#  Non-perturbative determination of the collisional broadening kernel and
medium-induced radiation in QCD plasmas††thanks: Presented at 29TH
INTERNATIONAL CONFERENCE ON ULTRA – RELATIVISTIC NUCLEUS - NUCLEUS COLLISIONS

Sören Schlichting
Ismail Soudi111Speaker. Fakultät für Physik, Universität Bielefeld
D-33615 Bielefeld, Germany.  Department of Physics and Astronomy, Wayne State
University
Detroit, MI 48201.

###### Abstract

Collisional broadening in QCD plasmas leads to the emission of medium induced
radiation, which governs the energy loss of highly energetic particles or
jets. While recent studies have obtained non-perturbative contributions to the
collisional broadening kernel $C(b_{\bot})$ using lattice simulation of the
dimensionally reduced long-distance effective theory of QCD, Electrostatic QCD
(EQCD), so far all phenomenological calculations of jet quenching rely on
perturbative determinations of the collisional broadening kernel. By matching
the short-distance behavior of the lattice extracted EQCD broadening kernel,
we determine the fully matched QCD broadening kernel non-perturbatively. We
present results for the collisional broadening kernel in impact-parameter
($C_{\rm QCD}(b_{\bot})$)) and momentum space ($C_{\rm QCD}(q_{\bot})$) and
employ them to determine the rates of medium induced radiation in infinite and
finite size QCD plasmas. By contrasting our results with leading and next-to-
leading order perturbative determinations as well as various approximations of
the splitting rates employed in the literature, we investigate the effect of
the non-peturbative determination of CQCD($q_{\bot}$) on medium-induced
radiation rates.

##  1 Introduction

During ultra-relativistic heavy-ion collisions, high energetic particles
originating from the initial hard partonic collisions, must traverse and
interact with the Quark-Gluon-Plasma (QGP) before reaching the detector.
Interactions of highly energetic parton with the medium have been studied
extensively, and it was shown that besides the elastic interactions with the
medium constituents, multiple soft scatterings between the hard partons and
the medium trigger the partons to radiate. These soft scatterings lead to an
infinite number of diagrams, which can be resummed to obtain an effective
$1\to 2$ in-medium radiation rates. Several formalisms for in-medium radiation
have been developed in the literature [1, 2, 3], which encapsulate the same
physics but differ on the level of approximations. However, all these
formalisms rely on a description of the elastic scattering with the medium,
obtained using the transverse momentum broadening kernel

$C(q_{\perp})\equiv\frac{(2\pi)^{2}\mathrm{d}^{3}\Gamma}{\mathrm{d}^{2}q_{\perp}\,\mathrm{d}L}\,,$
(1)

which describes the rate at which the hard partons exchange transverse
momentum $\boldsymbol{q}_{\perp}$ with the medium. To obtain the broadening
kernel three treatments of the medium are widely used in the literature :
Many random, static, screened color centers [1]:
$C(q_{\perp})\propto\frac{1}{(q_{\perp}^{2}+m_{\mathrm{D}}^{2})^{2}}$.
Dynamical moving charges at lowest order in perturbation theory [4]:
$C(q_{\perp})\propto\frac{1}{q_{\perp}^{2}(q_{\perp}^{2}+m_{\mathrm{D}}^{2})}$.
Many individually small scatterings, leading to transverse momentum diffusion
[5]: $C(b_{\perp})=\hat{q}\,b_{\perp}^{2}/4$.
Recently, there have been progress in obtaining non-perturbative contributions
to the broadening kernel [6, 7], using a dimensionally reduced long-distance
effective theory for QCD, 3D Electrostatic QCD (EQCD) [8]. After a Fourier
transform of the broadening kernel to impact parameter space, lattice
simulations of EQCD [9, 10, 6] can be performed to obtain the large distance
behavior of the broadening kernel $C_{\mathrm{EQCD}}(b_{\perp})$, which was
matched to the short distance behavior of QCD to obtain a non-perturbative
determination of $C_{\mathrm{QCD}}(b_{\perp})$ in QCD at all scales [7].
During these proceedings, we employ the broadening kernel extracted to obtain
the in-medium splitting rates of a highly energetic parton traversing a medium
with a finite extent and compare with results obtained using perturbative
results of the broadening kernels.

##  2 Non-perturbative broadening kernel

Figure 1: (left) Non-perturbative elastic broadening kernel $C_{\rm
QCD}(b_{\perp})$ in impact parameter space. Data points for two different
temperatures $T=250,500{\rm MeV}$ are shown alongside the interpolating
splines. (right) Elastic broadening kernel $C_{\rm
QCD}(\boldsymbol{q}_{\perp})$ in momentum space for $T=250,500$MeV. We also
compare the kernel to leading-order (LO) and next-to-leading order (NLO)
determinations at $T=500$MeV.

The non-perturbative (NP) broadening kernel $(C_{\mathrm{QCD}}(b_{\perp}))$ in
impact parameter space $b_{\perp}$ was obtained in [7], and shown in the left
panel of Fig. 1. However, in order to compute in-medium splitting rates in a
QCD medium of finite size, it is more preferable to work in momentum space.
Therefore, we perform a Fourier transform to obtain the non-perturbative
broadening kernel $(C_{\mathrm{QCD}}(q_{\perp}))$ in impact parameter space
$q_{\perp}$ which is shown for $T=250,500$MeV in the right panel of Fig. 1
(c.f. [11] for details of the procedure). We find that both data sets at
$T=250,500$MeV display very similar behavior when expressing
$TC_{\mathrm{QCD}}(q_{\perp})/C_{R}$ as a function of
$q_{\perp}/g_{\mathrm{s}}T$ in momentum space, or
$C(b_{\perp})/g_{\mathrm{s}}^{2}TC_{R}$ as a function of
$g_{\mathrm{s}}Tb_{\perp}$ in impact parameter space. When comparing with
leading order (LO) and next-to-leading order (NLO) pertubative broadening
kernels, we find that while they all follow the same ultraviolet (UV) behavior
$\sim 1/q_{\perp}^{4}$ and respectively $\sim b_{\perp}^{2}\log(b_{\perp})$
behavior, the infrared (IR) behavior of the NP and NLO kernels is markedly
different from the LO. The NP and NLO kernels in UV follow a $\sim
1/q_{\perp}^{3}$ and respectively $\sim b_{\perp}$. However, the slopes of the
kernels differ by a prefactor due to the difference in the string tension
$\sigma_{\rm EQCD}$.

##  3 In-medium splitting rates

Figure 2: Splitting rate for the medium-induced emission of a gluon from a
parent quark with energy $P=300T$ in an equilibrium plasma with temperature
$T=500$MeV as a function of the evolution time $t$ for different gluon
momentum fraction $z=0.05,0.5$. (top) Different curves in each panel show the
results for the different, leading order (LO), next-to-leading order (NLO) and
non-perturbative (NP) momentum broadening kernels in Fig. 1. (bottom) We
compare different approximations of the in-medium splitting rate, namely the
opacity expansion at $N=1$ [1], the resummed opacity rate of [12] ($N=X$) and
the NLO expansion around the Harmonic Oscillator [13] (NLO-HO) to the full
result ($T=500{\rm MeV}$). Note that all results are obtained with the non-
perturbative collision kernel. The lower panel of each plot displays the ratio
to the full rate.

Employing the different broadening kernels, and following [3], we obtain the
rate of medium-induced radiation $\frac{d\Gamma(P,z;t)}{dz}$, the rate of
quark with momentum $P$ radiating a gluon with momentum $zP$ after a time $t$
spent in the medium. The results for momenta fraction $z=0.05$ and $z=0.5$ are
shown in Fig. 2 as a function of time $t$ where we used the LO, NLO and NP
broadening kernels to obtain the orange, green and purple curves, and we show
the ratio to the LO curve in the lower inset of each panel. Initially we
observe a linear behavior of the rate which saturates rather quickly at later
times and recovers the infinite medium rates which were obtained earlier [7].
We find that the non-perturbative result starts lower than the LO rates before
it settles above the LO and below the NLO, which can be explained by the fact
that the early times radiation are driven mainly by single hard scatterings
with large momentum exchange $(q_{\perp}\gg m_{D})$ where the NP kernel is
below the LO. At late times, multiple soft scatterings with small momentum
exchange $(q_{\perp}\sim m_{D})$ are more important, and the NP kernel in this
IR behaves similarly to the NLO kernel. Overall, the results obtained using
the NP kernel do not depart from a band of $\pm 50\%$ around LO, while the NLO
results can become over $2\times$ larger than the LO result.

