Figure 11 also presents the DL throughput of passengers (dashed lines) and
pedestrians (solid lines), but for three different packet sizes and
considering $50\%$ of the UE in the system as passengers. Similar to the
analyses of Fig. 10, we can conclude here that the deployment of mIAB nodes
inside the buses improved the passengers DL throughput, while keeping the
pedestrians with a good DL throughput. Furthermore, we highlight that for
higher percentages of passengers in the system and higher packet size, i.e.,
more loaded systems, the gains of using mIAB relative to the only macros and
macros-picos scenarios were higher.

(a) Packet size = 3072 bits.

(b) Packet size = 2048 bits.

(c) Packet size = 1024 bits.

Figure 11: Downlink throughput – Impact of packet size, considering percentage
of passengers equal to $50\%$.

#### Downlink Data Transmission Quality

Figures 11a and 11b present the passengers’ MCS usage in DL for the only
macros and mIAB scenarios, respectively for the configuration with $50\%$ of
passengers in the system and packet size of $3072$ bits. The red bars
represent the percentage of transmissions with error and the blue bars
represent the percentage of successful transmissions. First, we remark that
Fig. 11b does not refer to the passengers end-to-end link quality (IAB donor -
passengers), it only refers to the quality of the link between the mIAB nodes
and their served passengers. The quality of the link between the IAB donors
and the mIAB nodes will be analyzed in Section V-B3.

On the one hand, notice that in Fig. 11a, in the only macros scenario, the
passengers received data mainly with lower MCS. This was mainly due to the bus
penetration loss, which degraded the link between passenger and macro gNB. On
the other hand, notice that in Fig. 11b, in the mIAB scenario, the passengers
received data from the mIAB node with the highest MCS and with almost zero
transmission error. This is a benefit of deploying the DU part of the mIAB
node inside the buses.

(a) Only macros scenario.

(b) mIAB scenario.

Figure 12: Histogram of passengers’ MCS usage in DL in a scenario with $50\%$
of passengers and packet size of $3072$ bits.

The pedestrians’ MCS usage in DL is presented in Fig. 13. Figure 12a is
related to the only macros scenario, while Fig. 12b concerns pedestrians
connected to an IAB donor in the mIAB scenario and Fig. 12c concerns
pedestrians connected to an mIAB node also in the mIAB scenario. First, notice
that the histograms in Fig. 12a and Fig. 12b are similar with small
differences, meaning that the SINR of a pedestrian connected to a macro gNB in
the only macros scenario and the SINR of a pedestrian connected to an IAB
donor in the mIAB scenario were similar. Furthermore, considering that the
pedestrian signal strength was also similar in both cases, since the macro gNB
and the IAB donor had similar characteristics, we conclude that the
interference in both cases was also similar.

In the mIAB scenario, $15\%$ of the pedestrians were served by an mIAB node
for at least a couple of TTI. Particularly, their connection lasted less than
$1\text{\,}\mathrm{s}\text{/}$ in $90\%$ of the cases, as illustrated in Fig.
14, which is not good in practice. They connected to a mIAB node due to a
higher RSRP. However, when they were receiving data from the DU part of the
mIAB node, the MT part was transmitting upstreaming data to the IAB donor in
the backhaul, as described in Table III slots $3$, $5$ and $8$. Thus, the MT
part of the mIAB node caused a high interference to the pedestrians connected
to an mIAB node, since both were active at the same time and both were outside
the bus, meaning that the interfering link did not suffer attenuation due to
the crossing of the bus body. This inference caused low values of SINR and so
the usage of low MCS as shown in Fig. 12c. Important to remark that this
interference was not a problem for the passengers, since they were inside the
bus and the signal from the MT part suffered from the bus body penetration
loss.

Based on these results, one can conclude that pedestrians should avoid
connecting to an mIAB node, unless the signal between them is strong enough to
compensate the interference from the MT part and the connection is expected to
last longer than a given threshold. For this, one could consider an admission
policy to allow a UE to connect to a mIAB cell. The admission criteria could
be: a minimum measured RSRP value; a maximum measured interference level (RSRQ
instead of RSRP); the relative UE and bus geographical position in a given
time interval; etc..

(a) Only macros scenario.

(b) mIAB scenario - pedestrians connected to an IAB donor.

(c) mIAB scenario - pedestrians connected to an mIAB node.

Figure 13: Histogram of pedestrians’ MCS usage in DL in a scenario with $50\%$
of passengers and packet size of $3072$ bits. Figure 14: CDF of connection
duration between a pedestrian and a mIAB node.

Figures 14a and 14b present the CDF of SINR and SNR in the DL of pedestrians
connected to a macro gNB in the only macros and mIAB scenarios, respectively.
Comparing these two figures, notice that the difference between the SINR and
SNR curves, i.e., the interference, is similar (the difference neither
increased nor decreased when changing the scenario). Thus, we can conclude
that the deployment of mIAB nodes may not impact the interference that the
pedestrians connected to a macro gNB suffered. This is an important result,
since the possible dynamic interference caused by mIAB nodes in the UE served
by IAB donors is one of the concerns regarding the deployment of mIAB cells.
The interference avoidance was obtained due to the adopted TDD scheme (Table
III), which did not allow transmissions in the access links of mIAB nodes when
IAB donors were operating in DL in their access links. The topic of exploiting
a TDD scheme for interference handling in mIAB networks is addressed in [118].

(a) Only macros scenario.

(b) mIAB scenario - pedestrians connected to an IAB donor.

Figure 15: SINR and SNR of pedestrians in a scenario with $50\%$ of passengers
and packet size of $3072$ bits.

Figures 15a and 15b present the CDF of SINR and SNR in the DL of passengers
connected to a macro gNB in the only macros scenario and connected to an mIAB
node in the mIAB scenario, respectively. In Fig. 15a, the SINR and SNR curves
are superposed, meaning that in the only macros scenario the interference was
not a problem for the passengers. This was a consequence of the bus body loss
that filtered possible interference links coming from neighbor gNB. The
drawback of the bus body loss is that it filters not only the interference but
also the signals serving the passengers, as we can see by the low values of
the SNR. In this scenario, $80\%$ the passengers had an SNR lower than
$10\text{\,}\mathrm{d}\mathrm{B}$. On the other hand, in the mIAB scenario,
their SNR had a boost when deploying the mIAB nodes in the buses (Fig. 15b).
In this scenario, when the backhaul (MT part of the mIAB nodes) was operating
in the UL, they interfered with the passengers receiving in the DL from the DU
part of the mIAB nodes. However, as we can see in Fig. 15b, the SINR was still
high enough to allow transmissions with the highest MCS.

(a) Only macros scenario.

(b) mIAB scenario - passengers connected to a mIAB.

Figure 16: SINR and SNR of passengers in a scenario with $50\%$ of passengers
and packet size of $3072$ bits.

