#  Scalar decay into pions via Higgs portal

Dmitry Gorbunov<EMAIL_ADDRESS>Institute for Nuclear Research of the
Russian Academy of Sciences, 117312 Moscow, Russia  Moscow Institute of
Physics and Technology, 141700 Dolgoprudny, Russia  Ekaterina Kriukova
<EMAIL_ADDRESS>Institute for Nuclear Research of the Russian
Academy of Sciences, 117312 Moscow, Russia  Lomonosov Moscow State University,
119991 Moscow, Russia  Oleg Teryaev<EMAIL_ADDRESS>Lomonosov Moscow
State University, 119991 Moscow, Russia  Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

###### Abstract

In extensions of the Standard Model (SM) of particle physics a light scalar
from a hidden sector can interact with known particles via mixing with the SM
Higgs boson. If the scalar mass is of GeV scale, this coupling induces the
scalar decay into light hadrons, that saturates the scalar width. Searches for
the light scalars are performed in many ongoing experiments and planned for
the next generation projects. Applying dispersion relations changes the
leading order estimate of the scalar decay rate into pions by a factor of
about a hundred indicating the strong final state interaction. This subtlety
for about thirty years prevented any reliable inference of the model
parameters from experimental data. In this letter we use the gravitational
form factor for neutral pion extracted from analysis of
$\gamma^{*}\gamma\to\pi^{0}\pi^{0}$ processes to estimate the quark
contribution to scalar decay into two pions. We find a factor of two
uncertainty in this estimate and argue that the possible gluon contribution is
of the same order. The decay rate to pions smoothly matches that to gluons
dominating for heavier scalars. With this finding we refine sensitivities of
future projects to the scalar-Higgs mixing. The accuracy in the calculations
can be further improved by performing similar analysis of $\gamma^{*}\gamma\to
KK$ and $\gamma^{*}\gamma\to\eta\eta$ processes and possibly decays like
$J/\psi\to\gamma+\pi\pi$.

††preprint: INR-TH-2023-003

1. New physics required to address neutrino oscillations, baryon asymmetry of the Universe, dark matter and other phenomena unexplained within the Standard Model of particle physics, can be confined in a hidden sector, so that the new particles are sterile with respect to the SM gauge interactions. They still can couple to the SM particles not only via gravity. There can be interactions via contact terms constructed by specific field products, invariant under the SM and hidden gauge groups.

One of the intriguing examples follows from the so-called scalar Higgs-field
portal Patt and Wilczek (2006), which combines the SM Higgs field $H$ and a
scalar $S$, singlet with respect to the SM gauge group, into the interaction

${\cal L}=\mu SH^{\dagger}H+\lambda S^{2}H^{\dagger}H\,.$ (1)

When the SM Higgs field gets non-zero vacuum expectation value $v=246$ GeV,
the first term in eq. (1) yields mixing between the scalar and the SM Higgs
boson $h$. Note, that if $S$ is charged under the hidden sector gauge group,
this term is absent. However, the mixing can still arise, if the hidden sector
gauge group is spontaneously broken, similar to the electroweak gauge group of
the SM. In this case the mixing between the Higgs boson and its analog in the
hidden sector comes from the second term in (1). Without loss of generality in
both cases the induced interaction between the SM Higgs boson $h$ and the
hidden scalar $S$ can be described as the mixing mass term in the scalar
sector

${\cal L}_{s}=\frac{1}{2}m_{h}^{2}h^{2}+\mu vSh+\frac{1}{2}M_{S}^{2}S^{2}\,.$
(2)

Thus, if kinematically allowed, the hidden scalar can be produced in
scatterings and decays of SM particles and can decay into the SM particles
through the virtual Higgs boson provided $\mu\neq 0$.

Hereafter we are interested in the models, where the hidden scalar is lighter
than the Higgs boson. This case may naturally be favored Vissani (1998); de
Gouvea  _et al._ (2014), because the heavy scalars coupled to the SM Higgs
field would induce large quantum corrections to its mass. Moreover, we
concentrate on the situation, where the scalar is at a GeV mass scale and so
it can be produced in particle collisions, including accelerator experiments.
While this choice may look ad hoc, there are particular extensions of the SM
which actually predict new scalars in this mass range, see e.g. Bezrukov and
Gorbunov (2010); Chen  _et al._ (2016); Dev  _et al._ (2017).

Searches for such light scalars have been performed in beam-dump experiments,
accelerator experiments on neutrino oscillations, collider experiments,
precision measurements, hunting for rare processes, etc, see Ref. Batell  _et
al._ (2022) for the most recent summary. So far, the negative results imposed
constraints on the scalar production and decay rates, which could not be
reliably transferred to limits on the model parameters because of a factor of
hundred uncertainties in estimates of the scalar decay rates into light
hadrons Donoghue  _et al._ (1990); Bezrukov and Gorbunov (2010); Winkler
(2019). In this Letter we perform the calculation of the scalar decay rate
into a couple of pions and argue that its uncertainty is only about a factor
of two.

To begin with, we note that the Lagrangian (2) can be diagonalized. The
resulting scalar couplings to the SM fields are those of the SM Higgs boson
couplings multiplied by the corresponding mixing angle $\xi$. The latter is
strongly constrained by negative results of the experimental searches, $\xi\ll
1$. In this regime we can safely use the same notations $S$, $h$ and names for
the true mass states in the scalar sector. The scalar effective interaction
with quarks $q$ and gluons $g$ is described by the Lagrangian

${\cal
L}_{qg}=-\xi\,S\sum_{q}\frac{m_{q}}{v}\bar{q}q+\xi\,S\frac{\alpha_{s}\,N_{h}}{12\pi\,v}G_{\mu\nu}^{a}G^{\mu\nu\,a}\,,$
(3)

where $m_{q}$ are quark masses. The first term in (3) is from the Yukawa
couplings of the SM Higgs boson. Then $N_{h}$ heavy quarks, i.e. $m_{q}\gg
M_{S}/2$, induce the second term by quantum corrections, there
$G_{\mu\nu}^{a}$ is gluonic field tensor, $a=1,\dots,8$, and strong coupling
$\alpha_{s}$ (being the QCD analogue of the fine structure constant) is
evaluated at the scale of the order of $M_{S}$. This Lagrangian allows one to
estimate the scalar decay rates to quarks,

$\Gamma(S\to\bar{q}q)=\xi^{2}\frac{N_{c}}{8\pi}\frac{m_{q}^{2}\,M_{S}}{v^{2}}\left(1-\frac{4\,m_{q}^{2}}{M_{S}^{2}}\right)^{3/2}$
(4)

(where $N_{c}=3$ is the number of quark color states) and to gluons,

$\Gamma(S\to
gg)=\xi^{2}\frac{N_{c}^{2}-1}{8}\frac{N_{h}^{2}\alpha_{s}^{2}}{32\pi^{3}}\frac{M_{S}^{3}}{v^{2}}$
(5)

(where $N_{c}^{2}-1=8$ is the number of gluon states). Numerically, quark
modes dominate over the gluon mode for heavy scalars. However, at GeV scale
only $u$, $d$ and $s$ quarks are relevant, $N_{h}=3$, and gluons become
important.

