# Arbitrary-phase locking of fiber Mach-Zehnder interferometers

Ruiyang Chen  International Quantum Academy, Shenzhen 518048, China  Shenzhen
Institute for Quantum Science and Engineering, Southern University of Science
and Technology, Shenzhen 518055, China  Yi-Han Luo<EMAIL_ADDRESS>International Quantum Academy, Shenzhen 518048, China  Jinbao Long
International Quantum Academy, Shenzhen 518048, China  Junqiu Liu
International Quantum Academy, Shenzhen 518048, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei 230088, China

Optical interferometers are extensively used in fundamental physics test,
gravitational wave detection, quantum metrology, topological photonics, and
quantum information processing. Fiber-based interferometers are compact,
robust and cheap, thus are ubiquitously deployed. However, the optical phase
in fiber interferometers is sensitive to ambient perturbation, resulting in
compromised phase sensing precision. Therefore, phase control, shifting and
stabilization of fiber interferometers is essential. Methods to create stable
interference patterns and to lock a fiber interferometer at arbitrary phase
have been shown, which however are sophisticated, bulky and delicate,
preventing wider application in harsh environment outside laboratories or in
space. Here we demonstrate a new method for arbitrary-phase locking of fiber
unbalanced Mach-Zehnder interferometers. Compared to existing method, our
method is simpler, more robust and more compact. We showcase the preparation
and characterization of narrow-band energy-time-entanglement photon state
generated in integrated nonlinear microresonators, where two-photon
interference visibility reaching $0.993(6)$ is enabled. Our method constitutes
a critical building block for photonic quantum network, and is useful to
emerging single-photon interference in curved space-time that facilitates
exploration of the interface of quantum mechanics and general relativity.

Optical phase is extensively used for coding information in modern
telecommunication systems. As is sensitive to environment perturbation,
optical phase also provides a route for sensing and measurement, e.g. to
precisely characterize temperature, pressure and vibration along the fiber
path. Unlike optical intensity, direct detection of optical phase is
challenging. Thus optical interferometers, mediating optical phase and
intensity, are developed and employed. Today optical interferometers are
equally essential in emerging applications including fundamental physics test
Vallone  _et al._ (2016); Wu  _et al._ (2024), gravitational wave detection
Amaro-Seoane  _et al._ (2017); Armano  _et al._ (2018), quantum metrology
Nagata  _et al._ (2007); Afek  _et al._ (2010), topological photonics
Regensburger  _et al._ (2012); Chen  _et al._ (2018, 2023), and quantum
information processing Marcikic  _et al._ (2004); Gisin and Thew (2007);
Farrera  _et al._ (2018); Tchebotareva  _et al._ (2019); Yu  _et al._ (2020);
Zhong  _et al._ (2020); Madsen  _et al._ (2022). In these applications, stable
interference is critical. For example, the quantum interference of single
photon travelling through curved space-time can be exploited to identity the
interface of quantum mechanics and general relativity Zych  _et al._ (2011);
Xu  _et al._ (2019); Terno  _et al._ (2020). To detect gravitational red-
shift-introduced tiny phase difference $\delta\phi$, the interferometer’s
phase fluctuation must be suppressed orders of magnitude lower than
$|\delta\phi|$. In quantum communication, the fidelity of quantum information
encoding and decoding critically rely on the interferometers’ stability at the
transmitter and receiver ends.

Fiber interferometers featuring compactness, robustness and low cost are
commonly used. However, contrast to interferometers based on free-space
optics, the optical interference in fiber interferometers can be severely
corrupted by thermal and mechanical disturbances. Consequently, active
feedback control and stabilization of optical phase are critical in these
interferometers and their applications. Unbalanced Mach-Zehnder
interferometers (UMZI) are the most widely used fiber interferometers. The
phase control and stabilization in a UMZI is illustrated in Fig. 1a. A laser
of frequency $f_{0}$ enters Port 1, while Port 2 is idle. The laser is evenly
splitted into two branches by a beam splitter (BS). The fiber length in the
upper branch is longer than that of the lower branch by $L$. The two branches
recombine on another BS and create optical interference. The phase difference
$\Phi$ between the two branches linearly depends on $L$, as $\Phi=2\pi
nLf_{0}/c=\pi L/L_{\pi}$, where $n$ is the refractive index of optical fibers,
$c$ is the speed of light in vacuum, and $L_{\pi}=c/(2nf_{0})$ is the unit
length for $\pi$-phase shift. Neglecting optical loss in optical fibers and
BS, we apply $L=L_{0}+\Delta L$, where $L_{0}=2NL_{\pi}\leavevmode\nobreak\
(N\in\mathbb{N}_{+})$ and $|\Delta L|<L_{\pi}\ll L_{0}$. Thus

$\Phi=2\pi N+\pi\Delta L/L_{\pi},$ (1)

The normalized output optical intensity at Port 3 varies as
$I_{3}=(1+\sin\Phi)/2=[1+\sin(\pi\Delta L/L_{\pi})]/2$, as shown in Fig. 1b
solid curve.

Experimentally, locking $\Phi$ is equivalent to stabilize $\Delta L$. However,
due to ambient thermal and mechanical perturbation, $\Delta L$ fluctuates
temporally as $\Delta L(t)$. Commonly, a feedback loop is required to
stabilize $\Delta L(t)$. Figure 1b dashed line shows the normalized output
optical intensity $I_{4}=[1-\sin(\pi\Delta L/L_{\pi})]/2$ at Port 4 probed by
a photodetector (PD). Due to the unitary nature of BS Zeilinger (1981),
$I_{4}$ has a $\pi$-phase shift to $I_{3}$. For example, the stabilization of
$\Delta L=0$ requires locking $I_{4}$ to the set-point $I_{0}=0.5$, marked as
the star in Fig. 1b. To do so, the electric signal from the PD is fed into a
PID module, which generates a control signal based on the error signal
$\epsilon=I_{4}(\Delta L)-I_{0}$. The control signal drives the fiber
stretcher (FS), which varies the fiber length of the lower branch, such that
$\epsilon=0$ is maintained, resulting in $\Delta L=0$.

Figure 1:  Principle of a fiber unbalanced Mach-Zehnder interferometer (UMZI).
a. The common phase-locking method of a fiber UMZI. FS, fiber stretcher to add
a fiber length $\delta L$. b. The UMZI’s output optical intensity $I_{3}$
(from Port 3) and $I_{4}$ (from Port 4) as a function of $\Delta L$ For
example, locking $I_{3}=0.5$ (red star) requires locking $I_{4}=0.5$ via
reducing the error signal $\epsilon$. There are two locking regions, the
green-shaded region where $I_{4}$ has negative slope and negative feedback is
enabled; and the region where $I_{4}$ has positive or near-zero slope and the
negative feedback fails.

However, the above method has two limitations. First, the PID’s output control
signal enables negative feedback if and only if $I_{4}(\Delta L)$ is
monotonic. Exemplified in Fig. 1b, if the PID allows negative feedback on the
negative slope of $I_{4}(\Delta L)$ (green-shaded region), the feedback
becomes positive on the positive slope of $I_{4}(\Delta L)$ and thus invalid
for locking. Second, for locking near $\Delta L\sim\pm L_{\pi}/2$ marked with
crosses in Fig. 1b, $I_{4}(\Delta L)$ is insensitive to $\Delta L$ variation,
nullifying phase locking. Consequently, once the PID is configured, the
locking range is limited, e.g. the green-shaded region in Fig. 1b.

Figure 2:  Illustration of our frequency-shifted method for arbitrary-phase
locking of a fiber UMZI. a. Principle of our phase-locking method. To lock
$\Delta L$ at any value, the locking laser is frequency-shifted by $-\Delta
f$, such that $I_{4}$($\Delta L$) curve is translated by $L_{\mathrm{offs}}$,
i.e. from the red curve to the blue curve. Stabilizing $I_{4}$ at $I_{0}=0.5$
leads to locking $\Delta L=L_{\mathrm{offs}}$, corresponding to the
translation from the red star to the blue star. The slope at the red star is
identical to that at the blue star, thus the feedback remains equally
effective. b. Experimental setup. c. Comparison of phase locking performance.
As the AOM’s modulation frequency $\Delta f$ is increased stepwise, $I_{4,H}$
is measured over 1 second for each stepped $\Delta f$. Our method (red dots)
allows locking $I_{4,H}$ to any value, while the common method (blue dots)
fails in the region near $I_{4,H}=0$ and $1$ (red-shaded region). d. Extracted
phase $\Phi$ as a function of $\Delta f$, and the linear fit. Error bar is
plotted as the standard deviation of $\Phi$ at each step, which is however
much smaller than the dot size and thus is invisible.

