# FedRAD: Federated Robust Adaptive Distillation

Stefán Páll Sturluson
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Samuel Trew
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Luis Muñoz-González
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Matei Grama
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Jonathan Passerat-Palmbach
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Daniel Rueckert
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>&Amir Alansary
Department of Computing
Imperial College London, UK
<EMAIL_ADDRESS>

###### Abstract

The robustness of federated learning (FL) is vital for the distributed
training of an accurate global model that is shared among large number of
clients. The collaborative learning framework by typically aggregating model
updates is vulnerable to model poisoning attacks from adversarial clients.
Since the shared information between the global server and participants are
only limited to model parameters, it is challenging to detect bad model
updates. Moreover, real-world datasets are usually heterogeneous and not
independent and identically distributed (Non-IID) among participants, which
makes the design of such robust FL pipeline more difficult. In this work, we
propose a novel robust aggregation method, Federated Robust Adaptive
Distillation (FedRAD), to detect adversaries and robustly aggregate local
models based on properties of the median statistic, and then performing an
adapted version of ensemble Knowledge Distillation. We run extensive
experiments to evaluate the proposed method against recently published works.
The results show that FedRAD outperforms all other aggregators in the presence
of adversaries, as well as in heterogeneous data distributions.

##  1 Introduction

In recent years, deep learning models have shown themselves to be powerful
prediction tools, often outperforming simpler models Krizhevsky et al. (2012).
They have been applied in a wide variety of fields, such as healthcare Jiang
et al. (2017), machine translation Bahdanau et al. (2014), and autonomous
driving Bewley et al. (2019). However, such methods are data hungry, and
require large datasets to perform well Obermeyer and Emanuel (2016). In
practice, data is usually expensive and time consuming to collect and
annotate. Besides, privacy-sensitive data cannot be directly shared globally,
as it violates the current laws stated by regulatory bodies such as the
general data protection regulations (GDPR)111https://gdpr-info.eu/ and the
health insurance portability and accountability act of
(HIPAA)222https://www.hhs.gov/hipaa/index.html.

Federated Learning (FL) enables model training collaboratively in a
decentralized manner without the need for data owners to hand over their data
McMahan et al. (2017). A typical FL pipeline consists of two main steps: (1)
training a copy of the global model locally on the client’s private data, and
(2) aggregating the local parameters into an updated global model. These two
steps are repeated until converging to a final model. Model aggregation is
usually performed by averaging the model updates (FedAvg). However, some of
the participants could be faulty or malicious by sharing bad parameters’
updates, which can ruin the global model’s performance and prevent it from
converging Yin et al. (2018).

Recent works Grama et al. (2020); Yin et al. (2018); Blanchard et al. (2017);
Muñoz-González et al. (2019); Hu et al. (2020) have been published to mitigate
the aforementioned limitations of FedAvg in the presence of malicious users
and noisy adversaries. However, these methods fail to achieve the same good
performance on heterogeneous data Zhao et al. (2018). In practice, data
heterogeneity is very common because of the different geographical locations
or tools used to acquire the data. For example, the frequency of species of
animals in natural image datasets collected in different continents will vary
a lot, leading to clients from different continents to have label-skewed
datasets.

In this paper, we propose a novel robust aggregation method that achieves
state-of-the-art performance on heterogeneous client data, while still being
able to detect adversaries or bad users and effectively defend against their
attacks. The main contributions of this paper can be summarized as:

  * •

A novel robust aggregator called Federated Robust Adaptive Knowledge
Distillation (FedRAD), which utilizes a median-based scoring system along with
knowledge distillation.

  * •

Extensive evaluations of different aggregation methods with different types of
attacks and levels of data heterogeneity.

  * •

FedRAD outperformed all other methods in the presence of adversaries, both in
IID and non-IID settings.

  * •

The code is open source and publicly available, which can accelerate
significantly the research in robust FL.

##  2 Background

Standard aggregation: Deep learning models are trained using stochastic
gradient descent (SGD), where the parameters’ updates are computed on a mini-
batch sampled from a local dataset, and back-propagated through the model.
FedSGD Shokri et al. (2017) is a federated leaning pipeline based on sampling
a fraction of the gradients for every client, which are then sent to and
averaged by a global server proportionally to the number of the samples on
each client. Federated Averaging (FedAvg) McMahan et al. (2017) is a more
efficient and generalized version of FedSGD by granting the clients to train
their local model parameters on multiple batches of the local data before
sharing the parameters of the model with the global server, instead of sharing
the gradients as in FedSGD. Hence, FedAvg can be defined as the weighted
average of the local updated models.

Model poising attacks: In distributed learning setup, the global model is
aggregated from local models trained by honest participants. However, in a
real world scenario, data may contain biases, such as wrongly annotated or
missing labels or missing data. Such faulty clients sabotage the learning
process by sending bad local models to the global server. In this work, we
highlight two main strategies for bad clients, and assess their impact on the
performance of the FL pipeline, namely:

  * •

_Faulty or Byzantine_ : Clients that add random noise to the model parameters
before sending them back to the aggregator.

  * •

_Malicious_ : Clients that train their models in the same way as healthy
clients, but instead of using correct labels they purposely train on incorrect
labels, known as a label flipping attack.

Standard aggregation methods such as FedAvg are vulnerable to these bad local
models.

Robust aggregation: Several works have been published to tackle the
aforementioned drawbacks of FedAvg against adversaries. These methods rely
mainly on the distribution of the local models or the distance between local
and global models. For example, COMED Yin et al. (2018) utilises the
coordinate-wise median of the client model parameters to construct the global
model. Other methods, such as Krum, Multi-Krum Blanchard et al. (2017), AFA
Muñoz-González et al. (2019) and FedMGDA+ Hu et al. (2020) assigns a score to
each client depending on the similarity of the models. These often consist of
using a metric, such as cosine similarity or L2 distance between local and
global model parameters. These scores are then used to determine which models
are averaged and in what proportions. AFA and FedMGDA+ can also block clients
permanently if they consistently are assigned a low score every federated
round.

Data heterogeneity: In real-world scenario, participants of a distributed
learning process are independent and have different distribution
characteristics. The majority of robust aggregation methods methods are
optimized to work on independently and identically distributed (IID) data
between the clients. Although such methods can detect and block bad
participants from the model aggregation step, benign clients with different
data or labels than the majority will be also blocked and excluded during
model aggregation. Inspired by Kairouz et al. (2019), we adapt _label skewed_
and _quantity skewed_ data for the non-IID participants, simulated from IID
datasets using the Dirichlet distribution.

Knowledge distillation: In this work, we utilize knowledge distillation for
robust model aggregation. Hinton et al. Hinton et al. (2015) showed that soft
predictions from a teacher model, or an ensemble of teacher models, can be
used to train a student model. Recently, Lin et al. Lin et al. (2020) proposed
a method, FedDF, that uses Knowledge distillation in federated setting to
learn a global model from an ensemble of client models using model fusion. A
similar work by Chen et al. Chen and Chao (2020), called FedBE, also uses
knowledge distillation to teach a student model to match predictions from a
Bayesian model ensemble. One drawback of FedDF and FedBE is that the
aggregation server needs to have its own unlabeled dataset. However, this can
be overcome using zero-shot learning, where synthetic data can be generated
for this purpose Micaelli and Storkey (2019); Nayak et al. (2019).

##  3 Federated robust adaptive distillation (FedRAD)

In this paper, knowledge distillation is performed using an ensemble of models
to construct soft pseudolabels for an unlabeled dataset on the server side. A
set of logits is predicted using the unlabeled data as an input of each model,
from which the pseudolabels are constructed. FedBE applies the softmax
function first to get the pseudo-probabilities from each model and then
average those to make the pseudolabels, while FedDF instead averages the
logits before applying a softmax function, resulting in soft pseudo-
probabilities that can be used as labels during training. This works well when
there are no attacks. However, when one or more classifiers are confidently
predicting the incorrect label, it can affect the average of the logits, as
shown in Table 3. Since logits outputs are unbounded, a single attacker can
shift the average significantly. To tackle this drawback, we make use of the
robust statistical characteristics of the median instead of the mean. This
improves the robustness of the pseudolabels against outliers from the
predicted models’ logits.

The median logits pseudolabels are analysed empirically against attackers
using the server-side unlabelled dataset. This can be done by counting how
often each client’s model has a logit output that is selected as the median
value. Histograms of how often each client’s model had the median output are
shown in Figure 1. This is a 10-class prediction task on $10,000$ images from
the server-side dataset, so there total count is $10\cdot 10,000=100,000$. The
Faulty clients, i.e. ones that send back noisy updates, have a count close to
$0$. The malicious clients are on average chosen less often than the healthy
clients, even in non-IID settings. This suggests that the pseudo-labels which
are constructed using median logits will be less affected by faulty and
malicious clients. By training the averaged model to match these pseudolabels,
the effects of adversarial models is dampened.

Figure 1: Median-counting histograms for prediction on $10,000$ images from
the MNIST dataset. There are 30 clients, 10 of which are attacking. Attackers
are clients number $2,5,8,11,14,17,20,23,26,29$. Left: IID, 10 Malicious.
Middle: Non-IID, 10 Malicious. Right: Non-IID, 10 Faulty.

Scoring system using medians: The median-based scoring algorithm assigns each
client a score based on the frequency of which it outputs the median logit for
a class prediction. This is done by counting how many times each client gave
the median logit prediciton, and then normalizing the counts such that the
scores add up to $1$, see Algorithm 1. These scores are then used for the
weighted average of the client models. Since the median-based scores are on
average lower for bad clients than healthy ones, this should decrease the
effect of the bad model updates.

Algorithm 1 Median-based scores (MedianScores)

An ensemble $E=\\{m_{i}\\}_{i=1}^{M}$ of $M$ models for $C$-dimensional
classification problem. A dataset $D$ (unlabeled).

MedianScores$(E,D)$:

$P=\\{0\\}_{i=1}^{M}$ $\triangleright$ Counter

for each datapoint $d\in D$ do

for model $m\in E$ do

$\hat{y}_{m}\leftarrow m(d)$ $\triangleright$ $(1\times C)$-dimensional logits
output of model $m$ for input $d$.

end for

$med,idx\leftarrow median(\\{\hat{y}_{m}\\}_{m\in E},dim=0)$ $\triangleright$
Median in ensemble dimension

for $i$ in $idx$ do

$P_{i}\leftarrow P_{i}+1$ $\triangleright$ Count how often model number $i$
was the median

end for

end for

$P\leftarrow P/sum(P)$ $\triangleright$ Normalizing the scores such that
$\sum_{i}P_{i}=1$

return $P$

FedRAD: The aggregation server starts by initializing the weights of a global
model. Then in each round, a subsample of clients are chosen and the global
model is sent to them. Each client trains the global model using their data,
then sends the model back to the aggregation server. Once the aggregation
server has all the models, the unlabelled dataset is used to compute the
median-based scores using Algorithm 1. These scores can be adjusted by the
size of each client’s dataset. A weighted average is then taken of the client
models to construct a new student model. Finally, the student model is trained
using median-based knowledge distillation, and assigned to the new global
model. These steps are repeated in federated rounds until convergence.
Algorithm 2 demonstrates the pseudocode of the proposed FedRAD.

