# Off-shell Higgs Couplings in $H^{*}\to ZZ\to\ell\ell\nu\nu$

Dorival Gonçalves  Department of Physics, Oklahoma State University,
Stillwater, OK, 74078, USA  Tao Han  Sze Ching Iris Leung  Han Qin  PITT PACC,
Department of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara
St., Pittsburgh, PA 15260, USA

###### Abstract

We explore the new physics reach for the off-shell Higgs boson measurement in
the ${pp\to H^{*}\rightarrow Z(\ell^{+}\ell^{-})Z(\nu\bar{\nu})}$ channel at
the high-luminosity LHC. The new physics sensitivity is parametrized in terms
of the Higgs boson width, effective field theory framework, and a non-local
Higgs-top coupling form factor. Adopting Machine-learning techniques, we
demonstrate that the combination of a large signal rate and a precise
phenomenological probe for the process energy scale, due to the transverse
$ZZ$ mass, leads to significant sensitivities beyond the existing results in
the literature for the new physics scenarios considered.

††preprint: OSU-HEP-20-13, PITT-PACC-2012

##  I Introduction

After the Higgs boson discovery at the Large Hadron Collider (LHC)
Higgs:1964ia ; PhysRevLett.13.508 ; PhysRevLett.13.321 ; Aad_2012 ;
Chatrchyan_2012 , the study of the Higgs properties has been one of the top
priorities in searching for new physics beyond the Standard Model (BSM).
Indeed, the Higgs boson is a unique class in the SM particle spectrum and is
most mysterious in many aspects. The puzzles associated with the Higgs boson
include the mass hierarchy between the unprotected electroweak (EW) scale
($v$) and the Planck scale ($M_{PL}$), the neutrino mass generation, the
possible connection to dark matter, the nature of the electroweak phase
transition in the early universe, to name a few. Precision studies of the
Higgs boson properties can be sensitive to new physics at a higher scale.
Parametrically, new physics at a scale $\Lambda$ may result in the effects of
the order $v^{2}/\Lambda^{2}$.

So far, the measurements at the LHC based on the Higgs signal strength are in
full agreement with the SM predictions. However, these measurements mostly
focus on the on-shell Higgs boson production, exploring the Higgs properties
at low energy scales of the order $v$. It has been argued that if we explore
the Higgs physics at a higher scale $Q$, the sensitivity can be enhanced as
$Q^{2}/\Lambda^{2}$. A particularly interesting option is to examine the Higgs
sector across different energy scales, using the sizable off-shell Higgs boson
rates at the LHC Kauer:2012hd ; Caola:2013yja ; Campbell:2013una ;
Aaboud:2018puo ; Sirunyan:2019twz . While the off-shell Higgs new physics
sensitivity is typically derived at the LHC with the $H^{*}\to ZZ\to 4\ell$
channel Gainer:2014hha ; Cacciapaglia:2014rla ; Azatov:2014jga ;
Englert:2014aca ; Buschmann:2014sia ; Corbett:2015ksa ; Goncalves:2017iub ;
Goncalves:2018pkt , we demonstrate in this work that the extension to the
channel $ZZ\to\ell\ell\nu\nu$ can significantly contribute to the potential
discoveries. This channel provides two key ingredients to probe the high
energy regime with enough statistics despite of the presence of two missing
neutrinos in the final state. First, it displays a larger event rate by a
factor of six than the four charged lepton channel. Second, the transverse
mass for the $ZZ$ system sets the physical scale $Q^{2}$ and results in a
precise phenomenological probe to the underlying physics.

In this paper, we extend the existing studies and carry out comprehensive
analyses for an off-shell channel in the Higgs decay

$pp\to H^{*}\to ZZ\to\ell^{+}\ell^{-}\ \nu\bar{\nu},$ (1)

where $\ell=e,\mu$ and $\nu=\nu_{e},\nu_{\mu},\nu_{\tau}$. Because of the
rather clean decay modes, we focus on the leading production channel of the
Higgs boson via the gluon fusion. First, we phenomenologically explore a
theoretical scenario with additional unobserved Higgs decay channels leading
to an increase in the Higgs boson width, $\Gamma_{H}/\Gamma_{H}^{SM}>1$. The
distinctive dependence for the on-shell and off-shell cross-sections with the
Higgs boson width foster the conditions for a precise measurement for this key
ingredient of the Higgs sector. We adopt the Machine-learning techniques in
the form of Boosted Decision Tree (BDT) to enhance the signal sensitivity.
This analysis sets the stage for our followup explorations. Second, we study
the effective field theory framework, taking advantage of the characteristic
energy-dependence from some of the operators. Finally, we address a more
general hypothesis that features a non-local momentum-dependent Higgs-top
interaction Goncalves:2018pkt , namely, a form factor, that generically
represents the composite substructure. Overall, the purpose of this paper is
to highlight the complementarity across a multitude of frameworks
Englert:2014aca ; Azatov:2014jga ; Englert:2014ffa ; Buschmann:2014sia ;
Corbett:2015ksa ; Goncalves:2017iub ; Goncalves:2018pkt  via the promising
process at the LHC $H^{*}\rightarrow Z(\ell\ell)Z(\nu\nu)$, from models that
predict invisible Higgs decays, passing by the effective field theory, and a
non-local form-factor scenario. Our results demonstrate significant
sensitivities at the High-Luminosity LHC (HL-LHC) to the new physics scenarios
considered here beyond the existing literature.

The rest of the paper is organized as follows. In Sec. II, we derive the Higgs
width limit at HL-LHC. Next, in Sec. III, we study the new physics sensitivity
within effective field theory framework. In Sec. IV, we scrutinize the effects
of a non-local Higgs-top form-factor. Finally, we present a summary in Sec. V.

##  II Higgs Boson Width

Figure 1: Representative Feynman diagrams for the DY $q\bar{q}\rightarrow ZZ$
(left), GF $gg\rightarrow ZZ$ continuum (center), and $s$-channel Higgs signal
$gg\rightarrow H^{*}\rightarrow ZZ$ (right).

Figure 2: Normalized distributions for the missing transverse momentum
$E_{T}^{\mathrm{miss}}$ (left panel), azimuthal $\phi$ (central panel) and
polar $\theta$ angles (right panel) of the charged lepton $\ell^{-}$ in the
$Z$ boson rest frame.

The combination of on-shell and off-shell Higgs boson rates addresses one of
the major shortcomings of the LHC, namely the Higgs boson width measurement
Kauer:2012hd ; Caola:2013yja . This method breaks the degeneracy present on
the on-shell Higgs coupling studies

$\sigma_{i\rightarrow H\rightarrow f}^{\text{on-
shell}}\propto\frac{g_{i}^{2}(m_{H})g_{f}^{2}(m_{H})}{\Gamma_{H}}\,,$ (2)

where the total on-shell rate can be kept constant under the transformation
$g_{i,f}(m_{H})\rightarrow\xi g_{i,f}(m_{H})$ with
$\Gamma_{H}\rightarrow\xi^{4}\Gamma_{H}$. The off-shell Higgs rate, due to a
sub-leading dependence on the Higgs boson width $\Gamma_{H}$

$\sigma_{i\rightarrow H^{*}\rightarrow f}^{\text{off-shell}}\propto
g_{i}^{2}(\sqrt{\hat{s}})g_{f}^{2}(\sqrt{\hat{s}})\,,$ (3)

breaks this degeneracy, where $\sqrt{\hat{s}}$ is the partonic c.m. energy
that characterizes the scale of the off-shell Higgs. In particular, if the new
physics effects result in the same coupling modifiers at both kinematical
regimes Englert:2014aca ; Azatov:2014jga ; Buschmann:2014sia ; Corbett:2015ksa
, the relative measurement of the on-shell and off-shell signal strengths can
uncover the Higgs boson width, $\mu_{\text{off-shell}}/\mu_{\text{on-
shell}}=\Gamma_{H}/\Gamma_{H}^{SM}$.

In this section, we derive a projection for the Higgs boson width measurement
at the ${\sqrt{s}=14}$ TeV high-luminosity LHC, exploring the $ZZ\rightarrow
2\ell 2\nu$ final state. We consider the signal channel as in Eq. (1). The
signal is characterized by two same-flavor opposite sign leptons, $\ell=e$ or
$\mu$, which reconstruct a $Z$ boson and recoil against a large missing
transverse momentum from $Z\to\nu\bar{\nu}$. The major backgrounds for this
search are the Drell-Yan (DY) processes $q\bar{q}\rightarrow ZZ,ZW$ and gluon
fusion (GF) $gg\rightarrow ZZ$ process, see Fig. 1 for a sample of the Feynman
diagrams. While the Drell-Yan component displays the largest rate, the gluon
fusion box diagrams interfere with the Higgs signal, resulting in important
contributions mostly at the off-shell Higgs regime Kauer:2012hd .

In our calculations, the signal and background samples are generated with
MadGraph5_aMC@NLO Alwall:2014hca ; mg5loop . The Drell-Yan background is
generated at the NLO with the MC@NLO algorithm Frixione:2002ik . Higher order
QCD effects to the loop-induced gluon fusion component are included via a
universal $K$-factor Campbell:2013una ; Bonvini:2013jha . Spin correlation
effects for the $Z$ and $W$ bosons decays are obtained in our simulations with
the MadSpin package Artoisenet:2012st . The renormalization and factorization
scales are set by the invariant mass of the gauge boson pair $Q=m_{VV}/2$,
using the PDF set nn23nlo Ball:2013hta . Hadronization and underlying event
effects are simulated with Pythia8 Sjostrand:2014zea , and detector effects
are accounted for with the Delphes3 package Ovyn:2009tx .

We start our analysis with some basic lepton selections. We require two same-
flavor and opposite sign leptons with $|\eta_{\ell}|<2.5$ and $p_{T\ell}>10$
GeV in the invariant mass window $76~{}\text{GeV}<m_{\ell\ell}<106$ GeV. To
suppress the SM backgrounds, it is required large missing energy selection
$E_{T}^{\text{miss}}>175$ GeV and a minimum transverse mass for the $ZZ$
system ${m_{T}^{ZZ}>250}$ GeV, defined as

$m_{T}^{ZZ}=\sqrt{\left(\sqrt{m_{Z}^{2}+p_{T(\ell\ell)}^{2}}+\sqrt{m_{Z}^{2}+(E_{T}^{\mathrm{miss}})^{2}}\right)^{2}-\left|\overrightarrow{p}_{TZ}+\overrightarrow{E}_{T}^{\mathrm{miss}}\right|^{2}}\,.$
(4)

The consistency of our event simulation and analysis setup is confirmed
through a cross-check with the ATLAS study in Ref. Aaboud:2018puo .