Light scalars decay directly into meson pairs, and these hadronic decay rates
can be described directly via effective interaction between the scalar and
light mesons. It can be obtained by making use of the renorminvariance of the
hadronic contribution to the trace of the energy-momentum tensor Vainshtein
_et al._ (1980),

$T^{\mu}_{\mu}\equiv\sum_{q=u,d,s}m_{q}\bar{q}q-\frac{9\alpha_{s}}{8\pi}G_{\mu\nu}^{a}G^{\mu\nu\,a}\,,$
(6)

which we present to the leading order in $\alpha_{s}$. The last term in (6)
comes from violation of the scale invariance by the trace anomaly due to the
running of $\alpha_{s}$ with energy. It is generated by one-loop triangle
diagrams with virtual $u,d,s$ quarks, and for heavy quarks the contributions
of two terms cancel. This relation allows one to recast the light scalar
interaction (3) in terms of quarks and $T^{\mu}_{\mu}$ as

${\cal
L}_{T}=-\xi\,\frac{S}{v}\left(\left(1-\frac{2N_{h}}{27}\right)\\!\\!\\!\sum_{q=u,d,s}m_{q}\bar{q}q+\frac{2\,N_{h}}{27}T^{\mu}_{\mu}\right).$
(7)

Therefore, to calculate the scalar decay rates to, say, pions, one must
evaluate the matrix elements ($a,b=1,2,3$)

$\displaystyle\langle\pi^{a}(p)\pi^{b}(p^{\prime})|m_{u}\bar{u}u+m_{d}\bar{d}d|0\rangle$
$\displaystyle\equiv\delta^{ab}\Gamma_{\pi}(s)\,,$ (8)
$\displaystyle\langle\pi^{a}(p)\pi^{b}(p^{\prime})|m_{s}\bar{s}s|0\rangle$
$\displaystyle\equiv\delta^{ab}\Delta_{\pi}(s)\,,$ (9)
$\displaystyle\langle\pi^{a}(p)\pi^{b}(p^{\prime})|T^{\mu}_{\mu}|0\rangle$
$\displaystyle\equiv\delta^{ab}T_{\pi}(s)$ (10)

and the similar elements for decays into kaons, $\eta$-mesons, etc.

The form factors entering eqs. (8)-(10) depend on the invariant mass of pion
states, $s=(p+p^{\prime})^{2}$. They can be calculated within the Chiral
Perturbation Theory (ChPT). The leading order terms read

$\displaystyle\Gamma_{\pi}(s)$ $\displaystyle=m_{\pi}^{2}\,,$ (11)
$\displaystyle\Delta_{\pi}(s)$ $\displaystyle=0\,,$ (12) $\displaystyle
T_{\pi}(s)$ $\displaystyle=s+2m_{\pi}^{2}\,,$ (13)

where $m_{\pi}$ stands for the pion mass. Hence the amplitude of the scalar
decay into pions is proportional to Voloshin (1986)

$\displaystyle G_{\pi}(s=M_{S}^{2})$ $\displaystyle\equiv
2\,T_{\pi}(s)+7\,\Gamma_{\pi}(s)+7\,\Delta_{\pi}(s)$ (14)
$\displaystyle=11\,m_{\pi}^{2}+2M_{S}^{2}\,.$ (15)

However, adopting these formulas for evaluation of the scalar decay rates was
argued to be unreliable Donoghue  _et al._ (1990) due to the strong
interaction of pions in the final states. The arguments were based on the
usage of dispersion relations and extracted from $\pi\pi\to\pi\pi$ data
$S$-matrix elements. The corresponding corrections strongly depend on $s$ and
change the leading-order estimate of the hadronic decay rate by a factor upto
a hundred Winkler (2019). Later the usage of dispersion relations has been
questioned in literature, e.g. Clarke  _et al._ (2014); Bezrukov  _et al._
(2018), but no credible alternative estimate of the hadronic decay rates were
presented.

2. In this letter we make use of the $q=u,d$ quark contributions to the gravitational Teryaev (2016) form factors of a pion, considered Kumano  _et al._ (2018) in the timelike domain. They are defined via quark energy-momentum tensor as

$\begin{split}&\langle\pi^{a}(p)\pi^{b}(p^{\prime})|T^{\mu\nu}_{q}(0)|0\rangle\\\
&\equiv\frac{\delta^{ab}}{2}\left(\left(s\,\eta^{\mu\nu}-P^{\mu}P^{\nu}\right)\Theta_{1,q}(s)+\Delta^{\mu}\Delta^{\nu}\Theta_{2,q}(s)\right),\end{split}$
(16)

where $P\equiv p+p^{\prime}$ and $\Delta\equiv p^{\prime}-p$. Convolution of
(16) with metric $\eta_{\mu\nu}$ and summation over quarks,
$\Theta_{i}\equiv\Theta_{i,u}+\Theta_{i,d}$, gives for the quark form factor
(8)

$\Gamma_{\pi}(s)=s\left(\frac{3}{2}\Theta_{1}(s)-\frac{1}{2}\Theta_{2}(s)\right)+2m^{2}_{\pi}\Theta_{2}(s)\,.$
(17)

The form factors $\Theta_{1(2),q}(s)$ have been inferred by fitting the
experimental data from Belle on $\gamma^{*}\gamma\to\pi^{0}\pi^{0}$ scattering
within the technique of Generalized Distribution Amplitudes Diehl  _et al._
(1998); Diehl (2003). The fitting formulas read (we use the original notations
from Kumano  _et al._ (2018) correcting the obvious typo: extra
$\beta^{2}$-factor in the resonance term in $\tilde{B}_{20}$):

$\displaystyle\Theta_{1,q}(s)$
$\displaystyle=-\frac{3}{5}\tilde{B}_{10}(s)+\frac{3}{10}\tilde{B}_{20}(s)$
(18) $\displaystyle\Theta_{2,q}(s)$
$\displaystyle=\frac{9}{10\beta^{2}}\tilde{B}_{20}(s)$ (19)

where $\beta^{2}=\beta^{2}(s)\equiv 1-4m_{\pi}^{2}/s$ and

$\displaystyle\tilde{B}_{10}(s)=$
$\displaystyle-\frac{10}{9}\left[\left(1+\frac{2m_{\pi}^{2}}{s}\right)M_{2(q)}^{\pi}F_{q}^{\pi}(s)\right.$
$\displaystyle\left.+\frac{3g_{f_{0}\pi\pi}\bar{f}_{f_{0}}}{2\sqrt{2}\sqrt{(M_{f_{0}}^{2}-s)^{2}+\Gamma_{f_{0}}^{2}M_{f_{0}}^{2}}}\right]\mathrm{e}^{i\delta_{0}(\sqrt{s})},$
$\displaystyle\tilde{B}_{20}(s)$
$\displaystyle=\frac{10}{9}\beta^{2}\left[M_{2(q)}^{\pi}F_{q}^{\pi}(s)\right.$
$\displaystyle+\left.\frac{g_{f_{2}\pi\pi}f_{f_{2}}M^{2}_{f_{2}}}{\sqrt{2}\sqrt{(M_{f_{2}}^{2}-s)^{2}+\Gamma_{f_{2}}^{2}M_{f_{2}}^{2}}}\right]\mathrm{e}^{i\delta_{2}(\sqrt{s})}$