Algorithm 2 Federated Robust Adaptive Distillation (FedRAD)

A set of participating clients $S=\\{k_{i}\\}_{i=1}^{M}$, each of which has a
dataset of size $n_{k}$. Unlabelled server-side dataset $D_{s}$.

FedRAD:

Initialise a global model with weights $\boldsymbol{w}_{0}^{g}$.

for each round $t=1,\dots,T$ do

$S_{t}\leftarrow$ Sample a subset of $C$ fraction of clients from $S$

Send global model weights $\boldsymbol{w}_{t-1}^{g}$ to clients in $S_{t}$

for each client $k\in S_{t}$ in parallel do

$\boldsymbol{w}_{t}^{k}\leftarrow$ Updated global model using client data

end for

$\boldsymbol{p}_{t}\leftarrow$
MedianScores$(\\{\boldsymbol{w}_{t}^{k}\\}_{k\in S_{t}},D_{s})$
$\triangleright$ See algorithm 1

$\boldsymbol{p}_{t}\leftarrow\left\\{\frac{n_{k}\cdot
p_{t}^{k}}{\sum_{k}n_{k}\cdot p_{t}^{k}}\right\\}_{k\in S_{t}}$
$\triangleright$ Adjust by size of datasets

$\boldsymbol{w}_{t}^{g}\leftarrow\sum_{k\in
S_{t}}p_{t}^{k}\cdot\boldsymbol{w}_{t}^{k}$ $\triangleright$ Element-wise
weighted average

$\boldsymbol{w}_{t}^{g}\leftarrow$
KnowledgeDistillation$(\boldsymbol{w}_{t}^{g},\\{\boldsymbol{w}_{t}^{k}\\}_{k\in
S_{t}},D_{s})$ $\triangleright$ Median-based KD

end for

##  4 Experiments and results

Simulating non-IID data: Here, we use the Dirichlet distribution which is a
multivariate continuous probability distribution. It is parameterized by a
K-dimensional vector $\boldsymbol{\alpha}=\\{\alpha_{i}\\}_{i=1}^{K},\forall
i:\alpha_{i}>0$. Its Probability Density Function (PDF) is

$f(\boldsymbol{X};\boldsymbol{\alpha})=\frac{1}{B(\boldsymbol{\alpha})}\prod_{i=1}^{K}x_{i}^{\alpha_{i}-1}$

where $B(\boldsymbol{\alpha})$ is a normalizing constant and
$\Gamma(z)=\int_{0}^{\infty}x^{z-1}e^{-x}dx$ is the gamma function.

Samples from the Dirichlet distributions are $K$-dimensional vectors. The
support is the open standard $(K-1)$-simplex,
$\\{x\in\mathbb{R}^{K}:x_{i}\in(0,1)\text{ for all
}i=1,2,...,K;\sum_{i=1}^{K}x_{i}=1\\}$.

Figure 2: Support and Contour maps of PDFs of Dirichlet distributions (K=3).
Left: Support. Middle: Symmetrical $\alpha>1$. Right: Symmetrical $\alpha<1$.

A symmetrical Dirichlet distribution can be used to simulate a non-IID split
of datasets between agents. This is done by distributing the samples in each
class unequally between agents, in proportions sampled from a symmetrical
Dirichlet distribution. If we have a dataset with $C$ classes and want to
divide the data among $K$ different agents, we can define a symmetrical
$K$-dimensional Dirichlet distribution $Dir(\boldsymbol{\alpha})$ where
$\boldsymbol{\alpha}=\alpha\cdot\boldsymbol{1}=\\{\alpha\\}_{i=1}^{K}$. We
then sample $C$ vectors from the distribution, one for each class, and divide
the data in each class between the agents in proportion with the sampled
values. The $\alpha$ hyperparameter can be tuned to control how unequally the
data classes are divided between agents. A high value of $\alpha$ leads to a
low variance in proportion, resulting in more equal splits between agents,
whereas a low $\alpha$ increases the variance of the proportions, leading to a
more non-IID data split.

Implementation: In order to thoroughly assess the performance of FedRAD, we
have implemented several aggregation methods and two types of adversarial
attacks. _Faulty_ clients are simulated by adding Gaussian noise with a
variance of $20$ to the model parameters. _Malicious_ clients are simulated by
setting all their data labels to $0$ before training.

Non-IID data distributions are simulated with three levels of heterogeneity:
IID data split with slight quantity skews, a slightly non-IID split sampled
from a symmetrical Dirichlet distribution with $\alpha=0.5$ and a very non-IID
split which uses $\alpha=0.1$, see Figure 3.

Figure 3: Examples of simulated non-IID data splits using samples from a
symmetrical Dirichlet distribution. Left: $\alpha=0.5$. Right: Symmetrical
$\alpha=0.1$.

A knowledge distillation module is implemented by training a student model to
emulate predictions from an ensemble of models. Two aggregation methods which
use Knowledge Distillation, FedDF Lin et al. (2020) and FedBE Chen and Chao
(2020), are also implemented.

For our experiments, we use the MNIST dataset LeCun (1998), which is a labeled
image dataset of handwritten digits from 0 to 9. It has a total of 70 thousand
images of size $28\times 28$. The training set includes 60 thousand images,
and the test set has the remaining 10 thousand. There are roughly the same
number of images in each class. Our classifier consists of a 3-layer fully-
connected neural network that outputs the logits it assigns to each of the 10
classes. A Softmax function is then used to get the predicted probabilities.
We ran our experiments for 30 communication rounds, with 5 local epochs and a
local learning rate of $0.05$. Performance is measured as the Error rate
calculated on the test set.

FedRAD code and implementation is publicly available on
GitHub333https://github.com/stefansturlu/FederatedMedical.

Results: The novel aggregators proposed in this project, median-based FedDF
(FedDFmed) and FedRAD, outperformed all other aggregators in the presence of
attacks, both in IID and non-IID settings. Using medians instead of average
logits for knowledge distillation is effective to combat malicious label-
flipping attacks. Furthermore, the median-based scoring system enhances the
robustness of the trained models by detecting faulty attackers and assigning
them very low scores. It is also successful in giving lower scores to
malicious attackers than healthy agents. FedRAD combines the novel median-
based knowledge distillation with the novel median-based scoring system.

Table 1 shows that it is more difficult to detect malicious attacks than
faulty ones by examining the model parameters, as malicious agents use the
same training procedure as healthy clients and aren’t extremely noisy like the
faulty models. The model weights of malicious agents are similarly distributed
to healthy ones, and in non-IID settings it becomes difficult to detect
malicious agents using similarity metrics. Table 1 also demonstrates that
FedRAD under a variety of attacks and data heterogeneity outperforms all the
other aggregation methods.

Attacks |  Aggregator | Error rates ($\%$)
---|---|---
IID | non-IID
| $\alpha=0.5$ | $\alpha=0.1$
No attacks |  FA |  $\boldsymbol{5.24\pm 0.07}$ |  $\boldsymbol{5.68\pm 0.11}$ |  $\boldsymbol{9.13\pm 0.82}$
COMED |  $\boldsymbol{5.30\pm 0.08}$ |  $6.38\pm 0.18$ |  $81.15\pm 12.94$
MKRUM |  $5.68\pm 0.16$ |  $7.63\pm 0.62$ |  $40.93\pm 13.51$
AFA |  $\boldsymbol{5.28\pm 0.17}$ |  $5.94\pm 0.16$ |  $\boldsymbol{10.36\pm 0.96}$
FedMGDA++ |  $6.28\pm 0.25$ |  $12.15\pm 5.39$ |  $21.12\pm 3.84$
FedDF |  $5.43\pm 0.10$ |  $\boldsymbol{5.82\pm 0.15}$ |  $\boldsymbol{9.17\pm 0.40}$
FedDFmed |  $5.45\pm 0.08$ |  $\boldsymbol{5.86\pm 0.25}$ |  $\boldsymbol{9.24\pm 0.43}$
FedRAD |  $\boldsymbol{5.29\pm 0.16}$ |  $\boldsymbol{5.82\pm 0.15}$ |  $\boldsymbol{9.38\pm 0.57}$
10 Faulty |  FA |  $90.03\pm 0.96$ |  $89.58\pm 0.50$ |  $90.10\pm 0.63$
COMED |  $\boldsymbol{5.39\pm 0.14}$ |  $6.44\pm 0.28$ |  $69.22\pm 22.75$
MKRUM |  $89.66\pm 0.63$ |  $90.37\pm 0.37$ |  $90.03\pm 0.65$
AFA |  $70.97\pm 32.56$ |  $90.12\pm 0.47$ |  $89.70\pm 0.33$
FedMGDA++ |  $89.82\pm 0.71$ |  $89.37\pm 0.28$ |  $89.89\pm 0.43$
FedDF |  $90.06\pm 0.82$ |  $89.65\pm 1.01$ |  $89.71\pm 0.62$
FedDFmed |  $81.02\pm 3.56$ |  $90.14\pm 0.54$ |  $90.11\pm 0.61$
FedRAD |  $\boldsymbol{5.44\pm 0.07}$ |  $\boldsymbol{5.91\pm 0.31}$ |  $\boldsymbol{12.63\pm 3.65}$
10 Malicious |  FA |  $17.33\pm 3.64$ |  $74.87\pm 30.66$ |  $38.09\pm 27.79$
COMED |  $25.11\pm 1.39$ |  $48.51\pm 21.52$ |  $90.08\pm 0.40$
MKRUM |  $31.15\pm 25.15$ |  $10.82\pm 2.19$ |  $60.78\pm 15.24$
AFA |  $17.68\pm 4.46$ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$
FedMGDA++ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$
FedDF |  $65.20\pm 28.69$ |  $76.12\pm 28.16$ |  $61.49\pm 31.78$
FedDFmed |  $8.34\pm 0.25$ |  $9.24\pm 1.23$ |  $16.74\pm 3.87$
FedRAD |  $\boldsymbol{5.89\pm 0.20}$ |  $\boldsymbol{6.55\pm 0.43}$ |  $\boldsymbol{13.09\pm 1.97}$
5 Faulty, 5 Malicious |  FA |  $89.64\pm 0.55$ |  $89.75\pm 0.75$ |  $89.62\pm 0.63$
COMED |  $8.00\pm 0.42$ |  $10.21\pm 0.93$ |  $89.10\pm 0.90$
MKRUM |  $89.60\pm 0.36$ |  $89.66\pm 0.43$ |  $89.82\pm 0.67$
AFA |  $67.62\pm 6.64$ |  $67.82\pm 30.59$ |  $80.97\pm 17.03$
FedMGDA++ |  $90.20\pm 0.00$ |  $90.21\pm 0.00$ |  $89.31\pm 1.61$
FedDF |  $90.39\pm 0.57$ |  $90.17\pm 0.67$ |  $89.97\pm 0.18$
FedDFmed |  $55.11\pm 5.41$ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$
FedRAD |  $\boldsymbol{5.63\pm 0.20}$ |  $\boldsymbol{6.15\pm 0.40}$ |  $\boldsymbol{10.95\pm 1.22}$
Table 1: Test set error rate for MNIST after 30 rounds. Average and standard
deviation of test-set error rates shown for $5$ different random seeds. The
best performing methods in each category, and those not statistically
significantly different according to a t-test, are in bold.