To further control the large Drell-Yan background, a Boosted Decision Tree
(BDT) is implemented via the Toolkit for Multivariate Data Analysis with ROOT
(TMVA) tmva . The BDT is trained to distinguish the full background events
from the $s$-channel Higgs production. The variables used in the BDT are
missing transverse energy, the momenta and rapidity for the leading and sub-
leading leptons $(p_{T}^{\ell 1},\eta^{\ell 1},p_{T}^{\ell 2},\eta^{\ell 2})$,
the leading jet $(p_{T}^{j1},\eta^{j1})$, the separation between the two
charged leptons $\Delta R_{\ell\ell}$, the azimuthal angle difference between
the di-lepton system and the missing transverse energy
$\Delta\phi(\vec{p}_{T}^{~{}\ell\ell},\vec{E}^{\mathrm{miss}}_{T})$, and the
scalar sum of jets and lepton transverse momenta $H_{T}$. Finally, we also
include the polar $\theta$ and azimuthal $\phi$ angles of the charged lepton
$\ell^{-}$ in the $Z$ rest frame Goncalves:2018fvn ; Goncalves:2018ptp . We
choose the coordinate system for the $Z$ rest frame following Collins and
Soper (Collins-Soper frame) Collins:1977iv . The signal and background
distributions for these observables are illustrated in Fig. 2. We observe
significant differences between the $s$-channel signal and background in the
$(\theta,\phi)$ angle distributions. These kinematic features arise from the
different $Z$ boson polarizations for the signal and background components at
the large di-boson invariant mass $m_{T}^{ZZ}$ Buschmann:2014sia ;
Goncalves:2017gzy . Whereas the $s$-channel Higgs tends to have $Z_{L}$
dominance, the DY background is mostly $Z_{T}$ dominated.

Figure 3: BDT distribution for the $s$-channel Higgs signal (red) and
background (blue).

We would like to illustrate the power of the implemented BDT analysis to
separate the $s$-channel Higgs from the background contributions in Fig. 3.
The BDT discriminator is defined in the range $[-1,1]$. The events with
discriminant close to $-1$ are classified as background-like and those close
to 1 are signal-like. The optimal BDT score selection has been performed with
TMVA. To estimate the effectiveness of the BDT treatment, we note that one can
reach $S/\sqrt{S+B}=5$ at an integrated luminosity of $273~{}\text{fb}^{-1}$
with signal efficiency 88% and background rejection of 34%, by requiring
${\text{BDT}_{\text{response}}>-0.26}$. Now that we have tamed the dominant
backgrounds ${q\bar{q}\rightarrow ZZ,ZW}$, we move on to the new physics
sensitivity study.

To maximize the sensitivity of the Higgs width measurement, we explore the
most sensitive variable, $m_{T}^{ZZ}$ distribution, and perform a binned log-
likelihood ratio analysis. In Fig. 4, we display the 95% CL on the Higgs width
$\Gamma_{H}/\Gamma_{H}^{SM}$ as a function of the $\sqrt{s}=14$ TeV LHC
luminosity. To infer the relevance of the multivariate analysis, that
particularly explore the observables $(E_{T}^{\mathrm{miss}},\theta,\phi)$
depicted in Fig. 2, we display the results in two analysis scenarios: in blue
we show the cut-based analysis and in red the results accounting for the BDT-
based framework. The significant sensitivity enhancement due to the BDT
highlights the importance of accounting for the full kinematic dependence,
including the $Z$-boson spin correlation effects. Whereas the Higgs width can
be constrained to $\Gamma_{H}/\Gamma_{H}^{SM}<1.35$ at 95% CL level following
the cut-based analysis, $\Gamma_{H}/\Gamma_{H}^{SM}<1.31$ in the BDT-based
study assuming $\mathcal{L}=3~{}\text{ab}^{-1}$ of data. Hence, the BDT limits
result in an improvement of $\mathcal{O}(5\%)$ on the final Higgs width
sensitivity. These results are competitive to the HL-LHC estimates for the
four charged lepton final state derived by ATLAS and CMS, where the respective
limits are $\Gamma_{H}/\Gamma_{H}^{SM}<\mathcal{O}(1.3)$ and
$\mathcal{O}(1.5)$ at 68% CL TheATLAScollaboration:2015stq ; CMS:2018qgz .

Figure 4: 95% CL bound on the Higgs width $\Gamma_{H}/\Gamma_{H}^{SM}$ as a
function of the $\sqrt{s}=14$ TeV LHC luminosity. We display the results for
the cut-based study (blue) and BDT-based analysis (red).

##  III Effective Field Theory

The Effective Field Theory (EFT) provides a consistent framework to
parametrize beyond the SM effects in the presence of a mass gap between the SM
and new physics states. In this context, the new physics states can be
integrated out and parametrized in terms of higher dimension operators
Appelquist:1974tg . In this section we parametrize the new physics effects in
terms of the EFT framework Buchmuller:1985jz ; Grzadkowski:2010es . Instead of
performing a global coupling fit, we will focus on a relevant subset of higher
dimension operators that affect the Higgs production via gluon fusion. This
will shed light on the new physics sensitivity for the off-shell
$pp\rightarrow H^{*}\rightarrow Z(\ell\ell)Z(\nu\nu)$ channel. Our effective
Lagrangian can be written as

$\displaystyle\mathcal{L}\supset$ $\displaystyle c_{g}\frac{\alpha_{s}}{12\pi
v^{2}}|\mathcal{H}|^{2}G_{\mu\nu}G^{\mu\nu}+c_{t}\frac{y_{t}}{v^{2}}|\mathcal{H}|^{2}\bar{Q}_{L}\tilde{\mathcal{H}}t_{R}+\text{h.c.}\,\,,$
(5)

where $\mathcal{H}$ is the SM Higgs doublet and $v=246$ GeV is the vacuum
expectation value of the SM Higgs field. The couplings are normalized in such
a way for future convenience. If we wish to make connection with the new
physics scale $\Lambda$, we would have the scaling as $c_{g},c_{t}\sim
v^{2}/\Lambda^{2}$. After electroweak symmetry breaking, Eq. (5) renders into
the following interaction terms with a single Higgs boson

$\displaystyle\mathcal{L}\supset$
$\displaystyle\kappa_{g}\frac{\alpha_{s}}{12\pi
v}HG_{\mu\nu}G^{\mu\nu}-\kappa_{t}\frac{m_{t}}{v}H\left(\bar{t}_{R}t_{L}+\text{h.c.}\right)\,\,,$
(6)

where the coupling modifiers $\kappa_{g,t}$ and the Wilson coefficients
$c_{g,t}$ are related by $\kappa_{g}=c_{g}$ and
$\kappa_{t}=1-\text{Re}(c_{t})$. We depict in Fig. 5 the $gg\rightarrow ZZ$
Feynman diagrams that account for these new physics effects.

Whereas Eq. (5) represents only a sub-set of high dimensional operators
affecting the Higgs interactions Buchmuller:1985jz ; Grzadkowski:2010es , we
focus on it to highlight the effectiveness for the off-shell Higgs
measurements to resolve a notorious degeneracy involving these terms. The
gluon fusion Higgs production at low energy regime can be well approximated by
the Higgs Low Energy Theorem Shifman:1979eb ; Kniehl:1995tn , where the total
Higgs production cross-section scales as
${\sigma_{\text{GF}}\propto|\kappa_{t}+\kappa_{g}|^{2}}$. Therefore, low
energy measurements, such as on-shell and non-boosted Higgs production
Baur:1989cm ; Harlander:2013oja ; Banfi:2013yoa ; Azatov:2013xha ;
Grojean:2013nya ; Azatov:2014jga ; Buschmann:2014twa ; Buschmann:2014sia ;
Azatov:2016xik , are unable to resolve the
${|\kappa_{t}+\kappa_{g}|=\text{constant}}$ degeneracy. While the combination
between the $t\bar{t}H$ and gluon fusion Higgs production have the potential
to break this blind direction Plehn:2015cta , we will illustrate that the
Higgs production at the off-shell regime can also result into relevant
contributions to resolve this degeneracy.

Figure 5: Feynman diagrams for the GF $gg\rightarrow ZZ$ process. The new
physics effects from Eq. (6) display deviations on the coefficients
$\kappa_{t}$ and $\kappa_{g}$ from the SM point
$(\kappa_{t},\kappa_{g})=(1,0)$.  Figure 6: Transverse mass distributions
$m_{T}^{ZZ}$ for the DY and GF $Z(\ell\ell)Z(\nu\nu)$ processes. The new
physics effects are parametrized by deviations from SM point
$(\kappa_{t},\kappa_{g})=(1,0)$. We follow the benchmark analysis defined in
Sec II.

Since the Higgs boson decays mostly to longitudinal gauge bosons at the high
energy regime, it is enlightening to inspect the signal amplitude for the
longitudinal components. The amplitudes associated to each contribution
presented in Fig. 5 can be approximated at $m_{ZZ}\gg m_{t},m_{H},m_{Z}$ by
Glover:1988rg ; Azatov:2014jga ; Buschmann:2014sia

$\displaystyle\mathcal{M}_{t}^{++00}$
$\displaystyle\approx+\frac{m_{t}^{2}}{2m_{Z}^{2}}\log^{2}\frac{m_{ZZ}^{2}}{m_{t}^{2}}\,,$
$\displaystyle\mathcal{M}_{g}^{++00}$
$\displaystyle\approx-\frac{m_{ZZ}^{2}}{2m_{Z}^{2}}\,,$
$\displaystyle\mathcal{M}_{c}^{++00}$
$\displaystyle\approx-\frac{m_{t}^{2}}{2m_{Z}^{2}}\log^{2}\frac{m_{ZZ}^{2}}{m_{t}^{2}}\,.$
(7)

Two comments are in order. First, both the $s$-channel top loop
$\mathcal{M}_{t}$ and the continuum $\mathcal{M}_{c}$ amplitudes display
logarithmic dependences on $m_{ZZ}/m_{t}$ at the far off-shell regime. In the
SM scenario the ultraviolet logarithm between these two amplitudes cancel,
ensuring a proper high energy behavior when calculating the full amplitude.
Second, it is worth noting the difference in sign between the $s$-channel
contributions $\mathcal{M}_{t}$ and $\mathcal{M}_{g}$. This results into a
destructive interference between $\mathcal{M}_{t}$ and $\mathcal{M}_{c}$,
contrasting to a constructive interference between $\mathcal{M}_{g}$ and
$\mathcal{M}_{c}$. In the following, we will explore these phenomenological
effects pinning down the new physics sensitivity with a higher precision.

Figure 7: 95% CL bound on the coupling modifiers $\kappa_{t}$ and $\kappa_{g}$
when accounting for the off-shell Higgs measurement in the
$Z(\ell\ell)Z(\nu\nu)$ channel. We assume the 14 TeV LHC with
$3~{}\text{ab}^{-1}$ of data.

