# Graph-Assisted Communication-Efficient
Ensemble Federated Learning

Pouya M. Ghari and Yanning Shen  Department of Electrical Engineering and
Computer Science
University of California, Irvine
Irvine, CA, USA
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Communication efficiency arises as a necessity in federated learning due to
limited communication bandwidth. To this end, the present paper develops an
algorithmic framework where an ensemble of pre-trained models is learned. At
each learning round, the server selects a subset of pre-trained models to
construct the ensemble model based on the structure of a graph, which
characterizes the server’s confidence in the models. Then only the selected
models are transmitted to the clients, such that certain budget constraints
are not violated. Upon receiving updates from the clients, the server refines
the structure of the graph accordingly. The proposed algorithm is proved to
enjoy sub-linear regret bound. Experiments on real datasets demonstrate the
effectiveness of our novel approach.

###### Index Terms:

federated learning, ensemble learning, graphs

##  I Introduction

Prevalence of distributed networks consisting of devices such as mobile phones
and sensors with growing computational and storage capability enables pushing
more network computations to the edge. _Federated learning_ has emerged as a
promising framework to train machine learning models under orchestration of a
central server while training data remains distributed among the edge devices
which are called _clients_ [1]. In federated learning, a central server sends
the current model to a set of clients at each learning round. Participating
clients then compute updates of the current model based on their local data
and send these updates to the server instead of their local data. The server
then update the model. This procedure continues until convergence. In this
context, communication efficiency is of utmost importance. To this end,
clients-to-server communication efficiency in federated learning has been
studied extensively in the context of model updates compression, see e.g., [2,
3]. On the other hand, server-to-clients communication bottleneck arises if
the learning task involves large model, such that the required bandwidth to
transmit the model exceeds the available bandwidth for server-to-clients
communication. For example, base stations can be employed as an aggregating
server in certain applications [4]. Often times the base station may only
assign a limited portion of available bandwidth for server-to-client
communication, while reserving most of the available bandwidth for other
service required by users. Meanwhile, large models trained at the central
server may exceed the clients’ memory capacity. These challenges all motivate
the study of server-to-clients communication-efficient federated learning,
which is also the focus of the present work.

Ensemble learning methods are known to be effective for learning large-scale
models, which combine several base predictors or experts to generate more
accurate ensemble model. However, conventional ensemble learning methods (see
e.g. [5]) are not directly applicable for communication efficient federated
learning. To adapt ensemble learning to federated setting, FedBoost has been
proposed by [6], where the server constructs a model by combining a subset of
pre-trained models. In this way, the server only needs to transmit a subset of
pre-trained models to the clients at each learning round. Pre-trained models
can be trained on publicly available data without observing clients’ data.
FedBoost imposes a budget constraint on the number of pre-trained models that
can be transmitted to the clients; however it cannot guarantees that the
budget constraint is not violated at each learning round. Instead, it only
guarantees the expected cost of model transmission satisfies the budget
constraint. Moreover, ensemble learning techniques has been employed in
vertical federated learning [7].

The present paper studies server-to-clients communication efficiency in
federated learning. Specifically, we aim at selecting a subset of pre-trained
models to construct an ensemble model. To this end, each pre-trained model is
viewed as an expert lying on a graph. At each learning round, the server
chooses a subset of pre-trained models based on the structure of the graph.
Upon receiving the updates from the clients, the server refines the structure
of the graph. In this context, the prediction provided by each pre-trained
model can be viewed as feedback given by the expert. Hence, the constructed
graph is named _feedback graph_. The cost of each pre-trained model is
proportional to its parameter size and a budget constraint is set to transmit
the models in order to construct the ensemble model. We develop an algorithm
called ensemble federated learning with feedback graph (EFL-FG) which selects
a subset of pre-trained models to be transmitted to clients, and guarantees
the resulting communication cost does not exceed budget constraint at each
learning round. EFL-FG is proved to enjoy sub-linear regret. Experiments on
real datasets showcase the effectiveness of our proposed algorithms compared
with state-of-art ensemble federated learning alternatives.

##  II Problem Statement and Preliminaries

This section introduces the problem of federated learning with ensemble
method. Let there are a set of $N$ clients that the server interacts with them
to perform a learning task. Moreover, ${\mathcal{X}}$ and ${\mathcal{Y}}$
denote the input space and output space, respectively such that a data sample
$({\bm{x}},y)\in{\mathcal{X}}\times{\mathcal{Y}}$. Furthermore, there are a
set of $K$ pre-trained models $f_{1}(\cdot),\ldots,f_{K}(\cdot)$ at the
server. Each model $f_{k}(\cdot):{\mathcal{X}}\rightarrow{\mathcal{Y}}$,
$\forall k\in[K]$ is a mapping from the input space to the output space where
$[K]$ denotes the set $\\{1,\ldots,K\\}$. At each round of learning, the
server uniformly chooses a random subset of clients to send them the current
model. The number of clients chosen by the server depends on the available
bandwidth for clients-to-server communication. Specifically, at each learning
round, the server constructs a model using pre-trained models and send it to
the chosen subset of clients. Then, upon receiving new data samples, clients
perform the learning task using the current model received from the server.

Let ${\mathbb{C}}_{t}$ denote the set of clients selected by the server at
learning round $t$, and
${\mathcal{S}}_{t}:=\\{({\bm{x}}_{i,t},y_{i,t}),\forall
i\in{\mathbb{C}}_{t}\\}$ represents a set of data samples observed by the
chosen subset of clients at learning round $t$. In this case, the problem of
federated learning can be viewed as a function approximation problem.
Specifically, given data samples $\\{{\mathcal{S}}_{t}\\}_{t=1}^{T}$, the goal
is to find the function $\hat{f}(\cdot)$, such that the difference between
$\hat{f}({\bm{x}}_{i,t})$ and $y_{i,t}$ is minimized. When the $i$-th client
performs the learning task on the received datum ${\bm{x}}_{i,t}$, it incurs
the loss ${\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})$ where
${\mathcal{L}}(\cdot,\cdot)$ denotes the loss function. And
${\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})$ denote the loss of each model
$f_{k}(\cdot)$, $\forall k\in[K]$ for the datum ${\bm{x}}_{i,t}$ associated
with the $i$-th client. In this context, the goal of the server is to
construct the function $\hat{f}(\cdot)$ using pre-trained models in a way that
the cumulative loss is minimized. In order to build the function
$\hat{f}(\cdot)$ using the pre-trained models, the server employs the ensemble
method. Using the conventional ensemble method, the model at the server is
constructed as

$\displaystyle\hat{f}({\bm{x}})=\sum_{k=1}^{K}{w_{k}f_{k}({\bm{x}})},\sum_{k=1}^{K}{w_{i}}=1.$
(1)

The ensemble method in (1) requires that the server sends all models to the
clients. However, this may not be feasible due to e.g., insufficient bandwidth
for server-to-clients communication, and limited computational capability and
memory of clients to store all models. The present paper proposes a novel
algorithmic framework to choose a data-driven subset of models in a way that
these limitations are taken into account.

##  III Ensemble Federated Learning with Graphs

The present section first introduces a disciplined way to construct a graph
based on the performance of pre-trained models. Then, a novel algorithm is
proposed to construct an ensemble model employing a subset of pre-trained
models chosen by the server based on the graph.

Let $c_{k}$ be the cost incurred when the $k$-th pre-trained model is
transmitted. Specifically, $c_{k}$ can be the bandwidth required for
transmission of the $k$-th model to clients. Let $B_{t}$ denote the budget of
the server which denotes the cumulative cost the server can afford for
transmission at learning round $t$, e.g., available bandwidth for server-to-
clients communication. In what follows, a principled algorithm to construct a
graph is proposed which assists the learner to obtain an ensemble model.