Figure 4 displays the learning curves in experiments with both faulty and
malicious attackers in IID and non-IID settings. With 5 faulty and 5 malicious
attacks, COMED did well in IID scenarios, but as usual failed for non-IID
scenarios. However, FedRAD consistently performed well under both types of
attacks at the same time. More results can be found in Appendix D.

Figure 4: Effects of non-IID data with $5$ faulty and $5$ malicious attackers.

##  5 Conclusions

In this work, we propose FedRAD for robust federated aggregation using a
scoring system based on the median statistic and network distillation. In
contrary to previous robust aggregation methods that use the high-dimensional
vectors of parameters of the models, our method use the output predictions
from the models for detecting adversaries in the presence of heterogeneous
client data.

Extensive experiments were conducted using the MNIST dataset to evaluate the
performance of different baseline methods in the presence of attackers, for
both IID and non-IID client data. Existing aggregators, like COMED and Multi-
Krum, were shown to break down when client data is heterogeneous. Existing
adaptive aggregators, AFA and FedMGDA+, were also shown to be unable to detect
and block adversaries in non-IID settings.

To conclude, FedRAD outperforms all other aggregation methods in the presence
of attacks, and is the only aggregator to successfully combat adversaries in
both IID and non-IID settings. We will investigate the performance of FedRAD
with privacy preserving techniques such as differential privacy in the future.

## Acknowledgments and Disclosure of Funding

Acknowledgments This work is supported by the UK Research and Innovation
London Medical Imaging and Artificial Intelligence Centre for Value Based
Healthcare, and in part by the Intramural Research Program of the National
Institutes of Health Clinical Center.

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  * Hinton et al. [2015] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean.  Distilling the knowledge in a neural network.  _arXiv preprint arXiv:1503.02531_ , 2015.
  * Lin et al. [2020] Tao Lin, Lingjing Kong, Sebastian U Stich, and Martin Jaggi.  Ensemble distillation for robust model fusion in federated learning.  _arXiv preprint arXiv:2006.07242_ , 2020.
  * Chen and Chao [2020] Hong-You Chen and Wei-Lun Chao.  Fedbe: Making bayesian model ensemble applicable to federated learning.  _arXiv preprint arXiv:2009.01974_ , 2, 2020.
  * Micaelli and Storkey [2019] Paul Micaelli and Amos Storkey.  Zero-shot knowledge transfer via adversarial belief matching.  _arXiv preprint arXiv:1905.09768_ , 2019.
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  * LeCun [1998] Yann LeCun.  The mnist database of handwritten digits.  _http://yann. lecun. com/exdb/mnist/_ , 1998.

##  Appendix A Federated learning pipeline

Figure 5: Federated Learning pipeline. A: The global model is sent to a random
subset of clients. B: The clients train the model locally using their data. C:
The local models are sent to an aggregation server. D: The client models are
aggregated to make a new global model. This process is repeated until
convergence.

##  Appendix B Robust aggregation methods

Name |  Description | Robust | Server data
---|---|---|---
FA (or FedAvg) |  Federated Averaging McMahan et al. [2017] | No | No
COMED |  Coordinate-wise median Yin et al. [2018] | Yes | No
MKRUM |  Multi-Krum Blanchard et al. [2017] | Yes | No
AFA |  Adaptive Federated Averaging Muñoz-González et al. [2019] | Yes | No
FedMGDA++ |  FedMGDA+ Hu et al. [2020] | Yes | No
FedDF |  FedDF Lin et al. [2020] | No | Yes
FedDFmed |  FedDF with median-based Knowledge Distillation | Yes | Yes
FedBE |  FedBE Chen and Chao [2020] | No | Yes
FedBEmed |  FedBE with median-based Knowledge Distillation | Yes | Yes
FedADF |  AFA Muñoz-González et al. [2019] with an additional median-based Knowledge Distillation | Yes | Yes
FedMGDA+DF |  FedMGDA+ Hu et al. [2020] with an additional median-based Knowledge Distillation | Yes | Yes
FedRAD |  Our novel aggregator, see Algorithm 2 | Yes | Yes
FedRADnoise |  FedRAD using uniform noise instead of server-side data | Yes | No
Table 2: List of aggregation methods. Methods in bold are novel. "Robust" is
used to describe whether steps have been taken to detect or defend against
adversaries. "Server data" is used to indicate whether an _unlabelled_ dataset
is required on the server side.

##  Appendix C Robustness of the median

Table 3 illustrates the advantage of using a robust statistic such as the
median instead of mean for model aggregation.

| Logits predictions from 10 clients | Average | Median
---|---|---|---
No attacker | $[1,1,2,2,3,3,4,4,5,5]$ | $3$ | $3$
One weak attacker | $[1,1,2,2,3,3,4,4,5,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}15}]$ | $4$ | $3$
Four weak attackers | $[1,1,2,2,3,3,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}14},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}14},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}15},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}15}]$ | $7$ | $3$
One strong attackers | $[1,1,2,2,3,3,4,4,5,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1005}]$ | $103$ | $3$
Four strong attackers | $[1,1,2,2,3,3,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1004},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1004},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1005},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1005}]$ | $403$ | $3$
Table 3: An illustrative example demonstrating the robustness of the median
against attackers with confidently incorrect predictions. The logits values
are hypothetical outputs from 10 different models for a particular class.
Attackers are coloured in red.

##  Appendix D Learning curves of attacks

###  D.1 No attacks

Results from experiments with no attacks for different levels of data
heterogeneity can be seen in Figure 6.

Figure 6: Effects of non-IID data for no attacks.

All the methods perform roughly the same on IID data when there are no
attackers. With increased heterogeneity, the learning starts to slow down for
all aggregators. When data is very non-IID, i.e. $\alpha=0.1$, the performance
of aggregators such as COMED, MKRUM, AFA and FedMGDA+ declines significantly.
Both FedMGDA+ and AFA slow down quite a bit, and both of them block some
healthy clients which effectively decreases the size of the training set.

Federated Averaging still performs well in non-IID circumstances when there
are no attackers. Methods that use Knowledge Distillation also perform
similarly to FedAvg.

###  D.2 Faulty Attacks

Faulty attackers, also known as Byzantine, are attackers which update their
models by adding a lot of random noise to the model parameters. Results for 1,
5 and 10 faulty attackers can be seen in Figures 7, 8 and 9 respectively.

Figure 7: Effects of non-IID data with $1$ Faulty attacker.

Just $1$ faulty attacker completely ruins performance for non-robust
aggregators. Both AFA and FedMGDA+ are able to effectively block the attacker
and learn, even with non-IID data.

Figure 8: Effects of non-IID data with $5$ Faulty attackers.

Once more attackers are added, AFA and FedMGDA+ start to fail. COMED still
works for homogeneous data. The only aggregators that work against 5 faulty
clients in the most heterogeneous case are our novel methods.

Figure 9: Effects of non-IID data with $10$ Faulty attackers.

With 10 attackers and non-IID data FedRAD is the only aggregator that manages
to learn anything. Even FedRADnoise, which uses only random noise for scoring
and distillation, is able to learn well. The good performance of aggregators
which use our novel median-scoring mechanism against is explained by Figure 1,
which shows that Faulty models rarely ever give the median logit prediction
for a given class, and thus often get a score of $0$, which is equivalent to
leaving faulty models out completely during the weighted averaging.

###  D.3 Malicious Attacks

Malicious attackers are agents who train the models with incorrect labels. In
our case, all labels are set to $0$ before training. Results for 1, 5, and 10
attackers can be seen in Figures 10, 11 and 12 respectively.

Figure 10: Effects of non-IID data with $1$ malicious attacker.

One malicious attacker does not impact learning very much.

Figure 11: Effects of non-IID data with $5$ malicious attackers.

With 5 malicious attackers, many aggregators start to perform badly and the
training becomes very noisy. This is where the benefits of median-based
Knowledge distillation starts to shine through: FedDF training is very noisy,
but FedDFmed is among the best performing methods.

Figure 12: Effects of non-IID data with $10$ malicious attackers.

With 10 malicious attackers, even more aggregators start to fail. Robust
methods which rely on comparing model parameters, such as AFA and FedMGDA+,
don’t learn anything at all in the non-IID case.

The best performing aggregators against malicious attacks in both IID and non-
IID scenarios are our novel methods which utilize median-based Knowledge
Distillation: FedDFmed and FedRAD. This is explained by the median-counting
histograms in Figure 1, which show that median logits are less contaminated by
malicious agents.

MKRUM, AFA and FedMGDA+ fail because models from malicious agents are quite
similar to healthy models when comparing their high-dimensional parameters. In
non-IID situations, it becomes difficult to detect malicious models among the
non-IID healthy models by using distance metrics on their parameters.

###  D.4 Both Types of Attacks

Experiments are done using both faulty and malicious attackers in IID and non-
IID settings. The results from these can be seen in Figures 4, and 13.

Using 10 faulty, 10 malicious and only 10 honest agents, we finally reached
the breaking point. But even with this many attakcers, our FedRAD aggregator
was the only one that showed any progress and managed to reach a $60\%$ error
rate on IID data, which is respectable considering only a third of the agents
are honest.

Figure 13: Effects of non-IID data with $10$ faulty and $10$ malicious
attackers.

