# Simultaneous broadband radio and optical emission of meteor trains imaged by
LOFAR / AARTFAAC and CAMS

Tammo Jan Dijkema1, Cees Bassa1, Mark Kuiack2, Peter Jenniskens3,
Carl Johannink3, Felix Bettonvil4, Ralph Wijers2, Richard Fallows1

(1ASTRON, Netherlands Institute for Radio Astronomy
2Anton Pannekoek Institute, University of Amsterdam
3SETI Institute
4Leiden University / NOVA
November 18, 2021)

###### Abstract

We report on simultaneous 30 - 60 MHz LOFAR / AARTFAAC12 radio observations
and CAMS low-light video observations of +4 to -10 magnitude meteors at the
peak of the Perseid meteor shower on August 12/13, 2020. 204 meteor trains
were imaged in both the radio and optical domain. Aside from scattered
artificial radio sources, we identify broadband radio emission from many
persistent trains, one of which lingered for up to 6 minutes. Unexpectedly,
fewer broadband radio meteor trains were recorded when the experiment was
repeated during the 2020 Geminids and 2021 Quadrantids. Intrinsic broadband
radio emission was reported earlier by the Long Wavelength Array, but for much
brighter meteors and observed with lower spatial resolution. The new results
offer insight into the unknown radio emission mechanism.

##  1 Introduction

Meteors are well-known to reflect artificial radio emission in forward and
backward scattering. Combined optical and radio meteor scatter observation of
meteors go back to (Prentice et al., 1947), showing that meteor trains are
overdense for most visible meteors. Forward meteor scatter, where transmitter
and receiver are not at the same location, can be used to study meteors as is
done in, e.g., the Belgian RAdio Meteor Stations (BRAMS) network (Lamy et al.,
2011).

Less well established is the detection of intrinsic natural radio emission
from the meteor or meteor train itself. Early efforts to detect intrinsic
emisison relied on temporal coincidences (e.g., (Price & Blum, 2000)). The
first more substantive reports based on radio imaging came from all-sky
imaging at low radio frequencies ($\sim$40 MHz) with the Long Wavelength
Array, detecting intrinsic non-thermal radio emission from fireballs
(Obenberger et al., 2014, 2016a, 2016b). The emission persisted well after the
meteor itself had faded. The emission mechanism of these meteor radio
afterglows is not fully understood, and so far has not been independently
confirmed with other instruments. A survey with the Murchison Widefield Array
(Zhang et al., 2018) did not show this intrinsic emission at higher 72–103 MHz
frequencies.

Several mechanisms for broadband radio emission from meteors have been
suggested in the literature, namely reflected broadband terrestrial radio
emission, intrinsic emission from chemically produced suprathermal electrons
(Obenberger et al., 2020), Langmuir waves (Obenberger et al., 2015), free-free
emission (Filonenko, 2018) and transition radiation (Obenberger et al., 2020).
Also, bright celestial radio emission shines on the meteor trains and that
radiation may be scattered, as suggested in Obenberger et al. (2015).

Here we report on low-frequency all-sky observations with the LOFAR (“Low
Frequency Array”) radio telescope (van Haarlem et al., 2013) during the 2020
Perseid meteor shower (Jenniskens, 2006). The radio data are complemented with
simultaneous, both temporally and spatially, optical video observations from
the CAMS BeNeLux network (Jenniskens et al., 2011) in an effort to study the
altitude dependence of the proposed intrinsic radio emission.

##  2 Observations

The LOFAR radio telescope consists of thousands of dipole antennas grouped
into stations spread over Europe, with a dense core of stations located in the
North of the Netherlands. Each LOFAR station has low-band antennas (LBAs)
which can observe from 10 to 90 MHz and high-band antennas (HBAs) operating
from 110 to 250 MHz. During regular LOFAR LBA observations radio signals are
combined (beamformed) hierarchically, by first combining radio signals from
dipole antennas in a station, and then combining or correlating the combined
radio signals from stations to increase sensitivity at the expense of field-
of-view.

The AARTFAAC piggyback instrument (Prasad et al., 2016; Kuiack et al., 2019,
2021) provides a special observing mode whereby LOFAR can create all-sky
images of the radio sky. It does so by correlating the radio signals from all
576 LBA dipoles from the inner 12 LOFAR stations located in the dense core
(with a 1.2 km diameter) surrounding the ”super terp” in Drenthe, the
Netherlands, see Fig. 1.

LOFAR/AARTFAAC observations were obtained during the Perseid meteor shower
(2020 August 12-13) for 64 frequency channels of 48.8 kHz bandwidth (grouped
into 16 subbands of 4 channels each), spread over radio frequencies between 30
to 60 MHz. These observations were used to create all-sky images with
$3\hbox{${}^{\prime}$}$ spatial resolution and 2 s time resolution.

Figure 1: Top: map of the LOFAR core with the location of all 512 dipole
antennas used for AARTFAAC. Bottom: map with CAMS camera locations (red), the
location of the LOFAR core (blue) and co-observed meteors.

Figure 2: AARTFAAC image (integrated over all observing bands) of the meteors
at 2020-08-12 21:25:48 UTC (top left) and 2020-08-13 01:41:24 UTC (bottom
left). The large-scale diffuse emission is the Galactic plane, the bottom left
image shows some residuals of the subtracted sources Cassiopeia A and Cygnus
A. The right column shows zoomed in meteors, from top to bottom corresponding
to numbers 144, 235, 249, 317, 333, and 704 in the CAMS data set. In red, the
trajectory as computed from the CAMS optical observations is overlaid.

All-sky radio images were created for each of the individual AARTFAAC subbands
(a set of 4 channels). This analysis used a newer version of the pipeline
described in Shulevski et al. (2021). As part of the pipeline, the brightest
radio sources Cassiopeia A and Cygnus A were subtracted, to minimize imaging
artefacts across the all-sky images. Typically, in subbands containing strong
narrow-band terrestrial radio emission reflected by meteor ionization trails,
such as emission from the BRAMS meteor forward scatter network transmitter in
Belgium near 50 MHz (Lamy et al., 2016), the imaging failed, and images from
these subbands were rejected. Images from the remaining subbands were averaged
in radio frequency across subbands. This resulted in 10713 all-sky radio
images at a 2 s cadence, corresponding to 5 h of data. Example all-sky images
obtained with AARTFAAC are shown in Fig. 2. A full time-lapse of these images
is available as Dijkema et al. (2021b).

The CAMS BeNeLux low-light video network is part of the global (”Cameras for
Allsky Meteor Surveillance”) network and uses close to 100 low-light video
cameras spread over the BeNeLux to triangulate optical meteors +4 and brighter
using methods described in Jenniskens et al. (2011). Weather was clear during
the night of 2020 August 12-13 and the trajectory and orbits of 720 meteors
were measured (Roggemans, 2020).

Figure 2 shows a subset of these meteor trajectories overlaid on the all-sky
radio images from the perspective of the ”super terp”. All all-sky radio
images within a few seconds of CAMS detections were inspected manually for
radio emission coincident with the reconstructed meteor trajectory. This
resulted in 204 meteors where radio emission was coincident with a CAMS
detection. Of these meteors, 59 had a discernible trail in the radio images,
the others showed up as point-like in the radio images. There were also dozens
of radio meteors without a counterpart in the CAMS data, mostly due to missing
sky coverage of CAMS in Germany at the time, see Fig. 1.

For each radio meteor coincident with a CAMS detection we have determined the
begin and end point of the radio trail in the all-sky radio images and the
duration for which the meteor was visible in the radio images.

##  3 Results

The coincidence of optical and radio trails demonstrates that meteors are a
source of radio emission, be it scattering or intrinsic emission. That
emission can last up to minutes; the longest radio train we have observed
lasted 6.5 min, coincident with a CAMS meteor detection with optical magnitude
of $-9$. In general, we find that brighter optical meteors result in radio
trains that remain visible for longer, as shown in Fig. 3a.

The detected radio emission is integrated over a 2-second time interval. Based
on the decay of brightness in subsequent 2-s intervals, the short trails are
mostly emission from the meteor’s persistent train, rather than from the
meteor itself.

The optical trails of CAMS detected meteors start and end at higher altitudes
for brighter meteors, and while the radio trains show a similar dependency
(Fig. 3b and c), the radio emission is first detected at lower altitudes
($h=101\pm 4$ km) compared to optical emission ($h=107\pm 6$ km). The altitude
at which the optical and radio emission ends is comparable for optical
($h=94\pm 6$ km) and radio ($h=92\pm 6$ km).

Figure 3: The dependency of the radio visibility and the altitude of the
beginning and end of the meteor trail with optical brightness and entry speed.
Note that the CAMS-derived peak optical brightness is less reliable below -5
magnitude due to video blooming.

Some of the radio trains that were persistent for several minutes were
distorted by high altitude winds, much like optical persistent trains. Figure
4 shows an example of the spatial evolution of the radio train with time. The
peak intensity of the radio meteor train decreased over time.

Figure 4: (Left:) The temporal and spatial evolution of a persistent radio
meteor train, showing distortion by high altitude winds. The time is
referenced to the beginning of the optical meteor trail. (Right:) Frequency
evolution of a meteor train from a bright optical meteor. In channels where
the emission from the meteor was very bright, the calibration has failed. That
is the case in the bands where narrow-band terrestrial radio emission is
reflected from the ionization train. Known transmitters operate at 32.54 MHz
(TV transmitter), 49.97 and 49.99 MHz (reflection of the BRAMS meteor radars;
Lamy et al. 2016) and 50.00 MHz (amateur radio transmissions).

Some frequency channels show reflected narrow-band artificial radio sources,
but all channels contain a broadband radio component that is not likely from
artificial radio sources. For bright radio meteors, the signal-to-noise ratio
was sufficient to detect this broadband emission in almost all individual
channels for frequencies below $\sim 50$ MHz. An example is shown in Fig. 4.

Further insight was obtained when repeating the AARTFAAC observations during
the 2020 Geminid meteor and the 2021 Quadrantid meteor shower. Unfortunately,
cloudy weather prevented simultaneous CAMS video observations. During that
campaign, the all-sky radio images showed significantly fewer radio meteors
compared to the observations of the Perseids.

