# Bianisotropic Susceptibility Theory
of Artificial Magnetic Conductors

Oscar Céspedes Vicente  Christophe Caloz

###### Abstract

Over the past 30 years, Artificial Magnetic Conductors (AMCs) have generated
immense interest in the microwave community. However, their description has
been restricted to crude RLC design or subsidiary transmission-line network
modelling. This paper introduces an _electromagnetic_ theory of AMCs in terms
of bianisotropic susceptibilities.

Bianisotropic Susceptibility Theory
of Artificial Magnetic Conductors

Faculty of Engineering Science Katholieke Universiteit (KU) Leuven 3001
Leuven, Belgium

##  I Introduction

Due to the inexistence of magnetic charges, magnetic conductors, the dual of
electric conductors, do not exist in a natural (atomic or molecular) form.
However, artificial types of magnetic conductors, dubbed Artificial Magnetic
Conductors (AMCs) and exhibiting quasi Perfect Magnetic Conductor (PMC)
responses [1, 2] at specific frequencies, have been realized using resonant
“meta-particles” of diverse shapes, including iris [3], mushroom [4], patch
[5], uniplanar cross potent [6] and grounded cross potent [7]. Such AMCs have
attracted immense attention due to diverse applications, such as low-profile
antennas [8], compact packaging [9], SAR reduction [10], etc.

Unfortunately, the theory of AMCs has been so far mostly limited to crude RLC
circuit modeling [4, 6] or subsidiary transmission-line network modeling [11,
12], and a more precise and fundamental theory – namely a _real
electromagnetic theory_! – has been seriously lacking.

This paper, leveraging the latest advances in metasurfaces [13], provides such
a theory by describing AMCs in terms of bianisotropic surface
susceptibilities, with application to the mushroom AMC and a double-layer
cross potent AMC. It paves the way for an even more complete electromagnetic
theory that will include a precise explanation of generic AMC structures in
terms of their dipolar responses.

##  II Selected AMC Structures

The theory presented in this paper is completely general. It applies to any
AMC structure. However, we shall consider the two particular structures shown
in Fig. 1 as representative examples in order both to make the theory more
concrete and provide study cases for the full-wave illustration of Sec. IV.
The first structure, shown in Fig. 1(a), is the mushroom (or Sievenpiper) AMC
structure, while the second structure, shown in Fig. 1(b), is a double-layer
cross potent AMC structure.

[width=0.4]MushroomVsCrossPotentV2

Figure 1: Unit cell of representative AMC structures considered in the paper.
(a) Mushroom structure. (b) Double-layer (asymmetric) cross potent structure.

##  III Susceptibility Analysis

We assume, by duality with a conventional PEC, a PMC [1, 2] AMC that is both
reciprocal and non-gyrotropic. Moreover, we restrict our attention, due to
lack of space, to the case of s-polarization with scattering in the $zx$-plane
(field components $E_{y}$, $H_{z}$ and $H_{x}$) and illumination under an
angle $\theta$ with respect to the normal of the structure. Finally, we write
the “metasurface” incident, reflected and transmitted electric and magnetic
field expressions and enforce the PMC boundary condition as $R=1$ and $T=0$,
where $R$ and $T$ are the reflection and transmission coefficients,
respectively, on the incident side of the structure.

Inserting the average and difference fields that incorporate all these
conditions into generalized sheet transition conditions (GSTCs) [13] placed at
in the top layer of the structure leads, after some algebraic manipulations,
to the general AMC susceptibility condition

$\displaystyle\chi_{\text{ee}}^{yy}$
$\displaystyle=\chi_{\text{mm}}^{zz}\sin^{2}(\theta),$ (1a)
$\displaystyle\chi_{\text{me}}^{xy}$ $\displaystyle=\frac{2j}{k},$ (1b)
$\displaystyle\chi_{\text{mm}}^{xx}\in\mathbb{R}\quad$
$\displaystyle\text{and}\quad\chi_{\text{em}}^{yx}=-\chi_{\text{me}}^{xy}.$
(1c)

This condition contains two difficulties: 1) it is undetermined, because it
results from a system of two equations with four unknowns, and 2) it is
restricted to a specific angle, whereas we would ideally wish to obtain an
angle-independent AMC response. Therefore, we shall adopt the alternative
strategy consisting in first relaxing the AMC condition, next finding the
general formulas for the angle-dependent reflection and transmission
coefficients, and finally using the previous underdeterminacy to enforce the
PMC condition at two separate angles.

Using the same procedure as above, but with arbitrary $R$ and $T$ parameters
and solving the resulting two equations for these two parameters yields

$\begin{split}R(\theta)=&-2\frac{k\chi_{\text{ee}}^{yy}+2k\chi_{\text{em}}^{yx}c_{\theta}}{2k\chi_{\text{ee}}^{yy}+2k\chi_{\text{mm}}^{xx}c_{\theta}^{2}}\cdots\\\
&\
\frac{-k\chi_{\text{mm}}^{xx}c_{\theta}^{2}}{-j\left(k^{2}\left(\chi_{\text{ee}}^{yy}\chi_{\text{mm}}^{xx}+(\chi_{\text{em}}^{yx})^{2}\right)-4\right)c_{\theta}}\cdots\\\
&\ \
\frac{-k_{x}\chi_{\text{mm}}^{zz}s_{\theta}}{+jkk_{x}\chi_{\text{mm}}^{xx}\chi_{\text{mm}}^{zz}s_{\theta}c_{\theta}-2k_{x}\chi_{\text{mm}}^{zz}s_{\theta}}\end{split}$
(2a) and
$\begin{split}T(\theta)=&j\frac{4c_{\theta}+k^{2}\chi_{\text{ee}}^{yy}\chi_{\text{mm}}^{xx}c_{\theta}}{4jc_{\theta}-jk^{2}\chi_{\text{ee}}^{yy}\chi_{\text{mm}}^{xx}c_{\theta}}\cdots\\\
&\
\frac{k^{2}(\chi_{\text{em}}^{yx})^{2}c_{\theta}}{-jk^{2}(\chi_{\text{em}}^{yx})^{2}c_{\theta}+2k\chi_{\text{ee}}^{yy}+2k\chi_{\text{mm}}^{xx}c_{\theta}^{2}}\cdots\\\
&\ \
\frac{-kk_{x}\chi_{\text{mm}}^{xx}\chi_{\text{mm}}^{zz}s_{\theta}c_{\theta}}{-2k_{x}\chi_{\text{mm}}^{zz}s_{\theta}+jkk_{x}\chi_{\text{mm}}^{xx}\chi_{\text{mm}}^{zz}s_{\theta}c_{\theta}},\end{split}$
(2b)

where $c_{\theta}=\cos\theta$, $s_{\theta}=\sin\theta$ and
$k_{x}=k\sin\theta$. Successively enforcing these two relations as $R=1$ and
$T=0$ for both $\theta=0$ and $\theta=\theta_{0}$, where $\theta_{0}$ is an
arbitrary angle in $]0,\pi/2]$, and solving the resulting system of (four)
equations for the four susceptibilities, yields then

