# Noninvasive cavity-based charge diagnostic for plasma accelerators

S. Bohlen1,2 O. Kononenko1 J.-P. Schwinkendorf1,2 F. Grüner1,2 D. Lipka1 M.
Meisel1,2 C.A.J. Palmer1 T. Staufer2 K. Põder1 and J. Osterhoff1,2 1Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
2Universität Hamburg, Luruper Chaussee 149, 22607 Hamburg, Germany
<EMAIL_ADDRESS>
###### Abstract

The charge contained in an electron bunch is one of the most important
parameters in accelerator physics. Several techniques to measure the electron
bunch charge exist. However, many conventional charge diagnostics face serious
drawbacks when applied to plasma accelerators. For example, integrating
current transformers (ICTs or toroids) have been shown to be sensitive to the
electromagnetic pulses (EMP) originating from the plasma, whereas
scintillating screens are sensitive to background radiation such as betatron
radiation or bremsstrahlung and only allow for a destructive measurement of
the bunch charge. We show measurements with a noninvasive, cavity-based charge
diagnostic (the DaMon), which demonstrate its high sensitivity, high dynamic
range and resistance towards EMP. The measurements are compared to both an ICT
and an absolutely calibrated scintillating screen.

##  1 Introduction

Plasma acceleration driven by laser pulses (LPA) [1] or particle beams (PWFA)
[2] enables the acceleration of electron bunches with gradients several orders
of magnitude higher than in conventional radio-frequency (RF) accelerators.
Thus, electron energies of hundreds of MeV can be achieved after a few
millimetres of acceleration. While a lot of progress has been made in terms of
stabilisation and control of plasma accelerators [3, 4, 5, 6], the level of
stability can not yet compete with their RF counterparts. As such, the
accurate single-shot detection and characterisation of electron bunches from
plasma accelerators is often desirable. Furthermore, accurate and non-invasive
diagnostics are required when plasma accelerators are used in applications
such as advanced light sources based on Thomson scattering [7] or when driving
free electron lasers [8]. Integrating current transformers (ICTs), which are
commonly used in conventional accelerators, ‘have become the instrument of
choice for measuring total charge from plasma-based accelerators’ [9], even
though they have early been identified to drastically overestimate the charge
in plasma accelerators by up to an order of magnitude [9, 10, 11]. The
overestimation of the measured charge using ICTs originates from
electromagnetic pulses (EMP) present in plasma accelerators. As such, accurate
determination of the charge in proximity of strong EMP has often only been
possible using destructive methods, such as absolutely calibrated
scintillating screens or Faraday cups.

Here, we demonstrate the use of a non-invasive, absolutely calibrated, cavity-
based charge diagnostic called DaMon, which offers a precise measurement of
the electron bunch charge over a wide range. In addition, the DaMon is shown
to be resilient to EMP, even in the presence of a high-voltage discharge in an
active plasma lens (APL). It is further compared to an ICT and an absolutely
calibrated scintillating screen (type DRZ-high). In these comparisons, we find
an overestimation of the charge by the ICT, especially at low electron bunch
charges, which is in line with previous findings. In contrast, the DaMon is
shown to be able to measure the bunch charge over a wide dynamic range of
several orders of magnitude in a non-invasive way. Due to its low noise level,
it was found to be more sensitive to low-charge electron bunches than the
other two diagnostics in the comparison and enabled the detection of bunches
with charges as low as tens of femtocoulomb.

##  2 Cavity-based charge detector: The DaMon

The DaMon is a cavity that non-invasively measures the charge of the electron
bunch. Originally, it was developed to measure bunch charges and dark currents
at the accelerators FLASH and XFEL at DESY [12, 13]. The DaMon is a resonator
made from stainless steel with the frequency of the first monopole mode (TM01)
at 1.3 GHz. An electron bunch passing through the cavity will induce the TM01
mode with a voltage of:

$U=U_{0}\;\sin(\omega t)\;e^{-\frac{t}{\tau}},$ (1)

where $\omega$ = 2$\pi f$ with $f$ being the resonance frequency of
$1.3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, and $\tau$ being the decay time
given by $\tau=Q_{L}/(\pi f)$ where $Q_{L}$ is the resonator loaded quality
factor. The amplitude of the induced voltage is given by:

$U_{0}=q\;S.$ (2)

Here, q is the electron bunch charge and S is the sensitivity defined as
$S=\pi f\sqrt{\frac{Z}{Q_{ext}}(\frac{R}{Q})}$, where $Z$, $Q_{ext}$ and $R/Q$
are the line impedance, external quality factor and simulated normalized shunt
impedance, respectively. The values for $Z$ and $Q_{ext}$ can be determined by
independent calibration measurements such that the electron bunch charge can
be calculated using Eq. 2 by measuring the amplitude of the induced TM01 mode
without further calibration. Two antennae are used for coupling-out the signal
from the cavity. The signal from each of the two antennae is processed by
electronics which consist of circulator, band-pass filter, logarithmic
detector and offset and gain control. One channel is equipped with limiter and
down conversion to an intermediate frequency, which enables higher sensitivity
for this branch. Due to the logarithmic output and two channels with different
sensitivity, the DaMon enables charge detection with a dynamic range of seven
orders of magnitude. More detailed information on the DaMon can be found in
the original publications [12, 13] and theses which describe its first use
with plasma accelerators [14, 15, 16].

##  3 Experimental setup

The experiments for the tests and comparison of charge diagnostics were
conducted at the FLARE facility at DESY [4, 17]. A schematic of the
experimental setup is depicted in Fig. 1. The laser was focused onto a
$1\text{\,}\mathrm{mm}$ supersonic gas jet using an f/12 off-axis parabola.
The gas used was 99.5 % helium with 0.5 % nitrogen doping to accelerate
electrons via ionisation injection (II) [18, 19, 20, 21, 22].

Figure 1: Schematic of the experimental setup.

To measure the electron bunch charge the DaMon, an ICT, and a scintillating
screen (type DRZ-High) were used. The DRZ screen was absolutely calibrated at
the Elbe accelerator at Helmholtz-Zentrum Dresden-Rossendorf prior to the
experiments [23, 24] and cross-calibrated using tritium sources [16]. While
both tritium sources and the used DRZ screen emit most of their light in a
similar wavelength regime, differences in the emitted spectrum prevented the
use of narrow-band interference, notch or edge-band filters for imaging of the
DRZ screen as this would have affected the absolute calibration. A background
subtraction was applied for measurements using the DRZ screen, which consisted
of 100 background images without any laser light.

The three charge diagnostics were installed at distances of approximately
$1.05\text{\,}\mathrm{m}$, $1.3\text{\,}\mathrm{m}$ and
$1.4\text{\,}\mathrm{m}$ from the plasma source for DaMon, ICT and DRZ screen,
respectively. To ensure that all three diagnostics measured the same amount of
electrons, a copper collimator was installed between the electron source and
the charge diagnostics. The collimator reduced the available beam aperture to
$30\text{\,}\mathrm{m}\mathrm{m}$ at a distance of approximately
$0.9\text{\,}\mathrm{m}$. The diagnostic with the smallest aperture of
$40.5\text{\,}\mathrm{m}\mathrm{m}$ was the DaMon, whereas ICT and DRZ screen
had an aperture of $72\text{\,}\mathrm{m}\mathrm{m}$ and
$65\text{\,}\mathrm{m}\mathrm{m}$, respectively. As such, the maximum beamsize
was limited by the aperture of the copper collimator and not the apertures of
any of the diagnostics. For parts of the experiments an active plasma lens
(APL) [25] was placed $0.15\text{\,}\mathrm{m}$ behind the source to study the
influence of strong EMP on the diagnostics.

A measurement of the electron spectra during the charge calibration was not
possible, due to the detection of the electron bunches on the DRZ screen.
Previously, broadband electron bunches with peaks around
$50\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}80\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
were measured at this setup. At these energies, the response of the detectors
should be energy independent. However, electrons with energies of less than a
few MeV could deposit more energy in the scintillating screen compared to the
more energetic electrons [10]. More information on the general setup and
further details on electron parameters previously measured can be found in
other publications [4, 17].

##  4 Comparison of DaMon, ICT and DRZ screen

In this section, we provide a comparison of the three independently absolutely
calibrated diagnostics DaMon, ICT, and DRZ screen. In the first part, the
sensitivity to noise from plasma creation and its effect on measuring low
charges is investigated. In the second part, we further explore the impact of
intense EMP on the charge measurement of the non-invasive diagnostics DaMon
and ICT. In the end, an overview of the performance of all three diagnostics
is presented to investigate their capability to detect charges spanning
multiple orders of magnitude.

###  4.1 Sensitivity to low charges

To investigate the behaviour of the diagnostics in an environment with EMP and
at low charges, studies were performed below and around the ionisation
injection threshold. The results are depicted in Fig. 2.

Figure 2: a) Comparison of the three charge diagnostics at low laser energies.
b) Same as panel a), but without the ICT. c) Zoom into shots 325-353 of panel
b). The grey lines in panels a) and b) indicates the laser energy.