Exploiting the larger rate for $ZZ\rightarrow\ell\ell\nu\nu$ than that for
$ZZ\rightarrow 4\ell$ Azatov:2014jga ; Englert:2014aca ; Buschmann:2014sia ,
we explore the off-shell Higgs physics at the HL-LHC. To simulate the full
loop-induced effects, we implemented Eq. (6) into FeynRules/NLOCT
Alloul:2013bka ; Degrande:2014vpa  through a new fermion state, and adjusting
its parameters to match the low-energy Higgs interaction
$HG_{\mu\nu}G^{\mu\nu}$ Shifman:1979eb ; Kniehl:1995tn . Feynman rules are
exported to a Universal FeynRules Output (UFO) Degrande_2012  and the Monte
Carlo event generation is performed with MadGraph5aMC@NLO Alwall:2014hca .

In Fig. 6, we present the Drell-Yan (DY) and the gluon-fusion (GF)
$m_{T}^{ZZ}$ distributions for different signal hypotheses. In the bottom
panel, we display the ratio between the GF beyond the SM (BSM) scenarios with
respect to the GF SM. In agreement with Eq. (7), we observe a suppression for
the full process when accounting for the $s$-channel top loop contributions
and an enhancement when including the new physics terms associated to
$\mathcal{M}_{g}$ at high energies.

We follow the benchmark analysis defined in Sec. II. After the BDT study, the
resulting events are used in a binned log-likelihood analysis with the
$m_{T}^{ZZ}$ distribution. This approach explores the characteristic high
energy behavior for the new physics terms highlighted in Eq. (7) and
illustrated in Fig. 6. We present in Fig. 7 the resulting 95% CL sensitivity
to the $(\kappa_{t},\kappa_{g})$ new physics parameters at the high-luminosity
LHC. In particular, we observe that the LHC can bound the top Yukawa within
$\kappa_{t}\approx[0.4,1.1]$ at 95% CL, using this single off-shell channel.
The observed asymmetry in the limit, in respect to the SM point, arises from
the large and negative interference term between the $s$-channel and the
continuum amplitudes. The upper bound on $\kappa_{t}$ is complementary to the
direct Yukawa measurement via $ttH$ Cepeda:2019klc  and can be further
improved through a combination with the additional relevant off-shell Higgs
final states. The results derived in this section are competitive to the CMS
HL-LHC prediction that considers the boosted Higgs production combining the
$H\to 4\ell$ and $H\to\gamma\gamma$ channels CMS:2018qgz . The CMS projection
results into an upper bound on the top Yukawa of $\kappa_{t}\lesssim 1.2$ at
95% CL.

##  IV Higgs-Top Form Factor

Figure 8: Transverse mass distribution $m_{T}^{ZZ}$ for $gg(\rightarrow H^{*})\rightarrow Z(2\ell)Z(2\nu)$ in the Standard Model (black) and with a new physics form factor (red). We assume $n=2,3$ and $\Lambda=1.5$ TeV for the form factor scenario.  | $\Gamma_{H}/\Gamma_{H}^{SM}$ | $\Lambda_{EFT}$ | $\Lambda_{Composite}^{n=2}$
---|---|---|---
$H^{*}\rightarrow ZZ\to\ell\ell\nu\nu$ | 1.31 | 0.8 TeV | 1.5 TeV
$H^{*}\rightarrow ZZ\to 4\ell$ | 1.3 (68% CL) TheATLAScollaboration:2015stq  | 0.55 TeV CMS:2018qgz  | 0.8 TeV Goncalves:2018pkt
Table 1: Comparison of the sensitivity reaches between $H^{*}\rightarrow
ZZ\to\ell\ell\nu\nu$ in this study and $H^{*}\rightarrow ZZ\to 4\ell$ in the
literature as quoted. All results are presented at 95% CL except for the Higgs
width projection derived by ATLAS with 68% CL TheATLAScollaboration:2015stq .
We assume that the Wilson coefficient for the EFT framework is given by
${c_{t}=v^{2}/\Lambda_{EFT}^{2}}$. Besides the $H\to 4\ell$ channel, Ref.
CMS:2018qgz  also accounts for the $H\to\gamma\gamma$ final state with a
boosted Higgs analysis.

The fact that the observed Higgs boson mass is much lighter than the Planck
scale implies that there is an unnatural cancellation between the bare mass
and the quantum corrections. Since the mass of the Higgs particle is not
protected from quantum corrections, it is well-motivated to consider that it
may not be fundamental, but composite in nature Pomarol:2012qf ; Panico:2011pw
; Panico:2015jxa ; Liu:2017dsz . In such a scenario, the Higgs boson is
proposed as a bound state of a strongly interacting sector with a composite
scale $\Lambda$. In addition, the top quark, which is the heaviest particle in
the SM, can also be composite. In this case, the top Yukawa coupling will be
modified by a momentum-dependent form factor at a scale $q^{2}$ close to or
above the new physics scale $\Lambda^{2}$. It is challenging to find a general
construction for such form factor without knowing the underlying dynamics.
Here, we will adopt a phenomenological ansatz motivated by the nucleon form
factor Punjabi:2015bba . It is defined as

$\Gamma(q^{2}/\Lambda^{2})=\frac{1}{(1+q^{2}/\Lambda^{2})^{n}}\,,$ (8)

where $q^{2}$ is the virtuality of the Higgs boson. For $n=2$, it is a dipole-
form factor and corresponds to an exponential spacial distribution. Building
upon Ref. Goncalves:2018pkt , we study the impact of this form factor on
$gg\rightarrow H^{*}\rightarrow ZZ$ process now with the complementary final
state $\ell^{+}\ell^{-}\nu\overline{\nu}$.

Figure 9: 95% CL sensitivity on the new physics scale $\Lambda$ as a function
of the LHC luminosity. We assume the form factor in Eq. (8) with $n=2$ (dashed
line) and $n=3$ (solid line) at the 14 TeV LHC.

In Fig. 8, we illustrate the $m_{T}^{ZZ}$ distribution for the full gluon
fusion $gg(\rightarrow H^{*})\rightarrow ZZ$ process. We show the Standard
Model (black) and the form factor scenario (red). We assume $n=2$ or $3$ and
$\Lambda=1.5$ TeV for the depicted form factor scenarios. The differences
between Standard Model and form factor cases become larger when the energy
scales are comparable or above $\Lambda$ due to the suppression of destructive
interference between Higgs signal and continuum background. Thus, we perform
the same BDT procedure introduced in Sec. II followed by a binned log-
likelihood ratio test in the $m_{T}^{ZZ}$ distribution to fully explore this
effect. In Fig. 9, we display the sensitivity reach for the LHC in the Higgs-
top form factor. We observe that the LHC can bound these new physics effects
up to $\Lambda=1.5$ TeV for $n=2$ and $\Lambda=2.1$ TeV for $n=3$ at 95% CL.
The large event rate for the $H^{*}\rightarrow ZZ\rightarrow\ell\ell\nu\nu$
signal results in a more precise probe to the ultraviolet regime than for the
$H^{*}\rightarrow ZZ\rightarrow 4\ell$ channel, where the limits on the new
physics scale are $\Lambda=0.8$ TeV for $n=2$ and $\Lambda=1.1$ TeV for $n=3$
at 95% CL Goncalves:2018pkt .

##  V Summary

We have systematically studied the off-shell Higgs production in the
$pp\rightarrow H^{*}\rightarrow Z(\ell\ell)Z(\nu\nu)$ channel at the high-
luminosity LHC. We showed that this signature is crucial to probe the Higgs
couplings across different energy scales potentially shedding light on new
physics at the ultraviolet regime. To illustrate its physics potential, we
derived the LHC sensitivity to three BSM benchmark scenarios where the new
physics effects are parametrized in terms of the Higgs boson width, the
effective field theory framework, and a non-local Higgs-top coupling form
factor.

The combination of a large signal rate and a precise phenomenological probe
for the process energy scale, due to the transverse $ZZ$ mass, renders strong
limits for all considered BSM scenarios. A summary table and comparison with
the existing results in the literature are provided in Table 1. Adopting
Machine-learning techniques, we demonstrated in the form of BDT that the HL-
LHC, with $\mathcal{L}=3~{}\text{ab}^{-1}$ of data, will display large
sensitivity to the Higgs boson width, $\Gamma_{H}/\Gamma_{H}^{SM}<1.31$. In
addition, the characteristic high energy behavior for the new physics terms
within the EFT framework results in relevant bounds on the
$(\kappa_{t},\kappa_{g})$ new physics parameters, resolving the low energy
degeneracy in the gluon fusion Higgs production. In particular, we observe
that the LHC can bound the top Yukawa within $\kappa_{t}\approx[0.4,1.1]$ at
95% CL. The upper bound on $\kappa_{t}$ is complementary to the direct Yukawa
measurement via $ttH$ and can be further improved in conjunction with
additional relevant off-shell Higgs channels. Finally, when considering a more
general hypothesis that features a non-local momentum-dependent Higgs-top
interaction, we obtain that the HL-LHC is sensitive to new physics effects at
large energies with $\Lambda=1.5$ TeV for $n=2$ and $\Lambda=2.1$ TeV for
$n=3$ at 95% CL. We conclude that, utilizing the promising $H^{*}\to
Z(\ell^{+}\ell^{-})Z(\nu\bar{\nu})$ channel at the HL-LHC and adopting the
Machine-Learning techniques, the combination of a large signal rate and a
precise phenomenological probe for the process energy scale renders improved
sensitivities beyond the existing literature, to all the three BSM scenarios
considered in this work.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy under grant No. DE-
FG02- 95ER40896 and by the PITT PACC. DG was supported by the U.S. Department
of Energy under grant number DE-SC 0016013.

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# Integrating Local Real Data with Global Gradient Prototypes for Classifier
Re-Balancing in Federated Long-Tailed Learning

Wenkai Yang1 Deli Chen2 Hao Zhou2 Fandong Meng2 Jie Zhou2 &Xu Sun3 1Center for
Data Science, Peking University
2Pattern Recognition Center, WeChat AI, Tencent Inc., China
3MOE Key Lab of Computational Linguistics, School of Computer Science, Peking
University<EMAIL_ADDRESS><EMAIL_ADDRESS>
{delichen, tuxzhou, fandongmeng<EMAIL_ADDRESS>
###### Abstract

Federated Learning (FL) has become a popular distributed learning paradigm
that involves multiple clients training a global model collaboratively in a
data privacy-preserving manner. However, the data samples usually follow a
long-tailed distribution in the real world, and FL on the decentralized and
long-tailed data yields a poorly-behaved global model severely biased to the
head classes with the majority of the training samples. To alleviate this
issue, decoupled training has recently been introduced to FL, considering it
has achieved promising results in centralized long-tailed learning by re-
balancing the biased classifier after the instance-balanced training. However,
the current study restricts the capacity of decoupled training in federated
long-tailed learning with a sub-optimal classifier re-trained on a set of
pseudo features, due to the unavailability of a global balanced dataset in FL.
In this work, in order to re-balance the classifier more effectively, we
integrate the local real data with the global gradient prototypes to form the
local balanced datasets, and thus re-balance the classifier during the local
training. Furthermore, we introduce an extra classifier in the training phase
to help model the global data distribution, which addresses the problem of
contradictory optimization goals caused by performing classifier re-balancing
locally. Extensive experiments show that our method consistently outperforms
the existing state-of-the-art methods in various settings.