#### Downlink Latency

(a) 75% of passengers.

(b) 50% of passengers.

(c) 25% of passengers.

Figure 17: Downlink latency – Impact of percentage of passengers, considering
packet size equal to $3072$ bits.

DL transmissions with low MCS, thus low DL throughput, to the passengers in
the only macros and macros-picos scenarios impacted the passengers DL latency
as it can be seen in Fig. 17 and Fig. 18. These figures present the CDF of the
UE end-to-end DL latency for different values of the percentage of passengers
and of the packet size, respectively. Dashed and solid curves represent the DL
latency of passengers and pedestrians, respectively. On the one hand, one can
see that passengers using delay sensitive services will suffer in the only
macros and macros-picos scenarios, since in the simulations their DL latency
achieved values up to $6\text{\,}\mathrm{s}\text{/}$. On the other hand, in
the mIAB scenario, DL packet latency was negligible. In that scenario, for the
majority of the passengers, in the worst case, the latency was equal to
$2\text{\,}\mathrm{ms}\text{/}$. It corresponded to the case where packets
were generated in the IAB donor at slot $5$ of Table III and needed to wait
until slot $10$ to be transmitted in the backaul from the IAB donor to the
mIAB node and only at the next slot $3$ of the next frame it was transmitted
from the mIAB node to the passengers, totalizing $8$ slots, i.e.,
$2\text{\,}\mathrm{ms}\text{/}$, of delay. Furthermore, similar to the
analyses of DL throughput, we can see that for higher percentages of
passengers in the system and higher packet size, i.e., more loaded systems,
the gains in the mIAB scenario for the passengers DL latency were higher
compared to the other scenarios.

(a) Packet size = 3072 bits.

(b) Packet size = 2048 bits.

(c) Packet size = 1024 bits.

Figure 18: Downlink latency – Impact of packet size, considering percentage of
passengers equal to $50\%$.

####  V-B2 Uplink

Until now, we have analyzed the mIAB performance in the DL. Now, let’s focus
on the UL. In general, in the UL we came to similar conclusions as the ones
drawn in the DL analyses. However, due to different transmit power and
interference pattern in DL and UL, some points were different and will be
highlighted in the following.

Figure 19 presents the passengers’ MCS usage in UL for the only macros and the
mIAB scenarios for the configuration with $50\%$ of passengers in the system
and packet size of $3072$ bits. It is equivalent to Fig. 12, but for the UL.
The red bars represent the percentage of transmissions with error and the blue
bars represent the percentage of successful transmissions. Comparing Fig. 11a
and Fig. 18a, one can notice that in the DL of the only macros scenario, even
if the majority of the transmissions were with lower MCS, $5\%$ of the
transmissions were successful with the highest MCS. However, in the UL, less
than $1.5\%$ of the transmissions were successful with the highest MCS. The
main reason for this was the lower transmit power in the UL compared to the
DL. As a consequence, the SINR of the passengers was lower, allowing them to
mostly use only the lowest MCS.

(a) Only macros scenario.

(b) mIAB scenario.

Figure 19: Histogram of passengers’ MCS usage in UL in a scenario with $50\%$
of passengers and packet size of $3072$ bits.

(a) Only macros scenario.

(b) mIAB scenario - pedestrians connected to an IAB donor.

(c) mIAB scenario - pedestrians connected to an mIAB node.

Figure 20: Histogram of pedestrians’ MCS usage in UL in a scenario with $50\%$
of passengers and packet size of $3072$ bits.

The pedestrians’ MCS usage in UL is presented in Fig. 20. Figure 19a is
related to the only macros scenario, while Fig. 19b concerns pedestrians
connected to an IAB donor in the mIAB scenario and Fig. 19c concerns
pedestrians connected to an mIAB node also in the mIAB scenario. Figure 20 is
equivalent to Fig. 13, but for the UL. Comparing Fig. 12c and Fig. 19c, one
can notice that, for the pedestrians connected to an mIAB node, in the DL,
only $5\%$ of the transmissions were successful with the highest MCS, while,
in the UL, $13\%$ of the transmissions were successful with the highest MCS.
This difference is explained by the fact that, even though the transmit power
in the UL was lower, the interference was even lower. More precisely,
analyzing Table III, we can see that a pedestrian connected to an mIAB node
receiving data in the DL suffered interference from the MT part of the mIAB
node, which was transmitting in the UL to the IAB donor and which did not
suffer from the bus penetration loss since it was placed outside the bus.
However, when the same pedestrian was transmitting in the UL to the DU part of
an mIAB node, the DU part suffered interference from other pedestrians
connected to the IAB donor and transmitting data to it in the upstreaming.
Thus, the interference that they caused in the DU also suffered from the bus
penetration loss.

The highlighted differences regarding the MCS distribution impacted the UL
throughput. Figure 21 presents the UL throughput of passengers (dashed lines)
and pedestrians (solid lines) considering packet size equal to $3072$ bits and
percentage of passengers equal to $50\%$. It is equivalent to Fig. 9b, but for
the UL. As we have seen, the link quality of passengers in the only macros
scenario was worse in the UL than in the DL due to the lower transmit power.
As a consequence, in the only macros scenario, passengers UL throughput was
worse than their DL throughput, as one can see comparing Fig. 21 and Fig. 9b.
Thus, for the passengers, the gains of deploying mIAB, in terms of throughput,
were even higher in the UL. Concerning the pedestrians, in the mIAB scenario,
their UL throughput was lower than their corresponding DL throughput. It is
explained by the fact that, in the adopted TDD scheme for the mIAB scenario,
less time slots are used for UL transmissions than for DL transmissions.

Figure 21: Uplink throughput considering packet size equal to $3072$ bits and
percentage of passengers equal to $50\%$.

Figure 22 presents the UL latency results. It is equivalent to Fig. 16b, which
presents the DL latency results. Conclusions similar to the ones obtained
frrom the UL throughput can be drawn regarding the UL latency.

Figure 22: Uplink latency considering packet size equal to $3072$ bits and
percentage of passengers equal to $50\%$.

####  V-B3 Backhaul Quality

In a mIAB scenario, besides the quality of the links of passengers and
pedestrians, one should also be careful with the quality of the backhaul link,
since when a bus suffered from a bad link to their serving IAB donor, all the
passengers connected to it also suffered. Figure 23 presents the histogram of
DL and UL backhaul’s MCS usage in a scenario with $50\%$ of passengers and
packet size of $3072$ bits. Comparing Fig. 22a and Fig. 22b with Fig. 11a and
Fig. 18a, respectively, one can again conclude that it was better for a
passenger to communicate with the IAB donor through an mIAB node instead of
being directly connected to it. The backhaul better link quality was mainly
due to the deployment of the mIAB node MT part outside the bus, thus
overcoming the bus penetration loss, and also due to the utilization of a ULA
with $64$ antenna elements instead of a single antenna as the passengers did.

