# Acceleration of a Positron Bunch in a Hollow Channel Plasma

Spencer Gessner<EMAIL_ADDRESS>SLAC National Accelerator Laboratory,
Menlo Park, California 94025, USA  Erik Adli  Department of Physics,
University of Oslo, 0316 Oslo, Norway  James M. Allen  SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Weiming An
Department of Astronomy, Beijing Normal University, Beijing 100875, China
Christine I. Clarke  SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA  Chris E. Clayton  Department of Electrical Engineering,
University of California Los Angeles, Los Angeles, California 90095, USA
Sebastien Corde  LOA, ENSTA ParisTech, CNRS, Ecole Polytechnique, Université
Paris-Saclay, 91762 Palaiseau, France  Antoine Doche  LOA, ENSTA ParisTech,
CNRS, Ecole Polytechnique, Université Paris-Saclay, 91762 Palaiseau, France
Joel Frederico  SLAC National Accelerator Laboratory, Menlo Park, California
94025, USA  Selina Z. Green  SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA  Mark J. Hogan  SLAC National Accelerator Laboratory,
Menlo Park, California 94025, USA  Chan Joshi  Department of Electrical
Engineering, University of California Los Angeles, Los Angeles, California
90095, USA  Carl A. Lindstrøm  Department of Physics, University of Oslo, 0316
Oslo, Norway  Michael Litos  University of Colorado Boulder, Boulder, CO
80309, USA  Kenneth A. Marsh  Department of Electrical Engineering, University
of California Los Angeles, Los Angeles, California 90095, USA  Warren B. Mori
Department of Physics and Astronomy, University of California Los Angeles, Los
Angeles, California 90095, USA  Brendan O’Shea  SLAC National Accelerator
Laboratory, Menlo Park, California 94025, USA  Navid Vafaei-Najafabadi
Stonybrook University, Stony Brook, NY 11794, USA  Vitaly Yakimenko  SLAC
National Accelerator Laboratory, Menlo Park, California 94025, USA

###### Abstract

Plasmas are a compelling medium for particle acceleration owing to their
natural ability to sustain large electric fields. Plasmas are also unique
amongst accelerator technologies in that they respond differently to beams of
opposite charge. The asymmetric response of a plasma to highly-relativistic
electron and positron beams arises from the fact that plasmas are composed of
light, mobile electrons and heavy, stationary ions. Hollow channel plasma
acceleration is a technique for symmetrizing the response of the plasma, such
that it works equally well for high-energy electron and positron beams. In the
experiment described here, we demonstrate the generation of a positron beam-
driven wake in an extended, annular plasma channel, and acceleration of a
second trailing witness positron bunch by the wake. The leading bunch excites
the plasma wakefield and loses energy to the plasma, while the witness bunch
experiences an accelerating field and gains energy, thus providing a proof-of-
concept for hollow channel acceleration of positron beams. At a bunch
separation of 330 $\upmu\textrm{m}$, the accelerating gradient is 70 MV/m, the
transformer ratio is 0.55, and the energy transfer efficiency is 18% for a
drive-to-witness beam charge ratio of 5:1.

##  I Introduction

Plasma wakefield acceleration (PWFA) is a novel technology which can be used
to accelerate particle beams with large gradients and high efficiency Litos et
al. (2014); Lindstrøm et al. (2021). The most ambitious application of plasma
accelerator technology is the generation of ultra-high energy, low-emittance
beams for a plasma-based linear collider (PLC). There are several challenges
on the path to the PLC. Chief among them is the acceleration of positron beams
in plasma Cao et al. (2023). For linear collider designs, such as the ILC
Aryshev et al. (2022), CLIC Brunner et al. (2022), and C3 Vernieri et al.
(2023), which use radio-frequency (RF) waves to accelerate particles, electron
and positron beams are accomodated by adjusting the phase of the RF wave. By
contrast, the plasma accelerator reacts differently to beams of opposite
charge. The plasma is composed of light, mobile electrons, and heavy, sluggish
ions. When operating in the blowout regime Lu et al. (2006), an electron beam
propagating into a plasma expels plasma electrons, creating a plasma bubble
with strong ion focusing, which is advantageous for transporting and
accelerating the electron beam, while maintaining the beam’s emittance. On the
other hand, a positron beam attracts plasma electrons toward the beam axis,
which creates a complicated wakefield structure Hogan et al. (2003); Blue et
al. (2003); Muggli et al. (2008); Corde et al. (2015). Recently, we have
demonstrated that nonlinear wakes driven by positron beams in a uniform plasma
can be used to accelerate trailing witness positron bunches with modest
gradients but with high energy-extraction efficiency from the wake Doche et
al. (2017). However, due to the complex focusing field structure of these
wakes, the positron beam emittance, which is a measure of beam quality, is not
preserved using this technique. In the moderately non-linear regime, it is
possible to preserve the positron beam emittance in a uniform plasma, but this
technique does not scale well to collider-quality emittances Hue et al.
(2021); Cao et al. (2023). We are therefore motivated to pursue other
approaches which in principle preserve the positron beam emittance. The hollow
channel plasma wakefield accelerator is an appealing concept for accelerating
positron beams in plasma because it can avoid the complicated wake structure
that arises from non-linear plasma wakefields in uniform plasmas.

Figure 1: (a). Schematic of the hollow channel experiment carried out at
FACET. All relevant diagnostics are labeled. Inset (b) The drive-witness
energy spectrum measured upstream of the hollow channel plasma. Inset (c)
Sample images on the YAG screen downstream of the hollow channel plasma
showing the transverse beam profile for different experimental conditions.
Inset (d) Sample image of the ionizing high-order Bessel pulse used to ionize
the plasma channel. Note that the image is deformed by the downstream optics
and is primarily used for shot-to-shot jitter measurements. Inset (e)
Comparison of beam energy spectra with and without the plasma channel present.

The hollow channel plasma wakefield accelerator was originally conceived as a
technique for guiding high-intensity laser pulses for laser wakefield
acceleration Chiou et al. (1995), and later recognized to have advantageous
properties for beam-driven wakefield acceleration Chiou and Katsouleas (1998).
In this scenario, an electron or positron beam propagates through a hollow
tube of plasma and drives a high-amplitude electromagnetic field in its wake.
The longitudinal field inside the channel is radially uniform, so that
particles with different initial offsets with respect to the channel axis all
gain energy at the same rate. The absence of plasma in the channel implies
that there are no transverse forces from on-axis plasma electrons. Finally,
for drive and witness beams propagating on-axis through the channel, there are
no transverse forces and the beam emittance $\varepsilon$ is preserved through
the acceleration process. However, for long acceleration lengths
($L\gg\beta_{x,y}$) an external focusing magnetic field may be necessary to
guide both the drive and witness beam. Also, small perturbations of either the
drive or witness beam centroids from the channel axis lead to the appearance
of transverse wakefields that can seed and amplify the beam breakup
instability Schroeder et al. (1999) which has recently been observed
experimentally Lindstrøm et al. (2018). Additionally, asymmetries in the shape
of the hollow plasma channel may lead to fields which deviate from the ideal
scenario.

In previous work Gessner et al. (2016), we described the excitation of a
hollow channel wakefield by a single positron bunch. Here we describe the
subsequent acceleration of a witness positron bunch in the hollow channel
wakefield. The hollow channel plasma is generated by ionizing a lithium vapor
Muggli et al. (1999) with a ring-shaped, high-intensity laser pulse using a
phase plate that transforms an initial Gaussian beam into a high-order Bessel
beam Kimura et al. (2011). The approximately annular plasma channel is
produced by multi-photon ionization of lithium. We propagate a positron drive
beam through the channel, creating a longitudinal wakefield that is used to
accelerate a witness positron bunch. By scanning the separation of the drive
and witness beams, we map the longitudinal shape of the wakefield and
determine the accelerating phase Schröder et al. (2020). From the energy
gained and lost by the witness and drive bunches, respectively, we deduce the
net energy transfer efficiency and the transformer ratio at the phase
corresponding to the maximum accelerating gradient.

##  II Experimental Overview

The experiment was carried out at the Facility for Advanced aCcelerator
Experimental Tests (FACET) at the SLAC National Accelerator Laboratory Hogan
et al. (2010). SLAC is the only laboratory with the infrastructure required
for providing high-energy, high-intensity positron beams for PWFA experiments.
A low-emittance positron beam is extracted from the damping ring (see Appendix
A) and accelerated in the linac to 20 GeV energy. Along the linac, the
particle beam is sent through a series of bunch compression chicanes which
reduce the longitudinal size of the bunch $\sigma_{z}$ from several
millimeters to a few hundred microns. In the final chicane, the beam is sent
through a notch-collimation system Litos et al. (2014) which exploits the
time-energy correlation of the beam to convert the beam from a single-bunch
structure to a two-bunch structure with variable separation. The collimation
system is also used to adjust the total charge of the beam delivered to the
experiment. At the end of the final chicane, the beam is focused by a series
of quadrupole magnets before entering the lithium oven.

The schematic of the experiement is depicted in Figure 1. The longitudinal
shape of the two-bunch beam structure is characterized by an electro-optical
sampling (EOS) crystal Berden et al. (2007) with a resolution of 10 microns
(30 femtoseconds). The plasma source is a lithium heat-pipe oven Muggli et al.
(1999). The plasma source has several heating coils which can be turned on and
off to adjust the length of the vapor region. The lithium is ionized by a
Terawatt-class, Ti:Sapphire laser with 800 nm central wavelength. For this
experiment, the laser is operated at low intensity with a maximum pulse energy
of 40 mJ in a 75 fs-long pulse. In order to ionize the 25 cm-long annular
plasma, the laser is shaped by a diffractive kinoform optic Andreev et al.
(1996); Fan et al. (2000); Kimura et al. (2011); Gessner et al. (2016). The
optic forms the laser into a high-order Bessel intensity profile with the
first maximum occurring at a radius of $250~{}\upmu\textrm{m}$ (see Appendix
B). The Bessel intensity profile has the attractive feature that the
transverse shape of the profile does not depend on the $z$-coordinate; the
laser ionizes a channel with a fixed radius along the lithium oven. The optic
is mounted on a 2D stage which allows for control of the transverse position
of the laser focus.

