# Entropy-based Active Learning of Graph Neural Network Surrogate Models for
Materials Properties

Johannes Allotey  School of Physics, University of Bristol, BS8 1TL, UK  Keith
T. Butler<EMAIL_ADDRESS>Scientific Machine Learning Research Group,
Scientific Computing Department, Rutherford Appleton Laboratory, Science and
Technology Facilities Council, Didcot, OX11 0DQ, UK
Jeyan Thiyagalingam  Scientific Machine Learning Research Group, Scientific
Computing Department, Rutherford Appleton Laboratory, Science and Technology
Facilities Council, Didcot, OX11 0DQ, UK

###### Abstract

Graph neural networks, trained on experimental or calculated data are becoming
an increasingly important tool in computational materials science. Networks,
once trained, are able to make highly accurate predictions at a fraction of
the cost of experiments or first-principles calculations of comparable
accuracy. However these networks typically rely on large databases of labelled
experiments to train the model. In scenarios where data is scarce or expensive
to obtain this can be prohibitive. By building a neural network that provides
a confidence on the predicted properties, we are able to develop an active
learning scheme that can reduce the amount of labelled data required, by
identifying the areas of chemical space where the model is most uncertain. We
present a scheme for coupling a graph neural network with a Gaussian process
to featurise solid-state materials and predict properties including a measure
of confidence in the prediction. We then demonstrate that this scheme can be
used in an active learning context to speed up the training of the model, by
selecting the optimal next experiment for obtaining a data label. Our active
learning scheme can double the rate at which the performance of the model on a
test data set improves with additional data compared to choosing the next
sample at random. This type of uncertainty quantification and active learning
has the potential to open up new areas of materials science, where data are
scarce and expensive to obtain, to the transformative power of graph neural
networks.

††preprint: APS/123-QED

##  I Introduction

Machine learning (ML) has become an important tool in almost every modern
scientific discipline, and materials’ science is no exception [1]. In
particular, the proliferation of data in recent years has given rise to the
advent of deep learning approaches, where neural networks with hundreds of
thousands or even millions of parameters learn to infer trends from relatively
unstructured data [2, 3, 4, 5, 6]. In materials’ science, network
architectures based on graphs, Graph Neural Networks (GNNs), have proved to be
successful both for molecules and condensed matter systems[3, 2, 6]. GNNs
allow the encoding of domain knowledge about connections (bonds) into the
topography of the underlying neural network architecture. Despite the great
success and promise of GNNs for materials’ science applications a number of
questions remain open. In this paper we address two of these questions (i) how
much can we trust the results of a GNN on a previously unseen sample, and (ii)
how can we train a GNN if data is scarce or labelled data is expensive to
obtain?

Figure 1: The structure of the convolution fed Gaussian process (CFGP). The
graph network part of megnet learns representations for materials, in the
standard model these are fed to a multi-layer perceptron for function
approximation, in the CFGP these are used as input for a Gaussian process for
function approximation.

Uncertainty quantification (UQ) of machine learning models, and neural
networks in particular, is currently an area of intense research. Conventional
neural networks return a single output value per property that they are
trained on (classes or values in a classification or a regression scenario).
These values contain neither information about how confident the network is,
nor how the output varies with variation of the input. In a recent high-
profile example, a network trained to classify COVID-19 infection from lung
X-rays classified a cat as COVID-19 infected lung[7]. In this case an
estimation of certainty would have been very beneficial.

Bayesian neural networks (BNNs) provide NNs with a principled UQ[8]. However,
BNNs are not scalable to most practical problems. Although various
approximations, eg Monte Carlo dropout and mean field variational inference,
[9, 10] improve scalability of BNNs, this comes at the cost of UQ reliability.
Deep ensembles provide generally good estimates of UQ but are not a priori
reliable.

Alternative machine learning methods, rooted in Bayesian statistics, such as
Gaussian processes do provide principled, and scalable UQ in many scenarios
[11]. GPs have proved very successful in materials modelling with a range of
studies recently employing them for applications such as calculating energies
and dynamics [12, 13]. The major barrier in applying GPs to practical problems
is the necessity to construct meaningful representations (or features) of the
data as an input to the GP. Several elegant examples, such as the SOAP
descriptor, have been proposed, but still need to be derived for each new
system[14].

One avenue to addressing the problem of feature engineering is to rely on
representation learning or feature learning, an area of machine learning that
focuses on automatically discovering the features that best describe the
underlying (possibly hidden) characteristics of the data. Representation
learning thus helps to not only better understand a dataset, but also to
produce better outcomes. Although specialised architectures can be built to
extract minimal representations of data [15], even the hidden layers of simple
neural networks progressively capture these representations
[16].Representation learning has been instrumental in a number of areas[17,
18].

In this work we combine a GNN with a GP by performing feature learning with
the GNN and using the features learned by the GNN as input to a GP, thus
circumventing the need for manual feature engineering. This approach has
recently been demonstrated to work well for calculating adsorption energies
for catalysis [19] and builds on a number of notable recent efforts towards
uncertainty quantification for machine learning of molecular and materials
properties[20, 21]. We demonstrate that the recently published megnet
architecture can be used to learn representations that can be applied in a GP
to predict formation energy of a crystal.

We first show that the latent space of the megnet model is structured in such
a way that information about the target properties can be inferred. We then
demonstrate a well-calibrated GP with reliable UQ on the prediction of
formation energies. We use this UQ to address the issue of using GNNs in
scenarios where labelled data is scarce. We use the uncertainties calculated
on previously unseen examples to choose the next training point for the ML
model. This so-called entropy sampling-based active learning procedure is
applied to training a model for formation energy prediction and is shown to
significantly outperform random sampling of the next training point.

We are confident that the methods and models presented in this paper can be
applied to a range of properties and scenarios in materials’ science, for
example when training data is difficult to obtain such as dielectric
properties, phonons or high quality electronic structures [22]. Active
learning has already demonstrated transformative power in the field of
molecular quantum chemistry[23], we hope to contribute towards its application
in condensed matter materials.

##  II Latent space and predicted properties

We first investigate whether a latent space representation of a material can
be useful as an input vector for a Gaussian process. In order to do this we
have extracted the outputs from the graph network section of megnet . To be
precise we use the values from the layer labelled “Dense 32” from [2], this
network was trained for 1,000 epochs on 1,800 randomly chosen materials from
the full dataset, before being used to extract the activations for the same
1,800 materials. After a material is input to the network, we extract the
values of the activations in this layer, this 32-dimensional vector serves as
a compressed representation of the material. Visualisation of higher
dimensional data is well-known to be difficult for human cognition, thus we
reduce the dimensions of the latent space for visulaisation purposes, using
the t-distributed stochastic neighbour embedding (t-SNE)[24]. t-SNE is a non-
linear and stochastic dimensionality reduction technique that projects the
original data points (32-dimensions) to two-dimensional space.

The t-SNE-derived 2D distribution of the activation vectors for the dataset
(see Dataset section for more details) are presented in Figure 2. In this
figure we have coloured the points according to their formation energy. From
this Figure, it is clear that there is a structure in the latent space
relating to formation energy; points tend to have similar formation energies
to those clustered close by. This gives us confidence that the graph network
is learning a representation (or features) that contains the relevant
information required for building a subsequent model for approximating the
formation energy.

Figure 2: A reduced dimensionality plot of the activations extracted from the
first dense layer of the megnet model. The activations are projected onto a 2D
space using the t-SNE algorithm. The points are coloured according to the
formation energy of the material.

##  III A network-fed Gaussian process

Having established that there is a structure of the latent space that is
related to the output property of interest, we now develop a Gaussian process
(GP) model to infer the relationship. Using the notation of Tran and co-
workers[19] we refer to this as a convolution-fed Gaussian process (CFGP). The
GP uses the 32-dimensional vector from the GNN as the input vector - full
details of the GP are given in the Models section.

To characterise the similarity between samples, we use the Laplacian kernel

$k(x,y)=a^{2}\exp\left(\frac{-||x-y||}{l}\right)$ (1)

where the similarity $k$ between two data samples at points $x$ and $y$
depends on a function of the difference between these points and an amplitude
$a$ and length scale $l$ of the kernel. The full details of how values for $a$
and $l$ are obtained are given in the Methods section.

The CFGP is trained on 7,800 materials from the original dataset. Where 1,800
materials were previously used to train the megnet model and the additional
6,000 materials are identified during an active learning procedure (see
Methods for details). Note that during the active learning procedure the
weights of the megnet model are not updated, only the GP is trained, however
at the end of the active learning we re-optimized megnet and the GP on the new
training set of 7,800 materials. We then test the model on 1,460 previously
unseen materials to evaluate the performance.

Figure 3 plots the CFGP predicted formation energy against the DFT calculated
formation energy, for a test set of 1,460 materials not used to train the
model. We can observe a high degree of correlation between the predicted and
ground-truth (DFT) values. The test set data has a mean absolute error (MAE)
of 0.0277 eV/atom, a mean squared error of 0.002 eV/atom. This performance is
comparable to the performance of other GNN-based methods as reported on the
matbench benchmark dataset[25], where megnet had and MAE of 0.0417 eV/atom. We
note that the training and test sets that we apply here are somewhat more
restricted than the matbench data, explaining our lower MAE.

Figure 3: The performance of the CFGP model for predicting formation energies
on a test set of 1460 materials. The model has a mean absolute error of 0.0277
meV/atom, with an $r^{2}$ value of 0.99.

This demonstrates that the CFGP can perform the task of predicting properties
from materials structure with comparable fidelity to GNNs employing multi-
layer perceptron function approximators. The advantage of the GP is that we
also obtain UQ, or degree of confidence on the prediction. In the section that
follows this, we evaluate the resulting UQ performance.

##  IV Calibration of uncertainty quantification

To assess the quality of the UQ element in our CFGP we compare the uncertainty
estimates in the training set to the residuals between the predicted and true
values. The work of Tran and et al. [19] provides an excellent, lucid
introduction to benchmarking UQ methods, outlining recently proposed protocols
[26, 27]. We briefly outline the criteria for good UQ methods here but, highly
recommend the aforementioned work of Tran et al.[19].