(a) Downlink.

(b) Uplink.

Figure 23: Histogram of backhaul’s MCS usage in a scenario with $50\%$ of
passengers and packet size of $3072$ bits.

####  V-B4 Links Served by IAB Donors

We have seen that the adopted TDD scheme impacts the results. Thus, it is
interesting to try to understand how to better allocate the resources. For
this, it is important to understand the profile of the links served by the IAB
donors. Hence, in this part we analyze the number of links served by the IAB
donors, the percentage of them that are access links and how many links are
served by the IAB donors in the backhaul of mIAB nodes.

Figure 24 presents the histogram of the number of links served by IAB donors.
When serving a bus, i.e., an mIAB node, an IAB donor indirectly served through
the backhaul the UE served by the bus (almost always, only passengers). Thus,
when a bus disconnected from the IAB donor, the load in the donor suddenly
changed a lot, since the passengers went away together with the bus. Due to
this, as we can see in Fig. 24 the number of links served by an IAB donor
could be quite different from one donor to another, since if they served a
different number of buses, the difference in the number of served links was
approximately multiple of the number of passengers per bus.

This effect is well visible in Fig. 23a, which refers to the case of $75\%$ of
passengers in the system, i.e., nine passengers per bus. Notice that, in this
figure, there are six peaks (six was the number of buses in the simulations),
each one centered in a multiple of nine, which is the number of passengers in
a bus for this case. When the number of passengers was reduced from nine to
six and three as in Fig. 23b and Fig. 23c, respectively, the load variability
of the IAB donors decreased, since, in those cases, moving a bus from one IAB
donor to another did not heavily impact the load of those donors.

Based on these results, one should consider adopting topology adaptation
strategies that take into account the IAB donor load. This may avoid
overloading IAB donors when changing the serving donor of a given bus.

(a) 75% of passengers.

(b) 50% of passengers.

(c) 25% of passengers.

Figure 24: Histogram of the number of links served by macro gNB.

Figure 25 focuses only on the number of backhaul links served by IAB donors
and presents the histogram of this number. Since, in the backhaul, the IAB
donors were serving links from the buses, one could expect that through the
backhaul the donors would serve a number of links multiple of the number of
passengers in the buses. However, Fig. 25 shows that sometimes the backhaul
served a number of links a little bit higher than a number multiple of the
number of passengersper bus. For example, in Fig. 24a, which refers to the
case of $75\%$ of passengers in the system, i.e., nine passengers per bus,
there are six peaks, each one in multiples of nine, plus some bars
corresponding to numbers a little bit higher that the closest multiple of
nine. This is explained by the fact that buses could serve not only
passengers, but also pedestrians, totalizing a number of served links equal to
the number of passengers plus a small number of pedestrians.

(a) 75% of passengers.

(b) 50% of passengers

(c) 25% of passengers

Figure 25: Histogram of the number of links in the backhaul served by macro
gNB.

Finally, Fig. 26 presents the histograms of the percentage of links served by
the IAB donors that correspond to access links. The complementary of these
numbers corresponds to backhaul links. Remark that the percentage of links
that are access links varies a lot. As a consequence, one can conclude that
adopting fixed time/frequency resources multiplexing schemes between donors
and nodes may waste resources. For example, when the percentage of access
links served by the IAB donor was low, it could be better to schedule more
resources to the backhaul of connected mIAB nodes. In other words, one should
consider adopting dynamic time/frequency resource multiplexing between access
and backhaul based on the type of served links.

(a) 75% of passengers.

(b) 50% of passengers.

(c) 25% of passengers.

Figure 26: Histogram of the percentage of links served by the macro gNB that
are access links.

Based on the conclusions drawn in this section and on the concepts presented
until now, next section summarizes the main learned lessons and points out
open issues and possible directions to address them.

##  VI Lessons Learned, Open Issues and Future Directions

###  VI-A Cell Planning

The mIAB state-of-the-art works which consider transports with previously
known behavior, e.g., trains and buses, usually take advantage of that and
deploy IAB donors in strategic places, e.g., along the railway track. For
systems without a well-defined behavior, e.g., vehicles moving in a city, it
is more challenging to design the system in advance, e.g., to define where the
IAB donors should be deployed and how resource allocation and interference
management can be handled. A promising direction is to adopt AI-based
solutions to learn possible mIAB node trajectories and how the traffic varies
along the day in order to be able to deploy IAB donors in strategic positions.

###  VI-B Mobility Management

A challenge imposed by the mobility of an mIAB node is that it can suddenly
overload its target IAB donor if the donor is not prepared to receive this new
mIAB node. Similar to the previous topic, one adopted approach to deal with
the mobility of mIAB nodes is to exploit the knowledge of the transport
position and its trajectory, e.g., where a train or bus is and when they are
going to arrive at the station. It allows the system to configure in advance
the next IAB donor to receive the incoming UE. However, this strategy is more
challenging when we deal with vehicles without a well-defined trajectory. A
promising solution to investigate is also to use prediction-based methods in
those cases. For example, if the system is able to predict the arrival of an
mIAB node, and its corresponding additional traffic, instants before it
arrives, nearby BS can perform load balancing in advance in order to reserve
resources for the coming node.

###  VI-C Dynamic Interference Management

Whether interference management is required or not depends on different
parameters such as the mIAB transmit power, scheduling, power adaptation
capability of the nodes, etc. Many works considered the access link provided
by mIAB nodes deployed in a frequency spectrum different from the one used by
the rest of the system in order to eliminate the interference caused by the
mobility of those UE through the system. This is not an efficient solution
since the spectrum is expensive and scarce and might stay unused in a given
area as long as no mIAB node is there. On the other hand, it is still a
challenge to deploy access and backhaul links at the same frequency band due
to the dynamic interference caused by the mobility of the mIAB nodes. Thus,
solutions for in-band interference management must be further investigated.
For example, one can perform power control in the mIAB nodes access links in
order to reduce/avoid interfering with nearby UE connected to the IAB donor.
Another option is to insert slots of silence on the TDD frame pattern, as
investigated in [118].

###  VI-D Bandwidth Allocation for Access and Backhaul

Most of the works considering in-band deployment of mIAB adopted a fixed split
of the resource (either in frequency or in time) between access and backhaul
link. A strategy for sharing resources between backhaul link and access links
of inside and outside UE is still a challenge and must be further
investigated, since it can lead to more efficient use of the spectrum. For
example, dynamic spectrum partitioning could be performed taking into account
the load of data to be transmitted in the backhaul and in the access links of
IAB donor and mIAB nodes. This solution guarantees fairness and avoids the
waste of resources in case part of the spectrum is reserved for a given type
of link, e.g., backhaul, but there is not enough data to be transmitted in
that link.

