# Constraining Weil-Petersson volumes by universal random matrix correlations
in low-dimensional quantum gravity

Torsten Weber  Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany  Fabian Haneder  Institut für Theoretische Physik,
Universität Regensburg, D-93040 Regensburg, Germany  Klaus Richter  Institut
für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
Juan Diego Urbina  Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany

###### Abstract

Based on the discovery of the duality between Jackiw-Teitelboim quantum
gravity and a double-scaled matrix ensemble by Saad, Shenker and Stanford in
2019, we show how consistency between the two theories in the universal Random
Matrix Theory (RMT) limit imposes a set of constraints on the volumes of
moduli spaces of Riemannian manifolds. These volumes are given in terms of
polynomial functions, the Weil-Petersson volumes, solving a celebrated
nonlinear recursion formula that is notoriously difficult to analyze. Since
our results imply linear relations between the coefficients of the Weil-
Petersson volumes, they therefore provide both a stringent test for their
symbolic calculation and a possible way of simplifying their construction. In
this way, we propose a long-term program to improve the understanding of
mathematically hard aspects concerning moduli spaces of hyperbolic manifolds
by using universal RMT results as input.

##  1 Introduction

While initially, the methods and concepts of quantum chaos attempted to
explain how chaos in a classical system finds its way into the observed
universality of short-range spectral fluctuations in the corresponding
quantised version [1, 2, 3], since its precise formulation in the 1980’s [4]
the connection between dynamical chaos and Random Matrix Theory (RMT) has also
offered some deep insights into mathematical questions. A paradigmatic example
of how such an approach works is number theory. Here, one starts from the
conjectured RMT-like statistical distribution of the non-trivial zeroes of the
Riemann zeta function [5], supported by a huge amount of numerical evidence
[6], and uses it to obtain number-theoretical results [7], even defining a
whole research program.

The rationale of this approach can be exported to any field and regime of
parameters where fidelity to RMT is expected to hold. Then, the universal –
and usually tractable – RMT results can be considered as constraints, imposing
relations between physical objects of the theory, similar to how the
statistical correlations of prime numbers are constrained by RMT. This is
particularly the case for theories where the microscopic mechanism responsible
for classical chaotic dynamics is not well understood, and therefore the well-
developed machinery of periodic orbit theory [1, 8, 9, 10] cannot be invoked
to explain RMT-like features.

Remarkably, certain aspects of quantum gravity fall into this category. This
is because, although the precise connection between periodic orbit theory and
the conjectured chaotic character of important quantum gravitational models
[11, 12, 13, 14, 15, 16] is still an open problem, an exact mapping between 2D
dilaton gravity (so-called Jackiw-Teitelboim gravity [17, 18]) and a matrix
model [19] has recently been discovered, creating an explosion of interest
[20, 21, 22, 23, 24].

The present paper (and its companion [25]) follows the route depicted above.
We invoke the well justified _assumption_ that in a certain precise limit,
spectral correlations computed from the exact solution of JT gravity are
identically given by the universal RMT results. Imposing such an equivalence
then constrains the objects appearing on the JT side, which in this case turn
out to be related to the moduli space of two-dimensional manifolds. Our
objective is to make these constraints explicit, and to initiate the study of
their structure and consequences. Doing so, we provide further evidence for
the equivalence of JT gravity and universal RMT, and in particular the
operational meaning of the “universal limit” as proposed in [21] and further
developed in [26].

To begin with, we now briefly present the main features of Jackiw-Teitelboim
gravity, which is a two-dimensional theory of gravity coupled to a dilaton
$\phi$, described by the action

$S_{\text{JT}}=-\frac{S_{0}}{2\pi}\chi(\mathcal{M})-\frac{1}{2}\int_{\mathcal{M}}\sqrt{g}\phi(R+2)-\int_{\partial\mathcal{M}}\sqrt{h}\phi(K-1).$
(1.1)

The first term is the Euler characteristic of the manifold $\mathcal{M}$,
multiplied by a constant $S_{0}\sim\frac{1}{G_{N}}$, and causes the path
integral over (1.1) to decompose into a genus expansion of the form

$\expectationvalue{Z(\beta_{1})\ldots
Z(\beta_{n})}=\sum_{g=0}^{\infty}e^{(2-2g-n)S_{0}}\int_{0}^{\infty}b_{1}\differential{b_{1}}\ldots
b_{n}\differential{b_{n}}Z_{\text{tr}}(\beta_{1},b_{1})\ldots
Z_{\text{tr}}(\beta_{n},b_{n})V_{g,n}(b_{1},\ldots,b_{n}),$ (1.2)

where $Z_{\text{tr}}(\beta,b)$ is called the trumpet partition function (the
precise form of which is not important here), and
$V_{g,n}(b_{1},\ldots,b_{n})$ are the Weil-Petersson volumes, detailed further
below. Eq. (1.2) holds except for special cases111The $n=1$ result is the disk
$Z_{\text{disk}}(\beta)=e^{S_{0}}\frac{\gamma^{\frac{3}{2}}e^{\frac{2\pi^{2}\gamma}{\beta}}}{(2\pi)^{\frac{1}{2}}\beta^{\frac{3}{2}}}$,
while for $n=2$, one has the “double-trumpet”, which can be written as part of
(1.2) by formally defining
$V_{0,2}(b_{1},b_{2})=\delta(b_{1}^{2}-b_{2}^{2})$., the genus 0 contributions
for $n=1,2$.

This theory has received a lot of attention in recent years due to its utility
in studying black holes [27, 28, 29, 30, 31, 32, 33], but also owing to the
convenient computability of $n$-point functions, and therein the appearance of
wormhole geometries. The latter are of central importance to the factorisation
problem [34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Several limits,
deformations, generalisations and applications have also been studied [44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63] after
a duality between JT gravity and a certain double-scaled Hermitian matrix
model was established in [19]. To be more precise, the authors of Ref. [19]
found a matrix model defined by some potential $V(H)$, and performed the limit
$\dim H=N\to\infty$ in such a way that the leading spectral density remained
normalised, yielding

$\rho_{0}^{t}(E)=e^{S_{0}}\frac{\gamma}{2\pi^{2}}\sinh(2\pi\sqrt{2\gamma E}).$
(1.3)

In the course of taking this limit, the usual perturbative $1/N$ expansion of
the matrix model is replaced by an expansion in $e^{-S_{0}}$, e.g. for
correlators of the partition function $Z(\beta)=\tr e^{-\beta H}$,

$\expectationvalue{Z(\beta_{1})\ldots
Z(\beta_{n})}=\sum_{g=0}^{\infty}e^{(2-2g-n)S_{0}}Z_{g,n}(\beta_{1},\ldots,\beta_{2}).$
(1.4)

It is this genus expansion that has been shown to exactly compute (1.2).
However, establishing the translation of nonperturbative aspects of the matrix
model to JT gravity has proven more challenging (see however [64, 65, 66,
67]). A key reason for this is that the matrix model of [19] suffers from a
nonperturbative instability, meaning that the integration contour of the
matrix integral must be deformed. This process is not unique, however, and
thus leads to an ambiguity in the nonperturbative completion. Much work has
been done on trying to give JT gravity a rigorous nonperturbative definition
and to study its features, particularly by Johnson [68, 69, 70, 71, 72, 73,
74, 75, 76, 77], and more recently, a nonperturbative completion in terms of
Kodaira-Spencer theory has been given [78, 79].

In light of the amount of interest particularly the nonperturbative sector of
the JT matrix model has received, it seems useful to look at concrete
nonperturbative features of the matrix model and study how they are realised
in JT gravity. Particularly, we want to focus on what we call the universal
limit of the matrix model222We sometimes refer to this limit as the RMT limit,
or particularly when talking about JT, the $\tau$-scaling limit. (see e.g.
[80]). This is the limit of the matrix model in which correlation functions
are given by the ones obtained in the appropriate Altland-Zirnbauer ensemble
(in the present case, the Gaussian unitary ensemble). We will describe how to
access this limit in JT gravity below, but for now it is sufficient to recall
that in this limit, correlation functions of e.g. the level density are
described by universal, _finite_ functions. From this latter property alone,
we will be able to derive very nontrivial identities between coefficients of
the _Weil-Petersson volumes_ , i.e. the polynomials $V_{g,n}$ appearing in
(1.2).

The Weil-Petersson (WP) volumes [81] describe the volume of the moduli space
of hyperbolic surfaces with genus $g$ and $n$ geodesic boundaries with lengths
$b_{1},\ldots,b_{n}$. In principle, they are computable individually from
scratch by performing all possible decompositions of the manifold in terms of
three-holed spheres, parametrising these decompositions by Fenchel-Nielsen
coordinates $(l_{i},\tau_{i})$ and integrating the Weil-Petersson form in
these coordinates $\omega_{\text{WP}}=dl_{i}\wedge d\tau_{i}$ over the moduli
space, while modding out the mapping class group of the surface to account for
overcounting the decompositions.

In practice, however, this is not feasible and one uses Mirzakhani’s recursion
relation [81] satisfied by the $V_{g,n}$ instead. While this is doable, the
effort required to determine the volumes particularly for high $g$ or $n$ is
still substantial, and a method of relating the different terms in the WP
volumes might be useful to simplify their calculation.