##  4 Discussion

We can exclude an important contribution from forward scattered narrow band
terrestrial radio emission, which is frequency resolved in our observations.
Broadband emission would require a source of very broad band (30–50 MHz)
terrestrial radio emission, which we consider unlikely.

Of the proposed intrinsic emission scenarios, chemically produced suprathermal
electrons (Obenberger et al., 2020) could cause lingering radiation over
minutes timescale if those electrons are captured by atoms and molecules that
slowly diffuse into the train. Other proposed mechanisms such as Langmuir
waves (Obenberger et al., 2015), free-free emission (Filonenko, 2018) and
transition radiation (Obenberger et al., 2020), would be expected strong in
the meteor itself, but we see the radiation increase in intensity following
the meteor head before fading.

There are several possible reasons why fewer meteor trails were detected
during the Geminids and Quadrantids. Geminids and Quadrantids are known to
reach peak brightness at lower altitudes than the Perseids and show generally
weaker optical persistent trains (Jenniskens, 2006). In our case, the lower
entry velocities of the Geminids and Quadrantids in comparison to the Perseids
is perhaps not to blame. We note that several slower sporadic meteors of
similar brightness were observed in radio during the Perseids meteor shower,
see Fig. 3a.

The atmospheric conditions may be important. During the Perseids, the summer
weather provided hot and dry conditions, while the Geminids and Quadrantids
were observed during the rainy winter season.

Finally, it is possible that the local sidereal time was important. The
scattering of bright celestial radio sources, as suggested in Obenberger et
al. (2015) was rejected in that paper with an argument involving the fact the
earlier LWA detections are very bright. Since we have detected much weaker
radio meteors, that argument does not hold here.

During the Perseids, the Galactic center and Galactic plane, which is the
source of most low-frequency radio emission, was above the LOFAR horizon,
while this was not the case during the Geminids and Quadrantids.

##  5 Conclusions

The AARTFAAC observations presented here show that persistent radio emission
from meteors coincides with their optical trajectories, and can be detected
for meteors of magnitude $\sim 0$ and brighter. Aside from a narrow-band
scattering of artificial radio sources, there is also a generally broadband
emission detected for frequencies below 50 MHz. The continuum radio emission
is first detected at lower altitudes compared to the optical emission, while
both radio and optical emission end at similar altitudes. For the brightest
optical meteors, we find that the radio emission can persist for several
minutes. Persistent radio trains are affected by high altitude winds like the
optical persistent trains.

Analysis of the AARTFAAC observations is ongoing, so it is currently not
possible to distinguish between the suggested origins of the broadband radio
emission.

Acknowledgements: We thank Albert van Duin and Edwin van Dijk for the use of
facilities at the Edgar Getreuer Observatory in Burlage. We acknowledge
interesting discussions with the researchers in the BRAMS project. We thank
all camera operators and other contributors to CAMS-BeNeLux. PJ acknowledges
support from NASA grant 80NSSC19K0563 (Solar System Workings).

Supplemental materials: Properties of the radio meteors observed
simultaneously with CAMS are available as (Dijkema et al., 2021a).

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# AQUILA: Communication Efficient Federated Learning with Adaptive
Quantization of Lazily-Aggregated Gradients

Zihao Zhao∗, Yuzhu Mao∗, Zhenpeng Shi, Yang Liu, Tian Lan, Wenbo
Ding${\dagger}$, and Xiao-Ping Zhang ∗ These authors contribute
equally.${\dagger}$ Corresponding author.Z. Zhao, Y. Mao, Z. Shi, W. Ding and
X.-P. Zhang are with Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen
International Graduate School, Tsinghua University, China. W. Ding is the
corresponding author. E-mail: ({zhao-zh21, myz20, shizp22}
@mails.tsinghua.edu.cn<EMAIL_ADDRESS>W. Ding is also with
RISC-V International Open Source Laboratory, Shenzhen, China, 518055.)Y. Liu
is with the Institute for AI Industry Research (AIR), Tsinghua University,
China. E-mail: (liuy03@air.tsinghua.edu.cn). Y. Liu and W. Ding are also with
Shanghai AI Lab, Shanghai, China.T. Lan is with the Department of Electrical
and Computer Engineering, George Washington University, DC, USA. Email:
(tlan@gwu.edu)X.-P. Zhang is also with the Department of Electrical, Computer
and Biomedical Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada.
E-mail<EMAIL_ADDRESS>
###### Abstract

The widespread adoption of Federated Learning (FL), a privacy-preserving
distributed learning methodology, has been impeded by the challenge of high
communication overheads, typically arising from the transmission of large-
scale models. Existing adaptive quantization methods, designed to mitigate
these overheads, operate under the impractical assumption of uniform device
participation in every training round. Additionally, these methods are limited
in their adaptability due to the necessity of manual quantization level
selection and often overlook biases inherent in local devices’ data, thereby
affecting the robustness of the global model. In response, this paper
introduces AQUILA (adaptive quantization of lazily-aggregated gradients), a
novel adaptive framework devised to effectively handle these issues, enhancing
the efficiency and robustness of FL. AQUILA integrates a sophisticated device
selection method that prioritizes the quality and usefulness of device
updates. Utilizing the exact global model stored by devices, it enables a more
precise device selection criterion, reduces model deviation, and limits the
need for hyperparameter adjustments. Furthermore, AQUILA presents an
innovative quantization criterion, optimized to improve communication
efficiency while assuring model convergence. Our experiments demonstrate that
AQUILA significantly decreases communication costs compared to existing
methods, while maintaining comparable model performance across diverse non-
homogeneous FL settings, such as Non-IID data and heterogeneous model
architectures.

###### Index Terms:

Federated learning, communication efficiency, optimization.

††publicationid: pubid: 0000–0000/00$00.00 © 2023 IEEE

##  I Introduction

With proliferation of ubiquitous sensing and computing devices, the Internet
of things (IoT), as well as many other distributed systems, have gradually
grown from mere concepts to a reality, bringing dramatic convenience to
people’s daily lives [1, 2, 3]. In order to fully exploit these distributed
computing resources, distributed learning provides a promising framework that
parallels the performance of traditional centralized learning schemes.
Nevertheless, concerns about the privacy and security of sensitive data during
the updating and transmission processes persist. _federated learning (FL)_
[4], a methodology developed to address these issues, has been developed,
allows distributed devices to collaboratively learn a global model without
privacy leakage by keeping private data isolated and masking transmitted
information with secure approaches. On account of its potential for privacy-
preservation in privacy sensitive fields such as finance and health, FL has
garnered substantial from both academia and industry in recent years.

Unfortunately, in many FL applications, such as image classification and
objective recognition, the trained model tends to be high-dimensional,
resulting in considerable communication costs. Consequently, communication
efficiency has emerged as an imperative challenge in FL. In response, [5]
proposed the lazily-aggregated quantization (LAQ) method to bypass unnecessary
parameter uploads by estimating the gradient innovation: the difference
between the current unquantized gradient and the previously quantized
gradient. Moreover, [6] devised an adaptive quantized gradient (AQG) strategy
based on LAQ to dynamically select the quantization level within some
artificially given numbers during the training process. Nevertheless, AQG has
proven insufficiently adaptive due to the difficulty of manually selecting the
appropriate quantization levels in complex FL environments. Alternatively, [7]
introduced an adaptive quantization rule for FL named AdaQuantFL, which
searches in a given range for an optimal quantization level and achieves a
better error-communication trade-off.

Existing research on adaptive quantization primarily presumes that all devices
in the FL system participate in each training round. However, this assumption
is both unrealistic and impracticable. Despite the enormous alleviation in
communication overhead via adaptive quantization methods, bandwidth
constraints may still be surpassed if all devices transmit their model updates
to the server, due to sheer number of devices. Currently, [8] proposed a
doubly-adaptive quantization algorithm, DAdaQuant, that dynamically adjusts
the quantization level across time and clients, and randomly selects $K$
devices per round. Nonetheless, this random sampling provides no theoretical
guarantee and could neglect the biases inherent in local devices’ data,
potentially yielding to underrepresent or overfit to specific patterns and
resulting in a less robust global model [9]. In response to these limitations,
this paper introduces a superior adaptive framework, AQUILA, that resorts to a
sophisticated device selection method to take the quality and usefulness of
the devices’ updates into account. Specifically, instead of relying on the
estimation of the global gradient such as some existing selection criteria,
AQUILA adopts a more precise device selection criterion. This device selection
approach uses the exact global model stored by devices and necessitates fewer
hyperparameters adjustments. Moreover, we intend to minimize the model
deviation induced by the device selection to garner a novel quantization
criterion that significantly improves communication efficiency and still
offers a convergence guarantee. The contributions of this paper are trifold.

  * •

We propose an innovative FL procedure with adaptive quantization of lazily-
aggregated gradients termed AQUILA, which simultaneously adjusts the
communication frequency and quantization level in a synergistic fashion.

  * •

We derive an adaptive quantization strategy from a new perspective that
minimizes the model deviation introduced by the device selection.
Subsequently, we present a new device selection criterion that is more precise
and saves more device storage. Furthermore, we provide a convergence analysis
of AQUILA under the generally non-convex case and the Polyak-Łojasiewicz
condition.

  * •

Except for normal FL settings, such as independent and identically distributed
(IID) data environment, we experimentally evaluate the performance of AQUILA
in a number of non-homogeneous FL settings, such as non-independent and non-
identically distributed (Non-IID) local dataset and various heterogeneous
model aggregations. The evaluation results reveal that AQUILA considerably
mitigates the communication overhead compared to a variety of state-of-art
algorithms.