A good uncertainty estimate should be well-calibrated, sharp, and disperse. In
our context, well-calibrated means that the residuals generally fall within
the range of the error estimates (in our case twice the standard deviation of
the GP). Sharpness implies that the error estimates are not unnecessarily
large; conservative large error-estimates give good calibration, but are not
necessarily useful for assessing how the model will perform. Finally disperse
means that the error-estimates cover a range of values, so that the model does
not simply predict a uniform estimate for all samples.

In Figure 4 we show the performance of the resulting UQ. To provide more
clarity, the upper plot shows a randomly selected set of 100 samples from the
test set. Specifically, the CFGP energies are plotted against the DFT energies
with error bars equal to twice the standard deviation ($2\sigma$) of the GP;
in a standard distribution, 95% of the data should fall within $2\sigma$. We
can observe that, within the sample set, the great majority of points fall
within $2\sigma$ of the DFT energy. In the lower panel we plot the residuals
against $\sigma$. We use the the same data as in Figure 3 for 1,460 test
materials, not used in training. This plot demonstrates that validation
samples with larger residuals generally do have larger uncertainty estimates.
We also colour the points by the difference between the residual and
$2\sigma$, indicating which points fall outside the error estimates.

We note in Figure 4 that there are a few materials that stand out with
particularly high errors and also with a large discrepancy between the UQ and
the error in the prediction. Among these, three particularly poor predictions
are labelled in the figure, we note that all three systems have phosphate
motifs. The phosphate motif is relatively unusual in the context of an
inorganic materials database. In addition two of the materials, ZrP2(HO3)2 and
SnPO3F have an apical oxygen in the phosphate replaced with P-H and P-F bonds
respectively. These kinds of bonds are unusual and are therefore probably not
present in large amounts in the training set. The scarcity of these motifs can
explain why the model fails to predict them with good accuracy. We suggest
that our UQ here is possibly over-cautious, but does generally identify well
cases where the results of the model are likely to be unreliable. The
calibration of the UQ could potentially be improved by exploring alternative
kernels for the GP or using different layers of megnet as input vectors, this
will be the subject of further investigations, for now we assess how well the
model can perform in an active learning procedure.

Figure 4: The effectiveness of uncertainty quantification by the Gaussian
process. Top plot shows 100 randomly selected points, the CFGP energy is
plotted against the DFT energy, the error-bars are set to twice the standard
deviation obtained from the CFGP. Bottom plot shows the residuals (difference
between DFT energy and CFGP energy) versus the standard deviation obtained
from the CFGP.

##  V Active learning

We now explore how the UQ demonstrated in the previous section can be further
leveraged for active learning. In active learning, the ML model not only
learns the relationships in the underlying data from labelled samples, but
also chooses which unlabelled sample should be labelled next in order to best
improve the performance and generality of the model. There are a number of
techniques that can be used to select the points to be labelled next [28].
Recently an approach based on query by committee has been used to accelerate
training for a quantum chemistry dataset [23].

In our case, we apply an entropy-based sampling approach, which chooses the
unlabelled sample with the highest uncertainty and uses that as the next
sample to be labelled. The uncertainty represents the entropy of the samples
from an information theory perspective. Hence, opting to label the sample with
greatest uncertainty next maximises the entropy reduction in the sampling
process. In the context of the GP, a point with high uncertainty is likely to
be found in an area of feature space that is more sparsely labelled than other
areas, thus choosing this point for labelling favours a more representative
sampling of all regions of feature space.

A schematic illustration of our process is shown in the upper panel of Figure
5. First, a trained megnet is passed the labelled data and the activations of
the dense layer are used to feed a GP for training (a CFGP). Then the
unlabelled data is passed through the CFGP providing predictions with
uncertainties. Here, the sample with the greatest uncertainty is chosen to be
labelled. Then the new larger labelled dataset is used to re-train the
Gaussian process. The overall process continues until some predefined end
point. In our case we started with 1,800 labelled materials from the dataset
described in the Dataset section, we then run through 1,500 cycles of the
active learning procedure evaluating the performance of the CFGP at each cycle
model on all of the remaining materials whose labels were not used in training
(i.e. the full set of materials minus the training set at that cycle).

The results of the entropy-based active learning process are presented in
Figure 5 lower panel. For comparison we have also performed a sampling of new
materials based on random selection from the unlabelled data. We can see that
the maximum entropy-based sampling approach significantly out-performs the
random sampling-based approach. By the end of the 1,500 cycles of active
learning, the model from the maximum entropy based approach outperforms the
model from the random approach by $\sim$ 43%. The enhancement in performance
achieved by the random selection process after 1,500 additional samples is
achieved by the entropy based sampling in fewer than half the samples; 740.

(a)

(b)

Figure 5: The process and effectiveness of entropy sampling based active
learning. Upper; the active learning procedure, labelled data are passed
through megnet and activations at the dense layer extracted to train a
Gaussian process, unlabelled data are then passed through the same process and
predictions and uncertainties obtained, samples with the greatest uncertainty
are then labelled and the procedure loops back around. Lower; the mean
absolute error on the test set of the CFGP with different sampling methods.
The entropy-based sampling method chooses the next sample to label based on
the uncertainty in prediction of the unlabelled samples, the random sampling
method chooses at random.

##  VI Discussion

We have demonstrated that Gaussian process trained on the outputs of a GNN, a
CFGP, performs well for prediction, uncertainty quantification and active
learning for the formation energy of materials. The CFGP model that we
presented here has uses in a very wide range of materials science
applications. In fact, a similar architecture was recently demonstrated to
perform very well for prediction and uncertainty quantification in absorption
energy prediction for catalysis[19]. This approach is particularly appealing
as it combines the theoretical rigour of the Gaussian processes-based
approaches, with the flexibility of the GNN approach. GNNs can, in principle,
represent any materials or molecular system, with minimal user-defined input
required, and therefore this approach should be easily adaptable to other
domains of interest with minimal feature engineering .

The ability to place reliable confidence bounds on predicted properties is
crucial for the application of ML methods and building trust in the outputs of
models. For example, in the context of virtual high-throughput screening ML
surrogate models are often applied as a filtering step [29, 30, 31, 32, 33].
Access to reliable UQ in this context is very attractive as it can be used in
combination with threshold values to filter out prospective candidates.
Reliable UQ can also reduce the number of false positives passing through the
procedure or the number of false negatives being rejected, and thus could be
used on-the-fly to assess edge cases at a higher level of fidelity.

The potential application of active learning in materials and molecular
science is being increasingly recognised [34, 35]. In applications such as
deriving new inter-atomic potentials[36, 37] or searching through materials
space for optimal catalysts[38], the ability to optimise the choice of the
next experiment is important. Active learning is particularly appealing in
scenarios where data is scarce and expensive to obtain. For energy materials
design properties such as high-quality electronic structure [39], dielectric
constants[40, 22, 41, 42], effective masses[43] and defect properties[44, 45]
are some high-profile examples where data-driven approaches are possible, but
obtaining large sets of labelled data can be prohibitive to applying deep
learning approaches. In these scenarios the availability of a method for
reducing the number of samples required to train accurate, and reliable models
is rather critical to the application of deep learning approaches in
materials’ science.

We note that there is definite room for improvement in the active learning
strategy applied here. As demonstrated in the large errors on a family of
phosphate materials, our approach may not be sampling the chemical space as
effectively as possible. Improving the calibration of the UQ would be
beneficial in active learning. There is scope for introducing chemical and
statistical heuristics in order to improve the coverage. As an example, one
could sample several new materials to label per active learning cycle, but
ensure that each one comes from a distinct region of the latent space by
enforcing a cut-off (possibly based on the kernel length of the GP).
Additionally choice of kernel for the GP may affect how well calibrated the UQ
is, this is an ongoing area of research. We also note that the active learning
procedure was run for a further 4,500 cycles after the process described in
the section Active Learning. However the advantage of using the entropy-based
sampling diminishes as the process continues because the probability of
randomly choosing useful material for training approaches that of actively
choosing good materials by the entropy-based method as we sample larger
portions of the available additional data points (in our case 7,195
materials). Thus, it is clear that the degree of advantage to be gained by
applying the active learning approach will depend on the size of the non-
labelled data and will likely increase as the unlabelled space increases.
However, our approach already demonstrates the true potential of active
learning using an example where size of the unlabelled dataset is relatively
small. With the number of unexplored inorganic materials estimated in the
region of $10^{11}$ materials[29], active learning will have an important role
to play in navigating this space.

##  VII Conclusions

We have presented an active learning approach based on convolution-fed
Gaussian processes, which is capable of greatly enhancing the learning rate of
deep neural networks for predicting the properties of materials, when compared
to random sampling. Our approach uses the fact that the latent space of a
graph neural network model provides a compressed representation of a 3D
crystal in a space with meaningful correlations to the composition-structure-
property relationship which the GNN has been trained to reproduce. We use this
compressed representation to feed a Gaussian process model, which provide
property estimates and uncertainty quantification (UQ) on those estimates. We
demonstrate that the estimates obtained are competitive with state-of-the-art
GNNs and that the UQ obtained is well calibrated and sharp. We then take
advantage of this UQ to develop an active learning procedure where the
training dataset is augmented on-the-fly sampling from unlabelled data based
on the UQ obtained from the model. We show that this active learning procedure
results in a much more rapid improvement in model performance compared to
sampling the unlabelled data randomly. The methods presented in this paper
will help increasing the confidence in the output of ML models by providing
reliable estimates of (un)certainty. Furthermore, the active learning
procedure can be extremely useful for training models in scenarios where
labelled data is difficult or expensive to obtain. We hope that our methods
can help to accelerate the application of machine learning for materials
design.

##  VIII Dataset

The dataset for the formation energy models consists of 10455 oxide materials
and their DFT calculated energies from the Materials Project database[46] the
exact dataset is available to download and can be used in association with our
open repository of this code [47].

The dataset is initially split into three parts - 1800 samples for training
megnet , 7,195 samples available for updating the model during active learning
and 1,460 samples kept completely separate for model evaluation purposes.