In this paper then, we will show that the existence of a finite universal
limit of the JT matrix model implies constraints on the coefficients appearing
in the WP volumes $V_{g,2}(b_{1},b_{2})$ determining the JT gravity 2-point
function. To this end, we will begin in section 2 by computing the spectral
form factor in the universal limit, using the full, non-perturbative result
for the spectral two-point function. Then, we will redo the calculation in
section 3 using the genus expansion, and compare the two results. Demanding
that they are at least compatible (i.e. that the perturbative result does not
diverge more strongly than the nonperturbative one) will yield constraints
relating certain coefficients appearing in a given $V_{g,2}$. Finally, in
section 4, we discuss the possible relation of our results to other areas of
research. As a last note, the same cancellations as the ones we report have
been found independently using intersection theory computations in [25].

##  2 The late time spectral form factor from random matrix universality

In this section, we will compute the late time spectral form factor (SFF) of
the matrix model of [19]. To do so, recall that the two-point function of
$Z(\beta)=\tr e^{-\beta H}$ can be expressed as a Laplace transform,

$\expectationvalue{Z(\beta_{1},\beta_{2})}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\expectationvalue{\rho^{t}(E_{1})\rho^{t}(E_{2})}e^{-\beta_{1}E_{1}-\beta_{2}E_{2}}\differential{E_{1}}\differential{E_{2}},$
(2.1)

where $\rho^{t}(E)$ is the _total_ spectral density in the convention of [19],
i.e. it is of order $e^{S_{0}}$. This yields a useful form for the SFF,

$\kappa_{\beta}(t)=\int_{-\infty}^{\infty}\differential Ee^{-2\beta
E}\int_{-\infty}^{\infty}\differential{\Delta}e^{-it\Delta}\expectationvalue{\rho^{t}\quantity(E+\frac{\Delta}{2})\rho^{t}\quantity(E-\frac{\Delta}{2})}.$
(2.2)

Next, we study the universal behaviour of $\kappa_{\beta}(t)$ by evaluating it
at times scaling with the Heisenberg time $e^{S_{0}}$. We define a rescaled
time

$\tau:=e^{-S_{0}}t,$ (2.3)

and obtain

$\displaystyle\kappa_{\beta}(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}\differential Ee^{-2\beta
E}\int_{-\infty}^{\infty}\differential{\Delta}e^{-ie^{S_{0}}\tau\Delta}\expectationvalue{\rho^{t}\quantity(E+\frac{\Delta}{2})\rho^{t}\quantity(E-\frac{\Delta}{2})}$
(2.4) $\displaystyle=e^{2S_{0}}\int_{-\infty}^{\infty}\differential
Ee^{-2\beta
E}\int_{-\infty}^{\infty}\differential{\Delta}e^{-ie^{S_{0}}\tau\Delta}\expectationvalue{\rho\quantity(E+\frac{\Delta}{2})\rho\quantity(E-\frac{\Delta}{2})},$

making the dependence on $e^{S_{0}}$ explicit in the second line. The rapidly
oscillating factor $e^{-ie^{S_{0}}\tau\Delta}$ localises the integral near
small $\Delta$ (more precisely, of order $e^{-S_{0}}$), making it sufficient
to evaluate
$\expectationvalue{\rho(E+\frac{\Delta}{2})\rho(E-\frac{\Delta}{2})}$ for
small differences in the arguments.

To find an expression for this, we utilise the universal RMT limit of the
matrix model. Note that we leave the firm ground established in [19] here,
since the universal limit of RMT correlation functions is a nonperturbative
result, while the duality uncovered in [19] is only at the perturbative
level333A nonperturbative completion of this kind can however be interpreted
in the context of minimal string theory in terms of D-brane insertions [19,
82]. In particular, this is consistent with the nonperturbative completion in
terms of Kodaira-Spencer universe field theory [79, 78]. However, our
computation does not rely on a particular (stringy) UV completion of JT
gravity or a specific nonperturbative matrix model, but only on the existence
of a perturbatively dual matrix model of GUE symmetry class.. To finally
express the universal limit, we introduce a different, useful notation [80],
444Note that it is not possible to take the “usual” universal limit
$N\to\infty$, since this has already been performed in the course of
implementing the double scaling limit, where $N$ is replaced by $e^{S_{0}}$.
Hence, we identify the appropriate universal limit for the double scaled
theory as $e^{S_{0}}\to\infty$, and adapt the expressions in [80] accordingly.

$\displaystyle R_{1}(E)=e^{S_{0}}\expectationvalue{\rho(E)}$ (2.5)
$\displaystyle
R_{2}(E_{1},E_{2})=e^{2S_{0}}\expectationvalue{\rho(E_{1})\rho(E_{2})}-e^{S_{0}}\delta(E_{1}-E_{2})\expectationvalue{\rho(E_{1})}.$
(2.6)

Accordingly, the connected two-point correlation function is given by

$\displaystyle C_{2}(E_{1},E_{2})$
$\displaystyle=R_{1}(E_{1})R_{2}(E_{2})-R_{2}(E_{1},E_{2})$ (2.7)
$\displaystyle=e^{2S_{0}}\expectationvalue{\rho(E_{1})}\expectationvalue{\rho(E_{2})}+e^{S_{0}}\delta(E_{1}-E_{2})\expectationvalue{\rho(E_{1})}-e^{2S_{0}}\expectationvalue{\rho(E_{1})\rho(E_{2})}.$
(2.8)

The universal limit, which is valid for all matrix models of unitary symmetry
class, now takes the form

$\displaystyle\lim\limits_{e^{S_{0}}\to\infty}\frac{C_{2}(E_{1},E_{2})}{R_{1}(E_{1})R_{2}(E_{2})}=\frac{\sin[2](\pi\expectationvalue{\rho(E)}e^{S_{0}}\Delta)}{\pi^{2}\expectationvalue{\rho(E)}^{2}e^{2S_{0}}\Delta^{2}},$
(2.9)

where $e^{S_{0}}\Delta:=\tilde{\Delta}$ is kept finite in accordance with the
above requirement of evaluating the correlation function for energy difference
of the order of the inverse Heisenberg time555Note that the same limit is
considered, and the same universal result used in [83]. However, the
cancellation they observe for the 2-point function between perturbative and
nonperturbative divergent pieces is not the same as the cancellation we will
observe. In particular, the disconnected term in (2.6) vanishes on its own in
the SFF when taking the limit.. This also justifies the replacement
$\expectationvalue{\rho(E_{1})}\approx\expectationvalue{\rho(E_{2})}\approx\expectationvalue{\rho(E)}$.
Using the explicit expressions and taking the universal limit, we can solve
for the desired correlation function,

$\displaystyle\expectationvalue{\rho\quantity(E+\frac{\Delta}{2})\rho\quantity(E-\frac{\Delta}{2})}=\expectationvalue{\rho(E_{1})}\expectationvalue{\rho(E_{2})}+\delta\quantity(\tilde{\Delta})\expectationvalue{\rho(E)}-\frac{\sin[2](\pi\expectationvalue{\rho(E)}\tilde{\Delta})}{\pi^{2}\tilde{\Delta}^{2}}.$
(2.10)

We can now plug this back into the SFF eq. 2.2 to find

$\displaystyle\kappa_{\beta}(\tau)=$
$\displaystyle\expectationvalue{Z(\beta+ie^{S_{0}}\tau)}\expectationvalue{Z\quantity(\beta-
ie^{S_{0}}\tau)}$ (2.11)
$\displaystyle+e^{S_{0}}\int_{-\infty}^{\infty}\differential Ee^{-2\beta
E}\int_{-\infty}^{\infty}\differential{\tilde{\Delta}}e^{-i\tau\tilde{\Delta}}\quantity[\delta\quantity(\tilde{\Delta})\expectationvalue{\rho(E)}-\frac{\sin[2](\pi\expectationvalue{\rho(E)}\tilde{\Delta})}{\pi^{2}\tilde{\Delta}^{2}}]$
$\displaystyle=$
$\displaystyle\expectationvalue{Z(\beta+ie^{S_{0}}\tau)}\expectationvalue{Z\quantity(\beta-
ie^{S_{0}}\tau)}+e^{S_{0}}\int_{E_{0}}^{\infty}\differential Ee^{-2\beta
E}\min\quantity{\frac{\tau}{2\pi},\expectationvalue{\rho(E)}},$

with $E_{0}$ the left edge of the spectrum of $\rho(E)$. At this point, a
comment on our notion of taking the limit is in order. Although taken to
infinity in the strict universal limit, $e^{S_{0}}$ is still around. This can
be understood in the sense of isolating the most strongly divergent term in
the limit $e^{S_{0}}\to\infty$. By the same argument, one immediately
concludes that the disconnected term is subleading compared to the integral in
(2.11). Likewise, it is enough to consider the genus 0 part of
$\expectationvalue{\rho(E)}$. With these simplifications, it is possible to
give an explicit result for the JT gravity SFF:

$\kappa_{\beta}(\tau)=e^{S_{0}}\int_{0}^{\infty}\differential Ee^{-2\beta
E}\min\quantity{\frac{\tau}{2\pi},\rho_{0}(E)}.$ (2.12)

This result has been first reported in [21] and evaluated in [26]. Important
for us is that $\kappa_{\beta}(\tau)$ is finite at finite $\tau$.