##  Appendix E Median-based knowledge distillation for different aggregators

The median based knowledge distillation module is also tested with AFA and
FedMGDA, called FedADF and FedMGDA+DF. FedBE and FedBEmed (which uses median-
logits) were also tested. Results are shown in Table 4

Attacks |  Aggregator | Error rates ($\%$)
---|---|---
IID | non-IID
|  $\alpha=0.5$ |  $\alpha=0.1$
No attacks |  AFA |  $\boldsymbol{5.28\pm 0.17}$ |  $5.94\pm 0.16$ |  $\boldsymbol{10.36\pm 0.96}$
FedMGDA++ |  $6.28\pm 0.25$ |  $12.15\pm 5.39$ |  $21.12\pm 3.84$
FedBE |  $\boldsymbol{5.36\pm 0.13}$ |  $\boldsymbol{5.93\pm 0.22}$ |  $11.70\pm 1.38$
FedBEmed |  $5.39\pm 0.10$ |  $\boldsymbol{5.85\pm 0.28}$ |  $\boldsymbol{9.67\pm 0.29}$
FedADF |  $5.45\pm 0.14$ |  $6.38\pm 0.31$ |  $16.81\pm 4.55$
FedMGDA+DF |  $6.27\pm 0.32$ |  $8.29\pm 0.45$ |  $15.11\pm 0.57$
FedRAD |  $\boldsymbol{5.29\pm 0.16}$ |  $\boldsymbol{5.82\pm 0.15}$ |  $\boldsymbol{9.38\pm 0.57}$
FedRADnoise |  $\boldsymbol{5.32\pm 0.15}$ |  $5.93\pm 0.09$ |  $\boldsymbol{9.71\pm 0.57}$
10 Faulty |  AFA |  $70.97\pm 32.56$ |  $90.12\pm 0.47$ |  $89.70\pm 0.33$
FedMGDA++ |  $89.82\pm 0.71$ |  $89.37\pm 0.28$ |  $89.89\pm 0.43$
FedBE |  $90.24\pm 0.85$ |  $90.02\pm 1.14$ |  $89.87\pm 0.55$
FedBEmed |  $90.30\pm 0.86$ |  $90.27\pm 0.72$ |  $89.40\pm 0.68$
FedADF |  $\boldsymbol{5.44\pm 0.18}$ |  $6.45\pm 0.36$ |  $27.68\pm 30.61$
FedMGDA+DF |  $54.54\pm 8.31$ |  $90.11\pm 0.32$ |  $90.19\pm 0.46$
FedRAD |  $\boldsymbol{5.44\pm 0.07}$ |  $\boldsymbol{5.91\pm 0.31}$ |  $\boldsymbol{12.63\pm 3.65}$
FedRADnoise |  $\boldsymbol{5.29\pm 0.21}$ |  $\boldsymbol{5.99\pm 0.08}$ |  $\boldsymbol{13.07\pm 3.29}$
10 Malicious |  AFA |  $17.68\pm 4.46$ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$
FedMGDA++ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$ |  $90.20\pm 0.00$
FedBE |  $30.40\pm 3.75$ |  $33.50\pm 9.16$ |  $29.53\pm 3.10$
FedBEmed |  $25.14\pm 4.03$ |  $63.77\pm 32.99$ |  $54.21\pm 34.26$
FedADF |  $13.11\pm 2.67$ |  $90.20\pm 0.00$ |  $84.83\pm 10.74$
FedMGDA+DF |  $70.55\pm 7.44$ |  $67.68\pm 11.76$ |  $86.42\pm 5.78$
FedRAD |  $\boldsymbol{5.89\pm 0.20}$ |  $\boldsymbol{6.55\pm 0.43}$ |  $\boldsymbol{13.09\pm 1.97}$
FedRADnoise |  $90.20\pm 0.00$ |  $73.45\pm 33.50$ |  $59.49\pm 37.61$
5 Faulty, 5 Malicious |  AFA |  $67.62\pm 6.64$ |  $67.82\pm 30.59$ |  $80.97\pm 17.03$
FedMGDA++ |  $90.20\pm 0.00$ |  $90.21\pm 0.00$ |  $89.31\pm 1.61$
FedBE |  $90.36\pm 0.28$ |  $90.47\pm 0.51$ |  $90.25\pm 0.32$
FedBEmed |  $90.21\pm 1.21$ |  $90.25\pm 0.42$ |  $89.84\pm 0.31$
FedADF |  $6.01\pm 0.19$ |  $9.53\pm 3.76$ |  $19.47\pm 8.33$
FedMGDA+DF |  $47.88\pm 4.25$ |  $87.82\pm 2.87$ |  $89.91\pm 0.82$
FedRAD |  $\boldsymbol{5.63\pm 0.20}$ |  $\boldsymbol{6.15\pm 0.40}$ |  $\boldsymbol{10.95\pm 1.22}$
FedRADnoise |  $\boldsymbol{5.64\pm 0.17}$ |  $23.39\pm 33.41$ |  $12.84\pm 1.52$
Table 4: Test set error rate for MNIST after 30 rounds. Average and standard
deviation of test-set error rates shown for $5$ different random seeds.


11institutetext: Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, 4150-762 Porto, Portugal
Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá,
PR, Brazil

Dynamics of evolution Population dynamics and ecological pattern formation
Ecology and evolution

# Weak species in rock-paper-scissors models

P.P. Avelino 1122 B.F. de Oliveira 33 and R.S. Trintin 33112233

###### Abstract

In this letter, we investigate the population dynamics in a May-Leonard
formulation of the rock-paper-scissors game in which one or two species, which
we shall refer to as “weak”, have a reduced predation or reproduction
probability. We show that in a nonspatial model the stationary solution where
all three species coexist is always unstable, while in a spatial stochastic
model coexistence is possible for a wide parameter space. We find, that a
reduced predation probability results in a significantly higher abundance of
“weak” species, in models with either one or two “weak” species, as long as
the simulation lattices are sufficiently large for coexistence to prevail. On
the other hand, we show that a reduced reproduction probability has a smaller
impact on the abundance of “weak” species, generally leading to a slight
decrease of its population size — the increase of the population size of one
of the “weak” species being more than compensated by the reduction of the
other, in the two species case. We further show that the species abundances in
models where both predation and reproduction probabilities are simultaneously
reduced may be accurately estimated from the results obtained considering only
a reduction of either the predation or the reproduction probability.

###### pacs:

87.23.Kg

###### pacs:

87.23.Cc

###### pacs:

87.23.-n

##  1 Introduction

Ecology is the study of the interactions of living organisms with one another
and with their ecosystem. Intra-specific and inter-specific interactions are
essential for the coexistence of a wide variety of species and are, therefore,
also crucial for development and sustainability of biodiversity. Evolutionary
games usually seek to quantify this natural process through the simulation of
the dynamics of populations with different species subject to non-hierarchical
interactions.

The classic RPS model (see [1, 2, 3] for the pioneer work by Lotka and
Volterra, and May and Leonard) describes the dynamics of a population of three
species with cyclic predation, as well as mobility and reproduction
interactions. In spatial RPS models the coexistence of all three species can
be maintained in a significant parameter volume of predation, mobility and
reproduction probabilities [4, 5, 6, 7]. Corresponding biological examples
have been found in nature, including E. coli bacteria [8, 9] and lizards [10].

Population dynamics in spatial stochastic RPS type models has been
investigated thoroughly in the literature considering a wide variety of models
and perspectives [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]. Although such dynamics is usually strongly dependent on a
variety of model parameters — including predation, reproduction and mobility
probabilities — in most cases these are assumed to be the same for all
species. However, this simplifying assumption is not expected to constitute a
valid approximation in the modelling of most biological systems in nature.

The impact of the reduction of the predation probability of a single
(“weakest”) species has been investigated in the context of the spatial
stochastic three species RPS model [30, 31, 32, 33]. It has been shown that
coexistence of all three species prevails for sufficiently large simulation
lattices, with the “weakest” species being the most abundant [32] — for
smaller simulation lattices, large amplitude oscillations at the initial
stages of simulations with random initial conditions can lead to a significant
dependence of the chance of “weakest” species survival on the lattice size
[32, 33]. The crucial importance of the simulation lattice size has also been
recognized in the context of public goods games with punishment [34, 35, 36,
37]. Recent research [38] dealing with the inhibition of protein production,
the digestion of genomic DNA and the rupture of cellular membrane with three
strains of E. coli interacting cyclically in an asymmetric system also
concluded for the dominance of the “weakest” strain.

In this letter we extend previous work on the performance of “weak” species,
by investigating the dynamics of RPS models in which one or two species have
their predation and/or reproduction probability reduced. We start by
describing a nonspatial RPS model, showing that it does not allow for stable
stationary solutions where all three species coexist. We then describe our
May-Leonard implementation of the spatial stochastic RPS model and assess the
performance of “weak” and “strong” species in numerical lattice simulations
allowing for the coexistence of all three species. In particular, we shall
highlight the dependence of the results on whether the “weak” species are
characterized by a reduced predation or reproduction probability. We will also
describe an approximation allowing for the estimate of the species abundances,
in models where both predation and reproduction probabilities are
simultaneously reduced, from the results obtained considering only a reduction
of either the predation or the reproduction probability. Finally, we present
the main conclusions of our work.

##  2 Nonspatial RPS model

Here we shall consider a four-state cyclic predator-prey model, also known as
a May-Leonard implementation of the RPS model. The densities of the species
$i=1,..,3$ shall be denoted by $\rho_{i}$. For simplicity of notation, the
density associated to the additional state — denoted by a ‘0’ — shall be
referred to as the density of empty sites, even when considering a nonspatial
model. The total density is normalized to unity, so that

$\rho_{0}+\sum_{i=1}^{3}\rho_{i}=1\,.$ (1)

In a nonspatial RPS model there are two possible interactions: predation [$i\
(i+1)\to i\ 0$] and reproduction [$i\ 0\to i\ i$], with probabilities $p_{i}$
and $r_{i}$, respectively (we shall use modular arithmetic throughout the
paper, so that the integers $i$ and $j$ represent the same species whenever
$i=j\ {\rm mod}\ 3$, where mod denotes the modulo operation). With an
appropriate choice of time unit, the set of equations that represent the
evolution of the densities of the different species may be written as

$\displaystyle{\dot{\rho}_{i}}$ $\displaystyle=$ $\displaystyle
r_{i}\,\rho_{i}\,\rho_{0}-p_{i-1}\,\rho_{i-1}\,\rho_{i}\,,$ (2)

The stationary solutions to Eq. (2) are

$\displaystyle\rho_{i}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left[1+\dfrac{p_{i}}{r_{i+1}}+\dfrac{p_{i}r_{i-1}}{p_{i+1}r_{i+1}}+\dfrac{p_{i}r_{i}}{p_{i-1}r_{i+1}}\right]}\,,$
(3) $\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left[1+\dfrac{r_{1}}{p_{3}}+\dfrac{r_{2}}{p_{1}}+\dfrac{r_{3}}{p_{2}}\right]}\,.$
(4)

We have verified that there are no stationary solutions to Eq. (2) with
$\rho_{i}\neq 0$ for all $i=1,2,3$ in which the real part of all the
eigenvalues of the Jacobian matrix, defined by

$\begin{bmatrix}\frac{\partial\dot{\rho_{1}}}{\partial\rho_{1}}&\frac{\partial\dot{\rho_{1}}}{\partial\rho_{2}}&\frac{\partial\dot{\rho_{1}}}{\partial\rho_{3}}\\\
\frac{\partial\dot{\rho_{2}}}{\partial\rho_{1}}&\frac{\partial\dot{\rho_{2}}}{\partial\rho_{2}}&\frac{\partial\dot{\rho_{2}}}{\partial\rho_{3}}\\\
\frac{\partial\dot{\rho_{3}}}{\partial\rho_{1}}&\frac{\partial\dot{\rho_{3}}}{\partial\rho_{2}}&\frac{\partial\dot{\rho_{3}}}{\partial\rho_{3}}\end{bmatrix}=$
$\begin{bmatrix}r_{1}(\rho_{0}-\rho_{1})-p_{3}\rho_{3}&-r_{1}\rho_{1}&-r_{1}\rho_{1}-p_{3}\rho_{1}\\\
-r_{1}\rho_{1}-p_{3}\rho_{1}&r_{2}(\rho_{0}-\rho_{2})-p_{1}\rho_{1}&-r_{2}\rho_{2}\\\
-r_{3}\rho_{3}&-r_{3}\rho_{3}-p_{2}\rho_{3}&r_{3}(\rho_{0}-\rho_{3})-p_{2}\rho_{2}\end{bmatrix}\,,$

is negative. Therefore, we conclude that the stationary solutions to Eq. (2),
given by Eqs. (3) and (4), leading to the coexistence of all three species are
always unstable in a nonspatial RPS model[39].

