# Federated Covariate Shift Adaptation for Missing Target Output Values

Yaqian Xu  International Institute of Finance, School of Management,
University of Science and Technology of China,
Hefei, 230026, P. R. China
<EMAIL_ADDRESS>Wenquan Cui  International Institute of Finance,
School of Management,
University of Science and Technology of China,
Hefei, 230026, P. R. China
<EMAIL_ADDRESS>Jianjun Xu  International Institute of Finance, School of
Management,
University of Science and Technology of China,
Hefei, 230026, P. R. China
<EMAIL_ADDRESS>Haoyang Cheng  College of Electrical and Information
Engineering,
Quzhou University,
Quzhou, 324000, P. R. China
<EMAIL_ADDRESS>
((Day Month Year); (Day Month Year); (Day Month Year))

###### Abstract

The most recent multi-source covariate shift algorithm is an efficient
hyperparameter optimization algorithm for missing target output. In this
paper, we extend this algorithm to the framework of federated learning. For
data islands in federated learning and covariate shift adaptation, we propose
the federated domain adaptation estimate of the target risk which is
asymptotically unbiased with a desirable asymptotic variance property. We
construct a weighted model for the target task and propose the federated
covariate shift adaptation algorithm which works preferably in our setting.
The efficacy of our method is justified both theoretically and empirically.

###### keywords:

Federated learning; covariate shift; multi-source; hyperparameter
optimization.

(Day Month Year)

AMS Subject Classification: 62J07, 62-08

##  1 Introduction

Covariate shift adaptation has been a pivotal part of transfer learning.[25]
It is a prevalent setting for machine learning in which the feature
distributions differ between the train (source) and test (target) domains,
while the conditional distributions of the output variable given the feature
variable remain unchanged, see Ref. 2000Improving. In this setting, when the
output values of the target data are missing, the multi-source covariate shift
(MS-CS)[17] algorithm can use multiple relevant source datasets whose output
values are available to predict the target task.

One critical, but often overlooked assumption in MS-CS is that we can merge
data from all sources as a training set. However, this assumption is not
satisfied in many cases. For example, we need to predict the disease scores
for patients in a new (target) hospital, but we only have historical medical
outcome data from some source hospitals other than the target one.
Furthermore, to protect patient privacy, each source hospital does not allow
its data to leave its local area or to be disclosed, resulting in data
islands. In this case, the MS-CS algorithm is infeasible. Fortunately, this
issue can be well-solved in federated learning since it provides a privacy
protection mechanism and allows us to learn and save locally at each node
rather than share data or parameters, see Ref.
2018Aby,2017Practical,2017SecureML and 2019Federated.

In this paper, we extend the MS-CS algorithm to the federated learning
framework. The difficulty in conducting MS-CS under federated learning is that
the variance reduced estimate[17] of the target risk is inaccessible due to
data islands between the sources. It is thus essential to construct a new
estimate of the target risk in our setting.

To solve this problem, we first formulate the federated covariate shift
(FedCS) setting. For covariate shift adaptation in the FedCS setting, we
propose the federated importance weighting estimate (FedIWE) of the target
risk which is asymptotically unbiased. Based on FedIWE, we further propose the
federated domain adaptation estimate (FedDAE) of the target risk, which has
the smallest asymptotic variance in a class of asymptotically unbiased
estimates of the target risk. These estimates can well satisfy the requirement
that every source does not allow its data to leave its local area or be
disclosed. We provide theoretical analysis and proofs for all properties. The
estimate of the hyperparameter is obtained by minimizing FedDAE, and then all
source models are determined. We construct a weighted model by weighting all
source models for the target task and present the error bound of the weighted
model. In order to describe the steps of our proposed method in the specific
implementation, we propose the federated covariate shift adaptation (FedCSA)
algorithm. We execute the FedCSA algorithm on simulated data and real data,
and the experimental results demonstrate that our proposed method is
effective.

Related Work. Covariate shift adaptation aims to evaluate the performance of
models for the target task using only a relevant single source dataset, see
Ref. 2000Improving,2007Covariate,2010A,2019Towards and 2010Cross. Combining
additional source information can improve the efficiency of searching
hyperparameters and obtain a better solution with less computation, see Ref.
bonilla2007multi,feurer2018practical and NEURIPS2018_14c879f3. Elvira _et
al_.[6] and Sugiyama _et al_.[25] offer an estimate of ground-truth model
performance based on importance sampling that is guaranteed unbiased in
theory, see Ref. 2007Covariate and 2018Conditional. However, the variance is
unbounded. Deep embedded validation combines the control variates method to
reduce the variance, see Ref. 2019Towards and Lemieux2017. Nomura and
Saito[17] propose the variance reduced estimate in the case where the labels
of the target data are completely missing under the MS-CS setting. Previous
studies have often assumed only a single source or multi sources without data
islands. This paper, on the other hand, focuses more on the real situation
with the problem of data islands.

