A preamble detection random access scheme based on Markov Chain Monte Carlo
(MCMC) was proposed in [14]. This scheme establishes the maximum a posteriori
(MAP) estimation model [15] to detect preambles. However, the MAP estimation
needs the prior distribution of the estimated variables. Additionally, the
MCMC method increases the diversity of particles through the randomness of
sampling, which decreases the accuracy of preamble detection.

To detect preamble collision early and improve the utilization of wireless
resources, we establish a preamble detection model based on maximum likelihood
estimation [16] without using any prior distribution in the second step of
handshaking. Furthermore, we propose two efficient algorithms based on Stein
variational gradient descent (SVGD) [17, 18, 19, 20] to find an approximate
solution to the maximum likelihood estimation problem.

The contributions are summarized as follows: (1) We propose a maximum
likelihood estimation model based on the SVGD detector to detect preambles in
the dense user scenario. (2) Through error analysis, we propose the normalized
SVGD (NSVGD) detector with momentum. It has better robustness and a higher
preamble detection accuracy than the SVGD detector and MCMC-based methods.

##  2 Background

###  2.1 Stein Variational Gradient Descent (SVGD)

SVGD is a novel particle-based variational inference algorithm [17]. It
efficiently utilizes gradient information for approximating the target
distribution through deterministic updates of particle methods. As shown in
Algorithm 1, $n$ particles are generated by the uniform distribution or some
other distribution. Then, the particles are updated using an optimized
gradient ${\varphi}$. Liu et al. [17] proved that the perturbation direction
given by ${\varphi}$ is optimal because it corresponds to the steepest descent
on the Kullback–Leibler divergence. After sufficient iterations, the obtained
particles follow the target distribution. In this paper, we will apply the
SVGD algorithm to take samples from the complicated distribution.

Input: A target function $T(\mathbf{x})$ and a set of initial paticles
$\\{\mathbf{x}_{i}^{0}\\}_{i=1}^{n}$

Output: A set of particles $\\{\mathbf{x}_{i}\\}_{i=1}^{n}$ that approximates
the target function

1

2for  _iteration $t$_ do

3
$\mathbf{x}_{i}^{t+1}\leftarrow\mathbf{x}_{i}^{t}+\varepsilon\varphi(\mathbf{x}_{i}^{t})$
where
$\varphi(\mathbf{x})=\frac{1}{n}\sum_{j=1}^{n}[k(\mathbf{x}_{j}^{t},\mathbf{x})\bigtriangledown_{\mathbf{x}_{j}^{t}}logT(\mathbf{x}_{j}^{t})+\bigtriangledown_{\mathbf{x}_{j}^{t}}k(\mathbf{x}_{j}^{t},\mathbf{x})]$

4 where $\varepsilon$ is the step size. $k(x,\cdot)$ is kernel function.

5 end for

Algorithm 1 SVGD

###  2.2 System Model

Assume that there are $N$ active users in the cell. Each user is equipped with
an antenna and the BS is equipped with $K$ antennas. The number of preambles
is $M$, and the length of each preamble is $S$. Each active user randomly
selects a preamble in the pool. Then, the signal received by the BS through
antenna $j$ can be expressed as:

$\mathbf{y}_{j}=\sum_{i=1}^{N}\mathbf{p}_{(i)}H_{i,j}e_{i}+\mathbf{n}_{j},$
(1)

for $j=1,\dots,K$, where $\mathbf{n}_{j}\in\mathbb{C}^{S}$ is the background
noise at the $j$-th antenna. $H_{i,j}$ is the channel coefficient from $i$-th
active device to $j$-th antenna. $e_{i}$ indicates the data symbol sent by the
active user to the base station. $\mathbf{p}_{(i)}\in\mathbb{C}^{S}$ stands
for the preamble sequence selected by $i$-th active user.

Suppose $x_{m}$ indicates the number of active users who select the $m$-th
preamble. The numbers of users selecting each preamble are represented by a
vector $\mathbf{x}=[x_{1},\dots,x_{m},\dots,x_{M}]^{\mathrm{T}}$, where
$x_{m}\in[0,N]$. Since our model is designed to detect preamble collision as
mentioned in Section 1, we focus on estimating $\mathbf{x}$ in the following
steps: Firstly, we define the likelihood of $\mathbf{x}$:

${\rm{Lik}(\mathbf{x})}=f({\mathbf{y}_{j}}\mid\mathbf{x}),$ (2)

where $f({\mathbf{y}_{j}}\mid\mathbf{x})$ represents the Likelihood function
of $\mathbf{x}$ given for $\mathbf{y}_{j}$.