The data used in this study are available from reference [48].

##  IX Models

Our models in this paper are based on the megnet GNN architecture [2]. megnet
consists of an embedding layer, followed by a flexible number of graph neural
network units. The output from the graphs is then converted to a consistent
sized vector using the set2set transformer[49]. This vector is then fed into a
multi-layer perceptron feed-forward neural network to approximate the property
of interest. In the CFGP, the vector is fed into a GP for function
approximation.

The megnet model is initially trained on 1,800 randomly selected materials
from the database. The model is trained for 1,000 epochs, until the validation
error has equilibrated. The performance is shown in Figure 6.

Using the megnet model trained as above, we build input vectors for the GP.
Input vectors are obtained by passing an input to megnet and extracting the
activations from the “Dense 32” layer. The GP uses these vectors as input. The
GP we employ uses the Laplacian kernel of Equation 1. We have implemented the
GP using TensorFlow Probability [50]. The hyper-parameters $a$ and $l$ are
obtained by optimising the negative log likelihood of GP on the training
dataset of 1,800 materials. This process can be rather sensitive to the
initial values of the hyper-parameters, starting from 1.0 for both $a$ and
$l$, we obtain optimal values of 0.4281 and 2.3997 respectively.

During active learning one new material is chosen at each cycle and added to
the training set for the GP (note we do not re-train the megnet model or the
hyper-parameters of the GP at each step during the active learning). The GP is
re-trained at each cycle of the active learning and the process was repeated
for 6,000 cycles. At each cycle the performance of the GP on 1,460 independent
samples, never used in any of the training steps is assessed.

For reference - the megnet model took around 5 hrs to train; the hyper-
parameter optimisation of the GP took 10,000 steps and 12 hrs and the active
learning for 6,000 cycles took 7 hrs, all on a NVIDIA Quadro P5000 GPU.

The code for the models, and trained model architectures are available at [47]
and [48], respectively.

(a)
Figure 6: The training performance of the megnet model. The mean absolute
error on training and validation sets is plotted against epoch.

## Acknowledgements

This work was partially supported by Wave 1 of The UKRI Strategic Priorities
Fund under the EPSRC Grant EP/T001569/1, particularly the “AI for Science”
theme within that grant and The Alan Turing Institute. The ML models were
trained using computing resources provided by STFC Scientific Computing
Department’s SCARF cluster and the PEARL cluster.

## Data Access Statement

All of the training data, trained neural networks and code for generating the
training data for this study are openly available at
https://zenodo.org/record/4922828#.YMHksBIo-xI.

A git repository containing the code used to build and train the neural
networks, as well as notebooks to recreate them are available at
https://github.com/keeeto/gp-net

## Author Contributions

KTB conceived, planned and steered the project. JA built, trained and applied
the neural networks. KTB, JT and JA wrote the manuscript together. JT was
involved in conception and establishing of the project and facilitated the
work in the paper.

## References

  * Butler  _et al._ [2018] K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev, and A. Walsh, Machine learning for molecular and materials science, Nature 559, 547 (2018).
  * Chen  _et al._ [2019] C. Chen, W. Ye, Y. Zuo, C. Zheng, and S. P. Ong, Graph networks as a universal machine learning framework for molecules and crystals, Chem. Mater. 31, 3564 (2019).
  * Xie and Grossman [2018] T. Xie and J. C. Grossman, Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties, Phys. Rev. Lett. 120, 145301 (2018).
  * Park and Wolverton [2020] C. W. Park and C. Wolverton, Developing an improved crystal graph convolutional neural network framework for accelerated materials discovery, Phys. Rev. Mater. 4, 063801 (2020).
  * Lee and Asahi [2021] J. Lee and R. Asahi, Transfer learning for materials informatics using crystal graph convolutional neural network, Comp. Mater. Sci. 190, 110314 (2021).
  * Raza  _et al._ [2020] A. Raza, A. Sturluson, C. M. Simon, and X. Fern, Message passing neural networks for partial charge assignment to metal–organic frameworks, J. Phys. Chem. C 124, 19070 (2020).
  * Mallick  _et al._ [2020] A. Mallick, C. Dwivedi, B. Kailkhura, G. Joshi, and Y. Han,  _Sample Efficient Uncertainty Estimation using Probabilistic Neighborhood Component Analysis_ , Tech. Rep. (Lawrence Livermore National Lab.(LLNL), Livermore, CA (United States), 2020).
  * MacKay [1992] D. J. MacKay, A practical bayesian framework for backpropagation networks, Neural computation 4, 448 (1992).
  * Blundell  _et al._ [2015] C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra, Weight uncertainty in neural networks, arXiv preprint arXiv:1505.05424 (2015).
  * Gal and Ghahramani [2016] Y. Gal and Z. Ghahramani, Dropout as a bayesian approximation: Representing model uncertainty in deep learning, in  _international conference on machine learning_ (2016) pp. 1050–1059.
  * Bilionis and Zabaras [2017] I. Bilionis and N. Zabaras, Bayesian uncertainty propagation using gaussian processes, in  _Handbook of Uncertainty Quantification_, edited by R. Ghanem, D. Higdon, and H. Owhadi (Springer International Publishing, Cham, 2017) pp. 555–599.
  * Raimbault  _et al._ [2019] N. Raimbault, A. Grisafi, M. Ceriotti, and M. Rossi, Using gaussian process regression to simulate the vibrational raman spectra of molecular crystals, New J. Phys. 21, 105001 (2019).
  * Meyer and Hauser [2020] R. Meyer and A. W. Hauser, Geometry optimization using gaussian process regression in internal coordinate systems, J. Chem. Phys. 152, 084112 (2020).
  * Musil  _et al._ [2021] F. Musil, A. Grisafi, A. P. Bartók, C. Ortner, G. Csányi, and M. Ceriotti, Physics-inspired structural representations for molecules and materials, arXiv preprint arXiv:2101.04673 (2021).
  * Tschannen  _et al._ [2018] M. Tschannen, O. Bachem, and M. Lucic, Recent advances in autoencoder-based representation learning, arXiv preprint arXiv:1812.05069 (2018).
  * Bengio  _et al._ [2013] Y. Bengio, A. Courville, and P. Vincent, Representation learning: A review and new perspectives, IEEE transactions on pattern analysis and machine intelligence 35, 1798 (2013).
  * Gidaris  _et al._ [2018] S. Gidaris, P. Singh, and N. Komodakis, Unsupervised representation learning by predicting image rotations, arXiv preprint arXiv:1803.07728 (2018).
  * Kopf and Claassen [2021] A. Kopf and M. Claassen, Latent representation learning in biology and translational medicine, Patterns 2, 100198 (2021).
  * Tran  _et al._ [2020] K. Tran, W. Neiswanger, J. Yoon, Q. Zhang, E. Xing, and Z. W. Ulissi, Methods for comparing uncertainty quantifications for material property predictions, Machine Learning: Science and Technology 1, 025006 (2020).
  * Janet  _et al._ [2019] J. P. Janet, C. Duan, T. Yang, A. Nandy, and H. J. Kulik, A quantitative uncertainty metric controls error in neural network-driven chemical discovery, Chem. Sci. 10, 7913 (2019).
  * Scalia  _et al._ [2019] G. Scalia, C. A. Grambow, B. Pernici, Y.-P. Li, and W. H. Green, Evaluating scalable uncertainty estimation methods for dnn-based molecular property prediction, arXiv preprint arXiv:1910.03127 (2019).
  * Morita  _et al._ [2020] K. Morita, D. W. Davies, K. T. Butler, and A. Walsh, Modelling the dielectric constants of crystals using machine learning, arXiv preprint arXiv:2005.05831 (2020).
  * Smith  _et al._ [2019] J. S. Smith, B. T. Nebgen, R. Zubatyuk, N. Lubbers, C. Devereux, K. Barros, S. Tretiak, O. Isayev, and A. E. Roitberg, Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning, Nat. Commun. 10, 1 (2019).
  * Hinton and Roweis [2002] G. Hinton and S. T. Roweis, Stochastic neighbor embedding, in  _NIPS_ , Vol. 15 (Citeseer, 2002) pp. 833–840.
  * Dunn  _et al._ [2020] A. Dunn, Q. Wang, A. Ganose, D. Dopp, and A. Jain, Benchmarking materials property prediction methods: The matbench test set and automatminer reference algorithm, arXiv preprint arXiv:2005.00707 (2020).
  * Kuleshov  _et al._ [2018] V. Kuleshov, N. Fenner, and S. Ermon, Accurate uncertainties for deep learning using calibrated regression, arXiv preprint arXiv:1807.00263 (2018).
  * Levi  _et al._ [2019] D. Levi, L. Gispan, N. Giladi, and E. Fetaya, Evaluating and calibrating uncertainty prediction in regression tasks, arXiv preprint arXiv:1905.11659 (2019).
  * Del Rosario  _et al._ [2020] Z. Del Rosario, M. Rupp, Y. Kim, E. Antono, and J. Ling, Assessing the frontier: Active learning, model accuracy, and multi-objective candidate discovery and optimization, J. Chem. Phys. 153, 024112 (2020).
  * Davies  _et al._ [2016] D. W. Davies, K. T. Butler, A. J. Jackson, A. Morris, J. M. Frost, J. M. Skelton, and A. Walsh, Computational screening of all stoichiometric inorganic materials, Chem 1, 617 (2016).
  * Davies  _et al._ [2019a] D. W. Davies, K. T. Butler, and A. Walsh, Data-driven discovery of photoactive quaternary oxides using first-principles machine learning, Chem. Mater. 31, 7221 (2019a).
  * Nyshadham  _et al._ [2019] C. Nyshadham, M. Rupp, B. Bekker, A. V. Shapeev, T. Mueller, C. W. Rosenbrock, G. Csányi, D. W. Wingate, and G. L. Hart, Machine-learned multi-system surrogate models for materials prediction, npj Comp. Mater. 5, 1 (2019).
  * Bobbitt and Snurr [2019] N. S. Bobbitt and R. Q. Snurr, Molecular modelling and machine learning for high-throughput screening of metal-organic frameworks for hydrogen storage, Mol. Sim. 45, 1069 (2019).
  * Li  _et al._ [2017] Z. Li, S. Wang, W. S. Chin, L. E. Achenie, and H. Xin, High-throughput screening of bimetallic catalysts enabled by machine learning, J. Mater. Chem. A 5, 24131 (2017).
  * Lookman  _et al._ [2019] T. Lookman, P. V. Balachandran, D. Xue, and R. Yuan, Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design, npj Comp. Mater. 5, 1 (2019).
  * Zhang  _et al._ [2019] Y. Zhang _et al._ , Bayesian semi-supervised learning for uncertainty-calibrated prediction of molecular properties and active learning, Chem. Sci. 10, 8154 (2019).
  * Podryabinkin and Shapeev [2017] E. V. Podryabinkin and A. V. Shapeev, Active learning of linearly parametrized interatomic potentials, Comp. Mater. Sci. 140, 171 (2017).
  * Smith  _et al._ [2018] J. S. Smith, B. Nebgen, N. Lubbers, O. Isayev, and A. E. Roitberg, Less is more: Sampling chemical space with active learning, J. Chem. Phys. 148, 241733 (2018).
  * Tran and Ulissi [2018] K. Tran and Z. W. Ulissi, Active learning across intermetallics to guide discovery of electrocatalysts for co 2 reduction and h 2 evolution, Nature Catalysis 1, 696 (2018).
  * Pilania  _et al._ [2017] G. Pilania, J. E. Gubernatis, and T. Lookman, Multi-fidelity machine learning models for accurate bandgap predictions of solids, Comp. Mater. Sci. 129, 156 (2017).
  * Petousis  _et al._ [2017] I. Petousis, D. Mrdjenovich, E. Ballouz, M. Liu, D. Winston, W. Chen, T. Graf, T. D. Schladt, K. A. Persson, and F. B. Prinz, High-throughput screening of inorganic compounds for the discovery of novel dielectric and optical materials, Scientific data 4, 1 (2017).
  * Noda  _et al._ [2020] Y. Noda, M. Otake, and M. Nakayama, Descriptors for dielectric constants of perovskite-type oxides by materials informatics with first-principles density functional theory, Science and technology of advanced materials 21, 92 (2020).
  * Tawfik  _et al._ [2020] S. A. Tawfik, O. Isayev, M. J. Spencer, and D. A. Winkler, Predicting thermal properties of crystals using machine learning, Adv. Theo. Sim. 3, 1900208 (2020).
  * Davies  _et al._ [2019b] D. W. Davies, C. N. Savory, J. M. Frost, D. O. Scanlon, B. J. Morgan, and A. Walsh, Descriptors for electron and hole charge carriers in metal oxides, J. Phys. Chem. Letters 11, 438 (2019b).
  * Sharma  _et al._ [2020] V. Sharma, P. Kumar, P. Dev, and G. Pilania, Machine learning substitutional defect formation energies in abo3 perovskites, J. Appl. Phys. 128, 034902 (2020).
  * Gu  _et al._ [2019] G. H. Gu, J. Noh, I. Kim, and Y. Jung, Machine learning for renewable energy materials, J. Mater. Chem. A 7, 17096 (2019).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. a. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Materials 1, 011002 (2013).
  * Allotey and Butler [2021a] J. Allotey and K. T. Butler, Gp-net, https://github.com/keeeto/gp-net (2021a).
  * Allotey and Butler [2021b] J. Allotey and K. T. Butler, Data and models for: Entropy based active learning of graph neural networks for materials properties, https://zenodo.org/record/4922828#.YMHksBIo-xI (2021b).
  * Vinyals  _et al._ [2015] O. Vinyals, S. Bengio, and M. Kudlur, Order matters: Sequence to sequence for sets, arXiv preprint arXiv:1511.06391 (2015).
  * Dillon  _et al._ [2017] J. V. Dillon, I. Langmore, D. Tran, E. Brevdo, S. Vasudevan, D. Moore, B. Patton, A. Alemi, M. Hoffman, and R. A. Saurous, Tensorflow distributions, arXiv preprint arXiv:1711.10604 (2017).