Another related field is federated learning, see Ref.
2018Aby,2017Practical,2017SecureML and 2019Federated. CryptoNet[9] improves
the efficiency of data encryption and the performance of federated learning.
Bonawitz _et al_.[4] proposes the aggregation scheme that updates the machine
learning model under the federated learning framework. SecureML[16] supports
the cooperative privacy protection training in the multi-client federated
learning system. All of these methods learn a single global model. On the
contrary, federated multi-task learning[23] learns a separate model for each
node. Liu _et al._[13] propose a semi-supervised federated transfer learning
method under privacy protection. In their settings, the target output values
are available. Federated adversarial domain adaptation proposes an efficient
adaptation algorithm that can be applied to the federated setting based on
adversarial adaptation and representation disentanglement, see Ref.
2019FederatedP. These approaches are capable of resolving data island issues,
and we are concentrating our efforts on extending MS-CS to the federated
learning framework.

Contributions. Our contributions are summarized as follows:

  * •

We formulate the FedCS setting which extends the MS-CS setting to the
framework of federated learning.

  * •

We propose FedIWE which is asymptotically unbiased for the target risk in the
FedCS setting.

  * •

We further propose FedDAE which has the smallest asymptotic variance in a
class of asymptotically unbiased estimates of the target risk.

  * •

We construct a weighted model for the target task by weighting all source
models and present the error bound of the weighted model.

  * •

We propose the FedCSA algorithm, which we have empirically proven to work well
in the FedCS setting.

The remainder of this paper is organized as follows. In Sec. 2, we formulate
the FedCS setting and propose FedIWE for covariate shift. We futher propose
FedDAE which has the smallest asymptotic variance in a class of asymptotically
unbiased estimates of the target ris. We construct a weighted model for the
target task and propose the FedCSA algorithm. The relevant theoretical proofs
are provided in the appendix. Sec. 3 and Sec. 4 justify the efficacy of our
method through experiments on the simulation data and a real dataset,
respectively. Sec. 5 concludes.

##  2 Method

In this paper, we call the data owner whose output values are available the
source, and a new data owner whose output values are missing the target. We
consider the case of $K$ sources and one target. Each source and target has a
domain and a task, see Ref. 2010A.

###  2.1 FedCS setting

Before formulating the FedCS setting, we first introduce some notations. Let
$\mathcal{X}\subset\mathbb{R}^{d}$ be the d-dimensional continuous bounded
feature space and $\mathcal{Y}\subset\mathbb{R}$ be the bounded real-valued
output space. We use $P(\bm{x})$ to denote the probability distribution
function of the feature variable $\bm{X}\in\mathcal{X}$. Denote the $j$th
source and target domains as
$\mathcal{D}_{j}=\\{\mathcal{X},P_{j}(\bm{\bm{x}})\\}$ and
$\mathcal{D}_{T}=\\{\mathcal{X},P_{T}(\bm{x})\\}$. The $j$th source and target
tasks are denoted by $\mathcal{T}_{j}=\\{\mathcal{Y},f_{j}(\cdot)\\}$ and
$\mathcal{T}_{T}=\\{\mathcal{Y},f_{T}(\cdot)\\}$, where $f(\cdot)$ is the
function which predicts the output value using the feature instance. We use
$p_{j}(\bm{x})$, $p_{T}(\bm{x})$, $p_{j}(\bm{x},y)$ and $p_{T}(\bm{x},y)$ to
denote the corresponding feature probability density functions and joint
probability density functions, respectively. More specifically, we denote the
i.i.d. dataset of the $j$th source as
$D_{j}=\\{(\bm{x}_{j,i},y_{j,i})\\}_{i=1}^{n_{j}}$, where
$\bm{x}_{j,i}\in\mathcal{X}$ is the feature instance and
$y_{j,i}\in\mathcal{Y}$ is the corresponding output value. Similarly, we
denote the i.i.d. dataset of the target as
$D_{T}=\\{(\bm{x}_{T,i})\\}_{i=1}^{n_{T}}$. In most cases,
$0<n_{T}\ll\sum_{j=1}^{K}n_{j}$.

Next, we present the detail of the FedCS setting and two assumptions for
covariate shift are described as follows:

The source domains have support for the target domain, while the marginal
feature distributions are different, i.e., $p_{T}(\bm{x})>0\Rightarrow
p_{j}(\bm{x})>0$, $p_{T}(\bm{x})\neq p_{j}(\bm{x})$,
$\forall\bm{x}\in\mathcal{X}$, $\forall j=1,2,\cdots,K$.

Conditional output distributions remain the same between the target and all
the sources, i.e., $p_{T}(y|\bm{x})=p_{j}(y|\bm{x}),j=1,2,\cdots,K$, where
$p_{T}(y|\bm{x})$ and $p_{j}(y|\bm{x})$ denote the conditional output density
functions of the target and $j$th source, respectively.

Assumption 2.1 is a commonly adopted condition in covariate shift, and
Assumption 2.1 is required to ensure that the source data can be used to
predict the target output values, see Ref. 2000Improving. Privacy protection,
in addition to covariate shift, needs to be considered in the FedCS setting.
In our paper, every source does not allow its data to leave its local area or
to be disclosed, and the output values of the target data are completely
missing. The purpose of this paper is to learn a parametric model that can
accurately predict the output values of the target data in the FedCS setting.