##  1 Introduction

Federated Learning (FL) McMahan et al. (2017) is proposed as an effective
distributed learning framework to enable local clients to collaboratively
train a global model without exposing their local private data to each other.
In the real world, there are two data distribution phenomena that introduce
great challenges to the good convergence of FL algorithms. One is that the
data samples are not identically and independently distributed (non-i.i.d.)
across different clients. Furthermore, the other important phenomenon is that
the global data distribution (i.e., the data distribution of the training
samples merged from all clients’ local data) usually shows a long-
tailed/class-imbalanced pattern Zhang et al. (2021), where head classes occupy
a much larger proportion of the training samples than tail classes. Directly
applying FL on such long-tailed data will produce a global model with poor
generalization ability that is severely biased to the head classes Wang et al.
(2021). However, it is challenging to deal with FL on the non-i.i.d. and long-
tailed data due to two aspects. First, affected by the non-i.i.d. data
partitions, the local data distributions (i.e., local imbalance) show
inconsistent long-tailed patterns with that of the global data distribution
(i.e., global imbalance) Wang et al. (2021). Thus, tackling the local
imbalance problem only (e.g., Fed-Focal Loss Sarkar et al. (2020)) will not
help to address the global imbalance problem in FL. Second, considering the
data privacy concern, it is infeasible to obtain the imbalance pattern of the
global data distribution from the local data information. This further limits
the application of the global class re-weighting strategy Cui et al. (2019).

To deal with the above problems, some existing studies manage to estimate the
global imbalance pattern by utilizing either the uploaded gradients w.r.t. the
classifier Wang et al. (2021) or the values of local training losses Shen et
al. (2021). They then apply the class-level Wang et al. (2021) or client-level
Shen et al. (2021) re-weighting practice to focus more on the gradients
contributed by tail classes or poorly-learned clients. However, previous
studies Kang et al. (2019); Zhou et al. (2020) have shown that the re-
weighting practice will do harm to the representation learning phase.
Therefore, the improvement brought by this kind of method is limited.

Recently, some centralized long-tailed learning studies Kang et al. (2019);
Zhou et al. (2020) manage to decouple the model learning on long-tailed data
into the representation learning phase and the classifier learning phase, and
find that the instance-balanced training (i.e., uniform sampling on the entire
training set to make the contribution of each sample the same) leads to the
well-learned representations but a biased classifier. Therefore, centralized
decoupled training aims re-train the classifier on a small balanced dataset
after the instance-balanced training, and has achieved very promising results.
However, decoupled training is difficult to be implemented in FL due to the
lack of a public balanced dataset. Then, CReFF Shang et al. (2022b) proposes
to re-train the classifier on a set of pseudo features created on the server.
Nevertheless, the improvement brought by CReFF is restricted by the high
similarity of the pseudo features per class, and the fundamental problem –
lack of real balanced data still exists.

To better solve the lack of the real balanced data issue in the application of
decoupled training in FL, we propose a different yet more effective classifier
re-balancing algorithm, and achieve state-of-the-art results in federated
long-tailed learning. That is, we choose to take full advantage of the
abundant real data stored in the local clients, and allow the clients to re-
balance the classifier during local training. Specifically, we make each
client re-balance the classifier on a local balanced dataset that is mixed
with the local real data and the global gradient prototypes of the classifier
sent by the server, while the latter is supposed to address the issue of
missing classes in the local datasets. Additionally, we add an extra
classifier in the local training phase to jointly model the global data
distribution. This practice helps to overcome the optimization difficulty on
the global representation learning brought by the practice of local classifier
re-balancing. Compared with CReFF, we allow the clients to collaboratively
train a balanced classifier with their sufficient real data during local
training, which needs no extra requirements on the server and produces an
optimal classifier with better generalization ability. We conduct extensive
experiments on the three popular long-tailed image classification tasks, and
the results show that our method can significantly outperform all existing
federated long-tailed learning methods in various settings.

##  2 Related Work

###  2.1 Federated Learning

Federated Averaging (FedAvg) McMahan et al. (2017) is the most widely-used FL
algorithm, but it has been shown that the performance of FedAvg drops greatly
when the data is non-i.i.d. Karimireddy et al. (2020). Therefore, plenty of
existing FL studies target on dealing with the non-i.i.d. data partitions in
FL. For example, FedProx Li et al. (2018) and FedDyn Acar et al. (2020) manage
to make the local models converge to the same global optimum by adding the
regularization terms in the local training objectives, SCAFFOLD Karimireddy et
al. (2020) chooses to correct the local gradient in each step with the
gradients from other clients to reduce the gradient variance across different
clients. FedAvgM Hsu et al. (2019) and FedOPT Reddi et al. (2020) adopt the
server momentum and the adaptive server optimizer in the server aggregation
phase.

###  2.2 Long-Tailed/Imbalanced Learning

In the real world, the data points usually show a long-tailed distribution
pattern. Therefore, learning good models on the long-tailed/class-imbalanced
data has been widely studied Zhang et al. (2021) in the traditional
centralized learning, and attracts more and more attention in the FL setting.

##### Centralized Long-Tailed Learning

The methods to tackle the class imbalance problem in the centralized long-
tailed learning can be mainly divided into three categories: (1) Class-level
re-balancing methods that includes over-sampling training samples from tail
classes Chawla et al. (2002), under-sampling data points from head classes Liu
et al. (2008), or re-weighting the loss values or the gradients of different
training samples based on the label frequencies Cui et al. (2019); Cao et al.
(2019) or the predicted probabilities of the model Lin et al. (2017). (2)
Augmentation-based methods aim to create more data samples for tail classes
either from the perspective of the feature space Chu et al. (2020); Zang et
al. (2021) or the sample space Chou et al. (2020). (3) Classifier re-balancing
mechanisms are based on the finding that the uniform sampling on the whole
dataset during training benefits the representation learning but leads to a
biased classifier, so they design specific algorithms to adjust the classifier
during or after the representation learning phase Zhou et al. (2020); Kang et
al. (2019).

##### Federated Long-Tailed Learning

Recently, a few studies begin to focus on the class imbalance problem in FL,
as FL becomes a more practical and popular learning paradigm and the long-
tailed data distribution is unavoidable in the real world. Fed-Focal Loss
Sarkar et al. (2020) directly applies Focal Loss Lin et al. (2017) in the
clients’ local training, but it neglects the fact that the local imbalance
pattern is inconsistent with the global imbalance pattern. Ratio Loss Wang et
al. (2021) utilizes an auxiliary dataset on the server (which is usually
impractical in real cases) to estimate the global data distribution, and send
the estimated information to clients to perform class-level re-weighting
during local training. CLIMB Shen et al. (2021) is proposed as a client-level
re-weighting method to give more aggregation weights to the clients with
larger local training losses. However, both Ratio Loss and CLIMB bring
negative effects to the representation learning caused by the re-weighting
practice. FEDIC Shang et al. (2022a) also needs the impractical assumption to
own an auxiliary balanced dataset for fine-tuning the global model on the
server, and uses the fine-tuned model along with the local models as teachers
to perform knowledge distillation on the original global model. Most recently,
CReFF Shang et al. (2022b) adopts the decoupled training idea to re-train the
classifier on the server by creating a number of federated features for each
class, and achieves previously state-of-the-art performance. However, the low
quality and the limited number of federated features restrict its potential.

##  3 Methodology

###  3.1 Problem Definition

In the FL framework, each client $k$ ($k=1,\cdots,N$) has its own local
dataset $\mathcal{D}_{k}$, and all clients form a federation to jointly train
a good global model under the constraint that the local data is always kept in
the local devices. Then, the optimization goal of FL can be formulated as

$\begin{aligned}
\boldsymbol{\theta}^{*}=\mathop{\arg\min}\limits_{\boldsymbol{\theta}}L(\boldsymbol{\theta})=\mathop{\arg\min}\limits_{\boldsymbol{\theta}}\sum_{k=1}^{N}\frac{|\mathcal{D}_{k}|}{\sum_{i=1}^{N}|\mathcal{D}_{i}|}L(\boldsymbol{\theta};\mathcal{D}_{k}),\end{aligned}$

(1)

where $|\mathcal{D}_{k}|$ represents the total number of training samples in
$\mathcal{D}_{k}$, and $L(\cdot;\mathcal{D}_{k})$ is the local training
objective in client $k$.

Federated Averaging (FedAvg) McMahan et al. (2017) is the most popular FL
framework to solve the above optimization problem. Specifically, at the
beginning of each communication round $t$, the server sends the updated global
model $\boldsymbol{\theta}^{t-1}$ from the last round to all available/sampled
clients $k\in\mathcal{C}^{t}$ in the current round, and each client $k$ takes
$\boldsymbol{\theta}^{t-1}$ as the initial model to perform multiple updates
on its local dataset $\mathcal{D}_{k}$ and gets the new model
$\boldsymbol{\theta}^{t}_{k}$. Then the clients will only send the accumulated
gradients
$\boldsymbol{g}^{t}_{k}=\boldsymbol{\theta}^{t-1}-\boldsymbol{\theta}^{t}_{k}$
back to the server, the server aggregates the collected local gradients and
updates the global model as the following:

$\boldsymbol{\theta}^{t}=\boldsymbol{\theta}^{t-1}-\eta_{s}\frac{1}{|\mathcal{C}^{t}|}\sum\limits_{k\in\mathcal{C}^{t}}\frac{|\mathcal{D}_{k}|}{\sum_{i\in\mathcal{C}^{t}}|\mathcal{D}_{i}|}\boldsymbol{g}_{k}^{t},$
(2)

where $\eta_{s}$ is the server learning rate, $|\mathcal{C}^{t}|$ is the
number of clients participating in the current round.