To overcome these issues, several strategies have been implemented Freschi and
Frejlich (1995); Rogers  _et al._ (2016); Roztocki  _et al._ (2021), where a
phase-locking laser consisting of multiple frequency components is used. In
Ref. Freschi and Frejlich (1995); Rogers  _et al._ (2016), the phase-locking
laser is sinusoidally phase-modulated to generate sidebands. In Ref. Roztocki
_et al._ (2021), a portion of the phase-locking laser is frequency-shifted by
an acousto-optic modulator (AOM). The misaligned interference fringes
corresponding to different frequency components are measured, digitized, and
processed with micro-controllers to calculate the real-time phase. This phase
is compared to the set-point, and a digital control signal is generated. The
digital signal is then converted to an analogue signal driving the fiber
stretcher. In such a way, the phase is varied and stabilized.

Figure 3:  Characterization of narrow-band, energy-time entangled photon pairs
using our phase-locking method a. Experimental setup. b. Two-photon
correlation histogram. In the left/right panel, the peak at zero delay
appears/vanishes due to constructive/destructive two-photon interference. c.
Two-photon interference fringe. The fit indicates a raw visibility of
$V=0.777(8)$. By subtracting the background, the visibility is improved to
0.993(6). Error bar is plotted, however much smaller than the dot size and
thus invisible.

Here we demonstrate an innovative phase-locking method that is much simpler,
more robust and efficient compared with existing methods. The principle is
following. To stabilize $\Delta L$ at any value, instead of changing the set-
point from $I_{0}=0.5$, we shift the laser frequency $f_{0}$ to $f_{0}-\Delta
f$. Thus Eq. 1 becomes

$\displaystyle\Phi(f_{0}-\Delta f)$ $\displaystyle=2\pi n(L_{0}+\Delta
L)(f_{0}-\Delta f)/c$ $\displaystyle=2\pi n(L_{0}+\Delta
L-L_{\mathrm{offs}})f_{0}/c$ $\displaystyle=2\pi N+\pi(\Delta
L-L_{\mathrm{offs}})/L_{\pi}$ (2)

where $\Delta f\ll f_{0}$, $L_{\mathrm{offs}}=L_{0}\Delta f/f_{0}$, and the
secondary term $\Delta f_{0}\Delta L$ is neglected. A comparison of Eq. 2 with
Eq. 1 suggests that, the frequency shift of $-\Delta f$ translates
$I_{4}(\Delta L)$ by $L_{\mathrm{offs}}$. This is illustrated as the
translation from the red curve to the blue curve in Fig. 2a. Thus now, for the
laser of frequency $f_{0}-\Delta f$, stabilizing $I_{4}$ at $I_{0}=0.5$ leads
to locking $\Delta L=L_{\mathrm{offs}}$, as the translation from the red star
to the blue star in Fig. 2a. Note that, the blue curve’s slope at the blue
star ($\Delta L=L_{\mathrm{offs}}$) is identical to the red curve’s slope at
the red star ($\Delta L=0$), thus the feedback remains equally effective.
Meanwhile, sufficiently large $L_{0}$ enables $L_{\mathrm{offs}}=L_{0}\Delta
f/f_{0}>2L_{\pi}$. Therefore, with proper $\Delta f$, $\Delta L$ can be locked
to any value within $[-L_{\pi},L_{\pi}]$, as

$\displaystyle\Delta L=\frac{L_{0}}{f_{0}}\Delta f$ (3)

Figure 2b shows our experimental setup to quantify the phase-locking
performance. The output from an external-cavity diode laser (ECDL) is splitted
into two branches. In the probe branch, light is oriented to horizontal (H)
polarization and enters Port 1 of the UMZI. In the locking branch, light is
oriented to vertical (V) polarization, frequency-shifted by an AOM, and enters
Port 2. Meanwhile, a variable optical attenuator (VOA) is used to compensate
the AOM’s transmission variation with different modulation frequency $\Delta
f$. The UMZI has $L_{0}\approx 14$ m ($\sim 69$ ns time delay), and is placed
in a heat-insulated container to suppress high-frequency phase fluctuation.
Upon exiting Port 4, the light components for probe and locking are separated
by a polarizing beam splitter (PBS) and probed by two PDs.

Experimentally, the AOM’s modulation frequency $\Delta f$ is increased
stepwise from 78.0 MHz to 92.5 MHz, resulting in linearly increasing
$L_{\mathrm{offs}}$. Thus stabilizing the output optical intensity of V
polarization at Port 4, i.e. $I_{4,V}=I_{0}=0.5$ and $\Phi(f_{0}-\Delta
f)=2\pi N$, leads to locking $\Delta L=L_{\mathrm{offs}}$. With the linearly
increasing $L_{\mathrm{offs}}$ and thus $\Delta L$, the detected $I_{4,H}$ of
H polarization varies sinusoidally as $\Phi(f_{0})=2\pi N+\pi\Delta
L/L_{\pi}$. In sum, we have

$\left\\{\begin{aligned} I_{4,V}&=0.5\\\ I_{4,H}&=\frac{1+\sin(\frac{\pi
L_{0}}{L_{\pi}f_{0}}\Delta f)}{2}\\\ \end{aligned}\right.$ (4)

For each stepped value of $\Delta f$, Figure 2c red dots show the measured
$I_{4,H}$ over 1 second. Meanwhile, the locking performance of $I_{4,H}$ using
our method is compared with that using the common method (blue dots). The
latter is implemented by removing the AOM and VOA. Note that for negative and
positive slope of $I_{4,H}$ (white regions), different PID configurations are
optimized. It is clear that, while the common method fails in the region near
$I_{4,H}=0$ and $1$ (red-shaded region), corresponding to
$\Phi(f_{0})=\pi/2,3\pi/2$ and $5\pi/2$, our method remains equally effective
at any value. Each stepper $\Phi$ is extracted from measured $I_{4,H}$ using
Eq. 4, whose average value and standard deviation are calculated. Figure 2d
shows the extracted $\Phi$ as a function of $\Delta f$, and the linear fit. By
averaging the standard deviation for each stepped $\Phi$, we estimate the
phase-locking precision to be 0.011 rad. This corresponds to stabilizing
$\Delta L$ with precision of $\lambda_{0}/561$, where $\lambda_{0}=c/f_{0}$ is
the optical wavelength in vacuum. For example, we have $\lambda_{0}/561=2.76$
nm for $\lambda_{0}=1550$ nm.

To showcase an application of our phase-locking method, we characterize two-
photon interference visibility of narrow-band, energy-time entangled photon
pairs Franson (1989), which are critical quantum light sources for long-
distance quantum communications Marcikic  _et al._ (2004); Gisin and Thew
(2007); Farrera  _et al._ (2018); Tchebotareva  _et al._ (2019); Yu  _et al._
(2020). Figure 3a shows the experimental setup. The ECDL’s output is divided
into two branches. The lower branch is again the locking branch. In the upper
branch, the laser of frequency $f_{\mathrm{p}}$ is coupled into an integrated
silicon nitride (Si3N4) microresonator of intrinsic quality factor exceeding
$10^{7}$ Ye  _et al._ (2023); Chen  _et al._ (2024) and 100 GHz free spectral
range (FSR). Via cavity-enhanced spontaneous four-wave mixing (SFWM) Helt  _et
al._ (2010); Luo  _et al._ (2015) in the Si3N4 microresonator, two photons in
the pump laser annihilate, creating a pair of signal and idler photons aligned
to the microresonator’s resonance grid. The signal and idler photons have
$f_{\mathrm{s}}$ and $f_{\mathrm{i}}$ frequency. Energy conservation requires
$2f_{\mathrm{p}}=f_{\mathrm{s}}+f_{\mathrm{i}}$.

Due to random generation time of photon pairs, the photon pair
$|e_{\mathrm{s}}\rangle|e_{\mathrm{i}}\rangle$ created earlier is superposed
with that created later $|l_{\mathrm{s}}\rangle|l_{\mathrm{s}}\rangle$,
resulting in an entangled state
$|\Psi\rangle=(|e_{\mathrm{s}}\rangle|e_{\mathrm{s}}\rangle+|l_{\mathrm{s}}\rangle|l_{\mathrm{s}}\rangle)/\sqrt{2}$,
where s/i denotes signal/idler photon. To measure two-photon interference
fringe, the pump laser is filtered out and a UMZI is used to overlay photons
generated at different time Franson (1989). The signal and idler photons are
separated using a dense wavelength-division multiplexer (DWDM) after the UMZI,
and detected with superconducting nanowire single-photon detectors (SNSPD).

With the experimental setup, we project the signal/idler photon along the
state
$|\phi_{\mathrm{s/i}}\rangle=(|e_{\mathrm{s/i}}\rangle+e^{i\Phi(f_{\mathrm{s/i}})}|l_{\mathrm{s/i}}\rangle)/\sqrt{2}$,
where $\Phi(f_{\mathrm{s/i}})$ is the UMZI’s phase for the signal/idler
photon. The probability of measuring $|\Psi\rangle$ to be
$|\phi_{\mathrm{s}}\rangle|\phi_{\mathrm{i}}\rangle$ is

$\displaystyle p$
$\displaystyle=\frac{1}{4}+\frac{1}{4}\cos\left[\Phi(f_{\mathrm{s}})+\Phi(f_{\mathrm{i}})\right]$
(5) $\displaystyle=\frac{1}{4}+\frac{1}{4}\cos 2\Phi(f_{\mathrm{p}})$

Here we use $2\Phi(f_{\mathrm{p}})=\Phi(f_{\mathrm{s}})+\Phi(f_{\mathrm{i}})$
due to $2f_{\mathrm{p}}=f_{\mathrm{s}}+f_{\mathrm{i}}$ and $\Phi(f)=2\pi
nLf/c$. Experiment, we vary the AOM’s modulation frequency $\Delta f$ to
stabilize $\Phi(f_{\mathrm{p}})$ within the range $[-\pi,\pi]$.