In these scans, the laser energy was increased in steps with increasing
shotnumber, as indicated by a grey line in the panels Fig. 2a and Fig. 2b. The
threshold for ionisation injection in the used setup was at a laser energy of
around $400\text{\,}\mathrm{m}\mathrm{J}$. Therefore, the first 300 laser
shots enable plasma generation and its related background effects to be
studied on the charge diagnostics without acceleration of charged particles.
From Fig. 2a, it is visible that the ICT has a noise level of approximately
$\pm 2\text{\,}\mathrm{p}\mathrm{C}$ and therefore is not suited for the
reliable detection of low charges. The DaMon has a constant noise level of
$47\pm 1\text{\,}\mathrm{f}\mathrm{C}$ independent of laser energy,
demonstrating its resistance to EMP noise from the plasma generation as is
visible in Fig. 2b. The camera looking at the DRZ screen shows an increase in
signal with increasing laser energy, which could be interpreted as charges of
up to $250\text{\,}\mathrm{f}\mathrm{C}$ at the injection threshold of
$400\text{\,}\mathrm{m}\mathrm{J}$ laser energy. However, this signal likely
originates from laser light and/or plasma light as the more sensitive DaMon
does not see this step wise ramp up in charge and no accelerated electrons
would be expected at the used laser energies. This potential background signal
can not be accounted for in the background images without laser light as it is
dependent on laser energy. While the imaging system of the DRZ screen has been
carefully shielded, the additional use of notch filters was not possible due
to the cross calibration using tritium sources as described above.

When the laser energy reached the threshold of
$400\text{\,}\mathrm{m}\mathrm{J}$, the generation of electron bunches with
low charge was possible. The ability of the DaMon to clearly distinguish these
bunches from the very constant background is visible in Fig. 2c, where bunches
with charges of $66\text{\,}\mathrm{f}\mathrm{C}$ and
$81\text{\,}\mathrm{f}\mathrm{C}$ are clearly distinguishable form the
background level of $47\text{\,}\mathrm{f}\mathrm{C}$. At the same time, no
signal above the noise level gets detected by the DRZ screen for these two
shots. The same is true for the ICT, which is not able to resolve charges this
low. The ICT even fails to detect some bunches, where both DaMon and DRZ
screen have signals of about $300\text{\,}\mathrm{f}\mathrm{C}$ above noise
level, such as shot 355, as can be seen in Fig. 2a and Fig. 2b.

The noise level of the DaMon is currently limited by the electronics and could
be lowered to a few femtocoulomb or even attocoulomb levels with modified and
cooled electronics [26]. In the past, charges as low as
$25\text{\,}\mathrm{f}\mathrm{C}$ have been measured using the DaMon in a
different configuration at the same setup [14]. The sensitivity of the
scintillating screens could also be increased by optimising camera and imaging
system, while careful shielding from plasma and laser light would be required
as demonstrated here. The ICT used in this study is not suitable for reliable
detection of low charge electron bunches. For low charges, Turbo-ICTs could be
used which have a lower noise level of approximately
$1\text{\,}\mathrm{p}\mathrm{C}$ [27].

###  4.2 Sensitivity to EMP

The effect of plasma EMP on the two non-invasive diagnostics DaMon and ICT was
studied in more detail using an active plasma lens (APL). The raw analog-to-
digital converter (ADC) traces of an electron bunch charge measurement with
and without the APL discharging are depicted in Fig. 3 for both DaMon and ICT.

Figure 3: Comparison of the raw ADC traces for a) DaMon with APL off (dark
blue, dashed) and APL on (light blue, solid) and b) ICT with APL off (dark
red, dashed) and APL on (light red, solid). The two traces with and without
the APL discharging shown in each panel are simultaneous measurements of the
same bunch. The inset in panel b) shows the ADC trace from the ICT without the
APL. For the inset, the minimum trace value was subtracted to show the
magnitude of the raw signal without the APL.

In Fig. 3a, it is visible that the raw ADC trace of the DaMon does not seem to
be affected by the EMP signal that is induced from discharging the APL. In
contrast, the signal picked up by the ICT when using the APL is orders of
magnitude higher than the raw ICT signal of an electron bunch passing through
without the APL discharging as can be seen in Fig. 3b. While it is possible
that parts of the EMP noise visible in the ADC trace of the ICT are induced in
the cables rather the device itself, it is obvious that careful implementation
and shielding of an ICT is required in environments where EMP noise is
present. Without such shielding, ICTs can overestimate the charge when
installed close to the plasma source, as has been shown by other groups before
[10, 11]. In the past, shielding from EMP was for example achieved by placing
thin foils inside the beamline and installing the ICT several meters away from
the plasma source [27, 28]. The DaMon on the other hand is unaffected by EMP
noise, making it a suitable diagnostic for non-invasive charge measurements of
plasma accelerators even in close proximity to the plasma source.

###  4.3 High dynamic range charge detection

By changing the gas density and the laser power in the experiments, it was
possible to accelerate electron bunches with charges spanning several orders
of magnitude. A comparison of the measured charge for approximately 1500
electron bunches with a variety of charges is depicted in Fig. 4 together with
fits of the measured charge.

Figure 4: Comparison of charge measurements and fits. a) DRZ screen vs DaMon,
b) DRZ screen vs ICT, c) ICT vs DaMon. For comparisons including the ICT, only
datapoints where the ICT measured charges above
$50\text{\,}\mathrm{p}\mathrm{C}$ where included in the fit.

The diagnostics detected charges between $0\text{\,}\mathrm{p}\mathrm{C}$ and
$200\text{\,}\mathrm{p}\mathrm{C}$. In general a good correlation between the
charge measurements was found. However, the individual charge measurement was
highly dependent on the diagnostic itself. At low charges, the ICT seems to
overestimate the charge and with the exception of only a few data points,
charges of around $10\text{\,}\mathrm{p}\mathrm{C}$ or more were detected even
when both DaMon and DRZ screen measured charges $<$
$1\text{\,}\mathrm{p}\mathrm{C}$, as is visible in Fig. 4b and Fig. 4c.
Towards higher charges ($>$$50\text{\,}\mathrm{p}\mathrm{C}$) a linear
correlation between ICT and DRZ screen, as well as ICT and DaMon can be
observed. Therefore, only datapoints where the ICT measured charges above
$50\text{\,}\mathrm{p}\mathrm{C}$ where included for the fits with the ICT.

DaMon and DRZ screen show a linear correlation throughout most of the
measurement range, with small differences towards high charges as is visible
in Fig. 4a. The distribution of the points is also much lower in this panel,
indicating both DaMon and DRZ screen are less effected by noise in this setup
than the ICT. Overall, the following correlations of charge measurements were
found using first order polynomial fits:

$\displaystyle Q_{DaMon}$ $\displaystyle=0.55\times
Q_{DRZ}+$1.1\text{\,}\mathrm{p}\mathrm{C}$$ (3) $\displaystyle Q_{ICT}$
$\displaystyle=0.77\times Q_{DRZ}+$29.0\text{\,}\mathrm{p}\mathrm{C}$$
$\displaystyle Q_{DaMon}$ $\displaystyle=0.62\times
Q_{ICT}-$12.4\text{\,}\mathrm{p}\mathrm{C}$$

The highest charges were detected using the DRZ screen. Similar results were
found before and could partly be explained with low energy electrons [27],
which deposit vastly more energy in the screen and therefore lead to a non-
linear behaviour of the screen [10]. For the presented results, the effect
might be further amplified due to the high thickness of DRZ-high and the use
of ionisation injection which is likely to produce such low-energy electrons.
Another reason for high charge measurements using the DRZ screen could be
background radiation such as on-axis bremsstrahlung from the collimator,
betatron radiation or laser and/or plasma light as discussed earlier. All of
these effects would not be present in the background shots without the laser,
and therefore would not be accounted for in the background subtraction. The
accuracy of the ICT is brought under question by the non-linear behaviour of
the ICT at low charges and previously reported over-estimations of the charge
for ICTs which are installed in close proximity to a plasma source [10, 11,
9]. The DaMon measures the lowest charges, detecting nearly half the charge
compared to the DRZ screen. In addition to the background effects of the DRZ
screen discussed earlier, the differences could also originate from errors in
the calibration, e.g. the absolute calibration of the DRZ screen, the cross
calibration of the imaging systems using tritium sources or from the
calibration of the DaMon. To resolve the differences, an additional comparison
of DaMon and a Faraday cup at the presented setup might be required. In
previous and recent calibrations at other setups at DESY excellent agreement
between DaMon and Faraday cup was achieved [12, 29].

##  5 Summary and Outlook

In this study, we have presented a comparison of three charge diagnostics,
namely the non-invasive diagnostics DaMon and ICT and an absolutely calibrated
DRZ screen. The DaMon was shown to be capable of detecting electron bunches
with low bunch charges of less than $100\text{\,}\mathrm{f}\mathrm{C}$. The
imaging system of the DRZ screen detected laser and/or plasma light despite
careful shielding, which in experiments with plasma lead to an additional
error source. To resolve this, spectral filters should be implemented from the
start in absolute calibrations of scintillating screens. The ICT was unable to
detect electron bunches with low bunch charge. This could be resolved by using
a Turbo-ICT, which has shown to reduce the noise level to approximately
$1\text{\,}\mathrm{p}\mathrm{C}$ [27]. While this is still not at the level of
the DaMon, a comparison of DaMon and Turbo-ICT should be performed in the
future to further compare the two diagnostics and their sensitivity to
(plasma) EMP.