# Energy and Spectral Efficiency Tradeoff in OFDMA Networks via Antenna
Selection Strategy

Ata Khalili† and Derrick Wing Kwan Ng⋆ †Department of Electrical and Computer
Engineering, Tarbiat Modares University, Tehran, Iran
⋆School of Electrical Engineering and Telecommunications, The University of
New South Wales, Australia
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>D. W. K. Ng is supported by
funding from the UNSW Digital Grid Futures Institute, UNSW, Sydney, under a
cross-disciplinary fund scheme and by the Australian Research Council’s
Discovery Project (DP190101363).

###### Abstract

In this paper, we investigate the joint resource allocation and antenna
selection algorithm design for uplink orthogonal frequency division multiple
access (OFDMA) communication system. We propose a multi-objective optimization
framework to strike a balance between spectral efficiency (SE) and energy
efficiency (EE). The resource allocation design is formulated as a multi-
objective optimization problem (MOOP), where the conflicting objective
functions are linearly combined into a single objective function employing the
weighted sum method. In order to develop an efficient solution, the
majorization minimization (MM) approach is proposed where a surrogate function
serves as a lower bound of the objective function. Then an iterative
suboptimal algorithm is proposed to maximize the approximate objective
function. Numerical results unveil an interesting tradeoff between the
considered conflicting system design objectives and reveal the improved EE and
SE facilitated by the proposed transmit antenna selection in OFDMA systems.

##  I Introduction

The next generation of wireless networks, namely 5G, aim at providing high
data rate and system capacity. To enable sustainable 5G networks, new
communication technologies have been proposed to ameliorate the system energy
efficiency (EE). In particular, various 5G techniques have been proposed which
aim to enhance the network throughput while consuming less energy without
sacrificing the quality of service [1, 2]. In practice, EE maximization
problem is more conspicuous in the uplink of wireless networks where extending
the battery life of mobile users are of great significance. Moreover, reducing
the transmit power of users may also alleviate the interference power arising
from co-channel users. On the other hand, in cellular networks, spectral
efficiency (SE) is also an important performance measure. Moreover, nowadays
users are employing multiple-antenna handsets while performing antenna
selection can provide a higher flexibility to the system operator to strike a
balance between SE and EE.

Recently, the studies of EE has drawn much attention for wireless orthogonal
frequency division multiple access (OFDMA) communication networks, where the
available bandwidth is effectively divided into some orthogonal sub-channels
and each sub-channel can be individually assigned with a flexible manner [3,
4, 5, 6]. For instance, in [3], the joint antenna-subcarrier-power allocation
to maximize the EE of downlink multi-user was studied where the Dinkelbach
algorithm was proposed. However, to reduce the computational complexity of the
algorithm, an upper bound on the maximum allowed interference is introduced as
a constant to simplify the problem. The joint subcarrier assignment and power
allocation of downlink transmission in two-tier heterogeneous cellular
networks was investigated in [4] to maximize the system EE, where a two-step
iterative solution and multi-objective optimization problem (MOOP) based on
epsilon constraint was proposed. In [6], the authors demonstrated energy-
efficient power allocation scheme for downlink multi-user distributed antenna
systems in a single cell system. However, sub-channel assignment for improving
the system performance, as well as antenna selection, were not considered.

SE and EE are the key performance metrics for wireless communication systems,
especially for green communication networks [1]. The optimization of both
these metrics can enhance the system performance. Hence, it is important to
take into account the non-trivial tradeoff between two objectives in system
design [7, 8, 9]. In [7], the EE maximization in downlink multiuser
distributed antenna system was formulated as a MOOP which was solved via the
weighting sum method. The authors in [8] investigated the tradeoff between EE
and SE for a downlink single-cell and multi-cell scenario where the stochastic
geometry approach was proposed to facilitate the analysis. In [9], the authors
formulated the resource allocation design as a MOOP which maximizes both the
EE and SE, simultaneously. In addition, it was shown in [9] that MOOP of EE-SE
is equivalent to maximizing data rate and minimizing the total power
consumption, simultaneously.

To date, various low-complexity antenna selection methods have been developed
[10, 11, 12] to offer high flexibility to the system operator for throughput
enhancement. In particular, antenna selection has been proposed for the uplink
of 4G LTE-Advanced [10] due to its low implementation cost and reduced
feedback overhead compared to other techniques such as beamforming or
precoding techniques. Hence, the idea of antenna selection becomes an
interesting research problem in various networks when the computational
complexity of users is of great concern. Although the full activation of
multi-antenna systems enables a highly desirable gain, simple antenna
selection can extract a comparable gain of the multi-antenna systems with a
marginal decrease of the performance [14, 15, 16].

Motivated by the aforementioned observations, we investigate the problem of
joint resource allocation and antenna selection in two scenarios. In the first
scenario, the conventional antenna selection (CAS) selects only one antenna
and only one RF chain is employed whereas, in the second scenario, the
generalized antenna selection (GAS) selects a subset of the antennas for
signal transmission [16]. We focus on a multicell system and formulate a MOOP
to strike a balance between the SE and EE of a multi-cell uplink network, by
considering QoS requirement and feasibility of the transmit power level. The
two competing objective functions of this MOOP is handled by the weighting
coefficient method which linearly combines them into a single objective
function. The resultant mixed integer non-linear problem (MINLP) is addressed
by using an efficient suboptimal algorithm with low computational complexity.
To facilitate the solution design, a penalty function is employed to handle
the binary variable constraints. The underlying problem is solved suboptimally
via a method based on majorization minimization (MM) approach by constructing
a sequence of surrogate functions to iteratively approximate the non-convex
objective function [13].