The the photon arrival events detected by SNSPD are recorded and analyzed with
a time tagger. The two-photon correlation histogram, describing the two-photon
arrival time difference, is shown in Fig. 3b, where two-photon interference is
evidenced. When $\Phi(f_{\mathrm{p}})=0$, the central peak reaches the maximum
and is fourfold to the sidebands. When $\Phi(f_{\mathrm{p}})=\pm\pi/2$, the
central peak vanishes due to destructive interference.

To obtain interference fringe and extract interference visibility, the zero-
delay peak in Fig. 3b is post-selected. The coincidence count rate
$n_{\mathrm{cc}}$ is calculated by summing up the bins within the temporal
range $[-2.8,2.8]$ ns. Figure 3c shows the measured $n_{\mathrm{cc}}$ versus
$\Phi(f_{\mathrm{p}})$, where $\Phi(f_{\mathrm{p}})$ is calculated from
$\Delta f$ using Eq. 1 and 3. The raw interference visibility $V$ is extracted
by fitting $n_{\mathrm{cc}}$ with
$n_{\mathrm{cc}}=0.5A[1+V\cos\Phi(f_{\mathrm{p}})]$, where $V=0.777(8)$ at
$P=689\leavevmode\nobreak\ \mu$W pump power and $A$ is another fit parameter.
As shown in Fig. 3b, the background far from zero delay is substantial due to
spontaneous Raman scattering Karpov  _et al._ (2016), which heavily
deteriorates interference visibility. We calculate the background by averaging
the bin values far from the zero delay, and subtract the background for each
data point in Fig. 3c. Finally a visibility of 0.993(6) is achieved.

In conclusion, we demonstrate a simple and efficient method allowing
arbitrary-phase locking for fiber UMZI with a fully analog PID feedback. The
phase-locking system requires neither digital signal processing, nor analog-
digital conversion. In addition, upon shifting the PID set-point, its slope
remains maximized, resulting in equally efficient PID performance. For a UMZI
of arm length difference of $\leavevmode\nobreak\ 14$ meter, we experimentally
lock the phase difference $\Phi$ to any value within $[0,2\pi]$ and with 0.011
rad precision. The precision can be further improved by using a high-speed
fiber stretcher enabling larger feedback bandwidth.

We use the stabilized UMZI to showcase the preparation and characterization of
two-photon interference visibility of integrated, narrow-band, energy-time
entangled photon pairs. Our method enables two-photon interference visibility
of 0.993(6), evidencing the qualification to encode and decode information in
quantum communication systems. It also benefits emerging single-photon
interference experiments in curved space-time, which can facilitates
exploration of the interface of quantum mechanics and general relativity.
Furthermore, using ultralow-loss integrated waveguides Liu  _et al._ (2021);
Puckett  _et al._ (2021); Zhang  _et al._ (2017), integrated AOM Stanfield
_et al._ (2019); Tian  _et al._ (2020); Liu  _et al._ (2020) or EOM Wang  _et
al._ (2018); He  _et al._ (2019), and VOA Rickman (2014), our method can be
translated to integrated photonics, allowing photonic-chip-based
interferometers for integrated quantum systems and networks.

Acknowledgments: We thank Yuan Cao and Hui-Nan Wu for the fruitful discussion.
J. Liu acknowledges support from the National Natural Science Foundation of
China (Grant No.12261131503), Innovation Program for Quantum Science and
Technology (2023ZD0301500), Shenzhen-Hong Kong Cooperation Zone for Technology
and Innovation (HZQB-KCZYB2020050), and Shenzhen Science and Technology
Program (Grant No. RCJC20231211090042078). The silicon nitride chips were
fabricated by Qaleido Photonics.

Author contributions: Y.-H. L. conceived the experiment. R. C. and Y.-H. L.
built the experimental setup and performed the experiment, assisted with J.
Long. R. C., Y.-H. L. and J. Liu analyzed the data and wrote the manuscript.
J. Liu supervised the project.

Conflict of interest: The authors declare no conflicts of interest.

Data Availability Statement: The code and data used to produce the plots
within this work will be released on the repository Zenodo upon publication of
this preprint.

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# A Dwarf Galaxy Debris Stream Associated with Palomar 1 and the Anticenter
Stream

Yong Yang CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, People’s Republic
of China  School of Astronomy and Space Science, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China  Jing-Kun Zhao CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, People’s Republic of China  Xian-Hao Ye
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, People’s Republic of China
School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, People’s Republic of China  Gang Zhao CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, People’s Republic of China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, People’s Republic of China  Ke-Feng Tan CAS Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, People’s Republic of China

###### Abstract

We report the discovery of a new stream (dubbed as Yangtze) detected in $Gaia$
Data Release 3. The stream is at a heliocentric distance of $\sim$ 9.12 kpc
and spans nearly 27$\arcdeg$ by 1.9$\arcdeg$ on sky. The colour–magnitude
diagram of Yangtze indicates a stellar population of Age $\sim$ 11 Gyr and
[M/H] $\sim$ -0.7 dex. It has a number density of about 5.5 stars degree-2
along with a surface brightness of $\Sigma_{G}\simeq$ 34.9 mag arcsec-2. The
dynamics and metallicity estimate suggest that Yangtze may be closely related
to Palomar 1 and the Anticenter stream.

Milky Way stellar halo (1060) — Stellar streams (2166)

††journal: ApJL

, , , ,

##  1 Introduction

The Milky Way has proven to be full of substructures either embedded in the
disk (e.g., Zhao et al., 2009, 2018; Liang et al., 2017; Antoja et al., 2018;
Yang et al., 2021; Re Fiorentin et al., 2021; Ye et al., 2021; Zhao & Chen,
2021) or hidden in the halo (e.g., Ibata et al., 1994; Newberg et al., 2009;
Law & Majewski, 2010; Grillmair & Carlin, 2016; Helmi et al., 2018; Zhao et
al., 2020; Ibata et al., 2021; Yang et al., 2022c; Mateu, 2022). Identifying
and studying them can help us better understand the current status as well as
the past story of our Galaxy, which is especially true for stellar streams
formed from disruption of satellite galaxies. This type of streams records
accretion events that allow to reveal the formation history of the Milky Way.
They have also maintained spatial clustering properties that make it possible
to probe their kinematics and dynamics by tracing orbits (e.g., Chang et al.,
2020; Malhan et al., 2021). However, unlike globular-cluster-origin streams,
the number of known dwarf galaxy streams is still limited so far (e.g., Ibata
et al., 1994, Sagittarius,; Grillmair, 2006a, Orphan,; Newberg et al., 2009,
Cetus,; Li et al., 2018, Tucana III,; Yuan et al., 2020, LMS-1,). Thus
searching for this type of streams becomes of great importance and necessity.

In this letter, we report on the detection of a dwarf galaxy debris stream
which we designate Yangtze. The stream is exposed by weighting stars in color-
magnitude diagram (CMD) and proper motions (PMs) simultaneously using $Gaia$
Data Release 3 (DR3) (Gaia Collaboration et al., 2021; Lindegren et al., 2021;
Riello et al., 2021). Section 2 describes the detecting strategy and Section 3
characterizes the stream. A conclusion is given in Section 4.

##  2 A Stream Scanner

To search for streams of the Galactic halo, stars from $Gaia$ DR3 with
galactic latitude $|b|>15\arcdeg$ are retrieved. In order to ensure good
astrometric and photometric solutions, only stars with a renormalized unit
weight error (RUWE) $<$ 1.4 and $|C^{*}|<3\sigma_{C^{*}}$ are retained, where
$C^{*}$ is the corrected BP and RP flux excess factor that was introduced by
(Riello et al., 2021) to identify sources for which the $G$–band photometry
and BP and RP photometry are not consistent.

A modified matched-filter technique has been adopted in Grillmair (2019) and
Yang et al. (2022a) to search for the tidal streams extended from globular
clusters (GCs). The technique weights stars using their color differences from
the cluster’s locus in CMD. These weights are further scaled based on stars’
departures from PMs of the cluster’s orbit. Nevertheless, the limitation is
that we must know the location and velocity of the progenitor such that we can
integrate its orbit along which we assign weights to stars. In this work, we
improve this method to make it applicable even if there is no prior knowledge
about the stream progenitor.