Additionally, we computed the in-medium radiation rates using several
approximations developed in the literature: First order opacity expansion,
which considers a perturbative expansion in the number of re-scatterings with
the medium. Here we take only a single scattering ($N=1$) [1].
Resummed opacity expansion ($N=X$) which re-sums the opacity expansion by
employing a cut-off to the momentum exchange of the scattering, the collision
integral of subsequent scatterings can then be exponentiated (c.f. [12]).
Harmonic Oscillator expansion (NLO-HO) where using an opacity expansion, a
single hard scattering is considered as a perturbation on top of fully
resummed multiple soft-scatterings computed in the diffusion approximation
[13].
Extensive details of calculation of each approximation are given in [11]. We
employ these approximations to obtain the in-medium splitting rate for the
same NP broadening kernel $C_{\rm QCD}(q_{\perp})$ at $T=500$MeV, and show a
comparison in Fig. 2. While the naive first order opacity expansion reproduces
the linear behavior of the rate at early times, it breaks down later on when
the time is large enough to allow for subsequent scatterings. Conversely, the
resummed $(N=X)$ opacity expansion is able to reproduce the rate even at late
times. For the NLO expansion around the HO, we observe that it can be quite
accurate throughout the evolution especially for quasi-democratic splittings
($z\sim 1/2$).

##  4 Conclusion

In these proceedings, we presented our results for in-medium splitting rates
computed using the non-perturbative collision kernel and compared them to the
calculations using leading and next-to-leading order perturbative collision
kernel as well as to different approximations of the in-medium splitting
rates. We find that approximations to the splitting rate calculation are quite
effective in reproducing the rate within their respective range of validity.
Conversely, we observed considerable differences between the results for
different collision kernels. Since the differences between the non-
perturbative and the LO kernel which is usually used in phenomenological
studies of jet quenching, can be on the order of $30\%$ it will be interesting
to further explore the phenomenological consequences. In this context, it is
important to point out, that the collisional broadening kernel and in-medium
splitting rates obtained in this study can be easily integrated into models of
jet quenching using a kinetic approach or MonteCarlo simulations [14].

Acknowledgement: We thank Guy D. Moore and Niels Schlusser for insightful
discussions, collaboration on our previous work [7]. This work is supported in
part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through the CRC-TR 211 ’Strong-interaction matter under extreme conditions’–
project number 315477589 – TRR 211 and the German Bundesministerium für
Bildung und Forschung (BMBF) through Grant No. 05P21PBCAA. The authors also
gratefully acknowledge computing time provided by the Paderborn Center for
Parallel Computing (PC2).

## References

  * [1] M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B 571 (2000), 197-233 doi:10.1016/S0550-3213(99)00713-0 [arXiv:hep-ph/9907461 [hep-ph]].
  * [2] P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 06 (2002), 030 doi:10.1088/1126-6708/2002/06/030 [arXiv:hep-ph/0204343 [hep-ph]].
  * [3] S. Caron-Huot and C. Gale, Phys. Rev. C 82 (2010), 064902 doi:10.1103/PhysRevC.82.064902 [arXiv:1006.2379 [hep-ph]].
  * [4] P. Aurenche, F. Gelis and H. Zaraket, JHEP 05 (2002), 043 doi:10.1088/1126-6708/2002/05/043 [arXiv:hep-ph/0204146 [hep-ph]].
  * [5] P. B. Arnold, Phys. Rev. D 79 (2009), 065025 doi:10.1103/PhysRevD.79.065025 [arXiv:0808.2767 [hep-ph]].
  * [6] G. D. Moore and N. Schlusser, Phys. Rev. D 101 (2020) no.1, 014505 [erratum: Phys. Rev. D 101 (2020) no.5, 059903] doi:10.1103/PhysRevD.101.014505 [arXiv:1911.13127 [hep-lat]].
  * [7] G. D. Moore, S. Schlichting, N. Schlusser and I. Soudi, JHEP 10 (2021), 059 doi:10.1007/JHEP10(2021)059 [arXiv:2105.01679 [hep-ph]].
  * [8] S. Caron-Huot, Phys. Rev. D 79 (2009), 065039 doi:10.1103/PhysRevD.79.065039 [arXiv:0811.1603 [hep-ph]].
  * [9] M. Panero, K. Rummukainen and A. Schäfer, Phys. Rev. Lett. 112 (2014) no.16, 162001 doi:10.1103/PhysRevLett.112.162001 [arXiv:1307.5850 [hep-ph]].
  * [10] M. D’Onofrio, A. Kurkela and G. D. Moore, JHEP 03 (2014), 125 doi:10.1007/JHEP03(2014)125 [arXiv:1401.7951 [hep-lat]].
  * [11] S. Schlichting and I. Soudi, Phys. Rev. D 105 (2022) no.7, 7 doi:10.1103/PhysRevD.105.076002 [arXiv:2111.13731 [hep-ph]].
  * [12] C. Andres, F. Dominguez and M. Gonzalez Martinez, JHEP 03 (2021), 102 doi:10.1007/JHEP03(2021)102 [arXiv:2011.06522 [hep-ph]].
  * [13] Y. Mehtar-Tani, JHEP 07 (2019), 057 doi:10.1007/JHEP07(2019)057 [arXiv:1903.00506 [hep-ph]].
  * [14] R. M. Yazdi, S. Shi, C. Gale and S. Jeon, [arXiv:2206.05855 [hep-ph]].


$\mathscr{H}_{\phi}\geq 0$. Let $u=u(x)$ be any smooth positive bounded
solution to (7.4) such that $u_{\nu}\geq 0$ over $\partial M$. Suppose the
assumptions (7.6) and (7.7) hold, then $u$ must be constant. In particular
$au^{s}\log u+bu^{r}=0$. Furthermore, if $a\neq 0$ and $b=0$, then $u\equiv
1$.

###### Proof.

In consideration of the the assumptions (7.6) and (7.7) above we have

$\displaystyle\Gamma_{+}^{\varepsilon}[u]$
$\displaystyle=\left[2H^{\prime}(u)+(3\varepsilon-2)H(u)/u\right]_{+}$
$\displaystyle=\Big{[}a[2+(2(s-1)+3\varepsilon)\log
u]\Big{]}_{+}\sup_{M}{u^{s-1}}+\Big{[}b[2(r-1)+3\varepsilon]\Big{]}_{+}\sup_{M}\\{u^{r-1}\\}$
$\displaystyle=0.$

The claims of the proposition therefore follow at once from Theorem 7.3.

∎

## Declarations

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Availability of data and material

All data generated or analyzed during the study are included in this paper.

### Conflict of interests

The author declares that there is no competing interests.

### Funding

This work does not receive any funding.

### Author’s contributions

AA is the only contributor in the preparation of this manuscript.

### Acknowledgements

This paper was completed during the author research visit to Instituto de
Matematicas, Universidad de Granada, Spain.