##  II Background and Related Work

Consider an FL system with one central parameter server and a device set
$\mathcal{M}$ with $M=|\mathcal{M}|$ distributed devices to collaboratively
train a global model parameterized by $\boldsymbol{\theta}\in\mathbb{R}^{d}$.
Each device $m\in\mathcal{M}$ has a private local dataset
$\mathcal{D}_{m}=\\{(\boldsymbol{x}_{1}^{(m)},\boldsymbol{y}_{1}^{(m)}),\cdots,(\boldsymbol{x}_{n_{m}}^{(m)},\boldsymbol{y}_{n_{m}}^{(m)})\\}$
of $n_{m}$ samples. The federated training process is typically performed by
solving the following optimization problem

$\displaystyle\min_{\boldsymbol{\theta}\in\mathbb{R}^{d}}f(\boldsymbol{\theta})$
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}f_{m}(\boldsymbol{\theta})$ (1)

where $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ denotes the empirical risk, and
$f_{m}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ denotes the local objective based
on the private data $\mathcal{D}_{m}$ of the device $m$. The FL training
process is conducted by iteratively performing local updates and global
aggregation as proposed in [4]. First, at communication round $k$, each device
$m$ receives the global model $\boldsymbol{\theta}^{k}$ from the parameter
server and trains it with its local data $D_{m}$. Subsequently, it sends the
local gradient $\nabla f_{m}(\boldsymbol{\theta}^{k})$ to the central server,
and the server will update the global model with learning rate $\alpha$ by

$\displaystyle\boldsymbol{\theta}^{k+1}\coloneqq\boldsymbol{\theta}^{k}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}}\nabla
f_{m}(\boldsymbol{\theta}^{k}).$ (2)

######  Definition 1.

For more efficiency, each device only uploads the quantized deflection between
the full gradient $\nabla f_{m}(\boldsymbol{\theta}^{k})$ and the last
quantization value $\boldsymbol{q}_{m}^{k-1}$ utilizing a quantization
operator $\mathcal{Q}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, i.e.,

$\Delta\boldsymbol{q}_{m}^{k}=\mathcal{Q}(\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}).$ (3)

For communication frequency reduction, the original lazy aggregation strategy
allows the device $m\in\mathcal{M}$ to upload its newly-quantized gradient
innovation at epoch $k$ only when the change in local gradient is sufficiently
larger than a threshold. Hence, the quantization of the local gradient
$\boldsymbol{q}_{m}^{k}$ of device $m$ at epoch $k$ can be calculated by

$\boldsymbol{q}_{m}^{k}\coloneqq\begin{cases}\hfil\boldsymbol{q}_{m}^{k-1},&\text{if
}\begin{split}\left\|\mathcal{Q}(\nabla
f_{m}\left(\boldsymbol{\theta}^{k}\right)-\boldsymbol{q}_{m}^{k-1})\right\|_{2}^{2}\\\
\leqslant Threshold\end{split}\\\
\hfil\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k},&\text{ otherwise
}\end{cases}.$ (4)

If the device $m$ skips the upload of $\Delta\boldsymbol{q}_{m}^{k}$, the
central server will reuse the last gradient $\boldsymbol{q}_{m}^{k-1}$ for
aggregation. Therefore, the global aggregation rule can be changed from (2)
to:

$\displaystyle\boldsymbol{\theta}^{k+1}$
$\displaystyle=\boldsymbol{\theta}^{k}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}}\boldsymbol{q}_{m}^{k}$
(5)
$\displaystyle=\boldsymbol{\theta}^{k}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}^{k}}\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right)-\frac{\alpha}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{q}_{m}^{k-1},$

where $\mathcal{M}^{k}$ denotes the subset of devices that upload their
quantized gradient innovation, and
$\mathcal{M}_{c}^{k}=\mathcal{M}\setminus\mathcal{M}^{k}$ denotes the subset
of devices that skip the gradient update and reuse the old quantized gradient
at epoch $k$.

For AdaQuantFL, it is proposed to achieve a better error-communication trade-
off by adaptively adjusting the quantization levels during the FL training
process. Specifically, AdaQuantFL computes the optimal quantization level
$(b^{k})^{*}$ by
$(b^{k})^{*}=\lfloor\sqrt{{f(\boldsymbol{\theta}^{0})}/{f(\boldsymbol{\theta}^{k})}}\cdot
b_{0}\rfloor$, where $f(\boldsymbol{\theta}^{0})$ and
$f(\boldsymbol{\theta}^{k})$ are the global objective loss defined in (1).

However, AdaQuantFL transmits quantized gradients for all local devices at
every communication round. In order to skip unnecessary communication rounds
and adaptively adjust the quantization level for each communication jointly, a
naive approach is to quantize lazily aggregated gradients with AdaQuantFL.
Nevertheless, it fails to achieve efficient communication for several reasons.
First, given the descending trend of training loss, AdaQuantFL’s criterion may
lead to a high quantization bit number even exceeding 32 bits in the training
process (assuming a floating point is represented by 32 bits in our case),
which is too large for cases where the global convergence is already
approaching and makes the quantization meaningless. Second, a higher
quantization level results in a smaller quantization error, leading to a lower
communication threshold in the lazy aggregation criterion (4) and thus a
higher transmission frequency.

Consequently, it is desirable to develop a more efficient adaptive
quantization method in the lazily-aggregated setting to improve communication
efficiency in FL systematically.

##  III Adaptive Quantization of Lazily-Aggregated Gradients

Given the above limitations of the naive joint use of the existing adaptive
quantization criterion and lazy aggregation strategy, this paper aims to
design a unifying procedure for communication efficiency optimization where
the quantization level and communication frequency are considered
synergistically and interactively.

###  III-A Precise device selection criterion

First, we introduce the definition of a deterministic rounding quantizer and
its corresponding quantization error.

######  Definition 2.

(Deterministic mid-tread quantizer). Every element of the gradient innovation
of device $m$ at epoch $k$ is mapped to an integer
$[{\boldsymbol{\psi}}_{m}^{k}]_{i}$ as $\forall i\in\\{1,2,...,d\\}$

$\displaystyle\left[{\boldsymbol{\psi}}_{m}^{k}\right]_{i}=$
$\displaystyle\left\lfloor\frac{\left[\nabla
f_{m}(\boldsymbol{\theta}^{k})\right]_{i}-\left[\boldsymbol{q}_{m}^{k-1}\right]_{i}+R_{m}^{k}}{2\tau_{m}^{k}R_{m}^{k}}\\!+\\!\frac{1}{2}\right\rfloor,$
(6)

where $\nabla f(\boldsymbol{\theta}_{m}^{k})$ denotes the current unquantized
gradient, $R_{m}^{k}=\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\|_{\infty}$ denotes
the quantization range, $b_{m}^{k}$ denotes the quantization level, and
$\tau_{m}^{k}:=1/(2^{b_{m}^{k}}-1)$ denotes the quantization granularity.

######  Definition 3.

(Quantization error). The global quantization error
$\boldsymbol{\varepsilon}^{k}$ is defined by the subtraction between the
current unquantized gradient $\nabla f(\boldsymbol{\theta}^{k})$ and its
quantized value $\boldsymbol{q}^{k-1}+\Delta\boldsymbol{q}^{k}$, i.e.,

$\boldsymbol{\varepsilon}^{k}=\nabla
f(\boldsymbol{\theta}^{k})-\boldsymbol{q}^{k-1}-\Delta\boldsymbol{q}^{k},$ (7)

where the current global terms can be computed as $\nabla
f(\boldsymbol{\theta}^{k})\\!=\\!\sum_{m\in\mathcal{M}}\nabla
f_{m}(\boldsymbol{\theta}^{k}),\Delta\boldsymbol{q}^{k}\\!=\\!\sum_{m\in\mathcal{M}}\Delta\boldsymbol{q}_{m}^{k},\boldsymbol{q}^{k-1}\\!=\\!\sum_{m\in\mathcal{M}}\boldsymbol{q}_{m}^{k-1}.$

Motivated by LAQ, AQUILA possesses a new communication criterion to avoid the
potential expansion of the devices group being skipped: $\forall
m\in\mathcal{M}_{c}^{k}$,

$\displaystyle\left\|\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\left\|\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}$
$\displaystyle\leqslant\frac{\beta}{\alpha^{2}}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$
(8)

where $\beta\geqslant 0$ is a tuning factor. Note that this skipping rule is
employed at epoch $k$, in which each device $m$ calculates its quantized
gradient innovation $\Delta\boldsymbol{q}_{m}^{k}$ and quantization error
$\boldsymbol{\varepsilon}_{m}^{k}$, then utilizes this rule to decide whether
uploads $\Delta\boldsymbol{q}_{m}^{k}$.

Instead of storing a large number of previous model parameters as LAQ, the
strength of (8) is that AQUILA directly utilizes the global model for two
adjacent rounds as the skip condition, which does not need to estimate the
global gradient (more precise), requires fewer hyperparameters to adjust, and
considerably reduces the storage pressure of local devices. This is especially
important for small-capacity devices (e.g., sensors) in practical IoT
scenarios. Furthermore, with the given threshold, AQUILA has a good
theoretical property. The theoretical analysis of AQUILA is easier to follow
with no Lyapunov function introduced as in LAQ. And the result in IV also
shows that AQUILA can achieve a better convergence rate under the non-convex
case and the PL condition.

###  III-B Optimal quantization level

As mentioned before, AQUILA intends to minimize the model deviation induced by
the device selection to deduce how each local device chooses the optimal
quantization level. First, we introduce the definition of the fully-aggregated
model.

######  Definition 4.

(Fully-aggregated model). The fully-aggregated model
$\boldsymbol{\tilde{{\theta}}}$ without lazy aggregation at epoch $k$ is
computed by

$\boldsymbol{\tilde{{\theta}}}^{k+1}=\boldsymbol{\theta}^{k}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}}\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right).$
(9)

######  Lemma 1.

The influence of lazy aggregation at communication round $k$ can be bounded by

$\displaystyle\left\|\tilde{\boldsymbol{\theta}}^{k}\\!-\\!\boldsymbol{\theta}^{k}\right\|_{2}^{2}\\!$
$\displaystyle\leqslant\\!\frac{4\alpha^{2}|\mathcal{M}_{c}^{k}|}{M^{2}}\\!\sum_{m\in\mathcal{M}_{c}^{k}}\Big{(}\big{(}\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})\\!-\\!\boldsymbol{q}_{m}^{k-1}\right\|_{2}\\!$
$\displaystyle-\\!\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\big{)}^{2}\\!+\\!4(R_{m}^{k})^{2}d\\!+\\!\frac{d}{2}\Big{)}.$
(10)

###### Proof.

To prove this Lemma, we will use the following equality and inequalities.
Suppose $n\in\mathbb{N}^{+}$ and $\|\cdot\|_{2}$ denotes the $\ell^{2}-$norm.
For $p$ in
$\mathbb{R}^{+},\boldsymbol{x}_{i},\boldsymbol{a},\boldsymbol{b}\in\mathbb{R}^{d}$,
there holds:

  1. 1.