##  3 Scaling: Power counting considerations

In the previous section, we derived an expression for the late time SFF of JT
gravity from nonperturbative information about the dual matrix model. Another
way to compute this quantity would be to use the JT genus expansion (which is
equivalent to the perturbative expansion of the matrix model) and then perform
the universal and late time (“$\tau$-scaling”) limit
$e^{S_{0}}\to\infty,\,t\to\infty,\,\tau=e^{-S_{0}}t=\text{const.}$

In this limit, the SFF (as calculated from the genus expansion) takes the form

$e^{S_{0}}\left(a_{1}\tau+a_{3}\tau^{3}+a_{5}(\beta)\tau^{5}+\ldots\right),$
(3.1)

with coefficients independent of $e^{S_{0}}$. The $a_{i}(\beta)$ specify a
convergent series666Technically, a series with finite radius of convergence,
which can be smoothly continued to $\tau\to\infty$. which, importantly, agrees
with the Taylor expansion of the result (2.12) around $\tau=0$ and moreover
gives the exact SFF for JT gravity in the $\tau$-scaling limit, including the
plateau [26].

In order to systematically compute the contributions to (3.1), we employ the
polynomial structure of the WP volumes [81]:

$V_{g,2}(b_{1},b_{2})=\sum_{\vec{\alpha}}^{\absolutevalue{\vec{\alpha}}\leq
3g-1}C^{(g)}_{\vec{\alpha}}b_{1}^{2\alpha_{1}}b_{2}^{2\alpha_{2}}$ (3.2)

with non-negative constants777Here and thereafter, we will suppress the
dependence of the constants on $g$ for readability.
$C_{\vec{\alpha}}\in\pi^{6g-2-2\absolutevalue{\vec{\alpha}}}\cdot\mathbb{Q}$.
Here, $\vec{\alpha}=(n,m)\in\mathds{N}$ with
$\absolutevalue*{\vec{\alpha}}\coloneqq n+m$. The contributions to the SFF
from each term appearing in (3.2) can be straightforwardly evaluated (all one
has to do is integrate a monomial against a Gaussian), yielding contributions
to the SFF of the form

$C_{n,m}\quantity(\kappa_{\beta}^{g}(t))^{n,m}=C_{n,m}\frac{\sqrt{\beta^{2}+t^{2}}}{\pi}\frac{n!m!2^{n+m-1}}{\gamma^{n+m+1}}\sum_{k=0}^{n}\sum_{j=0}^{m}\binom{m}{j}\binom{n}{k}i^{j+k}\quantity(-1)^{j}\beta^{m+n-k-j}t^{j+k}.$
(3.3)

We now consider these contributions for some fixed $g$ in the $\tau$-scaling
limit. For given $(n,m)$, take for example the leading term in the sum (3.3),
scaling like

$t^{n+m+1}=e^{(n+m+1)S_{0}}\tau^{n+m+1}.$ (3.4)

Meanwhile, every $\quantity(\kappa_{\beta}^{g}(t))^{n,m}$ is suppressed by a
factor $e^{-2gS_{0}}$ due to the genus expansion. As an example, the highest
order term coming from $n+m=3g-1$ contributes to the SFF to the order

$\displaystyle e^{-2gS_{0}}e^{3gS_{0}}\tau^{3g}=e^{gS_{0}}\tau^{3g}.$ (3.5)

The universal part of the SFF emerges by performing the $\tau$-scaling limit
and extracting the leading term, which should be of order $e^{S_{0}}$. Clearly
however, there are contributions to the $g>1$ SFF (or rather, to
$e^{-S_{0}}\mathrm{SFF}$, for given $n,m$) that diverge in this limit. From
this observation, we pose the central claim of this work:

_The finiteness of the universal part of the SFF, identified by the
$\tau$-scaling limit, requires that all terms scaling as $e^{(1+n)S_{0}}$
mutually cancel._888It is conceivable that nonperturbative effects cancel
perturbative divergences instead. However, none of the nonperturbative
corrections known to the authors are simple powers of the expansion parameter,
but rather terms of order e.g. $e^{-e^{S_{0}}}$. Corrections of this type
could not cancel perturbative divergences (which are $\order{e^{-nS_{0}}}$).
This suggests that our prediction still holds, and indeed that the
perturbative and nonperturbative parts of the SFF are individually finite. All
results we have obtained until now are consistent with this claim.

Let us elaborate on this claim: In order for different terms of (3.3) to be
able to cancel, they must have the same order in $e^{S_{0}},$ $\beta$ and
$\tau$. It is easy to see that this can only happen for terms with common $g$
and $n+m\equiv\absolutevalue{\vec{\alpha}}$. Hence, we need to sum up the
contributions from all partitions $(n,m)$ of $\absolutevalue{\vec{\alpha}}$
for fixed relevant $g$, and indeed we need to do this separately for each
relevant999Relevant in this context means $g>1$ and $j+k>2g$ in the sum (3.3).
Note also that terms with $j+k<2g$ are suppressed in the $\tau$-scaling limit,
while $j+k=2g$ provides precisely the universal part. power of $\beta$ and
$\tau$. Doing so will provide constraint equations that some set of
coefficients $C_{n,m}$ of the WP volumes need to satisfy.

Consider for example any relevant fixed $g>1$ and
$\absolutevalue{\vec{\alpha}}$. The first equation we obtain from the above
procedure (taking $j+k=m+n$) is

$\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!(-1)^{m}=\sum_{\begin{subarray}{c}n\geq
m\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}\frac{1}{2}^{\delta(n,m)}C_{n,m}n!m!\left[(-1)^{n}+(-1)^{m}\right]=0.$
(3.6)

A well-known property of the WP volumes is that the coefficients are
symmetric, $C_{n,m}=C_{m,n}$. We can thus immediately conclude that (3.6) is
always satisfied for odd $n+m$, as then the $(n,m)$ term automatically cancels
with $(m,n)$.

However, for even $n+m$, this trivial cancellation will not occur, and (3.6)
is a nontrivial constraint on the WP coefficients. To check this claim for
plausibility, let us construct the first example of such a nontrivial
cancellation. For $g=2$, the leading equation for maximal $n+m=3g-1=5$ is
trivial, while the next-to-leading equations will contribute at order
$e^{(5-2g)S_{0}}=e^{S_{0}}$, and hence provide the universal part.

The first nontrivial example thus requires $g=3$, and again maximal
$n+m=3g-1=8$. Taking the contributions from $j+k=8$ also maximal, we find the
following constraint equation:

$280C_{8,0}-35C_{7,1}+10C_{6,2}-5C_{5,3}+2C_{4,4}=0$ (3.7)

We can check this by plugging in the coefficients given by [84]

$\displaystyle C_{8,0}=\frac{1}{856141332480}$
$\displaystyle,\leavevmode\nobreak\ C_{7,1}=\frac{1}{21403533312}$
$\displaystyle,\leavevmode\nobreak\ C_{6,2}=\frac{77}{152882380800}$ (3.8)
$\displaystyle C_{5,3}=\frac{503}{267544166400}$
$\displaystyle,\leavevmode\nobreak\ C_{4,4}=\frac{607}{214035333120},$

which indeed solve (3.7). One can now leave $g$ fixed and vary $n+m$,
obtaining a similar equation for a different set of coefficients. In the
example at hand however, we would only find the trivial cancellation for
$n+m=7$, as well as the result $n+m=6$, which provides the $\beta$-independent
part of the universal result.

However, we could also leave e.g. $n+m=8$ fixed, and simply take terms with
$j+k<8$, i.e. contributing at a lower – but still divergent – order in
$e^{S_{0}}$. In this way, we can find additional equations for the same set of
coefficients as before, taking the form

$\displaystyle\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!($
$\displaystyle-1)^{m}(n-m)=0$ (3.9)
$\displaystyle\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!($
$\displaystyle-1)^{m}(n-m)^{2}=0$ (3.10) $\displaystyle\vdots$
$\displaystyle\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!($
$\displaystyle-1)^{m}(n-m)^{l}=0$ (3.11) $\displaystyle\vdots$

If the leading order equation is nontrivial, it follows that (3.9) is
trivially satisfied by cancellation of the $(n,m)$ contribution with the
$(m,n)$ one, reminiscent of what happens for odd $n+m$ at leading order.
However, we can also see that (3.10) is a nontrivial constraint equation which
is linearly independent from (3.6)101010In particular, $C_{n,n}$ is not
involved in (3.10)..

For $g=3$, the combination in (3.10) provides the $\beta^{2}$ part of the
universal result. Likewise, the analogue of (3.9) for $n+m=7$ gives the
$\beta$-linear part. Generally however, we will obtain a hierarchy of
constraints for the coefficients belonging to a given $n+m$, requiring
alternately trivial and nontrivial cancellations, until we arrive at a
combination that contributes in the universal limit. The general form (3.11)
of these constraints is readily determined by summing up the contributions to
a given $n+m$, yielding

$\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!\sum_{\begin{subarray}{c}j,k=0\\\
j+k=\absolutevalue{\vec{\alpha}}-l\end{subarray}}\pmqty{undef}\pmqty(-1)^{j}=0,$
(3.12)

and then cleverly subtracting combinations of the constraints appearing
before, i.e. for smaller $l$. For a more rigorous proof, see appendix A.