The influence of strong EMP on DaMon and ICT was further investigated using an
active plasma lens. In our tests, an APL in close proximity to the DaMon did
not influence its measurement of the electron bunch charge. As such, the DaMon
shows great potential for the use as non-invasive charge diagnostic in plasma
accelerators.

Furthermore, the performance of the diagnostics over a wide parameter range
was investigated. In our measurements, we found that the ICT tends to
overestimate the charge of our electron bunches, especially at low charges.
These findings are in line with previous results, that indicate an
overestimation of the charge by ICTs in plasma accelerators [10, 11].
Throughout the entire measurement range, the charge detected by the DaMon is
approximately half of that on the DRZ screen. This could be partly explained
by the non-linear response of phosphor screens to low-energy electrons [10],
which are expected in ionisation injection and have been identified to play a
significant role in charge detection at plasma accelerators in the past [27].
In addition, errors in the absolute screen calibration, the cross calibration
using tritium sources or in the calibration of the DaMon could play a role. An
additional measurement using another absolutely calibrated diagnostic such as
a Faraday cup, which previously showed excellent agreement with the DaMon [12,
29] and measurements including cleaning magnets [27] could resolve the
differences between DRZ screen and DaMon. The small scattering of datapoints
and generally linear correlation between DaMon and DRZ shows that both
diagnostics are well suited for charge measurements in plasma accelerators if
precisely calibrated.

Overall, our measurements highlight the difficulties of precise charge
measurements in plasma accelerators and compare the cavity-based DaMon to an
ICT and a scintillating screen. The Damon provides a non-invasive charge
measurement, which is not influenced by EMP and therefore can be implemented
in close proximity to the plasma source. Due to its high sensitivity (tens of
femtocoulomb) and large dynamic range (seven orders of magnitude) it is well
suited for various applications in plasma acceleration.

## References

## References

  * [1] Tajima, T. and Dawson, J. M. (1979). Laser electron accelerator. Phys. Rev. Lett. 43, 267.
  * [2] Chen, P. et al., (1985) Acceleration of electrons by the interaction of a bunched electron beam with a plasma. Phys. Rev. Lett. 54, 693.
  * [3] Rovige, L. et al., (2020). Demonstration of stable long-term operation of a kilohertz laser-plasma accelerator. Phys. Rev. Accel. Beams 23, 093401.
  * [4] Bohlen, S. et al., (2022). Stability of ionization-injection-based laser-plasma accelerators. Phys. Rev. Accel. Beams 25, 031301.
  * [5] A. R. Maier et al., (2020). Decoding Sources of Energy Variability in a Laser-Plasma Accelerator. Phys. Rev. X 10, 031039 .
  * [6] Lindstrøm, C. A. et al., (2023) Preservation of beam quality in a plasma-wakefield accelerator. submitted.
  * [7] Brümmer, T. et al., (2022). Compact all-optical precision-tunable narrowband hard Compton X-ray source. Sci. Rep. 12, 16017.
  * [8] Wang, W. et al., (2021). Free-electron lasing at 27 nanometres based on a laser wakefield accelerator. Nature 595, 516–520.
  * [9] Downer, M. C., et al., (2018). Diagnostics for plasma-based electron accelerators. Rev. Mod. Phys. 90, 035002.
  * [10] Glinec, Y. et al., (2006). Absolute calibration for a broad range single shot electron spectrometer. Rev. Sci. Instrum., 77(103301).
  * [11] Hidding, B. et al., (2007). Novel method for characterizing relativistic electron beams in a harsh laser-plasma environment. Rev. Sci. Instrum., 78(083301).
  * [12] Lipka, D., et al., (2011). Dark current monitor for the European XFEL. Proc. DIPAC2011 (Hamburg), p 572-574, (Germany: JACow).
  * [13] Lipka, D. et al., (2013). Resonator for charge measurement at REGEA. Proc. IBIC2013 (Oxford), p 872-875(UK: JACow).
  * [14] Kononenko, O. (2018). Controlled injection into a Laser-driven wakefield accelerator. PhD thesis, Universität Hamburg.
  * [15] Schwinkendorf, J.-P. (2018). Electron-beam diagnostics for a laser-driven plasma wakefield accelerator in the framework of FLASHForward. PhD thesis, Universität Hamburg.
  * [16] Bohlen, S. (2020). Measurement of Electron Parameters using Thomson Scattering. PhD thesis, Universität Hamburg.
  * [17] Bohlen, S. et al., (2022). In Situ Measurement of Electron Energy Evolution in a Laser-Plasma Accelerator. Phys. Rev. Lett. 129, 244801.
  * [18] Chen, M. et al., (2006). Electron injection and trapping in a laser wakefield by field ionization to high-charge states of gases. J. Appl. Phys. 99(5), 056109.
  * [19] Oz, E. et al., (2007). Ionization-induced electron trapping in ultrarelativistic plasma wakes. Phys. Rev. Lett. 98(8), 084801.
  * [20] Pak, A. et al., (2010). Injection and trapping of tunnel-ionized electrons into laser-produced wakes. Phys. Rev. Lett. 104(2), 025003.
  * [21] McGuffey, C., et al., (2010). Ionization induced trapping in a laser wakefield accelerator. Phys. Rev. Lett. 104(2), 025004.
  * [22] Clayton, C. E., et al., (2010). Self-Guided Laser Wakefield Acceleration beyond 1 GeV Using Ionization-Induced Injection. Phys. Rev. Lett. 105(10), 105003.
  * [23] Kurz, T., et al., (2018). Calibration and cross-laboratory implementation of scintillating screens for electron bunch charge determination. Rev. Sci. Instrum. 89(9), 093303.
  * [24] Schwinkendorf, J.-P., et al., (2019). Charge calibration of DRZ scintillation phosphor screens. J. Instrum. 14(09), P09025-P09025.
  * [25] van Tilborg, J. et al., (2015). Active Plasma Lensing for Relativistic Laser-Plasma-Accelerated Electron Beams. Phys. Rev. Lett. 115, 184802.
  * [26] Lipka, D., private communications
  * [27] Nakamura, K., et al., (2016). Pico-coulomb charge measured at BELLA to percent-level precision using a Turbo-ICT. Plasma Phys. Control. Fusion 58, 034010.
  * [28] Nakamura, K., et al., (2011). Electron beam charge diagnostics for laser plasma accelerators. Phys. Rev. ST Accel. Beams 14, 062801 .
  * [29] Lensch, T., et al., (2023). Charge measurement with resonators at ARES. Proc. IBIC2023 (Saskatoon), p 277-280, (Canada: JACow).


###### Proof.

###### Proof.

.:

###### Explanation-Guided Fair Federated Learning for Transparent 6G RAN
Slicing

###### Abstract

Future zero-touch artificial intelligence (AI)-driven 6G network automation
requires building trust in the AI black boxes via explainable artificial
intelligence (XAI), where it is expected that AI faithfulness would be a
quantifiable service-level agreement (SLA) metric along with
telecommunications key performance indicators (KPIs). This entails exploiting
the XAI outputs to generate transparent and unbiased deep neural networks
(DNNs). Motivated by closed-loop (CL) automation and explanation-guided
learning (EGL), we design an explanation-guided federated learning (EGFL)
scheme to ensure trustworthy predictions by exploiting the model explanation
emanating from XAI strategies during the training run time via Jensen-Shannon
(JS) divergence. Specifically, we predict per-slice RAN dropped traffic
probability to exemplify the proposed concept while respecting fairness goals
formulated in terms of the _recall_ metric which is included as a constraint
in the optimization task. Finally, the _comprehensiveness score_ is adopted to
measure and validate the faithfulness of the explanations quantitatively.
Simulation results show that the proposed EGFL-JS scheme has achieved more
than $50\%$ increase in terms of comprehensiveness compared to different
baselines from the literature, especially the variant EGFL-KL that is based on
the Kullback-Leibler Divergence. It has also improved the recall score with
more than $25\%$ relatively to unconstrained-EGFL.

###### Index Terms:

6G, fairness, FL, game theory, Jensen-Shannon, proxy-Lagrangian, recall,
traffic drop, XAI, zero-touch

##  I Introduction

Future 6G networks are expected to be a _machine-centric_ technology, wherein
all the corresponding _smart things_ would operate intelligently yet as smart
black boxes [1] that lack transparency in their prediction or decision-making
processes and could have adverse effects on the 6G network’s operations. In
this concern, XAI provides human interpretable methods to adequately explain
the AI system and its predictions/decisions and gain the human’s _trust_ in
the loop, which is anticipated to be a quantifiable KPI [2] in future 6G
networks. Nonetheless, most of the research on XAI for deep neural networks
(DNNs) focuses mainly on generating explanations, e.g., in terms of feature
importance or local model approximation in a _post-hoc_ way that does not
bridge the trade-off that exists between explainability and AI model’s
performance [3] nor guarantee the predefined 6G trust KPI requirement. In this
regard, there is less contribution on aspects related to the exploitation of
the explanations during training, i.e. in an _in-hoc_ fashion, which would
orient the model towards yielding explainable outputs that fulfill a
predefined trust level in AI-driven automated 6G networks, as well as tackling
the explainability-performance trade-off [4]. This approach is called
_explanation-guided learning_ (EGL) [5, 6], which focuses on addressing the
accuracy of prediction and explanation jointly, and their mutual relation in
run-time. It is particularly relevant in the context of 6G, where the
complexity of the network and the diversity of the applications being served
make it essential to guarantee the target trust in AI-driven zero-touch
management. This paper aims to present a distinct approach for improving
federated learning models’ interpretability by introducing a new closed-loop
training procedure that natively optimizes explainability and fairness goals
along with the main use-case task of traffic drop probability prediction for
6G network slices.