with $F_{q}^{\pi}(s)=(1+\beta^{2}s/\Lambda^{2})^{-1}$ and the relative
contribution of quarks to the total pion momentum
$M^{\pi}_{2(u)}+M_{2(d)}^{\pi}=0.5$. The phase shifts for $S$ and $D$-waves
were taken from numerical fit of Ref. Bydžovský  _et al._ (2016), and the
first one was corrected above the kaon threshold as
$\delta_{0}(\sqrt{s})\to\delta_{0}(\sqrt{s})+a_{\delta}\left(\sqrt{s}-2m_{K}\right)^{b_{\delta}}\,.$
The values (at the scattering energy scale) of the resonance parameters used
in the fit are presented in Tab. 1.

Meson ($h$) |  $M_{h}$ (GeV) |  $\Gamma_{h}$ (GeV) | $g_{h\pi\pi}$ |  $f_{h}$ (GeV)
---|---|---|---|---
$f_{0}$(500) | 0.475 | 0.550 | 2.959 GeV | –
$f_{2}$(1270) | 1.275 | 0.185 | 1.953 GeV-1 | 0.0754
Table 1: Parameters of the hadronic resonances entering the fitting formulas
for $\tilde{B}_{i0}$.

These numbers coincide with those in Tab.I of Ref. Kumano  _et al._ (2018)
after correcting the typo in the value of $g_{f_{2}\pi\pi}$ which was
presented there being smaller by a factor of 12.44 (however not affecting the
code and final results).

Numerical fit to the Belle data on $\gamma^{*}\gamma\to\pi^{0}\pi^{0}$
revealed the values of the fitting parameters summarized in Tab. 2.

Parameter | set 1 | set 2
---|---|---
$\alpha$ | $0.801\pm 0.042$ | $1.157\pm 0.132$
$\Lambda$ (GeV) | $1.602\pm 0.109$ | $1.928\pm 0.213$
$\bar{f}_{f_{0}(500)}$ (GeV) | 0 (fixed) | $0.0184\pm 0.0034$
$a_{\delta}$ | $3.878\pm 0.165$ | $3.800\pm 0.170$
$b_{\delta}$ | $0.382\pm 0.040$ | $0.407\pm 0.041$
$\chi^{2}/$d.o.f. | 1.22 | 1.09
Table 2: Sets of fitting parameters adopted from Ref. Kumano  _et al._ (2018).

With formulas and parameters above we accurately restore the real and
imaginary parts of the form factors $\Theta_{i,q}$ presented in Fig. 19 of
Kumano  _et al._ (2018). There are two sets with similar accuracy of the
fitting, which provides an estimate of the uncertainty in the gravitational
form factors, and hence in the scalar decay rate to pions, associated with the
exploited methods and experimental data. Numerical results for real, imaginary
parts and the absolute value of $\Gamma_{\pi}$ for $\Theta_{i,q}$ determined
by these fits are shown in Fig. 1.

Figure 1: Real, imaginary parts and the absolute value of $\Gamma_{\pi}$ for
the fitting sets 1 and 2 of Tab. 2.

Remarkably, both fits at $s\to 0$ approach the leading order prediction of
ChPT (11). With $x\equiv M_{S}^{2}/1$ GeV2 we obtain a numerical approximation
for the average between set 1 and set 2:

$|\Gamma_{\pi}|=(0.13+0.39x-0.062x^{2}+0.0041x^{3})\text{ GeV}^{2}.$ (20)

3. The estimated form factor contributes to the total scalar decay rate to pions (the rate to neutral pions is half of the rate to charged ones) as

$\displaystyle\Gamma(S\to\pi\pi)=\frac{3}{32\pi}\frac{49\xi^{2}\,|\Gamma_{\pi}(M_{S}^{2})|^{2}}{81\,v^{2}M_{S}}\,\beta(M_{S}^{2})$
(21) $\displaystyle=(0.983+6.54x-1.12x^{2}+0.071x^{3})\cdot 10^{-8}\text{
GeV}\,,$

while the leading order ChPT result is obtained from (21) by replacing
$\Gamma_{\pi}\to G_{\pi}$, see (14). Both results are outlined in Fig. 2

Figure 2: The decay width of hidden scalar to pions calculated using
$\Gamma_{\pi}$ from Fig. 1 for the fitting sets 1 and 2 (light and dark blue
lines) and ChPT (green line) divided by $\xi^{2}$. Light green line shows the
decay width to pions divided by $\xi^{2}$ obtained using the NLO ChPT results
for $\Gamma_{\pi}$ and $T_{\pi}$ Gasser and Leutwyler (1984); Kubis and
Meissner (2000). The result of LO (NLO Djouadi  _et al._ (1991)) QCD
calculation for the decay width of the hidden scalar to gluons is shown in red
(dark orange), the renormalization scale is $\sqrt{s}\equiv M_{S}$.

along with the leading-order QCD calculations of the decay rate into gluons
(5) and the next-to-leading order ones calculated for the light Higgs boson at
the renormalization scale $\mu=M_{S}$ Djouadi  _et al._ (1991). There is also
shown the decay width obtained using the NLO ChPT result for $\Gamma_{\pi}$
and $T_{\pi}$ Gasser and Leutwyler (1984); Kubis and Meissner (2000). We
observe, that with our estimate of $\Gamma_{\pi}$ the decay rate to pions
reasonably matches that into gluons at $M_{S}\simeq$ 1.5-2 GeV (light quark
contribution (4) is negligible), and in this mass range the total ChPT leading
order estimate reveals similar result. One may therefore speak on ”gluon-
hadron” (or in some sense ”quark-gluon”, as the pions are obviously quark
states) duality. While the smaller fitting curve seems to be preferable, the
deviations by no means may substantially exceed a factor of two, still
consisting with uncertainties we expect in the fitting and calculations. One
should bear in mind that the ChPT expansion is valid only up to dark scalar
masses of order 1 GeV, although we extend the green line in Fig. 2 to higher
$s$ in order to demonstrate the overlapping of the results.

Note that the scalar form factor behaviour within dispersion approach and ChPT
were systematically compared in Sec. 2 of Colangelo (2002). While both NNLO
contribution of ChPT and the dispersion relation provide a decrease of
$\Gamma_{\pi}(s)$ at $s\sim(0.5\,\textrm{GeV})^{2}$, there is a quantitative
discrepancy due to an underestimate of phase in ChPT. Moreover, at $s\sim
1\,\textrm{GeV}^{2}$ the phase evaluation in the Omnes approach should be
strongly violated by the contributions of inelastic channels, in particular,
$KK$, making their account rather important. Indeed $\Gamma_{\pi}$ exhibits
much lower peak within 2-channels analysis Bezrukov  _et al._ (2018).