## References

  * [1] A. Abolarinwa, Gradient estimates for a weighted nonlinear elliptic equation and Liouville type theorems, J. Geom. Phy. 155 (2020), 103737.
  * [2] A. Abolarinwa, Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation, J. Math. Anal. Appl. 473 (2019), 297–312.
  * [3] A. Abolarinwa, Differential Harnack and logarithmic Sobolev inequalities along Ricci-harmonic map flow, Pacific J. Math. 278(2)(2015), 257–290.
  * [4] A. Abolarinwa, A. Ali, F. Mofarreh, Triviality of bounded solutions and gradient estimates for nonlinear $f$-heat equations on complete smooth metric measure spaces, J. Geom. Phy. 182 (2022) 104670.
  * [5] A. Abolarinwa, Differential Hanarck estimates for a nonlinear evolution equation of Allen-Cahn type, Mediterr. J. Math. 18, 200(2021).
  * [6] A. Abolarinwa, S. O. Salawu, C. A. Onate, Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications, Heliyon 5(2019).
  * [7] A. Abolarinwa, J. O. Ehigie, A. H. Alkhaldi, Harnack inequalities for a class of heat flows with nonlinear reaction terms, J. Geom. Phy. 170 (2021), 104382.
  * [8] A. Abolarinwa, A. Taheri, Elliptic gradient estimates for nonlinear $f$-heat equation on weighted manifolds with time dependent metrics and potentials, Chaos, Solitons & Fractals, 142 (2021), 110329.
  * [9] S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1085-1095.
  * [10] L. Alzaleq, V. Manoranjan, Analysis of Fisher-KPP with a time dependent Allee effect, IOP SciNotes 1 (2020) 025003.
  * [11] M. Bǎileşteanu, X. Cao, A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258 (2010), 3517–3542.
  * [12] D. Bakry, M. Émery Diffusions hypercontractives In: Azḿa J., Yor M. (eds) Séminaire de Probabilités XIX 1983/84. Lecture Notes in Mathematics, vol 1123. Springer, Berlin, Heidelberg
  * [13] D. Bakry, I. Genctil, M. Ledoux Analysis and Geometry of Markov Diffusion Operators, Springer International Publishing (2014).
  * [14] K. Brighton, A Liouville-type theorem for smooth metric measure spaces, J. Geom. Anal. 23 (2013) 562–570.
  * [15] E. Calabi, An extension of Hopf maximum principle with application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56.
  * [16] L. Calatroni, P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamics boundary conditions, Nonlinear Anal. 79 (2013), 12–27.
  * [17] H. D. Cao, Recent progress on Ricci solitons in: Recent Advances in Geometric Analysis, Adv. Lect. Math. (ALM), 11, International Press, Somerville (2010), 1–38.
  * [18] X. Cao, M. Cerenzia, D. Kazaras, Harnack estimates for the endangered species equation, Proc. Am. Math. Soc. 143(10) (2014), 4537–4545.
  * [19] X. Cao, B. Liu, I. Pendleton, A. Ward, Differential Harnack estimates for Fisher’s equation, Pac. J. Math. 290(2) (2017), 273–300.
  * [20] J. S. Case, A Yamabe-type problem on smooth metric measure spaces, J. Differential Geom. 101 (2015), 467–505.
  * [21] J. S. Case, Y-J. Shu, G. Wei, Rigidity of quasi-Einstein metrics, Diff. Geom. Appl. 29 (1) (2011), 93–100.
  * [22] D. Castorina, C. Mantegazza, Ancient solutions of semilinear heat equations on Riemannian manifolds, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (1)(2017), 85–101.
  * [23] D. Castorina, C. Mantegazza, Ancient solutions of superlinear heat equations on Riemannian manifolds, Commun. Contemp. Math. (2020).
  * [24] R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. Amer. Math. Soc. 108 (1990), 961–970.
  * [25] X. Chen, Generation and propagation of interface for reaction-diffusion equations, J. Differ. Equ. 96 (1) (1992), 116 – 141.
  * [26] X.Cheng,T.Mejia, D.T. Zhou, Eigenvalue estimate and compactness for closed $f$-minimal surfaces, Pacific J. Math. 271 (2014), 347–367.
  * [27] Q. Chen, G. Zhao, Li-Yau type and Souplet-Zhang type gradient estimates of a parabolic equation for the $V$-Laplacian, J. Math. Anal. Appl. (2018).
  * [28] S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (3)(1975), 333–354.
  * [29] H.T. Dung, Gradient estimates for a general type of nonlinear parabolic equations under geometric conditions and related problems, Nonlinear Anal. 226 (2023) 113135.
  * [30] H. T. Dung, Gradient estimates and Harnack inequalities of nonlinear heat equations for the $V$-Laplacian, J. Korean Math. Soc. 55(6) (2018), 1285-1305.
  * [31] H. T. Dung, N. T. Dung, J. Y. Wu, Sharp gradient estimates on weighted manifolds with compact boundary, Commun. Pure Appl. Anal.(2021).
  * [32] N. T. Dung, N. N. Khanh, Gradient estimates for a class of semilinear parabolic equations and their applications, Vietnam J. Math. (2021).
  * [33] N. T. Dung, N. N. Khanh, Q. A. Ngo, Gradient estimates for some $f$-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces, Manuscripta Math. 155, (2018), 471–501.
  * [34] N. T. Dung, J. Y. Wu, Gradient estimates for weighted harmonic function with Dirichlet boundary condition, Nonlinear Anal. 213, (2021), 112498.
  * [35] R.A. Fisher, The wave of advance of advantageous genes, Annu. Eugen. 7 (1937), 355–369.
  * [36] X. Fu, J. Y. Wu, Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition, Kodai Math. J. 45 (1)(2022), 96 –109.
  * [37] X. Geng, S. Hou, Gradient estimates for the Fisher-KPP equation on Riemannian manifolds, Bound. Val. Prob. 2018, 25.
  * [38] R. Hamilton, The formation of singularities in the Ricci flow, Surv. Diff. Geom. 2 (1993), 7–136.
  * [39] R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1(1993), 113–126.
  * [40] S. Hou, Gradient estimates for the Allen-Cahn equation on Riemannian manifolds, Proc. Am. Math. Soc. 147, (2019), 619–628.
  * [41] A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech. 1 (1937) 1–26.
  * [42] K. Kunikawa, Y. Sakurai, Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition, Proc. Amer. Math. Soc. 150 (2022), 1767–1777.
  * [43] J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal. 100 (1991), 233–56.
  * [44] X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 (2005) 1295–1361
  * [45] P. Li, S-T. Yau, _On the parabolic kernel of the Schrödinger operator_ , Acta Math. 156 (1986), 153–201.
  * [46] S. Li, X-D. Li, _On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows_ , Asian J. Math. 22 (2018), 577–598.
  * [47] S. Li, X-D. Li, _Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flows_, arxiv.org/abs/1412.7034v2
  * [48] S. Li, X-D. Li, _The $W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials_, Pac. J. Math., 278(1), (2015), 173–199
  * [49] J. Lott, Some geometric properties of the Bakry-Émery Ricci tensor, Comment. Math. Helv. 78 (2003) 865–883.
  * [50] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009), 903–991.
  * [51] L. Ma Gradient estimates for a simple elliptic equation on complete noncompact Riemannian manifolds, Journal of Functional Analysis, 241 (1), (2006), 374–382.
  * [52] L. Ma, S.-H. Du, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians, C. R. Math. Acad. Sci. Paris 348 (2010), 1203–1206.
  * [53] B. Ma, F. Zeng, Hamilton-Souplet-Zhang’s gradient estimates and Liouville theorems for a nonlinear parabolic equation, C. R. Math. Acad. Sci. Paris, Ser. I 356 (5)(2018), 550–557.
  * [54] G. Perelman, The entropy formula for the Ricci Flow and its geometric application, arXiv:math.DG/0211159v1 (2002).
  * [55] P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J. 139(3) (2007), 555–579.
  * [56] P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations, Indiana Univ. Math. J. 56(2) (2007), 879-908.
  * [57] X. Ramos Olivé, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc. 147 (2019), 411–426.
  * [58] R.C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), 459–472.
  * [59] Y. Sakurai, Rigidity of manifolds with boundary under a lower Bakry-Émery Ricci curvature bound, Tohoku Math. J. 71 (2019), 69–109.
  * [60] R. Schoen, S.T. Yau, Lectures on Differential Geometry, International Press, 1994.
  * [61] P. Souplet, Q. S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38 (6)(2006), 1045–1053.
  * [62] C. Villani, Optimal Transport: Old and New, A Series of Comprehensive studies in mathematics, vol. 338. Springer, Berlin
  * [63] J. P. Wang, Global heat kernel estimates, Pacific J. Math. 178 (1997), 377–398.
  * [64] L.-F. Wang, Z.-Y. Zhang, Y.-J.Zhou, Comparison theorems on smooth metric measure spaces with boundary, Adv. Geom. 16 (2016), 401–411.
  * [65] G. F. Wei, W. Wylie, _Comparison geometry for the Bakry-Émery Ricci tensor_ , J. Differ. Geom. 83 (2009), 377–405.
  * [66] J-Y. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z.(2015).
  * [67] J-Y. Wu, Gradient estimates for a nonlinear parabolic equation and Liouville theorems, Manuscript Math. 159 (2018), 511–547.
  * [68] Y. Yang, Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds, Proc. Am. Math. Soc. 136 (2008), 4095 – 4102.
  * [69] S-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228.
  * [70] S-T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Math. J. 25 (1976), 659–670.