In CMD, we use an isochrone of old and metal-poor stellar population (e.g.,
Age = 13 Gyr and [M/H] = -2.2 dex) extracted from Padova database (Bressan et
al., 2012) as the filter. Individual stars are assigned weights based on their
color differences from the isochrone, assuming a Gaussian error distribution:

$w_{\rm CMD}=\frac{1}{\sqrt{2\pi}\sigma_{color}}{\rm
exp}\left[-\frac{1}{2}\left(\frac{color-
color_{0}}{\sigma_{color}}\right)^{2}\right].$ (1)

Here $color$ and $\sigma_{color}$ denote $BP-RP$ and corresponding errors.
$\sigma_{color}$ is simply calculated through
$\sqrt{\sigma^{2}_{BP}+\sigma^{2}_{RP}}$ where $\sigma_{BP}$ and $\sigma_{RP}$
are obtained with a propagation of flux errors (see CDS
website111https://vizier.u-strasbg.fr/viz-
bin/VizieR-n?-source=METAnot&catid=1350&notid=63&-out=text.). $color_{0}$ is
determined by the isochrone at a given $G$ magnitude of a star. All stars have
been extinction-corrected using the Schlegel et al. (1998) maps as re-
calibrated by Schlafly & Finkbeiner (2011) with $RV$ = 3.1, assuming
$A_{G}/A_{V}=0.83627$, $A_{BP}/A_{V}=1.08337$, $A_{RP}/A_{V}=0.63439$222These
extinction ratios are listed on the Padova model site
http://stev.oapd.inaf.it/cgi-bin/cmd.. Therefore, $w_{\rm CMD}$ is a function
of distance modulus ($dm$) once the isochrone is selected. In terms of PMs,
weights are computed as:

$w_{\rm PMs}=\frac{1}{2\pi\sigma_{\mu^{*}_{\alpha}}\sigma_{\mu_{\delta}}}{\rm
exp}\left\\{-\frac{1}{2}\left[\left(\frac{\mu^{*}_{\alpha}-\mu^{*}_{\alpha,0}}{\sigma_{\mu^{*}_{\alpha}}}\right)^{2}+\left(\frac{\mu_{\delta}-\mu_{\delta,0}}{\sigma_{\mu_{\delta}}}\right)^{2}\right]\right\\}.$
(2)

Here $\mu^{*}_{\alpha}$, $\mu_{\delta}$, $\sigma_{\mu^{*}_{\alpha}}$ and
$\sigma_{\mu_{\delta}}$ are measured PMs and corresponding errors of stars.
$\mu^{*}_{\alpha,0}$ and $\mu_{\delta,0}$ are two variables of $w_{\rm PMs}$
varying in PM space.

Finally, stars weights are obtained by multiplying both $w=w_{\rm CMD}\times
w_{\rm PMs}$, which is a function of $dm$, $\mu^{*}_{\alpha,0}$ and
$\mu_{\delta,0}$ (i.e., $w$ = $w(dm,\mu^{*}_{\alpha,0},\mu_{\delta,0})$). We
calculate $w$ for all stars over a grid of the three variables: $dm$ from 11.5
to 17.5 mag with a step of 0.2 mag, $\mu^{*}_{\alpha,0}$ from -15 to 15 mas/yr
and $\mu_{\delta,0}$ from -20 to 10 mas/yr, both with a step of 1
mas/yr333These ranges and steps might be adjusted depending on specific goals
and computational abilities.. At each grid point, stars’ weights are summed in
sky pixels to expose structures. We refer to this method as
StreamScanner444Its Python code can be found at
https://doi.org/10.5281/zenodo.7604477. later because what it is doing is like
scanning streams in CMD and PM space.

Apparently, StreamScanner is a simple and direct way that relies on
measurements from observation and does not need too complicated assumptions or
models. We happen to find the signature of Yangtze in this work when
StreamScanner is placed at $(dm,\mu^{*}_{\alpha,0},\mu_{\delta,0})$ = (13.7,
-1, 1). We then refine these parameters along with choice of the isochrone as
elaborated in the next Section.

##  3 The Yangtze Stream

A weighted sky map is obtained as shown in the middle panel of Figure 1. This
is a coadded result of $w(14.8,-0.4,0.8)$ and $w(14.8,-1,1.4)$ using an
isochrone with Age = 11 Gyr and [M/H] = -0.7 dex, where the isochrone and its
$dm$ are determined by a CMD fit to the stream (see Section 3.1) and the
choices of $(\mu^{*}_{\alpha,0},\mu_{\delta,0})$ are based on Figure 3. The
sky pixel width is 0.2$\arcdeg$ and the coadded map is smoothed with a
Gaussian kernel of $\sigma$ = 0.4$\arcdeg$. The stretch is logarithmic, with
brighter areas corresponding to higher weight regions. The white area in the
bottom left corner is due to the proximity to the Galactic disk. Furthermore,
only stars fainter than $G$ = 18 mag (corresponding to the stream’s main
sequence turn-off, see Figure 3) are shown here because random brighter field
stars might introduce large noises due to their rather low photometric and
astrometric uncertainties. On the left and right panels, we further plot the
dust extinction map extracted from Schlegel et al. (1998) and $Gaia$’s
scanning pattern covered by the DR3, respectively, with higher values
represented by darker colors. The red dashed line of the three panels
indicates the trajectory of Yangtze.

Figure 1: The left and right panels present the dust extinction map extracted
from Schlegel et al. (1998) and $Gaia$’s scanning pattern covered by the DR3,
respectively. The middle panel presents the weighted sky map for stars fainter
than $G$ = 18 mag. The green areas are two moving masks used to generate the
stream’s lateral and longitudinal profiles in Figure 2. The red dashed line of
the three panels indicates the trajectory of the stream.

From the matched-filter map, the signature of Yangtze starts at $\delta$ =
-13$\arcdeg$ and ends at $\delta$ = 14$\arcdeg$. Besides, there are several
random noises appearing around the stream, which do not represent actual
physical overdensities. According to the other two panels, we can verify that
the stream does not follow any structures in the interstellar extinction and
is not aligned with any features in $Gaia$’s scanning pattern. Although there
is a small region at $\delta\sim$ 0$\arcdeg$ where dust might be heavy, it
does not resemble the stream pattern. To make sure that dereddening photometry
with the map from Schlegel et al. (1998) is appropriate for this region, we
compare the extinction to that extracted from 3D dust map (Bovy et al., 2016)
by assuming the stream distance of 9.12 kpc (see CMD fitting in Section 3.1)
and find that the typical difference in $A_{V}$ is only $\sim$ 0.04 mag,
meaning that the 2D dust map we used from Schlegel et al. (1998) is accurate
enough. The trajectory can be well described with a second-order polynomial:

$\alpha=5.194\times 10^{-3}\delta^{2}+4.685\times 10^{-2}\delta+146.141$ (3)

with -13$\arcdeg$ $<$ $\delta$ $<$ 14$\arcdeg$.

To estimate the stream’s width, we create a mask as shown with the left green
window in Figure 1. Here $\delta$ is between -3$\arcdeg$ and 8$\arcdeg$ and
this segment corresponds to the most prominent portion of the detected
features. The boundaries on left and right sides are parallel to Equation 3
and separated by 0.8$\arcdeg$ (the mask’s width). We then move the mask across
the stream from left to right and sum all weights of stars fainter than $G$ =
18 mag in the mask (again, to reduce large noises caused by bright stars), to
create a one-dimensional stream profile as shown with the red solid line in
the left panel of Figure 2. This is directly analogous to the T statistic of
Grillmair (2009). From Figure 2, the stream is almost enclosed within
$-2\arcdeg<$ offset $<2\arcdeg$, and its peak is 14.88$\sigma$ above the
background noise outside this range. From the profile, we find an estimate of
its width (full width at half maximum) to be $\sim 1.9\arcdeg$.

Figure 2: Left: the lateral distributions for weighted (red solid) and
unweighted (blue dashed) numbers of stars fainter than $G$ = 18 mag, obtained
by moving the left mask in Figure 1 across the stream. Right: similar to the
left panel but for the longitudinal profile obtained by moving the bottom mask
along the stream.

Furthermore, we create the lateral profile of unweighted stars in the same way
and overplot it with the blue dashed line in left panel of Figure 2. There is
a gradient in the distribution of stars along offset, with more stars
populating at lower offset (near the disk). It can be concluded that the
stream signature is not caused by contamination of the Milky Way’s field
population. Otherwise, it is more likely to detect strong signals close to
offset = -6$\arcdeg$ side.

Following the same way, the stream’s longitudinal profile can be obtained
using the bottom mask (5$\arcdeg$ by 1.6$\arcdeg$) in Figure 1, by moving it
along the declination. As displayed in the right panel of Figure 2, the red
solid and blue dashed lines are presenting the same meaning as those of the
left panel. From the profile of weights, several gaps are found at, for
example, $\delta\sim$ -5$\arcdeg$, 2$\arcdeg$ and 9$\arcdeg$, which also
appear obvious in Figure 1. This kind of signature may suggest a result of
non-continuous stripping (Grillmair, 2011), or collisions with massive
perturbers in the Milky Way (Bonaca et al., 2019).