###  III-A Feedback Graph Generation

Let ${\mathcal{G}}_{t}=({\mathcal{V}},{\mathcal{E}}_{t})$ be a directed graph
at learning round $t$ with a set of vertices ${\mathcal{V}}$ and a set of
edges ${\mathcal{E}}_{t}$. Each vertex $v_{k}\in{\mathcal{V}}$, $\forall
k\in[K]$ represents the pre-trained model $f_{k}(\cdot)$. Let ${\bm{w}}_{t}$
be a weight vector, where the $k$-th element $w_{k,t}$ is the weight
associated with the $k$-th model $f_{k}(\cdot)$, indicating the server’s
confidence about the performance of model $f_{k}(\cdot)$. At each learning
round, the server updates ${\bm{w}}_{t}$ based on the observed loss of
$f_{k}(\cdot)$ which will be specified later. Let
${\mathbb{N}}_{k,t}^{\text{out}}$, $\forall k\in[K]$ be the out-neighbor set
of $v_{k}$. In order to construct the set ${\mathbb{N}}_{k,t}^{\text{out}}$,
$\forall k\in[K]$, the server appends nodes $v_{j}$ to
${\mathbb{N}}_{k,t}^{\text{out}}$ based on both weights and costs of models
such that the cumulative cost of nodes in ${\mathbb{N}}_{k,t}^{\text{out}}$
does not exceed the budget $B_{t}$. At first, the server append $v_{k}$ to
${\mathbb{N}}_{k,t}^{\text{out}}$ which means that there is a self loop for
each $v_{k}\in{\mathcal{V}}$. Let

$\displaystyle{\mathbb{M}}_{k,t}:=\\{v_{i}$ $\displaystyle|\forall
i:\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{c_{j}}+c_{i}\leq B_{t},$
$\displaystyle\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{w_{j}}+w_{i}\leq\\!\\!\sum_{j\in{\mathbb{N}}_{k,t-1}^{\text{out}}}{\\!\\!w_{j}},v_{i}\notin{\mathbb{N}}_{k,t}^{\text{out}}\\}$
(2)

denote a set of vertices associated with $v_{k}$ at learning round $t$. At
learning round $t$, find

$\displaystyle
v_{d}=\arg\max_{v_{i}\in{\mathbb{M}}_{k,t}}{\frac{w_{i,t}}{\sum_{v_{j}\in{\mathbb{N}}_{k,t}^{\text{out}}}{c_{j}}+c_{i}}}.$
(3)

the set ${\mathbb{M}}_{k,t}$ then is updated by appending $v_{d}$ to
${\mathbb{N}}_{k,t}^{\text{out}}$. This procedure continues until
${\mathbb{M}}_{k,t}$ becomes an empty set, i.e., $|{\mathbb{M}}_{k,t}|=0$,
where $|\cdot|$ represents the cardinality of a set. This means there is no
more node that can be appended to ${\mathbb{N}}_{k,t}^{\text{out}}$ such that
the constraints in (2) are satisfied. Moreover, according to (3), the server
appends $v_{d}$ to ${\mathbb{N}}_{k,t}^{\text{out}}$ by considering the trade-
off between the performance of nodes in prior rounds and the amount of cost
that they might add to current cumulative cost of out-neighbors of $v_{k}$.
When the server constructs ${\mathbb{N}}_{k,t}^{\text{out}}$, $\forall
k\in[K]$, the set of edges $\mathcal{E}_{t}$ can be constructed. Specifically,
$(k,j)\in\mathcal{E}_{t}$ if $v_{j}\in{\mathbb{N}}_{k,t}^{\text{out}}$. The
procedure to construct the graph ${\mathcal{G}}_{t}$ is summarized in
Algorithm 1. At each learning round, the server draws one node in
${\mathcal{G}}_{t}$ and transmits models which are out-neighbors of the chosen
one. Then the learning task is carried out with a subset of models which are
considered as nodes in ${\mathcal{G}}_{t}$. Thus, output of the selected
models can be viewed as feedback collected from ${\mathcal{G}}_{t}$, which is
henceforth named as _feedback graph_.

Algorithm 1 Feedback Graph Generation

Input:Models $f_{k}(.)$, weights $w_{k,t}$, costs $c_{k}$, $\forall k\in[K]$
and the budget $B_{t}$.

for $k=1,...,K$ do

Append $v_{k}$ to ${\mathbb{N}}_{k,t}^{\text{out}}$.

while $|{\mathbb{M}}_{k,t}|>0$ do

Node $v_{k}$ appends $v_{d}$ as in (3) to ${\mathbb{N}}_{k,t}^{\text{out}}$.

Update ${\mathbb{M}}_{k,t}$ with respect to updated
${\mathbb{N}}_{k,t}^{\text{out}}$.

end while

end for

Output:Feedback Graph ${\mathcal{G}}_{t}=({\mathcal{V}},{\mathcal{E}}_{t})$.

###  III-B Ensemble Federated Learning

At each learning round $t$, the server selects one node in ${\mathcal{G}}_{t}$
and constructs the ensemble model with out-neighbors of the chosen node. To
this end, the server assigns weight $u_{k,t}$ to node $v_{k}$ which indicates
the confidence in the accuracy of the obtained ensemble model when node
$v_{k}$ is chosen. Then, the server draws one of the nodes
$v_{k}\in{\mathcal{V}}$ according to the probability mass function (PMF)
${\bm{p}}_{t}$ as follows

$\displaystyle
p_{k,t}=(1-\xi)\frac{u_{k,t}}{U_{t}}+\frac{\xi}{|{\mathbb{D}}_{t}|}{\mathcal{I}}(v_{k}\in{\mathbb{D}}_{t})$
(4)

where ${\mathcal{I}}(\cdot)$ denotes the indicator function, $\xi$ is the
exploration rate and $U_{t}:=\sum_{k=1}^{K}{u_{k,t}}$. The set
${\mathbb{D}}_{t}$ denotes a dominating set for the feedback graph
${\mathcal{G}}_{t}$. The PMF in (4) constitutes a trade-off between
exploitation and exploration. Let $I_{t}$ denote the index of the drawn node
at learning round $t$. Let ${\mathbb{S}}_{t}$ be a set of indices of nodes
which are out-neighbors of the chosen node $v_{I_{t}}$. In this case, the
server utilizes the weighting vector ${\bm{w}}_{t}$ to construct the ensemble
model using models whose indices are in ${\mathbb{S}}_{t}$ as follows

$\displaystyle\hat{f}_{t}({\bm{x}})=\sum_{k\in{\mathbb{S}}_{t}}{\frac{w_{k,t}}{W_{t}}f_{k}({\bm{x}})}$
(5)

where $W_{t}:=\sum_{k\in{\mathbb{S}}_{t}}{w_{k,t}}$. Then, the server
transmits the ensemble model along with the chosen subset of pre-trained
models to a subset of $N_{t}$ clients, which is chosen uniformly at random.
Upon receiving new datum the chosen subset of clients perform the learning
task using the ensemble model sent by the server. Then, the $i$-th client
where $i\in{\mathbb{C}}_{t}$, incurs loss
${\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})$ associated with the received
data sample $({\bm{x}}_{i,t},y_{i,t})$. Furthermore, the $i$-th client
($\forall i\in{\mathbb{C}}_{t}$) computes the loss
${\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})$, $\forall
k\in{\mathbb{S}}_{t}$. Then each client in ${\mathbb{C}}_{t}$ transmits the
losses associated with the ensemble model and the chosen subset of models to
the server. Upon receiving the losses, the server updates $w_{k,t}$ and
$u_{k,t}$, $\forall k\in[K]$. To this end, the server employs the importance
sampling loss estimate which results in an unbiased estimation of the incurred
loss. The importance sampling loss estimate for the $k$-th model can be
expressed as

$\displaystyle\ell_{k,t}=\frac{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}{q_{k,t}}{\mathcal{I}}(k\in{\mathbb{S}}_{t})$
(6)

where

$\displaystyle
q_{k,t}:=\sum_{v_{j}\in{\mathbb{N}}_{k,t}^{\text{in}}}{p_{j,t}}$ (7)

represents the probability of $k\in{\mathbb{S}}_{t}$, where
${\mathbb{N}}_{k,t}^{\text{in}}$ denotes the in-neighbor set of $v_{k}$ in
${\mathcal{G}}_{t}$. In addition, define the importance sampling loss estimate
associated with incurred loss of ensemble model when $v_{k}$ is drawn by the
server as

$\displaystyle\hat{\ell}_{k,t}=\frac{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})}}{p_{k,t}}{\mathcal{I}}(k=I_{t}).$
(8)

Using the importance sampling loss estimates in (6) and (8), the weights
$w_{k,t}$ and $u_{k,t}$ can the be updated as follows

$\displaystyle w_{k,t+1}=w_{k,t}\exp(-\eta\ell_{k,t})$ (9a) $\displaystyle
u_{k,t+1}=u_{k,t}\exp(-\eta\hat{\ell}_{k,t})$ (9b)

where $\eta$ is the learning rate. The procedure that the server sends subset
of models to clients is summarized in Algorithm 2. The algorithm is called
EFL-FG which stands for Ensemble Federated Learning with Feedback Graph. In
each learning round, clients only need to send their computed losses to the
server while they do not have to reveal the loss function
${\mathcal{L}}(\cdot,\cdot)$ and observed data samples to the server.
Specifically, in some applications data samples $({\bm{x}}_{i,t},y_{i,t})$ may
include some information about clients that they do not wish to share with the
server.

Algorithm 2 EFL-FG: Ensemble Federated Learning with Feedback Graph

Input:Models $f_{k}(\cdot)$, weights $w_{k,t}$, costs $c_{k}$, $\forall
k\in[K]$.