# Combined Learning of Neural Network Weights for Privacy in Collaborative
Tasks

Aline R. Ioste  Alan M. Durham  Marcelo Finger

( Universidade de São Paulo - Brazil )

###### Abstract

We introduce CoLN, Combined Learning of Neural network weights, a novel method
to securely combine Machine Learning models over sensitive data with no
sharing of data. With CoLN, local hosts use the same Neural Network
architecture and base parameters to train a model using only locally available
data. Locally trained models are then submitted to a combining agent, which
produces a combined model. The new model’s parameters can be send back to
hosts, and can then be used as initial parameters for a new training
iteration. CoLN is capable of combining several distributed neural networks of
the same kind, but is not restricted to any single neural architecture. In
this paper we detail the combination algorithm and present experiments with
feed-forward, convolutional and recurrent Neural Network architectures,
showing that the CoLN combined model approximates the performance of a
hypothetical ideal centralized model, trained using the combination of the
local datasets. CoLN can contribute for secure collaborative research, as
required in the medical area, where privacy issues preclude data sharing, but
where the limitations of local data demand information derived from larger
datasets.

##  1 Introduction

During the standard training of neural network models, it is usually assumed
that all data is readily available to be used by the application. However,
this assumption is not valid when handling sensitive data that is distributed
over several hosts with access restrictions [1]. Existing data protection
laws, in particular the General Data Protection Regulation (GDPR) [2], limit
data sharing and, as a consequence, the potential size of the training sets,
which in turn can reduce the accuracy of the models. In fact, several
applications of Deep Learning in Health have already pointed to the conflict
between the need to collaborate and the need to respect privacy constraints
[3, 4, 5]. The consequence is a restriction on collaborative research across
international borders, when private patient data is involved [6, 7, 8].

The machine learning community has proposed several approaches to provide data
protection while still using centralized training, including: data
anonymization, differential privacy [9, 10], homomorphic encryption to data
cryptography [11, 12], and combined data anonymization with random noise
generation, enhanced with homomorphic encryption [13]. However, these
approaches are not always feasible. First, anonymization may still expose
personal information [14]; second, some approaches may reduce the final
performance of the machine learning method [11, 9]; third, different privacy
and security restrictions in the data may still preclude exporting training
data to centralized locations, specially when this involves data crossing
international borders [15, 1].

Decentralized learning addresses these challenges [16, 17, 18, 19, 20]. In a
decentralized learning approach, separate models are trained locally by
separate hosts, and a combined model is produced by using the parameters and
metadata of the local models. Such an approach is more privacy compliant, as
no data is shared and it is not possible to reconstruct input data from the
parameters of locally trained models [21, 22].

In the context of decentralized learning, one of the most popular approached
is Federated Learning, initially proposed in the Federated Average (FedAvg)
algorithm, where local parameters are averaged to obtain the centralized
values [23, 24].

Several variations of this technique were proposed:

  * •

Federated Proximal (FedProx) [25], an approach designed to optimize network
traffic. This approach was proposed as a federated optimization structure for
heterogeneous networks. This techique limits the impact of local updates,
restricting them to be close to the current model.

  * •

Agnostic Federated Learning (AFL) [26], an approach designed to cope with with
non-uniform distribution of data points which is agnostic with respect to data
distribution, assuming that global loss is an unknown convex combination of
local data. AFL uses a stochastic optimization process which applies a
regularization that provides learning guarantees.

  * •

Probabilistic Federated Neural Matching (PFNM) [27] proposes another form of
parameter combination that identifies subsets of neurons in each of the local
models that match neurons in other local models, and then combine the matched
neurons to form a global model. However, the method only works with simple
network architectures, without applicability to to Convolutional Neural
Networks (CNNs) and Recurrent Neural Networks (RNNs).

  * •

Federated Matched Averaging (FedMA) [28] extends PFNM ideas to CNNs and LSTMs.
FedMA is a layers-wide federated learning algorithm that employs Bayesian non-
parametric methods to adapt to heterogeneity in the data.

We present Combined Learning of Neural network weights (CoLN), a new
Distributed Learning approach to combine parameter information inferred
autonomously by several distributed hosts engaged in collaborative work. Only
local model parameters and information on local dataset sizes are transmitted
over the network by the collaborating hosts. The method avoids traffic issues
present in some distributed learning algorithms, with very little information
being exchanged in the network and a reduced number of steps required in the
process. More important, the method is very resilient to variations in the
local data distributions and dataset sizes.

CoLN achieves the goal of obtaining, from a set of models locally trained on
distinct datasets, a combined model that approximates the performance of a
hypothetical ideal centralized model, which would be the result of using the
combined local datasets for training a single model.

##  2 Methods

CoLN is an interactive approach that, at each round, first performs training
with the models initialized by the same set of parameter values, and then
compute a new set of parameters combining the parameters resulting from each
local training. For the combination phase we use a linear combination of the
local values that avoids catastrophic forgetting and, at the same time,
ensures generalization.

###  2.1 Basic architecture

We assume there are $H$ host sites, each with a distinct data set and with a
host model trained using the local data. All sites should use the same Neural
Network architecture as the basic model and, before training, are initialize
it with the same parameters.

Model parameters consist of weights, biases and hyper parameters. In this view
a layer is viewed as an individual matrix or tensor, whose parameters are
called simply _weights_ ; biases are assumed to be elements of layers like any
other weights in the network.

For the method’s presentation, we refer to a synchronizing site in charge of
combining the locally trained host models. However, the parameter combination
process to obtain the combined model can be performed in one of the hosts, or
in parallel by all local hosts.

The CoLN central hypothesis is that, when we have H local hosts each with W
weights, we can combined the learning of all hosts usint a linear
transformation on local host weights ($w^{h}_{i}$):

$\displaystyle\overline{w}_{i}$
$\displaystyle=\sum_{h=1}^{H}\alpha^{h}w^{h}_{i}+\beta_{i},\qquad\qquad 1\leq
i\leq W$ (1)

A central hypothesis of this work, which we tried to empirically validate, is
that a linear combination is a good way to combine network weights.
Intuitively, linear combinations are a well-founded and simple method, which
allows for quite flexible and expressive combinations based on the
coefficients $\alpha$ and $\beta$. The coefficients $\alpha$ and $\beta$ are
refined below with reference to _network layers_ in Section 2.2. The value of
$\alpha^{h}$ reflects the relevance of host $h$ relative to other hosts; the
value of $\beta_{i}$ is empirically determined based on a notion of distance
between the weights of host $h$ and a hypothetical centralized model.
Empirically, the linear combination detailed in Section 2.2 satisfies the
following characteristics:

  1. ($i$)

The composed model is closer to the centrally trained model $w^{*}$, that is,
$\textit{dist}(\overline{w},w^{*})<\textit{dist}({w}^{h},w^{*})$ for every
$h\in[1,H]$. The distance is measured by Eq. 4, but it could be understood in
terms of improved accuracy, namely that the combined model is closer to the
accuracy of a hypothetical centralized model than any of the local hosts.

  2. ($ii$)

We can re-train the local parameters using initial weights $\overline{w}$ and
reiterate the process, obtaining obtaining $\overline{\overline{w}}$, a better
approximation of $w^{*}$-performance, that is,
$\textit{dist}(\overline{\overline{w}},w^{*})<\textit{dist}(\overline{w},w^{*})$.

The first characteristic would give us the possibility of producing a
strongly-decoupled distributed learning method with low communication
overhead, in which each host starts with the same initial model $w^{0}$ and,
with its own dataset, autonomously produces a model $w^{h}$ which is sent only
to a synchronizer, which computes $\overline{w}$. The second characteristic
would ensure that this process could be iterated to obtain an even better
model.

It is important to note that, different than that other approaches, we do not
require the linear coefficients to be convex, that is, for a fixed weight $i$,
$\alpha^{1},\ldots,\alpha^{H},\beta_{i}$ are not required to add up to 1. In
fact, previous experiments indicated that convexity does not guarantee
characteristics ($i$) and ($ii$), suggesting that the centralized optimal
model lies outside of the convex hull of the local optimum models. This fact
should not be a surprise. If the centralized model resulting from an
optimization using all the data were obtainable as a convex combination of
host models, this would indicate a linear nature of the neural models,
contrary to the existence of non-linear activation functions in almost all
non-trivial neural models.