Inner product equality.

$\langle\boldsymbol{a},\boldsymbol{b}\rangle=\frac{1}{2}\left(\|\boldsymbol{a}\|_{2}^{2}+\|\boldsymbol{b}\|_{2}^{2}-\|\boldsymbol{a}-\boldsymbol{b}\|_{2}^{2}\right).$
(11)

  2. 2.

Norm-summation inequality.

$\left\|\sum_{i=1}^{n}\boldsymbol{x}_{i}\right\|_{2}^{2}\leqslant
n\sum_{i=1}^{n}\left\|\boldsymbol{x}_{i}\right\|_{2}^{2}.$ (12)

  3. 3.

Young’s inequality.

$\left\|\boldsymbol{a}+\boldsymbol{b}\right\|_{2}^{2}\leqslant(1+p)\left\|\boldsymbol{a}\right\|_{2}^{2}+(1+p^{-1})\left\|\boldsymbol{b}\right\|_{2}^{2}.$
(13)

  4. 4.

Minkowski’s inequality.

$\left\|\boldsymbol{a}+\boldsymbol{b}\right\|_{2}\leqslant\left\|\boldsymbol{a}\right\|_{2}+\left\|\boldsymbol{b}\right\|_{2}.$
(14)

With lazy aggregation, the actual aggregated model at epoch $k$ is:

$\boldsymbol{\theta}^{k+1}=\boldsymbol{\theta}^{k}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}^{k}}\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right)-\frac{\alpha}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{q}_{m}^{k-1}.$
(15)

Suppose $\Delta_{m}^{k}$ denotes the quantization loss of device $m$ at epoch
$k$ and $\boldsymbol{\psi}_{m}^{k}$ denotes the quantization representation of
local gradient innovation as in Definition 1, i.e.,

$\Delta_{m}^{k}=\boldsymbol{\psi}_{m}^{k}-\frac{\nabla
f_{m}\left(\boldsymbol{\theta}^{k}\right)-\boldsymbol{q}_{m}^{k-1}+R_{m}^{k}\boldsymbol{1}}{2\tau_{m}^{k}R_{m}^{k}}-\frac{1}{2}\boldsymbol{1}$
(16)

With (9), (15), and (16), and for mathematical simplicity we denote
$\Gamma=\cfrac{\alpha^{2}|\mathcal{M}_{c}^{k}|}{M^{2}}$, the model deviation
$\|\tilde{\boldsymbol{\theta}}^{k}-\boldsymbol{\theta}^{k}\|_{2}^{2}$ caused
by skipping gradients can be written as:

$\displaystyle\left\|\tilde{\boldsymbol{\theta}}^{k}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
(17)
$\displaystyle=\left\|\frac{\alpha}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\left(2\tau_{m}^{k}R_{m}^{k}\boldsymbol{\psi}_{m}^{k}-R_{m}^{k}\boldsymbol{1}\right)\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{sum_ineq}}{\leqslant}\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left\|2\tau_{m}^{k}R_{m}^{k}\boldsymbol{\psi}_{m}^{k}-R_{m}^{k}\boldsymbol{1}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{def_Delta}}{\leqslant}\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}+R_{m}^{k}\boldsymbol{1}+\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}+\Delta_{m}^{k}-R_{m}^{k}\boldsymbol{1}\right\|_{2}^{2}\right)$
$\displaystyle\overset{\eqref{def_Delta}}{\leqslant}2\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}+\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}^{2}+\left\|\Delta_{m}^{k}\right\|_{2}^{2}\right)$
$\displaystyle\overset{(a)}{\leqslant}2\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}+\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}^{2}+d\right)$
$\displaystyle\overset{\eqref{Minko}}{\leqslant}2\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}+\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}+d\right)$
$\displaystyle=2\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}-\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}+2\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}+d\right)$
$\displaystyle\leqslant
4\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}-\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}+4\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}^{2}+\frac{d}{2}\right)$
$\displaystyle\overset{(b)}{\leqslant}4\Gamma\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}-\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}+4(R_{m}^{k})^{2}d+\frac{d}{2}\right),$

where $\boldsymbol{1}\in\mathbb{R}^{d}$ denotes the vector filled with scalar
value $1$, (a) $\Delta_{m}^{k}\in(-1,0]$, (b)
$R_{m}^{k}\geqslant\tau_{m}^{k}R_{m}^{k}\geqslant 0$. ∎

Corresponding to Lemma 1, since $R_{m}^{k}$ is independent of $\tau_{m}^{k}$,
we can formulate an optimization problem to minimize the upper bound of this
model deviation caused by update skipping for each device $m$:

$\begin{array}[]{cl}\underset{0<\tau_{m}^{k}\leqslant
1}{\text{minimize}}&\quad\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}-\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}\\\
\text{ subject to
}&\quad\tau_{m}^{k}=\cfrac{1}{\left(2^{b_{m}^{k}}-1\right)}\end{array}.$ (18)

Solving the below optimization problem gives AQUILA an adaptive strategy (19)
that selects the optimal quantization level based on the quantization range
$R_{m}^{k}$, the dimension $d$ of the local model, the current gradient
$\nabla f_{m}(\boldsymbol{\theta}^{k})$, and the last uploaded quantized
gradient $\boldsymbol{q}_{m}^{k-1}$:

$(b_{m}^{k})^{*}=\left\lfloor\log_{2}\left(\frac{R_{m}^{k}\sqrt{d}}{\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}}+1\right)\right\rfloor.$
(19)

###### Proof.

Since $R_{m}^{k}$ is independent of $\tau_{m}^{k}$, we can formulate an
optimization problem about $\tau_{m}^{k}$ for device $m$ at communication
round $k$ as follows:

$\displaystyle\min_{0<\tau_{m}^{k}\leqslant 1}\quad\left(\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}-\left\|\tau_{m}^{k}R_{m}^{k}\boldsymbol{1}\right\|_{2}\right)^{2}$
(20)

Therefore, the optimal solution of $\tau_{m}^{k}$ in (20) is

$(\tau_{m}^{k})^{*}=\cfrac{\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}}{R_{m}^{k}\sqrt{d}}.$
(21)

Then, the optimal adaptive quantization level $(b_{m}^{k})^{*}$ is equal to

$\displaystyle(b_{m}^{k})^{*}$
$\displaystyle=\left\lfloor\log_{2}(\frac{1}{(\tau_{m}^{k})^{*}}+1)\right\rfloor$
(22)
$\displaystyle=\left\lfloor\log_{2}\left(\frac{R_{m}^{k}\sqrt{d}}{\left\|\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{q}_{m}^{k-1}\right\|_{2}}+1\right)\right\rfloor.$

∎

######  Remark.

Unlike certain adaptive quantization algorithms, such as DAdaQuant [8], which
necessitate a maximization operation of the computed quantization level
results (e.g., $b_{m}^{k}=\max(1,\operatorname{round}(\sqrt{\frac{a}{b}}\times
w_{i}^{2/3}))$), AQUILA’s method of determining the optimal quantization level
is self-consistent, because $(b_{m}^{k})^{*}\geqslant 1$ is always true since
$(\tau_{m}^{k})^{*}\leqslant 1$.

Algorithm 1 Communication Efficient FL with AQUILA

1:the number of communication rounds $K$, the learning rate $\alpha$.

2:the initial global model parameter $\boldsymbol{\theta}^{0}$.

3:Server broadcasts $\boldsymbol{\theta}^{0}$ to all devices.

4:for each device $m\in\mathcal{M}$ in parallel do

5: Calculates local gradient ${\nabla f_{m}(\boldsymbol{\theta}}^{0})$.

6: Compute $(b_{m}^{0})^{*}$ by setting
$\boldsymbol{q}_{m}^{k-1}=\boldsymbol{0}$ in (19) and the quantized gradient
innovation $\Delta\boldsymbol{q}_{m}^{0}$, and transmits it back to the server
side.

7:end for

8:for $k=1,2,...,K$ do

9: Server broadcasts $\boldsymbol{\theta}^{k}$ to all devices.

10: for each device $m\in\mathcal{M}$ in parallel do

11: Calculates local gradient ${\nabla f_{m}(\boldsymbol{\theta}}^{k})$, the
optimal local quantization level $(b_{m}^{k})^{*}$ by (19), and the quantized
gradient innovation $\Delta\boldsymbol{q}_{m}^{k}$.

12: if (8) does not hold for device $m$ then

13: device $m$ transmits $\Delta\boldsymbol{q}_{m}^{k}$ to the server.

14: end if

15: end for

16: Server updates $\boldsymbol{\theta}^{k+1}$ by the saving previous global
quantized gradient $\boldsymbol{q}_{m}^{k-1}$ and the received quantized
gradient innovation $\Delta\boldsymbol{q}_{m}^{k}$:
$\boldsymbol{\theta}^{k+1}:=\boldsymbol{\theta}^{k}-\alpha\left(\boldsymbol{q}^{k-1}+1/M\sum_{m\in\mathcal{M}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right)$.

17: Server saves the average quantized gradient $\boldsymbol{q}^{k}$ for the
next aggregation.