##  3 May-Leonard spatial stochastic RPS model

In this letter we shall consider a May-Leonard implementation of the spatial
RPS model on a two-dimensional square lattice with $\mathcal{N}$ sites and
periodic boundary conditions. At each simulation time step, an individual
(active) is randomly chosen to interact with one of its four nearest neighbors
(passive) — also randomly selected. Then, one of three possible interactions
(predation, reproduction, or mobility) is randomly chosen (with probabilities
$p$, $m$ and $r$, respectively). In this paper we shall consider the so-called
von Neumann neighbourhood (or 4-neighbourhood) composed of a central cell (the
active one) and its four non-diagonal adjacent cells. Whenever mobility is
selected the active swaps place with the passive. On the other hand, if
reproduction is selected and the passive is an empty site (otherwise the
interaction cannot be completed, and the procedure is repeated once again),
then the site becomes occupied with an individual of the same species of the
active. Finally, if predation is selected and the passive is a prey of the
active (otherwise the interaction cannot be completed, and the procedure is
repeated once again), then the passive is removed from the lattice and an
empty site is left at its previous position. A time step is completed whenever
one interaction is carried out. A generation time (our time unit) corresponds
to $\mathcal{N}$ time steps.

Here we shall investigate the dynamical impact of a reduction of the predation
or reproduction probabilities of one or two species by a factor of
$\mathcal{P}_{w}\in\ ]0,1]$ or $\mathcal{R}_{w}\in\ ]0,1]$, respectively. We
will consider the following cases: Model $P_{1}$ \- one species with reduced
predation probability, $p_{1}=\mathcal{P}_{w}p$ ($p_{2}=p_{3}=p$); Model
$R_{1}$ \- one species with reduced reproduction probability
$r_{1}=\mathcal{R}_{w}r$ ($r_{2}=r_{3}=r$); Model $P_{2}$ two species with
reduced predation probabilities $p_{1}=p_{2}=\mathcal{P}_{w}p$ ($p_{3}=p$);
Model $R_{2}$ \- two species with reduced reproduction probabilities
$r_{1}=r_{2}=\mathcal{R}_{w}r$ ($r_{3}=r$). In the models $P_{1}$ and $P_{2}$
we take $r_{1}=r_{2}=r_{3}=r$. In the models $R_{1}$ and $R_{2}$ it is assumed
that $p_{1}=p_{2}=p_{3}=p$. We shall also consider two additional models,
$(PR)_{1}$ and $(PR)_{2}$ (with one and two “weak” species, respectively), in
which both the predation and reproduction probabilities are reduced.

Fig. 1 displays the scheme of cyclic predator-prey interactions of the $3$
species RPS models we investigate in this letter, with the filled and unfilled
circles representing “weak” and “strong” species, respectively — cases (a) and
(b) correspond to models with one and two “weak” species. As previously
mentioned, the six models considered in this letter (three with one “weak”
species and three with two “weak” species) are denoted by $P_{n}$, $R_{n}$ and
$(PR)_{n}$, depending, respectively, on whether only the predation rate is
reduced, only the reproduction rate is reduced or both — $n$ represents the
number of “weak” species ($n=1$ or $n=2$). Model $P_{1}$ has also been
investigated in [32].

Figure 1: Scheme of predation interactions of the $3$ species RPS model: (a)
represents the case with a single “weak” species (models $P_{1}$, $R_{1}$,
$(PR)_{1}$); (b) represents the case with two ‘weak” species (models $P_{2}$,
$R_{2}$, $(PR)_{2}$). Filled and unfilled circles represent “weak” and
“strong” species, respectively.

##  4 Results

Here we assess the performance of “weak” and “strong” species using spatial
stochastic simulations of the RPS models $P_{n}$, $R_{n}$ and $(PR)_{n}$, with
$n=1,2$.

Fig. 2 shows the value of the average density of the three species and empty
sites as a function of $\mathcal{P}_{w}$ for the model $P_{1}$ with a single
“weak” species. $\mathcal{P}_{w}$, represents the reduction of the predation
probability of the “weak” species with respect to the standard case in which
the predation probability is the same for all species. The data points result
from an average over the last $10^{4}$ generations of a simulation with a time
span equal to $1.5\times 10^{4}$ generations performed on a $1000^{2}$
lattice, large enough to guarantee the preservation of coexistence. The
results for $\mathcal{P}_{w}=1$ were computed starting from random initial
conditions. To ensure fast convergence we used the final conditions of the
simulations with $\mathcal{P}_{w}=1$ as initial conditions of new simulations
with $\mathcal{P}_{w}=1-0.01$. The same procedure was repeated until
$\mathcal{P}_{w}=0.3$ was reached. We verified that, with such conditions,
$5\times 10^{3}$ generations are sufficient for $\langle\rho_{i}\rangle$ and
$\langle\rho_{0}\rangle$ to attain their asymptotic value. Fig. 2 shows that
the “weak” species generally has an advantage, specially over its predator
(the “weak” species abundance is similar to that of its prey for
$\mathcal{P}_{w}\in[0.5,1]$). The lines represent the nonspatial stationary
analytical (unstable) solution, given in Eqs. (3) and (4), for the density of
the species 1 (solid line), and 2 and 3 and empty sites (dashed line).

Figure 2: The average density of the species ($i=1,2,3$) and empty sites
($i=0$) for the spatial model $P_{1}$, as a function of $\mathcal{P}_{w}$,
assuming $m=0.5$ and $p=r=0.25$. Notice that the “weak” species and its prey
are the most abundant. The grey lines represent the stationary nonspatial
solution, given in Eqs. (3) and (4), for the density of the species 1 (solid
line), 2 and 3 and empty sites (dashed line). Notice the significant
differences between spatial and nonspatial results (this is also be verified
in Figs. 3, 4, and 5).

Fig. 3 is analogous to Fig. 2, except that in Fig. 3 model $R_{1}$ is
considered, with $\mathcal{R}_{w}$ representing the reduction of the predation
probability of the “weak” species with respect to the standard case in which
the predation probability is the same for all species. Fig. 3 shows that in
this case the density of the “weak” species is always smaller than the average
density of the “strong” species. The relatively small advantage that the
“weak” species has over its predator does not compensate the disadvantage with
respect to its prey.

Figure 3: The average density of the species ($i=1,2,3$) and empty sites
($i=0$) for the spatial model $R_{1}$, as a function of ${\mathcal{R}}_{w}$,
assuming $m=0.5$ and $p=r=0.25$. In this case the performance of the “strong”
species surpasses, on average, that of the “weak” species. The grey lines
represent the stationary nonspatial solution, given in Eqs. (3) and (4), for
the density of the species 1 (solid line), 2 and 3 and empty sites (dashed
line).

Fig. 4 displays the results of the model $P_{2}$. In this model there are two
“weak” species which have a lower predation probability. Fig. 4 shows that, in
this model, the two “weak” species ($1$ and $2$, but specially species $2$)
generally have advantage over the “strong” species ($3$). This shows that the
advantage of the “weak” species found in model $P_{1}$ is also observed in
three species RPS models with two “weak” species.

Figure 4: The same as in Fig. 2, but now for the spatial model $P_{2}$ with
two “weak” species having a lower predation probability. Notice that both
“weak” species are more abundant than the “strong” species for
$\mathcal{P}_{w}\lesssim 0.6.$. The grey lines represent the stationary
nonspatial solution, given in Eqs. (3) and (4), for the density of the species
1 and 2 (dashed line), and 3 and empty sites (solid line).

Fig. 5 displays the results of the model $R_{2}$. This model has two “weak”
species characterized by a lower reproduction rate. Fig. 5 shows that, in this
model, only one of the two “weak” species ($2$, the predator of the “strong”
species) has an advantage over the “strong” species ($3$). However, contrary
to the case of models $P_{1}$ and $P_{2}$, and in agreement with model
$R_{1}$, for ${\mathcal{R}}_{w}>0.5$ there is on average no advantage in being
the “weak” species.

Figure 5: The same as in Fig. 3, but now for the spatial model $R_{2}$. In
this case one weak species is the most abundant and the other is the least
abundant. The grey lines represent the stationary nonspatial solution, given
in Eqs. (3) and (4), for the density of the species 1 and 3 (solid line), and
2 and empty sites (dashed line).

The discrepancies between spatial and nonspatial solutions shown in Figs. 2,
3, 4, 5 for models $P_{1}$, $R_{1}$, $P_{2}$ and $R_{2}$, respectively, are
significant. They are a demonstration of the importance of the spatial
structure, such as the spiral patterns that emerge on this type of models, on
the dynamics of population networks. It is this spatial structure which is
responsible for the stability of the population of three species in
sufficiently large spatial simulations, in sharp contrast with the instability
associated to the nonspatial stationary solutions.

Fig. 6 displays the results of the model $(PR)_{1}$ with one “weak” species
($1$). These results were obtained using the same procedure followed in Figs.
2 and Figs. 3 for models $P_{1}$ and $R_{1}$, respectively, but now reducing
both the predation and reproduction of the “weak” species ($1$) by the same
amount ($(\mathcal{PR})_{w}={\mathcal{P}}_{w}={\mathcal{R}}_{w}$). Fig. 6
shows that, in this case, one of the strong species the species ($2$) is the
most abundant followed by the “weak” species ($1$). The “weak” species is
therefore not the most abundant, but its population is always larger than that
of its predator.