##  II System Model

The uplink of a multi-carrier cellular network with the set of
$\mathcal{B}=\\{1,2,..,B\\}$ cells is considered where the set of available
sub-channel in each cell is represented by $\mathcal{S}=\\{1,2...,S\\}$111All
BSs are equipped with multiple antennas. However, antenna selection at BSs is
out of scope of this work and is assumed to be predefined at the BSs, i.e.,
each antenna is reserved for a subset of users.. The total number of users in
each cell is ${M}$ where the set of user which is associated to BS b is
denoted by $\mathcal{M}_{b}$ and each user is equipped with
$\mathcal{Q}=\\{1,2,...,Q\\}$ antennas. We assume that the perfect channel
state information (CSI) is available at the resource allocator to design
resource allocation [17]. The uplink channel gain from the $m^{th}$ user in
the $b^{th}$ cell to its desired BS, i.e., the $b^{th}$ BS, in the $s^{th}$
sub-channel from antenna $q$ is represented by $h^{sq}_{m{b},b}$. Furthermore,
$a^{sq}_{mb}$ and $x^{s}_{mb}$ are defined as binary indicators showing if the
$m^{th}$ user in the $b^{th}$ cell selects the $s^{th}$ sub-channel and the
$q^{th}$ antenna. Accordingly, $p^{sq}_{mb}$ is the corresponding transmitted
power from the $q^{th}$ antenna in the $s^{th}$ sub-channel for this user.

According to the Shannon capacity formula, the achievable rate of the $m^{th}$
user in the $b^{th}$ cell for the $s^{th}$ sub-channel, when the $q^{th}$
antenna is selected can be written as

$\small
R^{sq}_{mb}\triangleq\log_{2}\Bigg{(}1+\frac{p^{sq}_{mb}|h^{sq}_{m{b},b}|^{2}}{\sigma^{2}+\underset{b^{\prime}\neq
b,b^{\prime}\in\mathcal{B}}{\overset{}{\mathop{\sum}}}~{}\underset{k\in\mathcal{M}_{b^{\prime}}}{\overset{}{\mathop{\sum}}}\underset{q^{\prime}\in\mathcal{Q}}{\overset{}{\mathop{\sum}}}a^{sq^{\prime}}_{kb^{\prime}}x^{s}_{kb^{\prime}}p^{s{q^{\prime}}}_{kb^{\prime}}|h^{s{q^{\prime}}}_{kb^{\prime},b}|^{2}}\Bigg{)},\quad$
(1)

where $\sigma^{2}$ represents the additive white Gaussian noise (AWGN) power.
Let us define the vector of power allocation, sub-channel assignment, and
antenna selection variables as $\textbf{p}\in\mathbb{R}^{MBSQ\times 1}$,
$\textbf{x}\in\mathbb{Z}^{MBS\times 1}$, and
$\textbf{a}\in\mathbb{Z}^{MBSQ\times 1}$, respectively. In what follows we
present antenna selection schemes for two scenarios, namely generalized
antenna selection and conventional transmit antenna selection.

###  II-A Generalized Antenna Selection (GAS)

As shown in Fig. 1, for GAS approach, an OFDM symbol is transmitted from any
antenna that the subcarriers are assigned to it. The GAS222Through the problem
formulation it is assumed that each sub-channel for each user can be assigned
to one of the antennas and the number of available RF chains is more than one
[16]. scheme aims to mitigate harmful effects of deep fades experienced in
wireless channel by employing an optimal linear combining rule to a subset of
the strongest available diversity branches, thereby alleviating the receiver
complexity and cost [14, 15]. In this case, a separate RF chain is required
for each antenna. In other words, a subset of the antenna must be selected.

Figure 1: A block diagram of the generalized antenna selection scheme.

The power consumption of the GAS is defined as
$P^{c-GAS}_{m}=p^{\textrm{static}}_{m}+p^{\textrm{antenna}}_{m}\sum_{s}\sum_{q}a^{sq}_{mb}$
where $p^{\textrm{static}}_{m}$ is the static power required to support the
basic circuit operations of the system at the transmitter and
$p^{\textrm{antenna}}_{m}$ denotes the dissipated power per antenna [18].

###  II-B Conventional Antenna Selection (CAS)

As shown in Fig. 2, for CAS approach, a single transmit antenna is merely
chosen and an OFDM symbol is transmitted via the selected antenna. In fact, it
is assumed that all sub-channels for each user can be assigned to one of the
antennas and only one RF branch is available [16]. The model power consumption
of the CAS is presented as
$P^{c-CAS}_{m}=p^{\textrm{static}}_{m}+p^{\textrm{antenna}}_{m}$. In what
follows, we study the resource allocation design for these two scenarios to
strike a balance between EE and SE.

Figure 2: A block diagram of the conventional antenna selection scheme.

##  III Problem Formulation

Energy efficiency (EE) of a wireless communication system is defined as a
ratio between the total achievable rate and the total energy consumption given
as follows:

$\displaystyle\eta_{EE}=\frac{R^{\textrm{total}}}{P^{\textrm{total}}},$ (2)

where
$P^{\textrm{total}}=\frac{1}{\kappa}\underset{b\in\mathcal{B}}{\overset{}{\mathop{\sum}}}\underset{m\in\mathcal{M}_{b}}{\overset{}{\mathop{\sum}}}~{}\underset{s\in\mathcal{S}}{\overset{}{\mathop{\sum}}}\underset{q\in\mathcal{Q}}{\overset{}{\mathop{\sum}}}\
\
p^{sq}_{mb}+\underset{b\in\mathcal{B}}{\overset{}{\mathop{\sum}}}\underset{m\in\mathcal{M}_{b}}{\overset{}{\mathop{\sum}}}P^{c}_{mb}$
and
${R}^{\textrm{total}}=\underset{b\in\mathcal{B}}{\overset{}{\mathop{\sum}}}\underset{m\in\mathcal{M}_{b}}{\overset{}{\mathop{\sum}}}~{}\underset{s\in\mathcal{S}}{\overset{}{\mathop{\sum}}}\underset{q\in\mathcal{Q}}{\overset{}{\mathop{\sum}}}R^{sq}_{mb}$
are the total power consumption and data rate of the network, respectively.
Note that ${R}^{\textrm{total}}$, is related to SE since we have
${R}^{\textrm{total}}=W\eta_{SE}$ (where $W$ represents the transmission
bandwidth). Moreover, $P^{c}_{mb}$ denotes the consumed circuit power of the
$m^{th}$ user device which depends on the CPU processing and includes the
entire transmit RF chain which varies for the GAS and CAS scenarios. Based on
the configuration of the active links, the total circuit power consumption,
$P^{c}_{mb}$, can be split into static and dynamic terms. Also $0<\kappa<1$
denotes the efficiency of the power amplifier at each user. As observed, the
power consumption of the two mentioned scenarios is different. For the GAS
approach, all the assigned sub-channels can transmit a signal from different
antennas i.e., the GAS has more degrees of freedom to increase the system
throughput as compared to the CAS scenario. However, in the GAS scenario the
aggregate power consumption increases with the number of activated RF chains.
In particular, the optimization of both scenarios can lead to a challenging
tradeoff problem between SE and EE. In what follows, we aim at formulating two
optimization problems for the two aforementioned scenarios to maximize both EE
and SE through minimizing the inverse of EE and SE.

###  III-A EE-SE for the GAS Scenario:

The resource allocation and antenna selection for this scenario is formally
stated as:

$\displaystyle\mathcal{O}_{1}:~{}\min_{\textbf{a},\textbf{x},\textbf{p}}~{}$
$\displaystyle\eta_{EE}^{-1}(\textbf{a},\textbf{x},\textbf{p})$
$\displaystyle\min_{\textbf{a},\textbf{x},\textbf{p}}~{}$
$\displaystyle\eta_{SE}^{-1}(\textbf{a},\textbf{x},\textbf{p})$ s.t.
$\displaystyle{\textrm{C}_{1}}:\underset{m\in\mathcal{M}_{b}}{\overset{}{\mathop{\sum}}}x_{mb}^{s}=1,$
$\displaystyle{\textrm{C}_{2}}:\underset{q=1}{\overset{Q}{\mathop{\sum}}}\,a_{mb}^{sq}=1,$
$\displaystyle{\textrm{C}_{3}}:p_{mb}^{sq}\geq 0,$ (3)
$\displaystyle{\textrm{C}_{4}}:\underset{q\in{\mathcal{Q}}}{\overset{}{\mathop{\sum}}}\,\underset{s\in{\mathcal{S}}}{\overset{}{\mathop{\sum}}}\,a_{mb}^{sq}x_{mb}^{s}p_{mb}^{sq}\leq{{p}_{\max}},$
$\displaystyle{\textrm{C}_{5}}:~{}\underset{q\in{\mathcal{Q}}}{\overset{}{\mathop{\sum}}}\,\underset{s\in{\mathcal{S}}}{\overset{}{\mathop{\sum}}}\,a_{mb}^{sq}x_{mb}^{s}R_{mb}^{sq}\geq{{R}_{\min}},$
$\displaystyle{\textrm{C}_{6}}:~{}x_{mb}^{s}\in\left\\{0,1\right\\},~{}~{}\textrm{C}_{7}:a_{mb}^{sq}\in\left\\{0,1\right\\}.$

In $\mathcal{O}_{1}$, constraints $\textrm{C}_{1}$ and $\textrm{C}_{2}$ ensure
each sub-channel in each cell can only be assigned to one user, and also, each
user utilizes only one antenna in each sub-channel, respectively.
$\textrm{C}_{3}$ and $\textrm{C}_{4}$ indicate that the transmit power is non-
negative and that the total transmit power of each user is limited to
$p_{\max}$, respectively. $\textrm{C}_{5}$ guarantees a minimum data rate
requirement for each user. $\textrm{C}_{6}$ and $\textrm{C}_{7}$ indicate that
the sub-channel and the antenna indicators take only binary values.