$\displaystyle\chi_{\text{mm}}^{xx}=0,$ (3a)
$\displaystyle\chi_{\text{ee}}^{yy}=0,$ (3b)
$\displaystyle\chi_{\text{em}}^{yx}=-\frac{2j}{k}=-\chi_{\text{me}}^{xy},$
(3c) $\displaystyle\chi_{\text{mm}}^{zz}=0.$ (3d)

These heteroanisotropic conditions are expected to represent, by design,
conditions for an angle-independent PMC response.

##  IV Full-Wave Illustration

Figure 2 plots _optimized_ AMC results for the two structures in Fig. 1. The
mushroom structure fails to exhibit an angle-independent response. The reason
is that it fails to satisfy the condition (3d). In contrast, the double cross-
potent structure does exhibit an angle-independent AMC response. This is
because it satisfies the four conditions in (3).

[width=0.38]MushroomVsCrossPotentPhase

Figure 2: Phase of $R$ versus frequency for different illumination angles. (a)
Mushroom structure. (b) Cross potent structure.

## References

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#  $Q$-factor: A measure of competition between the topper and the average in
percolation and in SOC

Asim Ghosh1, S. S<EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding author),
<EMAIL_ADDRESS>and Bikas K. Chakrabarti3,4 1Department of
Physics, Raghunathpur College, Raghunathpur 723133, India
2B-1/16 East Enclave Housing, 02 Biswa Bangla Sarani, New Town, Kolkata
700163, India
3Saha Institute of Nuclear Physics, Kolkata 700064, India
4Economic Research Unit, Indian Statistical Institute, Kolkata 700108, India

###### Abstract

We define the $Q$-factor in the percolation problem as the quotient of the
size of the largest cluster and the average size of all clusters. As the
occupation probability $p$ is increased, the $Q$-factor for the system size
$L$ grows systematically to its maximum value $Q_{max}(L)$ at a specific value
$p_{max}(L)$ and then gradually decays. Our numerical study of site
percolation problems on the square, triangular and the simple cubic lattices
exhibits that the asymptotic values of $p_{max}$ though close, are distinctly
different from the corresponding percolation thresholds of these lattices. We
have also shown using the scaling analysis that at $p_{max}$ the value of
$Q_{max}(L)$ diverges as $L^{d}$ ($d$ denoting the dimension of the lattice)
as the system size approaches to their asymptotic limit. We have further
extended this idea to the non-equilibrium systems such as the sandpile model
of self-organized criticality. Here, the $Q(\rho,L)$-factor is the quotient of
the size of the largest avalanche and the cumulative average of the sizes of
all the avalanches; $\rho$ being the drop density of the driving mechanism.
This study has been prompted by some observations in Sociophysics.

##  I Introduction

Major investigations on critical fluctuations near the critical points of
phase transitions in different systems during the seventies through nineties
of the last century have helped establishing a strong foundation of their
statistical physics descriptions. For introductions on the scaling and
critical behavior in different physical systems, see for example [1] for
magnetic and fluid systems, [2] for polymer systems, [3] for percolating
systems, and [4] Self-Organizing Critical (SOC) systems. Essentially, the
order parameter of the corresponding systems vanish following in general a
singular power law or critical behavior at the critical point and beyond. Its
higher moments include susceptibilities, diverge again with singular or
critical power law exponent values at the respective critical points. For SOC
systems, these singular behaviors are seen from the pre-critical side and then
remains critical in the SOC state of the systems. For practical purposes,
these diverging susceptibilities help locating the critical point.

For social systems, scientists had studied for ages, starting with Pareto law
80-20 law [5], Lorenz function [6], Gini index [7], Hirsch index [8], etc.,
the extreme unequal distributions of income or wealth, votes, paper citations
respectively. Following some recent observations [9-11] of extreme inequality
level in citation statistics of successful individuals and even institutions /
universities / journals, with Gini and other inequality index values going
beyond the Pareto 80-20 limit, we studied and found [12] clear presence of
similar level of the inequality index values in the physical models of SOC
system, like the Bak, Tang, Wiesenfeld (BTW) sandpile [13] and the Manna [14]
sandpile. Particularly, in our recent study [15] of the citation statistics of
some very successful prize winning scientists and a few others not so
successful scientists it has been observed that their research dynamics is
clearly SOC like and most successful have achieved the critical level in their
citation inequalities, while others are still approaching that level, though
they have not reached there. All these studies showed, just the average high
level of citations per paper (reflected by the Hirsch index values, which are
determined by the effective network coordination or Dunbar number [16-17]) do
not reflect the success of the scientist, but the high level (beyond the
Pareto level) of critical fluctuations in citations from publication to
publication of the scientist. Indeed, it was seen in [15] that crossing a
threshold value of a simple quotient of the citation number of the highest
cited paper and the average citation of all the papers (including the highest
cited one) by the scientist gives very good correlation with the appreciations
by the respective communities.

Following this clue, we study here how the topper competes with the average in
the well known models of percolation processes and in the sandpile model of
self-organized criticality. In the percolation model we have defined the
$Q$-factor as the quotient of the largest cluster size and the average size
over all clusters for the percolation models. Similarly, the $Q$-factor in the
sandpile model has been defined as the quotient of the largest avalanche size
and the average size over all avalanches. As the control variables are tuned
in these problems $Q$-factors grow very sharply right before a specific value
of the control variable, reaches the maxima, and then decay very rapidly. The
locations of the maxima are distinctly different from the critical points of
these systems.

We have described our calculations and results of the percolation problem for
the square, triangular and the simple cubic lattices in the three sub-sections
of section II. Here, we have calculated the $Q$-factors for the entire range
of the occupation probability $p$. A nice finite-size extrapolation gives the
precise value of the percolation occupation probability $p_{max}$ in the
asymptotic limit which we found to be larger than their percolation
thresholds. In section III we have executed similar analysis for the BTW
sandpile where we used the drop density $\rho$ as the tuning parameter. The
value of $\rho_{max}$ in the asymptotic limit have been calculated. Finally,
we have summarized in section IV.