###  2.2 Estimation of the target risk function

To obtain an accurate parametric model, we need to find the optimal
hyperparameter $\bm{\theta}^{\star}$ with respect to the target distribution:

$\bm{\theta}^{\star}=\underset{\bm{\theta}\in\Theta}{\arg\min}f_{T}(\bm{\theta})=\underset{\bm{\theta}\in\Theta}{\arg\min}\mathbb{E}_{(\bm{X},Y)\sim
p_{T}(\bm{x},y)}\Big{[}L\Big{(}h\Big{(}\bm{X};\bm{\omega},\bm{\theta}\Big{)},Y\Big{)}\Big{]},$
(1)

where $\Theta$ is the hyperparameter search space and
$L:\mathcal{Y}\times\mathcal{Y}\rightarrow\mathbb{R}^{+}$ is a continuous loss
function which satisfies the Lipschitz condition. $f_{T}(\bm{\theta})$ is the
target risk function, which is defined as the risk function over the target
distribution.
$h\left(\bm{x};\bm{\omega},\bm{\theta}\right):\mathcal{X}\rightarrow\mathcal{Y}$
is a parametric model which predicts the output value using the feature
instance $\bm{x}$ with the model parameter $\bm{\omega}$ when the
hyperparameter is $\bm{\theta}\in\Theta$.

In a standard hyperparameter optimization setting, see Ref. NIPS2011_86e8f7ab
and Feurer2019, the output values of the target dataset are available. Let
$D_{T}^{op}=\\{(\bm{x}_{i},y_{i})\\}_{i=1}^{n_{T}^{op}}$ be the i.i.d. target
dataset in this situation. Then, we can split $D_{T}^{op}$ to two disjoint
subsets $D_{T}^{optr}\cup D_{T}^{opval}$. For a given hyperparameter
$\bm{\theta}\in\Theta$, we estimate the target risk in (1) by the following
empirical mean:

$\hat{f}_{T}(\bm{\theta};D_{T}^{opval})=\frac{1}{|D_{T}^{opval}|}\sum_{(\bm{x}_{i},y_{i})\in
D_{T}^{opval}}L\Big{(}h\Big{(}\bm{x}_{i};\hat{\bm{\omega}}(\bm{\theta}),\bm{\theta}\Big{)},y_{i}\Big{)},$

where $|D|$ denotes the sample size of the dataset $D$. The model parameter is
estimated on $D_{T}^{optr}$ by

$\hat{\bm{\omega}}(\bm{\theta})=\underset{\bm{\omega}\in\Omega}{\arg\min}\frac{1}{|D_{T}^{optr}|}\sum_{(\bm{x}_{i},y_{i})\in
D_{T}^{optr}}L\Big{(}h\Big{(}\bm{x}_{i};\bm{\omega},\bm{\theta}\Big{)},y_{i}\Big{)},$

where $\Omega$ is the model parameter space. Then, we can obtain the estimate
of $\bm{\theta}^{\star}$ by minimizing
$\hat{f}_{T}(\bm{\theta};D_{T}^{opval})$.

In contrast, in the FedCS setting of our paper, the output values of the
target data are completely missing. Instead, we can utilize multiple source
datasets whose output values are available, but there are covariate shifts
between the sources and target. For covariate shift adaptation, the natural
idea is to use importance sampling to approximate the target risk, see Ref.
2015Efficient and 2007Covariate.

####  2.2.1 FedIWE

Since every source does not allow the data to leave its local area or be
disclosed, for a given hyperparameter $\bm{\theta}\in\Theta$, let the $j$th
source learns locally to obtain the estimate of the model parameter
$\hat{\bm{\omega}}_{j}(\bm{\theta})$. Although all sources use the same
hyperparameter $\bm{\theta}$,
$\\{\hat{\bm{\omega}}_{j}(\bm{\theta})\\}_{j=1}^{K}$ will be different because
the source datasets are different.

Each source uses importance sampling in local learning, which yields a density
ratio. We define the following density ratio.

######  Definition 2.1.

For any $(\bm{x},y)\in\mathcal{X}\times\mathcal{Y}$ with a positive source
density $p_{j}(\bm{x},y)>0$, the density ratio between the target and $j$th
source is

$r_{j}(\bm{x})=\frac{p_{T}(\bm{x},y)}{p_{j}(\bm{x},y)}=\frac{p_{T}(\bm{x})}{p_{j}(\bm{x})},j=1,2,\cdots,K.$

The equalities are derived from Assumption 2.1.

In the FedCS setting, according to Assumption 2.1, we can derive that
$r_{j}(\bm{x})\in[0,C]$ for a positive constant $C$.