The laser is directed onto the particle beam axis by a gold mirror with a hole
for the positron beam to pass through. The laser pulse and positron beam co-
propagate into the lithium vapor source. A delay stage is used to set the
relative laser-positron timing and adjust the longitudinal position of the
laser focus. The delay was set such that the laser arrives at most 5
picoseconds ahead of the positron beam, and with the laser focus set to the
downstream end of the lithium vapor source, creating a 25$\pm$1 cm-long plasma
channel. The energy of the ionized plasma electrons is less than 1 eV and have
a velocity of $0.4~{}\upmu\textrm{m}$/s. The motion of the plasma electrons
has negligible effect on the shape of the channel prior to the arrival of the
positron beam.

Downstream of the lithium oven, a second gold mirror with a hole deflects the
laser pulse off of the beam axis. The spent laser pulse is re-imaged onto two
cameras which we use to track the position of the laser focus during the
experiment. The positron beam passes through the holed-mirror and through a
yttrium aluminum garnet (YAG) crystal which is used to track the transverse
position and size of the positron beam. Finally, the beam passes through a
spectrometer dipole and the beam energy is measured on a scintillating LANEX
screen (see Appendix E).

##  III Results and Discussion

Figure 2: Results of the longitudinal bunch separation scan and supporting
QuickPIC simulation scan. Witness bunch data is denoted by red circles, and
witness bunch simulation results shown with open red squares. Drive bunch data
is denoted by blue circles, and drive bunch simulation results shown with open
blue squares. For the first point in the scan, the drive and witness beams are
not clearly distinguished and the data point is placed at an upper limit of 95
$\upmu\textrm{m}$ bunch separation. Error bars represent the $1\sigma$
standard deviation of the measurement. The QuickPIC simulation use the
parameters measured for each scan point described in Appendix F.

In the first phase of the experiment, the transverse and longitudinal positron
beam profiles and energy spectrum are measured for the beam propagating in
vacuum. The lithium plasma source is heated to produce lithium vapor with a
density of $3\times 10^{16}$ cm-3. At high charge, the positron beam fields
are large enough to self-ionize the neutral lithium atoms within the laser-
produced hollow channel. The incoming positron charge was therefore reduced
until there was no evidence of plasma interaction. The procedure was performed
at peak bunch compression (maximum positron beam field strength). Table 1
shows the positron beam parameters at the end of the charge reduction
procedure. All parameters are measured at the interaction point, except for
the beam emittance which was measured upstream of the final bunch compression
chicane.

$N_{tot}$ |  $\sigma_{x}(\upmu\textrm{m}$) |  $\sigma_{y}(\upmu\textrm{m}$) |  $\beta_{x}$(m) |  $\beta_{y}$(m) |  $\varepsilon_{x}(\upmu\textrm{m}$) |  $\varepsilon_{y}(\upmu\textrm{m}$)
---|---|---|---|---|---|---
$3.5\times 10^{9}$ | 35 | 25 | 0.5 | 5.0 | 97 | 5
Table 1: Positron beam parameters of the combined drive and witness bunches
after the charge reduction procedure. All parameters are measured at the
interaction point, except for the beam emittance which was measured upstream
of the final bunch compression chicane.

At peak compression, the EOS diagnostic does not clearly distinguish between
the drive and witness bunches, but can be used to set an upper limit on the
longitudinal separation between the two bunches
$\Delta_{dw}<95~{}\upmu\textrm{m}$ . The drive and witness bunches are well
separated on the spectrometer screen, as seen in Figure 1b), with drive bunch
energy $E_{d}=20.487$ GeV and witness bunch energy $E_{w}=20.095$ GeV. The
energy resolution of the spectrometer is 3 MeV.

After confirming non-interaction between the beam and lithium vapor, the laser
was fired to generate the hollow channel plasma. The laser profile was
previously optimized according to the procedure described in Appendix C. Next,
we performed a longitudinal drive-witness separation scan to map out the
hollow channel plasma wakefield, according to the procedure described in
Appendix D. The drive-witness separation is measured by the EOS system on
every shot and the spectrometer measures the change in energy of the drive and
witness beams.

Figure 2 shows the results of the scan. In the ideal bunch separation scan,
all beam parameters would be kept constant as the bunch separation is varied.
In practice, the bunch charge, length, and separation are coupled together and
depend on the longitudinal phase space. We actively corrected the bunch charge
and beam trajectory throughout the scan, but were not able to control the
length of the drive bunch independently of the bunch separation. The length of
the drive bunch increases by a factor of five over the course of the scan
while the drive charge varies by $\pm 18\%$. The witness bunch charge and
length are stable at the level of $\pm 20\%$. We observed that the magnitude
of the energy loss experienced by the drive bunch decreased over the course of
the scan, which is consistent with increasing drive bunch length. From the
maximum energy gain of 17.5 MeV at a bunch separation of 330
$\upmu\textrm{m}$, we infer an accelerating gradient of $70\pm 30$ MV/m, and
the uncertainty is the statistical $1\sigma$ standard deviatio of the
measurement.

Figure 3: Efficiency of energy transfer from drive bunch to witness bunch for
the $\Delta_{dw}=330~{}\upmu\textrm{m}$ scan point as a function of witness-
to-drive charge ratio. The error bars denote the error on the mean of the
values in a given witness-to-drive charge ratio bin due to jitter in the
measurement. Inset: Normalized histogram of transformer ratios for the same
set of shots.

We analyzed the efficiency and transformer ratio of the wakefield at a bunch
separation of 330 $\upmu\textrm{m}$. The transformer ratio is defined as

$T=\left|\frac{\max(E_{z})}{\min(E_{z})}\right|.$ (1)

While we do not measure the accelerating field directly, the centroid energy
gain of the witness bunch is a good proxy for the peak field because the
witness bunch is much shorter than the wavelength of the wakefield. Measuring
the peak decelerating field is complicated by the fact that the drive bunch
experiences a changing field over its length. We can extract the peak
decelerating field from the drive bunch spectrum if the field is single-valued
(monotonically decreasing) over the length of the bunch Clayton et al. (2016).
This condition is satisfied for the scan step at 330 $\upmu\textrm{m}$
separation because the $\sigma_{zd}=122~{}\upmu\textrm{m}$ and the end of the
drive bunch sits in the peak decelerating phase of the wake. We exploit the
linear correlation between position and energy in the drive beam phase space
and measure the change in energy of the lower end of the spectrum to determine
the peak decelerating field. By comparison, efficiency is a simpler quantity
to measure

$\eta=\frac{\Delta E_{w}Q_{w}}{\Delta E_{d}Q_{d}},$ (2)

where $\Delta E_{w}$ and $\Delta E_{d}$ are the centroid change in energy of
the witness and drive bunches, respectively. Figure 3 shows the efficiency of
energy transfer from the drive beam to the witness beam as a function of the
ratio of drive charge to witness charge. There is large shot-to-shot variation
of the charge in the drive and witness bunches and the data is binned by the
charge ratio. A histogram of measured transformer ratio values is shown as an
inset, with ratios in excess of one observed on some shots. The median
observed transformer ratio was 0.55. The efficiencies measured in this
experiment are roughly a factor of 2 below the best efficiencies observed for
plasma acceleration in a uniform plasma Lindstrøm et al. (2021).

###  III.1 Comparison with Theory and Simulation

Figure 4: QuickPIC simulations of positron drive and witness beams propagating
in a hollow channel plasma channel. The beams propagate to the left. The blue
circles denote the one-sigma beam contours. The dotted black line denotes the
transverse beam offset. The solid black line is the accelerating field sampled
at the transverse position of the beam. The maximum and minimum longitudinal
field values are denoted on the field curves. Top: Both beams centered on
axis. Bottom: The drive and witness beams are offset from the axis by
$65~{}\upmu\textrm{m}$. The accelerating field is enhanced and the wavelength
is elongated for the case of the offset beam. Shot-to-shot jitter of the
channel axis with respect to the beam axis is approximately
$25~{}\upmu\textrm{m}$ at the downstream end of the channel.

The wavelength of a hollow channel plasma wakefield is given by
$\lambda=2\pi/\chi_{||}k_{p}$ with $k_{p}$ the plasma wavenumber. The
geometric factor $\chi_{||}$ is derived from the boundary conditions and given
by Gessner (2016)

$\chi_{||}=\sqrt{\frac{2B_{10}(a,b)}{2B_{10}(a,b)-k_{p}aB_{00}(a,b)}},$ (3)

with $B_{ij}$ the Bessel boundary function given by

$B_{i,j}(a,b)=I_{i}(k_{p}a)K_{j}(k_{p}b)+(-1)^{i-j+1}I_{j}(k_{p}b)K_{i}(k_{p}a),$
(4)

for inner plasma radius $a$ and outer plasma radius $b$. These equations are
derived under the assumption that there is a sharp transition between the
ionized and un-ionized regions.

For fully-ionized vapor at $n_{0}=3\times 10^{16}$ cm-3, we expect to observe
the accelerating phase of the wake for a bunch separation
$\Delta_{dw}=165~{}\upmu\textrm{m}$, but in the scan, the peak accelerating
phase is observed at $\Delta_{dw}=460~{}\upmu\textrm{m}$. The bulk of the
discrepancy is the result of reducing the laser intensity used to ionize the
hollow channel plasma according to the channel symmetrization procedure
described in Appendix C. We use a multi-photon ionization model Perelomov et
al. (1966) to estimate that the ionized fraction of the vapor is roughly 10%,
which is consistent with the measurement and a rough scaling of the hollow
channel plasma wavelength $\lambda\propto n^{-1/2}$, modulo the factor
$\chi_{||}$ in Equation 3.

The sharp-boundary model estimates that the peak accelerating phase should
occur at a bunch separation of $\Delta_{dw}=386~{}\upmu\textrm{m}$, which is
consistent with our observations. The model also predicts a return to the
decelerating phase at large bunch separations, but the data show a long
tapering-off of the energy gain at large bunch separation. We investigated a
variety of possibilities, including the presence of a small amount of plasma
inside the plasma channel Schroeder et al. (2013), but the best fit to the
data is achieved by modeling the channel wall with a gradual density
transition from the central unionized region to the fully-ionized region. We
refer to this configuration as the “soft boundary” model. We chose a linear
ramp function for our model (see Appendix F) with a width of one skin depth.
The exact shape of the transition from unionized inner region to fully-ionized
outer region is not important. Rather, it is the length scale over which the
transition occurs that is significant Shvets et al. (1996). In the sharp
boundary model, the walls of the plasma channel contain the electromagnetic
wake within the channel. In the soft boundary case, the electromagnetic wake
interacts with plasma at a range of densities and frequencies over the
transition region. This causes phase mixing and electrostatic separation of
plasma electrons and ions. These effects both damp and lengthen the
electromagnetic wake. Finally, the inclusion of transverse offsets of the
drive and witness beam in the channel contributes to an even faster phase-
mixing and wake-damping process because the outer edges of the beam propagate
in the transition region.