Figure 1:  Plot of the average number of distinct clusters per lattice site
$\langle n(p,L)\rangle/L^{d}$ against the percolation occupation probability
$p$ for the square (pink) and triangular lattices (red) of size $L$ = 256,
1024, and 4096. For the simple cubic (blue) lattice system sizes used are: $L$
= 32, 64, and 128. Plots for different system sizes overlapped completely,
only the colors used for the largest lattice are visible.

##  II Site Percolation

###  II.1 Square lattice

An initially empty square lattice of size $L\times L$ has been gradually
filled in by occupying the randomly selected lattice sites one by one. At any
arbitrary intermediate stage the fraction ‘$p$’ of occupied sites is referred
as the percolation occupation probability. A cluster is defined as the set of
occupied sites connected by nearest neighbour distances. Different distinct
clusters have been identified using the well known Hoshen Kopelman algorithm
Hoshen . Since we are not going to study any spanning property of the
percolation clusters we have used the periodic boundary conditions in all
simulations reported here. The number of distinct clusters $n(p,L)$ increases
from unity at $p\to 0+$, reaches a maximum at some intermediate $p$ value and
then finally goes down to unity again at $p=1$. We refer the entire process as
a ‘run’.

In Fig. 1 we have plotted the average number of clusters per lattice site
$\langle n(p,L)\rangle/L^{d}$ against $p$ for three different system sizes of
the square, triangular, and the simple cubic lattices, where $d$ represents
their dimensionalities. The collapse of the plots on top of one another for
three system sizes are extremely well. At an intermediate stage the average
size of all clusters including the largest one is therefore
$s_{av}(p,L)=pL^{d}/n(p,L)$. This is further averaged over a large number of
independent runs and we define the average cluster size $\langle
s(p,L)\rangle=\langle s_{av}(p,L)\rangle$. The peak positions of these curves
have coordinates: (0.26968, 0.12954) for the square lattice, (0.21192,
0.096306) for the triangular lattice and (0.17871, 0.086066) for the simple
cubic lattice.

In Fig. 2(a) we have plotted the scaled cluster size $\langle
s(p,L)\rangle/L^{2}$ against $p$ only for the square lattice and again for the
same three system sizes. The curves become sharper as $p\to 1$ and as the
system size becomes larger. In Fig. 2(b) we have inverted the $x$-axis and
plotted $\langle s(p,L)\rangle/L^{2}$ against $(1-p)L^{1/2}$ to observe a nice
collapse of the data over the entire range of $p$ values.

Figure 2:  (a) Plot of the average cluster size $\langle s(p,L)\rangle$ scaled
by the total number $L^{2}$ of lattice sites against the percolation
occupation probability $p$ for the square lattice of size $L$ = 256 (black),
1024 (blue), and 4096 (red). (b) The same data have been replotted against
$(1-p)L^{1/2}$ which yield nice collapse of the data.

As more and more sites are occupied, the growth of the size $s_{max}$ of the
largest cluster has been monitored. The order parameter $\Omega(p,L)$ of the
percolation transition is defined as the average fractional size of the
largest cluster, i.e., $\Omega(p,L)=\langle s_{max}(p,L)\rangle/L^{d}$. In
Fig. 3 we have plotted the order parameter $\Omega_{sq}(p,L)$ for the square
lattice for three different system sizes. Larger the system size, the growth
of the order parameter becomes sharper. For a particular system
$\Omega_{sq}(p,L)$ grows rapidly as the percolation occupation probability $p$
approaches from below the site percolation threshold of the square lattice
whose best value till date is $p_{c}(sq)=0.59274605079210(2)$ Jacobsen ; Ziff-
Wiki .

Figure 3:  Plot of the percolation order parameter $\Omega_{sq}(p,L)=\langle
s_{max}(p,L)\rangle/L^{2}$ against the site percolation occupation probability
$p$ for the square lattice of sizes $L$ = 256 (black), 1024 (blue), and 4096
(red).  Figure 4:  Plot of the average values of (i) the size of the largest
cluster $\langle s_{max}(p,L)\rangle$ (black), (ii) the sizes of all clusters
$\langle s_{av}(p,L)\rangle$ (blue), and (iii) the $Q$-factor $Q(p,L)L^{2}$
(red) for a system of size $L$ = 256 on the square lattice.  Figure 5:  Plot
of $Q_{sq}(p,L)$ against the percolation occupation probability $p$ for the
square lattice of size $L$ = 256 (black), 1024 (blue), and 4096 (red).

Now we define the $Q$-factor as the quotient of the average largest cluster
size $\langle s_{max}(p,L)\rangle$ and the average size $\langle
s(p,L)\rangle$ of all clusters for every value of $p$ for a certain system
size $L$ as:

$Q(p,L)=(\langle s_{max}(p,L)\rangle/\langle s_{av}(p,L)\rangle)/L^{d}\\\
=\Omega(p,L)/\langle s_{av}(p,L)\rangle.$ (1)

In Fig. 4 we have plotted these three quantities for a particular system size
$L$ = 256. They are, (i) the average size $\langle s_{max}(p,L)\rangle$ of the
largest cluster, (ii) the average size $\langle s_{av}(p,L)\rangle$ of all
clusters, and their $Q(p,L)$ multiplied by the system size $L^{2}$. It is seen
when $p\to\langle p_{max}\rangle$, the value of $\langle s_{max}(p,L)\rangle$
becomes increasingly larger than the average cluster size $\langle
s_{av}(p,L)\rangle$ and therefore $Q(p,L)$ increases very sharply. However
after reaching $\langle p_{max}\rangle$, the growth of $\langle
s_{max}(p,L)\rangle$ becomes slower but $\langle s_{av}(p,L)\rangle$ maintains
its fast growth. Consequently, their ratio decays gradually and the $Q$-factor
goes down. This is visible in Fig. 5 where we have plotted $Q_{sq}(p,L)$
against $p$ for three different system sizes. All three curves have single
peaks of nearly the same heights but their positions have systematic
variations.