Contrary to estimates of Donoghue  _et al._ (1990), our result for the decay
rate as a function of scalar mass does not exhibit any peak-like structures,
which might be attributed to the impacts of light scalar hadronic resonances.
We find that impacts of some, in particular $f_{0}(500)$ and $f_{2}(1270)$,
are small, while some, e.g. $f_{0}(980)$, are not seen in the fit Kumano  _et
al._ (2018) and hence do not interfere in (8) being most probably bound states
of four quarks. Note in passing that operator $m\bar{q}q$ contributes also to
$T_{\pi}$. Its account implies the replacement $49/81\to 1$ in eq. (21).

4. The calculated decay rate of the hidden scalar into pions can be used along the decay rates into photons, leptons, etc, see e.g. Bezrukov and Gorbunov (2010), to evaluate the light scalar lifetime and entire pattern of branching ratios of the light scalar decays into the SM particles, see Fig. 3.

Figure 3: Branching ratios of hidden scalar to leptons, photon pairs, pions
(using the result of this work), kaons (within the framework of ChPT),
$s\bar{s}$ pairs and gluons (NLO QCD calculation). The branchings are shown
with solid lines. The dashed line shows hidden scalar lifetime multiplied by
$\xi^{2}$.

Here for $\Gamma_{\pi}(s)$ we use (20). In the mass region $M_{S}\sim 1$ GeV
we expect uncertainties by a factor of 2-3 due to uncertainties in the
gravitational form factor we used and due to our disregard of the gluonic
contribution to the latter. It might be partially accounted along the quark
contribution in the analysis of Ref. Kumano  _et al._ (2018), which deserves
an elaboration. We do not expect any induced by gluons features in the scalar
decay rate to hadrons provided no light scalar resonances consisted of gluons.
We use the corrected pattern in Fig. 3 to refine the experimental reach in the
model parameter space as presented in Fig. 4.

Figure 4: Expected reach of existing and proposed experiments in the model
parameter space updated in accordance with the result for decay rate to pions.
Peaks of original curves from Batell  _et al._ (2022) corresponding to the
resonance enhancement of $\Gamma(S\to\pi\pi)$ have been smoothed accordingly.

To further improve the accuracy in calculation of the scalar decay rates to
hadrons, it is worth to infer other hadronic gravitational form factors,
possibly from analyses of $\gamma^{*}\gamma\to KK$,
$\gamma^{*}\gamma\to\eta\eta$ scatterings or $J/\psi\to\gamma+\pi\pi$, etc,
decays collected at $c$\- and $b$-factories.

###### Acknowledgements.

We thank I. Timiryasov for stimulating discussions. OT is indebted to S.
Kumano and Qin-Tao Song for helpful correspondence. The work is partially
supported by the Russian Science Foundation RSF grant 21-12-00379. The work of
EK is supported by the grant of “BASIS” Foundation no. 21-2-10-37-1.

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# A Contract Theory based Incentive Mechanism for Federated Learning

Mengmeng Tian1 Yuxin Chen1 Yuan Liu1111Corresponding Author Zehui Xiong2 Cyril
Leung3 Chunyan Miao34
1Software College, Northeastern University, China
2Pillar of Information Systems Technology and Design, Singapore University of
Technology Design, Singapore.
3Joint NTU-UBC Research Centre of Excellence in Active Living for the Elderly
(LILY), Nanyang Technological University, Singapore
4School of Computer Science and Engineering, Nanyang Technological University,
Singapore
{1901270,2001236}<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Federated learning (FL) serves as a data privacy-preserved machine learning
paradigm, and realizes the collaborative model trained by distributed clients.
To accomplish an FL task, the task publisher needs to pay financial incentives
to the FL server and FL server offloads the task to the contributing FL
clients. It is challenging to design proper incentives for the FL clients due
to the fact that the task is privately trained by the clients. This paper aims
to propose a contract theory based FL task training model towards minimizing
incentive budget subject to clients being individually rational (IR) and
incentive compatible (IC) in each FL training round. We design a two-
dimensional contract model by formally defining two private types of clients,
namely data quality and computation effort. To effectively aggregate the
trained models, a contract-based aggregator is proposed. We analyze the
feasible and optimal contract solutions to the proposed contract model.
Experimental results show that the generalization accuracy of the FL tasks can
be improved by the proposed incentive mechanism where contract-based
aggregation is applied.

##  1 Introduction

With the ubiquitous adoption of Internet connected smart devices and
applications, the volumes of private data is growing in an unprecedented
speed. In a traditional data driven machine learning paradigm, such large
volumes of data are stored and analyzed on a third-party cloud server
benefiting from its advantages of computing and storage capacities. However,
with the data privacy issue ever-rising in both academics and industry, this
centralized paradigm becomes unpractical. In this context, federated learning
(FL) was proposed in McMahan et al. (2017); Bonawitz et al. (2017) and it has
emerged as a potential solution in order to address this privacy issue, where
the private data is stored and used to train a model at end-devices locally.

In a classical FL framework, an FL server posts a target model referred to as
an FL task to be collaboratively trained by distributed FL clients. To attract
FL clients actively participating in training the FL task and compensate their
efforts in executing the task, the FL server is necessary to offer sufficient
economic incentives for FL clients Sarikaya and Erçetin (2019). There are many
studies that investigate the design of incentive mechanism for federal
learning Hu and Gong (2020); Feng et al. (2019); Khan et al. (2020), such as
contract theory based mechanisms Kang et al. (2019a, b). In these existing
contract based solutions for federated learning, the data quality towards
improving model generalization accuracy Jin et al. (2020) is rarely discussed,
which is an essential performance metric of a deep learning model. What’s
more, all these contract models only studies adverse selection issue where the
FL server offers tasks and contracts to be chosen by clients according to
their types. The clients may not always put their efforts in executing FL
tasks resulting in moral hazard issue. In this study, we aim to design a
multi-dimensional contract considering clients’ data quality in model
generation performance and effort willingness. The main contributions of this
study are summarized as follows.

Figure 1: Overview of the proposed contract theory based federated learning
procedure

  * •

A contract theory based federated learning procedure is proposed, including 8
main processes, to support contract base incentive mechanisms for FL
platforms.

  * •

A multi-dimensional contract model is designed by formally constructing the
two private types of FL clients, i.e., generation type and effort willingness.
The utilities of clients and FL server are formalized to solve an optimal
contract solution.

  * •

A contract based aggregation scheme is designed to improve the model
generalization accuracy. Experimental results based on MNIST and CIFAR-10
dataset show that the proposed contract based incentives and aggregation
scheme outperforms other schemes in a single FL training round.