In CoLN, model combination proceeds in a sequence of rounds, whose main
characteristics are as follows.

  1. 1.

Initialization: At the beginning of each round all hosts use _the same_ set of
parameters. This is a synchronization point. In the first round, the initial
weights could be generated randomly and then distributed, or obtained from a
previously trained model. In subsequent rounds, the initial weights of the
models are the result of the CoLN combination of previous round.

  2. 2.

Training: Each host trains its model based only on locally available data,
independently from all other hosts. The number of epochs used by each host
site may vary; locally available datasets can be of different sizes and
present different distributions.

  3. 3.

Synchronizing: After local training, local model weights are exported to a
synchronizing site, where a synchronized model is computed using the linear
transformation. The round finishes with the redistribution of the synchronized
model weights back to the hosts.

###  2.2 Combination Coefficients

We now will refine and detail the combination method described in equation
(1). For this we need to associate individual weights with a _layer_ in the
model.

The linear combination have the following formal parameters:

  * •

$L$, the number of layers of the common architecture;

  * •

$M_{\ell}$, the number of weights and biases in layer $\ell$, $\ell\in[1,L]$
of the common architecture;

  * •

$H$, the amount of host models;

  * •

$T_{h}$, the size of the training corpus at host site $h$; the total size of
training data is $T=\sum_{h=1}^{H}T_{h}$;

  * •

$r_{h}$, the relative size of the training corpus at $d$ with respect to the
total size, $r_{h}=\frac{T_{h}}{T}$.

  * •

$w_{i,\ell}^{h}$, the value of $i$th weight in layer $\ell$ for host model
$h$, $1\leq i\leq m_{\ell}$, $1\leq h\leq H$.

  * •

$\overline{w}_{i,\ell}$, the value of $i$th weight in layer $\ell$ in the
synchronized model.

Now the can refine the combination method (1) to:

$\displaystyle\overline{w}_{i,\ell}$
$\displaystyle=\sum_{h=1}^{H}\alpha^{h}w^{h}_{i,\ell}+\beta_{i,\ell},\qquad\qquad
1\leq i\leq M_{\ell},1\leq\ell\leq L$ (2)

Equation (2) involves both layer independent _linear combiners_ $\alpha^{h}$
and layer dependent _shift values_ $\beta_{i,\ell}$; described below.

Determining Combination Rate $\alpha^{h}$

The first experimental study aimed at evaluating the format of the linear
combination and its hyper-parameters. For that, we used the COVID-19 dataset
with the aim of predicting the presence of COVID-19 in patients based on a
series of other laboratory exams. From the original dataset, two hosts were
artificially built, one with 100% positive COVID-19 cases and one with 100%
negative cases, and a balanced global test with 50% positive and 50% negative
cases. The goal of using models with high discrepancy is to have a clear
experimental view of accuracy gain by combining the 50% accuracy local models
and determining the best linear combiners to achieve a combined model that can
achieve a high rating. Note that at this point, we did not compare with a
centralized model trained on all data which, for the record, reaches 99.64%.
The initial idea was to study the feasibility of initializing identically two
host models locally training them and then to combine them using linear
combiner value of $\alpha^{h}$ = 1 + $r_{h}$. In the model exposed in Equation
(2) to (8), this corresponds to c = 1 expanded only for the first two terms of
its Taylor-series format. Both models were trained with the number of epochs
necessary to achieve the best local accuracy. The models are submitted to
several rounds of CoLN-combination and the results are shown in figure 1.

Figure 1: Combinated accuracy in COVID-19 data with
$\mathrm{\alpha}^{h}=1+rh,h=1,2$.

These initial combined accuracy obtained in the first round is 94,803%, and it
keeps improving for the next three rounds to 97.31%, but then it starts
decreasing and never recovers until, at round 24, it reaches about 50%, which
was the accuracy of each model trained separately. At this point, we had two
alternatives, either establishing a method to stop at the appropriate
combination round, which in this case was round 4, or to devise a method which
would not suffer from this form of decay. We decided for the second
alternative, exploring linear combiners of the form
$\alpha^{h}=\mathrm{e}^{c\cdot r_{h}}$. For the case c = 1, the results
obtained are shown in figure 2.

Figure 2: Combinated accuracy in COVID-19 data with
$\mathrm{\alpha}^{h}=\mathrm{e}^{c.r_{h}}$, $c=+1,h=1,2$.

In this case, in only three rounds we reached maximum combined accuracy of
97.32%, just like before, but the decay in figure 2 was very much like that in
figure 1, and we noted that both the average size of the weights as well as
their standard deviation were increasing at each round, much like the
phenomenon of exploding gradients. So, we searched for methods to avoid this
explosion, or somehow revert it. Employing a value of c = -1 in
$\alpha^{h}={e}^{c\cdot r_{h}}$ led to a degradation of performance, that is,
a total failure of combination, but then we had the idea of alternating c = +1
with c = -1 in consecutive rounds, and the results of this alternation are
shown in figure 3.

Figure 3: Combinated accuracy in COVID-19 data with
$\alpha^{h}=\mathrm{e}^{c\cdot r_{h}}$, $c=+1/c={(-1)},h=1,2$.

Starting from a random model, we accuracy of 66.129% in the first iteration,
64.51% in the second, 98.92% in the third, 98.74% in the fourth and 99.10% in
the fifth. We can see that, in the alternative case, the phenomenon of sharp
accuracy decay in precision is not present. In this setting, we can decide to
halt the combination rounds once the accuracy increase between rounds stays
within some interval (another hyper-parameter). However, the alternation
process is inefficient and computationally expensive. In fact, as seen in
figure 3, when we alternate weight expansion rounds $c={1}$ with weight
contracting rounds $c={-1}$, the latter combination never increases accuracy
and is employed only to avoid weight degradation, so that half rounds are not
moving the combination toward maximum accuracy; but once this point is
reached, it stays there. We still do not consider it as the best solution, and
investigate other combination rate values. In traditional gradient descent
backpropagation method, the learning rate allows one to in- corporate a
fraction of the gradient at every training epoch. When the learning rate is
large, the learning process may diverge, so usually small values of learning
rate are used for better results. In analogy to this case, we experimented
with small combination rates, without alternation, to avoid weight decay. In
that spirit, we explored linear combiners of the form
$\alpha^{h}=\mathrm{e}^{c\cdot r_{h}}$, with $c=10^{(-3)}$; the results
obtained are shown in figure 4

Figure 4: Combinated accuracy in COVID-19 data with
$\mathrm{\alpha}^{h}=\mathrm{e}^{c.r_{h}}$, $c=10^{(-3)}$, h = 1,2

Combining the network weights in small hops avoids the degradation of the
central model without requiring expensive alternations, as seen in figure 4.
Starting from a random model, we accuracy of 49,64% in the first iteration,
78,32% in the second, 97,85% in the third, 99,28% in the fourth and 100,000%
in the fifth. In the following, we fix CoLN combination rate hyperparameter at
$c=10^{(-3)}$ employing linear combiners of the form equation 3. Experiments
showed that it is possible to start the combination process with an already
trained model by some host, with faster convergence. However, to avoid
searching for a volunteer to start the process, we always start with a random
model.

Linear CoLNbyLayerombiners are host-dependent exponential factors based on the
relative size of the host’s dataset:

$\displaystyle\alpha^{h}=\mathrm{e}^{c\cdot r_{h}}$ (3)

where $r_{h}$ is the host’s relative dataset size; and $c$ is a hyper-
parameter, the _combination rate_.

In the empirical validation of the linear combination of layer weights,
several alternatives for the value of $\alpha^{h}$ were considered. Due to the
non-linear nature of neural networks, a convex combination of weights was
ruled out, and a we initially considered using the value
$\alpha^{h}_{1}=1+r_{h}$. Then we realized that $1+r_{d}$ are just the first
two terms of the Taylor-series expansion of the more general formula
$\alpha^{h}_{2}=\mathrm{e}^{c.r_{h}}$, and noted that this form for the linear
combiner leads to a faster convergence of the CoLN algorithm; note that
$\alpha^{h}_{2}$ corresponds to (3) with combination rate $c=+1$.