For each run we have estimated the maximum value $Q_{max}(p_{max},L)$ and its
location $p_{max}$ and then estimated their averages over all runs. The
average $\langle p_{max}(L)\rangle$ values of six different system sizes of
the square lattices from $L$ = 128, …, 4096 have been extrapolated to their
asymptotic limit: $\langle
p_{max}(L)\rangle=p_{max}(\infty)-const.L^{-1.041}$. The asymptotic value
$p_{max}(\infty)$ = 0.6033 is approximately one percent different from the
percolation threshold $p_{c}$ of the square lattice, see Fig. 6(a).

Figure 6:  Extrapolation of $\langle p_{max}(L)\rangle$ values to their
asymptotic limit of $L\to\infty$ gives the estimates of $p_{max}(\infty)$: (a)
0.6033 for the square lattice, (b) 0.5088 for the triangular lattice, and (c)
0.3448 for the simple cubic lattice.

We have also tried a log correction to the finite-size correction as follows:

$\langle p_{max}(L)\rangle=p_{max}(\infty)-AL^{-\alpha}(1-B(\ln L/L)).$ (2)

From our best fit we obtained $p_{max}(\infty)$ = 0.60329, $A$ = 1.4323,
$\alpha$ = 1.047 and $B$ = 0.2608 which shows that the log correction has very
little effect on $p_{max}(\infty)$.

The collapse of the peak positions $Q_{max}(p_{max},L)$ of the curves in Fig.
5 on one another implies the maximal cluster size at this point $\langle
s_{max}(L)\rangle\sim L^{\eta}$ with $\eta$ = 2. This has been directly
verified by plotting (figure not shown) $\langle s_{max}(L)\rangle$ against
$L$ using the double logarithmic scales for the square and simple cubic
lattices. The values of the exponent $\eta$ have been estimated from the
slopes of the curves. For the square lattice $\eta_{sq}=2.0057$ has been
obtained and for the simple cubic lattice $\eta_{sc}=3.0075$ is found. This
implies that since $\langle p_{max}\rangle$ values are slighly larger than the
percolation thresholds, the largest clusters turn out to be compact, and not
fractals like the percolatiing clusters at the percolation thresholds.
Consequently, their dimensionalities are equal to their embedding space
dimensions.

An alternate definition of $Q$-factor is as follows:

$Q^{\prime}(p,L)=\langle(s_{max}(p,L)/s_{av}(p,L))\rangle/L^{d}.$ (3)

Here, for each value of $p$ of every run, one first calculates the quotient of
the largest cluster size $s_{max}(p,L)$ and the average cluster size
$s_{av}(p,L)$ and then takes an average of this quotient over a large number
of independent runs.

We have calculated both the $Q(p,L)$ and $Q^{\prime}(p,L)$ factors for the
same set of runs. When we plot these two $Q$ factors against $p$ on the same
graph, it appears with the naked eye that one curve completely overlaps the
other as if the two factors are equal. Actually this is not the case which
becomes apparent when we plotted the difference $\Delta
Q=Q(p,L)-Q^{\prime}(p,L)$ against $p$. It is observed that though the maximal
value of the difference is very small, there is a nice peak for $\Delta Q$
occuring at $\langle p_{max}(L)\rangle$. Extrapolating them we obtain
$p_{max}(\infty)=0.6032$. The maximal values of both $Q(p,L)$ and
$Q^{\prime}(p,L)$ do not depend much on $L$ when the system sizes are large
i.e., from $L$ = 1024 to 4096.

Figure 7:  A plot of $Q_{tr}(p,L)$ against $p$ for the triangular lattices of
system sizes, $L$ = 256 (black), 1024 (blue), and 4096 (red) have been shown.

###  II.2 Triangular lattice

A parallel set of calculations have been done on the triangular lattice. A
plot of $Q_{tr}(p,L)$ against $p$ for three different system sizes have been
shown in Fig. 7. The positions of the maxima are very close to the triangular
lattice percolation threshold $p_{c}$ = 1/2 but slightly larger than 1/2. For
each run we have estimated the maximum value of $Q_{max}(p_{max},L)$, the
corresponding $p_{max}$ values and then averaged over all runs. The average
$\langle p_{max}(L)\rangle$ values of six different system sizes from $L$ =
128, …, 4096 have been extrapolated to their asymptotic limit: $\langle
p_{max}(L)\rangle=p_{max}(\infty)-const.L^{-1.053}$ with $p_{max}(\infty)$ =
0.5088 which is approximately 0.9 percent different from the percolation
threshold $p_{c}=1/2$, see Fig. 6(b).

###  II.3 Simple Cubic lattice

For the simple cubic lattice we could study only small lattice sizes up to $L$
=256 and plot them in Fig. 8. The only difference for the simple cubic lattice
is the quotient of the maximal size and average size is scaled by $L^{3}$,
that is the total number of lattice sites in the system.

$Q_{sc}(p,L)=(\langle s_{max}(p,L)\rangle/\langle s(p,L)\rangle)/L^{3}.$ (4)

The data for the positions of the maximum of $Q_{sc}(p,L)$ for the lattice
sizes $L$ = 32 to 256 have been used to extrapolate $\langle
p_{max}(L)\rangle=p_{max}(\infty)-const.L^{-1.812}$ with $p_{max}(\infty)$ =
0.3448 (Fig. 6(c)).

Figure 8: The $Q_{sc}(p,L)$ factors for the site percolation on the simple
cubic lattice have been plotted against the site occupation probability $p$
for three different system sizes: $L$ = 32 (black), 64 (blue), and 128 (red).

Therefore, in each of the three lattices, namely, square, triangular and
simple cubic we see that the precise values of the probabilities
$p_{max}(\infty)$ are slightly higher than their corresponding percolation
thresholds $p_{c}$. Clear power laws for the finite-size extrapolations in
Fig. 6 in all three cases indicate that indeed these threshold values
$p_{max}(\infty)$ are distinctly different from their $p_{c}$ values. These
extrapolations are characterized by the exponents whose values are very are
close, namely, 1.041 for the square lattice and 1.053 for the triangular
lattice and widely different 1.812 for the simple cublic lattice which may be
the indication of universality of the finite-size correction exponent. It may
be that more extensive study in future with much larger systems would yield
values 1 and 2 for these exponents in two and three dimensions, a possibility
which we cannot rule out at this moment. Our conclusion that $p_{max}(\infty)$
are different, has also been supported by the independent measurements of the
mass of the largest clusters at $p_{max}(L)$ which yield that indeed these
clusters are of compact structures instead of being fractals at their
percolation thresholds. Here we like to recall another problem of percolation
connectivity between two points at distance of separation of the order of the
system size Manna-Ziff . There also enhanced thresholds for the percolation
connectivities of the modified structure have been observed.

