5G Air-to-Ground Network Design and Optimization: A Deep Learning Approach

Our goal is to replace the cumbersome system-level NR simulator with an agile entity to expedite the training process for Optimizer-NN. Approximator-NN, denoted as $\mathcal{N}_{A}(\cdot)$, is a DNN designed to model the behavior of an NR A2G network and estimate its performance. Specifically, it approximates the $X$th-percentile user throughput for a given set of input parameters consisting of ISD, antenna up-tilt angles and traffic load. All other system parameters such as the number of BSs and the number of sectors per site remain fixed. We use the data generated from a 5G NR system-level simulator for the training and validation of Approximator-NN. Formally,

denotes the approximated user throughput output by Approximator-NN, and ${\textbf{w}_{A}}$ denotes the neural network (NN) parameters which are initialized randomly. As illustrated in Fig. 3a, the DNN consists of 3 fully connected hidden layers with a rectified linear unit (ReLU) [13] as the activation function. For training, we select the mean squared error (MSE) based on the throughput outputs of $\mathcal{N}_{A}(\cdot)$ and the NR simulator as the loss function, and use Adam [14] as the optimizer.

Optimizer-NN, denoted as $\mathcal{N}_{D}(\cdot)$, is a DNN designed for solving A2G deployment optimization problem leveraging the results obtained by Approximator-NN. It outputs the throughput-optimal values for the A2G deployment parameters which are also fed as an input to Approximator-NN. A neural network is usually trained to minimize a certain error function. For Optimizer-NN, we set the error function equal to the negative of the average throughput of a batch of candidate deployment configurations. Consequently, minimizing the considered error function during the training process ends up maximizing the average throughput in the batch.

In this work, we implement Optimizer-NN using a word embedding algorithm [15]. Word embedding, usually used for solving natural language problems, aims to map the words in a word set to numerical vectors that represent the word features. In our model, we treat a batch consisting of $n$ candidate deployment configurations $\{[d_{i},\mathbf{\Theta}_{i},l_{i}]\}_{i=1}^{n}$ as the word set to be embedded, i.e., the embedding vector for the $i$th word contains the features $[d_{i},\mathbf{\Theta}_{i},l_{i}]$, same as the configuration parameters to be optimized. Hence, finding the optimal embedding vectors for the deployment configurations leads to throughput-optimal ISD and up-tilt angles.

The training process of Optimizer-NN is based on a modified version of word2vec [15]. The detailed architecture is shown in Fig. 3b. A Tanh layer is put before ReLU to normalize the output range from the network to [-1,1] so that after ReLU, the output is restricted to [0,1]. This enables proper processing of the outputs which further serve as inputs for Approximator-NN. After the convergence, the output from Optimizer-NN consists of a batch of well-tuned configurations, among which we can select the optimal deployment that maximizes the user throughput.

We next describe the modified version of word2vec algorithm for our use case. Traditionally, in a word embedding NN, the indices $\{n\}=\{0,...,n-1\}$ of the words in a given word set with cardinality $n$ constitute the input for the embedding layer. The desired outputs are the numerical embedding vectors representing either explicit or implicit features of each word in the word set. In this work, we treat a set of $n$ candidate configurations as a word set, i.e., Optimizer-NN takes the set of indices $\{n\}$ as an input, and outputs a matrix D containing vectors representing each deployment configuration:

where the matrix $\textbf{w}_{D}$ contains the NN parameters. Furthermore, $\textbf{D}=\left[D_{0},\cdots,D_{n-1}\right]^{\textrm{T}}\in\mathds{R}^{n\times(\textrm{S}+2-c)}$, where each row $D_{i}$ for $i\in\{n\}$ equals $[d_{i},\mathbf{\Theta}_{i},l_{i}]$ when $c=0$, and T denotes the matrix transpose. The deployment configuration $D_{i}$ that, when input to Approximator-NN, yields the maximum user throughput is chosen as the optimal solution, i.e.,

For training Optimizer-NN, we define the loss function $L(\mathbf{w}_{D})$ as the negative of the batch average of the approximated user throughput, i.e.,

We use the Adam optimizer [14] for minimizing $L(\mathbf{w}_{D})$ such that the output D from $\mathcal{N}_{D}(\{n\};\textbf{w}_{D})$ contains candidate configurations with potentially high throughputs. We describe the training process in Algorithm 1. We initialize the matrices $\mathbf{w}_{D}$ and D randomly and run $\tau$ iterations per epoch (lines 7-21). If we intend to fix (rather than optimize) any of the parameters $\{d,\mathbf{\Theta},l\}$, the fixed parameter(s) can be concatenated with the output from Optimizer-NN (line 8). We compute the mean error for the loss function in line 15. The batch $\textbf{D}^{\prime}$ with the minimum error is stored for every epoch (lines 17-20) and the batch $\textbf{D}^{*}$ with the minimum error across all the epochs $\epsilon^{*}$ is retained (lines 22-28). Optimizer-NN converges when $\epsilon^{*}$ remains unchanged for more than $k_{\max}$ iterations. Finally, the throughput-optimal parameter values are obtained using (8) (line 30). The algorithm has a linear time complexity, which makes it attractive for implementation.