###  VI-E Antenna Deployment

It is almost a consensus that at least for trains and buses antennas are going
to be deployed outside and at the top of these transports being connected to
inside AP which will provide connectivity for inside UE. However, practical
considerations must still be further evaluated, e.g., each operator will need
to deploy its own equipment or will the infrastructure be shared, e.g.,
computational resources, antennas and frequency spectrum. Moreover, since a
possible use case of mIAB nodes is to serve not only onboard UE but also
surrounding UE, it is important to evaluate and compare pros and cons of
deploying DU antennas either only inside or both inside and outside of mIAB
nodes.

###  VI-F Role of UAV

UAV-based communication is not considered in 3GPP Release 18 work-item on
mIAB. However, from an academic perspective, it is still interesting to
investigate its performance. Works which use UAV to enhance the coverage of
the network usually first define an optimal position for the UAV and fix them
there. It is still under evaluated how UAV perform when continually moving
either to provide access link for UE or to provide a wireless backhaul for
other IAB nodes.

###  VI-G Improved Reliability for C-plane

3GPP standard documents do not yet cover mIAB aspects. Different aspects
should still be investigated, e.g., the possibility of splitting user and
control planes as in DC scenarios. In the context of mIAB U-plane could be
multi-hop while C-plane would be one-hop [119]. This approach aims at reducing
latency and failure probability. Furthermore, U/C-planes could even be
deployed at different frequencies, where U-plane would be deployed at mmWave
for enhanced throughput while C-plane would be deployed at C-band for higher
reliability and coverage.

###  VI-H Topology Adaptation on mobile IAB

With inter-CU topology adaptation, an IAB node migrates from an old parent to
a new parent where both parent nodes are served by different IAB donor CU. The
current discussions are about the need for both MT and DU parts of the
migrating IAB node to handover. The main point is that DU migration requires
that all child nodes and UE should perform handover to the new target donor-
CU. This is not the case when only MT-part handover is executed. DU migration
involves context transfer of the child IAB nodes and UE, update of their
security keys and causes a large amount of signaling to be exchanged via the
Xn, F1 and radio interfaces [120]. This flood of signaling is called signaling
storm. The main concern is that this procedure may cause service interruption
as, currently, each DU can be connected simultaneously to only one CU. One of
the envisaged solutions is the possibility of the support of two logical DU,
but this may increase the complexity of IAB. Although these discussions on
3GPP are focused on fixed IAB, they certainly are of utmost relevance for
future discussions on mIAB where topology adaptation should be more frequent.

###  VI-I Signaling Optimization for Group Handover

Another important aspect that has been discussed on 3GPP meetings is about the
optimization of group handover when the DU part changes its parent in inter-CU
topology adaptation. If standard handover procedure is followed, all child
nodes and UE would perform Xn handover procedure including the RACH procedure.
However, note that the RACH procedure is not necessary in these cases since
there is no change in the UE’s strongest cell and child IAB nodes’ strongest
parent. Finally, group signaling could be tailored for this case where
handover messages of several UE could be concatenated/compressed [121]. As an
example, each UE’s RRC reconfiguration message could be significantly reduced
by updating only few parameters.

##  VII Conclusions

As presented in this work, mIAB is a candidate solution to address at least
two challenges of current networks. Firstly, it enhances the service of moving
UE, e.g., passengers of trains and buses, guaranteeing that they are always
connected and avoiding overloading the network with HO messages when a group
of moving UE enters simultaneously the coverage area of a new cell. Secondly,
it allows a quick deployment of new cells to provide/improve connectivity in
high demanding areas with the deployment of mIAB nodes.

The basis for the deployment of mIAB were presented. They are: current status
of IAB standardization in 5G NR and state-of-the-art works regarding fixed IAB
and mIAB. In order to help readers interested in this subject, the state-of-
the-art works were classified following different criteria. On the one hand,
fixed mobile related works were classified according to the studied
dimensions, system modeling assumptions/constraints, considered problem
objectives and KPI, solution approaches and adopted mathematical tools. On the
other hand, mIAB related works were grouped according to the type of mobility,
e.g., train, bus, UAV, etc.. It was noticed that works considering similar
type of mobility usually consider similar problems, e.g., UAV positioning, and
take advantage of specific characteristics of each type of mobility, e.g.,
previously known mobile trajectory of buses.

After the literature review, we have performed an extensive analysis of mIAB
based on computational simulations. It was considered an urban macro scenario
where IAB nodes were deployed in buses in order to improve the passengers’
connection. It was presented passengers and pedestrians throughput, latency
and link quality in both downlink and uplink transmission directions. Results
related to the wireless backhaul were also presented as well as the profile of
links served by the IAB donors. It was concluded that the deployment of the DU
part of an IAB node inside a bus and its MT part outside the bus remarkably
improves the passengers throughput and latency. This is due to the fact that
this deployment overcomes the significant bus penetration loss that a link
between a BS outside the bus and a passenger suffers. Moreover, it was also
concluded that an admission policy to allow a UE to connect to a mIAB cell
should be adopted. The admission criteria could be: a minimum measured RSRP
value; a maximum measured interference level (RSRQ instead of RSRP); the
relative UE and bus geographical position in a given time interval; etc..
Furthermore, it was concluded that the adopted topology adaptation and TDD
scheme play an important role directly impacting the interference management
strategy. Thus, it would be better if dynamic topology adaptation and
time/frequency resource scheduling were adopted taking into account the IAB
donor load, e.g., amount of served traffic from access and backhaul links.

Finally, we have summarized lessons learned, open issues and future directions
related to mIAB. One important challenge is how to deal with the dynamic
interference caused by mobile cells when moving around the system, specially
in the case with in-band deployment of backhaul and access. Besides, a
promising solution for this and other challenges is the use of predictive
solutions, e.g., AI-based, in order to anticipate what is going to happen and
prepare the system in advance.