In the supplementary material we describe the series of experiments on
different families of coefficients $\alpha^{1},\ldots,\alpha^{H},\beta$, which
led to CoLN’s final linear combination coefficients. In fact, these
experiments showed that both $\alpha^{h}_{1}$ and $\alpha^{h}_{2}$ led to an
explosive growth of weight values after a few combination rounds, which in
turn leads to a sharp drop in the combined accuracy, a phenomenon akin to the
exploding gradients in Recurrent Nerual Networks [29].

After experimenting with alternating values of $c=+1$ and $c=-1$ in successive
combination rounds, we settled for values of $c\in[10^{-6}$,$10^{-3}]$, which,
in our experiments, avoided the weight size explosion phenomenon (see
supplementary material).

The computation of shift values employs two auxiliary functions. The first is
a measure of a distance between the specific weight and its counterparts in
the other host models, as given by:

$\displaystyle\textit{WeightDistance}(i,\ell)$
$\displaystyle=\sqrt{\sum_{j=1}^{H-1}\sum_{k=j+1}^{H}\left(w_{i,\ell}^{j}\cdot
r_{j}-w_{i,\ell}^{k}\cdot r_{k}\right)^{2}}$ (4)

According to Eq. 4, all weights at host $h$ are scaled by the data size rate
$r_{h}$ prior to computing a summation of all squared euclidian distances
between all scaled layer weights, thus producing
$\textit{WeightDistance}(i,\ell)$.

The second is a layer distance:

$\displaystyle\textit{LayerDistance}(\ell)$
$\displaystyle=\frac{\sqrt{\sum_{i=1}^{M_{\ell}}\sum_{j=1}^{H-1}\sum_{k=j+1}^{H}(w_{i,\ell}^{j}-w_{i,\ell}^{k})^{2}}}{M_{\ell}}$
(5)

The $\textit{LayerDistance}(\ell)$ computes a ponderation over all weights of
the sum of the squared distances between all layer weights. The idea is to
produce a shift for all weights in a layer whose distance to the layer is
below average.

We can now define $\beta_{i,\ell}$, the shift value for weight $i$ in layer
$l$:

$\displaystyle\beta_{i,\ell}=\left\\{\begin{array}[]{ll}\textit{WeightDistance}(i,\ell),&\textrm{if
}WeightDistance(i,\ell)<LayerDistance(\ell)\\\
0,&\textrm{otherwise}\end{array}\right.$ (8)

Note that there is a shift only in the cases where the weight distance is
smaller than the layer distance; the shift is used to compensate cases in
which weights are negligible with respect to other weights in the same layer.

CoLN is summarized by algorithm Algorithm 1.

Algorithm 1 CoLN

Input A. A description of the common model consisting of: (i)$L$
layers;(ii)$m$, a vector containing the number of weights per layer. B. The
weights of all host models:
$w_{i,\ell,d},i\in[1,m[\ell]],d\in[1,D],\ell\in[1,L]$; C. $r$: vector of
training data size rates of per host site.
Output Weights of the combined model $\overline{W}$

1:procedure CoLN($D,L,m,w,r$)

2: for $\ell=1$ to $L$ do

3: Compute $\textit{LayerDistance}(\ell)$

4: for $i=1$ to $m[\ell]$ do

5: Compute $\textit{WeightDistance}(i,\ell)$

6: for $d=1$ to $D$ do

7:
$\overline{w}[i,\ell]\\!\\!={\displaystyle\sum_{d=1}^{D}w[i,\ell,d]*\mathrm{e}^{c\cdot
r_{h}}}$

8: if ($\textit{WeightDistance}(i,\ell)\leq\textit{LayerDistance}(\ell)$) then

9: $\overline{w}[i,\ell]\ +\\!\\!=\textit{WeightDistance}(i,\ell)$

10: return $\overline{W}$

##  3 Results

The methods and parameters of CoLN are detailed in Section 2. All experiments
were performed with the method’s hyper-parameters determined in a series of
calibration experiments described in the supplementary material.

We have applied CoLN to two sets of experiments. The aim of the first set was
to estimate the accuracy and robustness of CoLN for different types of neural
network architectures (Multi-layer feed forward, Convolutional, Recurrent
LSTM), different number of local hosts and distribution biases. The aim of the
second set of experiments was to compare CoLN’s performance with Federated
learning approaches.

In all experiments, we also trained a “centralized model”, using the
combination of all local datasets. The performance of this model was used as
the “golden standard” for comparison. In all cases models were initialized
with the same random values before the first training.

###  3.1 Validation experiments

####  3.1.1 Breast Cancer Data

The goal of this experiment was to test if the CoLN combination method can
compensate for distribution biases in the local models that create noise. For
this, we used a feed forward Neural Network with 785 parameters (details of
the architecture are explained in the supplementary material).

In this experiment, we used breast cancer anonymized patient data from the
University of Wisconsin Hospitals [30, 31, 32]. We split the original data,
which did not include gender information, into two “virtual hospitals,”
Dataset1 and Dataset2. Then we added to the original data an artificial gender
parameter, in such a way as to introduce an opposite gender bias in cancer
data for each virtual hospital. In Dataset1, the data for patients with
malignant cancer was predominantly labeled as “male”, and data for patients
with benign cancer was predominantly labeled as “female”. Data in Dataset1
contains information from 230 patients with cancer, 115 malignant and 115
benign. Of the 115 malignant cancer patients 80% were labeled as male and 20%
as female. In Dataset2, the distribution is skewed in the other direction. It
contains data from 354 other cancer patients, 177 malignant and 177 benign. Of
the 177 malignant cancer patients, 80% were labeled female and 20% male. Test
data was balanced, containing 136 samples, corresponding to 68 malignant
cancer and 68 benign cancer, with gender being assigned randomly with 50% for
each gender in each class.

We performed 30 rounds of combination. Fig. 5 shows the results achieved by
each model and by a centrally trained model using all 490 samples; test
results for the centrally trained model were computed only once.

Figure 5:  Comparison of accuracy obtained in breast cancer data with
distributional bias. Model1 identify models trained using Dataset1 and Model2
identify models trained using Dataset2. In Round 0 the combined model used
only the common initial random parameters, Model1 and Model2 used parameters
that resulted from the first round of training using local data. In subsequent
rounds, Model1 and Model2 were the result of another round of training with
local data and the combined model used parameters combined in the previous
round.

The CoLN Combination reached 95.58% accuracy in the second round, maintaining
this value in the remaining rounds with few oscillations. Local models Model1
and Model2 were initialized with the same random weights of the synchronizing
model, with Model1 achieving initial accuracy 61.77% and Model2 77.94%. Each
local model at each round was trained using with 50 epochs. The centralized
model achieved 95.59% accuracy after 200 epochs.

It is important to note that, even though, after the first round, local models
are initialized with the weights obtained from CoLN in the previous round
(thus with a model of higher accuracy), after retraining with local data only,
the result is a model of lower accuracy. Local models never achieve the same
accuracy as CoLN. This is expected as, at each round, the models are re-
trained with biased data and incorporate the apparent dependency between
gender and cancer. However, CoLN combined learning model does not seem to be
affected by this bias, and oscillates with very little variation near its
upper bound, which corresponds to almost the same accuracy as the centrally
trained model. Detailed results for each round are described in the
supplementary material.

####  3.1.2 Covid-19 Data

In the second experiment we used a covid-19 dataset consisting of 92 different
clinical and laboratory tests from 5466 patients admitted to the Albert
Einstein Hospital of São Paulo in 2020 [33]. Clinical data from tests were
used to predict whether the patient would be positive for covid-19, as mesured
by a SARS-CoV-2 RT-PCR test. In the prediction task, a feed forward Neural
Network was used containing 1,873 parameters. Details of the models are
described in the supplementary material.

We randomly selected 811 patients for training and 588 for testing. The
training set was randomly partitioned in two sub-sets with different
distributions of positive and negative samples: Dataset1 consisted of 223
positive and 107 negative samples (total 330), and Dataset2 consisted 56
positive and 425 negative samples (total 481). The test dataset was balanced,
with 279 positive and 279 negative cases.

We used CoLN in 30 rounds of combination. Again, the centralized model was
trained only once. Results are shown in Fig. 6.

Figure 6: Comparison of accuracy obtained in COVID-19 data with distributional
bias, in which CoLN reaches the 90% accuracy in five rounds, and was followed
by Model1 and Model2 in a few rounds. Model1 identify models trained using
Dataset1 and Model2 identify models trained using Dataset2. In Round 0 the
combined model used only the common initial random parameters, Model1 and
Model2 used parameters that resulted from the first round of training using
local data. In subsequent rounds, Model1 and Model2 were the result of another
round of training with local data and the combined model used parameters
combined in the previous round.