We train Approximator-NN to output the 50th percentile user throughput $t_{50}$. We use the parameter values shown in Table II for data collection. For $S=1$, the number of neurons in each hidden layer is $10$. For $S\in\{3,4\}$, each hidden layer has $16$ neurons to further improve the approximation accuracy. The training dataset consists of $760$, $48905$, and $30405$ entries for $S=1$, $S=3$, and $S=4$, respectively.

In Fig. 4, we show the convergence of the training process in terms of the MSE of the user throughput. In Fig. 5, we plot the cumulative distribution function (CDF) of the user throughput prediction error. We observe that for $S=1$, 95% of the prediction errors are 6.5 Mbps or lower. For $S\in\{3,4\}$, 95% of the prediction errors are within 4.5 Mbps thanks to larger datasets. We emphasize that our goal is to optimize A2G network parameters to achieve high throughput, which is typically on the order of hundreds of Mbps in the considered scenario. This suggests that the throughput approximation errors in Approximator-NN are relatively small compared to the anticipated throughput range of an A2G system. Therefore, Approximator-NN can provide satisfactory throughput approximations for a given input $[d,\mathbf{\Theta},l]$.

We obtain optimal deployment configuration using Algorithm 1 with parameter values $\tau=1000$, $k_{\max}=20$, $n=4$ for $S=1$, and $n=8$ for $S\in\{3,4\}$. For each $S\in\{1,3,4\}$, the optimal configuration based on Optimizer-NN has an ISD of 20 km and a traffic load of 4 Mbps, same as the minimum allowed values in (5). The optimal configurations in the dataset have an ISD of 20 km and a traffic load of 10 Mbps, same as the minimum possible values in the dataset. This trend is plausible since a smaller ISD helps improve the received signal power while a lower traffic load reduces the inter-cell interference.

In Table III, we provide the optimal values for the antenna up-tilt angles with the corresponding user throughput for both Optimizer-NN and the dataset. The reported results for throughput are obtained directly from the simulator by plugging in the optimized configuration parameters. We find that the antennas need to be up-tilted somewhat aggressively (around 60$\degree$ to 90$\degree$) due to a relatively small ISD of 20 km. For the 4-sector case, one antenna needs to be in an almost upright position to boost throughput. This is because three sectors are sufficient to cover the entire cell due to its small size. With an almost upright fourth sector, the inter-cell interference is reduced as more power is directed to the region above the BS.

We note that the proposed algorithm learns the benefit of operating in a lightly-loaded system, as evident from the optimal load of 4 Mbps based on Optimizer-NN despite the minimum value of 10 Mbps considered in the dataset. Consequently, the throughput for the optimal configurations based on Optimizer-NN is higher than that based on the dataset.

We next consider another practically relevant scenario where the ISD is large. In Table IV, we provide the optimal antenna up-tilt angles for different number of sectors for a fixed ISD of 80 km. We observe that due to a larger ISD, it is optimal to have only moderately up-tilted antennas (around $30\degree$) for coverage extension. This finding holds for all considered values for the number of sectors. With a small ISD, the network is interference-limited and using highly up-tilted antennas (as in Table III) helps direct the signal power towards the aerial region above the BS, thus reducing inter-cell interference. As the cell size expands, the network becomes coverage-limited and the antennas need to be slanted to provide coverage throughout the cell. Furthermore, the larger ISD also results in a lower throughput. For instance, compared to the case with 20 km ISD, $t_{50}$ is reduced by 49%, 35% and 32% for $S=1$, $S=3$, and $S=4$, respectively.

In Fig. 6, we examine the throughput variation for the optimal antenna configuration in Table III and the optimal ISD under different traffic conditions. A higher traffic load corresponds to a higher resource utilization. The plot reaffirms the intuition that a higher traffic load leads to a stronger inter-cell interference, thus reducing the average user throughput. We also observe that the throughput for the optimal configuration obtained using the proposed framework is almost indistinguishable from that obtained using the dataset. Another noticeable trend in Fig. 6 is the throughput variation for different sectors under varying traffic conditions. Under a low traffic load, the 1-sector case results in the lowest throughput among all cases. As the traffic load increases, the 1-sector case yields the highest throughput while the 4-sector case the lowest. This is because deploying more sectors per site reduces the cell size which increases the received signal power for a typical user. This, however, also increases the total radiated power per site, which manifests as inter-cell interference. Under a high traffic load, the degradation due to interference overshadows the improvement in the received signal power, thus reducing the SINR and throughput. Finally, in Fig. 7, we observe similar SINR CDFs for the optimal configurations based on Optimizer-NN and the dataset.