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  * [120] 3GPP “IAB topology update procedure” TSG-RAN WG3 meeting no. 112-e, 2021 URL: https://www.3gpp.org/ftp/TSG_RAN/WG3_Iu/TSGR3_112-e/Docs/R3-212413.zip
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|  Victor F. Monteiro received the double B.Sc. degree in General Engineering,
from the École Centrale Lyon, France and in Telecommunications Engineering
(magna cum laude), from the Federal University of Ceará (UFC), Fortaleza,
Brazil, in 2013. In 2015 and 2019, he received the M.Sc. and PhD degrees in
Telecommunications Engineering from UFC, respectively. He is currently a
Postdoctoral researcher at the Wireless Telecommunications Research Group
(GTEL), UFC, where he works in projects in cooperation with Ericsson Research.
In 2016, he was an invited researcher at Ericsson Research in Lulea, Sweden,
where he worked for 6 months on the topic of LTE-NR Dual Connectivity.
Besides, in 2017/2018, he spent one year at Ericsson Research in Stockholm,
Sweden, where he worked on the topics of mobility management and channel
hardening. From 2010 to 2012, he took part, in France, of the Eiffel
Excellence Scholarship Program, established by the French Ministry of Foreign
Affairs. His research interests include machine learning, 5G architecture and
protocols, 5G measurement and reporting procedures, mobility management and
radio resource allocation.
---|---
|  Fco. Rafael M. Lima received the B.Sc. degree with honors in Electrical
Engineering in 2005, and M.Sc. and D.Sc. degrees in Telecommunications
Engineering from the Federal University of Ceará, Fortaleza, Brazil, in 2008
and 2012, respectively. In 2008, he has been in an internship at Ericsson
Research in Lulea, Sweden, where he studied scheduling algorithms for LTE
system. Since 2010, he has been a Professor of Computer Engineering Department
of Federal University of Ceará, Sobral, Brazil. Prof. Lima is also a senior
member of IEEE and senior researcher at the Wireless Telecom Research Group
(GTEL), Fortaleza, Brazil, where he works in projects in cooperation with
Ericsson Research. He has published several conference and journal articles as
well as patents in the wireless telecommunications field. His research
interests include radio resource allocation algorithms for QoS guarantees for
5G and beyond 5G networks in scenarios with multiple services, resources,
antennas and users.
---|---
|  Darlan C. Moreira received B.Sc. degree in electrical engineering in 2005
from the Federal University of Ceará (UFC), Fortaleza, Brazil, in 2005. In
2007 and 2020 he received the M.S. and Ph.D. degrees, respectively, in
Teleinformatics Engineering from the same institution. In 2007, he was an
invited researcher at Ericsson Research in Kista, Sweden, where he worked for
3 months. In 2010, he was an invited researcher at Supélec in Gif-sur-Yvette,
France, for 6 months. Since 2005 he has been a researcher at Wireless
Telecommunications Research Group (GTEL), Brazil, where he has been working in
projects within the technical cooperation between GTEL and Ericsson Research.
Some topics of his research interests include machine learning, MIMO
transceiver design, channel estimation, interference management, integrated
access and backhaul, and modeling and simulation of cellular communication.
---|---
|  Diego A. Sousa received the B.Sc. degree in Computer Engineering in
University of Ceará (UFC), Sobral, Brazil, in 2011. He received M.Sc. and
Ph.D. degree in Teleinformatics Engineering from the UFC, Fortaleza, Brazil,
in 2013 and 2018, respectively. Since 2013, he has been a researcher at the
Wireless Telecom Research Group (GTEL), UFC, participating of projects in a
technical and scientific cooperation with Ericsson Research. Also, since 2013,
he has been a Professor of the Federal Institute of Education, Science, and
Technology of Ceará (IFCE), Paracuru, Brazil. His research interests include
numerical optimization, 5G networks, coordinated scheduling, radio resource
management for QoS/QoE provisioning.
---|---
|  Tarcisio F. Maciel received his B.Sc. and M.Sc. degrees in Electrical
Engineering from the Federal University of Ceará (UFC) in 2002 and 2004,
respectively, and his Dr.Ing. degree from Technische Universitat Darmstat
(TUD), Germany, in 2008, also in Electrical Engineering. Since 2001, he has
actively participated in several projects in a technical and scientific
cooperation between the Wireless Telecommunications Research Grupo (GTEL),
UFC, and Ericsson Research. From 2005 to 2008, he was a research assistant
with the Communications Engineering Laboratory, TUD. Since 2008, he has been a
member of the Post-Graduation Program in Teleinformatics Engineering, UFC. In
2009, he was a Professor of computer engineering with UFC-Sobral and sine
2010, he has been a Professor with the Center of Technology, UFC. His research
interests include radio resource management, numerical optimization and multi-
user/multi-antenna communications.
---|---
|  Behrooz Makki (Senior Member, IEEE) received the Ph.D. degree in
communication engineering from Chalmers University of Technology, Gothenburg,
Sweden. He was a Postdoctoral Researcher with the Chalmers University of
Technology from 2013 to 2017. He is currently working as a Senior Researcher
with Ericsson Research, Gothenburg, Sweden. He has co-authored 67 journal
papers, 47 conference papers, and 90 patent applications. His current research
interests include integrated access and backhaul, green communications,
millimeter wave communications, NOMA, and backhauling. Behrooz is a recipient
of the VR Research Link Grant, Sweden, in 2014, the Ericsson’s Research Grant,
Sweden, in 2013, 2014, and 2015, the ICT SEED Grant, Sweden, in 2017, as well
as the Wallenbergs Research Grant, Sweden, in 2018. Also, he is a recipient of
the IEEE Best Reviewer Award for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
in 2018, and the IEEE Best Editor Award for IEEE WIRELESS COMMUNICATIONS
LETTERS in 2020. He is currently working as an Editor for IEEE TRANSACTIONS ON
COMMUNICATIONS, IEEE WIRELESS COMMUNICATIONS LETTERS, and IEEE COMMUNICATIONS
LETTERS. He was a member of the European Commission projects “mm-Wave Based
Mobile Radio Access Network for 5G Integrated Communications” and “ARTIST4” as
well as various national and international research collaborations.
---|---
|  Hans Hannu is a Master Research Engineer at Ericsson Research since 2011.
He holds a M. Sc. in Electrical Engineering / Signal Processing from Luleå
University of Technology, 1998, Sweden. Hans works in areas as service
realization and performance over cellular networks, including radio access and
network transport protocol, with algorithm development, such as data
transmission scheduling, and project management. He has co-authored 47 granted
patents.
---|---


# Modifications to a classic BFGS library for use with SIMD-equipped hardware
and an AAD library

Evgeny Goncharov, Alexandre Rodrigues Department of Pure Mathematics and
Mathematical Statistics, University of Cambridge, UK., and MatLogica, London,
UK<EMAIL_ADDRESS><EMAIL_ADDRESS>of Physics,
University of Aveiro, Portugal<EMAIL_ADDRESS>
###### Abstract

We introduce certain modifications of the BFGS method for functions that are
not parallelizable by nature (having consecutive operations only) taking
advantage of SIMD. We also provide a modified LBFGS++ library that takes
advantage of these modifications, and the use of AAD, and give an interface
for AAD users that takes advantage of the modified library automatically. We
give two examples to illustrate the performance. The modified library is up to
3.8 times faster for European Swaption curve calibration in ORE[16] (not
parallelizable) and 1.4 times faster for calibrating the LMM model by a set of
European options.