Figure 9:  For the BTW sandpile the values of $Q_{BTW}(\rho,L)$ have been
plotted against the average number of sand particles $\rho$ dropped per
lattice site. The four different system sizes $L$ are 64 (black), 128 (blue),
256 (green), and 512 (red).

##  III BTW sandpile

The Bak, Tang, and Wiesenfeld (BTW) sandpile btw  has been studied on the
square lattice of size $L\times L$ with open boundary condition. The dynamics
of sandpile starts from a completely empty lattice. Sand particles are dropped
one by one at randomly selected lattice sites. The system is allowed to relax
through the deterministic BTW sandpile dynamics btw . The avalanche created by
dropping one particle has size $s$, measured by the total number of sand
column topplings in the avalanche. At any arbitrary intermediate stage of the
sandpile dynamics, let $\rho$ be the average number of sand particles dropped
per lattice site. We refer it as the drop density which is a measure of the
net inflow of sand mass. This implies that an average number of $\rho L^{2}$
particles have been dropped on to the system many of which have left the
system by jumping outside through the boundary. Here, the drop density $\rho$
of particle addition is the control variable.

We have kept track of the maximal size $s_{max}(\rho,L)$ of all the avalanches
created till $\rho L^{2}$ particles have been dropped. At the same time we
have also calculated the cumulative average size $s_{av}(\rho,L)$ of all the
avalanches of size larger than zero, including the largest avalanche. Each run
consists of a sequence of particle drops up to the drop density $\rho$ = 2.5.
Quantities which are averaged over many such independent runs are denoted by
the angular brackets $\langle...\rangle$. Finally, we have defined a
$Q$-factor which is the quotient of the largest and the average avalanche
sizes of the sandpile:

$Q_{BTW}(\rho,L)=(\langle s_{max}(\rho,L)\rangle/\langle
s_{av}(\rho,L)\rangle)/L^{2}$ (5)

and plot this quantity against the drop density $\rho$ in Fig. 9. A nice peak
of $\lambda$-shape has been observed at the drop density $\rho_{max}(L)$. The
rise and fall of $Q_{BTW}(\rho,L)$ values on the two sides of $\rho_{max}(L)$
are found to asymmetric. We have measured the position $\rho_{max}(L)$ and the
height $Q_{max}(L)$ for the four different system sizes.

Figure 10:  For the BTW sandpile (a) the drop densities
$\langle\rho_{max}(L)\rangle$ for the maximal $Q$-factors and (b) the average
values of the maximal $Q$-factors $\langle Q_{max}(L)\rangle$ have been
plotted against different negative powers of $L$. The two colors, red and
blue, represent two different ways of calculations. The extrapolated values in
the asymptotic limit of $L\to\infty$ are consistent with each other.

For a single sequence of sand grain additions on a system of size $L$, let
$\rho_{max}(L)$ be the precise value of the average number of particles
dropped per site of the lattice corresponding to the maximum value
$Q_{max}(\rho,L)$ of the $Q$-factor of Eqn. 5. We have calculated
$\langle\rho_{max}(L)\rangle$ using two different methods and plotted them
against two different negative powers of $L$ in Fig. 10(a). (i) Red: For each
run we have we have picked up the maximum value $Q_{max}(\rho,L)$ of $Q$ and
its corresponding drop density $\rho_{max}$. These two quantities are then
averaged over a large number of independent runs. It has been observed that
both $\langle\rho_{max}(L)\rangle$ and $\langle Q_{max}(\rho,L)\rangle$ depend
on the system size $L$. The best values of the powers for extrapolation of
these two quantities are selected using the least square fit method and the
straight lines are then extrapolated to the $L\to\infty$ limit. We found
$\langle\rho_{max}(L)\rangle=2.127+5.95L^{-1.12}$ and $\langle
Q_{max}(L)=0.023+0.136L^{-0.44}$. (ii) Blue: Using a large number of runs and
we have calculated for each value of $\rho$ the value of $\langle
s_{max}(\rho,L)\rangle$ and $\langle s_{av}(\rho,L)\rangle$ and then
calculated $Q(\rho,L)$ using Eqn. 5. The maximum value $Q_{max}(\rho,L)$ have
been determined and its location $\rho_{max}(L)$ have been estimated. They are
again best fitted by the least square method and then extrapolated. We found
$\langle\rho_{max}(L)\rangle=2.126+5.78L^{-1.20}$ and $\langle
Q_{max}(L)\rangle=0.015+0.147L^{-0.56}$.

##  IV Summary

As mentioned in the introduction, the criticisms (see e.g., [21,22]) and the
extensive data analysis [11], suggested that statistically the Hirsch index
[8] of a researcher is strictly proportional to the square root of the total
citations or equivalently of the total number publications by the scientist.
These observations suggested that average citation per paper by a scientist
depends of the effective coordination number (or Dunbar number [16,17]) of the
scientist in the corresponding network and does not really measure the
individual researcher’s success. Observations reported in [15] suggested that
significantly large values of the quotient (equivalent to the $Q$ factor here)
of the citations of the highest cited paper to the average citation per paper
of the researcher clearly corresponds to the achievement of researcher and
precisely indicates the universal SOC state in their publication dynamics.

Inspired by these observations, we have studied the $Q(p,L)$-factor i.e., the
quotient of the size of the largest cluster and the average size of all
clusters in the site percolation models against the variation of the
occupation probability $p$. We find that the $Q(p,L)$-factor grows with $p$,
reaches sharply a maximum, and then decays down (see Figs. 5, 7 and 8 for
square, triangular and simple cubic lattices respectively). The average
occupation probability of the maximum $\langle p_{max}(L)\rangle$ nicely
approaches its asymptotic limit $\langle p_{max}\rangle$ with some negative
power $\alpha$ of the system size $L$ (see Fig. 6). The actual values of
$\langle p_{max}(L)\rangle$ and the value of $\alpha$ depend on the type of
lattice used. We have also observed that the other measures of mean cluster
size lead to different behaviors but only the one chosen here leads to a sharp
peak at the threshold. Similar studies have been executed for the BTW sandpile
model of self-organized criticality tuning the drop density $\rho$ of the
driving process, and obtained results similar to the percolation models (see
Figs. 9, 10).

We are very much thankful to Amnon Aharony and Robert Ziff for their
insightful comments and suggestions. BKC is grateful to the Indian National
Science Academy for their Senior Scientist Research Grant.