###  3.1 CMD and PMs

We display a background-subtracted binned CMD for the stream in the left panel
of Figure 3. The stream region is defined as the area around the trajectory
(Equation 3) $\pm 1\arcdeg$ in $\alpha$ direction, given the derived width of
1.9$\arcdeg$. The background is estimated through averaging two off-stream
regions parallel to the stream obtained by moving the stream region along
$\alpha$-axis by $\pm 3\arcdeg$, to eliminate the effect of the gradient.
Before the background subtraction, a PM selection is applied to both of the
stream and off-stream regions as illustrated with the red polygon in the right
panel of Figure 3, which corresponds to the stream’s distribution in PM space
(see below). We emphasize that this is a subtraction of star numbers, not
weighted counts.

The CMD bin size is 0.05 mag in color and 0.2 mag in $G$ magnitude. The
diagram is smoothed with a 2D Gaussian kernel of $\sigma$ = 1 pixel. The blue
dashed line represents the best-fit isochrone with Age = 11 Gyr and [M/H] =
-0.7 dex at $dm$ = 14.8 mag. To find this best combination, we explore an
isochrone grid that covers a metallicity range of -2.2 $\leq$ [M/H] $\leq$
-0.2 and an Age range of 10 $\leq$ Age $\leq$ 13 with 0.1 dex and 1 Gyr
spacing, along with a $dm$ varying from 13 to 16 with a step of 0.1 mag. The
stream’s strength is measured through the total weights between $-2\arcdeg<$
offset $<2\arcdeg$ in Figure 2 and the best-fit result is found when the
stream signals become the strongest. The PM term of StreamScanner is fixed at
(-0.4, 0.8) and (-1, 1.4) as mentioned before during this process. After the
PM selection and background subtraction, the stream’s main sequence along with
its turn-off is clearly seen and has a good match with the isochrone. The $dm$
= 14.8 mag corresponds to a heliocentric distance of $\sim$ 9.12 kpc.
Considering the width of 1.9$\arcdeg$, the physical width of Yangtze is about
302 pc, implying a dwarf galaxy origin.

Figure 3: The left panel is a 2D histogram of stars in CMD with PMs selected
and background subtracted. The blue dashed line represents the best-fit
isochrone with Age = 11 Gyr and [M/H] = -0.7 dex at $dm$ = 14.8 mag. The right
panel is a 2D histogram of PMs after CMD selection and background subtraction.
Both of the diagrams are smoothed with a 2D Gaussian kernel of $\sigma$ = 1
pixel. The red polygons represent the CMD and PM selections applied to the
stream and off-stream regions.

In the right panel of Figure 3, we present a 2D histogram of PMs. Similarly,
before the subtraction between the stream region and the mean of the off-
stream regions, a CMD selection is applied to them as shown with the red
polygon in the left panel. The diagram with bin size = 0.2 mas/yr is also
smoothed using a 2D Gaussian with $\sigma$ = 1 pixel. An overdensity at
$\mu^{*}_{\alpha}\sim$ -0.7 mas/yr and $\mu_{\delta}\sim$ 1.1 mas/yr is
discernable corresponding to the stream.

We further estimate the stream’s surface density and brightness. There are a
total of 296 stars within the PM polygon after the background subtraction.
This serves as an estimate to the number of the stream stars located in a
$27\arcdeg\times 2\arcdeg$ region. Thus the surface density is roughly 5.5
stars degree-2. For all stars in the stream region, each one is assigned a
weight by StreamScanner and this allows us to select the most likely members
of the stream based on the sorting of weights. We adopt stars with weights $>$
1.255 as the member candidates because the criterion leaves us 296 stars as
well. By combining their individual $G$ magnitudes, we get a surface
brightness of Yangtze to be $\Sigma_{G}\simeq$ 34.85 mag arcsec-2. We can put
this another way. We have 298 stars satisfying the CMD and PM selections
simultaneously, which can be treated as the member candidates as well. Those
stars give a surface brightness of $\Sigma_{G}\simeq$ 34.95 mag arcsec-2, very
close to the previous one.

###  3.2 Association with GCs and Streams

We aim to fit an orbit to Yangtze such that we can investigate whether it is
related to any GCs or known streams of the Milky Way. We adopt a Galactic
potential model used in Yang et al. (2022a, b), where the halo is taken from
McMillan (2017) and non-halo components come from Pouliasis et al. (2017,
model I). The position of the sun is set as $(R_{\sun},Z_{\sun})$ = (8.122,
0.0208) kpc (GRAVITY Collaboration et al., 2018; Bennett & Bovy, 2019), and
the solar velocities are set to $(V_{R,\sun},V_{\phi,\sun},V_{Z,\sun})$ =
(-12.9, 245.6, 7.78) km s-1 (Drimmel & Poggio, 2018), respectively. The
fitting parameters are position $\alpha$, $\delta$, heliocentric distance $d$,
PMs $\mu^{*}_{\alpha}$, $\mu_{\delta}$, and radial velocity $V_{r}$. We chose
to anchor the declination at $\delta$ = 0$\arcdeg$, near the midpoint of the
stream, leaving other parameters free to be varied. In a Bayesian framework,
sky positions and PMs of 298 member candidates are used to constrain the
parameters and the fitted results can be derived from their marginalized
posterior distributions through a Markov Chain Monte Carlo sampling. We have
cross-matched these candidates with spectroscopic catalogs like LAMOST or
SDSS, but unfortunately no common stars are found thereby no radial velocities
available. The best-fit parameters are $\alpha=146.14^{+0.06}_{-0.06}$
$\arcdeg$, $d=8.58^{+0.39}_{-0.53}$ kpc,
$\mu^{*}_{\alpha}=-0.76^{+0.02}_{-0.02}$ mas yr-1,
$\mu_{\delta}=0.96^{+0.02}_{-0.01}$ mas yr-1 and $V_{r}=8.78^{+5.16}_{-5.03}$
km s-1. We then obtain the stream’s orbit by integrating it for $\pm$ 1 Gyr
under the same potential.

We first examine possible connections between Yangtze and GCs by comparing
their angular momenta $L_{z}$ and energy $E_{\rm tot}$ as shown in the left
panel of Figure 4. The positions and velocities of GCs are taken from Vasiliev
& Baumgardt (2021). We note that the stream lies rather near Palomar 1 (Pal 1)
as marked by the blue triangle. Sakari et al. (2011) already showed that Pal
has unusual chemical characteristics including lower [$\alpha$/Fe] ratios than
Galactic stars of the same [Fe/H], and they concluded an extragalactic origin
for Pal 1. The metallicity of Pal 1 is -0.6 dex which is close to -0.7 dex of
Yangtze although the latter is estimated by an isochrone fitting. However,
Yangtze’s orbit does not pass through Pal 1 as displayed in the right panel of
Figure 4, and we have also verified that the cluster’s orbit does not match
with the stream’s track either. Thus it might be a better interpretation that
Pal 1 is a GC brought in by the progenitor of Yangtze.

Among known streams, we find that the trajectory of the Anticenter Stream
(ACS; Grillmair, 2006b) matches quite well with the orbit of Yangtze as shown
in the right panel of Figure 4. Here the footprint of ACS is based on results
of Ramos et al. (2021), which is extracted from galstreams library (Mateu,
2022). ACS is likely on the Yangtze’s orbit though they are separated by one
orbit around the Galaxy. Yangtze’s pericenter and apocenter are $R_{peri}$ =
12.1 kpc and $R_{apo}$ = 18.9 kpc, close to $R_{peri}$ = 15.4 kpc and
$R_{apo}$ = 19.0 kpc of ACS (Grillmair et al., 2008). It is also worth noting
that our metallicity estimate is nearly identical to [Fe/H] = -0.72 dex of ACS
(Zhang et al., 2022). All similarities connect these two streams together and
imply that they may share a common origin. Besides ACS, we also show the
Eastern Banded Structure (EBS; Grillmair, 2011), another substructure in the
anticenter region. The three streams are located rather close and nearly
parallel to one another on sky but for EBS and Yangtze, their inferred orbits
suggest that they are dynamically distinct.

Figure 4: The left panel displays GCs (black points) and Yangtze (red circle)
in angular momenta and energy space. The right panel presents projections of
streams and orbits in Galactic coordinate. The red track and black line
indicate Yangtze and its best-fit orbit. The blue and purple lines represent
ACS and EBS, respectively. The blue triangle denotes Pal 1.

##  4 Conclusion

With revised photometry and astrometry from $Gaia$ DR3, we show a new stream
probably formed from a dwarf galaxy’s disruption, which we dub Yangtze. The
stream is detected at a significance of $\sim 15\sigma$ with a StreamScanner
method that assigns weights to stars in CMD and PMs simultaneously. The
Yangtze is spanning 27$\arcdeg$ by 1.9$\arcdeg$ on sky at 9.12 kpc away from
the sun. Its stellar population is well fitted with an isochrone of Age = 11
Gyr and [M/H] = -0.7 dex. The stream has a low surface brightness of
$\Sigma_{G}\simeq$ 34.9 mag arcsec-2, with a density of about 5.5 stars
degree-2. We also demonstrate that Yangtze may be associated with ACS and Pal
1, and we suspect that these two streams are remnants of one dwarf galaxy that
brought the cluster into the Milky Way. Spectroscopic observations will be
required to give deeper insights to it.