Figure 4 shows positron drive and witness beams propagating through a hollow
channel plasma with a linear transition between the unionized and ionized
regions. In the bottom simulation, the beams are offset from the axis by
$65~{}\upmu\textrm{m}$ and which contributes to damping and lengthening of the
wake as compared to the on-axis case shown above. By including the soft
boundary model and transverse offsets, we find good agreement between data and
simulation for bunch separations up to $500~{}\upmu\textrm{m}$, as shown in
Figure 2.

##  IV Conclusion

We have demonstrated the acceleration of a trailing positron bunch by a
positron beam-driven wake in the hollow channel plasma. In this experiment, at
a bunch separation of 330 $\upmu\textrm{m}$, and a drive-to-witness charge
ratio of 5:1, we observe an accelerating gradient of $70\pm 30$ MV/m, a median
transformer ratio of 0.55, and an energy transfer eﬃciency of 18%. This
important proof-of-concept shows the promise of hollow channel acceleration
for positron beams, while also highlighting the engineering and physics
challenges that must be overcome to further advance this technology. The most
critical challenge is mitigation of the beam breakup instability Lindstrøm et
al. (2018) that will dilute the emittance of the accelerating beam. The wake
damping features of the soft boundary model affect both the $m=0$ longitudinal
mode and $m=1$ dipole mode. Further studies are warranted to see if the $m=1$
mode can be preferentially damped over the $m=0$ mode through a judicious
choice of transverse ramp parameters.

The acceleration gradient in this experiment was limited by the requirement
that the positron beam not self-ionize the residual neutral Li vapor inside
the hollow plasma channel. There are several approaches that would allow
experiments to reach GeV/m gradients in the hollow channel plasma. First, by
switching to a different gas species with a higher ionization threshold, such
as He, we will be able to increase the charge of the driving bunch without
ionizing the residual vapor in the channel. Without changing any other aspects
of the experiment, the accelerating gradient would increase to 300 MeV/m by
increasing the drive charge to 2 nC. Next, the accelerating gradient will be
increased by reducing the inner radius of the channel, which can be achieved
by selecting a different kinoform optic. A narrower channel will also require
greater beam stability from the accelerator in order to avoid excitation on
transverse modes in the channel. An alternative method for creating the plasma
channel may envoke self-guiding laser techniques Feder et al. (2020), but this
would result in a not-quite hollow channel plasma similar to the scenario
described by Schroeder Schroeder et al. (2013).

Future plasma-based positron acceleration experiments will use electron beam
drivers. The possibility of using electron beams to drive a wake with a
positron witness beam is a planned capability of FACET-II Yakimenko et al.
(2019). Recent work on hollow channel plasma wakefield acceleration suggests
that efficient beam loading can be achieved using an asymmetric drive electron
beam that mitigates the beam break up instability of the positron bunch Zhou
et al. (2021). Positron plasma acceleration remains a critical challenge on
the path to a Plasma Linear Collider, and hollow channel plasmas have
promising capabilities that require further development to be fully exploited.

##  V Acknowledgements

This work is supported by the U.S. Department of Energy under Contract DE-
AC02-76SF0051 and National Natural Science Foundation of China (NSFC) Grant
Nos. 12075030.

Step |  $Q_{d}~{}(1\times 10^{9})$ |  $\sigma_{d}~{}(\upmu\textrm{m}$) |  $Q_{w}~{}(1\times 10^{9})$ |  $\sigma_{w}~{}(\upmu\textrm{m}$) |  $\Delta_{dw}~{}(\upmu\textrm{m}$) |  $\Delta_{r}~{}(\upmu\textrm{m}$)
---|---|---|---|---|---|---
1 | 2.72 | 42 | 0.95 | 42 | 98 | 0
2 | 2.92 | 67 | 0.55 | 65 | 205 | 45
3 | 3.02 | 107 | 0.59 | 67 | 288 | 66
4 | 2.91 | 122 | 0.52 | 95 | 333 | 65
5 | 2.48 | 119 | 0.85 | 62 | 353 | 78
6 | 2.57 | 153 | 0.69 | 71 | 461 | 80
7 | 2.26 | 153 | 0.74 | 64 | 483 | 90
8 | 2.20 | 185 | 0.83 | 58 | 562 | 87
9 | 2.12 | 208 | 0.90 | 84 | 666 | 77
Table 2: Inputs to the QuickPIC simulations of the longitudinal beam
separation scan. $\Delta_{r}$ is the offset between the beam and channel
access.

##  Appendix A Positron Beam Generation

A 20 GeV electron beam is accelerated along a 2 kilometer-long S-band linac.
The positron beam is generated by colliding the electron beam with a tungsten
target. This produces a shower of high-emittance positrons that are captured
in a solenoidal horn and transported back to the start of the linac where they
are fed into a damping ring. The damping ring reduces the emittance of the
positron beam to roughly 50 mm mrad. After damping, it is injected into the
linac and accelerated to 20 GeV Clendenin et al. (1988).

##  Appendix B Hollow Laser Optics

The kinoform is a 1 mm thick piece of fused silica with an etched pattern that
approximates the spiral phase $\Phi=k_{\perp}r+m\phi$ that imprints a high-
order Bessel profile onto the laser pulse. Here, $r$ and $\phi$ are the radial
and azimuthal coordinates, $m$ is the Bessel order, and $k_{\perp}=\gamma k$
with $\gamma$ the angle of focused rays with respect to the axis and $k$ is
the wavenumber for 800 nm light. We chose $m=7$ and $\gamma=4.4$ mrad. This
produces a laser intensity profile of the form $I(r,z)=I_{0}2\pi
kz\gamma^{2}J_{7}^{2}(k_{\perp}r)$ with the first maximum occurring at a
radius of 250 $\upmu\textrm{m}$ Gessner et al. (2016). Here, $I_{0}$ is the
incident laser intensity on the kinoform, $z$ is the longitudinal coordinate
of laser propagation and $J_{7}$ is the seventh-order Bessel function of the
first kind.

##  Appendix C Channel Optimization

At the start of the experiment, the laser pulse used to create the plasma
channel is optimized for symmetry. Asymmetries may arise from phase errors
that accumulate during laser transport or nonuniform illumination of the
kinoform optic. Astigmatism is the most prominent asymmetry. For diagnostic
purposes, we reduce the intensity of the laser in order to view the focal spot
of the laser on a titanium foil just upstream of the plasma source. We adjust
a lens in the transport system to remove astigmatism and optimize for
uniformity at the focal point. This procedure is performed at low laser
intensity to avoid burning the titanium foil. We then remove the titanium
foils and increase the laser intensity to ionize lithium vapor.

We use a single positron beam to study the transverse shape of the hollow
channel plasma. Since we cannot view the laser focus directly when operating
with intensities large enough to ionize the plasma, we use beam-based
measurements to characterize the shape of the Bessel focus. We use the 2D
kinoform stage to scan the transverse position of the laser relative to the
positron beam axis, while monitoring the position of the laser focus using the
spent laser profiles. When the Bessel focus is set such that the beam
propagates inside the channel, but slightly off-axis, the beam feels a
transverse kick towards the wall of the plasma channel Lindstrøm et al.
(2018). We measure the deflection of the beam on the downstream YAG crystal.
If the focus is set such that the positron beam is propagating in the ionized
region, the beam experiences strong transverse focusing forces, but no
deflection. In this case, we measure an increase in the beam size on the YAG
screen due to the larger beam divergence. The scan reveals the center of
hollow channel, as well as any asymmetries that may be present. If the scan
reveals large asymmetries, we readjust the lens in the transport system and
repeat the scan.

##  Appendix D Longitudinal Phase Scans

In the longitudinal phase scan, the separation between the drive and witness
bunches is changed by adjusting the RF phase of the linac and by setting the
position of the notch-collimation system. The scan requires multiple
parameters to be tuned simultaneously, which affects the incoming energy
distribution of the drive and witness beams. Changes to the beam energy affect
the steering through the final bunch compression chicane, and that can lead to
mis-steering or mis-focusing of the beams if the accelerator lattice is not
compensated. During the longitudinal phase scan, the beam trajectory through
the IP area is monitored and adjusted as needed. Despite best efforts, there
is a net drift of the combined drive-witness transverse centroid of up to
$90~{}\upmu\textrm{m}$ over the course of the scan.

##  Appendix E Measuring Changes in the Beam Energy

The dumpline energy spectrometer is composed of a quadrupole doublet and a
dipole that bends the beam in the vertical plane. The doublet provides point-
to-point imaging of the end of the plasma to the LANEX screen where it is
viewed by a scientific-CMOS PCO Edge camera. The energy-per-pixel resolution
of the camera is 2.9 MeV for the lower-energy witness bunch and 3.0 MeV at the
higher-energy drive bunch. The calibration does not vary significantly over
the individual bunches.

With the quadrupole doublet imaging the exit plane of the plasma, there are
two first-order contributions to the position measured at the spectrometer
screen: the change in beam energy $\Delta E$ and the beam offset from nominal
trajectory at the end of the plasma $\Delta y_{plas}$. The change in the
position of the beam on the screen due to the plasma interaction is given by

$\Delta y_{screen}=\eta(\Delta E/E_{0})+R_{33}\Delta y_{plas},$ (5)

where $\eta$ is the dispersion due to the spectrometer dipole and $R_{33}$ is
the magnification of the position of the beam at the exit of the plasma
channel. With the laser off (no plasma interaction), we measure $R_{33}=2.35$.
A transverse vertical offset of $6~{}\upmu\textrm{m}$ in the image plane is on
the same scale as a 3 MeV energy change, and we therefore need to account for
beam orbit changes when measuring the energy.

When correcting for orbit changes, we are faced with the issue that the BPMs
do not distinguish between the drive and witness bunches. The YAG profile
monitor downstream of the plasma can in some instances distinguish the two
bunches, but not for the 5:1 drive:witness charge ratio used in this dataset.
We therefore assume that both the drive bunch and witness bunch have the same
offset as measured on the YAG screen and subtract off the orbital component
from the beam energy measurement.