## References

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# Open Problem:
Learning with Variational Objectives on Measures

Vivien Cabannes
Meta AI  Carles Domingo-Enrich
Meta AI

###### Abstract

The theory of statistical learning has focused on variational objectives
expressed on functions. In this note, we discuss motivations to write similar
objectives on measures, in particular to discuss out-of-distribution
generalization and weakly-supervised learning. It raises a natural question:
can one cast usual statistical learning results to objectives expressed on
measures? Does the resulting construction lead to new algorithms of practical
interest?

##  1 Introduction

Historically, machine learning has focused on learning functions. Statistical
learning theory has provided a relatively clear understanding of the error to
expect when utilizing those functions based on the number of samples seen at
training time. As machine learning applications have grown large-scale,
tackling more complex problems, the guarantees offered by this theory have
lost their sharpness (Xiang, 2023; Roose, 2023). In particular, utilization
environments can differ significantly from training and testing ones, those
latter sometimes hiding hard-to-mitigate biases (Caliskan et al., 2017;
Benjamin, 2019). More generally, the functional point-of-view is not ideal to
describe new machine-learned algorithms that generate content that resembles
the data it was fed with (see, e.g., Touvron et al., 2023, for technical
details).

In this paper, we suggest to cast the usual statistical learning theory that
deals with variational objectives on functions into a theory that deals with
variational objectives on measures. While we motivate such a construction, we
leave its practical construction as an open-problem. The motivations are
provided by several examples, such as distribution-shift, generative AI,
dealing with missing data, and dealing with weak supervision.

##  2 The Problem

### Classical Learning Theory.

The theory of statistical learning has focused on the learning of functions
(Vapnik, 1995; Steinwart and Christmann, 2008; Devroye et al., 2013; Mohri et
al., 2018; Bach, 2023). In essence, it models any algorithm as a rule to map
inputs (in a space $\mathcal{X}$) to outputs (in a space $\mathcal{Y}$).
Learning an algorithm consists in learning a mapping
$f:\mathcal{X}\to\mathcal{Y}$. Denoting by
$\rho_{\mathcal{X}}\in\Delta_{\mathcal{X}}$ the distribution of inputs that
the algorithm will be used for and by $\rho|_{x}$ a distribution that generate
the output $y$ that $f$ should map $x$ to, one can define a optimal mapping
through the variational objective

$f^{*}\in\operatorname*{arg\,min}_{f:\mathcal{X}\to\mathcal{Y}}{\mathcal{R}}(f):={\mathbb{E}}_{(X,Y)\sim\rho}[\ell(f(X),Y)],$
(1)

where $\rho\in\Delta_{\mathcal{X}\times\mathcal{Y}}$ is the resulting
distribution on input-output pairs, and
$\ell:\mathcal{Y}\times\mathcal{Y}\to{\mathbb{R}}$ quantifies the error cost
$\ell(f(x),y)$ associated to predicting $f(x)$ when the desired output was
$y$. To minimize such an objective it is usual to collect $n$ examples
$(x_{i},y_{i})$ that are thought of as independent realizations of the random
variable $(X,Y)\sim\rho$, and to minimize the approximate objective

$f_{n}\in\operatorname*{arg\,min}_{f\in{\mathcal{F}}}{\mathcal{R}}_{n}(f):=\frac{1}{n}\sum_{i=1}^{n}[\ell(f(x_{i}),y_{i})],$
(2)

where $f$ is constrained to belong to a class of function ${\mathcal{F}}$
which implicitly encodes how $f$ should generalize to unseen example
$X\sim\rho_{\mathcal{X}}$ based on the training examples
$(x_{i},y_{i})_{i\in[n]}$.

### Measure-based Statistical Learning?

In this open-problem presentation, we suggest retaking the skeleton of
statistical learning in order to apply it to measure-based variational
objectives. Rather than learning the mapping $f:\mathcal{X}\to\mathcal{Y}$,
one might be interested in learning a distribution $\mu\in{\mathcal{Z}}$ over
some space ${\mathcal{Z}}$ through a principled population objective as

$\mu^{*}\in\operatorname*{arg\,min}_{\mu\in\Delta_{{\mathcal{Z}}}}{\mathcal{E}}(\mu);\qquad{\mathcal{E}}:\Delta_{{\mathcal{Z}}}\to\bar{\mathbb{R}},$
(3)

where $\bar{\mathbb{R}}={\mathbb{R}}\cup\left\\{+\infty\right\\}$. It is
useful to cast any hard constraints $\mu\in{\mathcal{C}}$ into the objective
${\mathcal{E}}$ by adding a barrier regularization $\chi_{\mathcal{C}}$, where
$\chi_{\mathcal{C}}(\mu)=0$ if $\mu\in{\mathcal{C}}$ and
$\chi_{\mathcal{C}}(\mu)=+\infty$ otherwise.

In practice, one might not have access this population objective but only to
an approximation $\hat{\mathcal{E}}$ of it. This may be due to modeling
constraints, due to having only access to samples instead of distribution, or
due to optimization issues which force the usage of a surrogate objective. As
a consequence, one might only be able to practically optimize for

$\mu_{n}\in\operatorname*{arg\,min}_{\mu\in\Delta_{{\mathcal{Z}}}}{\mathcal{E}}_{n}(\mu);\qquad\hat{\mathcal{E}}:\Delta_{{\mathcal{Z}}}\to\bar{\mathbb{R}}.$
(4)

The open-problem is the following:

> Can we define abstract classes of objectives ${\mathcal{E}}$ and of their
> (random) approximations ${\mathcal{E}}_{n}$, such that
> ${\mathcal{E}}(\mu_{n})$ converges to ${\mathcal{E}}(\mu^{*})$ as $n$ goes
> to infinity? Can we give rates of convergence of such a convergence under
> some source condition, i.e. well-specified model assumptions? Do those
> abstract classes describe problems of practical interest?