We thank the referee for the thorough reviews that helped us to improve the
manuscript. This study is supported by the National Natural Science Foundation
of China under grant nos 11988101, 12273055, 11973048, 11927804, 11890694,
11873052 and 12261141689, and the National Key R&D Program of China, grant no.
2019YFA0405500. This work is also supported by the GHfund A (202202018107). We
acknowledge the support from the 2m Chinese Space Station Telescope project
CMS-CSST-2021-B05. This work presents results from the European Space Agency
(ESA) space mission Gaia. Gaia data are being processed by the Gaia Data
Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by
national institutions, in particular the institutions participating in the
Gaia MultiLateral Agreement (MLA). The Gaia mission website is
https://www.cosmos.esa.int/gaia. The Gaia archive website is
https://archives.esac.esa.int/gaia.

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#  Towards Energy Efficient Federated Learning over 5G+ Mobile Devices
††thanks: Dian Shi, Rui Chen, Pavana Prakash, and Miao Pan are with University
of Houston; Liang Li is with Xidian University; Yuguang Fang is with
University of Florida.

Dian Shi, Liang Li, Rui Chen, Pavana Prakash, Miao Pan, and Yuguang Fang

###### Abstract

The continuous convergence of machine learning algorithms, 5G and beyond (5G+)
wireless communications, and artificial intelligence (AI) hardware
implementation hastens the birth of federated learning (FL) over 5G+ mobile
devices, which pushes AI functions to mobile devices and initiates a new era
of on-device AI applications. Despite the remarkable progress made in FL, huge
energy consumption is one of the most significant obstacles restricting the
development of FL over battery-constrained 5G+ mobile devices. To address this
issue, in this paper, we investigate how to develop energy efficient FL over
5G+ mobile devices by making a trade-off between energy consumption for
``working'' (i.e., local computing) and that for ``talking'' (i.e., wireless
communications) in order to boost the overall energy efficiency. Specifically,
we first examine energy consumption models for graphics processing unit (GPU)
computation and wireless transmissions. Then, we overview the state of the art
of integrating FL procedure with energy-efficient learning techniques (e.g.,
gradient sparsification, weight quantization, pruning, etc.). Finally, we
present several potential future research directions for FL over 5G+ mobile
devices from the perspective of energy efficiency.

## Introduction

Machine learning (ML), particularly deep learning (DL), is one of the most
disruptive technologies the world has witnessed in the last few years.
Unfortunately, cloud-centric ML generates tremendous traffic and also causes
serious privacy concerns, which is not suitable for many resource-constrained
applications. In order to scale and move beyond the cloud-centric ML, Google
has introduced federated learning (FL), the currently popular distributed
machine learning paradigm, which aims to enable mobile devices to
collaboratively learn a joint global ML model without sharing their privacy
sensitive raw data [1]. With FL, distributed data stakeholders (e.g., mobile
devices) only need to periodically upload their updated local models to the
aggregation server for global updates, instead of uploading their potentially
private raw data, thus significantly lowering the risk of privacy leakage.
However, stakeholders in many IoT applications, like smart devices, are
resource constrained in terms of spectrum, energy, computing and storage,
which makes FL for such on-device applications highly challenging. Therefore,
another recent surging technology, 5G and beyond (5G+) [2], can further
facilitate the implementation of FL over mobile devices. First, due to the
advance of hardware design, 5G+ mobile devices are usually armed with ever-
increasingly high-performance computation units, such as the central
processing units (CPUs) and graphics processing units (GPUs), which enable
them to host computation-intensive learning tasks. Besides, 5G standard has
also embraced the computing capability, such as multi-access edge computing
(MEC), paving the way for performing computing for edge intelligence, and
hence building an effective wireless network architecture to support viable
FL. Moreover, 5G+ wireless transmissions are featured by a very high data rate
and ultra-low latency, which can be leveraged to tackle the communication
bottleneck issue for local model updates during training. Such a combination
of 5G+ and FL prompts tremendous successful applications over 5G mobile
devices, including keyboard prediction [3], cardiac event prediction [4],
financial risk management [5], etc.

While deploying FL over 5G+ mobile devices is promising to have so many
interesting applications, FL and 5G+ mobile devices cannot be easily married
for fruitful use without frictions. Severe challenges are foreseeable, of
which energy consumption is the dominant concern. On the one hand, executing
on-device computing and performing local model updates are both resource-
hungry, inducing a significant surge of energy consumption on mobile devices
and hence draining significant battery power. Thus, the first mountain we have
to climb is to improve the system energy efficiency in order to prolong the
lifetime of mobile devices during training, where the energy consumption
usually comes from both the local computing and wireless communications. On
the other hand, for FL over 5G+ mobile devices, there will be a trade-off
between computing and communication over resource-constrained mobile devices.
This stems from our observation on the comparable energy consumption for on-
device training with high-performance processors and wireless transmissions
with advanced communication techniques. For example, to transmit a ResNet-50
model, a commonly used deep network for image classification, with
approximately 100MB parameters via 100 Mbps 5G wireless uplinks typically
consumes 30J for Industry IoT devices [6]. This is comparable to the energy
consumption for performing a single-step local training on one GPU (e.g., 30J
for NVIDIA Tesla V100 on ImageNet dataset [7]). Therefore, how to make a
trade-off between computing and communications in order to accommodate
realistic computing environments is another critical issue when deploying FL
over 5G+ mobile devices.

Motivated by the aforementioned challenges, we plan to investigate the energy-
efficient FL over 5G+ mobile devices in this paper. Our goal is to enable
effective and efficient local training on 5G+ mobile devices while minimizing
the overall energy consumption for FL over 5G+ mobile devices for both
involved communications and computing. To this end, we first give an overview
on the FL over 5G+ mobile devices and discuss the energy consumption models.
Then, we study the local computing and wireless transmission co-design from
the long-term learning perspective, where we make a trade-off between the two
parts simultaneously. In addition, several advanced techniques, including
gradient sparsification, gradient quantization, weight quantization, model
pruning, and dynamic batch sizes, are integrated into the FL training
procedure to further reduce energy for 5G+ mobile devices. Finally, we
conclude this paper with discussions on potential research directions for
energy efficient FL over 5G+ mobile devices.

## Backgrounds and Energy Models

### FL over 5G+ Mobile Devices

As an emerging decentralized learning paradigm, FL has taken advantage of the
computing resources across massive participants. Specifically, all
participants collaboratively contribute to one global learning task in a
distributed manner with continuous interactions for model parameter updates.
With FL over 5G+ mobile devices, all 5G+ mobile devices (e.g., smartphones,
laptops, automatic vehicles, etc.) can serve as participants, and a server
such as gNodeB acts as the aggregator. In particular, a 5G server first
broadcasts a current global model to the participating 5G+ mobile devices in
FL. After receiving the global model, a 5G+ mobile device conducts the local
on-device training based on the local data and its computing capability.
Second, when a 5G+ mobile device finishes its local training in this round, it
will upload its local model updates (i.e., gradients) to the server via
wireless links with 5G+ techniques. Finally, the server does the aggregation
over all the received local gradients to update the global model and then
feeds it back to the participated mobile devices for the next-round training.
The above procedures are repeated until obtaining a converged global model,
which can be deployed by the 5G+ devices for future utilization. A typical
paradigm of the FL over 5G+ mobile devices, including local computation and
wireless communications parts, is shown in Fig. 1.

Figure 1: The illustration of the FL over 5G+ mobile devices.

FL over 5G+ mobile devices has become a natural way to implement the
artificial intelligence (AI) at the edge, like the keyboard prediction
introduced by Google [3]. Such a combination pushes AI functions to mobile
devices, which provides a flexible and convenient approach to conducting a
learning task, especially for some real-time and lifelong learning
applications. However, deploying FL over 5G+ mobile devices raises tremendous
challenges and difficulties, and energy consumption is one of the most
significant issues. Unlike other central servers with wired connections,
mobile devices have limited energy resources due to the limited battery power.
Besides, the energy consumption of wireless transmissions is not encountered
in learning scenarios with wired servers. Both make it extremely difficult for
5G+ mobile devices to handle the energy-hungry training tasks. In light of
this, it is worthwhile to investigate the FL's energy model over 5G+ mobile
devices to deal with energy-saving issue.

### Communication and Computation Energy Models

With FL, all 5G+ mobile devices contribute to one unified global model by
continuously transferring the local model updates with the centralized
aggregator. In this process, the energy consumption of the 5G+ mobile devices
mainly comes from the wireless transmissions of the model updates and the
local computation executed on them. Hence, brief descriptions of the
communication energy model and the computation energy model are given as
follows.