Motivated by the fact that, for fixed $p$, $r$ and $m$, and
$1-{\mathcal{P}}_{w}\ll 1$ and $1-{\mathcal{R}}_{w}\ll 1$

$\displaystyle\rho({\mathcal{P}}_{w},{\mathcal{R}}_{w})$ $\displaystyle\sim$
$\displaystyle\rho(1,1)+\left.\frac{\partial\rho}{\partial{\mathcal{P}}_{w}}\right|_{(1,1)}(\mathcal{P}_{w}-1)$
(5)
$\displaystyle+\left.\frac{\partial\rho}{\partial{\mathcal{R}}_{w}}\right|_{(1,1)}(\mathcal{R}_{w}-1)$
$\displaystyle\sim$
$\displaystyle\rho({\mathcal{P}}_{w},1)+\rho(1,{\mathcal{R}}_{w})-\rho(1,1)\,.$

We consider the following approximation, allowing for the estimation of the
abundances of the species and empty sites in models where both predation and
reproduction probabilities are simultaneously reduced from the results
obtained considering only a reduction of either the predation or the
reproduction probability:

$\rho_{i}^{\rm
approx}({\mathcal{P}}_{w},{\mathcal{R}}_{w})=\rho_{i}({\mathcal{P}}_{w},1)+\rho_{i}(1,{\mathcal{R}}_{w})-\rho_{i}(1,1)\,,$
(6)

$\rho_{0}^{\rm
approx}({\mathcal{P}}_{w},{\mathcal{R}}_{w})=\rho_{0}({\mathcal{P}}_{w},1)+\rho_{0}(1,{\mathcal{R}}_{w})-\rho_{0}(1,1)\,.$
(7)

The lines in Fig. 6 show the results obtained using the approximation given in
Eqs. (6) and (7) (i.e., summing the results of models $P_{1}$ and $R_{1}$ for
the corresponding values of $(\mathcal{P})_{w}$ and $(\mathcal{R})_{w}$ and
then subtracting either $\langle\rho_{i}((\mathcal{PR})_{w}=1)\rangle=0.3$
[species $1$, $2$ and $3$] or
$\langle\rho_{0}((\mathcal{PR})_{w}=1)\rangle=0.1$ [empty sites]). The quality
of the approximation is remarkable, especially for $(\mathcal{PR})_{w}>0.6$,
thus showing that an accurate estimate of the model $(PR)_{1}$ results may be
obtained directly from those found for models $P_{1}$ and $R_{1}$. The more
significant differences observed for $(\mathcal{PR})_{w}<0.6$ are a further
demonstration of the non-linear behaviour of these systems.

Figure 6: The average density of the species ($i=1,2,3$) and empty sites
($i=0$) for the model $(PR)_{1}$, as a function of
$(\mathcal{PR})_{w}=\mathcal{P}_{w}=\mathcal{R}_{w}$, assuming $m=0.5$ and
$p=r=0.25$ (in this model both the predator and reproduction probabilities are
reduced by the same amount). The lines representing the approximate solution
obtained using Eqs. (6) and (7), in combination with the results of models
$P_{1}$ and $R_{1}$, provide an excellent fit to the results of model
$(PR)_{1}$.

Fig. 7 is similar to Fig. 6, but now for the model $(PR)_{2}$ (in this model
the predation and reproduction probabilities are reduced by the same amount
for species $1$ and $2$). Fig. 7 shows that, in this case, one of the “weak”
species ($2$) is the most abundant followed by the “strong” species ($3$),
with the remaining “weak” species ($1$) being the least abundant. Again, it is
possible to verify that the approximate solution, obtained using Eqs. (6) and
(7) in combination with the results of models $P_{2}$ and $R_{2}$, provides an
excellent fit to the results of model $(PR)_{2}$. We have also verified that
the accuracy of this approximation is, in general, much better than $20\%$ for
models with reduced predation and reproduction probabilities (not necessarily
by the same amount) in the range $1\leq{\mathcal{P}}_{w}+{\mathcal{R}}_{w}\leq
2$.

Figure 7: The same as in Fig. 6, but now for the model $(PR)_{2}$. These
results show that the approximate solution, obtained using Eqs. (6) and (7) in
combination with the results of models $P_{2}$ and $R_{2}$, provides an
excellent fit to the results of model $(PR)_{2}$.

##  5 Conclusions

In this letter we investigated the performance of “weak” and “strong” species
in a May-Leonard formulation of a three species cyclic predator-prey model. We
generalized previous work by considering cases with two “weak” species and by
extending the definition of “weak” species to also include species with a
reduced reproduction rate. We have shown that the spatial dynamics is
essential for coexistence, with stationary solutions in which all three
species coexist being always unstable in nonspatial models. On the other hand,
we have found that, for sufficiently large simulation lattices, all three
species may naturally coexist for a wide range of parameters. We have shown
that the significantly higher abundance of “weak” species found in spatial
stochastic RPS models with a single “weak” species also occurs in models with
two “weak” species. However, we have shown that the predominance of the “weak”
species does not generally hold if the “weak” species have a reduced
reproduction rate. We have also developed a simple approximation which allows
for an accurate estimation of the species abundances in models where both
predation and reproduction probabilities of “weak” are simultaneously reduced
from the results obtained considering only a reduction of either the predation
or the reproduction probability.

###### Acknowledgements.

P.P.A. acknowledges the support by Fundação para a Ciência e a Tecnologia
(FCT) through the research grants UIDB/04434/2020, UIDP/04434/2020. B.F.O. and
R.S.T. thanks CAPES-Finance Code 001, Fundação Araucária, and INCT-FCx
(CNPq/FAPESP) for financial and computational support.

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# Systematic study of complete fusion suppression for 6,7Li nuclei using
artificial neural network

D. Chattopadhyay1<EMAIL_ADDRESS>1Department of Physics, The
ICFAI University Tripura, Agartala 799210, India

###### Abstract

In recent decades, there has been a significant increase in the measurement of
complete fusion cross-sections for various reactions, with particular emphasis
on the reactions involving weakly bound projectile. It has been well
established that the complete fusion cross-section involving weakly bound
nuclei is suppressed at above barrier energies due to the breakup effect.
Accurate determination of suppression factor is essential to understand the
effect of breakup on complete fusion suppression. In this study, Feedforward
Artificial Neural Network (ANN) methods based on Multilayer Perceptron is
utilized to estimate the complete fusion suppression factor for reactions
involving 6,7Li projectile from the comparison of ANN predicted reduced fusion
functions ($F(x)$) with the Universal Fusion Function($F_{0}(x)$). Average
suppression factor has been estimated as 0.68 and 0.74 for 6Li and 7Li induced
reactions respectively. Normalized Mean Squared Error(NMSE) has been
calculated as 1.85$\%$ for training and 1.92$\%$ for testing data of ANN for
6Li case, while for 7Li cases, the same is for 3.73$\%$ and 6.48$\%$
respectively. Present results have been compared with three other alternative
methods Support Vector Regression, Random Forest Regression and Gaussian
Process Regression. Relevant performance matrices has been estimated. It has
been observed that, ANN algorithm is giving the better accuracy as compared
with other. This results indicate that ANN might be an alternative tool for
estimation of complete fusion suppression factor for weakly bound projectile.

###### pacs:

25.70.Gh, 25.70.Jj, 25.60.Gc

## Introduction

Nuclear reactions occurring near the Coulomb barrier serve as an extensive
repository revealing diverse facets of fundamental quantum mechanics. This
phenomenon becomes even more intricate if the projectile is weakly bound in
nature. In nuclear systems with weak binding, correlations among nucleons and
pairing give rise to phenomena such as the emergence of strong clustering and
exotic shapes. Lithium isotopes such as 6,7Li offer a distinctive illustration
of nuclear clustering, characterized by a well-known $\alpha$ \+ $x$ cluster
structure. Indeed, 6Li exhibits a well-known $\alpha$ \+ $d$ cluster
structure, featuring a breakup threshold of approximately 1.47 MeV. Similarly,
7Li is characterized by a well-established $\alpha$ \+ $t$ cluster
configuration, with a breakup threshold of around 2.47 MeV. The investigation
of nuclear reactions involving weakly bound projectiles 6,7Li has garnered
significant interest, driven by the observation of numerous novel features
distinct from those associated with strongly bound projectiles. Suppression in
complete fusion (CF) cross sections tripathi02 ; liu05 ; shrivastava09 ;
dasgupta04 ; rath09 , breakup threshold anomaly in the optical potentials
obtained from elastic scattering santra11 , and high yield in $\alpha$
particle production chattopadhyay16 ; chattopadhyay18  are some of the
intriguing features in this context. The existence of projectile breakup
channels, along with other nonelastic channels, and their interaction with the
elastic channel, are the key factors contributing to the aforementioned
distinctions. Numerous measurements of complete fusion (CF) cross sections
have been conducted tripathi02 ; liu05 ; shrivastava09 ; dasgupta04 ; rath09 .

Various methods have been employed to explore the influence of breakup on
fusion reactions around the Coulomb barrier hagino00 ; diaz02 ; gomes11 . One
widely used approach is to compare experimental data either with predictions
from coupled channel (CC) calculations that do not consider breakup channels
kumawat12 ; rath12 ; pradhan11 ; palshetkar14 ; rath13  or with the
predictions of a single barrier penetration model (SBPM) tripathi02 ;
tripathi05 . Observations indicate that complete fusion (CF) cross sections
are suppressed at energies above the Coulomb barrier. A novel approach
proposed in Ref. canto09  involves assessing the dynamic impact of projectile
breakup on fusion by comparing a dimensionless reaction fusion function
derived from experimental data with a theoretical system-independent universal
fusion function (UFF). In this method, experimental fusion cross sections are
normalized to theoretical fusion cross sections obtained from coupled-channels
calculations, incorporating only the pertinent bound inelastic and transfer
channels. Any disparity between the two fusion functions is attributed to the
effect of projectile breakup. However, it is worth noting that the degree of
the suppression factor obtained through the aforementioned approaches is
observed to be contingent on the specific modeling choices employed kundu16 .

Artificial neural networks (ANN) serve as a powerful mathematical tool for
estimating values across various fields in science and technology, including
nuclear physics studies. This method is particularly effective in providing
accurate results for highly nonlinear relationships between dependent and
independent data points. Recently, ANN has found application in numerous areas
of nuclear physics, such as constructing consistent physical formulas for
detector counts in neutron exit channel selection akkoyun13 , determining one
and two proton separation energies athanassopoulos04 , developing nuclear mass
systematic athanassopoulos004 , determining ground state energies of nuclei
bayram14 , identifying impact parameters in heavy-ion collisions david95 ;
bass96 ; haddad97 , determining beta-decay energies akkoyun14 , and estimating
nuclear rms charge radius akkoyun013 .

Understanding the dynamical effect of fusion suppression through theoretical
formulation requires many physical factors like location of breakup, coupling
mechanism between breakup channel with the fusion channel. In machine learning
approaches for predicting cross-sections, the goal is not to understand or
define the physical mechanisms behind the fusion reaction but to achieve the
most accurate cross-section predictions. This means that cross-section
information can be obtained without relying on any physical formulations. If
only cross-section information is required, this is an advantage of using
machine learning. By training on existing experimental data, the machine can
generate new data independently of physical formulations. Since the most
reliable cross-section information comes from experiments, the potent tool of
Artificial Neural Networks (ANN) haykin99  is utilized in present
investigation to acquire complete fusion cross-section data involving of
6,7Li. The experimental datas were extracted from existing literature
tripathi02 ; liu05 ; shrivastava09 ; dasgupta04 ; rath09 ; tripathi05 ;
kumawat12 ; rath12 ; pradhan11 ; palshetkar14 ; rath13 ; mukherjee06 ;
shrivastava13 .