##  3 Some Research Directions

### Following the classical path.

A natural direction to build such guarantees is to follow the path of
classical statistical learning. First of all, by remarking that
${\mathcal{E}}_{n}(\mu^{*})\geq{\mathcal{E}}_{n}(\mu_{n})$, we have

${\mathcal{E}}(\mu_{n})-{\mathcal{E}}(\mu^{*})\leq{\mathcal{E}}(\mu_{n})-{\mathcal{E}}_{n}(\mu_{n})+{\mathcal{E}}_{n}(\mu^{*})-{\mathcal{E}}(\mu)$

When ${\mathcal{E}}_{n}$ is a random variable, the convergence of
${\mathbb{E}}[{\mathcal{E}}_{n}(\mu)]$ to ${\mathcal{E}}(\mu)$ is usually
simple to establish through variations of the law of large numbers; the speed
of the convergence can be derived with variations of central-limit theorem, or
more generally the tail of ${\mathcal{E}}_{n}(\mu)-{\mathcal{E}}(\mu)$ can be
studied with concentration inequality. The part in
${\mathcal{E}}(\mu_{n})-{\mathbb{E}}{\mathcal{E}}_{n}(\mu_{n})$ is more tricky
as $\mu_{n}$ is also a random variable. It is classical to bound this term
with
$\sup_{\mu\in{\mathcal{C}}}{\mathbb{E}}[{\mathcal{E}}(\mu_{n})-{\mathcal{E}}_{n}(\mu_{n})]$
for ${\mathcal{C}}\subset\Delta_{{\mathcal{Z}}}$ a set of measure that
$\mu_{n}$ is known to belong to. From there, one could cover the space
${\mathcal{C}}$ with some balls, and control the deviation of interest on each
ball, before performing some union bound, eventually with “chaining” to reduce
redundant events in the union bound.

### Some interesting facts on sigma-fields.

In essence, the fact that one can not directly get $\mu^{*}$ but has to
satisfy oneself with $\mu_{n}$ is due to some coarseness of the approximation
of ${\mathcal{E}}$ by ${\mathcal{E}}_{n}$ which does not allow to retrieve as
much information as in the ideal case. The notion of coarseness of information
is naturally captured by sigma-fields. Let $\Sigma_{n}$ be a sigma field on
$\mathcal{X}$. For $f:\mathcal{X}\to{\mathbb{R}}$, ideally we would like
${\mathbb{E}}\left[f\,\middle|\,\Sigma_{n}\right]$ to be $f$ itself. Yet in
some cases, $\Sigma_{n}$ might not be rich enough, and one might only retrieve
coarse information on $f$ through
${\mathbb{E}}\left[f\,\middle|\,\Sigma_{n}\right]$ (for example, this could be
due to missing information, or only accessing weak information as detailed in
the next section). As $\Sigma_{n}$ grows larger, this approximation will get
finer. A natural way to build ${\mathcal{E}}_{n}$ would be by considering
${\mathcal{E}}_{n}={\mathcal{E}}+\chi_{\Sigma_{n}}$ where $\Sigma_{n}$
converges to the Borel algebra on $\mathcal{X}$ in some sense (e.g. Kudo,
1974). In particular, if ${\mathcal{E}}$ displays some continuity properties
with respect to this convergence topology (see e.g. Björklund et al., 2020,
for concrete examples), this would ensure that convergence properties on the
$\Sigma_{n}$ can be cast on the difference
${\mathcal{E}}(\mu_{n})-{\mathcal{E}}(\mu)$.

### Looking at the abstract topological picture.

At an abstract level, one could try to leverage abstract topological facts on
measure spaces and on specific variational objectives in order to directly
draw conclusions on the good behavior of some methods that work with
variational objectives on measures. For example, when ${\mathcal{E}}_{n}$ is
deterministic, the consistency of the estimator $\mu_{n}$ is nothing but the
epi-convergence of ${\mathcal{E}}_{n}$ towards ${\mathcal{E}}$ (Rockafellar
and Wets, 1998; Feinberg et al., 2022).

##  4 Practical Examples

The goal of this open-problem is not only to build an abstract mathematical
framework, but also to make sure that such a framework actually helps advance
the state-of-the-art of machine learning by answering the need to formulate
better objectives on concrete practical problems. This section gives examples
of the variational formulations of important machine learning problems with a
measure-based objective.

### Generative modeling.

The first type of problems that are natural to express with a variational
objective on distributions are sampling problems, where one wants to learn to
generate samples according to a given distribution $\mu^{*}$ while only
accessing samples $(z_{i})\sim(\mu^{*})^{\otimes n}$. For example, one can
look for the following population principle and its empirical approximation

$\mu^{*}\in\operatorname*{arg\,min}_{\mu\in\Delta_{{\mathcal{Z}}}}D(\mu||\mu^{*}),\qquad\qquad\hat{\mu}\in\operatorname*{arg\,min}_{\mu\in\Delta_{{\mathcal{Z}}}}D(\mu||\frac{1}{n}\sum_{i=1}^{n}\delta_{z_{i}})+\chi_{\mathcal{C}}(\mu),$

where $D:\Delta_{{\mathcal{Z}}}\times{\mathcal{Z}}\to{\mathbb{R}}_{+}$ is some
divergence, the $(z_{i})$ are independent realization sampled accordingly to
$\mu^{*}$, and ${\mathcal{C}}\subset\Delta_{{\mathcal{Z}}}$ capture a class of
measure that $\hat{\mu}$ is searched in. The optimization of the population
formulation of this variational problem could be linked to diffusion models,
as it is well-known that gradient flow on the Kullback-Leibler divergence in
the $L^{2}$-Wasserstein space describes nothing but the evolution of the
critically damped Langevin dynamics from an initial measure $\mu_{0}$ towards
the stationary measure $\mu^{*}$ (Jordan et al., 1998; Otto, 2001).

### Uncertainty Quantification.

A second natural reason to look at the prediction of distributions instead of
functions is given by the practical approach to classification, where rather
than learning a function from $\mathcal{X}$ to a set ${\mathcal{Y}}$ of
classes, practitioners prefer to learn a “score” function from $\mathcal{X}$
to $\Delta_{{\mathcal{Y}}}$ (Friedman, 1994; Bartlett et al., 2006; Murphy,
2012). Indeed, even in regression settings, it might be preferable to learn
$\left(Y\,\middle|\,X\right)$, the law of $Y$ conditioned on $X$, instead of
only learning the conditional expectation
${\mathbb{E}}\left[Y\,\middle|\,X\right]$, since this could help to quantify
the uncertainty that the output associated with $X$ should be $f(X)$. Recent
advances in conformal prediction have provided interesting directions to
leverage when formulating the problem of quantifying a predictor uncertainty
from a measure point of view (Vovk and Shafer, 2008; Angelopoulos and Bates,
2022).