##  Appendix F Simulations

Simulations are performed with QuickPIC, a quasi-static Particle-in-Cell code
An et al. (2013). The simulations are performed on a 3D grid with $256\times
256\times 256$ cells and a box size of $15k_{p}^{-1}\approx
1500~{}\upmu\textrm{m}$ in each dimension. The plasma channel walls are
modeled as a trapezoidal function of the radius given by

$n_{p}(r)=\begin{cases}0,&r<a\\\ n_{0}\frac{r-a}{b-a},&a\leq r<b\\\
n_{0},&b\leq r<c\\\ n_{0}\frac{d-r}{d-c},&c\leq r<d\\\ 0,&d\leq r\end{cases}$
(6)

with $n_{0}=3\times 10^{15}$ cm-3, $a=115~{}\upmu\textrm{m}$,
$b=215~{}\upmu\textrm{m}$, $c=280~{}\upmu\textrm{m}$, and
$d=380~{}\upmu\textrm{m}$.

We used the beam parameters measured upstream of the hollow channel plasma
interaction as inputs to QuickPIC to simulate the longitudinal separation
scan. The inputs are shown in Table 2. For all points in the scan, both the
drive and witness beams have transverse size $\sigma_{x}=35~{}\upmu\textrm{m}$
and $\sigma_{y}=25~{}\upmu\textrm{m}$. Each point in the scan is a single-step
simulation using a non-evolving beam. The energy gain/loss of the
drive/witness beams are evaluated at the longitudinal centroid of the beams.
When the beams have a transverse offset from the propagation axis, the field
is sampled at the transverse centroid of the beams.

## References

  * Litos et al. (2014) M. Litos, E. Adli, W. An, C. I. Clarke, C. E. Clayton, S. Corde, J. P. Delahaye, R. J. England, A. S. Fisher, J. Frederico, et al., Nature 515, 92 (2014), URL https://doi.org/10.1038/nature13882.
  * Lindstrøm et al. (2021) C. A. Lindstrøm et al., Phys. Rev. Lett. 126, 014801 (2021), URL https://doi.org/10.1103/PhysRevLett.126.014801.
  * Cao et al. (2023) G. J. Cao, C. A. Lindstrøm, E. Adli, S. Corde, and S. Gessner, _Positron acceleration in plasma wakefields_ (2023), URL https://arxiv.org/abs/2309.10495.
  * Aryshev et al. (2022) A. Aryshev et al., _The international linear collider: Report to snowmass 2021_ (2022), URL https://arxiv.org/abs/2203.07622.
  * Brunner et al. (2022) O. Brunner, P. N. Burrows, S. Calatroni, N. C. Lasheras, R. Corsini, G. D’Auria, S. Doebert, A. Faus-Golfe, A. Grudiev, A. Latina, et al., _The clic project_ (2022), URL https://arxiv.org/abs/2203.09186.
  * Vernieri et al. (2023) C. Vernieri, E. A. Nanni, S. Dasu, M. E. Peskin, T. Barklow, R. Bartoldus, P. C. Bhat, K. Black, J. E. Brau, M. Breidenbach, et al., Journal of Instrumentation 18, P07053 (2023), ISSN 1748-0221, URL http://dx.doi.org/10.1088/1748-0221/18/07/P07053.
  * Lu et al. (2006) W. Lu, C. Huang, M. Zhou, W. B. Mori, and T. Katsouleas, Physical Review Letters 96 (2006), URL https://doi.org/10.1103/physrevlett.96.165002.
  * Hogan et al. (2003) M. J. Hogan, C. E. Clayton, C. Huang, P. Muggli, S. Wang, B. E. Blue, D. Walz, K. A. Marsh, C. L. O’Connell, S. Lee, et al., Physical Review Letters 90 (2003), URL https://doi.org/10.1103/physrevlett.90.205002.
  * Blue et al. (2003) B. E. Blue, C. E. Clayton, C. L. O’Connell, F.-J. Decker, M. J. Hogan, C. Huang, R. Iverson, C. Joshi, T. C. Katsouleas, W. Lu, et al., Physical Review Letters 90 (2003), URL https://doi.org/10.1103/physrevlett.90.214801.
  * Muggli et al. (2008) P. Muggli, B. E. Blue, C. E. Clayton, F. J. Decker, M. J. Hogan, C. Huang, C. Joshi, T. C. Katsouleas, W. Lu, W. B. Mori, et al., Physical Review Letters 101 (2008), URL https://doi.org/10.1103/physrevlett.101.055001.
  * Corde et al. (2015) S. Corde, E. Adli, J. M. Allen, W. An, C. I. Clarke, C. E. Clayton, J. P. Delahaye, J. Frederico, S. Gessner, S. Z. Green, et al., Nature 524, 442 (2015), URL https://doi.org/10.1038/nature14890.
  * Doche et al. (2017) A. Doche, C. Beekman, S. Corde, J. M. Allen, C. I. Clarke, J. Frederico, S. J. Gessner, S. Z. Green, M. J. Hogan, B. O’Shea, et al., Scientific Reports 7 (2017), URL https://doi.org/10.1038/s41598-017-14524-4.
  * Hue et al. (2021) C. S. Hue, G. J. Cao, I. A. Andriyash, A. Knetsch, M. J. Hogan, E. Adli, S. Gessner, and S. Corde, Physical Review Research 3 (2021), URL https://doi.org/10.1103/physrevresearch.3.043063.
  * Chiou et al. (1995) T. C. Chiou, T. Katsouleas, C. Decker, W. B. Mori, J. S. Wurtele, G. Shvets, and J. J. Su, Physics of Plasmas 2, 310 (1995), URL https://doi.org/10.1063/1.871107.
  * Chiou and Katsouleas (1998) T. C. Chiou and T. Katsouleas, Physical Review Letters 81, 3411 (1998), URL https://doi.org/10.1103/physrevlett.81.3411.
  * Schroeder et al. (1999) C. B. Schroeder, D. H. Whittum, and J. S. Wurtele, Physical Review Letters 82, 1177 (1999), URL https://doi.org/10.1103/physrevlett.82.1177.
  * Lindstrøm et al. (2018) C. Lindstrøm, E. Adli, J. Allen, W. An, C. Beekman, C. Clarke, C. Clayton, S. Corde, A. Doche, J. Frederico, et al., Physical Review Letters 120 (2018), URL https://doi.org/10.1103/physrevlett.120.124802.
  * Gessner et al. (2016) S. Gessner, E. Adli, J. M. Allen, W. An, C. I. Clarke, C. E. Clayton, S. Corde, J. P. Delahaye, J. Frederico, S. Z. Green, et al., Nature Communications 7 (2016), URL https://doi.org/10.1038/ncomms11785.
  * Muggli et al. (1999) P. Muggli, K. Marsh, S. Wang, C. Clayton, S. Lee, T. Katsouleas, and C. Joshi, IEEE Transactions on Plasma Science 27, 791 (1999), URL https://doi.org/10.1109/27.774685.
  * Kimura et al. (2011) W. D. Kimura, H. M. Milchberg, P. Muggli, X. Li, and W. B. Mori, Physical Review Special Topics - Accelerators and Beams 14 (2011), URL https://doi.org/10.1103/physrevstab.14.041301.
  * Schröder et al. (2020) S. Schröder, C. A. Lindstrøm, S. Bohlen, G. Boyle, R. D’Arcy, S. Diederichs, M. J. Garland, P. Gonzalez, A. Knetsch, V. Libov, et al., Nature Communications 11 (2020), ISSN 2041-1723, URL http://dx.doi.org/10.1038/s41467-020-19811-9.
  * Hogan et al. (2010) M. J. Hogan, T. O. Raubenheimer, A. Seryi, P. Muggli, T. Katsouleas, C. Huang, W. Lu, W. An, K. A. Marsh, W. B. Mori, et al., New Journal of Physics 12, 055030 (2010), URL https://doi.org/10.1088/1367-2630/12/5/055030.
  * Berden et al. (2007) G. Berden, W. A. Gillespie, S. P. Jamison, E.-A. Knabbe, A. M. MacLeod, A. F. G. van der Meer, P. J. Phillips, H. Schlarb, B. Schmidt, P. Schmüser, et al., Physical Review Letters 99, 164801 (2007), ISSN 0031-9007, URL http://link.aps.org/doi/10.1103/PhysRevLett.99.164801.
  * Andreev et al. (1996) N. E. Andreev, S. S. Bychkov, V. V. Kotlyar, L. Y. Margolin, L. N. Pyatnitskii, and P. G. Serafimovich, Quantum Electronics 26, 126 (1996), URL https://doi.org/10.1070/qe1996v026n02abeh000607.
  * Fan et al. (2000) J. Fan, E. Parra, I. Alexeev, K. Y. Kim, H. M. Milchberg, L. Y. Margolin, and L. N. Pyatnitskii, Physical Review E 62, R7603 (2000), URL https://doi.org/10.1103/physreve.62.r7603.
  * Clayton et al. (2016) C. E. Clayton, E. Adli, J. Allen, W. An, C. I. Clarke, S. Corde, J. Frederico, S. Gessner, S. Z. Green, M. J. Hogan, et al., Nature Communications 7 (2016), URL https://doi.org/10.1038/ncomms12483.
  * Gessner (2016) S. J. Gessner, Ph.D. thesis, Stanford University (2016), URL https://doi.org/10.2172/1340170.
  * Perelomov et al. (1966) A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Soviet Journal of Experimental and Theoretical Physics 23, 924 (1966).
  * Schroeder et al. (2013) C. B. Schroeder, E. Esarey, C. Benedetti, and W. P. Leemans, Physics of Plasmas 20, 080701 (2013), URL https://doi.org/10.1063/1.4817799.
  * Shvets et al. (1996) G. Shvets, J. Wurtele, T. Chiou, and T. Katsouleas, IEEE Transactions on Plasma Science 24, 351 (1996), URL https://doi.org/10.1109/27.509999.
  * Feder et al. (2020) L. Feder, B. Miao, J. E. Shrock, A. Goffin, and H. M. Milchberg, Physical Review Research 2 (2020), URL https://doi.org/10.1103/physrevresearch.2.043173.
  * Yakimenko et al. (2019) V. Yakimenko, L. Alsberg, E. Bong, G. Bouchard, C. Clarke, C. Emma, S. Green, C. Hast, M. Hogan, J. Seabury, et al., Physical Review Accelerators and Beams 22 (2019), URL https://doi.org/10.1103/physrevaccelbeams.22.101301.
  * Zhou et al. (2021) S. Zhou, J. Hua, W. An, W. B. Mori, C. Joshi, J. Gao, and W. Lu, Physical Review Letters 127 (2021), URL https://doi.org/10.1103/physrevlett.127.174801.
  * Clendenin et al. (1988) J. E. Clendenin et al., in _14th International Linear Accelerator Conference_ (1988).
  * An et al. (2013) W. An, V. K. Decyk, W. B. Mori, and T. M. Antonsen, Journal of Computational Physics 250, 165 (2013), URL https://doi.org/10.1016/j.jcp.2013.05.020.