Communication: All the participating 5G+ mobile devices transmit their
computed local model updates to the central aggregator through the wireless
transmissions, which corresponds to the communication energy consumption. Note
that conducting an FL task usually takes several minutes due to the heavy
computational load of training. In this situation, the channel condition may
not remain the same all the time and may suffer from fluctuations. Therefore,
one possible way to model the transmission rate for each device throughout the
entire training process is to do it in an expected manner, where the
expectation is taken over the channel variations. Besides, according to the
Shannon–Hartley theorem, both the bandwidth and the transmission power also
impact the transmission rate, and the power of additive white Gaussian noise
(AWGN) needs to be considered as well. Accordingly, the overall energy
consumed by each device for wireless transmissions during the training process
can be formulated as the product of the required number of global
communication rounds and the energy consumption in a single round. Here, the
one-round energy consumption is related to the transmission power, the
transmission rate, and the model size.

According to the characteristics of the energy model mentioned above, two
possible ways can be adopted to save the energy consumption on the
communication part in the entire training process, namely, decreasing the
required rounds of global communications and reducing the communication
workloads per round. Given this observation, several approaches can be
implemented to save the communication overhead. On the one hand, global
synchronizations can be taken after several local computing iterations to
decrease the communication frequency, that is, the federated average [1].
Similarly, the number of required global communication rounds can also be
reduced by gradually increasing the batch size throughout the training. On the
other hand, model compression technologies, e.g., model sparsification and
quantization, can greatly help reduce the size of the local model to be
transmitted, thus saving the communication energy in each round.

Computing: With the ever-increasing popularity of smart devices equipped with
high-performance GPUs, 5G+ mobile devices can undertake heavy computations
even for deep learning tasks. However, due to the powerful computational
capability and the massively parallel architecture of the GPU, the
computational energy consumption has been a significant burden for the
learning scenarios, especially for training tasks implemented on the mobile
devices with limited battery power. Thus, more research efforts are needed to
investigate the computing energy model and the corresponding computing energy-
saving strategies. Here, we assume that the 5G+ mobile devices are equipped
with GPUs, which are widely assumed in modern learning training. Specifically,
the GPU computation architecture involves the memory modules referring to data
fetching and the core modules referring to the data calculation. Under this
architecture, the voltage and the frequency of the corresponding modules can
be controlled independently.

The energy consumption for feeding a mini-batch of data in one local iteration
can be calculated as the product of the execution time and the runtime power.
Here, the execution time is determined by the device-dependent parameters,
like the memory frequency and the core frequency, and the task specific
information, such as the number of cycles for data fetching and calculation
[8]. Similarly, the runtime power is also affected by device-dependent
parameters, including the frequency and the voltage, and coefficients related
to the specific learning tasks. Hence, the total computation energy
consumption can be computed as the product of energy consumption for one local
iteration and the total number of iterations. Currently, model compression
techniques, such as pruning and weight quantization, together with the
corresponding hardware co-design, can alleviate the burden of computing energy
consumption in one local iteration because the needed number of cycles for
data fetching and calculation is decreased. The total number of local
iterations will increase mildly due to the falling of the model precision in
each local iteration, but in all, the total computational energy will be
saved.

## Energy-Efficient FL via Communication and Computing Co-Design

As illustrated in the previous section, the energy consumption of FL over 5G+
mobile devices mainly comes from two parts: on-device local computing and
wireless communications for model updates. Thus, in this section, we focus on
how to integrate various technologies (e.g., gradient sparsification, gradient
quantization, weight quantization, model pruning, etc.) and develop
corresponding communication and computing co-design to reduce overall energy
consumption for FL over 5G+ mobile devices. In this section, we will survey
some recent developments focusing on the communication and computing energy
consumption co-design, which are illustrated in Fig. 2.

Figure 2: Illustrative scenarios for energy-efficient FL over 5G+ mobile
devices.

### Gradient Sparsification to Reduce Communication Energy

FL communication energy consumption can be reduced by integrating the training
algorithm with two state-of-the-art communication compression strategies,
namely, local computations and gradient sparsification. The former allows more
local computations performed on the 5G+ mobile device between every two global
model updates, thereby reducing the total number of communication rounds. The
latter lets participants upload only a fraction of gradients with significant
magnitudes, thereby reducing the communication payload in each round. Besides,
error compensation is applied at each participant after every model update to
accelerate the global convergence by accumulating the error that arises from
only uploading sparse approximations of the gradient updates, ensuring all
gradient information gets eventually aggregated.

The convergence results for our FL algorithm indicate that, from the learning
perspective, the gradient sparsity magnitudes of all the participants jointly
make an impact on global convergence and communication complexity. Given a
target model accuracy, a lower sparsity results in a larger bound of
communication rounds. Besides, aggressively increasing local iterations can
also impair the learning efficiency since more communications may be involved.
In a realistic edge computing environment, these two types of compression
factors implicitly determine the energy consumption for participating 5G+
mobile devices by affecting the payload required for transmissions and the
workload required for processing, respectively. Thus, they are needed to be
well balanced for each participant to accommodate the specific environment.
This can be achieved by formulating a compression control problem using the
derived convergence bound from the long-term learning perspective [9], with
the goal for optimizing the overall energy efficiency for FL on 5G+ mobile
devices over wireless networks. As shown in Fig. 3(a), the flexible
sparsification based FL method (``FlexibleSpar''), which considers the
heterogeneity of participating 5G+ mobile devices and provides the flexible
sparsification strategies, consume less energy than the other methods (FedAvg;
``UnifiedSpar'': the unified sparsity approach) to reach a given target
accuracy. Furthermore, Fig. 3(b) shows the convergence rate in terms of
training epochs, which indicates that ``FlexibleSpar” exhibits very similar
behavior with ``UnifiedSpar” in terms of convergence rate and final accuracy,
both of which slightly underperform the baseline approach, i.e., FedAvg [1].

(a) Test error vs. energy consumption. (b) Training accuracy vs. epoch.

Figure 3: Performance evaluation for the FL with gradient sparsification.

### Gradient Quantization to Reduce Communication Energy

Similar to the gradient sparsification technique, another efficient
compression method to reduce the communication energy consumption is gradient
quantization. After completing the local training in one round, the computed
gradients should be uploaded to the aggregator for the global updates. With
gradient quantization, instead of uploading all raw gradients, we can also
quantify the computed local gradients with low precision, i.e., a small number
of bits, thus reducing the communication load per round. In this way, a high
gradient quantization level corresponds to a low communication energy
consumption per round. In contrast, the required global rounds increase due to
the precision reduction, thus increasing the total computing energy.
Therefore, the gradient quantization level needs to be controlled to make a
trade-off between the communication and computing for the overall energy
efficiency. It should be noted that local updates (i.e., gradients) refer to
the communication part, and thus gradient quantization can reduce the
communication energy consumption. However, when we download the global model
on the device, the weight quantization is related to the computing part, which
will be discussed in the following subsection.

### Weight Quantization to Reduce Computing Energy

Weight quantization is regarded as a promising solution to decrease the local
memory consumption in FL training. It reduces the model complexity and
computing energy consumption via representing the model parameters with small
bit-widths (e.g., 8-bit or 16-bit fixed point numbers) on 5G+ mobile devices.
Besides, considering device heterogeneity, the quantization selections for
different participants are varying. However, the quantization induces
information loss during training. The errors between the quantized and
original values make the FL model converge to a neighborhood of the optimal
solution. Smaller quantization levels (i.e., bit representations) lead to a
higher error and push the FL model further away from the minima. For
synchronous aggregation, given a fixed number of communication rounds, the
accuracy of the FL model depends on the average quantization errors of all the
participants. In this way, given a model accuracy, the participants, with
limited computing power and poor channel conditions, prefer more aggressive
quantization strategies (smaller bit-widths) to train the local models
efficiently. In contrast, others would coordinate with higher precision
training for the model performance.

As a result, one can consider a joint design for flexible quantization
selection and bandwidth allocation to capture the trade-off between the local
computing and communications and minimize the overall energy consumption for
FL training within allowed deadline [10]. One example of a stochastic
quantization scheme could first determine a quantization set based on
different quantization levels, and then map the model weights to the nearest
quantization point with high probability. The participating devices could
determine different quantization levels depending on their device capabilities
and the targeted model accuracy. Meanwhile, the server allocates the wireless
bandwidth to the participants considering both the channel conditions and
participants’ computing capabilities. This process terminates when it reaches
a certain global model accuracy level. Fig.4(a) shows the overall energy
consumption for the FL training procedure under different learning mechanisms.
For a fixed number of training iterations, those mechanisms equipped with
quantization (“UnifiedQnt” and “FWQnt”) consume less energy than ``FedAvg"
without quantization. Specifically, the scheme “FWQnt” considers the model
performance, device heterogeneity and wireless channel conditions and enhances
the energy efficiency with x2 - x100 less energy consumption than the other
schemes in the FL training process under the same accuracy level. Moreover,
the convergence rates for the corresponding schemes are also shown in
Fig.4(b), where both quantization based approaches slightly underperform
``FedAvg".