Then, let
$\mathbf{w}_{j}=[w_{1,k},w_{2,j},\dots,w_{m,j},\dots,w_{M,j}]^{\mathrm{T}}$,
and

$w_{m,j}=\sum_{i\in\mathcal{\mathcal{N}}_{m}}H_{i,j}e_{i},$ (3)

where $\mathcal{N}_{m}$ stands for the index set of active users who select
the $m$-th preamble. We assume that the signal transmitted by the active
device experiences independent Rayleigh fading:

$H_{i,j}\sim\mathcal{CN}(0,\delta^{2})$ (4)

Next, let $\mathbf{P}=[\mathbf{p}_{1},\dots,\mathbf{p}_{M}]$. The Eq. (1) can
be rewritten as:

$\mathbf{y}_{j}=\mathbf{P}\mathbf{w}_{j}+\mathbf{n}_{j},$ (5)

for $j=1,\dots,K$.

According to Eq. (3) and Eq. (4), $\mathbf{w}_{j}$ is a circularly-symmetric-
complex-Gaussian (CSCG) vector for a given $\mathbf{x}$. Furthermore, from Eq.
(4) and Eq. (5), $\mathbf{y}_{j}$ is also a CSCG vector for a given
$\mathbf{x}$. Its mean value is 0 and its covariance matrix is:

$\mathbb{E}[\mathbf{y}_{j}\mathbf{y}_{j}^{\mathrm{H}}\mid\mathbf{x}]=\delta^{2}\mathbf{P}\mathbf{V}_{\mathbf{x}}\mathbf{P}^{\mathrm{H}}+\beta\mathbf{I}=\phi(\mathbf{x}),$
(6)

where $\mathbf{V}_{\mathbf{x}}$ = diag$(x_{1}\dots x_{M})$.
$(\cdot)^{\mathrm{H}}$ represents conjugate transpose, and $\beta$ is the
noise power. According to Eq. (6), we have

$\mathbf{y}_{j}\mid\mathbf{x}\sim CN(\mathbf{0},\phi(\mathbf{x})),$ (7)

After that, $\ln f(\mathbf{y}_{j}\mid\mathbf{x})$ can be computed as:

$\begin{split}\ln
f(\mathbf{y}_{j}\mid\mathbf{x})=-\mathbf{y}_{j}^{\mathrm{H}}(\phi(\mathbf{x}))^{-1}\mathbf{y}_{j}-\ln\det(\phi(\mathbf{x}))+\xi,\end{split}$
(8)

where $\xi$ represents a constant. $\det(\cdot)$ stands for the determinant of
the matrix. According to
$f(\\{\mathbf{y}_{j}\\}\mid\mathbf{x})=\prod\nolimits_{j=1}^{K}f(\mathbf{y}_{j}\mid\mathbf{x})$,
log-likelihood function$\ln f(\\{\mathbf{y}_{j}\\}\mid\mathbf{x})$ can be
represented as:

$\begin{split}\ln
f(\\{\mathbf{y}_{j}\\}\mid\mathbf{x})=\sum\limits_{j=1}^{K}\ln
f(\mathbf{y}_{j}\mid\mathbf{x}),\end{split}$ (9)

Finally, the maximum likelihood estimation model can be derived by using the
log-likelihood function:

$\tilde{\mathbf{x}}=\mathop{\arg\max}f(\\{\mathbf{y}_{j}\\}\mid\mathbf{x}).$
(10)

The computational complexity can be expressed as $(N+1)^{M}$, which increases
exponentially with $M$. Therefore, the computational complexity of the maximum
likelihood detection increases significantly when $M$ becomes large.

##  3 Methodology

Since maximum likelihood detection is computationally prohibitive, the SVGD is
employed to find an approximate solution. Specifically, we use SVGD to take
samples of $\mathbf{x}$ from the distribution with the density function
$q(\mathbf{x})=f(\\{\mathbf{y}_{j}\\}\mid\mathbf{x})$.