In this paper, we study the optimization problem of FL in the setting where
the global data distribution $\mathcal{D}=\bigcup_{k}\mathcal{D}_{k}$ is long-
tailed. Previous studies in centralized long-tailed learning Kang et al.
(2019) propose to decouple the training on the long-tailed classification
tasks into the representation learning and classifier learning phases, and
point out that performing class-level re-weighting rather than instance-
balanced training brings negative impact on the representation learning, and
the imbalanced data distribution mainly affects the classifier learning. Thus,
our main motivation is to effectively re-balance the classifier when dealing
with the long-tailed global data. Specifically, in order to tackle the problem
of the lack of a global balanced dataset in FL, we manage to make each client
re-balance the classifier locally during training, by taking great advantage
of the abundant real data stored in the local datasets.

###  3.2 Our Optimization Target

We split the original model architecture $\boldsymbol{\theta}$ into two parts:
the representation encoder $\boldsymbol{P}$ and the classifier
$\boldsymbol{W}$, and aim to re-balance $\boldsymbol{W}$ during the local
training to make it behave well on the class-balanced data distribution
$\mathcal{D}^{bal}$. However, re-balancing classifier during (instead of
after) the representation learning phase leads to a contradictory optimization
target:

$\displaystyle(\boldsymbol{P}^{*},\boldsymbol{W}^{*})$
$\displaystyle=\mathop{\arg\min}\limits_{\boldsymbol{P},\boldsymbol{W}}L(\boldsymbol{P},\boldsymbol{W};\bigcup_{k}\mathcal{D}_{k})$
(3)
$\displaystyle=\mathop{\arg\min}\limits_{\boldsymbol{P},\boldsymbol{W}}\sum_{k=1}^{N}\frac{|\mathcal{D}_{k}|}{\sum_{i=1}^{N}|\mathcal{D}_{i}|}L(\boldsymbol{P},\boldsymbol{W};\mathcal{D}_{k}),$
$\displaystyle\text{s.t.}\qquad\boldsymbol{W}^{*}$
$\displaystyle=\mathop{\arg\min}\limits_{\boldsymbol{W}}L(\boldsymbol{P}^{*},\boldsymbol{W};\mathcal{D}^{bal}),$

As we can see, when the global data distribution
$\mathcal{D}=\bigcup_{k}\mathcal{D}_{k}$ is long-tailed, the above
optimization problem has no solution, caused by the contradictory goals when
updating $\boldsymbol{W}$. To address to negative impact of performing local
classifier re-balancing, we design an architecture of the two-stream
classifiers by adding a new classifier $\widehat{\boldsymbol{W}}$ in the
training phase, in order to help model the global data distribution
$\mathcal{D}$ and make re-balancing $\boldsymbol{W}$ possible. The full
illustrations of our model architecture and training process are in Figure 1,
and we re-formulate our global optimization target as:

$\begin{aligned}
(\boldsymbol{P}^{*},\boldsymbol{W}^{*},\widehat{\boldsymbol{W}^{*}})&=(\boldsymbol{\theta}^{*},\widehat{\boldsymbol{W}^{*}})\\\
&=\mathop{\arg\min}\limits_{\boldsymbol{\theta},\widehat{\boldsymbol{W}}}\sum_{k=1}^{N}\frac{|\mathcal{D}_{k}|}{\sum_{i=1}^{N}|\mathcal{D}_{i}|}L(\boldsymbol{\theta},\widehat{\boldsymbol{W}};\mathcal{D}_{k}),\\\
\text{s.t.}\qquad\boldsymbol{W}^{*}&=\mathop{\arg\min}\limits_{\boldsymbol{W}}L(\boldsymbol{P}^{*},\boldsymbol{W};\mathcal{D}^{bal}).\end{aligned}$

(4)

By making the combination of two classifiers model the global data
distribution in the first part of Eq. (4), we make sure that the
representation encoder is trained under the instance-balanced training
paradigm, which benefits the representation learning most. In the following,
we introduce our algorithm to solve Eq. (4) from three aspects, including the
local training stage, the server aggregation stage, and the inference stage.

###  3.3 Classifier Re-Balancing by Integrating Local Real Data with Global
Gradient Prototypes

Figure 1: The full illustration of our method. We add a new global classifier
$\widehat{\boldsymbol{W}}$, and perform the instance-balanced training on the
whole network (including the encoder and the two classifiers). Furthermore, we
propose a novel algorithm to re-balance the original classifier during the
local training by integrating the local real data and the global gradient
prototypes ($\boldsymbol{g}_{\boldsymbol{W},c}^{pro}$ in the figure) to form a
local balanced dataset for adjusting the original classifier. In the inference
phase, we only keep the global encoder $\boldsymbol{P}$ and the re-balanced
classifier $\boldsymbol{W}$. Algorithm 1 Local Training Process of RedGrape

Input: Round number $t$, local data $\mathcal{D}_{k}$ with local label set
$\mathcal{L}_{k}$, global model
$(\boldsymbol{P}^{t-1},\boldsymbol{W}^{t-1},\widehat{\boldsymbol{W}}^{t-1})$,
global gradient prototypes
$\\{\boldsymbol{g}_{\boldsymbol{W}^{t-2},c}^{pro}|c\in\mathcal{L}\\}$.

1: Calculate local gradient prototypes
$\\{\boldsymbol{g}_{\boldsymbol{W}^{t},k,c}^{pro}|c\in\mathcal{L}_{k}\\}$
based on Eq. (10).

2: for Local step $i=1,2,\cdots,I$ do

3: Update $\boldsymbol{P}$ and $\widehat{\boldsymbol{W}}$ based on Eq. (6).

4: Compute batch gradients for $\boldsymbol{W}$ based on Eq. (7).

5: Compute re-balancing gradients for $\boldsymbol{W}$ based on Eq. (9) and
Eq. (11).

6: Update $\boldsymbol{W}$ based on Eq. (12).

7: end for

8: return Local gradients
$(\boldsymbol{P}^{t}_{k}-\boldsymbol{P}^{t-1},\boldsymbol{W}^{t}_{k}-\boldsymbol{W}^{t-1},\widehat{\boldsymbol{W}}^{t}_{k}-\widehat{\boldsymbol{W}}^{t-1})$,
local gradient prototypes
$\\{\boldsymbol{g}_{\boldsymbol{W}^{t-1},k,c}^{pro}|c\in\mathcal{L}_{k}\\}$.

#### Local Training Stage

In the local training, each client aims to solve the sub-problem of Eq. (4) as

$\displaystyle(\boldsymbol{P}^{*},\boldsymbol{W}^{*},\widehat{\boldsymbol{W}^{*}})$
$\displaystyle=\mathop{\arg\min}\limits_{\boldsymbol{P},\boldsymbol{W},\widehat{\boldsymbol{W}}}L(\boldsymbol{P},\boldsymbol{W},\widehat{\boldsymbol{W}};\mathcal{D}_{k}),$
(5) $\displaystyle\text{s.t.}\qquad\boldsymbol{W}^{*}$
$\displaystyle=\mathop{\arg\min}\limits_{\boldsymbol{W}}L(\boldsymbol{P}^{*},\boldsymbol{W};\mathcal{D}^{bal}).$

It is a constrained optimization problem that is non-trivial to solve, we
choose to address it by considering it as a multi-target learning task and
optimizing all parameters concurrently. We briefly summarize the whole process
of local training in our method in Algorithm 1. To specific, the encoder
parameters $\boldsymbol{P}$ and the additional classifier
$\widehat{\boldsymbol{W}}$ will be trained under an instance-balanced manner
(Line 3). When updating $\boldsymbol{W}$, besides the gradients of the batch
samples from $\mathcal{D}_{k}$ (Line 4), our method creates a local balanced
dataset $\mathcal{D}^{bal}_{k}$ to help re-balancing $\boldsymbol{W}$ (Line
5-6) following the second target of Eq. (5). The detailed steps including the
following parts:

##### Updating $\boldsymbol{P}$ and $\widehat{\boldsymbol{W}}$.

In the $t$-th round, we perform the normal stochastic gradient decent
mechanism111We do not have the assumption about the local optimizer, and any
local optimizer (e.g., SGDM or Adam Kingma and Ba (2014)) is acceptable. Here,
we take SGD as an example. in which an instance-balanced dataloader is applied
to update $\boldsymbol{P}$ and $\widehat{\boldsymbol{W}}$.222For simplicity,
we omit the bias term here, while our method is still applicable when the bias
term exists. That is, for the local step $i=1,2,\cdots,I$, a random batch of
examples $\mathcal{B}_{k}^{i}$ is sampled from $\mathcal{D}_{k}$ to perform
that:

$\begin{aligned}
\boldsymbol{P}^{i}_{k}&=\boldsymbol{P}^{i-1}_{k}-\eta_{l}\nabla_{\boldsymbol{P}^{i-1}_{k}}L(\boldsymbol{P}^{i-1}_{k},\boldsymbol{W}^{i-1}_{k},\widehat{\boldsymbol{W}}^{i-1}_{k};\mathcal{B}_{k}^{i}),\\\
\widehat{\boldsymbol{W}}^{i}_{k}&=\widehat{\boldsymbol{W}}^{i-1}_{k}-\eta_{l}\nabla_{\widehat{\boldsymbol{W}}^{i-1}_{k}}L(\boldsymbol{P}^{i-1}_{k},\boldsymbol{W}^{i-1}_{k},\widehat{\boldsymbol{W}}^{i-1}_{k};\mathcal{B}_{k}^{i}),\end{aligned}$

(6)

in which the initial model
$(\boldsymbol{P}^{0}_{k},\boldsymbol{W}^{0}_{k},\widehat{\boldsymbol{W}}^{0}_{k})$
is chosen as the global model
$(\boldsymbol{P}^{t-1},\boldsymbol{W}^{t-1},\widehat{\boldsymbol{W}}^{t-1})$
received from the server in the current round, $\eta_{l}$ is the local
learning rate. One important thing is, when calculating the above loss $l$ on
each sample $(\boldsymbol{x},y)$, the representation vector
$\boldsymbol{h}:=f(\boldsymbol{x};\boldsymbol{P})$ will be first fed into both
two classifiers and get two logits $\boldsymbol{W}^{T}\boldsymbol{h}$ and
$\widehat{\boldsymbol{W}}^{T}\boldsymbol{h}$. Then we perform the element-wise
addition to get the final logits
$\boldsymbol{z}=\boldsymbol{W}^{T}\boldsymbol{h}+\widehat{\boldsymbol{W}}^{T}\boldsymbol{h}$,
and use $\boldsymbol{z}$ for the loss calculation.