# Federated Learning with Downlink Device Selection

Mohammad Mohammadi Amiri, Sanjeev R. Kulkarni, H. Vincent Poor  Department of
Electrical and Computer Engineering, Princeton University, Princeton, NJ
08544, USA

###### Abstract

We study federated edge learning, where a global model is trained
collaboratively using privacy-sensitive data at the edge of a wireless
network. A parameter server (PS) keeps track of the global model and shares it
with the wireless edge devices for training using their private local data.
The devices then transmit their local model updates, which are used to update
the global model, to the PS. The algorithm, which involves transmission over
PS-to-device and device-to-PS links, continues until the convergence of the
global model or lack of any participating devices. In this study, we consider
device selection based on downlink channels over which the PS shares the
global model with the devices. Performing digital downlink transmission, we
design a partial device participation framework where a subset of the devices
is selected for training at each iteration. Therefore, the participating
devices can have a better estimate of the global model compared to the full
device participation case which is due to the shared nature of the broadcast
channel with the price of updating the global model with respect to a smaller
set of data. At each iteration, the PS broadcasts different quantized global
model updates to different participating devices based on the last global
model estimates available at the devices. We investigate the best number of
participating devices through experimental results for image classification
using the MNIST dataset with biased distribution. 00footnotetext: This work
was supported by the U.S. National Science Foundation under Grants CCF-0939370
and CCF-1908308.

##  I Introduction

With the growing number of Internet of Things (IoT) devices and services, huge
amounts of data are being generated at the wireless edge. We also observe a
rapid increase in the Artificial Intelligence (AI)-driven applications
deploying increasingly complex machine learning (ML) models to gather
intelligence from data. While this is traditionally carried out by a cloud
server, offloading such massive amounts of data at the wireless network edge
to a cloud server is prohibitive in terms of communications cost and may
violate privacy concerns. Instead, a new paradigm, referred to as federated
learning (FL), offers pushing the network intelligence to the edge by
elevating edge computing capabilities in distributed networks. FL allows
distributed devices to train a global ML model collaboratively with the help
of a parameter server (PS) without sharing the raw data [1].

With FL, the PS keeps track of the global model and shares it with the devices
for training using their local data. After training the global model, the
devices transmit their updates to the PS, which uses them to update the global
model. FL comes with its own challenges in practical settings including
communications over unreliable links in both PS-to-device and device-to-PS
directions. Several studies have addressed this challenge through limiting the
communication requirements, particularly in the device-to-PS direction [1, 2,
3]. However, these studies ignore the underlying communication channels and
consider rate-limited bit-pipes as communication links.

Recently a new line of research, namely federated edge learning (FEEL), has
been created considering the physical layer characteristics of the underlying
wireless network in the FL framework [4, 5, 6, 7]. Several studies have
employed over-the-air computation by using the superposition property of the
multiple access channel (MAC) in the device-to-PS direction and have shown its
advantages in guaranteeing reliable communications of the updates from the
devices to the PS [4, 6, 7, 5, 8, 9, 10]. FEEL should be tolerant to the drop
of any devices during the training. This arises from the fact that the devices
share the channel resources, and typically a huge number of devices can be
potentially used for training, in which case each device may have access to
only a limited amount of channel resources for transmission. Also, wireless
devices may stop participating in the training at any time for various reasons
including lack of power, lack of network coverage, and privacy concerns.
Therefore, several studies consider FEEL with partial device participation
sharing the limited resources for various selection techniques [11, 12, 13,
14].

Recent studies have focused on noisy PS-to-device channel, as well as
transmission of a compressed global model to the devices to reduce the
communication footprint [15, 16, 17, 18, 19]. Papers [16, 15] investigated
analog transmission of the global model to the devices with [15] providing
convergence guarantees. Compressing the global model at the PS may lead to a
coarse estimate, which is due to the high average-to-peak ratio of the entries
of global model vector and its high empirical variance [19]. To overcome this
challenge, [17] employs a random linear projection of the global model at the
PS before the compression, [18] performs error compensation before compressing
the global model, and [19] introduces compression of the global model updates
with respect to the last global model estimate available at the devices, which
is shown to outperform the other two approaches.

In this paper, we build upon our approach in [19] and focus on digital
transmission of the global model updates at the PS. We extend the scheme in
[19] to the partial device participation scenario by considering transmission
of different messages to different devices in the PS-to-device direction,
where the devices are selected for participation in each training iteration
based on their downlink channels. Specifically, the PS transmits compressed
global model updates to the participating devices based on the last global
model estimates available at the devices where the rates are set to be within
the capacity region of the underlying wireless fading broadcast channel from
the PS to the devices. The number of participating devices directly impacts
the performance of FEEL, since the devices share the limited channel
resources, and fewer participating devices leads to better estimates of the
information exchanged in the network between different nodes. On the other
hand, fewer participating devices reduces the data usage in each training
iteration. Numerical results show that partial device participation based on
the downlink channels can significantly improve the performance of FEEL.

Notation: We denote the set of real and complex numbers by $\mathbb{R}$ and
$\mathbb{C}$, respectively. We let $[i]\triangleq\\{1,\dots,i\\}$. Vectors of
length $i$ with all zero and one entries are denoted by $\boldsymbol{0}_{i}$
and $\boldsymbol{1}_{i}$, respectively. We denote a circularly symmetric
complex Gaussian distribution with real or imaginary component with variance
$\sigma^{2}/2$ by $\mathcal{CN}\left(0,\sigma^{2}\right)$. We represent the
cardinality of a set or the magnitude of a complex value by
$\left|\cdot\right|$, and the $l_{2}$ norm of a vector $\boldsymbol{x}$ by
$\left\|\boldsymbol{x}\right\|_{2}$. For a set $\cal S$ of real values,
${\rm{indmax}}_{K}\mathcal{S}$ returns a $K$-element set of indices of
$\mathcal{S}$ with the largest values. We also let
$(a)^{+}\triangleq\max\\{a,0\\}$, and for any event $A$, $\mathds{1}(A)=1$, if
event $A$ is true, and $\mathds{1}(A)=0$, otherwise.

##  II System Model

We consider FEEL across $M$ devices, where device $m$ has access to dataset
$\mathcal{B}_{m}$ of size $B_{m}=\left|\mathcal{B}_{m}\right|$. The devices
use their local datasets to train a global model
$\boldsymbol{\theta}\in\mathbb{R}^{d}$ with the help of a parameter server
(PS) in order to minimize loss function
$F\left(\boldsymbol{\theta}\right)=\sum\nolimits_{m=1}^{M}\frac{B_{m}}{B}F_{m}(\theta)$,
where $B\triangleq\sum\nolimits_{m=1}^{M}B_{m}$, and
$F_{m}\left(\boldsymbol{\theta}\right)=\frac{1}{B_{m}}\sum\nolimits_{\boldsymbol{u}\in\mathcal{B}_{m}}f\left(\boldsymbol{\theta},\boldsymbol{u}\right)$
is the loss function at device $m$ with $f$ denoting an empirical loss
function.

###  II-A FEEL System

Minimization of loss function $F$ is carried out iteratively using the local
datasets at the devices with the PS having access to the global model. At each
iteration, the global model is shared with the devices and updated locally
using the data at the devices. The devices then transmit their local model
updates to the PS responsible for updating the global model. The algorithm
continues until the convergence of the global model, or lack of participating
devices for training.

The downlink (PS-to-device) and uplink (device-to-PS) connections are
wireless, and a noisy transmission over wireless medium takes place during
each downlink and uplink communication round. During iteration $t$, the PS
shares the global model $\boldsymbol{\theta}(t)$ with the devices, and they
perform a $\tau$-step stochastic gradient descent (SGD) to update the global
model using their local data. Let $\widehat{\boldsymbol{\theta}}_{m}(t)$
denote the noisy estimate of the global model $\boldsymbol{\theta}(t)$ at
device $m$, $m\in[M]$. For the $i$-th step SGD algorithm, device $m$ selects a
random subset of its local data, denoted by $\boldsymbol{\xi}_{m}^{i}(t)$, and
updates the model as follows:

$\displaystyle\boldsymbol{\theta}_{m}^{i+1}(t)=\boldsymbol{\theta}_{m}^{i}(t)-\eta^{i}_{m}(t)\nabla
F_{m}\left(\boldsymbol{\theta}_{m}^{i}(t),\boldsymbol{\xi}_{m}^{i}(t)\right),\,\mbox{$i\in[\tau]$},$
(1)

where $\boldsymbol{\theta}_{m}^{1}(t)=\widehat{\boldsymbol{\theta}}_{m}(t)$,
$\eta^{i}_{m}(t)$ denotes the learning rate, and $\nabla
F_{m}\left(\boldsymbol{\theta}_{m}^{i}(t),\boldsymbol{\xi}_{m}^{i}(t)\right)$
is the stochastic gradient which provides an unbiased estimate of the true
gradient $\nabla F_{m}\left(\boldsymbol{\theta}_{m}^{i}(t)\right)$. Device $m$
sends its model update
$\Delta\boldsymbol{\theta}_{m}(t)=\boldsymbol{\theta}_{m}^{\tau+1}(t)-\boldsymbol{\theta}_{m}^{1}(t)$
to the PS, computed after performing the $\tau$-step SGD, where we denote the
estimate of $\Delta\boldsymbol{\theta}_{m}(t)$ at the PS by
$\Delta\widehat{\boldsymbol{\theta}}_{m}(t)$. The PS then updates the global
model as follows:

$\displaystyle\boldsymbol{\theta}(t+1)=\boldsymbol{\theta}(t)+\frac{1}{M}\sum\nolimits_{m=1}^{M}\Delta\widehat{\boldsymbol{\theta}}_{m}\left(t\right).$
(2)

FEEL needs to address the partial device participation case, when only a
subset of the devices participate in the training at each iteration. First, we
need to consider the fact that some of the wireless devices might stop
participating in the training for various reasons. Furthermore, due to the
limited wireless resources shared among the participating devices, the
information that can be conveyed using each link may be limited after
allocating the channel resources. In this paper, we adopt partial device
participation, where at iteration $t$ only a subset of devices
$\mathcal{K}(t)\subset[M]$ with size $K=\left|\mathcal{K}(t)\right|$ is chosen
by the PS to participate in the training. Accordingly, the PS updates the
global model after receiving the local model updates from the participating
devices as

$\displaystyle\boldsymbol{\theta}(t+1)=\boldsymbol{\theta}(t)+\frac{1}{K}\sum\nolimits_{k\in\mathcal{K}(t)}\Delta\widehat{\boldsymbol{\theta}}_{k}\left(t\right).$
(3)

###  II-B Wireless Medium

Each iteration includes two communication phases over downlink, where the PS
shares the global model with the devices, and uplink, where the participating
devices transmit their local updates to the PS. Here we model the downlink and
uplink communication channels.