###### Contents

  1. 1 Introduction
  2. 2 The BFGS method
  3. 3 Modifications to the BFGS method and the LBFGS++ library
    1. 3.1 Parallelizing line search (LS)
    2. 3.2 Polynomial regression (Polyfit)
    3. 3.3 Computing derivatives
  4. 4 Implementation and testing
    1. 4.1 Implementation
    2. 4.2 Curve calibration in ORE
    3. 4.3 Calibrating the LMM model
  5. 5 Further improving the efficiency of LBFGS++

##  1 Introduction

The classical Newton’s method [2] uses the Hessian matrix $H$ (the matrix of
second derivatives) to find an extremum of a function
$f:\mathbb{R}^{n}\to\mathbb{R}$ by an iterative algorithm starting at some
initial point $x_{0}$. However, computing the Hessian can be too expensive,
which gave rise to the development of the less costly quasi-Newton methods.
The first one was proposed by William C. Davidon in [20] but it is rarely used
now as more efficient methods have been developed. Popular methods today
include the symmetric rank-one formula [7] and the BHHH method [9]. The BFGS
method [18, 14, 1, 6] and its low-memory extension LBFGS [13] are currently
the most used. We give an overview of the algorithms in Section 2. These
methods require little memory, have low computational complexity since no
matrix inversions are required, and can find the solution even when the
initial point $x_{0}$ is quite far from the extremum.

Parallelization is an important tool to improve computing efficiency. The two
main approaches to parallelization are multi-threading and Single Instruction
Multiple Data (SIMD). Multi-threading separates the calculation into parallel
independent computations (threads) whereas SIMD allows taking multiple data
points as input and processing them at the same time. It is straightforward to
implement multi-threading into the BFGS method for a parallelizable function
$f$. However, if $f$ is not parallelizable (e.g. it is a sequence of
consecutive operations), it is less clear how to use parallelization to
enhance performance. We review the BFGS method and introduce our
parallelization modifications utilizing SIMD in Section 3. We use Advanced
Vector Extensions (AVX) [4] in the process.

We implemented the modifications into the LBFGS++ library, the modified
library is available at [5]. The interface of the classical library assumes a
functor (an AAD library) that calculates both the value (forward mode) and the
gradient (reverse mode) of the target function $f$. However, straightforward
use of an AAD library may lead to unnecessary computations. We modify the
functions in the library to avoid them, see Section 3.3.

The AAD library that we use is Matlogica’s AADC [11]. It is compatible with
AVX and SIMD which makes it ideal for the project. We developed a C++ class
that allows AADC users to utilize the (modified version of) LBFGS++ library
automatically.

We discuss implementation in Section 4 and give two examples to illustrate
performance. In the first example, we take the Open Source Risk Project (ORE)
[16] code for the European Swaption curve calibration and pass it through the
AADC library to get optimized and vectorized binary kernels for the forward
(computing the function) and the reverse (computing the gradient) passes of
$f$. We achieve an up to 3.8 times acceleration for the optimized code with
the modified library. In the second example, we calibrate the Libor Market
Model (LMM) [19] using the recipe of [17] and achieve a 1.4 times acceleration
with the modified library. We note that in this case all the acceleration is
achieved just by avoiding the unnecessary gradient computations mentioned
above. Note also that the recent [3] gives alternative recipes for performing
AAD in this setup. All the code is available at [5].

Finally, we discuss other possible approaches to improving the performance of
the LBFGS++ library in Section 5.

Acknowlegements. We are grateful to Evgeny Lakshtanov for the useful advice,
discussions, and the suggestion to write up these ideas. We are also thankful
to Dmitri Goloubentsev for the idea to use polynomial fitting (Section 3.2),
and to Matlogica LTD for providing access to the AADC library.

##  2 The BFGS method

The BFGS method [18, 14, 1, 6] for computing a minimum of a function
$f:\mathbb{R}^{n}\to\mathbb{R}$ runs as follows.

######  Algorithm 1 (BFGS).

  1. Start with an initial guess $x_{0}$ and a Hessian matrix estimation $B_{0}$. Repeat the following steps for $k\geq 0$.

  2. 1.

Obtain the search direction: $p_{k}:=-B_{k}^{-1}\nabla f(x_{k})$.

  3. 2.

Perform a line search (one-dimensional optimization, see [12] for details) to
find the acceptable step $\alpha_{k}$ in the direction $p_{k}$. Line search is
terminated when a chosen termination condition (Armijo, Wolfe, or Strong
Wolfe) is reached.

Line search produces a sequence of steps $\alpha_{k}^{i}$ for $i\geq 0$ and
$x_{k}^{i}:=x_{k}^{i-1}+\alpha_{k}^{i}p_{k}$ for $k\geq 1$ (here
$x_{k}^{0}:=x_{k}$) until an acceptable step $\alpha_{k}$ is found (the
termination condition is reached).

  4. 3.

Use the displacement $s_{k}:=\alpha_{k}p_{k}$ to update the variable
$x_{k+1}:=x_{k}+s_{k}$.

  5. 4.

Set the change of gradient $y_{k}:=\nabla f(x_{k+1})-\nabla f(x_{k})$.

  6. 5.

Update the Hessian matrix estimation

$B_{k+1}:=B_{k}+{\frac{{y}_{k}{y}_{k}^{\mathrm{T}}}{{y}_{k}^{\mathrm{T}}{s}_{k}}}-{\frac{B_{k}{s}_{k}{s}_{k}^{\mathrm{T}}B_{k}^{\mathrm{T}}}{{s}_{k}^{{T}}B_{k}{s}_{k}}}.$

As $k\to\infty$, the $x_{k}$ converge to a minimum of $f$.

The LBFGS algorithm is an enhancement of Algorithm 1 that has similar steps
with certain modifications that allow to store less data in the memory (see
[13] for details). LBFGS++ is a classical library implementing the BFGS and
LBFGS methods. We make use of certain LBFGS++ parameters given to the solver
in one `LBFGSParam` object:

  * •

$\varepsilon_{\operatorname{rel}}$: we change it to guarantee convergence in
some applications;

  * •

`max_iterations`: limited to 100 to reduce computational time in LMM (Section
4.3);

  * •

`linesearch`: we tested for various line search termination conditions.

##  3 Modifications to the BFGS method and the LBFGS++ library

We shall now explain our modifications to the BFGS algorithm, and the LBFGS++
library. All our modifications only concern line search, that is Step 2 of
Algorithm 1.

###  3.1 Parallelizing line search (LS)

Suppose that we are at the line search step of Algorithm 1 and we are about to
compute $f(x_{k}^{i})=f(x_{k}^{i-1}+\alpha_{k}^{i}p_{k})$. A simple way to
improve performance is to instead calculate the function value at several
points at each call of the functor and choose the point with the smallest
value of the function.