###  3.1 SVGD Detector

Eq. (8) shows that for any $\mathbf{x}$, $f(\mathbf{y}_{j}\mid\mathbf{x})\geq
0$, which satisfies the SVGD requirement that the density function is positive
[17]. Then, we approximate $q(\mathbf{x})$ with a set of particles
$\\{\mathbf{x}_{i}\\}_{i=1}^{n}$, where $\mathbf{x}_{i}\in\mathbb{R}^{M}$ and
$n$ is the number of particles. Next, $\\{\mathbf{x}_{i}^{0}\\}_{i=1}^{n}$ is
used to initialize particles and SVGD updates the particles iteratively by

$\mathbf{x}_{i}^{t+1}\leftarrow\mathbf{x}_{i}^{t}+\varepsilon{\varphi}(\mathbf{x}_{i}^{t}),$
(11)

where $\varepsilon$ is a step size. $\varphi(\cdot)$ is a velocity field that
deterministically drives the distribution of particles toward the target. It
is computed as:

${\varphi}(\mathbf{x})=\frac{1}{n}\sum_{l=1}^{n}[k(\mathbf{x}_{l}^{t},\mathbf{x})\bigtriangledown_{\mathbf{x}_{l}^{t}}\log
q(\mathbf{x}_{l}^{t})+\bigtriangledown_{\mathbf{x}_{l}^{t}}k(\mathbf{x}_{l}^{t},\mathbf{x})],$
(12)

where $k(\mathbf{x}_{l}^{t},\mathbf{x}_{i}^{t})$ is the kernel function. The
selection of $k(\mathbf{x}_{l}^{t},\mathbf{x}_{i}^{t})$ will be introduced in
section 4. After a number of iterations, we obtain a set of points
$\\{\mathbf{\tilde{x}}_{i}\\}_{i=1}^{n}$ that approximate our target
distribution with the density function $q(\mathbf{x})$, so we can estimate the
number of users selecting each preamble using
$\\{\mathbf{\tilde{x}}_{i}\\}_{i=1}^{n}$. For example, as shown in Fig. 1,
given that $M=2,S=2,N=4,K=20,n=50$, the purpose is to estimate
$\mathbf{x}=[x_{1},x_{2}]$. At the beginning, fifty particles are initialized
with a random distribution $q_{0}(\mathbf{x})$. Then, the particles are
updated using Eq. (12). Repeating this iteration, a path of distributions
$\\{\mathbf{x}_{i}\\}_{i=1}^{50}$ between the $q_{0}(\mathbf{x})$ and
$q(\mathbf{x})$ is constructed. Finally, the particles converge to the ground
truth.

Fig. 1: Updating process of particles

###  3.2 Error Analysis of the SVGD Detector

Given a noisy environment, there will be a large noisy power $\beta$ in the
system. It is observed that if $\beta$ is significantly larger than each entry
of matrix
$\psi(\mathbf{x})=\delta^{2}\mathbf{P}\mathbf{V}_{\mathbf{x}}\mathbf{P}^{\mathrm{H}}$,
$\phi(\mathbf{x})=\delta^{2}\mathbf{P}\mathbf{V}_{\mathbf{x}}\mathbf{P}^{\mathrm{H}}+\beta\mathbf{I}$
will be independent to the change of $\psi(\mathbf{x})$. Then,
$\phi(\mathbf{x})\thickapprox\beta\mathbf{I}$. Therefore, Eq. (9) can be
computed as:

$\log
q(\mathbf{x}_{i}^{t})=-\sum_{j=1}^{K}\mathbf{y}_{j}^{\mathrm{H}}\mathbf{(}\beta\mathbf{I})^{-1}\mathbf{y}_{j}-K\ln\det(\beta\mathbf{I})+\xi,$
(13)

thus, $\bigtriangledown_{\mathbf{x}_{i}^{t}}\log q(\mathbf{x}_{i}^{t})=0$.
Then ${\varphi}(\mathbf{x})$ can be computed as:

${\varphi}(\mathbf{x})=\frac{1}{n}\sum_{i=1}^{n}[\bigtriangledown_{\mathbf{x}_{i}^{t}}k(\mathbf{x}_{i}^{t},\mathbf{x})],$
(14)

from Eq. (14), only
$\bigtriangledown_{\mathbf{x}_{i}^{t}}k(\mathbf{x}_{i}^{t},\mathbf{x})$ can be
used to update particles. However,
$\bigtriangledown_{\mathbf{x}_{i}^{t}}k(\mathbf{x}_{i}^{t},\mathbf{x})$ does
not include any information about the target density function $q(\mathbf{x})$.
As a consequence, errors are introduced during the update process.