Downlink channel. We assume that the downlink is a parallel fading broadcast
channel with $s^{\rm{dl}}$ sub-channels. We denote the channel gains vector
from the PS to device $m$ at iteration $t$ by
$\boldsymbol{h}^{\rm{dl}}_{m}(t)=[{h}^{\rm{dl}}_{m,1}(t),...,{h}^{\rm{dl}}_{m,{s^{\rm{dl}}}}(t)]\in\mathbb{C}^{s^{\rm{dl}}}$
with each entry independent and identically distributed (iid) according to
$\mathcal{CN}(0,(\sigma^{\rm{dl}})^{2})$, and the additive noise vector at
device $m$ by $\boldsymbol{z}^{\rm{dl}}_{m}(t)\in\mathbb{C}^{s^{\rm{dl}}}$
with each entry iid according to $\mathcal{CN}(0,1)$. We assume that the PS
has channel state information (CSI) about the downlink, and each device has
the CSI of its link, and we impose an average power constraint $P^{\rm{dl}}$
at the PS.

Uplink channel. We model the uplink channel with a parallel fading multiple
access channel (MAC) with $s^{\rm{ul}}$ sub-channels. The channel gains vector
from device $m$ to the PS at iteration $t$ is denoted by
$\boldsymbol{h}^{\rm{ul}}_{m}(t)=[{h}^{\rm{ul}}_{m,1}(t),...,{h}^{\rm{ul}}_{m,{s^{\rm{ul}}}}(t)]\in\mathbb{C}^{s^{\rm{ul}}}$
with each entry iid according to $\mathcal{CN}(0,(\sigma^{\rm{ul}})^{2})$. We
also denote the additive noise vector at the PS by
$\boldsymbol{z}^{\rm{ul}}_{m}(t)\in\mathbb{C}^{s^{\rm{ul}}}$ with each entry
iid according to $\mathcal{CN}(0,1)$. The channel input at each device is
limited to an average transmit power constraint $P^{\rm{ul}}$, and the PS has
full CSI, while each device knows its CSI.

##  III Communications Protocol

We consider digital communications of the information from the PS and the
devices, where each node compresses its message and transmits it at a rate
below its channel capacity. We first present the channel capacities of the
downlink and uplink, and then introduce the quantization technique used for
compression. For the ease of presentation, we drop the dependency of the
channels on the iteration count $t$.

###  III-A Downlink Channel Capacity

Recall that the downlink is a parallel fading broadcast channel with
$s^{\rm{dl}}$ sub-channels. Let us consider serving an arbitrary subset of the
devices $\mathcal{K}\subset[M]$ of size $K$. First, we focus on a fading
broadcast channel with a single sub-channel, i.e., $s^{\rm{dl}}=1$, where the
PS serves the $K$ devices. We denote the channel gain from the server to
device $k$ by ${h}^{\rm{dl}}_{k}$ and the channel gains vector by
$\boldsymbol{h}^{\rm{dl}}$. This yields a degraded broadcast channel with a
well known capacity [20]. Without loss of generality, let $\mathcal{K}=[K]$,
and
$|{h}^{\rm{dl}}_{1}|\leq|{h}^{\rm{dl}}_{2}|\leq\cdots\leq|{h}^{\rm{dl}}_{K}|$.
For some non-negative values $\alpha_{k}$, $\forall k\in\mathcal{K}$, such
that $\sum\nolimits_{k=1}^{K}\alpha_{k}=1$, the boundary of the capacity
region
$\mathcal{C}^{\rm{dl}}_{\mathcal{K},b}(\boldsymbol{h}^{\rm{dl}},P^{\rm{dl}})$
is given by

$\displaystyle\mathcal{C}^{\rm{dl}}_{\mathcal{K},b}(\boldsymbol{h}^{\rm{dl}},P^{\rm{dl}})=\Big{\\{}[C_{1},...,C_{K}]^{T}:$
$\displaystyle\;C_{k}=\log_{2}\Big{(}1+\frac{\alpha_{k}\left|{h}^{\rm{dl}}_{k}\right|^{2}P^{\rm{dl}}}{1+\sum\nolimits_{j=k+1}^{K}\alpha_{j}\left|{h}^{\rm{dl}}_{j}\right|^{2}P^{\rm{dl}}}\Big{)},\;k\in\mathcal{K}\Big{\\}}.$
(4)

Accordingly, the capacity region of the parallel fading broadcast channel is
given by [21]

$\displaystyle\mathcal{C}^{\rm{dl}}_{\mathcal{K}}(P^{\rm{dl}})=\bigcup\limits_{\\{P^{\rm{dl}}_{1},...,P^{\rm{dl}}_{s^{\rm{dl}}}\\}:\sum\limits_{i=1}^{s^{\rm{dl}}}P^{\rm{dl}}_{i}=P^{\rm{dl}}}\sum\limits_{i=1}^{s^{\rm{dl}}}\mathcal{C}^{\rm{dl}}_{\mathcal{K},b}(\boldsymbol{h}^{\rm{dl}}_{i},P^{\rm{dl}}_{i}),$
(5)

where
$\boldsymbol{h}^{\rm{dl}}_{i}\triangleq[{h}^{\rm{dl}}_{1,i},...,{h}^{\rm{dl}}_{K,i}]$,
for $i\in[s^{\rm{dl}}]$. The optimal rate and power allocation across sub-
channels to maximize the sum-capacity is given by [21]

$\displaystyle
C^{\rm{dl}}_{k,i}=\begin{cases}\log_{2}\Big{(}1+\Big{(}\frac{|h_{k,i}^{\rm{dl}}|^{2}}{\lambda}-1\Big{)}^{+}\Big{)},&\mbox{if
$|h_{k,i}^{\rm{dl}}|=\max\limits_{k^{\prime}\in\mathcal{K}}|h_{k^{\prime},i}^{\rm{dl}}|$},\\\
0,&\mbox{otherwise},\end{cases}$ (6a) $\displaystyle
P^{\rm{dl}}_{i}=\Big{(}\frac{1}{\lambda}-\frac{1}{\max\limits_{k\in\mathcal{K}}|h_{k,i}^{\rm{dl}}|^{2}}\Big{)}^{+},$
(6b)

where $\lambda$ is chosen to satisfy the power constraint
$\sum\nolimits_{i=1}^{s^{\rm{dl}}}{P}_{i}^{\rm{dl}}={P}^{\rm{dl}}$, and the
capacity of the channel to device $k$ is given by
$C_{\mathcal{K},k}^{\rm{dl}}=\sum\nolimits_{i=1}^{s^{\rm{dl}}}C^{\rm{dl}}_{k,i}$,
$k\in\mathcal{K}$. The optimal solution provided above dictates that in each
sub-channel the information for no more than a single user, the one with the
best channel quality in that sub-channel, is broadcast. Also, the power
allocation across different sub-channels follows a water-filling solution.

###  III-B Uplink Channel Capacity

The uplink is modelled by a parallel fading MAC with $s^{\rm{ul}}$ sub-
channels. We first consider $s^{\rm{ul}}=1$, where $K$ devices in
$\mathcal{K}$ share the wireless medium to communicate with the PS. We denote
the channel gains vector by
$\boldsymbol{h}^{\rm{ul}}=[{h}^{\rm{ul}}_{1},...,{h}^{\rm{ul}}_{K}]$, where
${h}^{\rm{ul}}_{k}$ is the channel gain from device $k$ to the PS. For an
average power $P^{\rm{ul}}_{k}$ at device $k$ and
$\boldsymbol{P}^{\rm{ul}}\triangleq[P_{1}^{\rm{ul}},...,P_{K}^{\rm{ul}}]$, the
capacity region of the above fading MAC is given by

$\displaystyle\mathcal{C}^{\rm{ul}}_{\mathcal{K},b}(\boldsymbol{h}^{\rm{ul}},\boldsymbol{P}^{\rm{ul}})=\Big{\\{}[C_{1},...,C_{K}]^{T}:$
$\displaystyle\sum\nolimits_{k\in\mathcal{S}}C_{k}\leq\log_{2}\Big{(}1+\sum\nolimits_{k\in\mathcal{S}}\left|{h}^{\rm{ul}}_{k}\right|^{2}P^{\rm{ul}}_{k}\Big{)},\;\forall\mathcal{S}\subset\mathcal{K}\Big{\\}}.$
(7)

The capacity region of the parallel fading MAC is then given by [22]

$\displaystyle\mathcal{C}^{\rm{ul}}_{\mathcal{K}}(P^{\rm{ul}})=\bigcup\limits_{\quad\scriptstyle\\{P_{k,1}^{\rm{ul}},...,P_{k,{s^{\rm{ul}}}}^{\rm{ul}}\\}:\hfill\atop\scriptstyle\sum\nolimits_{i=1}^{{s^{\rm{ul}}}}{P_{k,i}^{\rm{ul}}}={P^{\rm{ul}}},\forall
k\in\mathcal{K}\hfill}{\sum\limits_{i=1}^{{s^{\rm{ul}}}}}\mathcal{C}^{\rm{ul}}_{\mathcal{K},b}(\boldsymbol{h}^{\rm{ul}}_{i},\boldsymbol{P}_{i}^{\rm{ul}}),$
(8)

where
$\boldsymbol{h}^{\rm{ul}}_{i}\triangleq[{h}^{\rm{ul}}_{1,i},...,{h}^{\rm{ul}}_{K,i}]$
and
$\boldsymbol{P}^{\rm{ul}}_{i}\triangleq[{P}^{\rm{ul}}_{1,i},...,{P}^{\rm{ul}}_{K,i}]$,
$i\in[s^{\rm{ul}}]$. Assuming a total power constraint
$\sum\nolimits_{k=1}^{K}\sum\nolimits_{i=1}^{s^{\rm{ul}}}P_{k,i}^{\rm{ul}}=KP^{\rm{ul}}$,
the optimal power allocation across sub-channels to maximize the sum-capacity
is given by [22]