Since density ratio estimation, training the model and selecting optimal
hyperparameters are involved in our optimization process, we can split $D_{j}$
to three disjoint subsets $D_{j}^{de}\cup D_{j}^{tr}\cup D_{j}^{val}$, where

$D_{j}^{de}=\Big{\\{}(\bm{x}_{j,i}^{de},y_{j,i}^{de})\Big{\\}}_{i=1}^{n_{j}^{de}},D_{j}^{tr}=\Big{\\{}(\bm{x}_{j,i}^{tr},y_{j,i}^{tr})\Big{\\}}_{i=1}^{n_{j}^{tr}},D_{j}^{val}=\Big{\\{}(\bm{x}_{j,i}^{val},y_{j,i}^{val})\Big{\\}}_{i=1}^{n_{j}^{val}}.$

Let $n_{j}=n_{j}^{de}+n_{j}^{tr}+n_{j}^{val}$,
$n^{tr}=\sum_{j=1}^{K}n_{j}^{tr}$ and $n^{val}=\sum_{j=1}^{K}n_{j}^{val}$.

For a given hyperparameter $\bm{\theta}\in\Theta$, at the $j$th source,
empirical estimate of the target risk can be adjusted as

$\hat{f}_{IW}(\bm{\theta};D_{j})=\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\hat{r}_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)},$
(2)

where the estimate of the density ratio $\hat{r}(\cdot)$ can be obtained on
$D_{j}^{de}$ and $D_{T}$ using the unconstrained least-squares importance
fitting (uLSIF) method, see Ref. 2009A and Yamada2013Relative. The model
parameter is estimated on $D_{j}^{tr}$ as

$\hat{\bm{\omega}}_{j}(\bm{\theta})=\underset{\bm{\omega}\in\Omega}{\arg\min}\frac{1}{n_{j}^{tr}}\sum_{i=1}^{n_{j}^{tr}}L\Big{(}h\Big{(}\bm{x}_{j,i}^{tr};\bm{\omega},\bm{\theta}\Big{)},y_{j,i}^{tr}\Big{)},$
(3)

and the ideal model parameter is

${\bm{\omega}}_{j}(\bm{\theta})=\underset{\bm{\omega}\in\Omega}{\arg\min}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}L\Big{(}h\Big{(}\bm{x};\bm{\omega},\bm{\theta}\Big{)},y\Big{)}\Big{]}.$

We demonstrate that (2) is asymptotically unbiased for $f_{T}(\bm{\theta})$
and has an asymptotic variance as follows. The proof is provided in the
appendix.

######  Theorem 2.2.

For a given hyperparameter $\bm{\theta}\in\Theta$, suppose $h$ is continuous
for $\bm{\omega}$, we have [(ii)]

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}=f_{T}(\bm{\theta});$

$\lim\limits_{n_{j}^{de},n_{j}^{tr},\atop
n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}=\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}-f^{2}_{T}(\bm{\theta}).$

Because each source learns a parametric model locally, their learning process
does not involve other source data, satisfying the privacy protection in the
FedCS setting. Then, we propose FedIWE of the target risk as

$\hat{f}_{FedIW}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\frac{1}{n^{val}}\sum_{j=1}^{K}n_{j}^{val}\hat{f}_{IW}(\bm{\theta};D_{j}).$
(4)

Follow Theorem 2.2 and the linear property of the expectation operation, we
can derive that FedIWE is asymptotically unbiased for the target risk for any
given $\bm{\theta}\in\Theta$, i.e.,

$\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{FedIW}(\bm{\theta};\\{D_{j}\\}_{j=1}^{K})\Big{]}=f_{T}(\bm{\theta}).$

Then, the estimate of $\bm{\theta}^{\star}$ is obtained by

$\hat{\bm{\theta}}_{FedIW}=\underset{\bm{\theta}\in\Theta}{\arg\min}\hat{f}_{FedIW}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)},$

and the corresponding estimates of the model parameters are
$\\{\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}}_{FedIW})\\}_{j=1}^{K}$.

However, (2) is unstable, see Ref. 2007Covariate. From Theorem 2.2, when the
$j$th source has a distribution dissimilar to that of the target, i.e.,
$r_{j}(\bm{x}),\bm{x}\in\mathcal{X}$ can be large, the variance of (2) will
become large as $n_{j}^{de},n_{j}^{tr},n_{T}\to\infty$. However, the estimate
with large variance may not be accurate in the long run.

####  2.2.2 FedDAE

Lemieux[12] shows that the control variates method can reduce variance, so we
adjust (2) as follows, for a given hyperparameter $\bm{\theta}\in\Theta$,

$\hat{f}_{CV}(\bm{\theta};D_{j})=\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}\hat{r}_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(x_{j,i}^{val})-1\Big{)}\Big{]},$
(5)

where let
$\bm{x}_{j}^{val}=(\bm{x}_{j,1}^{val},\cdots,\bm{x}_{j,n_{j}^{val}}^{val})$
and $y_{j}^{val}=(y_{j,1}^{val},\cdots,y_{j,n_{j}^{val}}^{val})$,

$\hat{\eta}_{j}(\bm{\theta})=-\frac{\widehat{\operatorname{Cov}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\cdot
L\Big{(}h\big{(}\bm{x}_{j}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\big{)},y_{j}^{val}\Big{)},\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}}{\widehat{\operatorname{Var}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}},j=1,2,\cdots,K.$

For $j=1,2,\cdots,K$, we define

$\eta_{j}(\bm{\theta})=-\frac{\operatorname{Cov}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)},r_{j}(\bm{x})\Big{]}}{\operatorname{Var}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\Big{]}}.$

By Theorem 2.2 and the control variates method[12], we can obtain the
following corollary and the proof is provided in the appendix.