##  2 System Model

We consider a classical federated learning platform where an FL task is
proposed by a task requester and delegated to a trusted FL server, and the FL
server coordinates the FL task distributedly trained by a set of FL clients.
The FL clients participate in the FL task training under incomplete
information where FL clients privately train the task model according to their
private type and FL server cannot observe the clients’ behaviors or private
types but be aware of the private type distribution. To effectively
incentivize FL clients to execute tasks, we propose a contract theory based
federated learning procedure, as shown in Figure 1. It includes eight main
processes. Specifically, a task requester posts a model task to FL server, and
the server calculates a set of contracts for the task. The server then
publishes the task among the client network and clients can choose to sign a
contract through registering to the task according to the chosen contract. The
clients then take efforts in training the task model based on their private
datasets. Upon a qualified local model is trained, a client can submit the
model to the FL server and the server pays clients according to the
corresponding contracts. The server then aggregates the submitted model
according to a contract based aggregation scheme. In the following, we first
formulate the proposed contract, and then specify the utility functions of
clients and server, and finally introduce the design of contract based
aggregation.

###  2.1 Contract based Federated Learning

In the proposed system, FL clients have private local data and the FL server
cannot predict clients’ behavior. We aim to design a contract mechanism to
elicit such private information.

Suppose there are $I$ types of clients whose data coverage quality types are
sorted in an ascending order:
$\theta_{1}\leq\ldots\leq\theta_{i}\leq\ldots\leq\theta_{I}$. For each FL
iteration round, the FL server needs to propose a contract set
${\Phi}=\\{\phi_{i}=(f_{i},R_{i}(f_{i}))|i\in\\{1,\ldots,I\\}\\}$ to specify
the relationship between clients’ rewards and registration fee for each client
type, where $f_{i}$ is the registration fee for clients in $i-$th type to
accept a task and $R_{i}(f_{i})$ is the corresponding rewards. The design of
$f_{i}$ aims to ensure the participation of rational clients before accepting
a contact and the clients will not participate if they are unable to execute
the task. Then the server broadcasts the contract set among the clients, and
each client signs a contract according to its type. The clients then start
training the model based on their local data and finally submit the trained
model within the time requirement. The server makes a test about the
generalization accuracy of submitted models. The clients are rewarded with
$R_{i}(f_{i})$ if the model passes the test by reaching the corresponding
generalization accuracy $M_{i}$, and nothing otherwise. For the clients not
rewarded, their registration fee will not be returned and used as the clients’
penalty for breach of contract.

It is worth noting that the model aggregation of the server process can be
iterated for $n$ rounds, and the contract set should be set for each round.
For a consecutively following round of a task, the generalization accuracy
threshold $M_{i}$ should be set with a marginal increase. In this paper,
without loss of generality, we design the contracts for a single aggregation
round.

Next, we formalize the types of clients and their utility.

###  2.2 Data Coverage Quality with Adverse Selection

In the context of federated learning, to protect the data privacy of FL
clients, the clients are required to provide the trained models to the server
instead of directly sharing their local data. Because of the asymmetric
information, the quality of model uploaded by clients can not be verified,
which raises the adverse selection issue Akerlof (1970). We then model the
client private type based on the local data quality in the aspect of model
generalization capacity Jin et al. (2020).

Suppose the feature space with $d$ dimension is denoted by
$D=\left[0,1\right]^{d}$ which is a unit space and a subspace $\mathcal{A}\in
D$. Let $\mu(\mathcal{A})$ be the probability that a random sample in $D$
covered by $\mathcal{A}$ and $\mu(\mathcal{A})=1$ when $\mathcal{A}=D$.

######  Definition 1 ( $\epsilon$-Data Coverage).

With a certain radius $\epsilon$, the coverage of a data set $\mathcal{A}$
consisting of samples $x_{1},...x_{m}$ is measured by

$\mu(\mathcal{A},\epsilon)=D\cap\cup_{x_{i}\in\mathcal{A}}B(x_{i},\epsilon)$
(1)

where $B(x_{i},\epsilon)$ is a open ball space centered at $x_{i}$ with radius
$\epsilon$.

Suppose the data space is a Euclidean space, then the range of $\epsilon$ is
$[0,\sqrt{d}]$.

######  Definition 2 (Data Coverage Quality).

The data coverage quality of a local dataset $\mathcal{A}$ is denoted by
$\theta(\mathcal{A})$ to be measured as the expected coverage expected
coverage

$\theta(\mathcal{A})=\frac{1}{\sqrt{d}}\int_{0}^{\sqrt{d}}\mu(\mathcal{A},\epsilon){\rm
d}\epsilon$ (2)

We consider a concrete set of data coverage quality denoted by
$\Theta=\\{\theta_{1},...\theta_{I}\\}$ with $I$ types, where the clients with
$\theta(\mathcal{A})\in[\frac{i-1}{I},\frac{i}{I}]$ belong to type $i$.

###  2.3 Training Willingness with Moral Hazard

The client node consumes its local resource to complete an FL task and the
efforts in training is a piece of private information, bearing moral hazard
issue Holmstrom (1979). We model the second type of a client as training
willingness characterized by its efforts in training an FL task.

######  Definition 3 (Training Willingness).

The training willingness of a client, denoted by $e\in[0,1]$, is the extent of
the client taking their efforts in a task training.

The training cost of a client is especially determined by its training
willingness, which is measured by convex functionYe et al. (2020); Yu et al.
(2016) as

$\Phi(e)=\frac{c}{2}e^{2}$ (3)

where $c$ represents the unit cost in a given task training environment, such
as IoT devices, smart mobile, PC, or server. Without loss of generality, we
denote the training willingness of clients in $i$-th quality type as $e_{i}$.

###  2.4 Utility of Client

With a two-dimensional private information $\theta_{i}$ and $e_{i}$, the cost
of a client in the $i$-th type is

$C(\theta_{i},e_{i})=f_{i}+\frac{c}{2}e_{i}^{2}$ (4)

The client is rewarded with $R_{i}(f_{i})$ if the trained model pass the
generalization test benchmark $M_{i}$. The probability of passing the test is
determined by the data coverage quality and training willingness Zhang et al.
(2017). Therefore, the utility of the client is

$U_{i}=\theta_{i}e_{i}R_{i}-f_{i}-\frac{c}{2}e_{i}^{2}.$ (5)

###  2.5 Utility of Server

The local model uploaded by the client of type $i$ will generate a revenue for
the FL server, denoted by $G(M_{i})$ satisfying $G^{\prime}(M)>0$ and
$G^{\prime\prime}(M)>0$. Therefore, the utility of the server from enrolling
client in type $i$ is

$U_{s}^{i}=f_{i}+\theta_{i}e_{i}(G(M_{i})-R_{i})$ (6)

Given the type distribution of clients $\\{\beta_{i}\\}$ with
$i\in\\{1,...I\\}$ and $\sum_{i=1}^{I}\beta_{i}=1$, the expected utility of
the server is