Initialize: $w_{k,1}=1$, $u_{k,1}=1$, $\forall k\in[K]$.

for $t=1,...,T$ do

The server generates ${\mathcal{G}}_{t}$ using Algorithm 1.

The server draws one node in ${\mathcal{G}}_{t}$ according to the
${\bm{p}}_{t}$ in (4), with out-neighbors indexed by ${\mathbb{S}}_{t}$.

The server randomly selects a subset of clients ${\mathbb{C}}_{t}$.

The server sends models in ${\mathbb{S}}_{t}$ and the ensemble model to the
clients in ${\mathbb{C}}_{t}$.

Clients in ${\mathbb{C}}_{t}$ compute and send back to the sever the losses
${\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})$ and
${\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})$, $\forall
k\in{\mathbb{S}}_{t}$.

The server computes the importance sampling loss estimates $\ell_{k,t}$ and
$\hat{\ell}_{k,t}$, $\forall k\in[K]$.

The server updates $w_{k,t+1}$ and $u_{k,t+1}$, $\forall k$ as in (9).

end for

The present paper considers the case where the number of selected clients may
vary with learning rounds. According to Algorithm 2, clients in
${\mathbb{C}}_{t}$ need to send computed losses of sent models along with the
losses of ensemble model to the server. Therefore, according to available
bandwidth for clients-to-server communication and required bandwidth for
sending the updates, the server determines the number of selected clients
$N_{t}$ such that clients can send computed losses without interfering each
other. Specifically, the available bandwidth should be divided between clients
in ${\mathbb{C}}_{t}$ without overlap to prevent interference. Let $b_{t}$ be
the available bandwidth for clients-to-server communication at learning round
$t$. In this case, the server can obtain the number of clients as
$N_{t}\leq\left\lfloor{\frac{b_{t}}{b_{\ell}({|{\mathbb{N}}_{I_{t},t}^{\text{out}}|}+1)}}\right\rfloor$,
where $b_{\ell}$ is the required bandwidth for transmission of each loss.

Comparison with online learning. Online learning studies problems where a
learner interacts with a set of experts such that at each learning round the
learner makes decision based on advice received from the experts [8, 9]. The
learner may observe the loss associated with a subset of experts after
decision making, which can be modeled using a feedback graph [10, 11]. In EFL-
FG, each pre-trained model is also viewed as an expert. However, there is a
major innovative difference compared with online learning with feedback graph:
the proposed EFL-FG constructs and refines the feedback graph to improve the
performance while in online learning, the feedback graph is generated in an
adversarial manner.

###  III-C Regret Analysis

The present subsection studies the performance of EFL-FG in terms of
cumulative regret. The difference between the loss incurred by the ensemble
model and the loss of the best pre-trained model in the hindsight is defined
as the regret of the ensemble model. In this context, the best pre-trained
model in the hindsight is the one with minimum cumulative loss among all pre-
trained models. Therefore, the cumulative regret of EFL-FG can be written as

$\displaystyle{\mathcal{R}}_{T}=$
$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}$
$\displaystyle-\min_{k\in[K]}{\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}}$
(10)

where $\mathbb{E}_{t}[.]$ denotes the conditional expectation given observed
losses in prior learning rounds. In order to analyze the performance of EFL-
FG, we assume that the following conditions hold:
(a1) The loss function ${\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})$ is
convex with respect to $f_{k}({\bm{x}}_{i,t})$.
(a2) For each $({\bm{x}}_{i,t},y_{i,t})$, the loss is bounded
$0\leq{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})\leq 1$.
(a3) The budget satisfies $B_{t}\geq c_{k}$, $\forall k\in[K]$, $\forall t$.
The following Theorem presents the regret bound for EFL-FG with respect to the
best pre-trained model in hindsight.

######  Theorem 1.

Under (a1)–(a3), the expected cumulative regret of EFL-FG is bounded by

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}-{\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k^{*}}({\bm{x}}_{i,t}),y_{i,t})}}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln(K|{\mathbb{N}}_{k^{*},1}^{\text{out}}|)}{\eta}$
$\displaystyle+\sum_{t=1}^{T}{\left(\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}(K+\frac{1}{\bar{q}_{k^{*},t}})|{\mathbb{C}}_{t}|^{2}\right)}$
(11)

where
$k^{*}=\arg\min_{k\in[K]}{\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}}$
is the index of the best pre-trained model in the hindsight, and
$\frac{1}{\bar{q}_{k^{*},t}}:=\sum_{j\in{\mathbb{N}}_{k^{*},t}^{\text{out}}}\frac{w_{j,t}}{q_{j,t}W_{k^{*},t}}$.

###### Proof.

see Appendix. ∎

According to (4), $p_{k,t}>\frac{\xi}{|{\mathbb{D}}_{t}|}$, $\forall
k\in{\mathbb{D}}_{t}$. In addition, each node $v_{k}\in{\mathcal{V}}$ is in-
neighbor to at least one vertex in the dominating set ${\mathbb{D}}_{t}$.
Therefore, based on (7), it can be concluded that
$q_{k,t}>\frac{\xi}{|{\mathbb{D}}_{t}|}$, $\forall k\in[K]$ and as a result,
we have $\frac{1}{\bar{q}_{k^{*},t}}<\frac{|{\mathbb{D}}_{t}|}{\xi}$. If the
greedy set cover algorithm (see e.g. [12]) is employed to find a dominating
set for the feedback graph ${\mathcal{G}}_{t}$, EFL-FG obtains a dominating
set with $|{\mathbb{D}}_{t}|={\mathcal{O}}(\alpha({\mathcal{G}}_{t})\ln K)$
where $\alpha({\mathcal{G}}_{t})$ denotes the independence number of the
feedback graph ${\mathcal{G}}_{t}$ [13]. In this case, if the server sets
$\eta={\mathcal{O}}(\sqrt{\frac{\ln K}{T}})$, $\xi={\mathcal{O}}(\frac{(\ln
K)^{\frac{3}{4}}}{T^{\frac{1}{4}}})$ and $|{\mathbb{C}}_{t}|={\mathcal{O}}(1)$
from (11) it can be verified that EFL-FG obtains sub-linear regret of
${\mathcal{O}}(\sum_{t=1}^{T}{(\ln
K)^{\frac{3}{4}}\alpha({\mathcal{G}}_{t})T^{-\frac{1}{4}}})$. It is useful to
point out that $\alpha({\mathcal{G}}_{t})$ depends on the budget $B_{t}$.
Increase in $B_{t}$ can result in more connected feedback graph
${\mathcal{G}}_{t}$ and as a result $\alpha({\mathcal{G}}_{t})$ decreases.
Therefore, larger budget can assist EFL-FG to achieve tighter sub-linear
regret bound. For example, when the budget is large enough such that at each
learning round $t$, the feedback graph ${\mathcal{G}}_{t}$ is a densely
connected graph as $\alpha({\mathcal{G}}_{t})={\mathcal{O}}(1)$. In this case,
the EFL-FG can achieve regret of ${\mathcal{O}}((\ln
K)^{\frac{3}{4}}T^{\frac{3}{4}})$. By contrast, when the feedback graph
${\mathcal{G}}_{t}$ only includes self-loops, $\alpha({\mathcal{G}}_{t})=K$
and as a result the EFL-FG can achieve regret of ${\mathcal{O}}((\ln
K)^{\frac{3}{4}}KT^{\frac{3}{4}})$.

##  IV Experiments

We tested the performance of different ensemble federated learning methods:
our proposed EFL-FG and FedBoost [6] over the following real data sets
downloaded from UCI machine learning repository [14]:
Bias Correction: This dataset includes $7,750$ samples of air temperature
information with $21$ features such as maximum or minimum air temperatures in
the day. The goal is to predict the next-day minimum air temperature [15].
CCPP: The dataset contains $9,568$ samples, with $4$ features including
temperature, pressure, etc, collected from a combined cycle power plant. The
goal is to predict hourly electrical energy output [16].
Energy: This dataset contains $19,735$ samples of $27$ features of house
temperature and humidity conditions were monitored with a wireless sensor
network. The goal is to predict the energy use of appliances. [17].