######  Corollary 2.3.

For a given hyperparameter value $\bm{\theta}\in\Theta$, under the same
conditions in Theorem 2.2, suppose $\operatorname{Var}_{\bm{x}\sim
p_{j}(\bm{x})}\left[r_{j}(\bm{x})\right]\neq 0,j=1,2\cdots,K$. Then [(ii)]

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}=f_{T}(\bm{\theta});$

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}\leq\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}.$

It shows that $\hat{f}_{CV}(\bm{\theta};D_{j})$ is more stable than
$\hat{f}_{IW}(\bm{\theta};D_{j})$ as $n_{j}^{de},n_{j}^{tr},n_{T}\to\infty$,
so we further adjust the empirical estimate of the target risk as
$\hat{f}_{CV}(\bm{\theta};D_{j})$, $j=1,2,\cdots,K$. Then we define a class of
asymptotic unbiased estimates of the target risk.

######  Definition 2.4.

For a given hyperparameter $\bm{\theta}\in\Theta$, we define the federated
$\bm{\lambda}$ estimates (FedLE) for the target risk as

$\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}\cdot\hat{f}_{CV}(\bm{\theta};D_{j}),$

where $\bm{\lambda}=\\{\lambda_{1},\cdots,\lambda_{K}\\}$ is any set of
weights for sources that satisfies $\lambda_{j}\geq 0$, $j=1,2,\cdots,K$, and
$\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}=1$.

As shown in Ref. nomura2021efficient, we can construct an ingenious way to
integrate $\\{\hat{f}_{CV}(\bm{\theta};D_{j})\\}_{j=1}^{K}$. As a preliminary,
we first define a divergence measure, which quantifies the similarity between
the source and target.

######  Definition 2.5.

(Divergence Measure) For a given hyperparameter $\bm{\theta}\in\Theta$, the
divergence measure between the $j$th source and target is defined as

$\displaystyle\operatorname{Div}_{j}\left(\bm{\theta}\right)=$
$\displaystyle\mathbb{E}_{(\bm{X},Y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{X})\cdot
L\Big{(}h\Big{(}\bm{X};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},Y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{X})-1\Big{)}\Big{]}^{2}$
(6) $\displaystyle-f^{2}_{T}(\bm{\theta}),j=1,2,\cdots,K.$

This divergence measure is large when the corresponding source distribution
deviates significantly from the target distribution. Here, we estimate (6) on
the source data as

$\displaystyle\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)=\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}\hat{r}_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\Big{[}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{)}\Big{]}^{2}.$

Based on the divergence measure, we propose FedDAE of the target risk as

$\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}\cdot\hat{f}_{CV}(\bm{\theta};D_{j}),$
(7)

where the weight of the $j$th source is defined as

$\hat{\lambda}_{j}(\bm{\theta})=\Bigg{(}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}.$
(8)

Note that $\hat{\lambda}_{j}(\bm{\theta})\geq 0$, $j=1,2,\cdots,K$, and

$\displaystyle\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}=$
$\displaystyle\sum_{j=1}^{K}\Bigg{(}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}\cdot
n_{j}^{val}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{K}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}\Bigg{(}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}$
$\displaystyle=$
$\displaystyle\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}$
$\displaystyle=$ $\displaystyle 1.$

For $j=1,2,\cdots,K$, we define

$\lambda_{j}(\bm{\theta})=\Bigg{(}\operatorname{Div}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}.$

We investigate the asymptotic property of FedDAE below.

######  Theorem 2.6.

For a given hyperparameter $\bm{\theta}\in\Theta$, under the same conditions
in Corollary 2.3, suppose $\eta_{j}(\bm{\theta})$ is uniform bounded,
$j=1,2,\cdots,K$. Then, for any given set of weights for sources
$\bm{\lambda}$,

[(ii)]

$\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}=f_{T}(\bm{\theta});$

Especially, $\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}=f_{T}(\bm{\theta}).$

$\lim\limits_{\begin{subarray}{c}\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty\end{subarray}}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\begin{subarray}{c}\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty\end{subarray}}\mathbb{V}\Big{[}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}.$

The inequality in Corollary 2.6 demonstrates that FedDAE has the smallest
asymptotic variance in FedLE, that is a class of asymptotically unbiased
estimates of the target risk. Follow Corollary 2.3 and Corollary 2.6, we can
easily obtain the follow inequality and provide the proof in the appendix.