We use AVX as it allows us to use $4$ points (AVX-2) or $8$ points (AVX-512)
and is well-integrated into AADC. For AVX-2 we use the $4$ points:

$x_{k}^{i-1}+\frac{1}{2}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+\alpha_{k}^{i}p_{k},\ x_{k}^{i-1}+\frac{3}{2}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+2\alpha_{k}^{i}p_{k}.$

For AVX-512 we use the 8 points:

$\begin{split}&x_{k}^{i-1}+\frac{1}{4}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+\frac{1}{2}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+\frac{3}{4}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+\alpha_{k}^{i}p_{k},\\\
&x_{k}^{i-1}+\frac{5}{4}\alpha_{k}^{i}p_{k},\
x_{k}+\frac{3}{2}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+\frac{7}{4}\alpha_{k}^{i}p_{k},\
x_{k}^{i-1}+2\alpha_{k}^{i}p_{k}.\end{split}$

We let $x_{k}^{i}$ be the point for which the value of $f$ is the smallest
among the $4$ or $8$ points respectively, and proceed with line search (or do
a Hessian update and proceed with the algorithm if the termination condition
has been reached). This modification alone results in acceleration of the
European Swapton curve calibration by a factor of $2$ as we see in Section
4.2.

###  3.2 Polynomial regression (Polyfit)

We proceed with the same setup as above. However, instead of letting
$x_{k}^{i}$ be the point where $f$ is the smallest, we use polynomial fitting
as follows. We fit the function values at the $4$ or $8$ points using a
polynomial of degree $3$ or $7$ respectively as a least-square polynomial
fitting using the QR factorization from the Eigen library [8]. This gives a
polynomial:

$\displaystyle f(\alpha)$
$\displaystyle=p_{0}+p_{1}\alpha+p_{2}\alpha^{2}+p_{3}\alpha^{3}\text{ (if
using AVX-2)}$ $\displaystyle f(\alpha)$
$\displaystyle=p_{0}+p_{1}\alpha+p_{2}\alpha^{2}+p_{3}\alpha^{3}+p_{4}\alpha^{4}+p_{5}\alpha^{5}+p_{6}\alpha^{6}+p_{7}\alpha^{7}\text{
(if using AVX-512)}$

and we can find its roots. Let the smallest root be $\alpha_{k}^{\min}$ and
set $x_{k}^{\min}:=x_{k}^{i-1}+\alpha_{\min}p_{k}$.

We found the best acceptable step interval to be
$\alpha_{k}^{\min}\in[\frac{1}{4}\alpha_{k},4\alpha_{k}]$. If
$\alpha_{\min}\in[\frac{1}{4}\alpha_{k},4\alpha_{k}]$ and
$f(x_{k}^{\min})<f(x_{k}^{i-1})$ we let $x_{k}^{i}:=x_{k}^{\min}$. Otherwise,
we use the $x_{k}^{i}$ of Section 3.1. This turns out to be a more powerful
modification than the one of Section 3.1 and offers an acceleration of the
European Swapton curve calibration by a factor of $3.8$ (again, see Section
4.2).

###  3.3 Computing derivatives

By studying the line search implementation of the LBFGS++ library we realized
that we can improve performance by separating the interface of the
`operator()` function into 3 different functions. The `operator()` function of
LBFGS++ calculates both the target function value and the gradient. However,
none of the termination conditions use the gradient which is only required by
Step 4 of the LBFGS version of Algorithm 1. The termination conditions only
require directional derivatives at Step 2 of the LBFGS version of Algorithm 1.

Using `operator()`, the directional derivative is computed via the dot product

$\frac{\partial f}{\partial\alpha_{k}^{i}}=\nabla f(x_{k}^{i})\cdot p_{k}.$

We instead use the finite difference method implemented in a new function
`getDg()`. This change allows us to move the gradient computation from
`operator()` to a new function `getGrad()`, which is only called after line
search.

We now explain how `getDg()` works. Suppose that we are at some stage
$x_{k}^{i}$ of line search. To calculate the directional derivative of $f$ at
$x:=x_{k}^{i}$ we can use

$f^{\prime}(x)\approx\frac{f(x+hp_{k})-f(x)}{h},$

where $h>0$ is a parameter that one can set using an interface parameter
`delta`. This approach is simple but can cause a significant error for some
target functions $f$ and parameters $h$.

A better approach is to use AVX as before and define 4 or 8 points as follows:

$x_{-2}=x-2hp_{k},\ x_{-1}=x-hp_{k},\ x_{1}=x+hp_{k},\ x_{2}=x+2hp_{k}\text{
(if using AVX-2)}$ $\ \ \ \ \ \ \ \ \ x_{-4}=x-4hp_{k},\ x_{-3}=x-3hp_{k},\
x_{-2}=x-2hp_{k},\ x_{-1}=x-hp_{k},\ x_{1}=x+hp_{k},\\\ x_{2}=x+2hp_{k},\
x_{3}=x+3hp_{k},\ x_{4}=x+4hp_{k}\text{ (if using AVX-512)}$

Then one can approximate the derivative using:

$f^{\prime}(x)\approx\frac{\sum_{i}f(x_{i})c_{i}}{h}$ (1)

where $c_{i}$ are the central finite difference coefficients, see [10].

In the cases of $4$ and $8$ points we get:

$f^{\prime}(x)\approx\frac{1}{h}\left[\frac{1}{12}f(x-2hp_{k})-\frac{2}{3}f(x-hp_{k})+\frac{2}{3}f(x+hp_{k})-\frac{1}{12}f(x+2hp_{k})\right]$
$\ \ \ \ \ \ \ \ \ \ \ \
f^{\prime}(x)\approx\frac{1}{h}\left[\frac{1}{280}f(x-4hp_{k})-\frac{4}{105}f(x-3hp_{k})+\frac{1}{5}f(x-2hp_{k})-\frac{4}{5}f(x-hp_{k})\right.\\\
\left.+\frac{4}{5}f(x+hp_{k})-\frac{1}{5}f(x+2hp_{k})+\frac{4}{105}f(x+3hp_{k})-\frac{1}{280}f(x+4hp_{k})\right]$

This broader averaging usually gives better results.

##  4 Implementation and testing

We shall explain the changes that we made to the LBFGS++ library, introduce an
interface for AADC that takes advantage of the modified library automatically
and give two examples illustrating the efficiency of our modifications. All
the code is available at [5].

###  4.1 Implementation

We created a C++ class [5] that facilitates the use of the LBFGS++ library by
AADC users. The interface includes all the functions required by our
implementation. For most implementations we only need to pass the following
AADC data to the basic class: recorded function, workspace, input and output
variables.

One can set the value of $h$ in the computation of the directional derivative.
The implementations that need extra data or an iterative method to calculate
the gradient or function value require straightforward changes to the
interface. By default we calculate the function value as the sum of all the
outputs since most target functions are of the form
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$.

As our implementation is based on AVX we cannot disable the line search
parallelization but one can set the number of computation points with the
typedef template, i.e. initializing the interface object as
BFGSHelper<mmType>.

Polynomial regression is achieved with a polynomial of order $3$ (for AVX-2)
or $7$ (for AVX-512) by default as it uses avx_count points to compute it. The
order can be set and verified by the functions setPolyFitOrder(int ord) and
int getPolyFitOrder(). It can also be fully disabled by setting the order to
0.