In this experiment, the centralized model achieved 100% accuracy with 60
epochs. To demonstrate that CoLN can combine model features, the local
training rounds were executed with only 10 epochs, and Model1 and Model2
achieved initial accuracy of around 50%, a result similar to randomly flipping
a coin, and this behaviour was maintained in the next 7 rounds. However, just
after 5 rounds, CoLN already achieved more than 90% accuracy, and the weights
of this combined model were used initial state for both Model1 and Model2.
Model1, whose local data was less unbalanced than Model2, took 10 combination
rounds to achieve accuracy above 90%, while Model2 took 15 rounds to reach the
same level. This suggests that CoLN is not affected by unbalanced data, while
local models struggle to overcome that deficit even when initialized with more
accurate weights. The results the each round are described in the
supplementary material.

####  3.1.3 Text Sentiment Analysis

In the third experiment we used used a Long-short term (LSTM) recurrent neural
network [34], for the problem of textual sentiment analysis, this time using
five local hosts. Training for each host used as many epochs as necessary to
achieve minimum loss. The input consisted of a dataset of product reviews
containing a numeric evaluation of products (1–5), an evaluation text and a
binary-valued field indicating if the author of the review would recommend the
product to a friend [35]. The neural network used the product review text to
predict the recommendation. We used a single-layer LSTM with a total of of
105,060 parameters. Details of the architecture are described in the
supplementary material.

The training set consists of 11890 evaluation texts and the testing set
consisted of another 5096 evaluation texts. The training dataset was unevenly
split among five local subsets one for each local model: Dataset1 (3304
texts), Dataset2 (2726 texts), Dataset3 (4436 texts), Dataset4 (4582 texts)
and Dataset5 (1938 texts). The results are depicted in Fig. 7.

Figure 7: Comparison of accuracy obtained for product review sentiment
analysis employing LSTM models, with 5 distinct local hosts with distinct
dataset sizes. CoLN reach stable accuracy in three rounds. Model1 identify
models trained using Dataset1, Model2 identify models trained using Dataset2
and so forth until Model5. In Round 0 the combined model used only the common
initial random parameters, Model1 to Model5 used parameters that resulted from
the first round of training using local data. In subsequent rounds, Model1 to
Model5 were the result of another round of training with local data, but
initializing the parameters with those obteind by the CoLN combination of the
previous round, while the combination model results used the parameters from
the combination in the same round.

Here the centralized model achieved 60.85% accuracy after 180 epochs and the
CoLN combination reached 61.02% accuracy in the 13th round. Local models also
stabilized, but lower accuracy levels (between 57 and 59%).

CoLN performed better than all local models, but not by much, as all models
performed well, with differences mostly mirroring the sizes if each local
dataset.

####  3.1.4 Combining Convolutional Models for Image Classification

The goal of this experiment was to test the performance of the CoLN process on
a convolutional neural network (CNN) [36] in the task of multi-category image
classification. We designed the experiment so that each of the two local hosts
had unbalanced distributions of examples of the target classification
categories. The classification task was performed by a CNN with 1,658,700
parameters and an input matrix of 28x28x32. Details of the model are described
in the supplementary material.

We randomly selected 28,000 images from the CIFAR10 dataset [37] in four
categories: cat, frog, car and airplane. Training images were separated in two
local datasets: Dataset1 with the predominance of airplane and car images
(4,000 airplane, 4,000 car, 2,000 frog, and 2,000 cat) and Dataset2 with the
predominance of frog and cat images (4,000 frog, 4,000 cat, 2,000 car and
2,000 airplane). The test dataset was a balanced set of 4,000 images, 1,000 of
each category (airplane, car, cat and frog). We trained the centralized model
only once with all 28,000 images of the combined training sets, and we
performed 30 rounds of the CoLN algorithm. Centralized training was peformed
in 100 epochs and local training in 50 epochs. The results are shown in Figure
Fig. 8, and the combined accuracy in Figure Fig. 9.

Figure 8: Comparison of classification accuracy for class A: “Car” , B:
“Airplane” , C: “Cat” , D: “Frog” for image classification over CNN models
over four image classes.  Figure 9:  Comparison of classification accuracy
over all four classes of the for image classification employing CNN models.
Model1 was trained using Dataset1, a set of images biased towards cats and
dogs, Model2 was trained using Dataset2, a set of images biased towards planes
and automobiles. In Round 0 the combined model used only the common initial
random parameters, Model1 and Model2 used parameters that resulted from the
first round of training using local data. In subsequent rounds, Model1 and
Model2 were the result of another round of training with local data and the
combined model used parameters resulting from the CoLN combination of the
previous round.

As we can see, Model1 is an “expert” in cars and airplanes and performs even
better than the centralized model over these classes, while Model2 performs
poorly on theses categories, with a very low initial accuracy and only a small
improvement with CoLN. Similarly, Model2 is an “expert” in Cats and Frogs,
while Model1 performs poorly on those categories. The CoLN combination, with
respect to each category, achieves an intermediary performance in relation to
each “expert” model, but performs consistently below the centralized model
over classes “Airplane” and “Frog” and achieves an accuracy close to that of
the centralized model in classes “Cat” and “Car”.

The joint accuracy over all four classes is shown in Fig. 9, revealing a
picture more akin to previous results. The centralized obtained the accuracy
level of 81.65%, and CoLN achieves 77.18%, in the first round of combination,
while local model’s performance was 73.28% and 72.73%. CoLN obtains 78.85% in
the second round and oscillates around this value ($\pm 0.03\%$) in the
following rounds. The results show that clearly shows that CoLN ouperforms
local models, approaching the ideal centralized performance.

###  3.2 Comparing CoLN with Benchmark Federated Learning Methods over Image
Classification

In this experiment we compared CoLN with with the federated learning methods
using a Convolutional Neural Network Model in an image classification problem
using. For this, we used GitHub implementation of FedMA, FedAvg, and FedProx
available at https://github.com/IBM/FedMA, which was presented at FedMA
presentation [28]. We downloaded from CIFAR10 60,000 32x32 color images from
of 10 different classes (airplanes, cars, birds, cats, deer, dogs, frogs,
horses, ships, and trucks). From these, we created two ”local” training sets
with 25,000 images each and a test with 10,000 images (1,000 images of each
category). The centralized model was trained with the combined training sets
(50,000 images). The results are shown in Fig. 10.

Figure 10: Comparison of combination methods over CIFAR10 image recognition
task, employing two local hosts with fixed data. Five methods took part in the
comparison: Centralised, FedMA, FedAvg, FedProx and CoLN. The bars indicate
the final accuracy of each approach, the initial accuracy values for each of
the local models, and the accuracy for the centralized model, trained with the
combined local datasets.

CoLN presented the best performance of all approaches. It achieved higher
accuracy (77.8%) when compared with FedMA (69.4%), FedAvg (72.0%) and FedProx
(71.2%) and its final accuracy was very close to that of the centralized model
(77.8% vs 78.0%). Also, CoLN’ns maximum accuracy of 77.8% was reached in the
second round, and other approaches in 6 rounds. All experiments were executed
with 30 epochs and 6 rounds, and centralized in 120 epochs.

##  4 Conclusion

Combined Learning of Neural network weights (CoLN) is a method that addresses
privacy issues in neural network machine learning: only model parameters and
local dataset sizes are communicated, no data is ever exchanged. It uses a
efficient mathematical formulation to combine a set of identical neural models
whose parameters are autonomously trained over local data that is never
shared. Only local model parameters and information on local dataset sizes are
transmitted over the network by the collaborating hosts, maintaining privacy
over individual data. Our experiments in different Neural Network
architectures have shown that CoLN’s combined models improve on local host
performance, converges on small number of rounds, and approximates the ideal
situation where all data is available for training single, centralized model
(from two to 6 steps, depending on the number of categories in the
classification). CoLN also avoids several issues common to other distributed
learning algorithms [25, 26, 28]; it requires very little data traffic over
the network, makes no assumption on data distribution, and is resilient to
local data size imbalance.

Finally, CoLN compares favourably to other forms of federated learning models,
exceeding their performance on a standard benchmark.