In the present work, the data sets consisting of dimensionless variable
$x$[=$\dfrac{E_{c.m}-V_{B}}{\hbar\omega}$] as an input and Fusion Function
F(x) as output, where Ecm is the energy of the incident projectile in c.m.
frame, VB is the fusion barrier and $\hbar$$\omega$ is the width of the
barrier. The fusion barrier is nothing but the combination of coulomb barrier,
nuclear barrier and centrifugal barrier between the projectile and the target.
It is clear that, mass of the target nucleus is explicitly there inside the
expression of fusion barrier. Hence the effect is target mass is already been
included here. Through training the ANN with known Fusion Function F(x) with
$x$, the relationship between F(x) with $x$ is estimated. Any deviation of ANN
predicted Fusion Function F(x) from the UFF should be attributed to clustering
effects and the intensity of the effect could be estimated by the strength of
the deviation. For that, the average suppression factor for both 6Li and 7Li
induced reactions, has been calculated using the ratio of ANN model predicted
F(x) with UFF(x). The present result is also compared with the other
alternative interpolating methods Support Vector Regression, Random Forest
Regression and Gaussian Process Regression. The details of Artificial Neural
Network method will be discussed in next section followed by Calculations of
fusion suppression factor, Results and Discussion and Conclusions.

## Artificial Neural Network

An artificial neural network (ANN) is a computational model inspired by the
structure and function of biological neural networks in the human brain. It is
a key component of machine learning and artificial intelligence systems, and
it’s designed to mimic the way the human brain processes information. Neurons
are the basic building blocks of artificial neural networks. Each neuron
receives one or more inputs, performs a computation, and produces an output.
Neurons are organized into layers: an input layer, one or more hidden layers,
and an output layer. If information flows in one direction: from the input
layer to the output layer without cycles or loops, then the ANN is named as
feed-forward neural network. The network consists of multiple layers of nodes
(neurons). It has at least one hidden layer between the input and output
layers. Neurons are connected by edges, each associated with a weight. The
weight represents the strength of the connection between neurons. A bias term
is also often used to shift the output of a neuron. During training, the
weights and biases are adjusted to optimize the network’s performance. The
activation function determines the output of a neuron given its input. It
introduces non-linearity into the network, allowing it to learn complex
patterns. The network has a single output node, suitable for regression tasks,
as the goal is to predict a continuous variable. During the feed-forward
operation, the input data is propagated through the network, layer by layer.
The output of each neuron is computed based on the weighted sum of its inputs
and the application of the activation function. The training of a feed-forward
neural network involves adjusting the weights and biases based on the
difference between the predicted output and the actual target. This is
typically done using a supervised learning approach and an optimization
algorithm, such as gradient descent. The gradients of the loss with respect to
the weights and biases are calculated. This involves computing how much the
loss would change if each weight and bias were adjusted. The weights and
biases are adjusted in the direction that minimizes the loss, typically using
an optimization algorithm like stochastic gradient based descent,
Adam(Adaptive Moment Estimation) etc. In the present analysis, grid-search
will explore both ‘Adam’ and ‘Stochastic Gradient Desecent’ and the best-
performing solver will be determined during the hyperparameter tuning process.
The above processes are repeated for a specified number of training epochs
(maxiter) to iteratively improve the model.

The mathematical formulation of a simple feedforward neural network (Multi-
Layer Perceptron) used for regression is discussed in the following: Layer by
layer, the breakdown will be initiated, followed by the introduction of the
training process involving gradient descent.

### Forward Pass:

Figure 1:  Typical architecture of Feed Forward Neural Network

###  .1 Input Layer:

The input to the neural network is a vector X representing the features of a
single data point. If there are m features, X is a column vector of size m
$\times$ 1, often represented as

$\displaystyle X$ $\displaystyle=\begin{bmatrix}x_{1}\\\ x_{2}\\\ \vdots\\\
x_{m}\end{bmatrix}$ (1)

Here, each xi represents a feature of the input data point. The information
(x1,x2,…,xm) passes to the nodes in the hidden layer.

###  .2 Hidden Layer:

For a single hidden layer with n neurons, the output Zhidden is computed as:

$Z_{hidden}=(W_{hidden}\cdot X+b_{hidden})$ (2)

where,

Figure 2:  Flowchart of the ANN model

Whidden is the weight matrix of size n $\times$ m.

bhidden is the bias vector of size n $\times$ 1.

After computing Zhidden, it is passed through an activation function $\sigma$:

$A=\sigma(Z_{hidden})$ (3)

where A is the output(activation) of the hidden layer. If the weighted sum of
the inputs to a neuron is above the the threshold or activation layer, then
the neuron fires. In this analysis, the activation function for the hidden
layers is being explored with two options: ‘ReLU’ (Rectified Linear Unit) and
‘tanh’ (Hyperbolic Tangent). The model will be trained and evaluated with both
activation functions to determine which performs better for the given data.

So, the overall computation can be written as:

$A=\sigma(W_{hidden}\cdot X+b_{hidden})$ (4)

This represents the forward propagation through the first hidden layer of a
neural network. The activation function $\sigma$ introduces non-linearity to
the model, allowing it to learn complex patterns in the data.

If there are two hidden layers in the neural network, the computation of the
output at the first hidden layer (Zhidden1) and the subsequent hidden layer
(Zhidden2) can be expressed as follows:

For the first hidden layer:

$Z_{hidden1}=\sigma(W_{hidden1}\cdot X+b_{hidden1})$ (5)

Here:

Whidden1 is the weight matrix of the first hidden layer with dimensions n1
$\times$ m,

bhidden1 is the bias vector of the first hidden layer with dimensions n1
$\times$ 1,

$X$ is input vector with dimensions $m$ $\times$ 1,

$\sigma$ represents the activation function already mentioned above.

For the second hidden layer:

$Z_{hidden2}=\sigma(W_{hidden2}\cdot Z_{hidden1}+b_{hidden2})$ (6)

Here:

Whidden2 is the weight matrix of the second hidden layer with dimensions n2
$\times$ n1,

bhidden2 is the bias vector of the second hidden layer with dimensions n2
$\times$ 1.

Zhidden1 is output of the first hidden layer with dimensions n1 $\times$ 1.

This process extends the neural network’s capacity to learn hierarchical
features and complex representations from the input data.

###  .3 Output Layer:

For regression, there is typically no activation function in the output layer,
and the output Y is given by:

$Y=W_{output}\cdot Z_{hidden2}+b_{output}$ (7)

where, Woutput is a weight matrix of size 1 $\times$ n2

Zhidden is the output from the second hidden layer,

boutput is the bias term for the output layer.

### Training with Stochastic Gradient Descent:

###  .4 Loss Function:

The Mean Squared Error (MSE) is commonly used for regression:

$MSE=\dfrac{\sum_{i=1}^{N}(\hat{Y_{i}}-Y_{i})^{2}}{N}$ (8)

where, N is the data points. $\hat{Y_{i}}$ is the predicted output for data
point $i$. Yi is the actual output for data point i. Normalized Mean Squared
Error(NMSE) is defined as:

$NMSE=\dfrac{MSE}{F_{max}-F_{min}}$ (9)

where, Fmax and Fmin is the maximum value of $F(x)$ and minimum value of
$F(x)$. In addition to that, the regularization term $R$ is also added to the
loss function during training to prevent over-fitting by penalizing large
weights in the model. For L2 regularization, the regularization term is
proportional to the square of the weights:

$R_{L2}=\dfrac{\alpha}{2}\sum_{i=1}^{n}W_{i}^{2}$ (10)

Here, $\alpha$ is the regularization parameter, controls the strength of the
regularization effect. Higher values of $\alpha$ result in stronger
regularization.

###  .5 Back Propagation:

Backpropagation (short for ”backward propagation of errors”) is a supervised
learning algorithm commonly used for training artificial neural networks. It
is a gradient-based optimization algorithm that minimizes the error between
the predicted output and the actual target output. The key steps involved in
the ANN process are:

(1) Compute the gradient of the regularization term with respect to the output
layer weights: $\dfrac{\partial R}{\partial W_{output}}$.

(2) Compute the gradient of the mean squared error with respect to the output
layer weights: $\dfrac{\partial MSE}{\partial W_{output}}$.

(3) Compute the gradient of the mean squared error with respect to the output
layer biases: $\dfrac{\partial MSE}{\partial b_{output}}$.

(4) Propagate the gradients through the network to compute the gradient with
respect to the second hidden layer weights and biases: $\dfrac{\partial
R}{\partial W_{hidden2}}$,$\dfrac{\partial MSE}{\partial W_{hidden2}}$,
$\dfrac{\partial MSE}{\partial b_{hidden2}}$ .

(5) Propagate the gradients through the network to compute the gradient with
respect to the first hidden layer weights and biases: $\dfrac{\partial
R}{\partial W_{hidden1}}$,$\dfrac{\partial MSE}{\partial W_{hidden1}}$,
$\dfrac{\partial MSE}{\partial b_{hidden1}}$ .

The process involves backward propagation of the gradients through each layer
of the network. The chain rule is applied to compute the gradients at each
layer based on the gradients computed in subsequent layers.

###  .6 Update Weights and Biases:

Update the weights and biases using the gradients and a learning rate $\eta$:

$W_{output}\leftarrow W_{output}-\eta(\alpha\dfrac{\partial R}{\partial
W_{output}}+\dfrac{\partial MSE}{\partial W_{output}})$ (11)
$b_{output}\leftarrow b_{output}-\eta\dfrac{\partial MSE}{\partial
b_{output}}$ (12) $W_{hidden2}\leftarrow
W_{hidden2}-\eta(\alpha\dfrac{\partial R}{\partial
W_{hidden2}}+\dfrac{\partial MSE}{\partial W_{hidden2}})$ (13)
$b_{hidden2}\leftarrow b_{hidden2}-\eta\dfrac{\partial MSE}{\partial
b_{hidden2}}$ (14) $W_{hidden1}\leftarrow
W_{hidden1}-\eta(\alpha\dfrac{\partial R}{\partial
W_{hidden1}}+\dfrac{\partial MSE}{\partial W_{hidden1}})$ (15)
$b_{hidden1}\leftarrow b_{hidden1}-\eta\dfrac{\partial MSE}{\partial
b_{hidden1}}$ (16)

where, $R$ represents the regularization term added to the loss function
during training.

This process is repeated for maximum iterations until convergence. The
training with Stochastic Optimization using Adam(Adaptive Moment Estimation)
has been described in kingma14 . In the present calculations, the data sets
consisting of dimensionless variable $x$ as an input and Fusion Function F(x)
as output. For 6Li case, There are 114 data points and hence the size of the
input x is 114 X 1. Whereas, for 7Li case, the data points are 80 and hence
the size of the input here is 80 X 1. In the network structure, there are 1
input node in the input layer, 64 nodes in the hidden layer 1, 32 nodes in the
hidden layer 2 and 1 node in the output layer. The number of hidden layer and
hidden units are so chosen to get the minimum loss as much as possible. Hence,
The total number of adjustable weights are 2144 and the biases are 97.
Therefore, the total number of network parameters are 2241. The maximum
iteration is used as 5000. In the Adam algorithm, the initial learning rate is
0.0001 and decay constants are 0.8 and 0.98.