The first example of another constraint than (3.6) appearing is for $g=4$,
where the leading order term vanishes by symmetry. The first (and only)
nontrivial constraint for the coefficients with $n+m=3g-1=11$ is given by
(3.9), which can be simplified to

$\displaystyle
3630C_{11,0}-270C_{10,1}+42C_{9,2}-10C_{8,2}+3C_{7,4}-\frac{5}{7}C_{6,5}=0.$
(3.13)

This equation, as expected, is satisfied by the coefficients obtained by
solving the topological recursion,

$\displaystyle C_{11,0}=\frac{1}{650941377911193600}$
$\displaystyle,\leavevmode\nobreak\ C_{10,1}=\frac{1}{8453784128716800}$
$\displaystyle,\leavevmode\nobreak\ C_{9,2}=\frac{149}{59176488901017600}$
(3.14) $\displaystyle C_{8,3}=\frac{947}{46026158034124800}$
$\displaystyle,\leavevmode\nobreak\ C_{6,5}=\frac{487}{3287582716723200}.$

Furthermore, for $g=4$, we find the first constraint on coefficients
associated to a non-maximal value of $n+m=3g-2=10$. The constraint is of the
type (3.6) and reads

$\displaystyle-\frac{5}{56}C_{5,5}+\frac{3}{14}C_{6,4}-\frac{3}{8}C_{7,3}+C_{8,2}-\frac{9}{2}C_{9,1}+45C_{10,0}=0,$
(3.15)

which is indeed satisfied by the relevant coefficients,

$\displaystyle C_{5,5}=\frac{533\pi^{2}}{7134511104000}$
$\displaystyle,\leavevmode\nobreak\
C_{4,6}=\frac{1081\pi^{2}}{20547391979520}$
$\displaystyle,\leavevmode\nobreak\
C_{3,7}=\frac{16243\pi^{2}}{898948399104000}$ (3.16) $\displaystyle
C_{2,8}=\frac{53\pi^{2}}{19025362944000}$ $\displaystyle,\leavevmode\nobreak\
C_{1,9}=\frac{149\pi^{2}}{924632639078400}$
$\displaystyle,\leavevmode\nobreak\
C_{0,10}=\frac{23\pi^{2}}{9246326390784000}.$

It would be instructive to go to $g=5$ or higher to test these predictions
further, however the required coefficients prove very difficult to find both
by direct computation and in the literature. Let us endeavour to make some
general claims about the constraint hierarchies regardless.

First, in order for $n+m=\absolutevalue{\vec{\alpha}}$ to admit a nontrivial
constraint, we must have

$2g+1<\absolutevalue{\vec{\alpha}}\leq 3g-1.$ (3.17)

Hence, there are in total $3g-1-2g=g-1$ different sets of constrained
coefficients, i.e. $g-1$ different constraint hierarchies. In these
hierarchies, we will find in principle $n+m-2g$ linearly independent
constraints, but every other one of those will be trivially satisfied. This
leaves the number of nontrivial constraints for given $n+m$ as

$\displaystyle\frac{n+m}{2}-g$ $\displaystyle:n+m\mathrm{\leavevmode\nobreak\
even}$ (3.18) $\displaystyle\frac{n+m-1}{2}-g$
$\displaystyle:n+m\mathrm{\leavevmode\nobreak\ odd}.$

An interesting question might be how strongly these equations constrain the WP
coefficients. The answer, unfortunately, is ‘not very’. To this end, note that
the number of constrained coefficients in a given hierarchy is equivalent to
the number of (unordered) pairwise partitions of an integer
$n+m=\absolutevalue{\vec{\alpha}}$. This number is clearly $\frac{n+m}{2}+1$
for even $\absolutevalue{\vec{\alpha}}$ and $\frac{n+m+1}{2}$ for odd
$\absolutevalue{\vec{\alpha}}$. Hence, we observe that there are

$\\#\mathrm{\,variables}-\\#\mathrm{\,constraints}=1+g$ (3.19)

coefficients more per hierarchy than can be determined by the constraints.
Notably, this number is independent of $n+m$, but it still tells us that there
will never be enough equations to actually determine any of the WP
coefficients unambiguously, and indeed the problem gets worse as you increase
$g$.

##  4 Discussion

To summarise, we have shown that the Weil-Petersson volumes, which arise as
natural objects in the JT gravity path integral, obey some rather nontrivial
constraints on their coefficients. These constraints were predicted using
nonperturbative, universal information from the dual matrix model. It is this
universality that is one of the strengths of our result: we did not need to
rely on a particular nonperturbative completion either on the matrix model or
on the gravity side. All that was required is the universal limit of the
matrix model correlation function we studied, the spectral form factor.

The striking success of applying RMT universality in the JT/RMT context
suggests several further research directions. An obvious and interesting
generalisation of our results would be to compute higher $n$-point functions
and see if there are similar constraints on the WP volumes appearing therein.
However, identifying the correct analogue of the $\tau$-scaling limit that we
used for the SFF has proven somewhat difficult so far. A possible resolution
would be that the genus expansion for $n>2$ simply cannot capture the
universal limit, as it does in the $n=2$ case; a result that would be curious
since there does not seem to be anything special about the 2-point function
that makes our prediction hold there and not elsewhere.

Assuming that it is possible to generalise our results to higher $n$, another
obvious target would be to try and prove the cancellations directly from the
Mirzakhani recursion or (perhaps more simply, and certainly more generally)
the topological recursion. As evident from the coefficients cited above, the
WP volumes encode information about hyperbolic surfaces in a highly nontrivial
manner, and the resulting coefficients need to conspire very precisely to be
able to produce a finite result in the $\tau$-scaling limit. It is not far-
fetched then to suspect there to be a mechanism _at the level of the
recursion_ to ensure that the coefficients take values that are compatible
with a finite universal limit.

Possibly useful tools to investigate such a mechanism could be provided by
intersection theory, which has been used in [25] to identify the same
cancellations as we have reported in this work. Some attention has also been
directed towards resonance and resurgence of the JT gravity genus expansion
[85, 86], though mainly at the level of the free energy, and it is not
immediately clear whether such considerations could be profitably used for the
questions at hand.

Finally, since the physical argument that led us to identifying the constraint
relations – the existence of a finite universal limit of the matrix model dual
– isn’t particularly fine-tuned to JT gravity (see e.g. [46]), it seems
expedient to try to apply our reasoning to other gravitational models that are
often studied in conjunction with JT, for example Liouville quantum gravity or
minimal string models. Doing so might help better elucidate the relation of
universality and (quantum) gravity if indeed there is any. Another option
would be to simply work directly with matrix models defined by the topological
recursion, e.g. the one constructed from the spectral density $\tanh\sqrt{E}$.
This one is particularly interesting because it seems like the more natural
choice for JT gravity when using its description as an
$\text{SL}(2,\mathds{R})$ $BF$-theory111111We thank Thomas Mertens for
explaining this point to us.. Here, the metric information is encoded in a
flat $\text{SL}(2,\mathds{R})$ connection, for which the natural Plancherel
density is the hyperbolic tangent, rather than the hyperbolic sine. However,
restricting to smooth geometries (cf. [87]), one selects only one component of
the connection, exchanging the $\tanh$ for a $\sinh$ in the process (i.e.
going over to a description as a $\text{SL}^{+}(2,\mathds{R})$ $BF$-theory)
[88]. A similar issue appears in the “quantum particle in $AdS_{2}$”
description of [89], where summing over different $\text{SL}(2,\mathds{R})$
representations, the spectral density tracks a kind of winding number that
would correspond to singular geometries in JT gravity, hence requiring
regularisation in the form of an infinite imaginary magnetic field in the
quantum mechanical system to arrive at the JT result.

##  5 Conclusion

We have here proposed the systematic use of universal RMT results to uncover
new relations among functions defined over the moduli space of 2-dimensional
manifolds. The essence of this program is the conjectured existence – well
motivated on physical grounds – of a regime where a strict equivalence between
low-dimensional quantum gravity models and the universal correlations given by
RMT holds.

As a first key step in this direction, a particular form of this equivalence
has been made explicit in [26] by identifying the regime of parameters where
the 2-point functions of Jackiw-Teitelboim gravity are conjectured to be given
by the corresponding RMT correlators for the GUE ensemble. Assuming the
validity of this conjecture, that includes both perturbative and non-
perturbative contributions, we found hitherto unknown identities among the
numerical factors of the polynomial Weil-Petersson volumes on the JT side.

Since the non-trivial correctness of the identities for $n=2$ is now firmly
established and checked against exact results, the possible extension of this
program for higher order functions (and the corresponding Weil-Petersson
volumes) as well as to other types of models as discussed here is a promising
route to merge RMT with the theory of hyperbolic manifolds.

## Acknowledgements

We thank T. Mertens for illuminating discussions, and A. Blommaert, D.
Stanford, J. Kruthoff, T. Mertens and S. Yao for very helpful conversations
and for sharing some of their unpublished results during preparation of [25,
26].

We acknowledge financial support from the Deutsche Forschungsgemeinschaft
(German Research Foundation) through Ri681/15-1 within the Reinhart-Koselleck
Programme.