CoLN demonstrates a new direction for future work by combining the models in a
decentralized way using new mathematics capable of offering practical benefits
privacy, with security, as well as the ability to increase model accuracy
global without the need for each iteration to add new datasets and clients.
Experiments have shown that balanced learning can be a practical method that
allows collaborative research with datasets at each institution, mainly
especially when there is little data, significantly impacting research centers
smaller companies, start-ups and collaborative research across international
borders. Experiments and supplemental materials are available on the github
website: https://github.com/AlineIoste/CoLN

## References

  * [1] S. De Capitani Di Vimercati, S. Foresti, G. Livraga, and P. Samarati, “Data privacy: definitions and techniques,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 20, no. 06, pp. 793–817, 2012.
  * [2] C. O’Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy.  USA: Crown Publishing Group, 2016.
  * [3] D. Ravì, C. Wong, F. Deligianni, M. Berthelot, J. Andreu-Perez, B. Lo, and G.-Z. Yang, “Deep learning for health informatics,” IEEE journal of biomedical and health informatics, vol. 21, no. 1, pp. 4–21, 2016.
  * [4] W. Ning, S. Lei, J. Yang, Y. Cao, P. Jiang, Q. Yang, J. Zhang, X. Wang, F. Chen, Z. Geng, et al., “Open resource of clinical data from patients with pneumonia for the prediction of covid-19 outcomes via deep learning,” Nature biomedical engineering, pp. 1–11, 2020.
  * [5] S. B. Dias, S. J. Hadjileontiadou, J. Diniz, and L. J. Hadjileontiadis, “Deeplms: a deep learning predictive model for supporting online learning in the covid-19 era,” Scientific reports, vol. 10, no. 1, pp. 1–17, 2020\.
  * [6] C. J. Bennett, Regulating privacy: Data protection and public policy in Europe and the United States.  Cornell University Press, 1992.
  * [7] H. K. Patil and R. Seshadri, “Big data security and privacy issues in healthcare,” in 2014 IEEE international congress on big data, pp. 762–765, IEEE, 2014.
  * [8] W. N. Price and I. G. Cohen, “Privacy in the age of medical big data,” Nature medicine, vol. 25, no. 1, pp. 37–43, 2019.
  * [9] M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang, “Deep learning with differential privacy,” in Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 308–318, 2016.
  * [10] C. Dwork, “Differential privacy: A survey of results,” in International conference on theory and applications of models of computation, pp. 1–19, Springer, 2008.
  * [11] T. Graepel, K. Lauter, and M. Naehrig, “Ml confidential: Machine learning on encrypted data,” in International Conference on Information Security and Cryptology, pp. 1–21, Springer, 2012.
  * [12] A. Wood, K. Najarian, and D. Kahrobaei, “Homomorphic encryption for machine learning in medicine and bioinformatics,” ACM Computing Surveys (CSUR), 2019.
  * [13] T. Sharma and M. Bashir, “Use of apps in the covid-19 response and the loss of privacy protection,” Nature Medicine, pp. 1–2, 2020.
  * [14] L. Rocher, J. M. Hendrickx, and Y.-A. De Montjoye, “Estimating the success of re-identifications in incomplete datasets using generative models,” Nature communications, vol. 10, no. 1, pp. 1–9, 2019.
  * [15] P. Russom, “Big data analytics. tdwi best practices report, fourth quarter, 2011,” 2011.
  * [16] A. E. Williams, “Defining functional models of artificial intelligence solutions to create a library that an artificial general intelligence can use to increase general problem solving ability,” AfricArXiv, 2020.
  * [17] G. A. Montes and B. Goertzel, “Distributed, decentralized, and democratized artificial intelligence,” Technological Forecasting and Social Change, vol. 141, pp. 354–358, 2019.
  * [18] D. Helbing, B. S. Frey, G. Gigerenzer, E. Hafen, M. Hagner, Y. Hofstetter, J. Van Den Hoven, R. V. Zicari, and A. Zwitter, “Will democracy survive big data and artificial intelligence?,” in Towards digital enlightenment, pp. 73–98, Springer, 2019.
  * [19] J. Ahlbrandt, D. Brammen, R. W. Majeed, R. Lefering, S. C. Semler, S. Thun, F. Walcher, and R. Röhrig, “Balancing the need for big data and patient data privacy–an it infrastructure for a decentralized emergency care research database.,” Studies in health technology and informatics, vol. 205, pp. 750–754, 2014.
  * [20] G. Ras, M. van Gerven, and P. Haselager, “Explanation methods in deep learning: Users, values, concerns and challenges,” in Explainable and Interpretable Models in Computer Vision and Machine Learning, pp. 19–36, Springer, 2018.
  * [21] C. Angles-Tafalla, A. Viejo, and J. Castella-Roca, “Privacy-preserving and secure decentralized access control system for low emission zones,” in IEEE INFOCOM 2019-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), pp. 1–6, IEEE, 2019.
  * [22] A. Soliman, S. Girdzijauskas, M.-R. Bouguelia, S. Pashami, and S. Nowaczyk, “Decentralized and adaptive k-means clustering for non-iid data using hyperloglog counters,” in Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 343–355, Springer, 2020.
  * [23] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial Intelligence and Statistics, pp. 1273–1282, PMLR, 2017.
  * [24] J. Konečnỳ, H. B. McMahan, F. X. Yu, P. Richtárik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” arXiv preprint arXiv:1610.05492, 2016.
  * [25] T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith, “Federated optimization in heterogeneous networks,” arXiv preprint arXiv:1812.06127, 2018.
  * [26] M. Mohri, G. Sivek, and A. T. Suresh, “Agnostic federated learning,” arXiv preprint arXiv:1902.00146, 2019.
  * [27] M. Yurochkin, M. Agarwal, S. Ghosh, K. Greenewald, T. N. Hoang, and Y. Khazaeni, “Bayesian nonparametric federated learning of neural networks,” arXiv preprint arXiv:1905.12022, 2019.
  * [28] H. Wang, M. Yurochkin, Y. Sun, D. Papailiopoulos, and Y. Khazaeni, “Federated learning with matched averaging,” arXiv preprint arXiv:2002.06440, 2020\.
  * [29] R. Pascanu, T. Mikolov, and Y. Bengio, “On the difficulty of training recurrent neural networks,” in Proceedings of the 30th International Conference on International Conference on Machine Learning - Volume 28, ICML’13, p. III–1310–III–1318, JMLR.org, 2013.
  * [30] O. L. Mangasarian and W. H. Wolberg, “Cancer diagnosis via linear programming,” tech. rep., University of Wisconsin-Madison Department of Computer Sciences, 1990.
  * [31] W. H. Wolberg and O. L. Mangasarian, “Multisurface method of pattern separation for medical diagnosis applied to breast cytology.,” Proceedings of the national academy of sciences, vol. 87, no. 23, pp. 9193–9196, 1990.
  * [32] D. Dua and C. Graff, “UCI machine learning repository,” 2017.
  * [33] L. E. Mello, A. Suman, C. B. Medeiros, C. A. Prado, E. G. Rizzatti, F. L. S. Nunes, G. F. Barnabé, J. E. Ferreira, J. Sá, L. F. L. Reis, L. V. Rizzo, L. Sarno, R. de Lamonica, R. M. d. B. Maciel, R. M. Cesar-Jr, and R. Carvalho, “Opening Brazilian COVID-19 patient data to support world research on pandemics,” July 2020.  Preprint available at https://doi.org/10.5281/zenodo.3966427.
  * [34] H. Sak, A. W. Senior, and F. Beaufays, “Long short-term memory recurrent neural network architectures for large scale acoustic modeling,” Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH, pp. 338–342, 01 2014.
  * [35] L. Real, M. Oshiro, and A. Mafra, “B2w-reviews01: An open product reviews corpus,” in XII Symposium in Information and Human Language Technology (STIL2019) (C. A. Prolo and L. H. M. de Oliveira, eds.), pp. 200–208, 2019.  Dataset available at https://github.com/b2wdigital/b2w-reviews01.
  * [36] A. Vedaldi and K. Lenc, “Matconvnet: Convolutional neural networks for matlab,” in Proceedings of the 23rd ACM international conference on Multimedia, pp. 689–692, 2015.
  * [37] A. Krizhevsky, G. Hinton, et al., “Learning multiple layers of features from tiny images,” tech. rep., University of Toronto, 2009.