$U_{s}=\sum_{i=1}^{I}\beta_{i}U_{s}^{i}=\sum_{i=1}^{I}\beta_{i}(f_{i}+\theta_{i}e_{i}(G(M_{i})-R_{i}))$
(7)

###  2.6 Contract Optimization Problem

The contract optimization problem is formalized as

$\begin{split}\max&\sum_{i=1}^{I}\beta_{i}(f_{i}+\theta_{i}e_{i}(G(M_{i})-r_{i}))\\\
s.t.&\\\ (IR)&\theta_{i}e_{i}R_{i}-f_{i}-\frac{c}{2}e_{i}^{2}\geq 0\\\
(IC)&\theta_{i}e_{i}R_{i}-f_{i}-\frac{c}{2}e_{i}^{2}\geq\theta_{i}e_{i}^{j}R_{j}-f_{j}-\frac{c}{2}(e_{i}^{j})^{2}\\\
&\forall j\neq i,\ i,j\in\\{1,\dots,n\\}\end{split}$ (8)

where $e^{j}_{i}$ denotes the effort of type $\theta_{i}$ when selecting
contract ($f_{j},R_{j}$).

The first constraint ensures that each client can achieve non-negative
utility, which is also regarded as individual rationality property (IR). The
second constraint aims to ensure that each client can achieve their maximal
utility by choosing the contract corresponding to their truthful type, which
is regarded as incentive compatibility property (IC).

###  2.7 Contract-Based Model Aggregation

With a set of submitted models, the server should aggregate the models based
on their chosen contracts for the sake of better model generalization
performance. Suppose the total rewards paid by a server in a round is
$R_{total}$, then the weight assigned for a model trained by a client in type
$i$ is calculated according to Eq.(9).

$w_{i}=\frac{R_{i}}{R_{total}}$ (9)

##  3 Optimal Contract Solution

In this section, we solve the optimal contract solution to the problem defined
in Eq.(8). We first solve the optimal effort willingness made by clients and
then calculate the contract solution by maximizing the server utility.

Given the utility of a client in $i$-th type in Eq.(5), we compute the first
order derivative with respect to its effort willingness and we obtain

$\frac{\partial U_{i}}{\partial e_{i}}=\theta_{i}R_{i}-ce_{i}$ (10)

A rational client node should always maximize its utility by making the
optimal willingness which is denoted by $\hat{e}_{i}$ and $\hat{e}_{i}^{j}$ in
choosing contract $\phi_{i}=(f_{i},R_{i}(f_{i}))$ and contract
$\phi_{j}=(f_{j},R_{j}(f_{j}))$. Thus,

$\hat{e}_{i}=\frac{1}{c}\theta_{i}R_{i}\quad,\quad\hat{e}_{i}^{j}=\frac{1}{c}\theta_{i}R_{j}$
(11)

According to Eq.(11), we can know that a client’s willingness is positively
determined by the data quality and the chosen contract reward.

Bring Eq.(11) into Eq.(8), the objective function is updated as follows.

$\begin{split}\max&\sum_{i=1}^{I}\beta_{i}(f_{i}+\frac{1}{c}\theta_{i}^{2}R_{i}(G(M_{i})-R_{i}))\\\
s.t.&\\\ (IR)\ &\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq 0,\\\ (IC)\
&\frac{1}{2c}(\theta_{i}F_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{j})^{2}-f_{j},\\\
&\forall j\neq i,\ i,j\in\\{1,\dots,n\\}\end{split}$ (12)

Next, we will solve the optimal contract solution for each type $\theta_{i}$.
Some important conditions will be derived.

######  Lemma 1 (Monotonicity between $\theta$ and $R$).

For any feasible contract ($f_{i},R_{i}$), $R_{i}\geq
R_{j}\Leftrightarrow\theta_{i}\geq\theta_{j}$.

######  Proof of Lemma 1.

For clients of type $\theta_{i}$ and $\theta_{j}$, the following two IC
constrains should be satisfied

$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{j})^{2}-f_{j}$
(13)
$\displaystyle\frac{1}{2c}(\theta_{j}R_{j})^{2}-f_{j}\geq\frac{1}{2c}(\theta_{j}R_{i})^{2}-f_{i}$
(14)

By adding the above two inequalities, we have

$\displaystyle(\theta_{i}^{2}-\theta_{j}^{2})(R_{i}^{2}-R_{j}^{2})\geq 0$ (15)
$\displaystyle\Rightarrow$
$\displaystyle(\theta_{i}-\theta_{j})(R_{i}-R_{j})\geq 0$ (16)

for any $\theta_{i},\theta_{j}>0$ and $R_{i},R_{j}>0$.  $\square$

Lemma 1 imply that a client with a higher type $\theta$ fit for a higher
rewards $R$. Thus, the contract rewards should follow the order
$R_{1}<\dots<R_{I}$ with $\theta_{1}<\dots<\theta_{I}$.

######  Lemma 2 (Monotonicity between $R$ and $f$).

For any feasible contract ($f_{i},R_{i}$), $R_{i}\geq R_{j}\Leftrightarrow
f_{i}\geq f_{j}$.

######  Proof of Lemma 2.

The IC constrain holds when a client in type $\theta_{i}$ chooses contract
($f_{i},R_{i}$) compared with ($f_{j},R_{j}$).

$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{j})^{2}-f_{j}$
$\displaystyle\Rightarrow\left\\{\begin{array}[]{rl}f_{i}-f_{j}&\leq\frac{\theta_{i}^{2}}{2c}(R_{i}^{2}-R_{j}^{2})\\\
f_{j}-f_{i}&\geq\frac{\theta_{j}^{2}}{2c}(R_{j}^{2}-R_{i}^{2})\end{array}\right.$
(19)

According to Eq.(19), if $f_{i}\geq f_{j}$, then we have $R_{i}\geq R_{j}$,
and vice verse.  $\square$

Lemma 2 shows that $R$ and $f$ have the same trend, namely $R_{1}<\dots<R_{I}$
with $f_{1}<\dots<f_{I}$.

######  Corollary 1 (Monotonicity between $f$ and $\theta$).

For any feasible contract ($f_{i},R_{i}$), $f_{i}\geq
f_{j}\Leftrightarrow\theta_{i}\geq\theta_{j}$.

######  Proof of Corollary 1.

According to Lemma 1 and Lemma 2, both $\theta$ and $f$ monotonically increase
with $R$. Thus, we can derive that the positive correlation between $\theta$
and $f$.  $\square$

The above lemmas and corollary shows the monotonicity properties. Next, we
will find the optimal contract by reducing the IR constrains and IC
constrains.

######  Theorem 1 (IR transitivity).

All the IR constrains can be satisfied if the constrain of $\theta_{1}$ is
satisfied.