Consider the case where there are $100$ clients performing a regression task.
The server stores $22$ pre-trained models, including kernel based regression
models with $5$ Gaussian kernels, $5$ Laplacian kernels, $5$ polynomial
kernels, $5$ sigmoid kernels and $2$ feed-forward neural networks. The
bandwidth of Gaussian, Laplacian kernels, and the slope of sigmoid kernels are
$0.01,0.1,1,10,100$. Also, the degree of polynomial kernels are $1,2,3,4,5$.
The feedforward neural networks have $1$ and $2$ hidden layers respectively,
where each hidden layer consists of $25$ neurons with ReLU activation
functions. Each model is trained with $10\%$ of each dataset. Furthermore, the
cost of sending each model is considered to be the number of parameters
associated with model divided by the number of parameters associated with the
model with maximum number of parameters. Hence, the maximum cost of sending a
model is $1$. The budget for sending models is $B=3$. The learning rate $\eta$
and exploration rate $\xi$ for all methods are set to be $\frac{1}{\sqrt{T}}$.
The performance of ensemble federated learning methods is evaluated based on
mean square error (MSE) at learning round $t$ defined as
$\text{MSE}_{t}=\frac{1}{t}\sum_{\tau=1}^{t}{\frac{1}{|{\mathbb{C}}_{\tau}|}\sum_{i\in{\mathbb{C}}_{\tau}}{(\hat{y}_{i,\tau}-y_{i,\tau})^{2}}}$
where $\hat{y}_{i,t}$ denote the prediction made by the $i$-th client at
learning round $t$. In order to derive dominating sets for feedback graphs in
Algorithm 2, greedy set cover algorithm is employed [12]. Note that in this
experiments it is assumed that clients are not able to store their observed
data in batch. Therefore, FedBoost implementation is modified to cope with
this situation for fair comparison. Specifically, at learning round $t$, the
$i$-th client transmits its update with respect to newly observed sample
${\bm{x}}_{i,t}$ instead of the whole batch of data.

Table I: MSE ($\times 10^{-3}$) performance and percentage of budget violence. |  MSE($\times 10^{-3}$) |  budget violence ($\%$)
---|---|---
Algorithms | Bias | CCPP | Energy | Bias | CCPP | Energy
FedBoost | $69.11$ | $45.06$ | $10.19$ | $22.86\%$ | $13.79\%$ | $24.16\%$
EFL-FG | $4.81$ | $4.92$ | $8.36$ | $0\%$ | $0\%$ | $0\%$

Figure 1: MSE performance on Energy dataset.

Tables I shows the MSE and budget violence performance of different algorithms
over all datasets, where _budget violence_ indicates the proportion of
learning rounds when the instantaneous budget constraint is violated. As can
be seen from Table I, the proposed EFL-FG achieves lower MSE compared with
FedBoost. Table I shows that FedBoost violates the instantaneous budget in at
least $13\%$ of learning rounds for all datasets, while EFL-FG guaratees that
budget constraints is satisfied in every learning round.

Figure 1 illustrates the MSE versus learning rounds in Energy dataset. It can
be observed that our proposed EFL-FG outperforms FedBoost. Note that all
algorithms obtain different MSE performance starting from the first learning
round since different subsets of pre-trained models are chosen and combined by
different algorithms.

##  V Conclusion

The present paper developed a federated learning approach to learn ensemble of
pre-trained models when the server cannot transmit all models to clients due
to limitations in communication bandwidth and clients’ memory. Specifically,
the server generates a graph at each learning round and constructs an ensemble
model by choosing a subset of pre-trained models using the graph. This paper
provided the algorithm EFL-FG which constructs an ensemble model whose size
does not surpass a certain budget at each learning round. We proved that EFL-
FG achieves sub-linear regret. Experiments on several real datasets reveal the
merits of EFL-FG compared with other ensemble federated learning state-of-art.

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##  Appendix A Proof of Theorem 1

In order to proof Theorem 1, the following Lemmas are proved and used as step-
stone.

######  Lemma 2.

Let $\hat{f}_{k}(.)$ represents the ensemble model associated with
$v_{k}\in{\mathcal{V}}$. The regret of $\hat{f}_{k}(.)$ with respect to
$f_{k}(.)$ is bounded above as

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln|{\mathbb{N}}_{k,1}^{\text{out}}|}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}|{\mathbb{C}}_{t}|^{2}}{q_{j,t}W_{k,t}}}}$
(12)

which holds for all $v_{k}\in{\mathcal{V}}$.

###### Proof.

Let $W_{k,t}=\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{w_{j,t}}$. According
to definition of the set ${\mathbb{M}}_{k,t}$ and the Algorithm 1, it can be
written that
$W_{k,t+1}\leq\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{w_{j,t+1}}$. Thus,
for each $v_{k}\in{\mathcal{V}}$ we find

$\displaystyle\frac{W_{k,t+1}}{W_{k,t}}\leq\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t+1}}{W_{k,t}}}=\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}\exp(-\eta\ell_{j,t})}.$
(13)

Using the inequality $e^{-x}\leq 1-x+\frac{1}{2}x^{2},\forall x\geq 0$, from
(13), it can be concluded that

$\displaystyle\frac{W_{k,t+1}}{W_{k,t}}\leq\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}(1-\eta\ell_{j,t}+\frac{1}{2}\eta^{2}\ell_{j,t}^{2})}.$
(14)

Employing the inequality $1+x\leq e^{x}$ and taking logarithm from both sides
of (14), we obtain

$\displaystyle\ln\frac{W_{k,t+1}}{W_{k,t}}\leq\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}(-\eta\ell_{j,t}+\frac{1}{2}\eta^{2}\ell_{j,t}^{2})}.$
(15)

Summing (15) over learning rounds leads to

$\displaystyle\ln\frac{W_{k,T+1}}{W_{k,1}}\leq\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}(-\eta\ell_{j,t}+\frac{1}{2}\eta^{2}\ell_{j,t}^{2})}}.$
(16)

In addition, $\ln\frac{W_{k,T+1}}{W_{k,1}}$ can be bounded from below as

$\displaystyle\ln\frac{W_{k,T+1}}{W_{k,1}}\geq\ln\frac{w_{k,T+1}}{W_{k,1}}=-\eta\sum_{t=1}^{T}{\ell_{k,t}}-\ln|{\mathbb{N}}_{k,1}^{\text{out}}|.$
(17)

Combining (16) with (17), we get

$\displaystyle\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}\ell_{j,t}}}-\sum_{t=1}^{T}{\ell_{k,t}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln|{\mathbb{N}}_{k,1}^{\text{out}}|}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}\ell_{j,t}^{2}}}.$
(18)

According to (6) and considering the fact that
${\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})\leq 1$, taking expectation from
$\ell_{k,t}$ and $\ell_{k,t}^{2}$, $\forall v_{k}\in{\mathcal{V}}$, given
observed losses in prior learning rounds it can concluded that

$\displaystyle\mathbb{E}_{t}[\ell_{k,t}]$
$\displaystyle=\sum_{j\in{\mathbb{N}}_{k,t}^{\text{in}}}{p_{j,t}\frac{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}{q_{k,t}}}$
$\displaystyle=\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}$
(19a) $\displaystyle\mathbb{E}_{t}[\ell_{k,t}^{2}]$
$\displaystyle=\sum_{j\in{\mathbb{N}}_{k,t}^{\text{in}}}{p_{j,t}\frac{\left(\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}\right)^{2}}{q_{k,t}^{2}}}$
$\displaystyle=\frac{\left(\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}\right)^{2}}{q_{k,t}}\leq\frac{|{\mathbb{C}}_{t}|^{2}}{q_{k,t}}.$
(19b)

Taking the expectation from both sides of (18) results in

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}{\mathcal{L}}(f_{j}({\bm{x}}_{i,t}),y_{i,t})}}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln|{\mathbb{N}}_{k,1}^{\text{out}}|}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}|{\mathbb{C}}_{t}|^{2}}{q_{j,t}W_{k,t}}}}.$
(20)

Based on the Jensen’s inequality and the convexity of the loss function
${\mathcal{L}}(.,.)$, we can write

$\displaystyle\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}{\mathcal{L}}(f_{j}({\bm{x}}_{i,t}),y_{i,t})}\geq$
$\displaystyle{\mathcal{L}}(\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}}{W_{k,t}}f_{j}({\bm{x}}_{i,t})},y_{i,t})$
$\displaystyle={\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t}).$ (21)

It can then be concluded from (20) and (21) that

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln|{\mathbb{N}}_{k,1}^{\text{out}}|}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}|{\mathbb{C}}_{t}|^{2}}{q_{j,t}W_{k,t}}}}$
(22)

which proves the Lemma 2. ∎

######  Lemma 3.

The expected regret of EFL-FG with respect to $\hat{f}_{k}(.)$, $\forall
k\in[K]$ is bounded as

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$ $\displaystyle\frac{\ln
K}{\eta}+\sum_{t=1}^{T}{\left(\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}K|{\mathbb{C}}_{t}|^{2}\right)}$
(23)

###### Proof.

Recall $U_{t}=\sum_{k=1}^{K}{u_{k,t}}$. Based on the inequality $e^{-x}\leq
1-x+\frac{1}{2}x^{2},\forall x\geq 0$ we can write