###  I-A Related Works

In this section, we exhibit the state of the art of EGL to recognize
interdisciplinary open research opportunities in this domain. Indeed, the
works on EGL techniques are still in the early stage and facing technical
challenges in developing their frameworks [7]. According to current studies,
incorporating an additional explanation objective could potentially enhance
the usefulness of DNNs in various application domains, including computer
vision (CV) and natural language processing (NLP). However, its potential
benefits for future 6G networks have not been thoroughly investigated yet.
Despite this limited exploration, some researchers have started implementing
XAI techniques to address telecommunication issues in this field. In
particular, the authors of [8, 9] stress the importance of explainability and
identify challenges associated with developing XAI methods in this domain.
Additionally, the utilization of XAI for identifying the actual cause of
Service Level Agreement (SLA) violations has been studied in [10]. Meanwhile,
in [11, 12], the authors mention the necessity of XAI in the zero-touch
service management (ZSM) autonomous systems and present a neuro-symbolic XAI
twin framework for zero-touch management of Internet of Everything (IoE)
services in the wireless network. Moreover, [13] emphasizes the utilization of
the Federated Learning (FL) approach with XAI in 5G/6G networks. In [7], two
types of EGL termed global guidance and local guidance explanation are
mentioned, where both types refine the model’s overall decision-making
process. Another line of work [14] introduces the EGL framework that allows
the user to correct incorrect instances by creating simple rules capturing the
target model’s prediction and retraining the model on that knowledge. While
the authors of [15] define a very generic explanation-guided learning loss
called _Right for the Right Reasons_ loss (RRR). In addition, an approach to
optimize model’s prediction and explanation jointly by enforcing whole graph
regularization and weak supervision on model explanation is presented in [16].
Additionally, a novel training approach for existing few-shot classification
models is presented in [17], which uses explanation scores from intermediate
feature maps. Specifically, it leverages an adapted layer-wise relevance
propagation (LRP) method and a model-agnostic explanation-guided training
strategy to improve model’s generalization. So it is apparent that none of the
aforementioned works has considered EGL in telecommunication domains for
prediction and decision-making.

On the other hand, AI-native zero-touch network slicing automation has been
introduced to govern the distributed nature of datasets [18] of various
technological domains. Thus, a decentralized learning approach like federated
learning (FL) [8] has gained attraction for fulfilling the beyond 5G network
requirement. In this context, the authors of [19] proposed a method for
dynamic resource allocation in decentralized RAN slicing using statistical FL.
This approach enables effective resource allocation, SLA enforcement, and
reduced communication overhead. However, the lack of a clear understanding of
the underlying mechanisms in the paper may raise concerns about the trust and
transparency of the proposed model.

Table I: Notations Notation |  Description
---|---
$S_{\mu}(\cdot)$ |  Logistic function with steepness $\mu$
$L$ |  Number of local epochs
$T$ |  Number of FL rounds
$\mathcal{D}_{k,n}$ |  Dataset at CL $(k,n)$
$D_{k,n}$ |  Dataset size
$\ell(\cdot)$ |  Loss function
$\mathbf{W}_{k,n}^{(t)}$ |  Local weights of CL $(k,n)$ at round $t$
$\mathbf{x}_{k,n}^{(i)}$ |  Input features
$\mathbf{y}_{k,n}^{(i)}$ |  Actual Output
$\hat{y}_{k,n}^{(i)}$ |  Predicted Output
$\gamma_{n}$ |  Recall lower-bound for slice $n$
$\tilde{x}_{k,n}^{(i)}$ |  Masked input features
$\tilde{p}_{k,n}^{(i)}$ |  Masked predictions
$\mathcal{D}_{k,n}$ |  Samples whose prediction fulfills the SLA
$\pi_{k,n}^{(i,j)}$ |  Probability Distribution
$\rho_{k,n}$ |  Recall Score
$\alpha_{k,n}^{(i,j)}$ |  Weighted attribution of features
$\mathbf{JS}_{k,n}$ |  Jensen Shannon Divergence
$\lambda_{(\cdot)}$ |  Lagrange multipliers
$R_{\lambda}$ |  Lagrange multiplier radius
$\mathcal{L}_{(\cdot)}$ |  Lagrangian with respect to ($\cdot$)

From this state-of-the-art (SoA) review, it turns out that incorporating FL
and EGL strategies jointly can embrace into the network slicing of 6G RAN to
address the absence of a trustworthy and interpretable solution in the
prediction and decision-making process during learning. In this paper, we
present the following contributions:

  * •

To achieve transparent zero-touch service management of 6G network slices at
RAN in a non-IID setup, a novel iterative explanation-guided federated
learning approach is proposed, which leverages Jensen-Shannon divergence.

  * •

A constrained traffic drop prediction model and integrated-gradient based
_explainer_ exchange attributions of the features and predictions in a closed
loop way to predict per-slice RAN dropped traffic probability while respecting
fairness constraints formulated in terms of recall. By demonstrating the
traffic drop probability distribution and correlation heatmaps, we showcase
the guided XAI’s potential to support decision-making tasks in the telecom
industry.

  * •

We frame the abovementioned fair sensitivity-aware constrained FL optimization
problem under the _proxy-Lagrangian_ framework and solve it via a non-zero sum
two-player game strategy [20] and we show the comparison with state-of-the-art
solutions, especially the EGFL-KL. Further, to assess the effectiveness of the
proposed one, we report the recall and comprehensiveness performance, which
provides a quantitative measure of the sensitivity and EGFL explanation level.

  * •

More specifically, during the optimization process, we introduce a new
federated EGL loss function that encompasses Jensen Shannon divergence to
enhance model reliability and we also provide the corresponding results. We
carefully scrutinize the normalized loss over the no. of FL rounds of the
proposed EGFL model in the optimization process to support this phenomenon.

###  I-B Notations

We summarize the notations used throughout the paper in Table I.

Figure 1: Federated traffic drop probability prediction in 6G RAN NS.

##  II Explanation Guided Learning Concept

As mentioned before, the leading goal of EGL is to enhance both the model
performance as well as interpretability by jointly optimizing model prediction
as well as the explanation during the learning process. So, the main objective
function of Explanation-Guided Learning is defined in the existing research
work of [7, 21], which is as follows:

$\begin{split}&\min\,\underbrace{\mathcal{L}_{Pred}\left(\mathcal{F}\left(\mathcal{X}\right),\mathcal{Y}\right)}_{prediction}+\underbrace{\alpha\mathcal{L}_{Exp}\left({g}\left(\mathcal{F},\left(\mathcal{X,Y}\right)\right),\hat{M}\right)}_{explanation}\\\
&+\underbrace{\beta\ohm\left({g}\left(\mathcal{F},\left(\mathcal{X,Y}\right)\right)\right)}_{regularization},\end{split}$
(1)

From equation (1), we can observe three terms in the objective function of
EGL. Where the first term defines the prediction loss of the model, the second
term deals with the supervision of model explanation considering some explicit
knowledge, and the last term can introduce some properties for achieving the
right reason. These three terms can be specified and enforced differently,
pivoting on each explanation-guided learning method [7]. However, it is
noteworthy that we will adopt gradient based methods to explain the model’s
prediction in our proposed work. It generates attributions by computing the
gradient of the model’s output for its inputs, which indicates the input
feature contributions to the model’s final prediction. Theoretically, the
gradient values of all unimportant features should be close to zero, which may
not hold in some complex situations, giving biased results in the model
decision that is unexpected in the actual scenario [5]. So, even if we
introduce explainability during the learning process of any AI system for some
prediction or decision-making purpose, there is still a chance we will get a
final decision that is biased or different from actual results. Additionally,
this may hinder telecommunication service providers/operators from making
their system transparent, only considering XAI. Considering the above fact and
inspired by EGL, we came out with a solution to overcome this situation. We
aim to introduce EGL to our proposed framework of 6G RAN network slicing
during learning, ensuring that the irrelevant gradient will close to zero
without sacrificing the model performance.