##### Updating $\boldsymbol{W}$.

When updating $\boldsymbol{W}$, one part of gradients comes from the above
back propagation process on $\mathcal{B}_{k}^{i}$ as

$\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{local}=\nabla_{\boldsymbol{W}^{i-1}_{k}}L(\boldsymbol{P}^{i-1}_{k},\boldsymbol{W}^{i-1}_{k},\widehat{\boldsymbol{W}}^{i-1}_{k};\mathcal{B}_{k}^{i}),$
(7)

which corresponds to the first part in Eq. (5). For the second part of Eq.
(5), it needs to calculate the gradients of $\boldsymbol{W}^{i-1}_{k}$ on a
small balanced set $\mathcal{D}^{bal}_{k}$, that is supposed to be created in
the local. However, there exists difficulty in constructing
$\mathcal{D}^{bal}_{k}$ from $\mathcal{D}_{k}$, since it is very likely that
there are some classes missing in the local label set $\mathcal{L}_{k}$ of
$\mathcal{D}_{k}$ due to the non-i.i.d. data partitions. Then, we propose a
mixed gradient re-balancing mechanism to overcome this challenge by
integRating local rEal Data with Global gRAdient prototyPEs (RedGrape as our
method). Specifically, for each class $c$, (1) if the sample quantity of class
$c$ in $\mathcal{D}_{k}$ reaches a threshold $T$, we think client $k$ have
sufficient samples of class $c$ in its local dataset, and randomly sample $T$
training samples of class $c$ to form $\mathcal{D}^{bal}_{k,c}$ for
$\mathcal{D}^{bal}_{k}$.333In different rounds, client $k$ can choose
different $T$ samples of class $c$ for $\mathcal{D}^{bal}_{k,c}$, in order to
make fully use of the local real data. Then, the gradients contributed by
class $c$ in $\mathcal{D}^{bal}_{k}$ is

$\displaystyle\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k},c}=\nabla_{\boldsymbol{W}^{i-1}_{k}}L(\boldsymbol{P}^{i-1}_{k},\boldsymbol{W}^{i-1}_{k};D^{bal}_{k,c}).$
(8)

(2) If client $k$ does not have enough data of class $c$ in its local dataset,
we choose to estimate the gradient contribution of class $c$ with the global
gradient prototype of class $c$ in the $(t-1)$-th round, which is the averaged
gradient of training samples belonging to class $c$ w.r.t. the classifier
$\boldsymbol{W}^{t-2}$ across all available clients in the last round Shang et
al. (2022b):

$\displaystyle\boldsymbol{g}_{\boldsymbol{W}^{t-2},c}^{pro}=\frac{1}{|\mathcal{C}^{t-1}_{c}|}\sum_{k\in\mathcal{C}^{t-1}_{c}}\boldsymbol{g}^{pro}_{\boldsymbol{W}^{t-2},k,c},$
(9)
$\displaystyle\boldsymbol{g}_{\boldsymbol{W}^{t-2},k,c}^{pro}=\nabla_{\boldsymbol{W}^{t-2}}L(\boldsymbol{P}^{t-2},\boldsymbol{W}^{t-2};\mathcal{D}_{k,c}),$
(10)

where $\mathcal{C}_{c}^{t-1}$ represents the set of clients sampled in the
$(t-1)$-th round and have the training samples of class $c$, and
$\mathcal{D}_{k,c}$ denotes all training samples of class $c$ in
$\mathcal{D}_{k}$. Thus, it requires each client sampled in the previous round
to first calculate the local gradient prototype of each class
$c\in\mathcal{L}_{k}$ on the same model
$(\boldsymbol{P}^{t-2},\boldsymbol{W}^{t-2})$, return
$\\{\boldsymbol{g}_{\boldsymbol{W}^{t-2},k,c}^{pro}|c\in\mathcal{L}_{k}\\}$
back to the server along with other local gradients, and receive the global
gradient prototypes averaged and sent by the server. Then, the final gradients
on the local balanced dataset to optimize the second part of Eq. (5) is

$\displaystyle\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{bal}=\frac{1}{|\mathcal{L}|}(\sum\limits_{c\in\mathcal{L}_{k}^{bal}}\boldsymbol{g}_{\boldsymbol{W}_{k}^{i-1},c}+\sum\limits_{c\in\mathcal{L}\backslash\mathcal{L}_{k}^{bal}}\boldsymbol{g}_{\boldsymbol{W}^{t-2},c}^{pro}),$
(11)

where $\mathcal{L}^{bal}_{k}\subset\mathcal{L}_{k}$ is the label set in which
each classe consists of more than $T$ samples and $\mathcal{L}$ is the entire
label set. Finally, the updating rule for $\boldsymbol{W}^{i-1}_{k}$ is444The
local classifier re-balancing starts from the 2nd round.

$\begin{aligned}
\boldsymbol{W}^{i}_{k}=\boldsymbol{W}^{i-1}_{k}-\eta_{l}&\left[\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{local}+\lambda\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{bal}\frac{\|\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{local}\|}{\|\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{bal}\|}\right],\end{aligned}$

(12)

where $\lambda$ is a re-balance factor to control the re-balancing strength
for updating $\boldsymbol{W}$. In Eq. (12), we normalize the scale555Here,
$\|\cdot\|$ represents the Frobenius Norm. of
$\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{bal}$ at each step in order to
address the unstable training caused by the constant part of
$\boldsymbol{g}_{\boldsymbol{W}^{t-1},c}^{pro}$, by making its scale
consistent with the decreasing trend of the scale of real gradients during
training.

After training, the new model is
$(\boldsymbol{P}^{t}_{k},\boldsymbol{W}^{t}_{k},\widehat{\boldsymbol{W}}^{t}_{k})$,
and client $k$ sends the local gradients
$(\boldsymbol{g}_{\boldsymbol{P}^{t-1},k},\boldsymbol{g}_{\boldsymbol{W}^{t-1},k},\boldsymbol{g}_{\widehat{\boldsymbol{W}}^{t-1},k})=(\boldsymbol{P}^{t}_{k}-\boldsymbol{P}^{t-1},\boldsymbol{W}^{t}_{k}-\boldsymbol{W}^{t-1},\widehat{\boldsymbol{W}}^{t}_{k}-\widehat{\boldsymbol{W}}^{t-1})$
along with the local gradient prototypes
$\\{\boldsymbol{g}_{\boldsymbol{W}^{t-1},k,c}^{pro}|c\in\mathcal{L}_{k}\\}$ to
the server.

####  3.3.1 Server Aggregation Stage

The server first aggregates the gradients and updates the global model as

$\displaystyle(\boldsymbol{P}^{t},\boldsymbol{W}^{t},\widehat{\boldsymbol{W}}^{t})=(\boldsymbol{P}^{t-1},\boldsymbol{W}^{t-1},\widehat{\boldsymbol{W}}^{t-1})$
(13)
$\displaystyle-\eta_{s}\sum\limits_{k\in\mathcal{C}^{t}}{\frac{|\mathcal{D}_{k}|}{\sum_{i\in\mathcal{C}^{t}}|\mathcal{D}_{i}|}}(\boldsymbol{g}_{\boldsymbol{P}^{t-1},k},\boldsymbol{g}_{\boldsymbol{W}^{t-1},k},\boldsymbol{g}_{\widehat{\boldsymbol{W}}^{t-1},k}).$

Also, the server needs to update the global gradient prototypes as

$\boldsymbol{g}_{\boldsymbol{W}^{t-1},c}^{pro}=\left\\{\begin{aligned}
&\frac{1}{|\mathcal{C}^{t}_{c}|}\sum_{k\in\mathcal{C}^{t}_{c}}\boldsymbol{g}^{pro}_{\boldsymbol{W}^{t-1},k,c},\quad\mathcal{C}_{c}^{t}\neq\emptyset,\\\
&\boldsymbol{g}_{\boldsymbol{W}^{t-2},c}^{pro},\quad\mathcal{C}_{c}^{t}=\emptyset,\end{aligned}\right.$
(14)

in which the second case corresponds to the situation where all clients in
$C^{t}$ from the current round do not contain samples of class $c$. In this
case, we re-use the global gradient prototype of class $c$ from the previous
round. The updated global model and global gradient prototypes are broadcast
to the sampled clients in the next round.

### Inference Stage

After federated training, we only keep the re-balanced classifier
$\boldsymbol{W}$ and abandon $\widehat{\boldsymbol{W}}$ in the reference
stage:

$y_{\text{pred}}=\mathop{\arg\max}_{i}[\boldsymbol{W}^{T}f(x;\boldsymbol{P})].$
(15)

##  4 Experiments and Analysis

###  4.1 Experimental Settings

##### Datasets and Models

We conduct experiments on three popular image classification benchmarks: MNIST
LeCun et al. (1998), CIFAR-10 and CIFAR-100 Krizhevsky et al. (2009). We
follow existing studies Cao et al. (2019); Shang et al. (2022b) to create the
long-tailed versions of training sets of above three datasets (i.e., MNIST-LT,
CIFAR-10/100-LT), and keep the test sets as balanced. We first define the term
Imbalance Ratio: $\text{IR}=\frac{\max_{c}\\{n_{c}\\}}{\min_{c}\\{n_{c}\\}}$,
which is the ratio between the maximum sample number across all classes and
the minimum sample number across all classes, to reflect the imbalance degree
of the global data distribution. Then, the training sample quantity of each
class follows an exponential decay. We choose $\text{IR}=10,50,100$ in our
main experiments. Furthermore, we follow the existing studies Reddi et al.
(2020); Shang et al. (2022b) to adopt the Dirichlet distribution
$\text{Dir}(\alpha)$ for the non-i.i.d. data partitioning, in which $\alpha$
controls the non-i.i.d. degree. We set $\alpha=1.0$ in our main experiments,
and put the results of other $\alpha$s in the Appendix. We use the
convolutional neural network (CNN) McMahan et al. (2017) for MNIST, and use
ResNet-56 He et al. (2016) for CIFAR-10/100 datasets. More details can be
found in the Appendix.

##### Baseline Methods

We compare our method with the existing federated long-tailed learning
algorithms, including the traditional FedAvg algorithm with the CrossEntropy
Loss (FedAvg+CE) applied in the local training McMahan et al. (2017), Fed-
Focal Loss Sarkar et al. (2020), Ratio Loss Wang et al. (2021), CLIMB Shen et
al. (2021), and the state-of-the-art method CReFF Shang et al. (2022b).

##### Training Details

We conduct experiments in two popular FL settings based on the ratio of
clients participating in each round: (1) Full client participation setting:
all clients participate in updating the global model in each round, and the
total number of clients is 10 in this setting; (2) Partial client
participation setting: the total number of clients is 50 but only 10 clients
are randomly sampled in each round. We adopt SGDM as the optimizer for local
training. The local learning rate is 0.01 for MNIST-LT and 0.1 for
CIFAR-10/100-LT. The number of local epochs is 5 for all datasets. As for our
method, the re-balance factor $\lambda$ is fixed as 0.1 in all experiments,
and we explore the effect of different values of $\lambda$ in Section 5.1. The
quantity threshold $T$ for each class to create the local balanced dataset is
set as 8 for MNIST-LT and CIFAR-10-LT, and 2 for CIFAR-100-LT, and we put
further discussion in Section 5.2. Each experiment is run on 3 random seeds.
Complete training details (e.g., the number of communication rounds in each
setting, detailed hyper-parameters of other baselines) are in the Appendix.
Our code is implemented on the FedML He et al. (2020) platform.666Our code
will be released upon acceptance.