In the present case, Parameter grid search is employed in the provided code to
fine-tune the hyperparameters of the MLPRegressor model. The grid search space
encompasses various hyperparameters such as the hidden layer sizes, activation
function, alpha (regularization parameter), learning rate, solver, and others.
Each combination of hyperparameters is systematically evaluated using cross-
validation with five folds. RandomizedSearchCV from scikit-learn is utilized
for this purpose, with 20 iterations to randomly sample hyperparameter
configurations from the specified parameter grid. This approach allows for an
exhaustive search over the hyperparameter space while mitigating the
computational cost associated with evaluating all possible combinations. By
combining regularization techniques with cross-validation during grid search,
the code aims to find the optimal hyperparameters that not only minimize the
training error but also promote good generalization performance on unseen
data. This comprehensive approach helps in addressing overfitting by ensuring
that the selected model is robust and has good predictive power across
different data samples.

Table 1: Data table along with experimental suppression factor and ANN-predicted suppression factor Proj. | Target | FBu. | FBu.
---|---|---|---
|  | [expt.] | [ANN]
6Li |  90Zr kumawat12  | 0.66$\pm$0.08 | 0.68
6Li |  144Sm rath09  | 0.68 | 0.68
6Li |  152Sm rath12  | 0.72$\pm$0.04 | 0.68
6Li |  159Tb pradhan11  | 0.66$\pm$0.05 | 0.68
6Li |  197Au palshetkar14  | 0.65$\pm$0.23 | 0.68
6Li |  198Pt shrivastava09  |  | 0.68
6Li |  208Pb liu05  |  | 0.68
6Li |  209Bi dasgupta04  | 0.66$\pm$0.05 | 0.68
7Li |  144Sm rath13  | 0.75$\pm$0.04 | 0.74
7Li |  152Sm rath13  | 0.75$\pm$0.04 | 0.74
7Li |  159Tb mukherjee06  | 0.74 | 0.74
7Li |  165Ho tripathi02 ; tripathi05  | 0.70 | 0.74
7Li |  197Au palshetkar14  | 0.85$\pm$0.04 | 0.74
7Li |  198Pt shrivastava13  |  | 0.74
7Li |  209Bi dasgupta04  | 0.74$\pm$0.03 | 0.74
Table 2: Comparison of performance matrix of different machine learning regression models for 6Li induced reactions Model | FBu. | R2 | R2 | NMSE | NMSE
---|---|---|---|---|---
| [Model] | [Training] | [Testing] | [Training] | [Testing]
ANN | 0.680 | 0.98 | 0.98 | 1.85$\%$ | 1.92$\%$
GPR | 0.674 | 0.95 | 0.93 | 2.3$\%$ | 3.70$\%$
SVR | 0.701 | 0.975 | 0.96 | 1.93$\%$ | 2.28$\%$
RFR | 0.677 | 0.986 | 0.936 | 1.18$\%$ | 3.54$\%$
Table 3: Comparison of performance matrix of different machine learning regression models for 7Li induced reactions Model | FBu. | R2 | R2 | NMSE | NMSE
---|---|---|---|---|---
| [Model] | [Training] | [Testing] | [Training] | [Testing]
ANN | 0.740 | 0.970 | 0.952 | 3.73$\%$ | 6.48$\%$
GPR | 0.752 | 0.968 | 0.962 | 3.77$\%$ | 5.18$\%$
SVR | 0.720 | 0.960 | 0.950 | 3.91$\%$ | 6.61$\%$
RFR | 0.747 | 0.980 | 0.923 | 2.66$\%$ | 7.93$\%$
Figure 3:  Variation of complete fusion function F(x) with x for 6Li
projectile on different target nuclei.  Figure 4:  Variation of complete
fusion function $F(x)$ with $x$ for 7Li projectile on different target nuclei.
Figure 5:  Variation of complete fusion function $F(x)$ with $x$ for 6Li[top
panel] and 7Li[bottom panel] projectile on different target nuclei for Support
Vector Regression. Figure 6:  Variation of complete fusion function $F(x)$
with $x$ for 6Li[top panel] and 7Li[bottom panel] projectile on different
target nuclei for Random Forest Regression. Figure 7:  Variation of complete
fusion function $F(x)$ with $x$ for 6Li[top panel] and 7Li[bottom panel]
projectile on different target nuclei for Gaussian Process Regression.

## Calculations of Fusion Suppression Factor

To investigate the systematic patterns of breakup effects on CF cross
sections, it is imperative to entirely eliminate the geometric factors and
static influences arising from the potential between the two nuclei. According
to the method proposed by Canto et al. canto09 , the fusion cross section and
collision energy are transformed into a dimensionless fusion function, denoted
as $F(x)$, and a dimensionless variable, $x$.

$F(x)=\dfrac{2E_{c.m.}}{R_{B}^{2}\hbar\omega}\sigma_{F},x=\dfrac{E_{c.m}-V_{B}}{\hbar\omega}$
(17)

where, Ec.m is the energy of incident projectile in the c.m. frame,
$\sigma_{F}$ is the fusion cross-section and RB,VB and $\hbar\omega$ implies
the radius, fusion barrier and curvature of the barrier which is approximated
as parabolic in nature. The barrier parameters RB,VB and $\hbar\omega$ are
calculated from Single Barrier Penetration model using NRV nrv .

According to the prescription, of Wong, the fusion cross-section for parabolic
barrier is given by

$\sigma^{Wong}_{F}=\dfrac{\hbar\omega
R_{B}^{2}}{2E_{c.m.}}ln[1+exp(\dfrac{2\pi}{\hbar\omega}(E_{c.m.}-V_{B}))]$
(18)

If the Fusion Cross-section is described by the Wong’s formula, then
dimensionless quantity $F(x)$ reduces to,

$F_{0}(x)=ln[1+exp(2\pi x]$ (19)

The quantity $F_{0}$ is known as Universal Fusion Function($UFF(x)$). When
x$>$1, $F_{0}$(x) $\approx$ 2$\pi$x, therefore the fusion cross-section is
approximated as

$\sigma_{F}=\dfrac{\pi R_{B}^{2}}{E_{c.m.}}(E_{c.m.}-V_{B})$ (20)

Artificial Neural Network method has been applied to extract the reliable
Fusion Function $F(x)$ from the already available fusion cross-section data
involving 6,7Li nuclei. Once the $F(x)$ is available, the complete fusion
suppression factor due to the breakup has been estimated from the following
relation,

$F^{Bu}(x)=\dfrac{F(x)}{F_{0}(x)}$ (21)

## Results and Discussions

After conducting numerous trials in the initial phase of the study, the
optimal number of hidden layer neurons was identified and employed to yield
the most favorable outcomes. The empirical data for total fusion cross-
sections for 6Li and 7Li nuclei, sourced from different literature tripathi02
; liu05 ; shrivastava09 ; dasgupta04 ; rath09 ; tripathi05 ; kumawat12 ;
rath12 ; pradhan11 ; palshetkar14 ; rath13 ; mukherjee06 ; shrivastava13 ,
encompassed compound nuclei mass numbers ranging from 90 to 209 are used in
the present analysis. The dataset comprises of dimensionless quantity
$x$(independent variable) and Fusion function $F(x)$(dependent variable).
Subsequently, this dataset was randomly partitioned into distinct sets for
training (70$\%$) and testing (30$\%$) purposes. The neural network is
implemented using scikit-learn’s MLPRegressor, which stands for Multi-Layer
Perceptron Regressor. After making suitable adjustments, the final weights are
determined once a satisfactory level of error convergence between the
predicted and desired outputs is achieved.

Upon completion of the network training, the neural network utilized the final
weights to generate outputs for both training and test data values. The
Normalized Mean Squared Error(NMSE) is estimated as 1.85 $\%$ for 6Li induced
induced reactions and the same is 3.73 $\%$ for 7Li induced induced reactions
for the training datasets. The coefficient of determination has been obtained
using R2 value, it is found to be $\approx$ 0.98 for 6Li induced reactions
training data and $\approx$ 0.97 for 7Li induced reactions training data. Fig.
3 and Fig. 4 vividly illustrates the effective reproduction of the highly non-
linear experimental fusion function $F(x)$ with dimensionless quantity $x$ by
the ANN method.

The test stage holds significant importance in confirming the success of the
method. By employing the final weights of the ANN, the complete fusion
function $F(x)$ in the test data for both 6Li and 7Li induced reactions were
predicted. The Normalized Mean Squared Error (NMSE) for test data has been
computed as 1.92$\%$ for reactions induced by 6Li and 6.48$\%$ for those
induced by 7Li. R2 value is obtained as $\approx$ 0.98 and $\approx$ 0.95 for
test data of 6Li and 7Li induced reactions respectively.

In the final phase of the study, feed forward neural network model is used to
estimate the complete fusion function value $F(x)$ for finer $x$ for both 6Li
and 7Li induced reactions cases. Once the $F(x)$ is obtained, the fusion
suppression factor is calculated by taking the ratio with Universal Fusion
Function $F_{0}(x)$. The suppression factor is calculated as 0.68 and 0.74 for
6Li and 7Li induced reactions respectively which is quite impressive
suggesting the ANN method as an alternative tool for extraction of complete
fusion suppression factor. Present results has been compared with other three
alternative methods Support Vector Regression, Random Forest Regression and
Gaussian Process Regression. The performance matrix has been compared in the
Table II and Table III. It has been observed that, ANN algorithm is giving the
better accuracy as compared with other. Variation of F(x) with respect to $x$
has been manifested in Fig. 5, Fig. 6 and Fig. 7 for both 6Li and 7Li induced
reactions using Support Vector Regression, Random Forest Regression and
Gaussian Process Regression respectively. Subsequently, the average
suppression factor has been estimated which has been tabulated in Table II and
Table III. It has been observed that the ANN algorithm provides better
accuracy compared to the other methods.

## Conclusions

In this study, the utilization of Artificial Neural Network (ANN) methods to
estimate the complete fusion suppression factor for reactions involving 6,7Li
projectiles is demonstrated. This is achieved by comparing the Universal
Fusion Function ($F_{0}(x)$) with ANN-predicted reduced fusion functions
($F(x)$). The average suppression factors are found to be 0.68 and 0.74,
respectively. The Normalized Mean Squared Error (NMSE) is calculated as
1.85$\%$ for training and 1.92$\%$ for testing data of ANN in the case of 6Li.
For 7Li , the corresponding values are determined to be 3.73$\%$ and 6.48$\%$.
The correlation coefficient, determined through the R2 value, reveals a high
degree of correlation in the study. Specifically, the correlation coefficient
is approximately 0.98 for training data in 6Li induced reactions and
approximately 0.97 for testing data in the same reactions. The R2 values
obtained for the test data also indicate a strong correlation, with
approximately 0.98 for 6Li-induced reactions and approximately 0.95 for 7Li-
induced reactions. The current results have been compared with three
alternative methods: Support Vector Regression, Random Forest Regression, and
Gaussian Process Regression. Relevant performance metrics have been assessed.
It was observed that the ANN algorithm provides better accuracy compared to
the other methods. These findings suggest that ANN might be a viable tool for
estimating the complete fusion suppression factor for weakly bound
projectiles.

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