The computation of the directional derivative uses all the available avx_count
points. The options are using $2,\ 4,\ 6$ or 8 points (note that for AVX2 the
maximum is $4$). The user can use the functions setDgOrder(int ord) and int
getDgOrder() to set and verify the number of points used. If an invalid number
is set, the procedure is disabled by substituting it with the original
implementation where the directional derivative is computed as the dot product
of the gradient and the direction.

The user can also set the finite differences coefficients $c_{i}$ using the
functions
setDgCoeffs(std::vector<double> cs) and std::vector<double> setDgCoeffs()
(although the interface already has the default central finite difference
coefficients for any valid number of points).

###  4.2 Curve calibration in ORE

We used the third example in the Open Source Risk Project (ORE) user guide
[16] that shows the exposure evolution of European Swaptions with cash and
physical delivery. This example already had AADC integrated into the
underlying quantitative finance library QuantLib [15].

We made some minor changes to `reportwriter.cpp` and used as input the curve
“xois 0 EUR”. This curve has 14 outputs and 33 curve parameters so the curve
function is $g:\mathbb{R}^{33}\rightarrow\mathbb{R}^{14}$. Our target function
$f:\mathbb{R}^{33}\to\mathbb{R}$ for ORE is the quadratic loss function
relative to the desired outputs:

$f(x):=\sum_{i=1}^{14}\left(g(x)_{i}-g(x_{\min})_{i}\right)^{2}$ (2)

where $g(x)_{i}$ is the $i$-th output of the curve function at the point $x$
and $x_{\min}$ represents the vector of the target parameters.

We set $\varepsilon_{\operatorname{rel}}:=3\cdot 10^{-4}$ and $h:=10^{-14}$.
These parameters required modifications in certain specific cases. We take the
initial vector $x_{0}$ to be a vector of random numbers limited to the
intervals $[0.9c_{i},1.1c_{i}]$ where the $c_{i}$-s are the respective curve
parameters.

We have experimented with many termination conditions and line search
algorithms. Table 1 summarizes the results when using the Backtracking Wolfe
condition. Other conditions produce similar results (except for the number of
function calls).

ORE Results | Time(ms) | Iterations | Value | Forward | Reverse | LS It.
---|---|---|---|---|---|---
Before | 15213 | 65 | $7.18\cdot 10^{-19}$ | 81 | 1134 | 80
AVX2 | 9213 | 42 | $9.08\cdot 10^{-19}$ | 130 | 602 | 45
AVX2 Polyfit | 4624 | 18 | $6.23\cdot 10^{-19}$ | 79 | 266 | 21
AVX512 | 17142 | 36 | $1.23\cdot 10^{-18}$ | 111 | 518 | 38
AVX512 Polyfit | 9627 | 17 | $5.69\cdot 10^{-19}$ | 71 | 252 | 18
Table 1: ORE Results

We can see a significant reduction in the number of iterations, reverse calls,
and line search iterations. The final resulting value for the objective
function only improves when we use polynomial regression. AVX-512 has a
significant overhead cost thus not resulting in any speedup relative to AVX2.
We can also observe that Polyfit allowed us to eliminate the need for extra
forward function calls relative to the original implementation.

###  4.3 Calibrating the LMM model

We analyzed the simple example of calibrating the LMM model by a set of
European options. In this particular case, the loss function can be expressed
in the form

$G=\frac{1}{2}\sum_{i=1}^{m}(Ey_{i}-C_{i})^{2}.$ (3)

Applying AAD to a function of this form is not straightforward and was done in
[17].

We employed a calibration tool provided by Matlogica (available at [5]) with
the algorithm of [17] implemented that used Monte Carlo simulations,
multithreading, LBFGS++, AVX, and AADC. This tool has two major functions:
`simulate1()` and `simulate2()`. `Simulate1()` updates the values needed to
calculate the function value at the chosen point. `Simulate2()` calls
`simulate1()` to update the values and then computes the gradient at that
point. Therefore, we can use `simulate1()` when we only want the function
value (forward mode) and `simulate2()` to get the gradient (reverse mode).

It was not possible to implement our SIMD modifications of Sections 3.1 and
3.2 due to the calibration tool already using AVX and multithreading.
Implementing the changes would require a full refactoring of the tool.
However, we split the interface into $3$ functions as explained in Section
3.3. We also saved the reverse calls by deactivating the call of `simulate1()`
when we called `simulate2()` in `getGrad()`. This could be done since the
required outputs to compute the gradient were already calculated in
`operator()`.

We set $\varepsilon_{\operatorname{rel}}:=10^{-6}$, $h:=10^{-5}$ and take the
initial vector $x_{0}$ to be $90\%$ of the target parameters
$x_{0}:=0.9x_{\min}$.

We tested for unlimited LBFGS iterations and for LBFGS limited to 100
iterations (to reduce computational time). We have also experimented with many
termination conditions and line search algorithms. Table 2 summarizes the
results when using the Bracketing Wolfe condition with only one thread and an
unlimited number of LBFGS iterations.

LMM Results | Time(s) | Iterations | Value | Forward | Reverse | LS It.
---|---|---|---|---|---|---
AVX2 Before | 384 | 161 | $1.52\cdot 10^{-7}$ | $7.19\cdot 10^{6}$ | $3.59\cdot 10^{6}$ | 218
AVX2 After | 338 | 133 | $1.53\cdot 10^{-7}$ | $7.65\cdot 10^{6}$ | $2.20\cdot 10^{6}$ | 189
AVX512 Before | 341 | 167 | $1.48\cdot 10^{-7}$ | $3.58\cdot 10^{6}$ | $1.79\cdot 10^{6}$ | 217
AVX512 After | 261 | 117 | $1.49\cdot 10^{-7}$ | $3.26\cdot 10^{6}$ | $9.67\cdot 10^{6}$ | 160
Table 2: LMM Results

##  5 Further improving the efficiency of LBFGS++

It would be interesting to try to improve the method of approximating the
Hessian matrix in the case of the LBFGS version of Algorithm 1. The LBFGS++
implementation consists of a loop and it should be possible to improve
efficiency by using SIMD to calculate portions of the loop in parallel.

We have considered the Nocedal-Wright line search algorithm [12] but have not
been able to notably improve efficiency. Although we have implemented our
modifications to this method it would be interesting to try to improve
efficiency with specific changes to the interface.

LBFGS-B is an extension of LBFGS to simple box constraints [12]. It would be
interesting to see if our modifications are effective in this case. Extra
verification may be needed to guarantee that we do not exit the box
constraints.

This research was undertaken before more effective methods for computing
gradients of functions of the form (3) were found in [3] (expanding on the
work of [17]). It would be interesting to implement the modifications of
Section 3.3 for calibrating with these improved algorithms.

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