$\displaystyle\frac{U_{t+1}}{U_{t}}$
$\displaystyle=\sum_{k=1}^{K}{\frac{u_{k,t+1}}{U_{t}}}=\sum_{k=1}^{K}{\frac{u_{k,t}}{U_{t}}\exp(-\eta\hat{\ell}_{k,t})}$
$\displaystyle\leq\sum_{k=1}^{K}{\frac{u_{k,t}}{U_{t}}(1-\eta\hat{\ell}_{k,t}+\frac{\eta^{2}}{2}\hat{\ell}_{k,t}^{2})}.$
(24)

According to (4), the inequality in (24) can be rewritten as

$\displaystyle\frac{U_{t+1}}{U_{t}}\leq\sum_{k=1}^{K}{\frac{p_{k,t}-\frac{\xi}{|{\mathbb{D}}_{t}|}{\mathcal{I}}(k\in{\mathbb{D}}_{t})}{1-\xi}(1-\eta\hat{\ell}_{k,t}+\frac{\eta^{2}}{2}\hat{\ell}_{k,t}^{2})}$
(25)

Using the inequality $1+x\leq e^{x}$, taking the logarithm from both sides of
(25) and summing the result over learning rounds, it can be inferred that

$\displaystyle\ln\frac{U_{T+1}}{U_{1}}$ $\displaystyle\leq$
$\displaystyle\sum_{t=1}^{T}{\sum_{k=1}^{K}{\frac{p_{k,t}-\frac{\xi}{|{\mathbb{D}}_{t}|}{\mathcal{I}}(k\in{\mathbb{D}}_{t})}{1-\xi}(-\eta\hat{\ell}_{k,t}+\frac{\eta^{2}}{2}\hat{\ell}_{k,t}^{2})}}.$
(26)

Furthermore, $\ln\frac{U_{T+1}}{U_{1}}$ can be bounded from below as follows

$\displaystyle\ln\frac{U_{T+1}}{U_{1}}\geq\ln\frac{u_{k,T+1}}{U_{1}}=-\eta\sum_{t=1}^{T}{\hat{\ell}_{k,t}}-\ln
K.$ (27)

Combining (26) with (27) leads to

$\displaystyle\sum_{t=1}^{T}{\sum_{k=1}^{K}p_{k,t}\hat{\ell}_{k,t}}-\sum_{t=1}^{T}{\hat{\ell}_{k,t}}$
$\displaystyle\leq$ $\displaystyle\frac{\ln
K}{\eta}+\sum_{t=1}^{T}{\sum_{k=1}^{K}{\frac{\xi{\mathcal{I}}(k\in{\mathbb{D}}_{t})}{|{\mathbb{D}}_{t}|}\hat{\ell}_{k,t}}}$
$\displaystyle+\sum_{t=1}^{T}{\sum_{k=1}^{K}{\frac{\eta}{2}\left(p_{k,t}-\frac{\xi}{|{\mathbb{D}}_{t}|}{\mathcal{I}}(k\in{\mathbb{D}}_{t})\right)\hat{\ell}_{k,t}^{2}}}.$
(28)

Expected values of $\hat{\ell}_{k,t}$ and $\hat{\ell}_{k,t}^{2}$ given prior
observed losses can be expressed as

$\displaystyle\mathbb{E}_{t}[\hat{\ell}_{k,t}]$
$\displaystyle=p_{k,t}\frac{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}{p_{k,t}}$
$\displaystyle=\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}$
(29a) $\displaystyle\mathbb{E}_{t}[\hat{\ell}_{k,t}^{2}]$
$\displaystyle=p_{k,t}\frac{(\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})})^{2}}{p_{k,t}^{2}}$
$\displaystyle=\frac{(\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})})^{2}}{p_{k,t}}\leq\frac{|{\mathbb{C}}_{t}|^{2}}{p_{k,t}}.$
(29b)

Taking the expectation from both sides of the (28), we get

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\sum_{k=1}^{K}{p_{k,t}{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$ $\displaystyle\frac{\ln
K}{\eta}+\sum_{t=1}^{T}{\sum_{k=1}^{K}{\frac{\xi{\mathcal{I}}(k\in{\mathbb{D}}_{t})}{|{\mathbb{D}}_{t}|}{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle+\sum_{t=1}^{T}{\sum_{k=1}^{K}{\frac{\eta}{2}(p_{k,t}-\frac{\xi}{|{\mathbb{D}}_{t}|}{\mathcal{I}}(k\in{\mathbb{D}}_{t}))\frac{|{\mathbb{C}}_{t}|^{2}}{p_{k,t}}}}.$
(30)

Taking into account that
${\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})\leq 1$ and $p_{k,t}\leq
1$, we can conclude that

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\sum_{k=1}^{K}{p_{k,t}{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$ $\displaystyle\frac{\ln
K}{\eta}+\sum_{t=1}^{T}{\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}K|{\mathbb{C}}_{t}|^{2}}.$
(31)

Moreover, according to Algorithm 2, it can be inferred that

$\displaystyle\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]=\sum_{k=1}^{K}{p_{k,t}{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}.$
(32)

Thus, we can write

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(\hat{f}_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$ $\displaystyle\frac{\ln
K}{\eta}+\sum_{t=1}^{T}{\left(\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}K|{\mathbb{C}}_{t}|^{2}\right)}.$
(33)

which proves Lemma 3. ∎

Combining Lemma 2 with Lemma 3, the following inequality holds

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}-\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln(K|{\mathbb{N}}_{k,1}^{\text{out}}|)}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k,t}^{\text{out}}}{\frac{w_{j,t}|{\mathbb{C}}_{t}|^{2}}{q_{j,t}W_{k,t}}}}$
$\displaystyle+\sum_{t=1}^{T}{\left(\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}K|{\mathbb{C}}_{t}|^{2}\right)}$
(34)

which holds for all $v_{k}\in{\mathcal{V}}$. Therefore, we can conclude that

$\displaystyle\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{\mathbb{E}_{t}[{\mathcal{L}}(\hat{f}({\bm{x}}_{i,t}),y_{i,t})]}}-\min_{k\in[K]}\sum_{t=1}^{T}{\sum_{i\in{\mathbb{C}}_{t}}{{\mathcal{L}}(f_{k}({\bm{x}}_{i,t}),y_{i,t})}}$
$\displaystyle\leq$
$\displaystyle\frac{\ln(K|{\mathbb{N}}_{k^{*},1}^{\text{out}}|)}{\eta}+\frac{\eta}{2}\sum_{t=1}^{T}{\sum_{j\in{\mathbb{N}}_{k^{*},t}^{\text{out}}}{\frac{w_{j,t}|{\mathbb{C}}_{t}|^{2}}{q_{j,t}W_{k^{*},t}}}}$
$\displaystyle+\sum_{t=1}^{T}{\left(\xi(1-\frac{\eta}{2}|{\mathbb{C}}_{t}|^{2})+\frac{\eta}{2}K|{\mathbb{C}}_{t}|^{2}\right)}$
(35)

which proves the Theorem 1.


# Localised response retrieval from Hamamatsu H9500 for a coded aperture
neutron-gamma imaging system based on an organic pixelated plastic
scintillator (EJ-299-34)

Michal J. Cieslak  Engineering Department
Lancaster University
Lancaster, LA1 4YW
United Kingdom
Email<EMAIL_ADDRESS>Kelum A.A. Gamage  School of Engineering
University of Glasgow
Glasgow, G12 8QQ
United Kingdom
Email<EMAIL_ADDRESS>
###### Abstract

Localised response of a sensitive light detector, such as Hamamatsu H9500
multi anode photomultiplier tube (MAPMT), is of vital importance for coded-
aperture imaging systems. When coupled with a suitable sensitive detector
(e.g. EJ-299-34 plastic scintillator), output signals of the MAPMT can be
retrieved to infer the interaction location in the pixelated scintillator.
Given the number of anodes in H9500 (256), significant processing power would
be necessary to handle each pulse individually. Therefore, a readout
electronics system was developed, based on resistive network approach, which
reduces the number of output signals to individual X and Y coordinates, and
subsequently allows particle identification. Coordinates retrieved in this
manner can be analysed in real time and used to infer the two-dimensional
location. Particle type can be also exploited by pulse shape discrimination
(PSD) application to the scintillator’s response. In this study, 169 anodes
were used (due to coded-aperture design rules), and reduced to two X and Y
output signals. These have been digitised using a bespoke FPGA based two
channel 14-bit 150 MSPS digitiser. The digital data are transferred to a host
application using UART to USB converter operating at 12 Mbits/s. Promising
results have been observed when the scintillator’s response was tested in
single particle field of 137Cs. However, further tests performed in the mixed-
field environment of 252Cf suggest that faster digitiser may be required to
obtain the required PSD performance.