######  Proof of Theorem 1.

For any client in type $i\in\\{1,\dots,I\\}$ and $i\geq 1$, we have

$\begin{split}U_{i}=\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{1})^{2}-f_{1}\\\
\geq\frac{1}{2c}(\theta_{1}R_{1})^{2}-f_{1}=U_{1}\end{split}$ (20)

and its utility is monotonous.  $\square$

######  Theorem 2 (Tight IC Constrain).

The following IC constrain is sufficient for client in type $\theta_{i}$ to
achieve its maximal utility.

$\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}=\frac{1}{2c}(\theta_{i}R_{i-1})^{2}-f_{i-1}$
(21)

where $i\in\\{2,...,I\\}$.

######  Proof of Theorem 2.

The following proof is organized as three parts. First of all, we reduce the
redundant IC constrains in two direction: client in type $\theta_{i}$ select
the contract $\phi_{i+1}$ and $\phi_{i-1}$ respectively. And then, All
redundant constraints will be eliminated, leaving only tight constraints (21).

1) Downward Selection:

$\displaystyle\frac{1}{2c}(\theta_{i+1}R_{i+1})^{2}-f_{i+1}\geq\frac{1}{2c}(\theta_{i+1}R_{i})^{2}-f_{i}$
(22)
$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{i-1})^{2}-f_{i-1}$
(23)

Transfer (22) and (23),we have

$\displaystyle f_{i+1}-f_{i}$
$\displaystyle\leq\frac{1}{2c}\theta_{i+1}^{2}(R_{i+1}^{2}-R_{i}^{2})$ (24)
$\displaystyle f_{i}-f_{i-1}$
$\displaystyle\leq\frac{1}{2c}\theta_{i}^{2}(R_{i}^{2}-R_{i-1}^{2})$ (25)

Since $\theta_{i}<\theta_{i+1}$, according to (25) we have

$\displaystyle\frac{1}{2c}\theta_{i}^{2}(R_{i}^{2}-R_{i-1}^{2})$
$\displaystyle\leq\frac{1}{2c}\theta_{i+1}^{2}(R_{i}^{2}-R_{i-1}^{2})$
$\displaystyle\Rightarrow f_{i}-f_{i-1}$
$\displaystyle\leq\frac{1}{2c}\theta_{i+1}^{2}(R_{i}^{2}-R_{i-1}^{2})$ (26)

Add (24) and (26), then we have

$\displaystyle
f_{i+1}-f_{i-1}\leq\frac{1}{2c}\theta_{i+1}^{2}(R_{i+1}^{2}-R_{i-1}^{2})$ (27)
$\displaystyle\Rightarrow\frac{1}{2c}(\theta_{i+1}R_{i+1})^{2}-f_{i+1}\geq\frac{1}{2c}(\theta_{i+1}R_{i-1})^{2}-f_{i-1}$

Then, we can obtain all the downward IC constrains:

$\displaystyle\frac{(\theta_{i+1}R_{i+1})^{2}}{2c}\\!-\\!f_{i+1}\\!$
$\displaystyle\geq\\!\frac{(\theta_{i+1}r_{i-1})^{2}}{2c}\\!-\\!f_{i-1}\\!\geq\\!\cdots$
$\displaystyle\geq\frac{(\theta_{i+1}r_{1})^{2}}{2c}\\!-\\!f_{1}$

Therefore, all the adjacent type of downward IC selection is enough to drive
all the other downward selection.

2) Upward Selection: Similarly, we can drive all the upward selections by
adjacent selection like

$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}\\!-\\!f_{i}\\!\geq\\!\frac{1}{2c}(\theta_{i}R_{i+1})^{2}\\!-\\!f_{i+1}\\!\geq\\!\dots$
$\displaystyle\\!\geq\\!\frac{1}{2c}(\theta_{i}R_{I})^{2}\\!-\\!f_{I}$

Therefore, we can drive other IC constrains from adjacent upward type
selection.

3) Tight IC constrain: From the above two steps, the IC constrain of adjacent
is left. Then, we will further remove the redundant restrictions. The tight IC
constrain can be replaced by adjacent downward IC constrain and monotonicity
show as follows.

######  Proposition 1.

The downward IC constrains can grantee the upward IC constrains.

######  Proof of Proposition 1.

Accordind to the above steps 1) and 2), we have

$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{i-1})^{2}-f_{i-1}$
(28)
$\displaystyle\frac{1}{2c}(\theta_{i}R_{i})^{2}-f_{i}\geq\frac{1}{2c}(\theta_{i}R_{i+1})^{2}-f_{i+1}$
(29)

According to Eq.(29), we have

$\displaystyle
f_{i+1}\geq\frac{1}{2c}\theta_{i}^{2}(R_{i+1}^{2}-R_{i}^{2})+f_{i}$ (30)
$\displaystyle\Rightarrow
f_{i}\geq\frac{1}{2c}\theta_{i-1}^{2}(R_{i}^{2}-R_{i-1}^{2})+f_{i-1}$ (31)

and according to Eq.(28), we have

$f_{i}\leq\frac{1}{2c}\theta_{i}^{2}(R_{i}^{2}-R_{i-1}^{2})+f_{i-1}$ (32)

Comparing Eq.(31) and Eq.(32), given $\theta_{i}>\theta_{i-1}$, we can derive
that the server will achieve its maximal utility when $f_{i}$ is assigned with
its maximal value. So the tight IC constrain is Eq. (21).  $\square$

Taking the tight constrains into the objective function in Eq. (12), we can
formulate the Lagrange function to solve the final optimal contract:

$\begin{split}\mathcal{L}\\!=\\!&\\!\sum_{i=1}^{I}\left\\{\left[\beta_{i}(f_{i}+\frac{1}{c}\theta_{i}^{2}r_{i}(G(M_{i})-R_{i}))\right]+\right.\\\
&\left.\lambda_{i}\left[\frac{1}{2c}(\theta_{i}R_{i})^{2}\\!-\\!\frac{1}{2c}(\theta_{i}R_{i-1})^{2}\\!-\\!f_{i}\\!+\\!f_{i-1}\right]\right\\}\\\
+&\mu\left[\frac{1}{2c}(\theta_{1}R_{1})^{2}-f_{1}\right]\end{split}$ (33)

where $\lambda_{i}$ is Lagrange multiplier of IC constrain for $\theta_{i}$,
and $\mu$ is Lagrange multiplier of IR constrain for $\theta_{1}$.

Thus the optimal value of $R$ is

$R_{i}=G(M_{i}),\forall i\in\\{1,\dots,I\\}$

and the optimal value of $f$ can be driven according the IR transitivity and
IC transitivity.

$\displaystyle f_{1}$ $\displaystyle=\frac{1}{2c}(\theta_{1}R_{1})^{2}$
$\displaystyle f_{i}$
$\displaystyle=\frac{1}{2c}(\theta_{i}R_{i})^{2}-\frac{1}{2c}(\theta_{i}R_{i-1})^{2}+f_{i-1},\forall
i\in\\{2,\dots,I\\}$

The above solution is the optimal contract solution after relaxing the
constrains. We need to further verify whether the solution satisfies
monotonicity condition of $R$. If $R$ is not monotonic, the adjustment
algorithm Gao et al. (2011) can be applied.