Table II: Dataset Features and Output Feature | Description
---|---
Averege PRB | Average Physical Resource Block
Latency | Average transmission latency
Channel Quality | SNR value expressing the wireless channel quality
|
Output | Description
Dropped Traffic | Probability of dropped traffics (%)
Figure 2: Explanation Guided FL building blocks

##  III RAN Architecture and Datasets

A shown in Fig. 1, we consider a radio access network (RAN), which is composed
of a set of $K$ the base station (BSs), wherein a set of $N$ parallel slices
are deployed, driven by the concept of slicing-enabled mobile networks [22].
The deployment of these parallel slices within the RAN enables efficient
resource allocation, differentiation of services, and flexibility to meet
diverse user and application needs. Here, each BS runs a local control closed-
loop (CL) which collects monitoring data and performs traffic drop prediction.
The local control CL system proactively manages network resources by gathering
real-time data on network conditions and using predictive algorithms to
estimate the likelihood of traffic drops or disruptions. Specifically, the
collected data serves to build local datasets for slice $n\,(n=1,\ldots,N)$,
i.e.,
$\mathcal{D}_{k,n}=\\{\mathbf{x}_{k,n}^{(i)},y_{k,n}^{(i)}\\}_{i=1}^{D_{k,n}}$,
where $\mathbf{x}_{k,n}^{(i)}$ stands for the input features vector while
$y_{k,n}^{(i)}$ represents the corresponding output. In this respect, Table II
summarizes the features and the output of the local datasets. These
accumulated datasets are non-IID due to the different traffic profiles induced
by the heterogeneous users’ distribution and channel conditions. Moreover,
since the collected datasets are generally non-exhaustive to train accurate
anomaly detection classifiers, the local CLs take part in a federated learning
task wherein an E2E slice-level federation layer plays the role of a model
aggregator. In more specifics, each slice within the BS represents a diverse
use case of vertical applications and is characterized by input features such
as PRBs, channel quality, and latency. By leveraging these input features, the
FL optimization process can train a predictive model guided by XAI that learns
to predict the occurrence of traffic drops for each slice within the BS. This
enables proactive measures to be taken to mitigate potential disruptions and
optimize the performance of the communication system.

##  IV Explanation-Guided FL for Transparent Traffic Drop Prediction

In this section, we describe the different stages of the explanation-guided
fair FL as summarized in Fig. 2.

###  IV-A Closed-Loop Description

In our proposed architecture, Fig. 2 represents a block that is placed within
the local models shown in Fig. 1. This block is designed to facilitate
proposed explainable guided federated learning, which involves iterative local
learning with runtime explanation in a closed-loop manner. We design a simple
neural network FL model. For each local epoch, the Learner module feeds the
posterior symbolic model graph to the Tester block which yields the test
features and the corresponding predictions $\hat{y}_{k,n}^{(i)}$ to the
Explainer. The latter first generates the features attributions using
integrated gradients XAI method.

The _Featuere Masking_ block then selects, for each sample, the feature with
lowest attribution and masks it. The corresponding masked inputs,
$\tilde{x}_{k,n}^{(i)},(i=1,\ldots,D_{k,n})$ are afterward used to calculate
the masked predictions, $\tilde{p}_{k,n}^{(i)},(i=1,\ldots,D_{k,n})$ that is
fed back to the Divergence module at step 5. At the same time, predictions
$\hat{y}_{k,n}^{(i)}$ from the tester block are fed in the Divergence block as
shown in the step 3". Divergence module is the responsible to calculate
Jensen-Shannon divergence between model predictions of original and masked
inputs, $\mathbf{JS}(\hat{y}_{k,n}^{(i)}||\tilde{p}_{k,n}^{(i)})$. The
corresponding result, denoted as $\mathbf{JS}_{k,n}$ from this module is
included with cross entropy loss as the objective function of the local model,
shown at the stage 4". Similarly, the _Recall Mapper_ calculates the recall
score $\rho_{k,n}$ based on the predicated and true positive values at stage 6
and 6" to include it in the local constrained optimization in step 7. Our
focus lies in addressing imbalanced datasets within our traffic drop
prediction problem, where accurately identifying positive classes assumes
critical significance. By incorporating recall constraints into our EGFL
framework, the model is able to effectively capture positive instances, even
those belonging to the minority class. This constraint prioritizes reducing
false negatives, improving the overall detection of positive cases, and
mitigating potential costs associated with missed instances.

Indeed, for each local CL $(k,n)$, the predicted traffic drop probability
$\hat{y}_{k,n}^{(i)},\,(i=1,\ldots,D_{k,n})$, should minimize the main loss
function with respect to the ground truth $y_{k,n}^{(i)}$ and Jensen-Shannon
divergence score, while jointly respecting some long-term statistical
constraints defined over its $D_{k,n}$ samples and corresponding to recall
score. As shown in steps 1 and 7 of Fig. 2, the optimized local weights at
round $t$, $\mathbf{W}_{k,n}^{(t)}$, are sent to the server which generates a
global FL model for slice $n$ as,

$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\mathbf{W}_{k,n}^{(t)},$
(2)

where $D_{n}=\sum_{k=1}^{K}D_{k,n}$ is the total data samples of all datasets
related to slice $n$. The server then broadcasts the global model to all the
$K$ CLs that use it to start the next round of iterative local optimization.
Specifically, it leverages a two-player game strategy to jointly optimize over
the objective and original constraints as well as their smoothed surrogates
and detailed in the sequel.

###  IV-B Model Testing and Explanation

As depicted in stage 2 of Fig. 2, upon the reception of the updated model
graph, the Tester uses a batch drawn from the local dataset to reconstruct the
test predictions $\hat{\mathbf{y}}_{k,n}^{(i)}$. All the graph, test dataset
and the predictions are fed to the Explainer at stage 3. After that, at stage
4, Explainer generates the attributions by leveraging the low-complexity
Integrated Gradient (IG) scheme [23], which is based on the gradient variation
when sampling the neighborhood of a feature. Attributions are a quantified
impact of each single feature on the predicted output. Let
$\mathbf{a}_{k,n}^{(i)}\in\mathbb{R}^{Q}$ denote the attribution vector of
sample $i$, which can be generated by any attribution-based XAI method.

###  IV-C Feature Masking

To improve the faithfulness of the generated attributions, we introduce a
novel approach to guide the following training. Specifically, we generate
model’s masked predictions, $\tilde{p}_{k,n}^{(i)}$, using the masked inputs,
$\tilde{\mathbf{x}}_{k,n}^{(i)}$. Specifically, previously generated
attributions score are used to select the lowest important features of each
input samples and masking it with zero padding.

In this respect, the _explanation-guided Mapper_ at stage 5 of Fig. 2 starts
by selecting the lowest important features of each input sample based on their
attributions which is collected from stage 4 and replace them with zero
padding and after that new model output has generated. Finally this output are
directly going to Divergence module.

###  IV-D Divergence Module

As mentioned above, We use this module to calculate Jenson-Shannon divergence
(JSD) score based on the predicted and masked predicted values, which we can
define as
$\mathbf{JS}_{k,n}\triangleq\mathbf{JS}\left(\hat{y}_{k,n}^{(i)}||\tilde{p}_{k,n}^{(i)}\right)$.
It measures the similarity between two probability distributions while
ensuring that the model’s input or output data doesn’t drastically change from
a baseline. So, the idea behind including JSD along with the cross-entropy
loss as our objective function is to train and encourage the models to
generate samples similar to a target distribution considering both the
original input and masked input. In this way, our model will learn to allocate
low gradient values to irrelevant features in model predictions during its
learning process. It will result in faithful gradients, giving unbiased and
actual model decisions at the end of the learning.

###  IV-E Explanation-Guided Fair Traffic Drop Prediction

To ensure fairness, we invoke the _recall_ as a measure of the sensitivity of
the FL local predictor, which we denote $\rho_{k,n}$, i.e.,

$\rho_{k,n}=\pi^{+}\left(\mathcal{D}_{k,n}\left[\hat{y}_{k,n}^{(i)}=1\right]\right)$
(3)

Where, $\pi^{+}(\mathcal{D}_{k,n})$ defines the proportion of
$\mathcal{D}_{k,n}$ classified positive, and $\mathcal{D}_{k,n}[*]$ is the
subset of $\mathcal{D}_{k,n}$ satisfying expression *.

In order to trust the traffic drop anomaly detection/classification, a set of
SLA is established between the slice tenant and the infrastructure provider,
where a lower bound $\gamma_{n}$ is imposed to the recall score. This
translates into solving a constrained local regression problem for each local
epoch in FL rounds $t\,(t=0,\ldots,T-1)$ i.e.,

$\min_{\mathbf{W}_{k,n}^{(t)}}\,\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right)+\mathbf{JS}\left(\hat{y}_{k,n}^{(i)}||\tilde{p}_{k,n}^{(i)}\right)$
(4a)

$\mathrm{s.t.}\hskip 14.22636pt\rho_{k,n}\geq\gamma_{n},$ (4b)

which is solved by invoking the so-called _proxy Lagrangian_ framework [24],
since the recall is not a smooth constraint. This consists first on
constructing two Lagrangians as follows:

$\begin{split}\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}=&\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right)\\\
&+\mathbf{JS}\left(\hat{y}_{k,n}^{(i)}||\tilde{p}_{k,n}^{(i)}\right)+\lambda_{1}\Psi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right),\end{split}$
(5a)
$\mathcal{L}_{\lambda}=\lambda_{1}\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)$
(5b)

where $\Phi_{1}$ and $\Psi_{1}$ represent the original constraints and their
smooth surrogates, respectively. In this respect, the recall surrogate is
given by,