##  Appendix A Proof of the form of the constraint hierarchies

As stated above, for a chosen value of genus $g$ and
$\absolutevalue{\vec{\alpha}}:=n+m$ one finds a hierarchy of constraints,
given by

$\displaystyle C_{l}(\absolutevalue{\vec{\alpha}})$
$\displaystyle\coloneqq\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!\sum_{\begin{subarray}{c}j,k=0\\\
j+k=\absolutevalue{\vec{\alpha}}-l\end{subarray}}\pmqty{undef}\pmqty(-1)^{j}$
(A.1) $\displaystyle\eqqcolon\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!P_{l}(n,m),$ (A.2)

for values of $l$ such that $j+k>2g$. We proceed to prove that a linearly
independent combination of these constraints takes the form

$C_{l}^{s}(\absolutevalue{\vec{\alpha}}):=\sum_{\begin{subarray}{c}n,m=0\\\
n+m=\absolutevalue{\vec{\alpha}}\end{subarray}}C_{n,m}n!m!\quantity(-1)^{m}\quantity(n-m)^{l}.$
(A.3)

As a first step by using the property $\binom{n}{m}=\binom{n}{n-m}$ of the
binomial coefficient it is useful to rewrite the $P_{l}(n,m)$ into the more
convenient form

$\displaystyle P_{l}(n,m)$ $\displaystyle=\sum_{\begin{subarray}{c}j,k=0\\\
j+k=\absolutevalue{\vec{\alpha}}-l\end{subarray}}\binom{m}{m-j}\binom{n}{n-k}(-1)^{j}$
(A.4)
$\displaystyle=\quantity(-1)^{m}\sum_{\begin{subarray}{c}\delta,\gamma=0\\\
\delta+\gamma=l\end{subarray}}(-1)^{\gamma}\binom{m}{\gamma}\binom{n}{\delta},$
(A.5)

where $\gamma=m-j$ and $\delta=n-k$.
Next, we show that $P_{l}(n,m)$ is a polynomial of degree $l$ in m, as well as
in n. To do so, it suffices to show that each individual term in $P_{l}(n,m)$
is a polynomial. For $a<b$, it holds that

$\displaystyle\binom{b}{a}=\frac{b!}{a!\quantity(b-a)!}=\frac{1}{n!}\prod_{j=0}^{a-1}\quantity(b-j),$
(A.6)

which is a polynomial of degree $a$ in $b$. Hence, both binomial coefficients
appearing in $P_{l}(n,m)$ are polynomials in $n$ or $m$, and so is their
product. Evidently, the degree of these polynomials is $l$, whence

$\displaystyle P_{l}(n,m)$
$\displaystyle=\quantity(-1)^{m}\quantity[\binom{m}{0}\binom{n}{l}+\quantity(-1)^{l}\binom{m}{l}\binom{n}{0}+\quantity(\text{lower
order in $n,m$})]$ (A.7)
$\displaystyle=\quantity(-1)^{m}\quantity[\frac{1}{l!}n^{l}+\frac{1}{l!}\quantity(-m)^{l}+\quantity(\text{lower
order in $n,m$})]$
$\displaystyle=\quantity(-1)^{m}\quantity[\frac{1}{l!}\quantity(n-m)^{l}+\quantity(\text{lower
order in $n,m$})].$

From here, (A.3) follows immediately upon realising that the lower order
terms, plugged into (A.2), all reduce to linear combinations of constraints
for smaller $l$ (notice that $n+m$ is a global constant for each term and can
be pulled out of the sum). If the previous constraints are satisfied, they can
be safely subtracted, leaving the desired expression (A.3).

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###### Proof.

###### Proof.

.:

###### Towards Bridging the FL Performance-Explainability Trade-Off: A
Trustworthy 6G RAN Slicing Use-Case

###### Abstract

In the context of sixth-generation (6G) networks, where diverse network slices
coexist, the adoption of AI-driven zero-touch management and orchestration
(MANO) becomes crucial. However, ensuring the trustworthiness of AI black-
boxes in real deployments is challenging. Explainable AI (XAI) tools can play
a vital role in establishing transparency among the stakeholders in the
slicing ecosystem. But there is a trade-off between AI performance and
explainability, posing a dilemma for trustworthy 6G network slicing because
the stakeholders require both highly performing AI models for efficient
resource allocation and explainable decision-making to ensure fairness,
accountability, and compliance. To balance this trade off and inspired by the
closed loop automation and XAI methodologies, this paper presents a novel
explanation-guided _in-hoc_ federated learning (FL) approach where a
constrained resource allocation model and an _explainer_ exchange—in a closed
loop (CL) fashion—soft attributions of the features as well as inference
predictions to achieve a transparent 6G network slicing resource management in
a RAN-Edge setup under non-independent identically distributed (non-IID)
datasets. In particular, we quantitatively validate the faithfulness of the
explanations via the so-called attribution-based _confidence metric_ that is
included as a constraint to guide the overall training process in the run-time
FL optimization task. In this respect, Integrated-Gradient (IG) as well as
Input $\times$ Gradient and SHAP are used to generate the attributions for our
proposed in-hoc scheme, wherefore simulation results under different methods
confirm its success in tackling the performance-explainability trade-off and
its superiority over the unconstrained Integrated-Gradient _post-hoc_ FL
baseline.

###### Index Terms:

6G, closed-loop, federated learning, game theory, in-hoc, post-hoc, proxy-
Lagrangian, resource allocation, XAI, ZSM

##  I Introduction

6G networks aim to support numerous simultaneous and diverse slices for
various vertical use cases, resulting in increased complexity. Extensive
research efforts have been dedicated to AI-based autonomous management and
orchestration within the zero-touch network and service management (ZSM)
framework, standardized by the ETSI, which aims to effectively handle the end-
to-end network and services to accommodate the various services offered by 6G
networks [1, 2]. On the other hand, FL is a decentralized approach to machine
learning that allows for the training of models on distributed data without
the need to transfer the data to a central server. The ZSM framework can
incorporate FL as a key component for managing zero-touch distributed network
slices while preserving the confidentiality of sensitive data [3].
Furthermore, for the widespread adoption of AI in telecommunication networks,
addressing the lack of transparency, trust, and explainability in black-box AI
models becomes crucial [4]. The deployment of network automation requires a
thorough understanding of AI models’ decision-making and behavior. This drives
the search for AI approaches that offer understandable and explainable
outcomes. In this regard, explainable AI (XAI) methods have emerged as a means
to scrutinize the decisions made by black-box AI models. These methods build
white-box models and generate feature contributions to explain AI decisions,
promoting fairness, accuracy, and transparency. The ability to provide
confidence and trust in AI models is vital for businesses and organizations
deploying AI-enabled systems [5, 6]. Nonetheless, there exists a clear trade-
off between performance and explainability which is an ongoing challenge in
the field of machine learning. Simpler models like linear regression or
decision trees are more interpretable but may have limited predictive
performance with complex data. On the other hand, complex models like deep
neural networks can achieve higher accuracy but lack interpretability [5].
Additionally, minimizing the model loss conflicts with its explainability.

In real-world deployments, it is essential to tackle the trade-off between
performance and explainability towards ensuring trustworthy network
orchestration. To achieve this goals, we anticipate that _in-hoc_ XAI
approaches, that leverage the explanations during model’s training and
optimization, is a path to achieve that trade-off. We therefore introduce an
explanation-guided _in-hoc_ FL strategy for achieving transparent zero-touch
service management of 6G network slices at the RAN-Edge, specifically in a
non-independent identically distributed (non-IID) data.

###  I-A Related Work

This section discusses the state-of-the-art works on explainable AI (XAI) in
the telecommunications domain. AI transparency is now more critical than ever
with the advent of 6G networks, especially for zero-touch network management
[5, 7]. The "human-centric" character of 6G highlights how important it is for
XAI to win over humans and keep them in the loop [7]. The authors in [8]
specify the challenges of developing XAI methods and show causal inference in
6G technology. The importance of explainability in 5G for essential services,
device-to-device architectural security, and causal inference in 6G technology
has also been emphasized in [9, 10]. Moreover, The significance of
explainability in upcoming 6G networks has been acknowledged by researchers
[11], with applications to resolving handover and resource allocation issues.
A study [12] compares XAI techniques and suggests that SHAP is the best for
identifying the cause of SLA violations. However, the study lacks clear
explanations for the discrepancies among the results, casting doubt on the
model’s reliability. In [13], a human-understandable architecture is proposed
for a network automation task. However, no quantitative metrics are used to
evaluate the fidelity of the explanations. Thus, to resolve this gap, in [14],
an extensive range of XAI metrics is introduced for qualitative assessment,
but no integration-based XAI methods are considered, and there is also no
comparative analysis of different XAI techniques. In [5], it is emphasized
that the interpretability and transparency of AI/ML models are crucial for the
full automation of a ZSM framework in 6G networks. Recently, [15] proposes a
neuro-symbolic XAI twin framework to enhance reasoning and trustworthiness in
zero-touch IoE service management. However, they do not incorporate a
explainable-guided AI approach or address distributed learning with XAI in the
ZSM framework like us. Additionally, in [16], the concept of Federated
Learning (FL) of XAI is introduced, focusing on its importance in 5G/6G
networks. However, their work lacks quantitative validation of explanation
faithfulness. In [17], the authors utilize the SHAP method from XAI to apply a
trust-aware deep reinforcement learning-based device selection technique. This
technique is employed in an FL training process for an autonomous driving
scenario. On the other hand, several AI-based research works [18, 19, 20] have
focused on solving resource allocation issues in 6G network slices. However,
most of these works have not addressed the challenge of providing
explainability in their proposed solutions. While some research areas like NLP
and Healthcare have started recognizing the trade-off between model
performance and explainability [21], the field of telecommunication is still
in its early stages concerning this concept. However, in a fortunate
development, the authors [5] have emphasized the importance of _in-model_
explainable AI (XAI) methods for the beyond 5G/6G field. They discuss the
challenges and significance of achieving a better trade-off between
explainability and AI model performance. They advocate for developing
inherently explainable models and establishing quantifiable metrics to
evaluate the effectiveness of XAI. _In-model_ explainability aims to create
transparent models from the start, reducing reliance on post hoc XAI
techniques. This approach allows stakeholders to understand the decision-
making process or any decision while balancing performance and explainability.
In addition to the aforementioned research areas, a new research approach
called Explanation Guided Learning (EGL) has emerged, which aims to exploit
the explanations of machine learning models during the training to guide the
model towards fulfilling both trustworthiness and performance. Specifically,
EGL incorporates additional supervision signals or XAI-driven prior knowledge
into the model’s reasoning process, which helps improving the confidence of
model’s predictions [22], which might improve the balance between
explainability and AI model performance.