$\displaystyle
P_{k,i}^{\rm{ul}}=\begin{cases}\Big{(}\frac{1}{\lambda}-\frac{1}{\left|h_{k,i}^{\rm{ul}}\right|^{2}}\Big{)}^{+},&\mbox{if
$|h_{k,i}^{\rm{ul}}|=\max\limits_{k^{\prime}\in\mathcal{K}}|h_{k^{\prime},i}^{\rm{ul}}|$},\\\
0,&\mbox{otherwise},\end{cases}$ (9)

where $\lambda$ is chosen to satisfy
$\sum\nolimits_{k=1}^{K}\sum\nolimits_{i=1}^{s^{\rm{ul}}}P_{k,i}^{\rm{ul}}=KP^{\rm{ul}}$.
Accordingly, the sum capacity is given by

$\displaystyle
C^{\rm{ul}}_{\mathcal{K},\rm{sum}}=\sum\nolimits_{i=1}^{s^{\rm{ul}}}\log_{2}\Big{(}1+\Big{(}\frac{\max\nolimits_{k\in\mathcal{K}}|h_{k,i}^{\rm{ul}}|^{2}}{\lambda}-1\Big{)}^{+}\Big{)},$
(10)

and for a symmetric rate allocation, the capacity of each device is given by
$C_{\mathcal{K},k}^{\rm{ul}}=C^{\rm{ul}}_{\mathcal{K},\rm{sum}}/K$, $\forall
k\in\mathcal{K}$. Similar to the downlink transmission, with the uplink
transmission, at each sub-channel the power is allocated to only a single
device with the best channel condition in that sub-channel. We note that,
besides its CSI, each device needs to know the indices of the sub-channels
that it has the best channel conditions among all the devices (the sub-
channels that it uses for transmission), and we assume that this information
is provided by the PS.

######  Remark 1.

For the above capacity analysis, we have assumed slow fading and coding across
several blocks with invariant channel gains. This provides an upper bound on
the capacity of the downlink and uplink channels presented in Section II-B.

###  III-C Quantization Technique

We follow the stochastic quantization technique presented in [19]. Let
$\boldsymbol{x}\in\mathbb{R}^{d}$ be a vector with the $i$-th component
denoted as $x_{i}$. We define
$x_{\rm{max}}\triangleq\max\\{|\boldsymbol{x}|\\}$ and
$x_{\rm{min}}\triangleq\min\\{|\boldsymbol{x}|\\}$. For a quantization level
$q\geq 1$, we define, for $i\in[d]$,

$\displaystyle{Q}\left({x}_{i},q\right)\triangleq{\rm{sign}}\left({x}_{i}\right)\cdot\Big{(}x_{\rm{min}}$
$\displaystyle\qquad\qquad\quad+\left(x_{\rm{max}}-x_{\rm{min}}\right)\cdot\varphi\Big{(}\frac{\left|{x}_{i}\right|-x_{\rm{min}}}{x_{\rm{max}}-x_{\rm{min}}},q\Big{)}\Big{)},$
(11)

where $\varphi$ is a stochastic quantization function defined next. Let
$x\in\mathbb{R}$ such that $0\leq x\leq 1$ and $q\geq 1$, and
$l\in\\{0,1,...,q-1\\}$ be an integer such that $x\in[l/q,(l+1)/q)$. We define

$\displaystyle\varphi\left(x,q\right)\triangleq\begin{cases}l/q,&\mbox{with
probability $1-xq+l$},\\\ (l+1)/q,&\mbox{with probability $xq-l$}.\end{cases}$
(12)

We further define the vector quantization as
$\boldsymbol{Q}(\boldsymbol{x},q)\triangleq[Q({x}_{1},q),\cdots,Q({x}_{d},q)]^{T}$.
With this quantization approach, each vector $\boldsymbol{x}$ is represented
by a total number of

$\displaystyle R_{q}=64+d\left(1+\log_{2}(q+1)\right)\mbox{ bits}.$ (13)

##  IV Downlink Device Selection Technique

Here we present our FEEL approach with device selection in the downlink. We
extend our work in [19] by allowing the PS to broadcast different messages to
different devices that enables partial device participation at each training
iteration. At each iteration, the PS selects a subset of the devices and
broadcasts different versions of the global model to the selected devices with
a rate vector within the capacity region $\mathcal{C}^{\rm{dl}}(P^{\rm{dl}})$.
The global model estimate at device $m$ at iteration $t$ is denoted by
$\widehat{\boldsymbol{\theta}}_{m}(t)$, and if $m\in\mathcal{K}(t)$, i.e.,
device $m$ is selected to participate in the training at iteration $t$, it
performs the $\tau$-step SGD algorithm. It then quantizes the local model
update $\Delta{\boldsymbol{\theta}}_{m}\left(t\right)$, along with error
compensation, and transmits it to the PS with a rate no larger than
$C^{\rm{ul}}_{m}$, $m\in[M]$. In the following, we elaborate the downlink and
uplink transmission.

###  IV-A Downlink Transmission

At iteration $t$, the PS first selects $K$ devices with the largest
$\sum\nolimits_{i=1}^{s^{\rm{dl}}}\big{\|}\boldsymbol{h}^{\rm{dl}}_{m}(t)\big{\|}^{2}_{2}$,
$\forall m\in[M]$; that is,

$\displaystyle\mathcal{K}(t)={\rm{indmax}}_{K}\Big{\\{}\sum\nolimits_{i=1}^{s^{\rm{dl}}}\big{\|}\boldsymbol{h}^{\rm{dl}}_{1}(t)\big{\|}^{2}_{2},...,\sum\nolimits_{i=1}^{s^{\rm{dl}}}\big{\|}\boldsymbol{h}^{\rm{dl}}_{M}(t)\big{\|}^{2}_{2}\Big{\\}}.$
(14)

The PS aims to transmit
$\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}(t_{k})$ to device
$k\in\mathcal{K}(t)$ over the downlink, where $t_{k}$ denotes the iteration
index during which device $k$ was selected last time, and it is initialized
with $t_{k}=0$. We will show that, with the proposed approach, the PS knows
about $\widehat{\boldsymbol{\theta}}(t_{k})$ as side information and can
remove this information from the current global model
$\boldsymbol{\theta}(t)$, $\forall k\in\mathcal{K}(t)$.

For a set of selected devices $\mathcal{K}(t)$ with the downlink capacity
region $\mathcal{C}^{\rm{dl}}_{\mathcal{K}(t)}(P^{\rm{dl}})$, the PS quantizes
$\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}(t_{k})$ to a limited
rate and broadcasts it to device $k$, $\forall k\in\mathcal{K}(t)$, such that
the resultant rate vector is within
$\mathcal{C}^{\rm{dl}}_{\mathcal{K}(t)}(P^{\rm{dl}})$. Accordingly, the PS
broadcasts
$\boldsymbol{Q}(\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}_{k}(t_{k}),q^{\rm{dl}}_{k}(t))$
to device $k$, where $q^{\rm{dl}}_{k}(t)$ is chosen as the largest integer
satisfying $R_{q^{\rm{dl}}_{k}(t)}\leq C_{\mathcal{K}(t),k}^{\rm{dl}}$,
$\forall k\in\mathcal{K}(t)$. Therefore, the rate vector
$[R_{q^{\rm{dl}}_{1}(t)},...,R_{q^{\rm{dl}}_{K}(t)}]^{T}$ is within
$\mathcal{C}^{\rm{dl}}_{\mathcal{K}(t)}(P^{\rm{dl}})$, and device $k$ can
successfully decode
$\boldsymbol{Q}(\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}_{k}(t_{k}),q_{k}(t))$,
$\forall k\in\mathcal{K}(t)$. After receiving
$\boldsymbol{Q}(\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}_{k}(t_{k}),q^{\rm{dl}}_{k}(t))$,
device $k$ obtains an estimate of global model $\boldsymbol{\theta}(t)$
through, for $k\in\mathcal{K}(t)$,

$\displaystyle\widehat{\boldsymbol{\theta}}_{k}(t)=\widehat{\boldsymbol{\theta}}_{k}(t_{k})+\boldsymbol{Q}(\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}_{k}(t_{k}),q^{\rm{dl}}_{k}(t)),$
(15)

which is equivalent to
$\widehat{\boldsymbol{\theta}}_{k}(t)={\boldsymbol{\theta}}(0)+\sum\nolimits_{i=1}^{t}\mathds{1}(\mathcal{K}(i))\boldsymbol{Q}\big{(}\boldsymbol{\theta}(i)-\widehat{\boldsymbol{\theta}}_{k}(i_{k}),q^{\rm{dl}}_{k}(i)\big{)}$,
where we assumed that
$\widehat{\boldsymbol{\theta}}_{k}(0)=\boldsymbol{\theta}(0)$, $\forall k$,
and for any time index $i$, we denote the last time device $k$ was selected by
$i_{k}$. Accordingly, the PS knows about
$\widehat{\boldsymbol{\theta}}_{k}(t)$ since it has knowledge about
$\boldsymbol{Q}(\boldsymbol{\theta}(t)-\widehat{\boldsymbol{\theta}}_{k}(t_{k}),q^{\rm{dl}}_{k}(t))$,
$\forall k,t$.