##  I Introduction

Neutron radiation is an example of highly penetrating radiation, which if not
controlled, can pose a significant threat from nuclear safety and security
viewpoint. Therefore, effective neutron detection it is of vital importance in
areas, such as medicine [1], border control [2] and nuclear decommissioning
[3]. Moreover, the localisation of neutron emitting isotopes can be
particularly challenging in nuclear decommissioning applications, where large
areas may be contaminated and exact location of the radioactive substance
unknown. Collimator based imaging systems with a single opening in the
aperture, can be used to scan the large areas and create an image identifying
the location of the neutron emitters [4]. However, the drawbacks of such
system include the fact that process of image creation can be time consuming
(due to scanning requirement) and the effective field-of-view much limited.

Coded-aperture neutron imaging systems show promising results when considered
as an alternative to the single opening collimators. Relatively large scale
neutron detectors utilising coded-aperture approach have been previously
tested and characterised [5], [6]. Both examples exploit pulse shape
discrimination (PSD) capabilities of organic liquid (EJ-309) and plastic
(EJ-299) scintillators. Small scale implementation of the coded-aperture
imaging was previously realised with CLYC (Cs2LiYCl6:Ce) inorganic
scintillation block as the radiation sensitive detector [7].

In this study a pixelated organic plastic scintillator (Fig. 1) was coupled to
Hamamatsu H9500 MAPMT. The size of the single pixel has been matched to the
size of a single anode of Hamamatsu H9500 (2.8mm x 2.8 mm) to reduce the risk
of crosstalk between the anodes. Also, each scintillator block (2.8 mm x 2.8
mm x 15 mm) has been covered with 3M reflective tape to avoid interference
between adjoining blocks. Given the sensitivity of the scintillator to both
neutrons and gamma-ray photons, such configuration requires a bespoke readout
electronics system. It is required that the readout electronics can not only
infer the location of the interaction, but also enable particle classification
(either neutron or gamma-ray photon).

Figure 1: Solid pixelated organic plastic scintillator (EJ-299-34).

##  II Methodology

The readout electronics system, consists of resistive network and pulse
shaping circuit followed by Field Programmable Gate Array (FPGA) based signal
digitiser. The resistive network, first proposed by Popov [8], splits the
voltage incurred as a result of an interaction in the specific pixel into
equal parts on X and Y axis. This is then passed on to the weighted pre-amp
pulse shaping stage, where based on the voltage detected, X and Y coordinates
of the interaction can be computed.

This stage has been simulated using TI-TINA (SPICE based) simulation package.
Simulated neutron pulses were applied to three adjoining anodes and transient
response recorded at the relative outputs of the pre-amp stage. Fig. 2
presents the proportional response of the simulated circuit at selected
outputs, which advocates the claim that two-dimensional coordinates of the
interaction can be inferred using this method. Furthermore, the resistive
network does not appear to distort the shape of the applied pulse, presenting
the potential of PSD techniques application for particle identification.

Following the successful simulation of the circuit, resistive network was
designed and built into a printed circuit board (PCB). It contains 507
surface-mount resistors in 803 and 605 packages, split onto four printed
circuit boards (PCBs), which are directly attached to the output connectors of
the Hamamatsu H9500 MAPMT. Complete assembly attached to the MAPMT is
presented in Fig. 3. It reduces the number of output signals from 169 to 26.
This is further reduced to 2 signals (X and Y coordinates) through the shaping
pre-amp and summing op-amp circuitry.

Figure 2: Proportional response of the circuit simulated in TI-TINA software
package for pulses connected to three adjoining anodes on one axis. VF17-VF19
correspond to transient response to the simulated neutron pulse connected to
the three adjoining anodes. Figure 3: A network of 507 resistors that reduces
the number of output signals from 169 to 26 (13 on X and 13 in Y axis).

Analogue pulses present at the output stage of the summing op-amp circuitry
are digitised utilising two Analog Devices AD9254 150 MSps, 14-bit amplitude
resolution ADCs directly linked to Altera Cyclone IV EP4CE115 FPGA. Verilog
VHDL code has been developed that reads the outputs of the two 14-bit ADCs and
transfers the data at 12 Mbits/s to a host application (such as Python
application on a Laptop running Linux) via USB to UART converter. Each sample
is transmitted to the host application with a channel identifier (A or B),
e.g. A4087, B6743.

In order to further assess PSD capability of the pixelated scintillator, the
anode of the PMT was directly linked to a 10-bit, 500 MSps "raw" data
digitizer where the data were transferred from the digitizer to a host PC
using Ethernet protocol. The drawback of this system is single channel
operation. However, much higher sampling rate is expected to allow more
accurate pulse reconstruction despite the reduced ADC resolution.

##  III Results and Conclusions

Simulation of the resistive network deem a result (Fig. 2) that is
proportional to the location of the interaction of the incoming particle,
suggesting localised response of the MAPMT can be obtained in this way.
Additionally, pulse shape analysis of the response of the pixelated
scintillator to single particle field of 137Cs proposes PSD as a viable
technique for particle identification, given the comparison to modelled
pulses, as presented in Fig. 4.

(a)

(b)

Figure 4:  A comparison of a) A mathematical model of gamma-ray photon and
neutron induced pulses from organic scintillator, b) A recoded gamma-ray pulse
from 137Cs.

Initial simulation and experimental results obtained, based on the single
particle type detection, present a potential for good localisation and
particle separation capabilities of the complete system. Given the neutron-
gamma separation potential of the scintillator, the complete system was also
tested in dual-particle environment of the spontaneous fission field of 252Cf.

Two different digitizer systems were tested, as initial results obtained using
the FPGA based 150 MSps digitizer system failed to discriminate neutrons from
gamma-ray photons. Results obtained from the 500 MSps "raw" data digitizer
(Fig. 5) show somewhat promising PSD potential, given the modulated energy
spectrum of 252Cf at Lancaster University. However, the estimated figure-of-
merit (FOM) of 0.337 suggests considerably poorer performance when compared
with the FOM of 0.649 previously quoted in the literature for the cylindrical
plastic scintillator samples tested in the same environment [9]. Additional
tests and analysis with "bare" mixed-field sources would also be advantageous
to further assess the suitability of the system for similar applications.

Figure 5: PSD scatter plots using CCM for Pixelated EJ-299-34 plastic
scintillator).

## Acknowledgment

The authors would like to acknowledge the funding support from EPSRC via
Faculty of Science and Technology, Lancaster University, U.K. and Sellafield
Ltd.

## References

  * [1] S. D. Clarke, E. Pryser, B. M. Wieger, S. A. Pozzi, R. A. Haelg, V. A. Bashkirov, R. W. Schulte.  A scintillator-based approach to monitor secondary neutron production during proton therapy A scintillator-based approach to monitor secondary neutron production during proton therapy  Medical Physics, vol. 5915, no. 11, pp. 5915–5925, 2016, https://doi.org/10.1118/1.4963813.
  * [2] R. T. Kouzes, E. R. Siciliano, J. H. Ely, P. E. Keller, R. J. McConn.  Passive neutron detection for interdiction of nuclear material at borders  Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 584(2–3), 383–400, https://doi.org/10.1016/j.nima.2007.10.026.
  * [3] F. Jallu, A. Reneleau, P. Soyer, J. Loridon.  Dismantling and decommissioning: The interest of passive neutron measurement to control and characterise radioactive wastes containing uranium  Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 271, 48–54. https://doi.org/10.1016/j.nimb.2011.09.018.
  * [4] K. A. A. Gamage, M. J. Joyce, G. C. Taylor.  A digital approach to neutron-$\gamma$ imaging with a narrow tungsten collimator aperture and a fast organic liquid scintillator detector  Applied Radiation and Isotopes : Including Data, Instrumentation and Methods for Use in Agriculture, Industry and Medicine, 70(7), 1223–7, https://doi.org/10.1016/j.apradiso.2012.01.021.
  * [5] P. Hausladen, J. Newby, F. Liang, M. Blackston.  The Deployable Fast-Neutron Coded- Aperture Imager : Demonstration of Locating One or More Sources in Three Dimensions  Oak Ridge National Laboratory Report, ORNL/TM-2013/446, September 2013.
  * [6] C. V. Griffith, R. S. Woolf, B. F. Phlips.  64-Element Fast-Neutron, Coded-Aperture Imager.  IEEE International Symposium on Technologies for Homeland Security (HST), pages 1–5, 2017, https://doi.org/10.1109/THS.2017.7943453.
  * [7] C. M. Whitney, L. Soundara-Pandian, E. B. Johnson, S. Vogel, B. Vinci, M. Squillante, J. F. Christian.  Gamma–neutron imaging system utilizing pulse shape discrimination with CLYC  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 784, 346–351, https://doi.org/10.1016/j.nima.2014.09.022.
  * [8] V. Popov.  Advanced data readout technique for Multianode Position Sensitive Photomultiplier Tube applicable in radiation imaging detectors  Journal of Instrumentation , 6(1), C01061–C01061, https://doi.org/10.1088/1748-0221/6/01/C01061.
  * [9] M. J. Cieślak, K. A. A. Gamage, R. Glover,  Pulse shape discrimination characteristics of stilbene crystal, pure and 6Li loaded plastic scintillators for a high resolution coded-aperture neutron imager, Journal of Instrumentation, 12 (2017).