Table 1: Contract and Client Settings Of MNIST Task Parameters Type Index | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---|---|---|---
Client Type $\theta_{i}$ | 0.790 | 0.795 | 0.800 | 0.805 | 0.810 | 0.815 | 0.820 | 0.825 | 0.830 | 0.835
Client Data Size | 1000 | 1500 | 2000 | 2500 | 3500 | 5000 | 6500 | 8500 | 12000 | 16000
Optimal effort $e_{i}$ | 0.279 | 0.331 | 0.389 | 0.451 | 0.519 | 0.592 | 0.670 | 0.753 | 0.842 | 0.936
Test Generalization $M_{i}$ | 0.230 | 0.250 | 0.270 | 0.290 | 0.310 | 0.330 | 0.350 | 0.370 | 0.390 | 0.410

##  4 Experiment Results and Analysis

In this section, we evaluate the proposed contract based incentive mechanism
for FL in two classical datasets, i.e. MNIST and CIFAR-10. The proposed model
is compared with other two schemes to demonstrate the effectiveness of our
model in the aspect of generalization accuracy.

####  4.0.1 Experimental Settings

There are 10 types of contract and 10 types of clients are set up to
correspond to the corresponding contracts. The types of clients follow a
uniform distribution, i.e., $\lambda_{i}=0.1$. The contract and client
settings are shown in Table 1.

Based on the above settings, we complete the steps from 1 to 6 in Figure 1.
After the client uploading their local model, the server will test whether the
model meets the corresponding test generalization benchmark $M$ for each
client according to the chosen contract. Due to the fact that the data are
heterogeneous and follow non-IID, the data of the server used for testing is a
part of the whole dataset and chosen randomly in the whole data space, serving
as a sampling test for prevention of fraud. At the same time, through
experiments, we found that although our model training accuracy in the local
environment is 93% , it is only 46% in the server test. This fully shows the
test error caused by different data coverage between server and client.
Therefore, if the test baseline $M$ is set too high, the test results of the
model will deviate greatly. In this experiment, the $M$ we set is relatively
low, as shown in Table 1. We require that with the increase of client types,
our benchmark requirements increase by 2%.

The following three schemes are compared by setting different rewarding
methods and aggregation protocols.

  * •

Scheme-1: The clients are rewarded according to the proposed contract based
solution and the server aggregates the submitted model according to the
contracted based aggregation protocol.

  * •

Scheme-2: The clients are rewarded according to the proposed contract based
solution and the server aggregates the submitted model according to FedAvg
protocol Kairouz et al. (2019) in which the model aggregation weight is same
for all clients.

  * •

Scheme-3: The clients are rewarded equally and the server aggregates the
submitted model according to FedAvg protocol.

Notice that we set the same rewarding scheme with average effort and reward of
contract-based rewarding scheme for each client for fairness. In addition, we
also adjust parameter $c$ to show the impact of service environment on model
aggregation performance.

####  4.0.2 Experimental Results

This generalization accuracy of the three schemes under two different
parameter $c$ settings in two datasets is presented in Figure 2. In Figure 2,
under the same parameter $c$, the proposed method, i.e. Scheme-1 shows the
highest model generalization accuracy which is better that of Scheme-2 or
Scheme-3. By using the contract-based incentive mechanism in Scheme-1 and
Scheme-2, the clients work more hard and consistently perform more better than
the fixed incentive in Scheme-1. The reason of better performance of
Scheme-1over Scheme-2 is that Scheme-1 uses contract based aggregation which
can set a higher weight for the model trained on a high quality data source.
In addition,comparing the model accuracy under different parameter $c$, we can
observe that the smaller parameter $c$ setting brings the better
generalization accuracy, indicating that the clients with lower training cost
is more likely to be incentivied to improve the model generalization
performance.

Figure 2: Generalization accuracy of the proposed scheme under different
values of $c$

##  5 Related Work

At present, there are many researches on the design of incentive mechanisms
for federated learning based on game theoretical approaches Wahab et al.
(2021). The existing incentive mechanisms can be mainly divided into two main
categories: Stackelberg game-based and contract theory-based methods. In the
first categories, FL server offer a task associated with a price and clients
choose a task and take efforts in training the task to achieve the pricing
rewards Hu and Gong (2020); Feng et al. (2019); Khan et al. (2020). In Hu and
Gong (2020), a two-stage Stackelberg game is formalized for FL with private
data, a Nash equilibrium (NE) is solved with the optimal privacy budget of
clients and the optimal pricing scheme of server. In Feng et al. (2019), the
interactions between model requester and mobile user are formalized as a
Stackelberg game to analyze the NE composed by the optimal training data price
and data size. In Khan et al. (2020), a Stackelberg game based incentive
mechanism is designed for FL clients in strategically set the local iterations
and FL server maximizing task global accuracy. The shortcoming of the
Stackelberg game based methods is that it can only consider a single dimension
private strategical types.

In the second category, FL servers offer a set of contract items according to
clients types where the contract models can be single-dimensional or multi-
dimensional depending on the dimensions of the considered type. In Kang et al.
(2019b), the client training quality type is formalized based and computation
resources, and an optimal contract solution offers more rewards to clients
with higher type values. Considering clients’ communication delay and local
training cost, Ding et al. (2021) introduces a contract-based incentive
mechanism to maximize server aggregation accuracy and total payments, where
the local training cost is related to the clients’ network environment. Ye et
al. (2020) considers both data quality and model computation resources. A most
recent study Ding et al. (2021) investigates a two-dimensional contract model
considering data quality in terms of data sizes and communication time types
and analyze the optimal contract solutions in three scenarios: complete
information, weakly incomplete information, and strongly incomplete
information. There are two limitations in the existing contract based
methods:1) they neglect to consider the data quality in the aspects of
improving model generalization accuracy, which will be studied as an important
private type of clients; 2) they assume that the clients take their efforts in
executing FL tasks, which bears the designed contracts with the moral hazard
issue where their willingness may hinder the achievement of their optimal
solution.

##  6 Conclusion

In this paper, we propose a contract based incentive mechanism for federated
learning. A two-dimentional contract model is formally designed where we
consider the client’s data coverage quality and effort willingness. We also
propose a contract based FL aggregation protocol. The optimal contract
solution is theoretically analyzed. Finally the proposed incentive mechanism
is experimentally evaluated and the results show that our contract based
scheme achieves higher aggregation accuracy compared with the other two
schemes.

## Acknowledgements

This work was supported by Alibaba Group through Alibaba Innovative Research
(AIR) Program and Alibaba-NTU Singapore Joint Research Institute (JRI),
Nanyang Technological University, Singapore; Key-Area Research and Development
Program of Guangdong Province NO.2020B0101090005; National Natural Science
Foundation of China under Grant No.62032013, and No.U20B2046; 111 Project
(B16009); and the Fundamental Research Funds for the Central Universities
N182410001.

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