###  I-B Contributions

The main contributions of this paper are as follows:

  * •

We introduce a novel explanation-guided _in-hoc_ federated learning approach,
where a constrained slice-level CPU resource allocation model and an
_explainer_ exchange—in a closed loop way—explanations in the form of soft
feature attributions as well as predictions to achieve a transparent and
explainability-aware 6G network orchestration in a non-IID setup,

  * •

We adopt the integrated gradients (IG) XAI method to generate explanations in
terms of feature attributions, and map them to a soft probability space,

  * •

These soft attributions are then used to quantitatively evaluating the model’s
_confidence metric_ which is included as a constraint in the FL optimization
task.

  * •

We formulate the corresponding XAI-constrained optimization problem under the
_proxy-Lagrangian_ framework and solve it via a non-zero sum two-player game
strategy, while comparing with a vanilla unconstrained post-hoc IG FL
baseline.

  * •

We present a comparative analysis of additional XAI methods that are used to
generate attributions for the _in-hoc_ scheme and assess their confidence
metrics via distribution plots.

  * •

We showcase the impact of network conditions (channel quality indicator (CQI),
OTT traffics and MIMO full-rank usage) on the output CPU allocation.

###  I-C Notations

We summarize the notations used throughout the paper in Table I.

Table I: Notations Notation |  Description
---|---
$S_{\mu}(\cdot)$ |  Logistic function with steepness $\mu$
$F(\cdot)$ |  CDF
$\tilde{F}(\cdot)$ |  Complementary CDF
$L$ |  Number of local epochs
$T$ |  Number of FL rounds
$\mathcal{D}_{k,n}$ |  Dataset at CL $(k,n)$
$D_{k,n}$ |  Dataset size
$\ell(\cdot)$ |  Loss function
$\mathbf{W}_{k,n}^{(t)}$ |  Local weights of CL $(k,n)$ at round $t$
$\mathbf{x}_{k,n}$ |  Input features
$\hat{x}_{k,n}^{(i,j)}$ |  Mutated features
$\hat{y}_{k,n}^{(i)}$ |  Allocated resource by CL $(k,n)$
$\alpha_{n}$ |  Resource lower-bound for slice $n$
$\beta_{n}$ |  Resource upper-bound for slice $n$
$\nu_{n}$ |  Threshold of Confidence bound for slice $n$
$\mathcal{D}_{k,n}$ |  Samples whose prediction fulfills the SLA
$\hat{z}_{k,n}^{(i)}$ |  Predictions after mutation
$\pi_{k,n}^{(i,j)}$ |  Probability Distribution
$C_{k,n}$ |  Confidence Metric
$\alpha_{k,n}^{(i,j)}$ |  Weighted attribution of features
$\lambda_{(\cdot)}$ |  Lagrange multipliers
$R_{\lambda}$ |  Lagrange multiplier radius
$\mathcal{L}_{(\cdot)}$ |  Lagrangian with respect to ($\cdot$)

##  II System Model and Problem Statement

###  II-A System Model

As depicted in Fig. 1, a 6G RAN-Edge topology under a per-slice central unit
(CU)/distributed unit (DU) functional split is considered, wherein DUs are co-
located with the transmission/reception point (TRP), while each CU $k$ is a
virtual network function (VNF) running on top of a commodity hardware at the
Edge domain and implements a closed loop (CL) $k\,(k=1,\ldots,K)$ consisting
of a key performance indicators (KPIs) collection as well as AI-enabled slice
resource allocation functions. This CL concept adheres to the ZSM framework
ETSI [2], which considers autonomous feedback loops between data monitoring,
AI-driven data analytics, and decision-making functions to achieve particular
network management tasks. In this regard, the architecture entails
synthesizing data of several over-the-top (OTT) applications, which serves to
build local datasets for each slice $n\,(n=1,\ldots,N)$, i.e.,
$\mathcal{D}_{k,n}=\\{\mathbf{x}_{k,n}^{(i)},y_{k,n}^{(i)}\\}_{i=1}^{D_{k,n}}$,
where $\mathbf{x}_{k,n}^{(i)}$ stands for the input features vector, which
includes OTT traffics, channel quality indicator (CQI) and multiple-input
multiple-output (MIMO) full-rank usage, while $y_{k,n}^{(i)}$ represents the
supervised output, which is the CPU load as shown in Table II. The considered
slices include several OTTs each, namely,

  * •

eMBB: Netflix, Youtube and Facebook Video,

  * •

Social Media: Facebook, Whatsapp and Instagram,

  * •

Browsing: Apple, HTTP and QUIC,

where the corrsponding accumulated datasets are non-IID due to the different
traffic patterns induced by the heterogeneous users’ distribution and channel
conditions. Note that the intuitive link between the input features and the
output resides in the fact that when the radio conditions are enhanced, the
transmission and computing queues are relaxed, delivering thereby a higher
number of packets which also results in increasing the CPU load.

Figure 1: Decentralized closed loops (CLs) architecture

Next, to enhance model accuracy and preserve data privacy as well as reduce
transport overhead, the architecture adopts a federated learning (FL)
approach, where local closed loops (CLs) participate in training using their
respective synthesized datasets. The FL process is guided by XAI ensuring
transparency and interpretability during model training. The knowledge gained
from each local CL is then aggregated through an E2E slice-level server,
collectively improving the global AI model’s explainability and AI model
performance.

###  II-B Problem Statement

Unlike post-hoc XAI strategies, the aim of this paper is to design an
explanation-guided _in-hoc_ FL-based resource allocation scheme for
transparent zero-touch 6G network slicing at the RAN-Edge domain while
balancing the trade-off between the performance and explainability. To achieve
this, the proposed approach incorporates the XAI-based _confidence metric_ as
constraint in a closed-loop manner during the model learning process. This
allows for iterative optimization and adjustment of the model based on the
insights gained from XAI considerations. Indeed, this approach is needed to
strengthen the intelligence entities of our ZSM-based slice-level resource
management [15].

Table II: Dataset Features and Output Feature | Description
---|---
OTT Traffics per TRP |  Apple, Facebook, Instagram, NetFlix, HTTPS, Whatsapp, and Youtube
CQI |  Channel quality indicator reflecting the average quality of the radio link of the TRP
MIMO Full-Rank |  Usage of MIMO full-rank spatial multiplexing in (%)
|
Output | Description
CPU Load | CPU resource consumption (%)

##  III Proposed Approach

Towards bridging the AI performance-explainability trade-off for trustworthy
RAN slicing resource allocation, we detail in this section the proposed
architecture shown in Fig. 2.

###  III-A EGL-Driven Trustworthy Resource Allocation

Inspired by the CL and EGL principle, we propose an explanation-aided _in-hoc_
federated learning architecture where the local learning is performed
iteratively with run-time explanation. The overall working principle of the
proposed model is manifested in Fig. 2 by clearly indicating the steps. For
each local epoch, the Learner module feeds the posterior symbolic model graph
to the Tester block which yields the test features and the corresponding
predictions $z_{k,n}^{(i)}$ to the Explainer as shown from steps 1 to 3. The
latter first generates the features attributions using one of the feature
attribution XAI methods. It then converts these attributions to a soft
probability distribution, as indicated in step 4, which is translated
afterward into a confidence metric by the Confidence Mapper and fed back to
the Learner to include it in the local constrained optimization, as pointed
out in stages 5 and 6. Indeed, for each local CL $(k,n)$, the predicted amount
of resources $\hat{y}_{k,n}^{(i)},\,(i=1,\ldots,D_{k,n})$, should minimize the
main loss function with respect to the ground truth $y_{k,n}^{(i)}$ and guided
by the gained insights from XAI. Hence, as depicted in those mentioned steps 1
to 7, the optimized local weights at round $t$, $\mathbf{W}_{k,n}^{(t)}$, are
sent to the server which generates a global FL model for slice $n$ as,

$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\mathbf{W}_{k,n}^{(t)},$
(1)

where $D_{n}=\sum_{k=1}^{K}D_{k,n}$ is the total data samples of all datasets
related to slice $n$. The server then broadcasts the global model to all $K$
CLs that use it to start the next round of turbo local optimization.
Specifically, it leverages a two-player game strategy to jointly optimize over
the objective and original constraints as well as their smoothed surrogates
and detailed in the sequel.

Figure 2: _in-hoc_ FL for transparent resource allocation.