###  4.2 Main Results

Method | MNIST-LT | CIFAR-10-LT | CIFAR-100-LT
---|---|---|---
$\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$
FedAvg+CE | 97.99 | 95.98 | 92.71 | 76.21 | 68.41 | 59.83 | 49.08 | 36.47 | 33.28
Fed-Focal Loss | 97.90 | 96.14 | 92.97 | 77.92 | 61.21 | 59.86 | 48.14 | 35.51 | 30.05
Ratio Loss | 97.96 | 96.20 | 92.99 | 78.58 | 68.01 | 59.27 | 48.30 | 37.62 | 31.92
CLIMB | 97.89 | 95.87 | 92.71 | 78.95 | 66.25 | 57.67 | 49.27 | 36.13 | 32.18
CReFF | 97.68 | 96.49 | 93.85 | 83.18 | 73.46 | 69.36 | 46.58 | 35.82 | 33.46
Ours | 98.34 | 97.06 | 95.73 | 83.74 | 74.01 | 71.04 | 51.09 | 38.49 | 34.63
Table 1: Results under the full client participation setting. We report the overall test accuracy on the balanced testing set of each dataset. Method | MNIST-LT | CIFAR-10-LT | CIFAR-100-LT
---|---|---|---
$\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$
FedAvg+CE | 95.51 | 91.82 | 89.92 | 60.38 | 45.15 | 40.06 | 40.81 | 24.62 | 22.08
Fed-Focal Loss | 96.79 | 92.59 | 90.45 | 61.16 | 46.20 | 41.10 | 40.85 | 24.73 | 20.17
Ratio Loss | 95.17 | 91.10 | 89.64 | 63.97 | 44.22 | 42.11 | 40.96 | 24.12 | 23.06
CLIMB | 95.67 | 92.24 | 89.75 | 61.75 | 46.91 | 42.02 | 40.64 | 23.99 | 21.44
CReFF | 96.29 | 94.16 | 92.16 | 69.38 | 60.52 | 55.63 | 39.38 | 25.42 | 24.77
Ours | 97.54 | 95.17 | 93.61 | 71.68 | 61.42 | 57.11 | 42.97 | 27.73 | 25.64
Table 2: Results under the partial client participation setting. We report the
overall test accuracy on the balanced testing set of each dataset.

In the main paper, we report the averaged accuracy over the last 10 rounds on
the balanced testing set of each dataset following existing studies Reddi et
al. (2020). We also display the averaged test accuracy on tail classes in each
setting in the Appendix to show that our method can significantly bring
improvement to the model’s performance on tail classes. The results under the
full client participation setting are in Table 1, and Table 2 displays the
results under the partial client participation setting. We can draw the main
conclusion from these tables as: our method can consistently outperform the
existing algorithms in all settings.

As we can see, Fed-Focal Loss achieves lower performance than the FedAvg with
CE loss in some settings, which validates the claim that directly applying the
centralized long-tailed learning methods can not help to address the global
class imbalance problem in FL, as it ignores the mismatch between the global
and the local imbalance patterns. Ratio Loss and CLIMB apply the class-level
re-weighting and client-level re-weighting idea separately, and gain slight
improvement compared with FedAvg. We analyze that the reason for the limited
improvement lies in that though the re-weighting practice helps the model to
focus more on the learning of tail classes, it is not conducive to the
representation learning on the abundant data of the head classes Kang et al.
(2019). Moreover, the assumption of obtaining a global auxiliary dataset makes
Ratio Loss impractical in real cases.

Figure 2: The test accuracy curves on CIFAR-10-LT with $\text{IR}=100$ under
the full client participation setting. Our method achieves faster convergence
speed and better performance than all existing baselines.

The superior performance of CReFF helps to validates the effectiveness of the
classifier re-balancing on the final performance. However, the optimization of
the federated features requires massive computations on the server (especially
when the number of classes is large), and the federated features from the same
class may converge to be similar. Thus, the re-trained classifier faces the
problem that it may overfit on the highly similar and small amount of the
federated features (reflected in the poorer performance under smaller IR). Our
method instead takes full advantage of the local real data integrated with the
global gradient prototypes to locally re-balance the classifier while
maintaining the good effects of instance-balanced training on the
representation learning, and consistently outperforms all previous methods by
a large margin. Compared with CReFF and Ratio Loss, we do not have extra
requirements except for the aggregation procedures on the server, and produce
a re-balanced classifier that has better generalization ability with the help
of abundant real data.

We further display the evaluation accuracy curve after each round in
CIFAR-10-LT ($\text{IR}=100$) under the full client participation setting in
Figure 2. As we can see, our method not only has the best converged
performance, but also achieves much faster convergence speed than all baseline
methods. That is because our method re-balances the classifier at each local
training step, and this makes it converge faster to the optimal balanced
classifier.

###  4.3 Results in Another Class Imbalance Setting

Method | Full Participation | Partial Participation
---|---|---
MNIST | CIFAR-10 | MNIST | CIFAR-10
FedAvg+CE | 93.91 | 70.91 | 91.97 | 61.27
Fed-Focal Loss | 93.53 | 70.77 | 91.77 | 59.18
Ratio Loss | 94.23 | 73.22 | 92.32 | 62.86
CLIMB | 93.89 | 72.04 | 92.11 | 63.18
CReFF | 96.70 | 78.98 | 96.01 | 71.41
Ours | 96.86 | 79.88 | 96.43 | 73.31
Table 3: Results in the binary class imbalance setting. $\text{IR}=100$ and
the randomly chosen tail classes are 0, 7 and 8.

We also conduct experiments in a binary class imbalance setting in FL Shen et
al. (2021), in which three classes are randomly chosen as the tail classes,
and they are assigned with $1/\text{IR}$ number of sampled compared with other
normal/head classes. The experiments are conducted on MNIST and CIFAR-10
datasets with $\text{IR}=100$, and other experimental settings are kept as the
same as that in our main experiments. The results are in Table 3. The
conclusion remains the same that, our method achieves the best performance in
all cases.

##  5 Further Explorations

In this section, we make further explorations about the two crucial hyper-
parameters of our method. The following experiments are conducted on
CIFAR-10-LT with $\text{IR}=100$ under the full client participation setting.

###  5.1 Re-Balancing Strength Decides on The Convergence Trade-off

Figure 3: The test accuracy of using different re-balance factor $\lambda$ in
CIFAR-10-LT. Smaller $\lambda$ leads to slower convergence speed but
relatively better generalization ability.

In order to solve the optimization target of Eq. (4), we consider updating
$\boldsymbol{W}$ in a multi-task learning setting as Eq. (12), where a re-
balance factor $\lambda$ is used to control the re-balancing strength. Here,
we conduct experiments to explore the effect of different $\lambda$s on the
model’s performance, and the results are shown in Figure 3. We find that the
smaller $\lambda$ results in slower convergence speed but obtains relatively
better performance of the converged model. We analyze the reason lies in that,
the $\boldsymbol{g}_{\boldsymbol{W}^{i-1}_{k}}^{bal}$ contains a part of
global gradient prototypes calculated in the previous round and is a constant
when updating $\boldsymbol{W}$. It will adversely affect the model’s
convergence in the late stage of the training when we are still using a large
$\lambda$ to re-balance the classifier. An interesting direction to improve
our method is designing an adaptive $\lambda$ that decays along with the
training, which we leave to future work. When $\lambda=0.0$, the addition of
two classifiers equals to one normal classifier used in FedAvg, so FedAvg is a
special case of our method in this case.

###  5.2 Local Real Data Plays An Important Role in Re-Balancing The
Classifier

During creating the local balanced datasets, we set a threshold $T$ to decide
whether the local clients own the enough data of a specific class. Larger $T$
decreases the number of available classes in which the real data can be used
to calculate the gradients for classifier re-balancing, while smaller $T$
leads to the relatively unreliable gradients of class $c$. We then explore the
effect of different $T$s, and put the results in Figure 4. We indeed observe a
trade-off pattern as expected and find that $T=4,8$ are the most proper
choices. $T=\infty$ means we remove the role of local real data on the
classifier re-balancing and only use the global gradient prototypes instead,
and we find the performance degrades greatly, which verifies the large
benefits of using local real data to adjust the classifier.

Figure 4: The test accuracy of using different sample quantity thresholds for
each class when creating the local balanced datasets in CIFAR-10-LT.

##  6 Conclusion

In this paper, motivated by the decoupled training idea, we propose a novel
and effective classifier re-balancing algorithm for tackling federated long-
tailed learning. In order to overcome the lack of a public balanced dataset in
FL, we propose to re-balance the classifier during local training by
integrating local real data with global gradient prototypes. Furthermore, in
order to address the problem of contradictory optimization goals during
training brought by performing local classifier re-balancing, we introduce a
two-stream classifiers architecture to help model the global data
distribution. Thorough experiments verify the great effectiveness of our
method over strong baselines without extra data requirements.

## Ethical Statement

Our purpose is to address the optimization problem of FL on the non-i.i.d. and
long-tailed data and help to learn a better global model that has good
performance on all classes. The datasets used in our experiments are all
publicly available. Also, our method only requires the normal gradients
transmission between the server and the clients, which will not expose the
local data privacy and does not have any ethical concerns.

## Acknowledgments

This work was supported by a Tencent Research Grant. Xu Sun is the
corresponding author of this paper.

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##  Appendix A Detailed Experimental Settings

###  A.1 Datasets and Models

Here, we introduce the datasets and the backbone models we used in our
experiments. We choose three classical image classification tasks, including
MNIST LeCun et al. [1998], CIFAR-10 and CIFAR-100 Krizhevsky et al. [2009]. We
then follow existing centralized and federated long-tailed learning studies
Cao et al. [2019]; Shang et al. [2022b] to create the long-tailed versions of
the training sets of above datasets (i.e., MNIST-LT, CIFAR-10/100-LT).
Specifically, the long-tailed degree is controlled by a ratio called the
Imbalance Ratio: $\text{IR=}\frac{\max_{c}\\{n_{c}\\}}{\min_{c}\\{n_{c}\\}}$,
where $n_{c}$ represents the sample quantity of class $c$ (0-indexed). Then,
we manage to make the sample quantity of each class follow an exponential
decay trend:

$n_{c}=n_{0}\times\left(\frac{1}{\text{IR}}\right)^{\frac{c}{C-1}},\quad
c=0,\cdots,C-1.$ (16)

As for the non-i.i.d. data partitioning, we follow existing studies Reddi et
al. [2020]; Shang et al. [2022b] to adopt the Dirichlet distribution
$\text{Dir}(\alpha)$. Smaller $\alpha$ means the heavier non-i.i.d. degree. We
choose $\alpha=1.0$ in our main paper, and we also put the results on
different $\alpha$s in Appendix C.