Fig. 2: Estimation of active users in a dense user scenario

Fig. 2 shows the estimation of active users in a dense user scenario with
$(M,S,N,K)=(20,10,20,30)$. When the signal-to-noise ratio (SNR) is low, there
is a large error between the estimated value $||\tilde{\mathbf{x}}||_{1}$ and
the true value $||{\mathbf{x}}||_{1}$. With SNR increasing,
$\bigtriangledown_{\mathbf{x}_{i}^{t}}\log q(\mathbf{x}_{i}^{t})$ becomes non-
negligible. Therefore, the SGVD detector can extract effective gradient
information to update particles, which improves the detection accuracy.

Input: A target function $q(\mathbf{x})$ and a set of initial particles
$\\{\mathbf{x}_{i}^{0}\\}_{i=1}^{n}$

Output: A set of particles $\\{\tilde{\mathbf{x}}_{i}\\}_{i=1}^{n}$ that
approximates the target function

1for  _iteration $t$_ do

2 Calculate the error $\Delta$ using Eq. (15).

3 Update the gradient ${\varphi}(\mathbf{x}_{i}^{t})$ using Eq. (12).

/* Accumulate history gradients */

4 if  _$\alpha\neq 0$_ then

5 if  _$t >1$_ then

6
${\mathbf{g}}_{t}\leftarrow\alpha{\mathbf{g}}_{t-1}+(1-\alpha){\varphi}^{2}(\mathbf{x}_{i}^{t})$

7 else

8 ${\mathbf{g}}_{t}\leftarrow{\varphi}^{2}(\mathbf{x}_{i}^{t})$

9 end if

10

11 end if

12

13 Compute ${\varphi}(\mathbf{x}_{i}^{t})$ using Eq. (16).

/* Weight Decay */

14 if  _$\lambda\neq 0$_ then

15
${\varphi}(\mathbf{x}_{i}^{t})\leftarrow{\varphi}(\mathbf{x}_{i}^{t})-\lambda\mathbf{x}_{i}^{t}$

16 end if

17

/* Gradient with Momentum */

18 if  _$\gamma\neq 0$_ then

19 if  _$t >1$_ then

20
${\mathbf{r}}_{t}\leftarrow\gamma{\mathbf{r}}_{t-1}+(1-\gamma){\varphi}(\mathbf{x}_{i}^{t})$

21 else

22 ${\mathbf{r}}_{t}\leftarrow{\varphi}(\mathbf{x}_{i}^{t})$

23 end if

24 ${\varphi}(\mathbf{x}_{i}^{t})\leftarrow{\mathbf{r}}_{t}$

25 end if

26

27 Compute $\mathbf{x}_{i}^{t+1}$ using Eq. (17).

28 end for

Algorithm 2 Normalized SVGD Detector with Momentum

###  3.3 Normalized SVGD Detector with Momentum

The irregular gradients and noise have a negative influence on the update of
particles. To overcome these limitations of the SVGD detector, we propose a
normalized SVGD (NSVGD) detector with momentum. Firstly, as is shown in
Algorithm 2, a bias correction term $\Delta$ is added to the updating process
to improve the anti-interference performance. This bias term $\Delta$
calculates the error between the estimated and actual number of active users
using

$\Delta\leftarrow\mu\left(nN-\sum_{i=1}^{n}||\mathbf{x}_{i}^{t}||_{1}\right).$
(15)

This error corrects the updating direction of particles. In a noisy
environment, although $\bigtriangledown_{\mathbf{x}_{i}^{t}}\log
q(\mathbf{x}_{i}^{t})$ will disappear during the update process, the bias term
$\Delta$ can still provide useful information for the update of particles.
Therefore, the bias term $\Delta$ can improve the robustness of the detector
in different environments.