###  III-B Model Testing and Explanation

In this subsection, operational details of Model Tester and Explainer blocks
of Fig. 2 are discussed more descriptively. As depicted in stage 2 of Fig. 2,
upon the reception of the updated model graph, the Tester uses a batch drawn
from the local dataset to reconstruct the test predictions
$\mathbf{z}_{k,n}^{(i)}$. All the graph, test dataset and the predictions are
fed to the Explainer at stage 3 to generate the attributions, i.e., a
quantified impact of each single feature on the predicted output. Let
$\mathbf{a}_{k,n}^{(i)}\in\mathbb{R}^{Q}$ denote the attribution vector of
sample $i$, which can be generated by any attribution-based XAI method. In our
solution, for estimating the attributions of the considered features, we
mainly leverage the low-complexity Integrated Gradient (IG) scheme [23], which
is based on the gradient variation when sampling the neighborhood of a
feature. At stage 4, the Explainer then calculates what we call _soft
attributions_ by mapping the attributions into a probability space as follows,

${\pi_{k,n}^{(i,j)}=\frac{\exp\\{\absolutevalue{\alpha_{k,n}^{(i,j)}}\Bigr{\\}}}{\sum_{q=1}^{Q}{\exp\\{\absolutevalue{\alpha_{k,n}^{(i,q)}}\Bigr{\\}}}},\,j=1,\ldots,Q},$
(2)

where, $\alpha_{k,n}^{(i,j)}$ = $a_{k,n}^{(i,j)}/x_{k,n}^{(i,j)}$ stands for
the weighted attribution, since the sensitivity of the model’s output with
respect to an input feature in the neighborhood of the input is approximately
given by the ratio of the attribution for that input feature to the value of
that feature [24].

###  III-C Confidence Mapping

To characterize the trustworthiness of the local model, we invoke the
explanation-based confidence metric $C_{k,n}$ [24]. Its rationale lies in the
fact that slightly shifting the value of high-magnitude features (in the sens
of attributions) is an acceptable way to determine the model’s conformance.
Such an approach is vital because there will likely be no change in the SLA
group (SLA violation or non-violation) if we sample over low attribution
features space. In this respect, the _Confidence Mapper_ at stage 5 of Fig. 2
starts by performing a feature mutation, where it selects from the dataset
feature $x_{k,n}^{(i,j)}$ with probability $\pi_{k,n}^{(i,j)}$, and changes it
to the baseline value, i.e., zero,

$\hat{x}_{k,n}^{(i,j)}=x_{k,n}^{(i,j)}\times(1-p),\,\,p\sim\mathcal{B}\left(1,\pi_{k,n}^{(i,j)}\right),$
(3)

to force the change of the model’s prediction into the opposite category. In
classification tasks, the categories (or subsets) are merely the classes. In
contrast, we cast CPU resource allocation as a regression problem since it is
well-suited for predicting continuous variables, allowing the FL model to
estimate the CPU load as a numerical value. In this case, the subsets are
defined according to an SLA threshold, i.e.,

$\mathcal{D}_{k,n}=\mathcal{U}_{k,n}\cup\bar{\mathcal{U}}_{k,n},$ (4)

where $\mathcal{U}_{k,n}$ contains the samples whose prediction fulfills the
SLA, i.e., their CPU load lies in an interval $[\alpha_{n},\beta_{n}]$. The
aforementioned transformation leads to a mutated dataset
$\\{\hat{x}_{k,n}^{(i,j)}\\}$. The Confidence Mapper reports then the fraction
of samples in the neighborhood for which the decision of the model,

$\hat{z}_{k,n}^{(i)}=\mathcal{M}_{k,n}(\mathbf{W}_{k,n}^{(t)},\hat{\mathbf{x}}_{k,n}^{(i)}),$
(5)

does not move to the other set, that is, conforms to the original decision, as
the conservatively estimated confidence measure [24], i.e.,

$C_{k,n}=\frac{1}{{u_{k,n}}}\sum_{i=1}^{u_{k,n}}\max\Bigl{\\{}\mathds{1}_{\mathbb{R}^{-}}\left(\alpha_{n}-\hat{z}_{k,n}^{(i)}\right),\mathds{1}_{\mathbb{R}^{-}}\left(\hat{z}_{k,n}^{(i)}-\beta_{n}\right)\Bigl{\\}},$
(6)

where $\hat{z}_{k,n}^{(i)}$ are the predictions after mutation, while
$u_{k,n}$ stands for the size of the considered original category
$\mathcal{U}_{k,n}$.

###  III-D Explainability-Aware Resource Allocation

We cast the slice-level resource allocation as a confidence-constrained
regression task. For this purpose, we consider the datasets corresponding to
the different slices as summarized in the Table II and section II, where
resources at CU-level are dynamically allocated to slices according to their
traffic patterns and radio conditions (CQI and MIMO full-rank usage). In this
respect, we formulate the constrained optimization using the proxy-Lagrangian
framework and solve it via a non-zero sum two-player game strategy.
Specifically, the confidence metric should be higher than a predefined
threshold $\nu_{n}$. This translates into solving a statistically constrained
local resource allocation problem as shown at stage 6 of Fig. 2, i.e.,

$\min_{\mathbf{W}_{k,n}^{(t)}}\,\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right),$
(7a)

$C_{k,n}\geq\nu_{n},$ (7b)

which is solved by invoking the so-called _proxy Lagrangian_ framework [25].
This consists first on considering two Lagrangians as follows:

$\begin{split}\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}=&\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right)\\\
&+\lambda_{1}\Psi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right),\end{split}$ (8a)
$\mathcal{L}_{\lambda}=\lambda_{1}\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)$
(8b)

where $\Phi_{1}$ and $\Psi_{1}$ represent the original constraints and their
smooth surrogates, respectively. Specifically, the indicator term in (7b) are
replaced with Logistic functions and the soft maximum is used as surrogate of
the maximum as,

$\begin{split}\Psi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)=\nu_{n}-\frac{1}{u_{k,n}}\sum_{i=1}^{u_{k,n}}&\log\\{\exp[S_{\mu}\left(\hat{z}_{k,n}^{(i)}-\alpha_{n}\right)]\\\
&+\exp[S_{\mu}\left(\beta_{n}-\hat{z}_{k,n}^{(i)}\right)]\Bigl{\\}}\leq
0,\end{split}$ (9)

where $L_{\mu}$ stands for the Logistic function with steepness parameter
$\mu$, i.e.,

$S_{\mu}\left(\theta\right)=\frac{1}{1+e^{-\mu\theta}}.$ (10)

Input: $K$, $m$, $\eta_{\lambda}$, $T$, $L$. # See Table III

Server initializes $\mathbf{W}_{n}^{(0)}$ and broadcasts it to the
$\mathrm{CL}$s

for  _$t=0,\ldots,T-1$_ do

parallel for $k=1,\ldots,K$ do

Initialize $M=$ num_constraints and $\mathbf{W}_{k,n,0}=\mathbf{W}_{n}^{(t)}$

Initialize $\mathbf{A}^{(0)}\in\mathbb{R}^{(M+1)\times(M+1)}$ with
$\mathbf{A}_{m^{\prime},m}^{(0)}=1/(M+1)$

for  _$l=0,\ldots,L-1$_ do

Receive the graph $\mathcal{M}_{k,n}$ from the local model

# Test the local model and calculate the attributions

$a_{k,n}^{i,j}=$ Int. Gradient
$\left(\mathcal{M}_{k,n}\left(\mathbf{W}_{k,n,l},\mathbf{x}_{k,n}\right)\right)$

# Generate soft attributions

${\pi_{k,n}^{(i,j)}=\frac{\exp\\{\absolutevalue{\alpha_{k,n}^{(i,j)}}\Bigr{\\}}}{\sum_{p=1}^{L}{\exp\\{\absolutevalue{\alpha_{k,n}^{(i,p)}}\Bigr{\\}}}},\,j=1,\ldots,Q},$

# Mutate the test dataset

Mutate $x_{k,n}^{(i,j)}$ with probability $\pi_{k,n}^{(i,j)}$

# Calculate the confidence metric

$C_{k,n}=\frac{\sum_{i=1}^{u_{k,n}}\max\Bigl{\\{}\mathds{1}_{\mathbb{R}^{-}}\left(\alpha_{n}-\hat{z}_{k,n}^{(i)}\right),\mathds{1}_{\mathbb{R}^{-}}\left(\hat{z}_{k,n}^{(i)}-\beta_{n}\right)\Bigl{\\}}}{u_{k,n}}$

Let $\lambda^{(l)}$ be the top eigenvector of $\mathbf{A}^{(l)}$

# Solve problem (7) via oracle optimization

Let
$\hat{\mathbf{W}}_{k,n,l}=\mathcal{O}_{\delta}\left(\mathcal{L}_{\mathbf{W}_{k,n,l}}(\cdot,\hat{\lambda}^{(l)})\right)$

Let $\Delta_{\lambda}^{(l)}$ be a gradient of
$\mathcal{L}_{\lambda}(\hat{\mathbf{W}}_{k,n,l},\lambda^{(l)})$ w.r.t.
$\lambda$

# Exponentiated gradient ascent

Update
$\tilde{\mathbf{A}}^{(l+1)}=\mathbf{A}^{(l)}\odot\cdot\exp{\eta_{\lambda}\Delta_{\lambda}^{(l)}(\lambda^{(l)})}$

# Colunm-wise normalization

$\mathbf{A}_{m}^{(l+1)}=\tilde{\mathbf{A}}_{m}^{(l+1)}/\norm{\mathbf{A}_{m}^{(l+1)}}_{1},\,m=1,\ldots,M+1$

end for

return
_$\hat{\mathbf{W}}_{k,n}^{(t)}=\frac{1}{L^{\star}}\sum_{l=0}^{L-1}\hat{\mathbf{W}}_{k,n,l}$_

Each local CL $(k,n)$ sends $\hat{\mathbf{W}}_{k,n}^{(t)}$ to the server.

end parallel for

return
_$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\hat{\mathbf{W}}_{k,n}^{(t)}$_

and broadcasts the value to all local CLs.

end for

Algorithm 1 _In-hoc_ Explainable Federated Learning

This optimization task turns out to be a non-zero-sum two-player game in which
the $\mathbf{W}_{k,n}^{(t)}$-player aims at minimizing
$\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}$, while the $\lambda$-player wishes to
maximize $\mathcal{L}_{\lambda}$ [26, Lemma 8]. While optimizing the first
Lagrangian w.r.t. $\mathbf{W}_{k,n}$ requires differentiating the constraint
functions $\Psi_{1}(\mathbf{W}_{k,n}^{(t)})$, to differentiate the second
Lagrangian w.r.t. $\lambda$ we only need to evaluate
$\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)$. Hence, a surrogate is only
necessary for the $\mathbf{W}_{k,n}$-player; the $\lambda$-player can continue
using the original constraint functions. The local optimization task can be
written as,

$\begin{split}\min_{\mathbf{W}_{k,n}\in\Delta}\,\,\,\,\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}\end{split}$ (11a)
$\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\,\,\min_{\mathbf{W}_{k,n}\in\Delta}\mathcal{L}_{\lambda},$
(11b)

where thanks to Lagrange multipliers, the $\lambda$-player chooses how much to
weigh the proxy constraint functions, but does so in such a way as to satisfy
the original constraints, and ends up reaching a nearly-optimal nearly-
feasible solution [27]. These steps are all summarized in Algorithm 1.