18:end for

The superiority of (19) comes from the following three aspects. First, since
$R_{m}^{k}\geqslant[\nabla
f_{m}(\boldsymbol{\theta}^{k})]_{i}-[\boldsymbol{q}_{m}^{k-1}]_{i}\geqslant-
R_{m}^{k}$, the optimal quantization level $(b_{m}^{k})^{*}$ must be greater
than or equal to $1$. Second, AQUILA can personalize an optimal quantization
level for each device corresponding to its own gradient, whereas, in
AdaQuantFL, each device merely utilizes an identical quantization level
according to the global loss. Third, the gradient innovation and quantization
range $R_{m}^{k}$ tend to fluctuate along with the training process instead of
keeping descending, and thus prevent the quantization level from increasing
tremendously compared with AdaQuantFL.

The detailed process of AQUILA is comprehensively summarized in Algorithm 1.
At epoch $k=0$, each device calculates $b_{m}^{0}$ by setting
$\boldsymbol{q}_{0}^{k-1}=\boldsymbol{0}$ and uploads
$\Delta\boldsymbol{q}_{0}^{k}$ to the server since the (8) is not satisfied.
At epoch $k\in\left\\{1,2,...,K\right\\}$, the server first broadcasts the
global model $\boldsymbol{\boldsymbol{\theta}}^{k}$ to all devices. Each
device $m$ computes ${\nabla f(\boldsymbol{\theta}}_{m}^{k})$ with local
training data and then utilizes it to calculate an optimal quantization level
by (19). Subsequently, each device computes its gradient innovation after
quantization and determines whether or not to upload based on the
communication criterion (8). Finally, the server updates the new global model
$\boldsymbol{\boldsymbol{\theta}}^{k+1}$ with up-to-date quantized gradients
$\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}$ for those devices who
transmit the uploads at epoch $k$, while reusing the old quantized gradients
$\boldsymbol{q}_{m}^{k-1}$ for those who skip the uploads.

##  IV Theoretical Derivation and Analysis of AQUILA

As aforementioned, we bound the model deviation caused by skipping updates
with respect to quantization bits. Specifically, if the communication
criterion (8) holds for the device $m$ at epoch $k$, it does not contribute to
epoch $k$’s gradient. Otherwise, the loss caused by device $m$ will be
minimized with the optimal quantization level selection criterion (19). In
this section, the theoretical convergence derivation of AQUILA is based on the
following standard assumptions.

######  Assumption 1.

Each local objective function $f_{m}$ is $L_{m}$-smooth, i.e., there exist a
constant $L_{m}>0$, such that
$\forall\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{d}$,

$\left\|\nabla f_{m}(\mathbf{x})-\nabla f_{m}(\mathbf{y})\right\|_{2}\leqslant
L_{m}\left\|\mathbf{x}-\mathbf{y}\right\|_{2},$ (23)

which implies that the global objective function $f$ is $L$-smooth with
$L\leq\bar{L}=\frac{1}{m}\sum_{i=1}^{m}L_{m}$.

######  Assumption 2.

(Uniform lower bound). For all $\boldsymbol{x}\in\mathbb{R}^{d}$, there exist
$f^{*}\in\mathbb{R}$ such that $f(\boldsymbol{x})\geq f^{*}$.

######  Lemma 2.

Following the assumption that the function $f$ is L-smooth, we have

$\displaystyle f(\boldsymbol{\theta}^{k+1})-$ $\displaystyle
f(\boldsymbol{\theta}^{k})\leqslant-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$ (24)
$\displaystyle+\alpha\left(\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}\right)$
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}.$

######  Assumption 3.

All devices’ quantization errors $\boldsymbol{\varepsilon}^{k}$ will be
constrained by the total error of the omitted devices., i.e., $\forall\
k=0,1,\cdots,K$, if $\mathcal{M}_{c}^{k}\neq\varnothing$, $\exists\
\gamma\geqslant 1$, such that

$\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}=\left\|\frac{1}{M}\sum_{m\in\mathcal{M}}\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}\leqslant\frac{\gamma}{M^{2}}\left\|\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2},$
(25)

where K denotes the termination time, and
$\boldsymbol{\varepsilon}_{m}^{k}=\nabla
f_{m}(\boldsymbol{\theta}^{k})-\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right)$.

This assumption is easy to verify when $\mathcal{M}_{c}^{k}\neq\varnothing$, a
bounded variable (here is $\boldsymbol{\varepsilon}^{k}$) will always be
bounded by a part of itself
($\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{\varepsilon}_{m}^{k}$)
multiplied by a real number ($\gamma$). Note that there is another nontrivial
scenario that $\mathcal{M}_{c}^{k}\neq\varnothing$ but
$\boldsymbol{\varepsilon}_{m}^{k}=0$ for all $m\in\mathcal{M}_{c}^{k}$, which
implies that $\gamma=0$ or not exists and conflicts with our assumption.
However, this situation only happens when all entries of
$\boldsymbol{\varepsilon}_{m}^{k}=0$, i.e., $[\nabla
f_{m}(\boldsymbol{\theta}^{k})]_{i}=[\boldsymbol{q}_{m}^{k-1}]_{i}$ for all
$0\leqslant i\leqslant d$.

######  Lemma 3.

The summation of quantized gradient innovation and quantization error is
bounded by the global model difference:

$\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}\leqslant\frac{\beta\gamma}{\alpha^{2}}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$
(26)

###### Proof.

$\displaystyle\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$
(27) $\displaystyle\overset{(a)}{\leqslant}$
$\displaystyle\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\gamma\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{sum_ineq}}{\leqslant}$
$\displaystyle|\mathcal{M}_{c}^{k}|\sum_{m\in\mathcal{M}_{c}^{k}}\left\|\frac{1}{M}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\gamma|\mathcal{M}_{c}^{k}|\sum_{m\in\mathcal{M}_{c}^{k}}\left\|\frac{1}{M}\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}$
$\displaystyle{=}$
$\displaystyle\frac{|\mathcal{M}_{c}^{k}|}{M^{2}}\sum_{m\in\mathcal{M}_{c}^{k}}\left(\left\|\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\gamma\left\|\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}\right)$
$\displaystyle\overset{(b)}{\leqslant}$
$\displaystyle\frac{|\mathcal{M}_{c}^{k}|}{M^{2}}\sum_{m\in\mathcal{M}_{c}^{k}}\left(\gamma\left\|\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}+\gamma\left\|\boldsymbol{\varepsilon}_{m}^{k}\right\|_{2}^{2}\right)$
$\displaystyle\overset{(c)}{\leqslant}$
$\displaystyle\frac{\beta\gamma|\mathcal{M}_{c}^{k}|^{2}}{\alpha^{2}M^{2}}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2}$
$\displaystyle\leqslant$
$\displaystyle\frac{\beta\gamma}{\alpha^{2}}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$

where (a) follows 3, (b) follows $\gamma$ is larger than 1 by definition, and
(c) utilizes our novel trigger condition (8). ∎

######  Lemma 4.

From Definition 1, we can derive that the relationship between quantized
gradient innovation $\Delta\boldsymbol{q}_{m}^{k}$ and its quantization
representation $\boldsymbol{\psi}_{m}^{k}$ which applies $b_{m}^{k}$ bits for
each dimension:

$\Delta\boldsymbol{q}_{m}^{k}=2\tau_{m}^{k}R_{m}^{k}\boldsymbol{\psi}_{m}^{k}-R_{m}^{k}\boldsymbol{1},$
(28)

where $\boldsymbol{1}\in\mathbb{R}^{d}$ denotes a vector filled with scalar
value $1$.

###### Proof.

This Lemma can easily be derived by the definition of the deterministic mid-
tread quantizer 6. ∎

######  Remark.

We can utilize (28) to calculate the quantized gradient innovation in the
experimental implementation.

###  IV-A Convergence analysis for the generally non-convex case.

######  Theorem 1.

Suppose Assumptions 1, 2, and 3 be satisfied. If
$\mathcal{M}_{c}^{k}\neq\varnothing$, the global objective function $f$
satisfies

$\displaystyle
f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k}){\leqslant}-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
(29)
$\displaystyle+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$

###### Proof.

Suppose Assumptions 4.1, 4.2, and 4.3 are satisfied and
$M_{c}^{k}\neq\varnothing$. For the simplicity of the convergence proof, we
assume
$\Phi^{k}=\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}$.
First, we prove Lemma 4.

$\displaystyle f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k})$ (30)
$\displaystyle\leqslant$ $\displaystyle\left\langle\nabla
f(\boldsymbol{\theta}^{k}),\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\rangle+\frac{L}{2}\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle=$ $\displaystyle\left\langle\nabla
f(\boldsymbol{\theta}^{k}),-\alpha\left(\nabla
f(\boldsymbol{\theta}^{k})-\boldsymbol{\varepsilon}^{k}-\Phi^{k}\right)\right\rangle+\frac{L}{2}\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle=$ $\displaystyle-\alpha\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\alpha\left\langle\nabla
f(\boldsymbol{\theta}^{k}),\boldsymbol{\varepsilon}^{k}+\Phi^{k}\right\rangle+\frac{L}{2}\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{inner_pro}}{=}$
$\displaystyle-\alpha\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\frac{\alpha}{2}\left(\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\left\|\boldsymbol{\varepsilon}^{k}+\Phi^{k}\right\|_{2}^{2}\right.$
$\displaystyle\left.-\frac{1}{\alpha^{2}}\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}\right)+\frac{L}{2}\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle\leqslant$ $\displaystyle-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\frac{\alpha}{2}\left\|\boldsymbol{\varepsilon}^{k}+\Phi^{k}\right\|_{2}^{2}+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{sum_ineq}}{\leqslant}$
$\displaystyle-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\alpha\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}+\alpha\left\|\Phi^{k}\right\|_{2}^{2}$
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}.$

Hence, we have

$\displaystyle
f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k})\overset{\eqref{error_phi}}{\leqslant}-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
(31)
$\displaystyle+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$

which gives us Theorem 1. ∎

######  Corollary 1.

Let all the assumptions of Theorem 1 hold and
$\frac{L}{2}-\frac{1}{2\alpha}+\frac{\beta\gamma}{\alpha}\leqslant 0$, then
the AQUILA requires

$K=\mathcal{O}\left(\frac{2\omega_{1}}{\alpha\epsilon^{2}}\right)$ (32)

communication rounds with
$\omega_{1}\\!=\\!f\left(\boldsymbol{\theta}^{1}\right)\\!-\\!f\left(\boldsymbol{\theta}^{*}\right)\\!+\\!\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{1}\\!-\\!\boldsymbol{\theta}^{0}\right\|_{2}^{2}$
to achieve $\min_{k}\|\nabla
f(\boldsymbol{\theta}^{k})\|_{2}^{2}\leqslant\epsilon^{2}$.