###  III-B EE-SE for the CAS Scenario:

The resource allocation and antenna selection for this scenario can be written
as:

$\displaystyle\mathcal{O}_{2}:~{}\min_{\textbf{a},\textbf{x},\textbf{p}}~{}$
$\displaystyle\eta_{EE}^{-1}(\textbf{a},\textbf{x},\textbf{p})$
$\displaystyle\min_{\textbf{a},\textbf{x},\textbf{p}}~{}$
$\displaystyle\eta_{SE}^{-1}(\textbf{a},\textbf{x},\textbf{p})$ s.t.
$\displaystyle{\textrm{C}_{1}}:\underset{m\in\mathcal{M}_{b}}{\overset{}{\mathop{\sum}}}\underset{q\in\mathcal{Q}}{\overset{}{\mathop{\sum}}}x_{mb}^{sq}=1,$
$\displaystyle{\textrm{C}_{2}}:\underset{q=1}{\overset{Q}{\mathop{\sum}}}\,a_{mb}^{q}=1,$
$\displaystyle{\textrm{C}_{3}}:p_{mb}^{sq}\geq 0,$ (4)
$\displaystyle{\textrm{C}_{4}}:\underset{q\in{\mathcal{Q}}}{\overset{}{\mathop{\sum}}}\,\underset{s\in{\mathcal{S}}}{\overset{}{\mathop{\sum}}}\,a_{mb}^{q}x_{mb}^{sq}p_{mb}^{sq}\leq{{p}_{\max}},$
$\displaystyle{\textrm{C}_{5}}:~{}\underset{q\in{\mathcal{Q}}}{\overset{}{\mathop{\sum}}}\,\underset{s\in{\mathcal{S}}}{\overset{}{\mathop{\sum}}}\,a_{mb}^{q}x_{mb}^{sq}R_{mb}^{sq}\geq{{R}_{\min}},$
$\displaystyle{\textrm{C}_{6}}:~{}x_{mb}^{sq}\in\left\\{0,1\right\\},\textrm{C}_{7}:a_{mb}^{q}\in\left\\{0,1\right\\}.$

Constraints of $\mathcal{O}_{2}$ are similar to those of $\mathcal{O}_{1}$
except $\textrm{C}_{1}$ and $\textrm{C}_{2}$. Specifically, $\textrm{C}_{1}$
indicates that each sub-channel inside each cell is assigned to at most one
user and each user utilizes only one antenna for all its assigned sub-
channels, respectively. Other constraints are similar to problem
$\mathcal{O}_{1}$. In the sequel, we only focus on solving optimization
$\mathcal{O}_{1}$ which is a general form of $\mathcal{O}_{2}$. Furthermore,
we compare the results obtained from GAS and CAS via numerical results and
study the trade off between EE and SE for two aforementioned scenarios. Now,
we propose a MOOP framework, to strike a balance between EE and SE. In order
to handle the non-convex optimization problem $\mathcal{O}_{1}$, we define a
new MOOP that is a generalization of $\mathcal{O}_{1}$. It is proved that the
optimization problem $\mathcal{O}_{1}$ can be equivalently written as the
throughput is maximized while the aggregated power consumption is minimized
simultaneously as follow [9]

$\displaystyle\begin{aligned} &\max_{\textbf{a},\textbf{x},\textbf{p}}\
R^{\textrm{total}}\\\
&\min_{\textbf{a},\textbf{x},\textbf{p}}P^{\textrm{total}}~{}\text{s.t.}~{}{\textrm{C}_{1}}-{\textrm{C}_{7}}.\end{aligned}$
(5)

The weighting coefficients are employed to convert the MOOP at hand into a
SOOP that reflects the required preferences [19]. Therefore, we rewrite the
objective function of (5) as:

$\displaystyle\max_{\textbf{a},\textbf{x},\textbf{p}}~{}\textrm{EE-
SE}=\frac{\nu}{w_{R}}{R}^{\textrm{total}}-\frac{(1-\nu)}{w_{P}}{P}^{\textrm{total}}$
(6) $\displaystyle\text{s.t.}~{}{\textrm{C}_{1}}-{\textrm{C}_{7}},~{}~{}~{}$

where $w_{R}$, and $w_{P}$ indicate the normalization factors. Furthermore,
$0\leq\nu\leq 1$ denotes the weighting coefficient, capturing the importance
of different objectives (here EE and SE).

##  IV Proposed solution

This section aims at providing a suboptimal solution for the problem of (6).
To this end, we first exploit the fact that $x^{s}_{mb}$ and $a^{sq}_{mb}$
take binary values at the optimal point. Noting that the product term
$a^{sq}_{mb}x^{s}_{mb}$ and the coupling with an affine variable $p^{sq}_{mb}$
in $\textrm{C}_{4}$ are the obstacles for the design of an efficient resource
allocation algorithm. By applying the big-M formulation, we can rewrite
$\textrm{C}_{4}$ into the following equivalent constraints:

$\displaystyle\textrm{C}_{8}:~{}$ $\displaystyle p^{sq}_{mb}\leq
x^{s}_{mb}~{}p_{\max},$ $\displaystyle\textrm{C}_{9}:~{}$ $\displaystyle
p^{sq}_{mb}\leq a^{sq}_{mb}~{}p_{\max},$ $\displaystyle\textrm{C}_{10}:~{}$
$\displaystyle a^{sq}_{mb}\leq x^{s}_{mb}.$ (7)

In (7), it can be seen that $a^{sq}_{mb}x^{s}_{mb}$ is decoupled which
simplifies the problem at hand. Yet, even with equivalent constraint
$\textrm{C}_{8}-\textrm{C}_{10}$ the problem (6) is still a MINLP which is
complicated to solve. Thus, we take one step further to relax the binary
constraints. The first step is to equivalently express the binary constraints
$\textrm{C}_{6}$ and $\textrm{C}_{7}$ as the intersection of the following
regions [20, 21]

$\displaystyle\mathcal{R}_{1}:0\leq a^{sq}_{mb}\leq
1,~{}\,\mathcal{R}_{2}:0\leq x^{s}_{mb}\leq 1,$ (8)
$\displaystyle\mathcal{R}_{3}:\sum_{b}\sum_{m}\sum_{s}\sum_{q}\Big{(}a^{sq}_{mb}-(a^{sq}_{mb})^{2}+x^{s}_{mb}-(x^{s}_{mb})^{2}\Big{)}\leq
0.$

For notational simplicity, we define the feasible set spanned by all
constraints and $R_{3}$ denoted by $\mathcal{D}$. The optimization problem of
(6) can be written as follows:

$\displaystyle\begin{aligned}
~{}&\max_{\textbf{a},\textbf{x},\textbf{p}}~{}\textrm{EE-
SE}(\textbf{a},\textbf{x},\textbf{p})~{}&\text{s.t.}~{}\textbf{a},\textbf{x},\textbf{p}\in\mathcal{D},~{}\mathcal{R}_{1},~{}\mathcal{R}_{2}.\end{aligned}$
(9)

Then, we propose a method to obtain an integer solution for $a^{sq}_{mb}$ and
$x^{s}_{mb}$ as well as the power allocation policy. To achieve this goal, we
introduce a penalty term for the objective function in (9). Thus, the problem
is modified as:

$\displaystyle\max_{\textbf{a},\textbf{x},\textbf{p}}\
L(\textbf{a},\textbf{x},\textbf{p},\lambda)~{}~{}~{}\text{s.t.}~{}~{}\textbf{a},\textbf{x},\textbf{p}\in\mathcal{D},\mathcal{R}_{1},\mathcal{R}_{2},$
(10)

where $L(\textbf{a},\textbf{x},\textbf{p},\lambda)$ is the abstract Lagrangian
of (9), and is defined as

$\displaystyle\begin{aligned}
&L(\textbf{a},\textbf{x},\textbf{p},\lambda)\triangleq\textrm{EE-
SE}(\textbf{a},\textbf{x},\textbf{p})\\\
-&\lambda\Bigg{[}\sum_{b,m,s,q}\Big{(}a^{sq}_{mb}-(a^{sq}_{mb})^{2}\Big{)}+\sum_{b,m,s}\Big{(}x^{s}_{mb}-(x^{s}_{mb})^{2}\Big{)}\Bigg{]}.\end{aligned}$
(11)

Note that, $\lambda\gg 1$ acts as the penalty factor to penalize the objective
function [20, 21, 22]. Furthermore, we adopt a short hand notation
$\sum_{b,m,s,q}=\sum_{b\in\mathcal{B}}\sum_{m\in\mathcal{M}}\sum_{s\in\mathcal{S}}\sum_{q\in\mathcal{Q}}$
for the sake of presentation.

The optimization problem (10) is a continuous optimization problem with
respect to all variables. However, due to the interference in the rate
function, the resulting optimization problem in (10) is still non convex. To
facilitate the solution design, we first rewrite the optimization problem in
the form of difference of concave (DC) function. Mathematically speaking, we
have

$\displaystyle\frac{\nu}{w_{R}}\Big{(}F({\textbf{p}})-G({\textbf{p}})\Big{)}-\frac{(1-\nu)}{w_{P}}P^{\textrm{total}}$
(12)
$\displaystyle-\lambda\Big{(}D_{1}(\mathbf{a})-D_{2}(\mathbf{a})+E_{1}(\mathbf{x})-E_{2}(\mathbf{x})\Big{)},$

where

$\displaystyle
F({\textbf{p}})\\!\\!\triangleq\\!\\!\\!\\!\\!\\!\sum_{b,m,s,q}\\!\\!\Bigg{(}\log_{2}\Big{(}{{p}^{sq}_{mb}|h^{sq}_{m{b},b}|^{2}}+\sigma^{2}+\\!\\!\\!\\!\\!\\!\\!\\!\sum_{b^{\prime}\neq
b,k,q^{\prime}}{p}^{sq^{\prime}}_{kb^{\prime}}|h^{sq^{\prime}}_{k{b^{\prime}},b}|^{2}\Big{)}\Bigg{)},$
(13) $\displaystyle
G({\textbf{p}})\triangleq\sum_{b,m,s,q}\Big{(}\log_{2}\big{(}\\!\sigma^{2}\\!\\!+\\!\\!\sum_{b^{\prime}\neq
b,k,q^{\prime}}{p}^{sq^{\prime}}_{kb^{\prime}}|h^{sq^{\prime}}_{k{b^{\prime}},b}|^{2})\Big{)},$
(14) $\displaystyle
D_{1}(\mathbf{a})\triangleq\sum_{b,m,s,q}\Big{(}a^{sq}_{mb}\Big{)},~{}D_{2}(\mathbf{a})\triangleq\sum_{b,m,s,q}\Big{(}a^{sq}_{mb}\Big{)}^{2},$
(15) $\displaystyle
E_{1}(\mathbf{x})\triangleq\sum_{b,m,s}\Big{(}x^{s}_{mb}\Big{)},E_{2}(\mathbf{x})\triangleq\sum_{b,m,s}\Big{(}x^{s}_{mb}\Big{)}^{2}.$
(16)