(a) Training accuracy vs. energy consumption. (b) Test error vs. epoch.

Figure 4: Performance evaluation for the FL with weight quantization.

### Model Pruning to Reduce Computing Energy

Another efficient learning technique that can be integrated into the FL
process is model pruning, which can save the computing energy during training
with proper underlying code design. Generally speaking, model pruning
compresses the model by removing less contributing weights and connections,
while retaining the performance of the original dense models. This will
significantly affect the reduction of the computing energy per iteration
required. When utilizing gradual pruning schemes, with the model size
gradually reduced during training, energy consumption due to communications is
also decreased. Besides, reducing the model size also saves the inference time
and energy, enabling the pruning method to be much more suitable for FL
training over 5G+ mobile devices.

Moreover, the pruning can be coupled with the quantization technique in FL
training to address the restrictions in implementing deep neural networks on
resource-constrained 5G+ mobile devices. Essentially, quantization requires a
smaller number of bits to represent each pruned connection, thereby reducing
memory, bandwidth and energy consumption. To enable a communication efficient
and mobile device compatible FL process, we develop a three-fold compression
of double quantization along with model pruning approach [11]. In particular,
the gradients and weights of uplink and downlink models are appropriately
quantified, and the gradual pruning of the received model is utilized to
reduce the computing and communication loads. This has a dual advantage of
reducing the communication time and energy, and reducing the memory bandwidth
due to less memory accesses. Therefore, we can reduce the model redundancy and
make the FL process computation, storage, and communication sufficiently
efficient to deploy large-sized deep neural networks over 5G+ mobile devices.

### Other Techniques to Save Energy

The total energy consumption can also be reduced by capturing the intrinsic
training dynamics, such as dynamically adjusting the batch size [12].
Specifically, we can interpret the stochastic gradient descent (SGD) training
process as integrating a stochastic differential equation (SDE) whose ``noise
scale" is related to the batch size selection. Small batch size theoretically
corresponds to large-scale random fluctuations, which can help explore the
parameter space to avoid trapping in local minima at the initial stage in the
FL problem. At later stage, small-scale fluctuations (large batch size) are
more desirable to fine-tune the parameters when a promising region of
parameter space is reached. Therefore, gradually increasing batch size in the
training process with a well-designed increment strategy can help reduce the
communication rounds in FL training process.

Moreover, gradually increasing batch size also leads to positive effects on
computational energy-saving. Due to the GPU's parallelism property, the local
computing energy is no longer proportional to the batch size. Accordingly, the
energy consumption of unit data calculation is relatively small for large
batch training, especially executing on the 5G+ mobile devices with multiple
GPUs. Besides, it has been theoretically and experimentally demonstrated that
both the fixed batch size approach and the dynamic batch size approach need
similar data epochs. In this case, thanks to the benefit of large batch
training at later stage, the training approach with dynamical batch sizes is
more energy-efficient. Moreover, a batch size control scheme catering to the
GPU computing performances and wireless communication conditions of mobile
devices can be further developed to balance the computing and communications,
thus achieving energy-efficient FL over 5G+ mobile devices.

### Balancing Communications and Computing to Reduce Overall Energy
Consumption

As aforementioned, the trade-off always exists between communications and
computing. Therefore, it is widely expected that both computing and
communication parts need to be considered in the energy-efficient FL training
process. When integrating both in the FL training process, the needed number
of global communication rounds is a critical element. One possible way to
approximate the needed global rounds is to conduct the FL theoretical
convergence analysis. After identifying the specific dataset and the training
model, the bound of the required number of communications can be derived based
on the required training accuracy under some mathematical assumptions. Note
that some learning settings, e.g., batch size, local iteration numbers, etc.,
also impact the required number of communications. Accordingly, the total
energy consumption corresponding to the required number of global rounds can
be obtained, which exhibits a global view of the overall energy consumption in
the FL training process. Despite employing the computing or communication
efficient learning techniques, the energy efficiency cannot be significantly
improved without elaborately controlling the key training parameters from a
global perspective. This requires us to optimize the training parameters, so
that communications and computing can be well balanced. This is why, in the
above subsections, we firstly study the efficient communication (e.g.,
gradient sparsification, gradient quantization and dynamic batch sizes) or
efficient local computing (e.g., weight quantization and pruning [13]) methods
in order to find the effective schemes. We then integrate these efficient
methods with resource allocation strategies to strike a good balance between
communications and computing, thus minimizing the energy consumption for FL
over 5G+ mobile devices.

## Challenges and Future Research Directions

Although there are a few pioneering research works done on the energy
efficiency for FL over 5G+ mobile devices, the relevant study is still in its
infancy and requires more thorough investigation. In this section, we
summarize some existing challenges and potential future directions.

### Energy-Efficient FL over Heterogeneous Mobile Devices

Most of the existing energy-efficient FL algorithms assume certain homogeneity
of mobile devices, while they may be considerably different in practice. The
heterogeneity across participating devices may lie in local training data,
computing capability, and wireless channel condition, etc., which require
flexible and customizable training strategies for each participant. Thus, it
is expected to develop advanced design methodology for efficient FL over 5G+
mobile devices across multi-dimensional heterogeneity. Accordingly, the
quantization granularity, the pruning strategy, and compression levels for
different participants can vary to accommodate their realistic environments
for energy-saving. For example, the device with powerful GPUs but poor channel
conditions may choose to compute more and communicate less and vice versa. In
these situations, it is also significant to optimize the personalized
compression or pruning strategy, adapting to the heterogeneous devices'
capability for total energy saving by balancing communications and computing.

### Energy-Efficient FL under Flexible Aggregation

The practical scenarios with heterogeneous data and devices demand higher
requirements on the aggregation strategies. On the one hand, the updated local
models across devices may differ in size or even structure, which invalidates
the current FL aggregation schemes (e.g., FedAvg [1]) if averaging model
parameters directly. Thus, we need to investigate more powerful and more
flexible aggregation schemes for FL over heterogeneous 5G+ mobile devices,
while mitigating the energy consumption to the same extent as the baseline FL
algorithms. On the other hand, due to the Non-IID (Independent and Identically
Distributed) data sources and different precision for local models, each
participant may have different contributions to the global model in terms of
accuracy and convergence. Therefore, from the energy perspective, only a
proportion of the participants need to contribute their models in each
communication round, or the aggregation will be performed in an asynchronous
way, thus reducing more energy compared with the original inefficient
aggregation method. Overall, flexible and asynchronous aggregations are
efficient methods to further reduce energy consumption, which deserves further
investigation.

### Energy-Efficient FL with Privacy Preservation

One of the inherited features of FL over 5G+ mobile devices is the privacy
preservation of the users' sensitive raw data. Unfortunately, the private
information can still be inferred from local updates communicated between user
devices and the aggregation server with some recently developed attack
mechanisms. Therefore, such a distributed learning, which needs to exchange
intermediate model parameters, brings in a significant design challenge for
privacy protection. A common strategy to enhance the privacy and security for
participating users is to introduce perturbations into the FL training
framework, such as adding the noise. Fortunately, some early research works
have already shown that pruning, quantization, and other energy-efficient
methods can introduce randomness into the FL training process and provide the
enhanced privacy guarantee, i.e., intrinsic privacy preservation. Under such
conditions, energy-saving strategy and privacy protections are perfectly
combined, where the privacy will be preserved without additional noisy
computation. However, the related research is still in its initial stages and
needs deeper exploration, especially for the differential privacy
preservation, where the privacy level can be precisely quantified.

### Extensive Applications of Energy-Efficient FL

With the massive growth of personal data with end users and the rapid
popularization of the power-efficient mobile edge devices, FL over 5G+ mobile
devices can be applied to a large number of applications [14] and can be
involved in every area of daily life. With energy-efficient training
strategies, FL over 5G+ mobile devices is perfectly compatible with lifelong
on-device learning that requires a constant training process and is battery-
driven. For example, it can be applied to some real-time assisting services
like the voice UI, keyboard prediction, and some low-latency control scenarios
such as gaming and automated guided vehicles [15]. Moreover, along with the
exponential improvement in on-device AI capabilities, more sensing data from
smart sensors like the cameras, microphones, and compass, can be effectively
utilized in Industrial IoT, e-health, finance, and social networks, etc.

## Conclusion

This paper has studied the FL over 5G+ mobile devices to address the issue on
energy consumption during the FL training process. We have investigated how to
properly conserve energy and allocate resources during FL training. We start
with introduction of wireless communications and on-GPU computing models.
Then, we discuss several energy-efficient training techniques, including
gradient sparsification, gradient quantization, weight quantization, model
pruning, and dynamic batch sizes, to save energy. At the same time, the
resource allocation strategies are adapted to reasonably manage energy
resources by balancing communications and computing energy consumption. We
conduct extensive simulations to demonstrate the efficacy of the techniques
mentioned above for FL over 5G+ mobile devices. Finally, we have presented
some existing design challenges and the corresponding research directions.

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