###### Proof.

Sum (29) up for $k=1,2,\cdots,K$, we have

$\displaystyle
f(\boldsymbol{\theta}^{K+1})-f(\boldsymbol{\theta}^{1})\leqslant-\frac{\alpha}{2}\sum_{k=1}^{K}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{K+1}-\right.$
(33)
$\displaystyle\left.\boldsymbol{\theta}^{K}\right\|_{2}^{2}+\sum_{k=1}^{K-1}\left(\frac{L}{2}-\frac{1}{2\alpha}+\frac{\beta\gamma}{\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}.$

Notice that inequality (33) holds for both $M_{c}^{k}\neq\varnothing$ and
$M_{c}^{k}=\varnothing$. Therefore, for
$\left(\frac{L}{2}-\frac{1}{2\alpha}+\frac{\beta\gamma}{\alpha}\right)\leqslant
0$ and all hyperparameters are chosen properly, considering the minimum of
$\|\nabla f(\boldsymbol{\theta}^{k})\|_{2}^{2}$

$\displaystyle\min_{k=1,2,\cdots,K}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$
$\displaystyle\leqslant\frac{1}{K}\sum_{k=1}^{K}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$ (34)
$\displaystyle\overset{\eqref{descent}}{\leqslant}\frac{2}{\alpha
K}\left(f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{K})+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}\right).$

For
$\left(\frac{L}{2}-\frac{1}{2\alpha}+\frac{\beta\gamma}{\alpha}\right)\leqslant
0$ and all hyperparameters are chosen properly, we have that

$\displaystyle\min_{k=1,2,\cdots,K}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}\leqslant\frac{2}{\alpha
K}\left(f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{*})+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}\right)\leqslant\epsilon^{2},$
(35)

which demonstrates AQUILA requires
$K=\mathcal{O}\left(\cfrac{2\omega_{1}}{\alpha\epsilon^{2}}\right)$
communication round with
$\omega_{1}=f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{*})+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}$
to achieve $\min_{k=1,2,\cdots,K}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}\leqslant\epsilon^{2}$. ∎

######  Corollary 2.

As a specific case for Corollary 1, we also proof the feasibility of our
algorithm in an extreme condition: all clients skip for updating in round $k$,
i.e., $M_{c}^{k}=\varnothing$.

###### Proof.

Since the skipping subset of devices are the empty set, from (5), we have

$\displaystyle\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}$
$\displaystyle=-\frac{\alpha}{M}\sum_{m\in\mathcal{M}^{k}}\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right)-\frac{\alpha}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\boldsymbol{q}_{m}^{k-1}$
(36)
$\displaystyle{=}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}}\left(\boldsymbol{q}_{m}^{k-1}+\Delta\boldsymbol{q}_{m}^{k}\right)$
$\displaystyle\overset{\eqref{def_quanti_error}}{=}-\frac{\alpha}{M}\sum_{m\in\mathcal{M}}\left(\nabla
f_{m}(\boldsymbol{\theta}^{k})-\boldsymbol{\varepsilon}_{m}^{k}\right)$
$\displaystyle{=}-\alpha\left(\nabla
f(\boldsymbol{\theta}^{k})-\boldsymbol{\varepsilon}^{k}\right).$

From (24) we have:

$\displaystyle f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k})$ (37)
$\displaystyle\leqslant-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\alpha\left\|\frac{1}{M}\sum_{m\in\mathcal{M}_{c}^{k}}\Delta\boldsymbol{q}_{m}^{k}\right\|_{2}^{2}$
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}+\alpha\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$
$\displaystyle\leqslant-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}+\alpha\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{youngs}}{\leqslant}-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\alpha^{2}\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left((1+p)\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}\right.$
$\displaystyle\left.+\left(1+p^{-1}\right)\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}\right)+\alpha\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$
$\displaystyle=-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\frac{1}{2}\left(\alpha^{2}L-\alpha\right)(1+p)\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$
$\displaystyle+\frac{1}{2}\left(\alpha^{2}L-\alpha\right)\left(1+p^{-1}\right)\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}+\alpha\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$
$\displaystyle=\frac{\alpha}{2}\left(\left(\alpha
L-1\right)(1+p)-1\right)\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}$
$\displaystyle+\frac{\alpha}{2}\left((\alpha
L-1)\left(1+p^{-1}\right)+2\right)\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}.$

If the factor of $\left\|\boldsymbol{\varepsilon}^{k}\right\|_{2}^{2}$ in (37)
is less than or equal to $0$, that is,

$(\alpha L-1)\left(1+p^{-1}\right)+2\leqslant 0,$ (38)

then the factor of $\|\nabla f(\boldsymbol{\theta}^{k})\|_{2}^{2}$ will be
less than $-\frac{\alpha}{2}$, which indicates that

$f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k})\leqslant-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}.$ (39)

Note that it is not difficult to demonstrate that (38) and
$\frac{L}{2}-\frac{1}{2\alpha}+\frac{\beta\gamma}{\alpha}\leqslant 0$ can
actually be satisfied at the same time. For instance, we can set
$p=0.1,\alpha=0.1,\beta=0.25,\gamma=2,L=2.5$ that satisfies both of them. ∎

Compared to LAG. Corresponding to eq. (70) in [10], LAG defines a Lyapunov
function
$\mathbb{V}^{k}:=f(\boldsymbol{\theta}^{k})-f(\boldsymbol{\theta}^{*})+\sum_{d=1}^{D}\beta_{d}\|\boldsymbol{\theta}^{k+1-d}-\boldsymbol{\theta}^{k-d}\|_{2}^{2}$
and claims that it satisfies

$\mathbb{V}^{k+1}\\!-\\!\mathbb{V}^{k}\\!\leq\\!\\!-\\!\left(\frac{\alpha}{2}\\!-\\!\tilde{c}\left(\alpha,\beta_{1}\right)(1\\!+\\!\rho)\\!\alpha^{2}\right)\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2},$ (40)

where $\tilde{c}\left(\alpha,\beta_{1}\right)=L/2-1/(2\alpha)+\beta_{1}$,
$\beta_{1}=D\xi/(2\alpha\eta)$, $\xi<1/D$, and $\rho>0$. The above result (40)
indicates that LAG requires

$K_{LAG}=\mathcal{O}\left(\frac{2\omega_{1}}{\left(\alpha-2\tilde{c}\left(\alpha,\beta_{1}\right)(1+\rho)\alpha^{2}\right)\epsilon^{2}}\right)$
(41)

communication rounds to converge. Since the non-negativity of the term
$\tilde{c}\left(\alpha,\beta_{1}\right)(1+\rho)\alpha^{2}$, we can readily
derive that
$\alpha<\alpha-2\tilde{c}\left(\alpha,\beta_{1}\right)(1+\rho)\alpha^{2}$,
which demonstrates AQUILA achieves a better convergence rate than LAG with the
appropriate selection of $\alpha$.

###  IV-B Convergence analysis under Polyak-Łojasiewicz condition.

######  Assumption 4.

($\mathbf{\mu-}$PŁ condition). Function $f$ satisfies the PL condition with a
constant $\mu>0$, that is,

$\left\|\nabla f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}\geqslant
2\mu(f(\boldsymbol{\theta}^{k})-f(\boldsymbol{\theta}^{*})).$ (42)

######  Theorem 2.

Suppose Assumptions 4.1, 4.2, and 4.3 be satisfied and
$\mathcal{M}_{c}^{k}\neq\varnothing$, if the hyperparameters satisfy
$\frac{\beta\gamma}{\alpha}\leqslant(1-\alpha\mu)\left(\frac{1}{2\alpha}-\frac{L}{2}\right)$,
then the global objective function satisfies

$\displaystyle
f(\boldsymbol{\theta}^{k+1})\\!-\\!f(\boldsymbol{\theta}^{k}){\leqslant}\\!-\\!\alpha\mu(f(\boldsymbol{\theta}^{k})\\!-\\!f(\boldsymbol{\theta}^{*}))\\!$
(43)
$\displaystyle+\\!\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}\\!-\\!\boldsymbol{\theta}^{k}\right\|_{2}^{2}\\!+\\!\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}\\!-\\!\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$

and the AQUILA requires

$K=\mathcal{O}\left(-\frac{1}{\log(1-\alpha\mu)}\log\frac{\omega_{1}}{\epsilon}\right)$
(44)

communication round with
$\omega_{1}=f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}$
to achieve
$f(\boldsymbol{\theta}^{K+1})-f(\boldsymbol{\theta}^{*})+(\frac{1}{2\alpha}-\frac{L}{2})\|\boldsymbol{\theta}^{K+1}-\boldsymbol{\theta}^{K}\|_{2}^{2}\leqslant\epsilon$.