One can readily observe that the objective function can be written as the
difference of two concave functions. In a similar way, the left hand side of
the constraint $\textrm{C}_{5}$ can be written as:

$f_{mb}({\textbf{p}})-g_{mb}({\textbf{p}}),$ (17)

where

$\displaystyle
f_{mb}({\textbf{p}})\\!\triangleq\sum_{s,q}\log_{2}\bigg{(}{{p}^{sq}_{mb}}|h^{sq}_{m{b},b}|^{2}+\sigma^{2}+\\!\\!\\!\\!\\!\\!\\!\\!\sum_{b^{\prime}\neq
b,k,q^{\prime}}{p}^{sq^{\prime}}_{kb^{\prime}}|h^{sq^{\prime}}_{k{b^{\prime}},b}|^{2}\bigg{)},$
(18) $\displaystyle
g_{mb}({\textbf{p}})\triangleq\sum_{s,q}\log_{2}\bigg{(}\\!\sigma^{2}\\!\\!+\\!\\!\sum_{b^{\prime}\neq
b,k,q^{\prime}}{p}^{sq^{\prime}}_{kb^{\prime}}|h^{sq^{\prime}}_{k{b^{\prime}},b}|^{2}\bigg{)}.\quad\quad$
(19)

To solve the equivalent DC problem we apply the majorization minimization
approach [13] to obtain a locally optimal solution of (12). To this end, the
first order Taylor approximations for
$G({\textbf{p}}),~{}D_{2}(\textbf{a}),~{}E_{2}(\textbf{x})$, and
$g({\textbf{p}})$ are being exploited to convert the DC problem into a convex
problem which can be effectively solved iteratively. Since
$G({\textbf{p}}),~{}D_{2}(\textbf{a}),~{}E_{2}(\textbf{x})$, and
$g({\textbf{p}})$ are differentiable convex functions, for any feasible point
$\textbf{p}^{i}$, $\textbf{a}^{i}$, and $\textbf{x}^{i}$, we have

$\displaystyle{G}({\textbf{p}})\leq
G({\textbf{p}}^{i-1})+\nabla_{{\textbf{p}}}G^{T}({\textbf{p}}^{i-1})({\textbf{p}}-{\textbf{p}}^{i-1})\triangleq\tilde{G}({\textbf{p}}),$
(20) $\displaystyle{D}_{2}(\textbf{a})\geq
D_{2}(\textbf{a}^{i-1})+\nabla_{\textbf{a}}D_{2}^{T}(\textbf{a}^{i-1})(\textbf{a}-\textbf{a}^{i-1})\triangleq\tilde{D}_{2}(\textbf{a}),$
(21) $\displaystyle{E}_{2}(\textbf{x})\geq
E_{2}(\textbf{x}^{i-1})+\nabla_{\textbf{x}}E_{2}^{T}(\textbf{x}^{i-1})(\textbf{x}-\textbf{x}^{i-1})\triangleq\tilde{E}_{2}(\textbf{x}),$
(22) $\displaystyle{g}_{mb}(\textbf{p})\leq
g_{mb}({\textbf{p}}^{i-1})+\nabla_{{\textbf{p}}}g^{T}_{mb}({\textbf{p}}^{i-1})({\textbf{p}}-{\textbf{p}}^{i-1})\triangleq\tilde{g}_{mb}(\textbf{p}).$
(23)

Hence, for the $i^{th}$ iteration, noting $\tilde{G}({\textbf{p}})$,
$\tilde{E}_{2}(\textbf{a}),~{}D_{2}(\textbf{x})$, and
$\tilde{g}_{mb}({\textbf{p}})$ are all affine functions. Therefore, a
suboptimal solution of (12) can be obtained by solving the following convex
optimization problem:

$\displaystyle\max_{\textbf{a},\textbf{x},{\textbf{p}}}$
$\displaystyle\frac{\nu}{w_{R}}\Big{(}F({\textbf{p}})-\tilde{G}({\textbf{p}})\Big{)}-\frac{(1-\nu)}{w_{P}}P^{\textrm{total}}$
$\displaystyle-\lambda\Big{(}D_{1}(\mathbf{a})-\tilde{D}_{2}(\mathbf{a})+E_{1}(\mathbf{x})-\tilde{E}_{2}(\mathbf{x})\Big{)}$
$\displaystyle\text{s.t.}\quad
f_{mb}({\textbf{p}})-\tilde{g}_{mb}({\textbf{p}})\geq R_{\min},$
$\displaystyle
x^{s}_{mb},a^{sq}_{mb}\in[0,1],~{}\textrm{C}_{1}-\textrm{C}_{4},~{}\textrm{C}_{8}-\textrm{C}_{10}.$
(24)

The convex optimization problem of (IV) is solved based on method as presented
in Algorithm 1.

Algorithm 1 Proposed Method MOOP based on Successive Convex Approximation
(MOOP DC Programming)

1: Initialize $i=0$ and maximum number of iteration $I_{\max}$, penalty factor
$\lambda\gg 1$ , set appropriate weighting coefficient factor $(\nu)$ and
feasible set vector $\textbf{p}^{0}$, $\textbf{x}^{0}$, and $\textbf{a}^{0}$

2: Repeat

3: Update $\tilde{G}({\textbf{p}})$, $\tilde{D}_{2}(\textbf{a})$,
$\tilde{E}_{2}(\textbf{x})$, and $\tilde{g}({\textbf{p}})$ as presented in
(20), (21), (22), and (23), respectively.

4:Solve optimization problem of (24) and store the intermediate resource
allocation policy $\textbf{a}^{i}$, ${\textbf{x}^{i}}$, and $\textbf{p}^{i}$

5:Set $i=i+1$ and $\textbf{p}^{i}=\textbf{p}$, $\textbf{x}^{i}=\textbf{x}$,
and $\textbf{a}^{i}=\textbf{a}$

6: Until convergence or $i=I_{\max}$

7:$\textbf{a}^{*}=\textbf{a}^{i}$, $\textbf{x}^{*}=\textbf{x}^{i}$,
$\textbf{p}^{*}=\textbf{p}^{i}$

##  V Computational Complexity Analysis

In this section, we analyze the computational complexity of considered
optimization problems in the proposed algorithm in Algorithm 1 via DC
programming [20, 24]. For the optimization problem there are totally $BMSQ$
decision variables and $SB+BMS+BM+BM+BMS+2MBSQ$ convex and affine constraints
in the convex program (III-A). Therefore, solving the optimization problem via
the interior point method requires the order of
$\mathcal{O}(BMSQ)^{3}(SB+2BM+2BMS+2MBSQ)$ which is polynomial time
complexity. An exhaustive search would require the examination of all
$(M)^{BSQ}B^{Q}B^{S}$ possible sub-channel assignment and antenna selection
choices. Note that exhaustively searching over all possible choices of sub-
channels/antennas is a time consuming task even for moderate number of
antennas/users and sub-channels. Furthermore, when CVX is employed to solve
the problems in (24), it employs D.C. with the interior point method and the
number of required iterations for this approach is
$\frac{\log(c)/t^{0}\delta}{\log(\epsilon)}$, where $c$ is the total number of
constraints, $t^{0}$ is the initial point, $0<\delta\ll 1$ is the stopping
criterion, and $\epsilon$ is used for updating the accuracy of the method [25,
24, 27].

##  VI Numerical Results

In this section, the performance of our proposed approach is investigated. We
adopt the simulation parameters given in Table I, unless specified otherwise.
The wireless channel gains are Rayleigh fading set as
$h^{n}_{mb}=\phi^{n}d^{-\alpha}_{mb}$, where $d_{mb}$ is the distance between
users and the base stations. The carrier frequency and sub-channel bandwidth
are 2 GHz and 180 kHz, respectively. Monte Carlo simulation is performed by
generating random channel realizations and we compute the average system
performance.

TABLE I: simulation parameters Parameter | Value
---|---
Cell radius |  $500$ m
The number of cell | $4$
The number of user in each cell | $4$
$N$, subcarrier bandwidth | {$8,16$} and $180$ kHz
Noise power ($N_{0}$) |  $-174$ dBm/Hz
Path loss exponent ($\alpha$) | $3$
$P_{m}^{\textnormal{static}}$ |  $10$ dBm
$P_{m}^{\textnormal{antenna}}$ |  $7$ dBm
$p_{\textrm{max}}$ |  $23$ dBm
$R_{\textrm{min}}$ |  $5$ bps/Hz
Number of channel realization | $100$
Figure 3:  Average system EE versus different number of user’s antenna for the
three schemes. Figure 4: Average sum rate versus different number of user’s
antenna for the three schemes.

Fig. 3 and Fig. 4 illustrate the EE and SE versus number of user’s antenna for
different schemes namely, GAS, CAS, and without antenna selection scenarios,
respectively. It can be concluded from Fig. 3 that in the GAS scenario for a
large value of dynamic power, by increasing the number of user’s antenna, the
EE first increases and reaches to a maximum value and then decreases with the
number of user’s antenna. This is because when the number of user’s antenna
increases, the dynamic term of the power consumption grows linearly which
leads to a reduction of EE. Moreover, from this figure, we observe that the EE
increases as the dynamic power decreases. We also observe that the slope of
the curve decreases with the number of user’s antennas. However, the slope of
the curve varies from one case to another. For instance, for the GAS scenario
with $P_{m}^{\textnormal{antenna}}=0$ dBm, the slope of the curve starts to
decline for $Q>4$ while for GAS scenario with $P_{m}^{\textnormal{antenna}}=7$
dBm starts to decrease when $Q>3$. The reason for this trend is that once the
maximum EE of the system is achieved, a further increase in the number of
user’s antenna would result in a degradation in EE due to increasing the
dynamic term of power consumption.