$\Psi_{1}=\frac{\sum_{i=1}^{D_{k,n}}{y}_{k,n}^{(i)}\times\min\Bigl{\\{}\hat{y}_{k,n}^{(i)},1\Bigl{\\}}}{\sum_{i=1}^{D_{k,n}}{y}_{k,n}^{(i)}}-\gamma_{n}$
(6)

This optimization task turns out to be a non-zero-sum two-player game in which
the $\mathbf{W}_{k,n}^{(t)}$-player aims at minimizing
$\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}$, while the $\lambda$-player wishes to
maximize $\mathcal{L}_{\lambda}$ [20, Lemma 8]. While optimizing the first
Lagrangian w.r.t. $\mathbf{W}_{k,n}$ requires differentiating the constraint
functions $\Psi_{1}(\mathbf{W}_{k,n}^{(t)})$, to differentiate the second
Lagrangian w.r.t. $\lambda$ we only need to evaluate
$\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)$. Hence, a surrogate is only
necessary for the $\mathbf{W}_{k,n}$-player; the $\lambda$-player can continue
using the original constraint functions. The local optimization task can be
written as,

$\begin{split}\min_{\mathbf{W}_{k,n}\in\Delta}\,\,\,\,\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}\end{split}$ (7a)
$\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\,\,\min_{\mathbf{W}_{k,n}\in\Delta}\mathcal{L}_{\lambda},$
(7b)

where thanks to Lagrange multipliers, the $\lambda$-player chooses how much to
weigh the proxy constraint functions, but does so in such a way as to satisfy
the original constraints, and ends up reaching a nearly-optimal nearly-
feasible solution [25]. These steps are all summarized in Algorithm 1.

Input: $K$, $m$, $\eta_{\lambda}$, $T$, $L$. # See Table II

Server initializes $\mathbf{W}_{n}^{(0)}$ and broadcasts it to the
$\mathrm{CL}$s

for  _$t=0,\ldots,T-1$_ do

parallel for $k=1,\ldots,K$ do Initialize $M=$ num_constraints and
$\mathbf{W}_{k,n,0}=\mathbf{W}_{n}^{(t)}$

Initialize $\mathbf{A}^{(0)}\in\mathbb{R}^{(M+1)\times(M+1)}$ with
$\mathbf{A}_{m^{\prime},m}^{(0)}=1/(M+1)$

for  _$l=0,\ldots,L-1$_ do

Receive the graph $\mathcal{M}_{k,n}$ from the local model

# Test the local model and calculate the attributions

$a_{k,n}^{i,j}=$ Int. Gradient
$\left(\mathcal{M}_{k,n}\left(\mathbf{W}_{k,n,l},\mathbf{x}_{k,n}\right)\right)$

# Mask the lowest dataset based on the attributions with zero padding,
$\tilde{\mathbf{x}}_{k,n}^{(i)}$

# Calculate the new model’s predictions, $\tilde{p}_{k,n}^{(i)}$

# Calculate the Jensen-Shannon divergence,
$\mathbf{JS}(\hat{y}_{k,n}^{(i)}||\tilde{p}_{k,n}^{(i)})$

# Calculate the recall metric

$\rho_{k,n}=\pi^{+}\left(\mathcal{D}_{k,n}\left[\hat{y}_{k,n}^{(i)}=1\right]\right)$

Let $\lambda^{(l)}$ be the top eigenvector of $\mathbf{A}^{(l)}$

# Solve problem (4) via oracle optimization

Let
$\hat{\mathbf{W}}_{k,n,l}=\mathcal{O}_{\delta}\left(\mathcal{L}_{\mathbf{W}_{k,n,l}}(\cdot,\hat{\lambda}^{(l)})\right)$

Let $\Delta_{\lambda}^{(l)}$ be a gradient of
$\mathcal{L}_{\lambda}(\hat{\mathbf{W}}_{k,n,l},\lambda^{(l)})$ w.r.t.
$\lambda$

# Exponentiated gradient ascent

Update
$\tilde{\mathbf{A}}^{(l+1)}=\mathbf{A}^{(l)}\odot\cdot\exp{\eta_{\lambda}\Delta_{\lambda}^{(l)}(\lambda^{(l)})}$

# Column-wise normalization

$\mathbf{A}_{m}^{(l+1)}=\tilde{\mathbf{A}}_{m}^{(l+1)}/\norm{\mathbf{A}_{m}^{(l+1)}}_{1},\,m=1,\ldots,M+1$

end for

return
_$\hat{\mathbf{W}}_{k,n}^{(t)}=\frac{1}{L^{\star}}\sum_{l=0}^{L-1}\hat{\mathbf{W}}_{k,n,l}$_

Each local CL $(k,n)$ sends $\hat{\mathbf{W}}_{k,n}^{(t)}$ to the server.

end parallel for

return
_$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\hat{\mathbf{W}}_{k,n}^{(t)}$_

and broadcasts the value to all local CLs.

end for

Algorithm 1 Explanation Guided Federated Deep Learning

##  V EGFL Convergence Analysis

Within this section, we examine the probability of convergence for the
explanation-guided federated learning method. To accomplish this, we establish
a closed-form formula that determines the minimum probability of convergence,
considering various factors such as the Lagrange multiplier radius, the
optimization oracle error, and the violation rate, while incorporating the
Jensen-Shannon divergence. As a preliminary, we recall the following
definition,

######  Definition 1 (Approximate Bayesian Optimization Oracle).

An oracle $\mathcal{O}{\delta}$ for $\delta$-approximate Bayesian optimization
refers to a procedure that takes a loss function/Lagrangian $\mathcal{L}$ as
input and outputs quasi-optimal weights $\mathbf{W}{k,n,l}$, satisfying,

$\mathcal{L}\left(\mathcal{O}_{\delta}\left(\mathcal{L}\right)\right)\leq\inf_{\mathbf{W}_{k,n,l}^{\star}}\mathcal{L}\left(\mathbf{W}_{k,n,l}^{\star}\right)+\delta.$
(8)

The key theorem can therefore be formulated as,

######  Theorem 1 (EGFL Convergence Analysis).

We consider that EGFL follows a geometric failure model and fails to meet the
recall constraint with an average violation rate $0<\nu<1$. It is also assumed
that an oracle $\mathcal{O}_{\delta}$ with error—$\delta$ excluding the JS
divergence—optimizes $\mathcal{L}_{\mathbf{W}{k,n}^{(t)}}$, while
$R_{\lambda}$ and $B_{k,n}$ represent the Lagrange multipliers radius and the
maximum subgradient norm, respectively. In such a scenario, the convergence
probability of EGFL can be expressed as,

$\displaystyle\mathbf{Pr}\bigg{[}\frac{1}{T}\sum_{t=1}^{T}$
$\displaystyle\Big{(}\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{\star}\right)$
(9)
$\displaystyle-\inf_{\mathbf{W}_{n}^{\star}}\mathcal{L}\left(\mathbf{W}_{n}^{\star},\lambda^{(t)}\right)\Big{)}<\epsilon\bigg{]}\geq\Delta\left(\nu,\epsilon\right),$

where

$\Delta\left(\nu,\epsilon\right)=1-\frac{\nu}{1+(\nu-1)\exp{\frac{-(D_{n}\epsilon)^{2}}{2(2R_{\lambda}\sum_{k=1}^{K}D_{k,n}B_{k,n}+D_{n}\alpha)^{2}}}}.$
(10)

and $\alpha=\delta+\ln(1-V^{2}/4)$ while $V$ stands for the total variation
distance between $\hat{y}_{k,n}^{(i)}$ and $\tilde{p}_{k,n}^{(i)}$.

###### Proof.

First, relying on the subgradient inequality, we have at round $t$,

$\mathcal{E}^{(t)}=\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{*}\right)-\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{(t)}\right)\leq\langle\nabla\mathcal{L}^{(t)},\lambda^{*}-\lambda^{(t)}\rangle.$
(11)

Then, by decomposing the By means of Cauchy-Schwarz inequality, we get,

$\mathcal{E}^{(t)}\leq\norm{\nabla\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{(t)}\right)}\norm{\lambda^{\star}-\lambda^{(t)}}.$
(12)

By invoking the federated learning aggregation (2), we can write

$\nabla\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{(t)}\right)=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\nabla\mathcal{L}\left(\mathbf{W}_{k,n}^{(t)},\lambda^{(t)}\right).$
(13)

Therefore, from (12) and (13) and by invoking the triangle inequality we have,

$\displaystyle\mathcal{E}^{(t)}$
$\displaystyle\leq\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\norm{\nabla\mathcal{L}\left(\mathbf{W}_{k,n}^{(t)},\lambda^{(t)}\right)}\norm{\lambda^{\star}-\lambda^{(t)}}$
(14) $\displaystyle\leq
2R_{\lambda}\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}B_{k,n}.$

At this level, let

$\mathcal{U}^{(t)}=\mathcal{L}\left(\mathbf{W}_{n}^{(t)},\lambda^{\star}\right)-\inf_{\mathbf{W}_{n}^{\star}}\mathcal{L}\left(\mathbf{W}_{n}^{\star},\lambda^{(t)}\right).$
(15)

On the other hand, from [26, 27], we have

$\mathbf{JS}_{k,n}>-\ln(1-\frac{V^{2}}{4}),$ (16)