#  TriSol: a major upgrade of the TwinSol RNB facility

P.D. O’Malley  T. Ahn, D.W. Bardayan, M. Brodeur, S. Coil  J.J.
Kolata111Emeritus professor<EMAIL_ADDRESS>Dept. of Physics and Astronomy,
University of Notre Dame, Notre Dame, IN 46556, United States

###### Abstract

We report here on the recent upgrade of the TwinSol radioactive nuclear beam
(RNB) facility at the University of Notre Dame. The new TriSol system includes
a magnetic dipole to provide a second beamline and a third solenoid which acts
to reduce the size of the radioactive beam on target.

###### keywords:

Radioactive ion beams, Solenoid spectrometer

††journal: Journal of LaTeX Templates

##  1 Introduction

The first radioactive nuclear beam setup [1] at the University of Notre Dame,
used from 1987-1995, was replaced by the TwinSol facility [2, 3, 4, 5, 6, 7]
which became operational in 1998. Both of these were the result of a
collaboration between the University of Notre Dame and a group at the
University of Michigan headed by Prof. F. D. Becchetti.

Figure 1: (Color online) The TwinSol radioactive nuclear beam facility. Figure
from Ref. [7].

TwinSol (Fig.1) consists of a pair of 30 cm bore, 6T superconducting solenoids
contained within low-loss cryostats ($<$0.1 LHe/h) having a holding time of
$>$2 months. The solenoids are air-core (hence no iron yokes) and are operated
in persistent mode. As a result, their magnetic fields do not suffer from
hysteresis and scale exactly with the applied currents. However, the fields
extend to a large distance (up to 0.1 T at 2 m on axis at maximum field) and
this must be accounted for both in computing the required magnet current and
in safety concerns for those working near the magnets when they are energized.

Referring to Fig.1 above, a variable aperture (0-30 mm dia.) at the crossover
point between the two solenoids provides momentum analysis and therefore a
degree of isotope separation. In some cases, an absorber foil was also located
there to provide further separation via differential energy loss. Most of the
early experiments were carried out in the small scattering chamber close to
the second solenoid. In this location, the beam spot size was typically 5-6 mm
full width at half maximum (FWHM). However, in later experiments involving
neutron or $\gamma$-ray detection, a shielding wall consisting of borated
water followed by concrete blocks was inserted as shown. Experiments were also
carried out in the larger scattering chamber illustrated, or in other
apparatus such as the prototype active-target time projection chamber (pAT-
TPC) [8] from Michigan State University (MSU) or the neutron array described
in [9].

The primary target was initially situated within an ISO-100 cross, followed by
an ISO-200 cross containing an entrance aperture and a Faraday cup having an
outer diameter of 2.5 cm. Typically this setup accepted particles scattered
between 3∘ and 6∘ in the laboratory (lab) frame, although larger collimators
accepting up to 11∘ were available if needed. However, more recently the
ISO-100 cross was replaced by a 30 cm dia. chamber to provide room for a
moveable target system holding 4 gas targets [10], and also the ability to
place silicon (Si) detectors around the target for certain experiments. In
this configuration, the angular acceptance of the system is between 2∘ and
5.25∘ in the lab. For most experiments, the primary target is a 2.5 cm long
gas cell containing 2H or 3He gas at atmospheric pressure (though solid
targets are occasionally used). The gas-cell windows are typically 4 $\mu$m Ti
foils. A list of the many publications carried out with TwinSol during its
nearly 25 years of existence can be found here:
http://notredame.box.com/s/9w79ctrizr3ol8ac1pow9bckm5sa73bu.

##  2 TriSol

A further recent change in the setup shown in Fig.1 was the removal of the
neutron shielding wall shown and extension of the beam line through a 1.25 m
thick high-density concrete shielding wall into an adjacent area. This
modification provided additional isolation from neutron and $\gamma$-ray
background coming from the primary target and more room to attach ancillary
equipment such as the $\beta$-decay station described in Ref.[11].
Unfortunately, at this location the ion optics resulted in a beam spot size of
25 mm FWHM or greater. While acceptable for some experiments, this was
problematic for others which resulted in the first impetus to design an
upgrade which would produce improved RNBs. Additionally, the “St. Benedict”
ion trap [12] requires a dedicated beam line. Hence, reaction studies with
exotic beams or other experiments including additional lifetime measurements
could only be continued if two lines were available. Therefore, a small XY
steering magnet followed by a magnetic dipole were inserted immediately after
the shielding wall. The dipole, which was originally part of the ORRUBA
beamline [13] at Oak Ridge National Laboratory (ORNL), has ports at 0∘ and 15∘
to the TwinSol beam line. The latter was used for TriSol to make use of its
magnetic dispersion as discussed below. Since the expected radioactive beams
were much larger in extent than those used at ORNL, the clear aperture of this
dipole was increased to 12 cm by the insertion of soft iron sections into its
yoke. Afterwards, the magnetic field was extensively mapped as a function of
position and current, including the fringe field out to 65 cm. The maximum
central field of the modified magnet easily exceeded requirements for the most
rigid beams that could be produced at TriSol. The third superconducting
solenoid [14] has a clear, warm bore of 15 cm dia., a maximum central field of
5T, and a length of 1 m. As was the case with the original two solenoids, the
magnetic field was measured by the vendor and a file of the results was
supplied with the magnet. Its location was determined according to its clear
aperture and the desired overall magnification of the system. The 100 liter
LHe supply of this solenoid is maintained within a recondensing cryostat [15]
by a cryocooler [16].

###  2.1 Parameter calculation

Since the TriSol ion optics are more complex than those of TwinSol, it was
necessary to use a simulation package such as LISE++ [17]
https://lise.nscl.msu.edu/lise.html from MSU to design the spectrometer and
enable calculation of its parameters. A number of important operational
considerations for this package are outlined in this section:

(1) LISE++ currently has no ability to directly simulate a gas target. As a
result, the energy, energy dispersion, and angular dispersion of the incident
primary beam after the entrance window of the gas cell (typically a 4 $\mu$m
Ti foil) must be calculated using the embedded “Physical calculator” module,
then entered into the “Projectile” window. The “Stripper” is the exit foil.
The gas target between these two segments is treated as if it were a foil
located at the center of the gas cell. This introduces a small but not very
serious error into the simulation. For example, moving the target location by
$\pm$1.25 cm (the length of the gas cell) results in a $\pm 4\%$ change in the
calculated rate, a $\pm 3\%$ change in the spot size, and a $\pm 6\%$ change
in the FWHM of the energy resolution. The actual error will be of the order of
one-half of these extreme values.

(2) The suggested method to determine the required B-fields for the magnetic
elements is to scale the settings from a previous calculation according to the
magnetic rigidity:

$R=B\rho=K\sqrt{\frac{ME}{Q^{2}}}.$ (1)

Here, M is the mass of the ion (generally taken to be the mass number A), E is
its energy, Q is its charge state, $\rho$ is the radius of curvature of the
ion’s path, and K is a constant. This will usually give initial settings
within a few percent of the required values. The final values can then be
obtained by making small changes in the parameters while observing the spot
size and/or rate of the beam at various locations along the beam path.

(3) The production method is selected using “Physics Models”, three of which
are relevant for TriSol calculations: (a) Fusion residuals, (b) Two-Body
Reactions, and (c) ISOL mode. Option (a) requires no additional input since
the relevant cross sections and angular distributions are computed from an
evaporation model. Option (b) requires the input of cross sections, either
directly from experimental data or from a reaction calculation using, e.g., a
Distorted Wave Born Approximation (DWBA) code. This is the most-used method.
The accuracy of the LISE++ beam-rate predictions then depends on that of these
cross sections, and will be best if there is experimental data at nearby
energies. The case of 8B, discussed below, is one of the more difficult ones.
The two-nucleon transfer process is calculated in a cluster model and there is
no “spectroscopic factor” to determine the absolute cross section.
Furthermore, there is only limited data at one energy far below that of
interest to compare with. The experimental yield was only about one third of
the computed value. However, the agreement with the calculation is much better
for single-nucleon-transfer reactions. Option (c) has only been used to for
calibration purposes with an $\alpha$-particle source.