###  IV-B Uplink Transmission

After computing $\widehat{\boldsymbol{\theta}}_{k}(t)$, device
$k\in\mathcal{K}(t)$ performs the $\tau$-step SGD algorithm as given in (1)
and obtains $\Delta\boldsymbol{\theta}_{k}(t)$, which is quantized and
transmitted over the MAC through error compensation. Specifically, device
$k\in\mathcal{K}(t)$ transmits
$\boldsymbol{Q}\big{(}\Delta\boldsymbol{\theta}_{k}(t)+\boldsymbol{\delta}_{k}(t),q^{\rm{ul}}_{k}(t)\big{)}$,
where $q^{\rm{ul}}_{k}(t)$ is chosen as the largest integer satisfying
$R_{q^{\rm{ul}}_{k}(t)}\leq C_{\mathcal{K}(t),k}^{\rm{ul}}$. The error
accumulation vector $\boldsymbol{\delta}_{k}(t)$ is then updated as

$\displaystyle\boldsymbol{\delta}_{k}(t+1)=\Delta\boldsymbol{\theta}_{k}(t)+\boldsymbol{\delta}_{k}(t)-\boldsymbol{Q}\big{(}\Delta\boldsymbol{\theta}_{k}(t)+\boldsymbol{\delta}_{k}(t),q^{\rm{ul}}_{k}(t)\big{)},$
(16)

where $\boldsymbol{\delta}_{k}(0)=\boldsymbol{0}_{d}$. After receiving
$\boldsymbol{Q}\big{(}\Delta\boldsymbol{\theta}_{k}(t)+\boldsymbol{\delta}_{k}(t),q^{\rm{ul}}_{k}(t)\big{)}$,
$\forall k\in\mathcal{K}(t)$, the PS updates the global model as

$\displaystyle\boldsymbol{\theta}(t+1)=\sum\nolimits_{m=1}^{M}\frac{B_{m}}{B}\big{(}\boldsymbol{Q}\big{(}\Delta\boldsymbol{\theta}_{k}(t)+\boldsymbol{\delta}_{k}(t),q^{\rm{ul}}_{k}(t)\big{)}+\widehat{\boldsymbol{\theta}}_{k}(t)\big{)}.$
(17)

######  Remark 2.

By selecting a larger number of devices, each device obtains a coarser
estimate of the global model from the PS and sends a less accurate estimate of
its local model update to the PS. This is due to the resource sharing nature
of digital transmission over both downlink and uplink. On the other hand,
having a larger number of devices participating in the training, the global
model is updated using a larger portion of the entire dataset. Therefore, we
need to optimize the number of participating devices to obtain the best
performance.

######  Remark 3.

In [19], the same version of the global model update with respect to the last
model estimate available at the devices is broadcast to all the devices. This
requires that all the devices know about the last global model estimate in
order to recover an estimate of the global model. We have relaxed this
requirement by allowing the devices to receive different versions of the
global model updates with respect to the last global model estimate available
at each device which enables partial device participation.

##  V Numerical Experiments

We evaluate the performance of the proposed algorithm and compare it with the
approach in [19] for image classification of the MNIST dataset [23] using ADAM
optimizer [24]. We train a convolutional neural network with 5 layers
including two $5\times 5$ convolutional layers with ReLU activation and same
padding and 32 and 64 channels, respectively, each followed by a $2\times 2$
max pooling layer, and a softmax output layer.

We assume non-iid data distribution across the devices where each device has
access to the samples from only a single class. We first split the data
samples at each class to $M/10$ disjoint groups (assuming that $M$ is
divisible by 10), and allocate each group to a unique device. We measure the
performance as the accuracy with respect to the test samples, referred to as
the test accuracy, versus the iteration count $t$.

We consider $M=100$ devices, $s^{\rm{dl}}=10^{7}$ and $s^{\rm{ul}}=5\times
10^{6}$ downlink and uplink sub-channels, respectively, and channel variances
$(\sigma^{\rm{dl}})^{2}=(\sigma^{\rm{ul}})^{2}=10$. We set the power
constraints for downlink and uplink to relatively high values
$P^{\rm{dl}}=10^{5}$ and $P^{\rm{ul}}=10^{3}$, respectively, which is to make
sure that in the case of full device participation, i.e., when $K=M$, the PS
and the devices can transmit at least one bit of information.

Figure 1: Performance comparison of the proposed scheme with different $K$
values and the approach in [19] for non-iid data.

In Fig. 1, we investigate the impact of the number of selected devices on the
performance for $K\in\\{5,10,40,100\\}$. We observe that selecting all the
devices leads to unstable accuracy performance and low convergence rate and
accuracy level. This is due to the fact that the information in both downlink
and uplink are communicated less accurately when all the devices participate
in the training. On the other hand, when a small subset of the devices is
selected, i.e., $K=5$, the performance suffers from lack of data usage for
updating the global model at each iteration, which in particular deteriorates
the performance in the case of non-iid data. As a result, we observe that
selecting a moderate number of devices $K=40$ leads to a relatively good
performance in terms of final accuracy level, convergence rate, and stability.
We have further included the performance of the scheme introduced in [19],
where all the devices are selected at each iteration, and the quantized global
model update is broadcast to all the devices with the same rate accommodating
the worst device’s channel capacity. We observe that, in the full device
participation case, sending the global model update with different rates to
different devices based on their channel qualities, as performed by the
proposed scheme, can slightly improve the performance.

##  VI Conclusions

We have studied FEEL with noisy bandwidth-limited downlink and uplink channels
and investigated the advantages of device selection based on downlink channel
conditions. For the downlink, we have adopted digital transmission of the
global model updates from the PS to the devices, where the PS broadcasts
different messages to different devices. Accordingly, we have designed a FEEL
framework with partial device participation based on the downlink channel
conditions. The PS broadcasts different global model updates to different
participating devices according to the last global model estimate available at
the devices. The participating devices update the global model estimate
according to their local data and send the quantized local model updates to
the PS, which updates the global model using the local model updates received
from the devices. Partial device participation enhances the resources
allocated to the underlying links and leads to a more accurate information
exchange over both the downlink and uplink. On the other hand, the data usage
reduces when fewer devices participate in the training. Numerical experiments
have shown that partial device participation based on the downlink channel
improves the performance while the full device participation case suffers
significantly from inaccurate information exchange.

## References

  * [1] J. Konecny, H. B. McMahan, F. X. Yu, P. Richtarik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” in _Proc. Adv. Neural Inf. Proc. Syst._ , Barcelona, Spain, 2016.
  * [2] H. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in _Proc. AISTATS_ , 2017.
  * [3] J. Konecny, B. McMahan, and D. Ramage, “Federated optimization: Distributed optimization beyond the datacenter,” [online]. Available: https://arxiv.org/pdf/1511.03575.pdf, Nov. 2015.
  * [4] M. M. Amiri and D. Gündüz, “Machine learning at the wireless edge: Distributed stochastic gradient descent over-the-air,” _IEEE Trans. Signal Process._ , vol. 68, pp. 2155–2169, Apr. 2020.
  * [5] ——, “Federated learning over wireless fading channels,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 5, pp. 3546–3557, May 2020.
  * [6] G. Zhu, Y. Wang, and K. Huang, “Broadband analog aggregation for low-latency federated edge learning,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 491–506, Jan. 2020.
  * [7] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via over-the-air computation,” _IEEE Trans. on Wireless Commun._ , vol. 19, no. 3, pp. 2022–2035, Mar. 2020.
  * [8] M. M. Amiri, T. M. Duman, D. Gündüz, S. R. Kulkarni, and H. V. Poor, “Blind federated edge learning,” _IEEE Trans. on Wireless Commun._ , (Early Access), Mar. 2021.
  * [9] T. Sery and K. Cohen, “On analog gradient descent learning over multiple access fading channels,” _IEEE Trans. Signal Process._ , vol. 68, pp. 2897–2911, Apr. 2020.
  * [10] W.-T. Chang and R. Tandon, “Communication efficient federated learning over multiple access channels,” [online]. Available: https://arxiv.org/pdf/2001.08737.pdf, Jan. 2020.
  * [11] H. H. Yang, A. Arafa, T. Q. S. Quek, and H. V. Poor, “Age-based scheduling policy for federated learning in mobile edge networks,” in _Proc. IEEE ICASSP_ , Singapore, Sep. 2014, pp. 1058–1062.
  * [12] Y. Sun, S. Zhou, and D. Gündüz, “Energy-aware analog aggregation for federated learning with redundant data,” in _Proc. IEEE ICC_ , Dublin, Ireland, Jun. 2020, pp. 1–7.
  * [13] H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor, “Scheduling policies for federated learning in wireless networks,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 317–333, Jan. 2020.
  * [14] M. M. Amiri, D. Gündüz, S. R. Kulkarni, and H. V. Poor, “Convergence of update aware device scheduling for federated learning at the wireless edge,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 6, pp. 3643–3658, Jun. 2021\.
  * [15] ——, “Convergence of federated learning over a noisy downlink,” [online]. Available: https://arxiv.org/pdf/2008.11141.pdf, Aug. 2020.
  * [16] J.-H. Ahn, O. Simeone, and J. Kang, “Cooperative learning via federated distillation over fading channels,” in _Proc. IEEE ICASSP_ , Barcelona, Spain, May 2020, pp. 8856–8860.
  * [17] S. Caldas, J. Konecny, H. B. McMahan, and A. Talwalkar, “Expanding the reach of federated learning by reducing client resource requirements,” [online]. Available: https://arxiv.org/pdf/1812.07210.pdf.
  * [18] H. Tang, X. Lian, C. Yu, T. Zhang, and J. Liu, “DoubleSqueeze: Parallel stochastic gradient descent with double-pass error-compensated compression,” in _Proc. PMLR_ , Long Beach, CA, 2019.
  * [19] M. M. Amiri, D. Gündüz, S. R. Kulkarni, and H. V. Poor, “Federated learning with quantized global model updates,” [online]. Available: https://arxiv.org/pdf/2006.10672.pdf, Jun. 2020.
  * [20] P. Bergmans, “Random coding theorem for broadcast channels with degraded components,” _IEEE Trans. Inform. Theory_ , vol. 19, no. 2, pp. 197–207, Mar. 1973.
  * [21] D. N. Tse, “Optimal power allocation over parallel Gaussian broadcast channels,” in _Proc. IEEE ISIT_ , Ulm, Germany, Jun. 1997, p. 27.
  * [22] D. N. Tse and S. V. Hanly, “Multiaccess fading channels–part i: Polymatroid structure, optimal resource allocation and throughput capacities,” _IEEE Trans. Inform. Theory_ , vol. 44, no. 7, pp. 2796–2815.
  * [23] Y. LeCun, C. Cortes, and C. Burges, “The MNIST database of handwritten digits,” Available: http://yann.lecun.com/exdb/mnist/, 1998.
  * [24] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” [online]. Available: https://arxiv.org/pdf/1412.6980.pdf, Jan. 2017.