Combining (14), (15) and (16) with Definition 1 yields,

$\mathcal{U}^{(t)}\leq
2R_{\lambda}\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}B_{k,n}+\delta+\ln(1-\frac{V^{2}}{4})=C.$
(17)

By means of Hoeffding-Azuma’s inequality [28], we have,

$\Pr[\frac{1}{T}\sum_{t=1}^{T}\mathcal{U}^{(t)}<\epsilon\mid T=\tau]\geq
1-\exp{-\frac{\tau\varepsilon^{2}}{2C^{2}}},,$ (18)

where we consider that the federated learning model is fair, i.e., respecting
the recall constraint up to and including time $T=\tau$. Therefore, by
recalling the geometric failure probability mass function $P_{\tau}$ given by,

$P_{\tau}=\nu\left(1-\nu\right)^{\tau},$ (19)

and combining it with (18), yields

$\Pr[\frac{1}{T}\sum_{t=1}^{T}\mathcal{U}^{(t)}<\epsilon]\geq\sum_{\tau=0}^{+\infty}\nu\left(1-\nu\right)^{\tau}\times\left(1-\exp{-\frac{\tau\epsilon^{2}}{2C^{2}}}\right).$
(20)

Finally, by noticing that $\nu<1$ and following some algebraic manipulations,
we obtain the target result as in (9) and (10). ∎

(a)  Analysis of convergence with two approaches (b)  Analysis of convergence
with additional approaches for uRLLC slice

Figure 3: Analysis of FL training loss vs FL rounds of EGFL with lower bound
of recall score, $\gamma=[0.82,0.85,0.84]$

##  VI Results

This section scrutinizes the proposed Closed loop EGFL framework.The dataset
used in this study was generated through extensive simulations conducted
within the framework described in the referenced paper [29]. The simulations
involved generating diverse slice traffic profiles using heterogeneous Poisson
distributions with time-varying intensities. These profiles were designed to
accurately capture the distinctive behavior of each network slice. We use
feature attributions generated by the XAI to build a fair sensitivity-aware,
explanation-guided, constrained traffic drop prediction model. We also present
the results corresponding to the fairness recall constraint and discuss its
impact in detail. Moreover, we show FL convergence results and, finally, we
analyze the comprehensiveness score of all slices to show the faithfulness of
our proposed model. Notably, we invoke DeepExplain framework, which includes
state-of-the-art gradient and perturbation-based attribution methods [30] to
implement the Explainer for generating attribution scores of the input
features. We integrate those scores with our proposed fair EGFL framework in a
closed-loop iterative way.

###  VI-A Parameter Settings and Baseline

Three primary slices eMBB, uRLLC and mMTC are considered to analyze the
proposed EGFL. Here, the datasets are collected from the BSs and the overall
summary of those datasets are presented in Table II. We use vector $\alpha$
for the lower bound of recall score corresponding to the different slices. As
a baseline, we adopt a vanilla FL [31] with post-hoc integrated gradient
explanation, that is, a posterior explanation performed upon the end of the FL
training.

Table III: Settings Parameter | Description | Value
---|---|---
$N$ | # Slices | $3$
$K$ | # BSs | $50$
$D_{k,n}$ | Local dataset size |  $1500$ samples
$T$ | # Max FL rounds | $40$
$U$ | # Total users (All BSs) | $15000$
$L$ | # Local epochs | $40$
$R_{\lambda}$ | Lagrange multiplier radius | Constrained: $10^{-5}$
$\eta_{\lambda}$ | Learning rate | $0.12$

###  VI-B Result Analysis

In this scenario, resources are allocated to slices according to their traffic
patterns and radio conditions.In this section, we thoroughly compare our
proposed EGFL-JS scheme to other compatible state-of-the-art baselines from
various perspectives, such as convergence, fairness, and faithfulness
analysis, to show its superior efficacy.

Figure 4: Analysis of recall score with lower bound $\gamma=[0.82,0.85,0.84]$
Figure 5: Analysis of comprehensiveness score with lower bound of recall
score, $\gamma=[0.82,0.85,0.84]$

(a)  Distribution plot for URLLC network slice

(b)  Attribution-based feature impact on the model output for URLLC network
slice

Figure 6: Root cause analysis of EGFL-JS predictions in a telecommunication
network. Figure 7: Correlation Heatmap for URLLC network slice

  * •

Convergence: First, we have plotted FL normalized training loss vs. FL rounds
of EGFL-JS, and shown the comparison with another compatible strategy termed
EGFL-KL. The key distinction between the two techniques is the divergence
measure used to guide their respective loss functions. The JSD guides the
cross-entropy loss function in our proposed solution, whereas the EGFL-KL
model uses the Kullback-Leibler Divergence (KL Divergence) to guide its loss
function. It is known that JSD, a symmetrized variant of KL Divergence, is
less sensitive to severe discrepancies across distributions, which may help
speed up convergence. We visually depict both models’ convergence behavior
across several slices in Fig. 3a. Notably, compared to the EGFL-KL model, our
suggested model consistently shows a higher rate of convergence. This finding
is consistent with what we predicted since JSD is naturally less susceptible
to being impacted by severe distribution discrepancies, which results in a
smoother optimization landscape and, as a result, faster convergence.
Additionally, we broaden the scope of our research beyond the EGFL-KL model by
including two additional methods: unconstrained EGFL (without recall
constrained) and recall-constrained FL (without Jensen-Shannon Divergence). As
depicted in Fig. 3b, we can conclude that the proposed constrained EGFL
prediction model of the uRLLC slice has converged faster than the
unconstrained EGFL. Moreover, compared to constrained FL, EGFL has given lower
training losses. Overall, the empirical results shown in Fig. 3a and 3b
support our proposed model’s improved convergence characteristics and
demonstrate the benefits of including JSD in the loss function. These findings
further strengthen the prospect of our concept as a highly efficient and
successful solution in different real-world situations.

  * •

Fairness Analysis: As illustrated in Fig. 4, the recall metric has been chosen
to analyze our proposed model’s fairness. From that figure, we can observe
that the recall score of the proposed EGFL-JS for all slices is closer to the
target threshold $\lambda$ (i.e., around 0.80 $\%$), which would be an
acceptable value for slices’ tenants. In summary, the recall scores of EGFL-
JS, EGFL-KL and constrained FL for all slices are almost identical and higher
than the unconstrained EGFL. In comparison, the training loss value of EGFL is
lower than other approaches. The model with divergence included is better at
generalizing to new, unseen data. However, having the same recall score
indicates that the model’s ability to identify positive samples has not been
affected by the inclusion of divergence, which is a favorable sign. This can
be fruitful in cases where the model will be deployed in real-world
applications, as it suggests that the model is more likely to perform well on
new data that it has not seen during training. It gives us an approximate idea
of our model’s reliability and trustworthiness.

  * •

Faithfulness of Explanation: Comprehensiveness evaluates if all features
needed to make a prediction are selected [7]. The model comprehensiveness is
calculated as: Comprehensiveness =
$\hat{y}_{k,n}^{(i)}-\tilde{p}_{k,n}^{(i)}$. Here, the higher score implies
that the explanation provided is complete, comprehensive, and so helpful to
the user. In Fig. 5 we have plotted the comprehensiveness score of all the
slices, where we notice that the proposed EGFL has a higher score than the
EGFL-KL and baseline one. Moreover, in Fig. 5b, we monitor the effect of the
changing value of the p$\%$ features removal on the comprehensiveness score,
considering the proposed model for all slices. Specifically, during the
calculation of comprehensiveness score based on the removal of $p\%$ of
features with lowest attributions, we have created a new masked input, denoted
as $\tilde{\mathbf{x}}_{k,n}^{(i)}$ for each p$\%$ value and also have
generated the corresponding masked predictions, $\tilde{p}_{k,n}^{(i)}$. Here,
the score increases versus the lowest $p\%$, which means that the masked
features are decisive and the corresponding explanation in terms of
attribution is comprehensive.