###### Proof.

Based on the intermediate result of Theorem 1 and 4 ($\mathbf{\mu-}$PŁ
condition), we have

$\displaystyle f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{k})$ (45)
$\displaystyle\leqslant-\frac{\alpha}{2}\left\|\nabla
f(\boldsymbol{\theta}^{k})\right\|_{2}^{2}+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2}$
$\displaystyle\overset{\eqref{PL_condtion}}{\leqslant}-\alpha\mu(f(\boldsymbol{\theta}^{k})-f(\boldsymbol{\theta}^{*}))+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2},$

which is equivalent to

$\displaystyle f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{*})$
$\displaystyle\overset{\eqref{PL_condtion}}{\leqslant}(1-\alpha\mu)(f(\boldsymbol{\theta}^{k})-f(\boldsymbol{\theta}^{*}))$
(46)
$\displaystyle+\left(\frac{L}{2}-\frac{1}{2\alpha}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}+\frac{\beta\gamma}{\alpha}\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2}.$

Suppose
$\frac{\beta\gamma}{\alpha}\leqslant(1-\alpha\mu)\left(\frac{1}{2\alpha}-\frac{L}{2}\right)$,
we can show that

$\displaystyle
f(\boldsymbol{\theta}^{k+1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\left\|\boldsymbol{\theta}^{k+1}-\boldsymbol{\theta}^{k}\right\|_{2}^{2}$
(47)
$\displaystyle\leqslant(1-\alpha\mu)\left(f(\boldsymbol{\theta}^{k})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\left\|\boldsymbol{\theta}^{k}-\boldsymbol{\theta}^{k-1}\right\|_{2}^{2}\right).$

Therefore, after multiply $k=1,2,\cdots,K$, we have

$\displaystyle
f(\boldsymbol{\theta}^{K+1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\left\|\boldsymbol{\theta}^{K+1}-\boldsymbol{\theta}^{K}\right\|_{2}^{2}$
(48)
$\displaystyle\leqslant(1-\alpha\mu)^{K}\left(f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\left\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\right\|_{2}^{2}\right)\leqslant\epsilon,$

which demonstrates that our proposed AQUILA requires
$K=\mathcal{O}\left(-\frac{1}{\log(1-\alpha\mu)}\log\frac{\omega_{1}}{\epsilon}\right)$
communication round with
$\omega_{1}=f(\boldsymbol{\theta}^{1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\|\boldsymbol{\theta}^{1}-\boldsymbol{\theta}^{0}\|_{2}^{2}$
to achieve
$f(\boldsymbol{\theta}^{K+1})-f(\boldsymbol{\theta}^{*})+\left(\frac{1}{2\alpha}-\frac{L}{2}\right)\|\boldsymbol{\theta}^{K+1}-\boldsymbol{\theta}^{K}\|_{2}^{2}\leqslant\epsilon$.
∎

Compared to LAG. According to eq. (50) in [10], we have that

${\mathbb{V}^{K}}\leq\left(1-\alpha\mu+\alpha\mu\sqrt{D\xi}\right)^{K}{\mathbb{V}^{0}},$
(49)

where $\xi<1/D$. Thus, we have that LAG requires

$K_{LAG}=\mathcal{O}\left(-\frac{1}{\log(1-\alpha\mu+\alpha\mu\sqrt{D\xi})}\log\frac{\omega_{1}}{\epsilon}\right)$
(50)

communication rounds to converge. Compared to Theorem 2, we can derive that
$\log(1-\alpha\mu)<\log(1-\alpha\mu+\alpha\mu\sqrt{D\xi})$, which indicates
that AQUILA has a faster convergence than LAG under the PŁ condition.

######  Remark.

We want to emphasize that LAQ introduces the Lyapunov function into its proof,
making it extremely complicated. In addition, LAQ can only guarantee that the
final objective function converges to a range of the optimal solution rather
than an accurate optimum $f(\boldsymbol{\theta}^{*})$. Nevertheless, as
discussed in Section III-A, we utilize the precise model difference in AQUILA
as a surrogate for the global gradient and thus simplify the proof.

##  V Experiments and Discussion

TABLE I: Numerical numbers of total communication bits in the homogeneous
environment. Acc denote the text accuracy (%), PP denotes the Perplexity, and
Cost is the value of total communication bits in the entire training process
for all clients.

Total Comm Bits (GB) | QSGD | AdaQ | LAQ | LAdaQ | LENA | MARINA | AQUILA
---|---|---|---|---|---|---|---
Dataset | Data split | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost
CF-10 | IID-100 | 69.26 | 156.07 | 69.67 | 226.33 | 69.26 | 153.26 | 70.9 | 226.36 | 69.67 | 160.2 | 69.26 | 162.84 | 70.49 | 138.35
IID | 93.38 | 15.61 | 94.85 | 34.19 | 92.65 | 15.22 | 92.65 | 34.18 | 94.12 | 15.95 | 94.12 | 16.28 | 96.32 | 4.59
Non-IID | 92.65 | 15.61 | 91.91 | 20.39 | 94.85 | 14.48 | 94.85 | 19.86 | 93.38 | 17.64 | 94.12 | 16.28 | 94.12 | 11.53
CF-100 | IID-100 | 47.4 | 165.55 | 49.4 | 224.02 | 51.6 | 164.11 | 50.4 | 223.64 | 50.8 | 166.87 | 49.4 | 167.71 | 49. | 142.55
IID | 67.65 | 16.56 | 64.71 | 28.68 | 68.38 | 16.28 | 63.97 | 14.41 | 68.38 | 16.63 | 68.38 | 16.77 | 75.74 | 3.98
Non-IID | 83.09 | 8.28 | 83.82 | 14.54 | 81.62 | 8.27 | 80.15 | 14.25 | 84.56 | 9.19 | 80.88 | 8.49 | 79.41 | 6.12
WT-2 | IID-80 | 3.85 | 470.95 | 4.87 | 711.49 | 5.73 | 513.07 | 4.87 | 710.17 | 4.87 | 341.17 | 5.68 | 338.38 | 3.76 | 218.59
IID | 1.68 | 134.56 | 1.68 | 340.97 | 1.72 | 106.92 | 1.68 | 170.40 | 1.68 | 150.07 | 1.68 | 136.31 | 1.75 | 71.91

TABLE II: Numerical numbers of total communication bits in the heterogeneous environment. Acc denote the text accuracy (%), PP denotes the Perplexity, and Cost is the value of total communication bits in the entire training process for all clients. Total Comm Bits (GB) | QSGD | Ada | LAQ | Ada+LAQ | LENA | MARINA | AQUILA
---|---|---|---|---|---|---|---
Dataset | Data split | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost | Acc/PP | Cost
CF-10 | IID | 96.32 | 9.76 | 94.85 | 21.99 | 94.85 | 9.55 | 94.12 | 10.98 | 94.85 | 9.97 | 94.85 | 10.18 | 95.59 | 2.65
Non-IID | 97.06 | 9.76 | 97.06 | 16.15 | 97.79 | 9.25 | 95.59 | 14.67 | 97.06 | 11.19 | 97.06 | 10.18 | 97.79 | 7.16
CF-100 | IID | 75. | 10.56 | 72.79 | 19.42 | 75. | 10.56 | 75.74 | 9.7 | 77.94 | 10.61 | 73.53 | 10.7 | 83.82 | 2.51
Non-IID | 81.62 | 5.28 | 84.56 | 10.07 | 85.29 | 5.28 | 86.03 | 5.02 | 87.5 | 5.56 | 85.29 | 5.42 | 86.03 | 3.66
WT-2 | IID | 1.26 | 99.09 | 1.26 | 248.87 | 1.26 | 92.74 | 1.26 | 124.47 | 1.26 | 119.83 | 1.26 | 100.38 | 1.46 | 53.84

###  V-A Experiment setup

In this paper, we evaluate AQUILA on CIFAR-10, CIFAR-100 [11], and WikiText-2
dataset [12], considering IID, Non-IID data scenario, and heterogeneous model
architecture (which is also a crucial challenge in FL) simultaneously.

The FL environment is simulated in Python 3.9 with PyTorch 11.1 [13]
implementation. For the diversity of the neural network structures, we train
ResNet-18 [14] at CIFAR-10 (CF-10) dataset, MobileNet-v2 [15] at CIFAR-100
(CF-100) dataset, and Transformer [16] at WikiText-2 (WT-2) dataset.

As for the FL system setting, considering the large-scale feature of FL, we
validate AQUILA on a large system with $M=100/80$ total devices for CIFAR $/$
WikiText-2 dataset. The hyperparameters and additional details of our
experiments are revealed in Appendix.C (the supplementary file).

Figure 1: Comparison of AQUILA with other communication-efficient algorithms
on IID and Non-IID settings with homogeneous model structure. (a)-(c):
training loss v.s. total transmitted bits, (d)-(f): transmitted bits per epoch
v.s. global epoch. Figure 2: Comparison of AQUILA with other communication-
efficient algorithms on IID and Non-IID settings with heterogeneous model
structure. (a)-(c): training loss v.s. total transmitted bits, (d)-(f):
transmitted bits per epoch v.s. global epoch.

###  V-B Homogeneous environment

We first evaluate AQUILA with homogeneous settings where all local models
share the same model architecture as the global model. To better demonstrate
the effectiveness of AQUILA, its performance is compared with several state-
of-the-art methods, including AdaQuantFL, LAQ with fixed levels, LENA [17],
MARINA [18], and the naive combination of AdaQuantFL with LAQ. Note that based
on this homogeneous setting, we conduct both IID and Non-IID evaluations on
CIFAR-10 and CIFAR-100 dataset, and an IID evaluation on WikiText-2. To
simulate the Non-IID FL setting as [19], each device is allocated two classes
of data in CIFAR-10 and 10 classes of data in CIFAR-100 at most, and the
amount of data for each label is balanced.

The experimental results are presented in Figure 1, where 100% implies all
local models share a similar structure with the global model (i.e.,
homogeneity), 100% (80 devices) denotes the experiment is conducted in an 80
devices system, and LAdaQ represents the naive combination of AdaQuantFL and
LAQ. For better illustration, the results have been smoothed by their standard
deviation. The solid lines represent values after smoothing, and transparent
shades of the same colors around them represent the true values. For the
simplicity of the figure, we only display the quantization level change of
AQUILA, and the comprehensive experimental results are established in
Appendix.C (in a separated file). Additionally, Table I shows the total number
of bits transmitted by all devices throughout the FL training process.

###  V-C Non-homogeneous scenario

In this section, we also evaluate AQUILA with heterogeneous model structures
as HeteroFL [19], where the structures of local models trained on the device
side are heterogeneous. Suppose the global model at epoch $k$ is
$\boldsymbol{\theta}^{k}$ and its size is $d=w_{g}*h_{g}$, then the local
model of each device $m$ can be selected by
$\boldsymbol{\theta}_{m}^{k}=\boldsymbol{\theta}^{k}\left[:w_{m},\,:h_{m}\right]$,
where $w_{m}=r_{m}w_{g}$ and $h_{m}=r_{m}h_{g}$, respectively. In this paper,
we choose model complexity levels $r_{m}=0.5$.