######  Corollary 2.7.

For a given hyperparameter $\bm{\theta}\in\Theta$, under the same conditions
in Corollary 2.6, then

$\lim\limits_{\begin{subarray}{c}\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty\end{subarray}}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\begin{subarray}{c}\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty\end{subarray}}\mathbb{V}\Big{[}\hat{f}_{FedIW}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}.$

It shows that the asymptotic variance of FedDAE is smaller than that of FedIW,
so we futher adjust the estimate of the target risk as FedDAE. Then, we
estimate $\bm{\theta}^{\star}$ by

$\hat{\bm{\theta}}_{FedDA}=\underset{\bm{\theta}\in\Theta}{\arg\min}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}.$

The corresponding estimates of the model parameters and the weights of the
sources are $\\{\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}}_{FedDA})\\}_{j=1}^{K}$
and $\\{\hat{\lambda}_{j}(\hat{\bm{\theta}}_{FedDA})\\}_{j=1}^{K}$,
respectively. Thus, we obtain the model learned locally by the $j$th source,
i.e.,
$h(\cdot;\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}}_{FedDA}),\hat{\bm{\theta}}_{FedDA})$,
$j=1,2,\cdots,K$. Next we have to consider how to use all source models for
the execution of the target task.

###  2.3 FedCSA algorithm

As shown in Ref. peng2019federated, we can integrate these models as a
weighted model. Choosing the reasonable weights $\alpha_{1},\cdots,\alpha_{K}$
for the source models can prevent negative transfer[18] from occurring and the
empirical research shows that if the distribution difference between two
domains is too significant, the rough transfer may affect the implementation
of the target task, see Ref. smith2001transfer.

We define the weight of the $j$th source model based on the weight of the
$j$th source in FedDAE, and use the following weighed model

$\hat{h}_{T}(\cdot)=\sum_{j=1}^{K}\alpha_{j}h(\cdot;\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}}_{FedDA}),\hat{\bm{\theta}}_{FedDA}),$
(9)

where $\alpha_{j}=\hat{\lambda}_{j}(\hat{\bm{\theta}}_{FedDA})n_{j}^{val}$ is
the weight of the $j$th source model. Note that $\alpha_{j}\geq
0,j=1,2\cdots,K$ and $\sum_{j=1}^{K}\alpha_{j}=1$. The experimental results in
the next two sections verify the weighted model (9) works well. Refer to Peng
_et al_.[20] and Blitzer _et al_.[3], we have the following lemma.

######  Lemma 2.8.

Let $\epsilon_{T}(h)$ and $\hat{\epsilon}_{S}(h)$ be the error of the target
task and the empirical error on the mixture of source datasets with size
$n=\sum_{j=1}^{K}n_{j}$. Then,
$\forall\alpha_{j}>0,\sum_{j=1}^{K}\alpha_{j}=1$, with probability at least
$1-\delta$ over the choice of samples, for a given hyperparameter
$\bm{\theta}\in\Theta$,

$\displaystyle\epsilon_{T}(\hat{h}_{T}(\cdot))\leq$
$\displaystyle\hat{\epsilon}_{S}\Big{(}\sum_{j=1}^{K}\alpha_{j}h(\cdot;\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta})\Big{)}+\sum_{j=1}^{K}\alpha_{j}\Big{(}\frac{1}{2}\widehat{d}_{\mathcal{H}\Delta\mathcal{H}}\left(D_{j},D_{T}\right)+t_{i}\Big{)}$
(10) $\displaystyle+4\sqrt{\frac{2\log(2n)+\log(4/\delta)}{n}},$

where $t_{i}$ is the error of the optimal model on the mixture of $D_{j}$ and
$D_{T}$, and
$\widehat{d}_{\mathcal{H}\Delta\mathcal{H}}\left(D_{j},D_{T}\right)$ denotes
the divergence involving $\mathcal{H}\Delta\mathcal{H}$ discrepancy[1] between
the $j$th source and target.

The details of the proof are provided in Peng _et al_.[20] and Blitzer _et
al_.[3]. Especially, (10) holds for $\hat{\bm{\theta}}$.

We propose the FedCSA algorithm which use FedDAE for the target risk. This
algorithm can also use other estimates. Later we will compare FedIWE and
FedDAE. For a given hyperparameter $\bm{\theta}\in\Theta$, we write a
shorthand
$\hat{f}(\bm{\theta})=\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}\hat{f}_{j}(\bm{\theta})$.
For FedDAE, $\hat{f}$ is $\hat{f}_{FedDA}(\bm{\theta};\\{D_{j}\\}_{j=1}^{K})$
and $\hat{f}_{j}$ is $\hat{f}_{CV}(\bm{\theta};D_{j})$. For FedIWE, $\hat{f}$
is $\hat{f}_{FedIW}(\bm{\theta};\\{D_{j}\\}_{j=1}^{K})$, $\hat{f}_{j}$ is
$\hat{f}_{IW}(\bm{\theta};D_{j})$ and $\hat{\lambda}_{j}(\bm{\theta})$ is
$1/n_{j}^{val}$.