Next, we compute the gradient using Eq. (12). After that, we accumulate the
history gradients and normalize the accumulated gradient:

${\varphi}(\mathbf{x}_{i}^{t})\leftarrow\frac{{\varphi}(\mathbf{x}_{i}^{t})}{\epsilon+\sqrt{{\mathbf{g}}_{t}}},$
(16)

where $\sqrt{(\cdot)}$ represents element-wise square root. These operations
solve the problem of continuous decline in learning rate in comparison to the
AdaGrad method [21]. Additionally, the weight decay [22] is applied to adjust
the gradient to find an optimal result. Moreover, we apply momentum, a
strategy that assists in navigating high error and low curvature regions [23].
Finally, the particles are updated according to the gradient and estimated
error:

$\mathbf{x}_{i}^{t+1}\leftarrow\mathbf{x}_{i}^{t}+\varepsilon{\varphi}(\mathbf{x}_{i}^{t})+\Delta.$
(17)

##  4 Experimental Results

In this section, we compare our methods with the MCMC method using Gibbs
sampling [14]. We consider non-orthogonal preambles. The elements of
$\mathbf{P}$ are independent circularly-symmetric-complex-Gaussian (CSCG)
random variables with zero-mean and variance $\frac{1}{S}$, i.e.,
$[{\rm{P}}]_{u,v}\sim\mathcal{CN}(0,\frac{1}{S})$. We initialize vector
$\mathbf{x}$ in a uniform distribution on $[1,1.1]$ in both the SVGD detector
and the normalized SVGD detector with momentum. In addition, we use Gaussian
radial basis function (RBF) kernel [24]. It is defined as
$k(\mathbf{x},\mathbf{x}^{\prime})=\exp(-\frac{1}{h}\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}^{2})$,
where bandwidth $h=\frac{\rm{med}^{2}}{\log n}$, and med is the median of the
pairwise distance between the current points $\\{\mathbf{x}_{i}\\}^{n}_{i=1}$.
The parameters $n$, $\mu$, $\alpha$, ${\epsilon}$, $\gamma$, $\varepsilon$,
and $\lambda$ are chosen as 6, 0.01, 0.9, 1, 0.9, 0.01, and 0.1, respectively.
We use $N_{iteration}$ to represent the number of iterations to obtain stable
particles. The sample mean can be obtained by all particles:

$\bar{\mathbf{x}}=\frac{1}{n}\sum_{i=1}^{n}\tilde{\mathbf{x}}_{i}=[\bar{x}_{0},\dots,\bar{x}_{m},\dots,\bar{x}_{M}].$
(18)

We use the rounded sample mean to estimate $x_{m}$, i.e.,
$\hat{x}_{m}=\lfloor\bar{x}_{m}\rceil\in\\{0,\dots,N\\}$, where $\lfloor
x\rceil$ represents the nearest integer of $x$.

We consider two performance measures. The first one is the mean square error
(MSE). The second one is to test the accuracy of different methods named the
probability of activity detection error:

$\displaystyle\begin{array}[]{l}{\rm{P_{ADE}}}={\rm{Pr}}(x_{m}\neq\hat{x}_{m}).\end{array}$
(20)

Fig. 3: Probability of ADE for different SNR when $M=20,S=10,K=30$.
Fig. 4: MSE for different SNR when $M=20,S=10,K=30$.

In Fig. 3 and Fig. 4, with $\rm{P_{ADE}}$ and MSE, we compare the performance
of the NSVGD detector with momentum, the SVGD detector, and the MCMC-based
detector for different values of SNR in a dense user scenario. The NSVGD
detector with momentum outperforms the SVGD detector and the MCMC approach.
The improvement of performance at a low SNR shows that the NSVGD detector can
reduce the impact of external environmental noise on the SVGD detector. The
improvement of performance at a high SNR proves that the NSVGD detector has
better robustness and self-adaptation than the SVGD detector. In addition,
with the number of active users decreasing from 30 to 20, the possibility of
preamble collision decreases gradually and the detection accuracy of the three
algorithms also improves.

The computation complexities of the MCMC-based detector, SVGD detector and
NSVGD detector are $O(MS(S+K))$, $O(MKS^{3})$ and $O(M(KS^{3})+16M)$
respectively. It is noticed that the computational complexities of the three
detectors are all linearly proportional to the number of preambles $M$ and do
not change when the number of active users increases. Therefore, three
detectors have a similar computation complexity.

##  5 Conclusion

In this paper, we propose a novel preamble detection algorithm based on SVGD,
which efficiently leverages deterministic updates of particles to complete
preamble detection. Through the error analysis, the performance of the SVGD
detector degrades due to the noise. To solve this problem, we propose the
NSVGD detector with momentum, which adds a bias correction term to enhance the
robustness. Simulation results show that the proposed algorithm performs
better than MCMC-based approaches in a dense user scenario. Moreover, the
NSVGD detector has a low computation complexity, which is linearly
proportional to the number of preambles.

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