Most of the symbols in Figure 2 are identical to the Figure 1. 100%-50% is a
newly introduced symbol that implies half of the devices share the same
structure with the global model while another half only have 50% * 50%
parameters as the global model.

Figure 3: Loss comparison of AQUILA with various selections of the tuning
factor $\beta$ in three datasets. Figure 4: Accuracy (Perplexity) comparison
of AQUILA with various selections of the tuning factor $\beta$ in three
datasets.

Performance Analysis. First of all, AQUILA achieves a significant transmission
reduction compared to the naive combination of LAQ and AdaQuantFL in all
datasets, which demonstrates the superiority of AQUILA’s efficiency.
Specifically, Table I indicates that AQUILA saves 57.49% of transmitted bits
in the system of 80 devices at the WikiText-2 dataset and reduces 23.08% of
transmitted bits in the system of 100 devices at the CIFAR-100 dataset,
compared to the naive combination. And other results in Table II also show an
obvious reduction in terms of the total transmitted bits required for
convergence.

Second, in Figure 1 and Figure 2, the changing trend of AQUILA’s communication
bits per each round clearly verifies the necessity and effectiveness of our
well-designed adaptive quantization level and skip criterion. In these two
figures, the number of bits transmitted in each round of AQUILA fluctuates a
bit, indicating the effectiveness of AQUILA’s selection rule. Meanwhile, the
value of transmitted bits remains at quite a low level, suggesting that the
adaptive quantization principle makes training more efficient. Moreover, the
figures also inform that the quantization level selected by AQUILA will not
continuously increase during training instead of being as AdaQuantFL. In
addition, based on these two figures, we can also conclude that AQUILA
converges faster under the same communication costs.

Finally, AQUILA is capable of adapting to a wide range of challenging FL
circumstances. In the Non-IID scenario and heterogeneous model structure,
AQUILA still outperforms other algorithms by significantly reducing overall
transmitted bits while maintaining the same convergence property and objective
function value. In particular, AQUILA reduces 60.4% overall communication
costs compared to LENA and 57.2% compared to MARINA on average. These
experimental results in non-homogeneous FL settings prove that AQUILA can be
stably employed in more general and complicated FL scenarios.

###  V-D Ablation study on the impact of tuning factor $\beta$

One key contribution of AQUILA is presenting a new lazy aggregation criterion
(8) to reduce communication frequency. In this part, we evaluate the effects
of the loss performance of different tuning factor $\beta$ value in Figure 3.
As $\beta$ grows within a certain range, the convergence speed of the model
will slow down (due to lazy aggregation). Still, it will eventually converge
to the same model performance while considerably reducing the communication
overhead. Nevertheless, increasing the value of $\beta$ will lead to a
decrease in the final model performance since it skips so many essential
uploads that make the training deficient. The accuracy (perplexity) comparison
of AQUILA with various selections of the tuning factor $\beta$ is shown in
Figure 4, which indicates the same trend.To sum up, we should choose the value
of factor $\beta$ to maintain the model’s performance and minimize the total
transmitted amount of bits. Specifically, we select the value of
$\beta=0.1,0.25,1.25$ for CIFAR-10, CIFAR-100, and WikiText-2 datasets for our
evaluation, respectively.

##  VI Conclusions and Future Work

This paper proposes AQUILA, an innovative strategy for adaptive quantization
level selection and device selection in FL scenarios. Leveraging a novel
combination of these strategies, AQUILA has been demonstrated to be capable of
reducing the transmitted costs while maintaining the convergence guarantee and
model performance compared to existing methods. The evaluation with Non-IID
data distribution and various heterogeneous model architectures demonstrates
that AQUILA is compatible in a non-homogeneous FL environment.

## Acknowledgement

This work was supported by the National Key R&D Program of China under Grant
No.2022ZD0160504, by Tsinghua Shenzhen International Graduate School-Shenzhen
Pengrui Young Faculty Program of Shenzhen Pengrui Foundation (No.
SZPR2023005), and by Tsinghua-Toyota Joint Research Institute inter-
disciplinary Program and Tsinghua University (AIR)-Asiainfo Technologies
(China) Inc. Joint Research Center under grant No. 20203910074. We would also
like to thank anonymous reviewers for their insightful comments.

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|  Zihao Zhao received his B.S. degree in University of Electronic Science and
Technology of China (UESTC) in 2021. He is currently pursuing his M.S. degree
in Data Science and Information Technology at Smart Sensing and Robotics (SSR)
group, Tsinghua University. His research interests include Internet of Things
(IoTs), Federated Learning, and Machine Learning.
---|---
|  Yuzhu Mao received the B.E. degree in computer science from Wuhan
University, Wuhan, China, in 2020. Yuzhu Mao is currently pursuing her M.S.
degree in Data Science and Information Technology at Smart Sensing and
Robotics (SSR) group, Tsinghua University. Her research interests include
Federated Learning, Internet of Things (IoTs), and Multi-agent Systems.
---|---
|  Zhenpeng Shi received the B.S. degree in Statistics from Wuhan University
in 2021. Zhenpeng Shi currently pursuing his M.S. degree in Data Science and
Information Technology at Smart Sensing and Robotics(SSR) group, Tsinghua
University. His research interest include Federated Learning, Reinforcement
Learning and Multi-agent Systems.
---|---
|  Yang Liu is an associate professor with the Institute for AI Industry
Research, Tsinghua University. Before joining Tsinghua, she was the principal
researcher and research team lead at WeBank. Her research interests include
federated learning, machine learning, multi-agent systems, statistical
mechanics and AI industrial applications. Her research work was recognized
with multiple awards, such as AAAI Innovation Award and CCF Technology Award.
She is also named as Innovators on Privacy-Preserving Computation by MIT
Technology Review China.
---|---
|  Tian Lan received the B.A.Sc. degree from the Tsinghua University, China in
2003, the M.A.Sc. degree from the University of Toronto, Canada, in 2005, and
the Ph.D. degree from the Princeton University in 2010. Dr. Lan is currently a
full Professor of Electrical and Computer Engineering at the George Washington
University. His research interests include network optimization, algorithms,
and machine learning. He received the Meta Research Award in 2021, SecureComm
Best Paper Award in 2019, SEAS Faculty Recognition Award in 2018, Hegarty
Faculty Innovation Award in 2017, AT&T VURI Award in 2015, IEEE INFOCOM Best
Paper Award in 2012, Wu Prizes for Excellence at Princeton University in 2010,
IEEE GLOBECOM Best Paper Award in 2009, and IEEE Signal Processing Society
Best Paper Award in 2008.
---|---
|  Wenbo Ding received the BS and PhD degrees (Hons.) from Tsinghua University
in 2011 and 2016, respectively. He worked as a postdoctoral research fellow at
Georgia Tech under the supervision of Professor Z. L. Wang from 2016 to 2019.
He is now an associate professor and PhD supervisor at Tsinghua-Berkeley
Shenzhen Institute, Tsinghua Shenzhen International Graduate School, Tsinghua
University, where he leads the Smart Sensing and Robotics (SSR) group. His
research interests are diverse and interdisciplinary, which include self-
powered sensors, human-machine interfaces, wearable devices for health and
robotics with the help of signal processing, machine learning, and mobile
computing. He has received many prestigious awards, including the Gold Medal
of the 47th International Exhibition of Inventions Geneva and the IEEE Scott
Helt Memorial Award.
---|---
|  Xiao-Ping Zhang received B.S. and Ph.D. degrees from Tsinghua University,
in 1992 and 1996, respectively, both in Electronic Engineering. He holds an
MBA in Finance, Economics and Entrepreneurship with Honors from the University
of Chicago Booth School of Business, Chicago, IL. He is a Professor with
Tsinghua-Berkeley Shenzhen Institute, Tsinghua University, and with the
Department of Electrical, Computer and Biomedical Engineering, Toronto
Metropolitan University (Formerly Ryerson University), Toronto, ON, Canada,
where he is the Director of the Communication and Signal Processing
Applications Laboratory. He is cross-appointed to the Finance Department at
the Ted Rogers School of Management, Toronto Metropolitan University. He was a
Visiting Scientist with the Research Laboratory of Electronics, Massachusetts
Institute of Technology. His research interests include sensor networks and
IoT, image and multimedia content analysis, machine learning, statistical
signal processing, and applications in big data, finance, and marketing. Dr.
Zhang is Fellow of the Canadian Academy of Engineering, Fellow of the
Engineering Institute of Canada, Fellow of the IEEE, a registered Professional
Engineer in Ontario, Canada, and a member of Beta Gamma Sigma Honor Society.
He is the general Co-Chair for the IEEE International Conference on Acoustics,
Speech, and Signal Processing, 2021. He is the general co-chair for 2017 and
2019 GlobalSIP Symposium on Signal, Information Processing and AI for Finance
and Business. He was an elected Member of the ICME steering committee. He is
the General Chair for the IEEE International Workshop on Multimedia Signal
Processing, 2015. He is Editor-in-Chief for the IEEE JOURNAL OF SELECTED
TOPICS IN SIGNAL PROCESSING. He is Senior Area Editor for the IEEE
TRANSACTIONS ON IMAGE PROCESSING. He served as Senior Area Editor the IEEE
TRANSACTIONS ON SIGNAL PROCESSING and Associate Editor for the IEEE
TRANSACTIONS ON IMAGE PROCESSING, the IEEE TRANSACTIONS ON MULTIMEDIA, the
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, the IEEE
TRANSACTIONS ON SIGNAL PROCESSING, and the IEEE SIGNAL PROCESSING LETTERS. He
is the Chair of the IEEE Signal Processing Society Technical Committee on
Image, Video, and Multidimensional Signal Processing (IVMSP). He received
Sarwan Sahota Ryerson Distinguished Scholar Award – the Ryerson University
highest honor for scholarly, research and creative achievements. He is an IEEE
Distinguished Lecturer of the IEEE Signal Processing Society, and of the IEEE
Circuits and Systems Society.
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