For any given hyperparameter $\bm{\theta}\in\Theta$, let the $j$th source
learns a parametric model locally, then transmits $\hat{f}_{j}(\bm{\theta})$,
$\hat{\bm{\omega}}_{j}(\bm{\theta})$, $\hat{\lambda}_{j}(\bm{\theta})$ and
$n_{j}^{val}$ to the target. The target estimate $\bm{\theta}^{\star}$ by

$\hat{\bm{\theta}}=\underset{\bm{\theta}\in\Theta}{\arg\min}\hat{f}(\bm{\theta})=\underset{\bm{\theta}\in\Theta}{\arg\min}\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}\hat{f}_{j}(\bm{\theta}).$

The corresponding estimate of the model parameter and the source weight are
$\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}})$ and
$\hat{\lambda}_{j}(\hat{\bm{\theta}})$. Finally, the target obtains the
weighted model as

$\hat{h}_{T}(\cdot)=\sum_{j=1}^{K}\hat{\alpha}_{j}h\Big{(}\cdot;\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}});\hat{\bm{\theta}}\Big{)},\hat{\alpha}_{j}=\hat{\lambda}_{j}(\hat{\bm{\theta}})n_{j}^{val},j=1,2,\cdots,K.$

Algorithm 1 FedCSA algorithm

0: Target data $D_{T}=\\{(\bm{x}_{T,i})\\}_{i=1}^{n_{T}}$; Source data
$D_{j}=\\{(\bm{x}_{j,i},y_{j,i})\\}_{i=1}^{n_{j}},j=1,2,\cdots,K$;
Hyperparameter search space $\Theta$; Parametric model $h$; Estimate of the
target risk
$\hat{f}(\bm{\theta})=\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}\hat{f}_{j}(\bm{\theta})$.

0: The prediction model of the target task, $\hat{h}_{T}(\cdot)$.

1: The target transmits $D_{T}$ to each source separately.

2: for $j=1$ to $K$ do

3: The $j$th source:

  1. 1.

Split $D_{j}$ as three disjoint folds: $D_{j}^{de},D_{j}^{tr},D_{j}^{val}$;

  2. 2.

Estimate density ratio by uLSIF[10] on $D_{T}$ and $D_{j}^{de}$, record as
$\hat{r}_{j}(\cdot)$ ;

  3. 3.

Learn locally on $D_{j}^{tr}$ and estimate the model parameter as
$\hat{\bm{\omega}}_{j}(\bm{\theta})$ by (3);

  4. 4.

Calculate $\hat{f}_{j}(\bm{\theta})$ and $\hat{\lambda}_{j}(\bm{\theta})$ on
$D_{j}^{val}$ by (2) or (5) and (8);

  5. 5.

Transmit $\hat{f}_{j}(\bm{\theta})$, $\hat{\bm{\omega}}_{j}(\bm{\theta})$,
$\hat{\lambda}_{j}(\bm{\theta})$ and $n_{j}^{val}$ to the target.

4: end for

5: The target:

  1. 1.

Obtain
$\hat{f}(\bm{\theta})=\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})n_{j}^{val}\hat{f}_{j}(\bm{\theta});$

  2. 2.

Estimate
$\hat{\bm{\theta}}=\underset{\bm{\theta}\in\Theta}{\arg\min}\hat{f}(\bm{\theta});$

  3. 3.

Determine $\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}})$ and
$\hat{\lambda}_{j}(\hat{\bm{\theta}})$, $j=1,2,\cdots,K$;

  4. 4.

Derive $\hat{\alpha}_{j}=\hat{\lambda}_{j}(\hat{\bm{\theta}})n_{j}^{val}$,
$j=1,2,\cdots,K$.

6: Return
$\hat{h}_{T}(\cdot)=\sum_{j=1}^{K}\hat{\alpha}_{j}h\Big{(}\cdot;\hat{\bm{\omega}}_{j}(\hat{\bm{\theta}});\hat{\bm{\theta}}\Big{)}$.

##  3 Simulation Studies

In this section, we perform some simulations to illustrate the performance of
our proposed method. We consider two cases, one is to study the performance of
the method under different sample sizes, and the other is to study the
performance of the method under different covariate shifts between the sources
and target. The data generation progress is as follows:

Case 1: The input variable $\bm{x}=(\bm{x}_{1},\bm{x}_{2},\cdots,\bm{x}_{10})$
is 10-dimensional. There are two sources, their distributions are
$N(\bm{1}_{10},3\times\bm{I}_{10})$ and
$N(5\times\bm{1}_{10},0.5\times\bm{I}_{10})$. $\bm{1}_{10}$ is the
10-dimensional vector with all ones and $\bm{I}_{10}$ is the 10-dimensional
identity matrix. The sample sizes of two sources are $n_{1}$ and $n_{2}$. The
distribution of the target is $N(\bm{0}_{10},\bm{I}_{10})$, $\bm{0}_{10}$ is
the 10-dimensional vector with all zeros. The sample size of the target is
$n_{T}$. The output variate $y$ is 1-dimensional. The conditional output
distribution is $y|\bm{x}\sim N(\tilde{\bm{x}},1)$,
$\tilde{\bm{x}}=\frac{1}{10}\sum^{10}_{i=1}\bm{x}_{i}$.