On the other hand, as illustrated in Fig. 4 in the GAS scenario, the system
sum rate improves by increasing the number of user’s antenna. It should be
noted that although GAS is an effective method to improve the systems
throughput, it increases the power consumption of the network. This is because
a subset of the antenna, must be selected which increases the power
consumption of user per RF chain. Moreover as illustrated in Fig. 3, for the
CAS scenario when the number of user’s antenna increases, the EE increases
monotonically and becomes saturated. As shown in Fig. 4, although CAS improves
system throughput (in comparison to the scenario where no antenna selection
scheme is employed), its data rate is still less than that of GAS. On the
other hand, the aggregated power consumption of CAS is less than GAS. This is
due to the fact in CAS, we have only one RF chain which reduces the dynamic
term of the power consumption model, leading to EE enhancement (in comparison
to the GAS scenario). In fact, there is a non-trivial trade off between the
data rate of the system and its total power consumption.

##  VII Conclusion

This paper investigated the tradeoff between EE and SE of uplink multi-cell
networks via joint sub-channel assignment, power control, and antenna
selection for two scenarios. In the first scenario, known as CAS, there is
only one RF chain available which all sub-channels for each user can be
assigned to one of the antennas. For the second scenario, known as GAS, the
number of RF chains is equal to the number of antennas in which each sub-
channel for each user can be assigned to one of the antennas and subset of the
antenna must be selected. It is shown that the resource allocation design can
be formulated as a MOOP, which the conflicting objective functions are
linearly combined into a single objective function employing the weighted sum
method. The considered problem MINLP is generally intractable. In order to
obtain a computationally efficient suboptimal solution, the majorization
minimization approach is proposed where a surrogate function serves as a lower
bound of the objective function. Simulation result demonstrated the
superiority of the proposed method. Furthermore, the proposed antenna
selection scheme can strike an excellent balance of improving SE and EE.

## References

  * [1] Q. Wu, G. Y. Li, W. Chen, D. W. K. Ng, and R. Schober, “An overview of sustainable green 5G networks,” IEEE Wireless Commun., vol. 24, no. 4, pp. 72-80, Aug. 2017.
  * [2] V. W. Wong, R. Schober, D. W. K. Ng, and L.-C. Wang, Key Technologies for 5G Wireless Systems. Cambridge University Press, 2017.
  * [3] X. Li, X. Ge, X. Wang, J. Cheng, and V. C. M. Leung, “Energy efficiency optimization: joint antenna-subcarrier-power allocation in OFDM-DASs,” IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7470-7483, Nov. 2016.
  * [4] J. Tang, D. K. C. So, E. Alsusa, K. A. Hamdi, A. Shojaeifard, and K. Wong, “Energy-efficient heterogeneous cellular networks with spectrum underlay and overlay access,” IEEE Trans. Veh. Technol., vol. 67, no. 3, pp. 2439-2453, Mar. 2018\.
  * [5] H. Zhang, H. Liu, J. Cheng, and V. C. M. Leung, “Downlink energy efficiency of power allocation and wireless backhaul bandwidth allocation in heterogeneous small cell networks,” IEEE Trans. Commun, vol. 66, no. 4, pp. 1705-1716, Apr. 2018\.
  * [6] M. Tham, S. F. Chien, D. W. Holtby, and S. Alimov, “Energy-efficient power allocation for distributed antenna systems with proportional fairness,” IEEE Trans. Green Commun. Netw., vol. 1, no. 2, pp. 145-157, Jun. 2017.
  * [7] H. Chunlong, B. Sheng, P. Zhu, X. You, and G. Y. Li, “Energy- and spectral-efficiency tradeoff for distributed antenna systems with proportional fairness,” IEEE J. Select. Areas Commun., vol. 31, no. 5, pp. 894-902, May. 2013.
  * [8] D. Tsilimantos, J. Gorce, K. Jaffrs-Runser, and H. Vincent Poor, “Spectral and energy efficiency trade-offs in cellular networks,” IEEE Trans. Wireless Commun., vol. 15, no. 1, pp. 54-66, Jan. 2016
  * [9] O. Amin, E. Bedeer, M. Hossam, and O.A. Dobre, “Energy efficiency-spectral efficiency tradeoff: a multiobjective optimization approach”, IEEE Trans. Veh. Technol., vol. 65, no. 4, pp. 1975-1981, Apr. 2016.
  * [10] N. B. Mehta, A. F. Molisch, and S. Kashyap, “Antenna selection in LTE: from motivation to specification,” IEEE Communications Magazine, vol. 50, no. 10, pp. 144-150, Oct. 2012
  * [11] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Commun. Mag., vol. 42, no. 10, pp. 68-73, Oct. 2004
  * [12] N. P. Le, F. Safaei, and L. C. Tran, “Antenna selection strategies for MIMO-OFDM wireless systems: An energy efficiency perspective,” IEEE Trans. Veh. Technol., vol. 65, no. 4, pp. 2048-2062, Apr. 2016.
  * [13] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” IEEE Trans. Signal Process., vol. 65, no. 3, pp. 794-816, Feb. 2017.
  * [14] A. Annamalai, G. Deora, and C. Tellambura, “Analysis of generalized selection diversity systems in wireless channels”, IEEE Trans. Veh. Technol.,vol. 55, no. 6, pp. 1765-1775, Nov. 2006.
  * [15] A. F. Molisch, M. Z. Win, Y.-S. Choi, and J. H. Winters, “Capacity of MIMO systems with antenna selection”, IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1759-1772, Jul. 2005.
  * [16] A. Khalili, M. Robat Mili, M. Rasti, S. Parsaeefard, and D. W. K. Ng, “Antenna selection strategy for energy efficiency maximization in uplink OFDMA networks: A multi-objective approach,” IEEE Trans. Wireless Commun., vol. 19, no. 1, pp. 595-609, Jan. 2020.
  * [17] A. Khalili, S. Akhlaghi, H. Tabassum, and D. W. K. Ng “Joint user association and resource allocation in the uplink of heterogeneous networks,” arXiv:1911.12697, 2019.
  * [18] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks,” IEEE J. Select. Areas Commun., vol. 22, no. 6, pp. 1089-1098, Aug. 2004.
  * [19] Y. Collette and P. Siarry, Multiobjective optimization: principles and Case Studies. Springer, 2003.
  * [20] E. Che, H. D. Tuan and, H. H. Nguyen, “Joint optimization of cooperative beamforming and relay assignment in multi-user wireless relay networks,” IEEE Trans. Wireless Commun., vol. 13, no. 10, pp. 5481-5495, Oct. 2014.
  * [21] A. Khalili, S. Zarandi, and M. Rasti, “Joint resource allocation and offloading decision in mobile edge computing,” IEEE Communications Letters. vol. 23, no. 4, pp. 684-687, Apr. 2019.
  * [22] Y. Sun, D. W. K. Ng, Z. Ding, and R. Schober, “Optimal joint power and subcarrier allocation for full-duplex multicarrier non-orthogonal multiple access systems,” IEEE Trans. Commun., vol. 65, no. 3, pp. 1077-1091, Mar. 2017.
  * [23] H. H. Kha, H. D. Tuan, and H. H. Nguyen, “Fast global optimal power allocation in wireless networks by local D.C. programming,” IEEE Trans. Wireless Commun., vol. 11, no. 2, pp. 510-515, Feb. 2012.
  * [24] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
  * [25] M. R. Mili, A. Khalili, D. W. K. Ng and H. Steendam, “A novel performance tradeoff in heterogeneous networks: A multi-objective approach,” IEEE Wireless Commun Letts. vol. 8, no. 5, pp. 1402-1405, Oct. 2019.
  * [26] A. Khalili, S. Zarandi, M. Rasti, and E. Hossain “Multi-objective optimization for energy-and spectral-efficiency tradeoff in in-band full-duplex (IBFD) communication” arXiv:1907.08250v1, Accepted by IEEE Global Communications Conference, 2019.
  * [27] R. Aslani, M. Rasti, and A. Khalili, “Energy efficiency maximization via joint sub-carrier assignment and power control for OFDMA full duplex networks,” IEEE Trans. Veh Technol., vol. 68, no. 12, pp. 11859-11872, Dec. 2019.


* [23] Ashish Vaswani et al.  “Attention is All you Need”  In _Advances in Neural Information Processing Systems_ 30 Curran Associates, Inc., 2017  URL: https://proceedings.neurips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf
  * [24] Xiaofeng Yuan, Lin Li and Yalin Wang  “Nonlinear Dynamic Soft Sensor Modeling With Supervised Long Short-Term Memory Network”  In _IEEE Transactions on Industrial Informatics_ 16.5, 2020, pp. 3168–3176  DOI: 10.1109/TII.2019.2902129

##  Appendix A Hyperparameters used

In this chapter, we consolidate the optimal hyperparameter settings used in
all our architectures.

Table A.1: Hyperparameter setting for arch-i:Mott (Section 3.1), arch-ii:Adaptive architecture (Section 3.2), arch-iii:Spatio-Temporal sequential (Figure 3.8 Left) and arch-iv:Spatio-Temporal one-shot architecture using actor query(Figure 3.8 Right) parameter |  arch-i |  arch-ii |  arch-iii |  arch-iv
---|---|---|---|---
unroll_length |  160 |  239 |  239 |  239
chunk_size |  NA |  80 |  80 |  80
num_buffers |  60 |  60 |  60 |  60
num_actors |  32 |  32 |  50 |  50
batch_size |  12 |  16 |  12 |  12
Table A.2: Hyperparameter setting for Divided Space-Time (Figure 3.10 Right) and Joint Space-Time (Figure 3.10 Left) parameter |  Divided |  Joint
---|---|---
unroll_length |  239 |  239
chunk_size |  80 |  10
num_buffers |  40 |  40
num_actors |  32 |  32
batch_size |  24 |  4
embed_dim |  16 |  16
patch_size |  7 |  7
height |  84 |  42
width |  84 |  42
n_layer |  1 |  1