### Out-Of-Distribution Generalization.

While machine learning usually needs curated data in order to be trained
efficiently, one may be interested in using the learned algorithm under a
distribution which is quite different from the training one. There are two
potential sources of variations between training to utilization settings:
either $\rho|_{x}$ is changing, which might be due to a change of confounding
variables (e.g. $x$ corresponds to geographic locations, $y$ to some
atmospheric conditions, but training took place in winter only), or
$\rho_{\mathcal{X}}$ is changing (e.g. training took place in the Western
world, but utilization is done worldwide), which is known as covariate shift
(Gretton et al., 2008). Eventually, one could have a prior
$\mu\in{\mathcal{C}}$ on future distributions in utilization settings, and try
to minimize the worse performance over utilization environment,

$\mu\in\operatorname*{arg\,min}_{\mu\in\Delta_{{\mathcal{Z}}}}-\chi_{\mathcal{C}}(\mu)^{-1}\inf_{f:\mathcal{X}\to\mathcal{Y}}{\mathbb{E}}_{(X,Y)\sim\rho}\left[\ell(f(X),Y)\right]=\operatorname*{arg\,max}_{\mu\in{\mathcal{C}}}{\mathcal{R}}^{*}(\rho),$

where ${\mathcal{R}}^{*}(\rho)$ is the minimum risk in (1) one can suffer when
the data are sampled accordingly to $\rho$.

### Weak Information Disambiguation.

A final example is provided by a situation where one only has access to weak
information on the real data distribution $\rho$. This could be due to missing
data (Little and Rubin, 1987), or to only accessing label proportions
(Quadrianto et al., 2009), weak supervision on the labels (Ratner et al.,
2020), or partial labels (Cabannes et al., 2021). Once again, this problem can
be formulated as needing to find a candidate measure for what $\rho$ should
be, given only underdetermined constraints
$\rho\in{\mathcal{C}}\subset\Delta_{{\mathcal{Z}}}$ provided by the weak
information. In this setting, it is natural to fill the missing information by
looking at the simplest distribution in the set of potential distributions.
This notion of simplicity can eventually be captured by the entropy of the
distribution, leading to

$\mu^{*}\in\operatorname*{arg\,min}_{\mu\in{\mathcal{C}}}H(\mu):={\mathbb{E}}_{Z\sim\mu}[\log\left(\frac{\mathop{}\\!\mathrm{d}\mu}{\mathop{}\\!\mathrm{d}\tau}(Z)\right)],$

for some base measure $\tau\in\Delta_{{\mathcal{Z}}}$, with
$\mathop{}\\!\mathrm{d}\mu/\mathop{}\\!\mathrm{d}\tau$ denoting the
Radon–Nikodym derivative.

##  5 Conclusion

In this note, we have explained how it could be beneficial to derive a new
theory of statistical learning, where instead of focusing on the input-output
mappings, one focuses on distributions. In particular, it would be convenient
to derive a mathematical framework to cast principled population objectives on
measures into approximate objectives (the approximation being eventually due
to limited access to empirical data samples or to computation constraints),
while making sure that as the approximation gets finer, one retrieves an
optimal measure. We have motivated such a theory with several examples where
it could have a deep impact to solve some of key challenges of nowadays
machine learning.

### Acknowledgement.

VC would like to thank Léon Bottou for raising a question on out-of-
distribution generalization, and mentioning Vladimir Vapnik’s interest to
foster his theory with a measure oriented point-of-view.

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##  7 Appendix

Benchmark point | Cross-section, $\sigma$ (fb) |  $\sigma^{\prime}=\sigma\times 3l^{\prime}$ (fb)
---|---|---
BP1(400,350) | 144.26 | 4.76
BP2(500,430) | 56.42 | 1.86
BP3(800,100) | 6.19 | 0.20
BP4(1200,100) | 0.6 | 0.02
Table 7: The NLL+NLO cross-section of the signal benchmark points are listed in this table. Background | Cross-section, $\sigma$ (fb) |  $\sigma^{\prime}=\sigma\times nl^{\prime}$ (fb) | Total event generated
---|---|---|---
$WZ+jets$ | $3.619\times 10^{4}$ | 1196 ($n=3$) | 3.7$\times 10^{6}$
$ZZ+jets$ | $1.067\times 10^{4}$ | 111 ($n=4$) | 1.6$\times 10^{5}$
$t\bar{t}+jets$ | $5.930\times 10^{5}$ | 62250 ($n=2$) | 1.2$\times 10^{7}$
$WWZ+jets$ | $1.488\times 10^{2}$ | 3.3251 ($n=3$) | 9.8$\times 10^{4}$
$WWW+jets$ | $1.925\times 10^{2}$ | 6.546 ($n=3$) | 5.3$\times 10^{4}$
$t\bar{t}W+jets$ | $5.510\times 10^{2}$ | 18.74 ($n=3$) | 1.6$\times 10^{5}$
$t\bar{t}Z+jets$ | $6.893\times 10^{2}$ | 15.4 ($n=3$) | 1.3$\times 10^{5}$
$Z+jets$ | $4.516\times 10^{7}$ | 4606000 ($n=2$) | 1.3$\times 10^{6}$
Table 8: The matched cross-sections and the total event generated
corresponding to each SM background are displayed in this table.

Figure 25: The signal and background events distribution with the probability
cut in the test dataset corresponding to the BP1 for the Random Forest
classifier (upper left), AdaBoost (upper right), XGBoost classifier (lower
left) and LightGBM (lower right).

Figure 26: The distributions of the signal significance with the probability
cut are shown for all the benchmark points corresponding to each different
algorithm. Optimized hyperparameters
---
Random Forest | AdaBoost | XGBoost | LightGBM
n_estimators = 300 |  n_estimators = 30 |  num_trees = 500 |  boosting_type = ‘gbdt’
max_depth = 5 |  max_depth = 6 |  max_depth = 8 |  num_leaves = 70
max_leaf_nodes = 150 |  learning_rate = 0.03 |  learning_rate = 0.07 |  max_depth = 8
max_features = ‘sqrt’ | algorithm=“SAMME.R” |  subsample = 0.5 |  learning_rate = 0.1
criterion = ‘gini’ |  |  min_child_weight = 2.5 |  n_estimators = 200
|  |  alpha = 0.21 |  subsample = 0.5
|  |  gamma = 0.68 |  min_child_weight = 1.5
|  |  |  colsample_bytree = 0.8
Table 9: The values of optimized hyperparameters that lead to the best
performance of the model corresponding to different algorithms are mentioned
in this table.