(4) TriSol produces a “cocktail” beam. The most intense contaminant typically
arises from scattering of the primary beam somewhere near the production
target. This component cannot be accurately predicted since the location of
the source is distributed and its intensity is uncertain. As a result, the
contamination level of the beam must be experimentally determined.
Instructions on how to use LISE++ to simulate TriSol beams, and some sample
input files, are available at this site. Comparisons between computed and
experimental yields will be added as they become available.

###  2.2 Initial Results

The beam line to St. Benedict is in the process of being completed up to the
gas catcher, the first of its elements. This is simply an extension of the
existing TwinSol line and so the beam transmission efficiency is well known,
which has been confirmed by a LISE++ calculation. The calculated spot size is
well within the acceptance of the gas catcher.

The TriSol beam line has been completed. A diagram of this line, taken from
LISE++ output, is shown in Fig. 2. A test run was carried out with a 8B beam
to verify its performance. The reaction was 3He(6Li, 8B) at an incident energy
of 36 MeV, resulting in a 27.6 MeV 8B5+ beam with a calculated intensity of
2.5x104 ions per second for 500 electrical nanoamperes (enA) of 6Li. The
LISE++ Monte-Carlo calculation of the beam envelope is illustrated in Fig. 3.

Figure 2: (Color online) The TriSol beam line. The two bars on the left denote
the primary target, the three solenoids are designated by green rectangles,
and the dipole, located after the shielding wall, is illustrated by the dark-
green wedge. Collimators are shown as white lines and detector elements by red
lines. The two white diamonds are steering elements which do not physically
exist but are included in the simulation to test the influence of off-centered
beam on the prediction. In addition, two X-Y steerers (not shown) are situated
just before the dipole, and just after the dipole on the 15∘ line, to trim the
beam position in the two lines as mentioned above. (See Fig. 3 for the scale.)
Figure 3: (Color online) The calculated 8B beam envelope. This is a scale
diagram showing the placement of the various elements of TriSol. The dipole is
indicated by the blue rectangle centered at 10.5 m

The corresponding hit patterns at the focal point are shown in Fig. 4, with
and without current in the third solenoid. The detector used was a 5 cm x 5 cm
double-sided Si strip detector (DSSD) having a 3.3 mm x 3.3 mm pixel size. In
these images, the 8B beam is not exactly centered on the detector. The
solenoid-off hit pattern would otherwise extend beyond the left edge of the
DSSD. The observed beam compression is compatible with the predictions of the
Monte-Carlo simulation, thus verifying the TriSol concept. Referring to Fig.
3, it can be seen that the divergence angle of the beam has been increased in
the process, as must happen due to conservation of the transverse emittance.
However, the maximum divergence is only $\pm$0.8∘, and there is sufficient
room behind the solenoid to decrease this by at least a factor of two (at the
expense of spot size) by moving the secondary target location downstream for
those experiments that require smaller divergence.

Figure 4: (Color online) The measured hit patterns. On the left is the pattern
with the third solenoid turned off. On the right is the hit pattern with this
solenoid turned on. These plots were taken for 965 8B events on the detector
with the solenoid off and 913 events with the solenoid on. The circle is the
two-standard deviation diameter of the predicted hit pattern for 913 events.
Figure 5: (Color online) The timing spectrum from the 8B run; time-of-flight
on the Y-axis (0.39 ns/channel) vs. ion energy on the X-axis (66.2
keV/channel). For each ion, the charge state Q is equal to Z.

Finally, several TwinSol experiments (see, e.g., Ref. [18]) required the use
of time-of-flight (TOF) to select the relevant ion from a cocktail beam. The
Notre Dame FN tandem accelerator facility has the ability to produce a bunched
beam with a time resolution of 2 ns (FWHM). The resulting TOF spectrum for the
8B beam from TriSol is shown in Fig. 5. The main isotopes in this spectrum are
identified there. The TOF is repeated twice due to the way the timing hardware
operates. As a result, the time difference between the two groups of similar
ions, which is 101 ns, provides a calibration for this variable. Observe that
the TOF is highly correlated with energy so the ion’s energy resolution can be
improved using this feature, as discussed in Ref. [19]. In addition, note the
strong 6Li groups from scattered primary beam particles. In general, the TOF
spectrum from TriSol is improved from that observed for TwinSol due to the
longer flight path.

Recently, we have been working to suppress contamination from scattered
primary beam. One method that seems to be effective in favorable cases
involves charge-changing using a thin (400 $\mu$g/cm2 Mylar) foil placed after
the crossover collimator shown in Fig. 3. In the case of 14O production that
we have investigated, the yields of the 7+ and 8+ charge states at 30 MeV are
comparable in intensity. As a result, only a small amount of beam rate is lost
when the first solenoid is set to pass 7+ ions while the rest of the magnetic
elements are set for the 8+ charge state. However, the contaminant 12C ions
are focused into a ring around the 14O ions immediately in front of the third
solenoid. A set of X-Y slits placed there can selectively pass 14O while
eliminating some of the 12C contaminant. The reaction used was 3He(12C,14O) at
an incident energy of 54 MeV, with a 12C current of 1 e$\mu$A which is easily
achievable. In this case, there exists experimental data for the 3He reaction
at an energy near the required value of 12.7 MeV [21]. The result is an 14O
beam energy of 29.5 MeV and a rate of 4.3x104/s (extrapolated from a
measurement with a 1 enA primary beam current). This is 85% of the calculated
rate when no slits are used. However, the beam purity under these conditions
is only 6%. The purity can be increased at the expense of beam rate by using
the slits in front of the third solenoid to selectively intercept the 12C
contaminant as discussed above. Fig. 6 shows the result. The fraction of 14O
in the spectrum increased from 6% to 18.5%, but the rate was reduced by about
a factor of three. However, this is a much better option for active target
detectors, such as the ND-Cube [20], which have high efficiency but are rate
limited due to pulse-pileup considerations.

Figure 6: Energy spectra for 14O production, without (left) and with (right)
slits set to intercept the 12C contaminant. The 12C peaks in the right
spectrum are doubled due to scattering from two different locations.

##  3 Summary

The TriSol project, nearing completion, provides separate beam lines for ion-
trap experiments and nuclear reaction studies with radioactive nuclear beams.
The addition of a third solenoid, acting as a beam compressor, has resulted in
beam spot sizes that are typically about 1 cm FWHM with a divergence angle of
less than $\pm$1∘. In addition, space exists behind the third solenoid to
allow for some reduction of the divergence, at the expense of spot size, by
moving the secondary target downstream. Time-of-flight measurement of the
ions, often an important experimental parameter, has been improved due to the
longer flight path of the system, and the location of the final focus,
completely behind thick shielding walls, has essentially eliminated background
from the primary target, thus improving the prospects for neutron and
$\gamma$-ray experiments with radioactive beams.

The additional complexity of this system, compared with TwinSol, required the
use of LISE++ to simulate the spectrometer and thereby determine initial
settings for the magnetic elements, which are typically within a few percent
of their final values. This also provided the ability to test new methods to
improve the purity of the beam by reducing the scattered primary beam. As an
example, some TwinSol experiments introduced an absorber foil after the mid-
plane collimator to purify the beam via differential energy loss. However,
LISE++ simulations have shown that a thinner foil, which changes the charge of
the ions between the solenoids, is a far better purification method. Under
favorable circumstances, this can be accomplished with only a small reduction
in the intensity of the desired beam.

With the completion of the TriSol project, coupled with an array of new
instruments such as the ND-Cube active-target time projection chamber [20] and
the St. Benedict ion trap, the faculty and staff of the Notre Dame Nuclear
Science Laboratory, and its national and international collaborators, will be
well-positioned to continue the ongoing studies of nuclear astrophysics,
reaction mechanisms of exotic nuclear beams, and fundamental interactions.

## Acknowledgements

The TriSol project was supported by a grant from the University of Notre Dame,
and by the US National Science Foundation under Grant $\\#$2011890\. We also
acknowledge important contributions from William von Seeger, Chevelle
Boomershine, and Scott Carmichael, graduate students at Notre Dame, while co-
author Sydney Coil is an undergraduate student. Professor Kolata especially
wishes to thank Dr. Oleg Tarasov of Michigan State University for his help in
implementing the LISE++ simulation package for use with TriSol.

## Authorship credit contribution statement

P.D. O’Malley: Supervision, Investigation, Writing $\&$ Editing,
Visualization, Validation. T. Ahn: Investigation, Writing $\&$ Editing,
Visualization. D.W. Bardayan: Project Management, Investigation, Writing $\&$
Editing, Visualization. M. Brodeur: Investigation, Writing $\&$ Editing,
Visualization. S. Coil: Magnetic dipole implementation, Magnetic field
measurement. J.J. Kolata: Conceptualization, Investigation, Writing - Initial
draft, Editing, Visualization, Validation, Software implementation and
validation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this manuscript.

## References

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