We use the same convolutional neural network (CNN) used in McMahan et al.
[2017] for experiments on MNIST-LT, and adopt the ResNet-56 He et al. [2016]
as the backbone model for CIFAR-10/100-LT.

###  A.2 Complete Training Details

#### Local Training Settings

We utilize SGDM as the local optimizer in all experiments. The local learning
rate is $0.01$ for MNIST-LT, and $0.1$ for CIFAR-10/100-LT. For all three
datasets, the batch size for local training is 64, and the number of local
training epochs is 5. As mentioned in main paper, we perform experiments in
both full client participation and partial client participation settings. We
set different total communication rounds in different setting considering the
different convergence speeds of the global models: (1) In the full client
participation setting, the number of communication rounds is 200 for MNIST-LT,
and 500 for CIFAR-10/100-LT. (2) In the full client participation setting, the
number of communication rounds is 500 and 1000 for MNIST-LT and
CIFAR-10/100-LT separately.

#### Server Aggregation Settings

During server aggregation, we follow the same procedure as that in FedAvg to
aggregate the collected local gradients in the current round, and update the
global model with the averaged gradients. The server learning rate is tuned as
$1.0$ for all experiments. Furthermore, as for CReFF and our method, the
server needs to update the global gradient prototypes (refer to Section 3.3 in
our main paper) by averaging local gradient prototypes. However, compared with
CReFF, we do not have extra requirements on the server to make it perform
further optimization and training process.

#### Hyper-Parameters of Each Method

Here, we introduce the choices of hyper-parameters used in each method in
detail.

##### Fed-Focal Loss:

Fed-Focal Loss Sarkar et al. [2020] directly applies Focal Loss Lin et al.
[2017] to the local training. The form of Focal Loss is

$L_{focal}=-(1-p_{T})^{\gamma}\log(p_{T}),$ (17)

where $p_{T}$ is the predicted probability of the sample corresponding to the
ground truth class. We set $\gamma=2$ in our experiments.

##### Ratio Loss:

Ratio Loss Wang et al. [2021] applies the class-level re-weighting practice by
first estimating the global imbalance pattern on the server with an auxiliary
balanced dataset. Its form can be written as

$L_{ratio}=(\alpha+\beta\mathbb{R})L_{CE},$ (18)

where $L_{CE}$ is the traditional CrossEntropy Loss, $\mathbb{R}$ is the ratio
vector that contains the relatively estimated sample quantity of each class on
the server, $\alpha$ and $\beta$ are two hyper-parameters. Thus, we follow the
original study Wang et al. [2021] to set the sample number of each class on
the auxiliary balanced dataset to be 32, $\alpha=1.0$, $\beta=0.1$.

##### CLIMB:

CLIMB Shen et al. [2021] aims to perform the client-level re-weighting to up-
weight the aggregation weights for the local gradients with larger local
training losses, as the global model behaves poorly on these clients’ local
data. The hyper-parameters in CLIMB includes a tolerance constant $\epsilon$
and a dual step size $\eta_{D}$. In our experiments, we follow the original
setting to set $\epsilon=0.01$ for MNIST-LT and $\epsilon=0.1$ for
CIFAR-10/100-LT, set $\eta$ as 2.0 and 0.1 for MNIST-LT and CIFAR-10/100-LT
separately.

##### CReFF:

CReFF Shang et al. [2022b] needs to create a set of federated features on the
server, of which the number per class is 100. Following the original setting,
the optimization steps on the federated features is 100, the classifier re-
training steps is 300. Further, the learning rate of optimizing the federated
features is 0.1 for all datasets, and the learning rate of classifier re-
training is kept as the same as that used in the local training on that
dataset.

##### RedGrape (Ours):

Our method introduces two hyper-parameters: $\lambda$ for the classifier re-
balancing strength, and $T$ for the sample quantity threshold of each class on
creating local balanced datasets. We put the detailed discussions about these
two hyper-parameters in our main paper. The recommended search grids for
$\lambda$ are $\\{1.0,0.1,0.01\\}$, and $\\{2,4,8\\}$ for $T$.

#### Code and Infrastructure

Our code is implemented based on the open-sourced FL platform FedML He et al.
[2020]. We will release our code upon acceptance. Our experiments are
conducted on 8 * GeForce RTX 2080 Ti.

Method | MNIST-LT | CIFAR-10-LT | CIFAR-100-LT
---|---|---|---
$\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$
FedAvg+CE | 96.05 | 90.23 | 82.21 | 67.87 | 55.26 | 29.24 | 36.25 | 13.37 |  08.50
Fed-Focal Loss | 95.84 | 90.47 | 82.62 | 74.94 | 47.37 | 33.86 | 34.40 | 12.53 |  06.49
Ratio Loss | 96.04 | 90.76 | 83.01 | 71.27 | 55.79 | 33.28 | 34.55 | 15.35 |  07.98
CLIMB | 95.70 | 89.93 | 82.01 | 73.68 | 55.86 | 30.95 | 35.81 | 13.31 |  08.65
CReFF | 95.98 | 91.98 | 86.62 | 82.87 | 66.10 | 57.03 | 38.18 | 22.77 | 18.98
Ours | 96.78 | 92.97 | 89.59 | 83.86 | 69.74 | 60.41 | 40.11 | 20.19 | 15.58
Table 4: Averaged evaluation accuracy on the tail classes under the full client participation setting in main experiments. Method | MNIST-LT | CIFAR-10-LT | CIFAR-100-LT
---|---|---|---
$\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$ | $\text{IR}=10$ | $\text{IR}=50$ | $\text{IR}=100$
FedAvg+CE | 89.75 | 80.05 | 74.35 | 51.99 | 24.03 |  02.64 | 29.65 |  08.55 |  05.74
Fed-Focal Loss | 92.65 | 82.43 | 75.46 | 49.78 | 27.43 |  05.68 | 29.51 |  08.89 |  04.08
Ratio Loss | 89.83 | 77.93 | 73.35 | 55.54 | 23.81 |  04.87 | 29.28 |  08.36 |  05.86
CLIMB | 90.09 | 81.49 | 74.58 | 54.45 | 23.90 |  04.14 | 29.85 |  08.37 |  04.97
CReFF | 92.87 | 88.94 | 83.75 | 74.81 | 62.67 | 45.30 | 34.38 | 17.46 | 16.58
Ours | 94.82 | 90.06 | 86.38 | 75.71 | 63.58 | 43.92 | 35.34 | 17.91 | 13.39
Table 5: Averaged evaluation accuracy on the tail classes under the partial
client participation setting in main experiments.

##  Appendix B Results on The Tail Classes in Main Experiments

In our main paper, we put the results of the overall accuracy on the balanced
testing sets of each method, and we have that our method consistently
outperform all other methods in all settings. Here, we put the averaged
accuracy on the tail classes of each method. Specifically, we define the tail
classes as the last 30% classes the minimum sample quantity.

The results are in Table 4 and Table 5. As we can see, our method brings
significant improvement on the tail classes in most cases. Also, we find that
CReFF tends to achieve better performance on the tail classes when the
imbalance degree is larger. However, the performance of CReFF on the overall
testing sets is worse than our method according to the results in our main
paper. This partly validates our analysis that, re-training the classifier on
a set number of federated features on the server can indeed re-balance the
classifier to some extent, but it is likely to produce a sub-optimal
classifier that overfits on these limited number of pseudo features.

##  Appendix C Experiments on Different Non-I.I.D. Degrees

Method | MNIST-LT | CIFAR-10-LT | CIFAR-100-LT
---|---|---|---
$\alpha=0.1$ | $\alpha=1.0$ | $\alpha=10.0$ | $\alpha=0.1$ | $\alpha=1.0$ | $\alpha=10.0$ | $\alpha=0.1$ | $\alpha=1.0$ | $\alpha=10.0$
FedAvg+CE | 93.42 | 92.71 | 94.09 | 59.46 | 59.83 | 63.01 | 31.18 | 33.28 | 31.74
Fed-Focal Loss | 94.16 | 92.97 | 94.21 | 53.47 | 59.86 | 61.06 | 31.72 | 30.05 | 30.18
Ratio Loss | 93.50 | 92.99 | 94.34 | 59.30 | 59.27 | 61.82 | 32.60 | 31.92 | 31.93
CLIMB | 93.80 | 92.71 | 94.23 | 61.10 | 57.67 | 61.46 | 31.64 | 32.18 | 32.45
CReFF | 93.94 | 93.85 | 94.76 | 63.25 | 69.36 | 70.30 | 32.20 | 33.46 | 31.60
Ours | 95.34 | 95.73 | 95.93 | 64.30 | 71.04 | 71.82 | 32.86 | 34.63 | 34.42
Table 6: The overall accuracy on the balanced testing sets under different
non-i.i.d. degrees. Our method consistently outperforms all baselines.

In our main experiments, we fix the non-i.i.d. degree $\alpha=1.0$, in order
to mainly explore the effects of different imbalance degrees. Here, we conduct
extra experiments with $\alpha=0.1$ and $\alpha=10.0$, and fix the imbalance
ratio $\text{IR}=100$. We conduct experiment on MNIST-LT and CIFAR-10 under
the full client participation setting, and other experimental settings are
kept as the same as that in our main experiments.

We put the results in Table 6. We can draw the main conclusion from the table
that, our method can achieve the best performance under different non-i.i.d.
degrees.

##  Appendix D Explorations on The Role of The Additional Global Classifier

As we discussed in Section 3.2, in order to address the issue of the
contradictory optimization goals on updating $\boldsymbol{W}$ caused by re-
balancing the classifier during local training, we add an extra classifier
$\widehat{\boldsymbol{W}}$ to help model the global data distribution and make
re-balancing $\boldsymbol{W}$ possible. Here, we want to draw comparisons
between our proposed two-stream classifier architecture and using one
classifier only through experiments on CIFAR-10/100-LT datasets
($\text{IR}=100$, $\alpha=1.0$) under the full client participation situation.
The results in the partial client participation setting are similar.

The results are displayed in Figure 5. We observe that if we do not add the
extra classifier in the training phase, the model will converge to a bad local
optimum and behave much worse than that if we adopt the two-stream classifier
architecture. This helps to validate our motivation and the great
effectiveness of introducing a new global classifier to address the
optimization difficulty brought by the local classifier re-balancing practice.

(a) Results on CIFAR-10-LT.

(b) Results on CIFAR-100-LT.

Figure 5: The positive effect of adding an extra classifier in the training
phase to help model the global data distribution when locally re-balancing the
classifier by our method.