  * •

Root cause analysis in wireless networks and Interpretation: Feature
attributions generated by XAI methods reflect the contribution of the input
features to the output prediction. From a telecommunication point of view,
this can be leveraged to perform root cause analysis of different network
anomalies and KPIs degradation. In this respect, we investigate the factors
that impact slice-level packet drop by plotting, for instance, the
distribution of the URLLC slice feature attributions in Fig. 6a and their
average in Fig. 6b. The first subfigure demonstrates that EGFL traffic drop
predictions are negatively influenced by PRB and the channel quality, since
the attributions are distributed and concentrated towards negative values,
implying that higher values of these features reduce the likelihood of traffic
drops. It is noted that PRB has a slightly higher impact compared to channel
quality. In this regard, Fig. 6b further supports these observations by
confirming that in average PRB has the most influence on the model’s output,
reaffirming the importance of PRB allocation in reducing traffic drops. This
observation aligns with the notion that higher PRB allocation implies
dedicating more resources to the URLLC slice, which in turn relaxes the
transmission queues and reduces the packet drop. Lastly, the distribution and
average plots reveal that latency has a narrower and more focused distribution
near zero. This suggests that latency may have lesser significance in
predicting traffic drops compared to other features. Furthermore, the
correlation heatmap of Fig. 7 indicates the strength and direction of the
linear relationship between the predicted traffic drops and each input
feature, namely PRB, channel quality, and latency with a correlation of 81
$\%$, 81 $\%$ and 69 $\%$ respectively., which corroborates the previous
analysis. Note that to plot correlation matrix heatmap, we consider one
matrix, $\mathbf{R}_{k,n}$ = [$\mathbf{a}_{k,n},\hat{\mathbf{y}}_{k,n}$],
where, $\mathbf{a}_{k,n}$ denotes the attribution score of features with
dimensions $D_{k,n}\times Q$ and $\hat{\mathbf{y}}_{k,n}$ is the predicted
output variable with dimensions $D_{k,n}\times 1$. A correlation of 81 $\%$
between the predicted traffic drops and PRB as well as channel quality
suggests a strong positive relationship. This means that higher values of PRB
and channel quality are associated with a higher likelihood of lower traffic
drops in the URLLC slice. On the other hand, the correlation coefficient of 69
% between the predicted traffic drops and latency indicates a moderate
positive relationship. This means that higher values of latency are generally
associated with a higher likelihood of traffic drops in the URLLC slice. Based
on these observations, it can be inferred that when dealing with real-time
traffic in the telecom industry, maintaining an appropriate allocation of PRB
and monitoring channel quality in the network slice becomes crucial for
ensuring better services to users.

##  VII Conclusion

In this paper, we have proposed a novel fair explanation guided federated
learning (EFGL) approach to fulfill transparent and trustworthy zero-touch
service management of 6G network slices at RAN in a non-IID setup. We have
proposed a modified objective function where we consider Jenson-Shannon
divergence along with the cross-entropy loss and a fairness metric, namely the
recall, as a constraint. The underlying optimization task has been solved
using a proxy-Lagrangian two-player game strategy. The simulated results have
validated that the proposed scheme can give us a new direction for
implementing trustworthy and fair AI-based predictions.

##  VIII Acknowledgment

This work has been supported in part by the project SEMANTIC (861165). It has
also been supported in part by the MINECO (Spain) and the EU
NextGenerationEU/PRTR (Call UNICO I+D 5G 2021, ref. number TSI-063000-2021-10)

## References

  * [1] C. Benzaid and T. Taleb, “AI-driven zero touch network and service management in 5G and beyond: Challenges and research directions,” _IEEE Network_ , vol. 34, no. 2, pp. 186–194, 2020.
  * [2] S. Jha and S. R. and others, “Attribution-based confidence metric for deep neural networks,” in _Neural Information Processing Systems_ , 2019.
  * [3] M. Naylor, C. French _et al._ , “Quantifying explainability in NLP and analyzing algorithms for performance-explainability tradeoff,” in _Proceedings of the Thirty-eighth International Conference on Machine Learning_ , 2021.
  * [4] S. Roy, H. Chergui, and C. Verikoukis, “TEFL: Turbo Explainable Federated Learning for 6G Trustworthy Zero-Touch Network Slicing,” 2022. [Online]. Available: https://arxiv.org/abs/2210.10147
  * [5] A. A. Ismail, H. Corrada Bravo, and S. Feizi, “Improving deep learning interpretability by saliency guided training,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 26 726–26 739, 2021.
  * [6] K. Li, Z. Wu, and a. Peng, “Tell me where to look: Guided attention inference network,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 9215–9223.
  * [7] Y. Gao, S. Gu, and a. Junji Jiang, “Going beyond XAI: A systematic survey for explanation-guided learning,” _ArXiv_ , vol. abs/2212.03954, 2022.
  * [8] B. Brik and A. Ksentini, “On predicting service-oriented network slices performances in 5G: A federated learning approach,” in _45th IEEE Conference on Local Computer Networks, LCN 2020, Sydney, Australia, November 16-19, 2020_ , H. P. Tan, L. Khoukhi, and S. Oteafy, Eds. IEEE, 2020, pp. 164–171. [Online]. Available: https://doi.org/10.1109/LCN48667.2020.9314849
  * [9] C. Li, W. Guo, and a. Sun, “Trustworthy deep learning in 6G-enabled mass autonomy: From concept to quality-of-trust key performance indicators,” _IEEE Vehicular Technology Magazine_ , vol. 15, no. 4, pp. 112–121, 2020\.
  * [10] A. Terra and a. Inam, “Explainability methods for identifying root-cause of SLA violation prediction in 5G network,” in _GLOBECOM 2020 - 2020 IEEE Global Communications Conference_ , 2020, pp. 1–7.
  * [11] S. Wang, M. Qureshi, and a. Luis Miralles-Pechua’an, “Explainable AI for B5G/6G: Technical aspects, use cases, and research challenges,” _ArXiv_ , vol. abs/2112.04698, 2021.
  * [12] M. Munir, K. T. Kim, and a. Adhikary, “Neuro-symbolic explainable artificial intelligence twin for zero-touch IoE in wireless network,” _arXiv preprint arXiv:2210.06649_ , 2022.
  * [13] A. Renda and o. Ducange, Pietro and, “Federated learning of explainable AI models in 6G systems: Towards secure and automated vehicle networking,” _Information_ , vol. 13, no. 8, 2022. [Online]. Available: https://www.mdpi.com/2078-2489/13/8/395
  * [14] T. Popordanoska, M. Kumar, and S. Teso, “Machine guides, human supervises: Interactive learning with global explanations,” _ArXiv_ , vol. abs/2009.09723, 2020.
  * [15] A. S. Ross and M. C. H. and others, “Right for the right reasons: Training differentiable models by constraining their explanations,” in _International Joint Conference on Artificial Intelligence_ , 2017.
  * [16] Y. Gao and T. a. Sun, “Gnes: Learning to explain graph neural networks,” in _2021 IEEE International Conference on Data Mining (ICDM)_ , 2021, pp. 131–140.
  * [17] J. Sun and S. L. and others, “Explanation-guided training for cross-domain few-shot classification,” _2020 25th International Conference on Pattern Recognition (ICPR)_ , pp. 7609–7616, 2020.
  * [18] W. Wu, C. Zhou, and a. Li, “AI-native network slicing for 6G networks,” _IEEE Wireless Communications_ , vol. 29, no. 1, pp. 96–103, 2022.
  * [19] H. Chergui and L. a. Blanco, “Statistical federated learning for beyond 5G SLA-constrained RAN slicing,” _IEEE Transactions on Wireless Communications_ , vol. 21, no. 3, pp. 2066–2076, 2022.
  * [20] A. Cotter, H. Jiang, and K. Sridharan, “Two-player games for efficient non-convex constrained optimization,” _CoRR_ , vol. abs/1804.06500, 2018\. [Online]. Available: http://arxiv.org/abs/1804.06500
  * [21] M. Glockner _et al._ , “Why do you think that? exploring faithful sentence-level rationales without supervision,” in _Findings of the Association for Computational Linguistics: EMNLP 2020_. Online: Association for Computational Linguistics, Nov. 2020, pp. 1080–1095. [Online]. Available: https://aclanthology.org/2020.findings-emnlp.97
  * [22] Y. K. Tun and N. H. T. and others, “Wireless network slicing: Generalized kelly mechanism-based resource allocation,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, pp. 1794–1807, 2019.
  * [23] M. Sundararajan, A. Taly, and Q. Yan, “Axiomatic attribution for deep networks,” in _Proceedings of the 34th International Conference on Machine Learning - Volume 70_ , ser. ICML’17. JMLR.org, 2017, pp. 3319–3328.
  * [24] A. Cotter and a. Maya R. Gupta, “Training well-generalizing classifiers for fairness metrics and other data-dependent constraints,” in _ICML_ , 2019\.
  * [25] G. J. Gordon, A. Greenwald, and C. Marks, “No-regret learning in convex games,” in _Proceedings of the 25th International Conference on Machine Learning_ , ser. ICML ’08. New York, NY, USA: Association for Computing Machinery, 2008, pp. 360–367. [Online]. Available: https://doi.org/10.1145/1390156.1390202
  * [26] T. Yamano, “Some bounds for skewed $\alpha$-jensen-shannon divergence,” _Results in Applied Mathematics_ , vol. 3, p. 100064, 2019. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2590037419300640
  * [27] J. Lin, “Divergence measures based on the shannon entropy,” _IEEE Transactions on Information Theory_ , vol. 37, no. 1, pp. 145–151, 1991.
  * [28] W. Hoeffding, “Probability inequalities for sums of bounded random variables,” _Journal of the American Statistical Association_ , vol. 58, no. 301, pp. 13–30, 1963. [Online]. Available: http://www.jstor.org/stable/2282952
  * [29] F. Rezazadeh and L. a. Zanzi, “On the specialization of FDRL agents for scalable and distributed 6G ran slicing orchestration,” _IEEE Transactions on Vehicular Technology_ , vol. 72, no. 3, pp. 3473–3487, 2023.
  * [30] [Online]. Available: https://github.com/marcoancona/DeepExplain
  * [31] H. B. McMahan and a. Eider Moore, “Communication-efficient learning of deep networks from decentralized data,” in _International Conference on Artificial Intelligence and Statistics_ , 2016.