At line 1 of Algorithm 1, we declare the baseline parameters. Then, at line 2,
the aggregation server initializes the weights and broadcasts them to all
participating CLs. Lines 3 to 4 describe the execution of the FL optimization
task, as mentioned earlier at each FL round.

##  IV Results

In this section we evaluate the proposed _in-hoc_ FL framework by first
justifying the use of feature attributions as a pillar to build the
explainability-aware constrained resource allocation model. We then present
the FL convergence, confidence score and performance metrics and showcase
their underlying trade-offs. We afterward study the correlation between
features attributions and draw some important conclusions. Finally we present
a time complexity analysis to assess the computational efficiency of the
proposed framework. To implement the model Tester and Explainer, we invoke
DeepExplain framework, which includes state-of-the-art gradient and
perturbation-based attribution methods [28]. It provides an attribution score
based on the feature’s contribution to the model’s output, which we integrate
with our proposed constrained resource allocation FL framework in a closed-
loop iterative way.

###  IV-A Parameter Settings and Baseline

Here, we define the parameters to tackle the FL optimization task. The details
of the considered slices and associated KPIs are mentioned in section II. We
use vectors $\alpha$, $\beta$ for the cpu resource bounds corresponding to the
different slices for a particular resource and $\nu$ for the explainability
confidence metrics threshold. The parameters settings are presented in Table
III. Recently, there has been growing interest in combining FL and XAI
approaches. In this work, the considered baseline is the state-of-the-art
unconstrained FL with post-hoc explanation [29]. The authors [29] investigate
the feature importance challenge in vertical FL scenarios and propose a
method, the Sharp Federated algorithm, which utilizes Shapley values to
determine feature importance in a post-hoc way.

Table III: Settings Parameter | Description | Value
---|---|---
$N$ | # Slices | $3$
$K$ | # CLs | $100$
$D_{k,n}$ | Local dataset size |  $1000$ samples
$\alpha_{n}$ | SLA lower bound |  $0$%
$\beta_{n}$ | SLA upper bound |  $3$%
$T$ | # Max FL rounds | $30$
$L$ | # Local epochs | $100$
$R_{\lambda}$ | Lagrange multiplier radius | Constrained: $10^{-5}$
$\eta_{\lambda}$ | Learning rate | $0.02$

###  IV-B Bridging the Performance-Explainability Trade-off

To study the performance-explainability trade-off of the proposed strategy, we
plot both the convergence curves and the confidence metric vs. the FL rounds
in Fig. 3 and Fig. 4, respectively, considering our _in-hoc_ FL scheme and the
baseline unconstrained IG post-hoc. It shows that the confidence metric of the
proposed in-hoc approach for the different slices conserves a higher value
above 85$\%$, while presenting a similar convergence trend as the post-hoc FL
baseline. In contrast, the latter fails to ensure the confidence of the model,
since its confidence metric decreases as we gradually approach the
convergence. This behavior conveys that the in-hoc strategy addresses the
trade-off successfully, and guarantees explainability and trust in the
training phase. For analysis completeness, the Explainer block in Fig. 2 is
also implemented using both the perturbation-based XAI method SHAP and
gradient-based method Input$\times$Gradient [30], where Fig. 4-(b) confirms
that our proposed _in-hoc_ FL algorithm preserves the same behavior during the
testing phase. Moreover, the confidence score has remained almost the same
even if the attribution scores have been generated by various XAI methods,
which makes our proposed algorithm more reliable.

Overall, based on the presented results in Fig. 3 and Fig. 4, the constrained
_in-hoc_ FL ensures a trade-off between convergence and confidence, i.e.,
while the model loss decreases within an allowable confidence threshold, the
model’s confidence increases and significantly outperforms the post-hoc
baseline. This indicates that the model becomes more confident in its
predictions as it converges. This is achieved thanks to the explanation-guided
_in-hoc_ constrained FL optimization. On the contrary, in the state-of-the-art
post-hoc scenario, the model confidence degrades as it starts to converge.

Figure 3: FL training MSE loss vs. FL rounds with $\nu=[0.82,0.83,0.85]$

(a) Model confidence metric vs. FL rounds with slices’ confidence bounds
$\nu={[}0.82,0.83,0.85{]}$. (b) Confidence metric with the proposed in-hoc
scheme on a test dataset.

Figure 4: Confidence metric using different XAI methods.

###  IV-C Explaining the Impact of Network Parameters on Slice Resource
Allocation

In this subsection, we first examine the post-hoc feature attributions plots
of the eMBB slice, which are generated via SHAP as illustrated in Fig 5. The
analysis reveals that CQI has the highest impact on the CPU allocation, as it
is widely concentrated towards positive values. This concentration indicates
that better channel quality is leading to lower CPU load, which can be
interpreted by the reduction of retransmissions and queuing time.
Consequently, the model can anticipate efficient data transmission, increased
network capacity, and potential resource optimization. Additionally, the
concentration of OTT traffic per TRP around the lowest positive values
highlights varying traffic levels that impact CPU load. Lower levels of OTT
traffic typically require less processing resources, leading to decreased CPU
load. The FL model can also leverage this information to optimize resource
allocation and guide network planning and scaling efforts. Furthermore, the
negative concentration of MIMO full rank suggests signal degradation and
interference, which could potentially lead to higher CPU load predictions.
However, in our specific case, where the value is very small, the impact of
MIMO full rank on CPU load can be considered negligible. Nonetheless, this
analysis still allows the FL model to identify and address potential issues
related to signal degradation and interference, providing valuable insights
for network optimization in other scenarios where MIMO full rank plays a more
significant role. By considering these features, the FL model can make
accurate predictions and enable resource optimization, leading to improved
network performance and enhanced management of CPU load in network slices in a
transparent and explainable way.

Figure 5: Distribution plot of attribution for emBB slice

###  IV-D Time complexity

Attention has been also paid to the training time having in mind future actual
deployment of our solution. The efficiency of the solution’s training process
directly impacts its real-world usability and effectiveness. Faster training
times enable the system to respond promptly to dynamic changes, ensuring real-
world efficiency and cost-effectiveness. Additionally, efficient training
times facilitate scalability, allowing the system to handle increasing data
volumes and user demands without compromising performance. For evaluating the
time complexity of the proposed solution, the _in-hoc_ FL explainer can be
implemented with various feature attribution generators, we use IG as the
primary method. Fig. 6 shows the convergence time of the browsing slice under
our _in-hoc_ FL strategy with SHAP, IG and Input $\times$ Grad as attribution
scores generators as well as the baseline unconstrained post-hoc method. Since
Fig. 3 indicates that the browsing slice with _in-hoc_ FL starts to converge
at around round 13, the computation time has been calculated until that
convergence round. It can observe from the Fig. 6 that _in-hoc_ FL with IG
takes lower time than others but nearly close to _post-hoc_ one while
presenting the highest confidence. This finding suggests that utilizing IG as
the primary feature attribution method in the _in-hoc_ FL explainer enables
faster convergence without compromising the confidence of the predictions.
This is a significant advantage for real-world deployment, as it indicates
efficient allocation of computational resources while maintaining the
trustworthiness of the model.

Figure 6: Convergence time of the proposed _in-hoc_ FL under different
explainers vs. the post-hoc baseline.

##  V Conclusion

In this paper, we have presented a novel explanation-guided _in-hoc_ federated
learning approach to achieve a transparent zero-touch service management of 6G
network slices while bridging the trade-off between performance and
explainability. We have considered the XAI-based confidence metric in a closed
loop way to solve the underlying joint optimization task using a proxy-
Lagrangian two-player game strategy. In particular, we have used both
integrated-gradient and perturbation-based XAI schemes to generate the
attributions consumed by our _in-hoc_ FL, which present almost the same
superior performance compared to an unconstrained _post-hoc_ FL baseline. We
have also provided a post-hoc analysis of the impact of networks parameters on
the slice-level resource allocation, which points out that CQI is the key
influencing feature. Finally, computational complexity (up to the convergence
round) has been assessed which demonstrates that our _in-hoc_ FL has less
complexity than the _post-hoc_ while presenting a clearly superior confidence.

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