Case 2: Two sources, the distributions are $N(c\times\bm{1}_{10},\bm{I}_{10})$
and $N((c+1)\times\bm{1}_{10},\bm{I}_{10})$.
$c\in\\{1,1.5,2,2.5,3,3.5,4,4.5,5\\}$. Their sample sizes are 50 and 40. The
distribution of the target domain is $N(\bm{0}_{10},\bm{I}_{10})$, and its
sample size is 20. The others are the same as that of Case 1.

We use ridge regression model and the square loss function:
$L(\hat{y},y)=(\hat{y}-y)^{2}$. The hyperparameter is the regularization
parameter $\lambda$, and the hyperparameter search space is $\Theta=[0,1]$. We
report the prediction error with mean absolute error:
$\epsilon=\frac{1}{n}\sum_{i=1}^{n}|\hat{y}_{i}-y_{i}|$. We compare four
methods and FedDA represents our proposed method: [(ii)]

FedIW: use (4) as $\hat{f}(\bm{\theta})$ in Algorithm 1. Especially,
$\hat{\lambda}_{j}=1/n_{j}^{val}$ and $\alpha_{j}=n_{j}^{val}/n^{val}$,
$j=1,2,\cdots,K$. They don’t rely on the hyperparameter $\bm{\theta}$.

FedDA: use (6) as $\hat{f}(\bm{\theta})$ in Algorithm 1.

Naive: select the parametric model directly without considering covariate
shifts between the source and target domains.

Reference: use the target data with output values to learn a parametric model,
that are not feasible in the FedCS setting and report its performance as a
reference.

In Case 1, we run 100 experiments with different random seeds for different
sample sizes. The mean and the standard error of the 100 experimental results
for every method are shown in Table 3. In Case 2, for any
$c\in\\{1,1.5,2,2.5,3,3.5,4,4.5,5\\}$, we run 100 experiments with different
random seeds and the experimental results are shown in (Fig. 1). Fig. 1(a)
shows the approximate 95% confidence interval for the average performance of
each method. Fig. 1(b) shows the ratio of the average performance of FedIW and
FedDA.

Performances (Mean$\pm$StdErr) of methods with diﬀerent sample sizes  $n_{T}$
$n_{1}$ $n_{2}$ FedIW FedDA Naive Reference 20 30 20 1.3548$\pm$0.3994
0.8649$\pm$0.3862 1.6049$\pm$0.4620 1.1222$\pm$0.2961 40 30 5.5673$\pm$0.9708
1.0530$\pm$0.2413 4.6059$\pm$0.4027 1.5219$\pm$0.3845 50 40 1.7531$\pm$0.4173
0.5635$\pm$0.1646 1.7646$\pm$0.4096 0.9178$\pm$0.2866 60 50 2.4793$\pm$0.4965
0.8353$\pm$0.2340 2.5488$\pm$0.4832 1.2125$\pm$0.2962 70 60 1.8884$\pm$0.4445
1.1029$\pm$0.2694 1.9818$\pm$0.4622 1.9644$\pm$0.4675 30 40 30
3.3341$\pm$0.3237 0.7470$\pm$0.1815 3.9530$\pm$0.3446 0.8630$\pm$0.1847 50 40
1.7427$\pm$0.2686 0.8093$\pm$0.1530 1.9016$\pm$0.2760 1.2516$\pm$0.2745 60 50
1.9798$\pm$0.2976 0.8922$\pm$0.2017 2.1204$\pm$0.3008 1.1003$\pm$0.2598 70 60
4.1626$\pm$0.2907 0.6271$\pm$0.1230 4.5362$\pm$0.2966 0.7376$\pm$0.1413 80 70
3.0442$\pm$0.2679 0.8102$\pm$0.1511 3.2368$\pm$0.2808 1.0502$\pm$0.2951 40 50
40 1.8087$\pm$0.3199 0.7836$\pm$0.1462 1.9628$\pm$0.3363 0.8702$\pm$0.1836 60
50 5.2020$\pm$0.2335 0.8926$\pm$0.1479 5.7482$\pm$0.2241 0.8993$\pm$0.2119 70
60 4.9135$\pm$0.3448 0.9632$\pm$0.1878 5.3687$\pm$0.3548 1.1801$\pm$0.2110 80
70 1.5635$\pm$0.2199 0.7066$\pm$0.1217 1.7052$\pm$0.2222 0.7929$\pm$0.1563 90
80 1.9881$\pm$0.2415 0.8646$\pm$0.1760 2.0885$\pm$0.2468 0.9280$\pm$0.2259 50
60 50 2.8928$\pm$0.3238 0.7966$\pm$0.1375 3.1662$\pm$0.3347 0.8658$\pm$0.1386
70 60 1.4436$\pm$0.1979 0.6825$\pm$0.1176 1.5184$\pm$0.2033 0.8381$\pm$0.1441
80 70 2.2632$\pm$0.2607 0.6893$\pm$0.1185 2.1614$\pm$0.2255 0.8567$\pm$0.1229
90 80 2.2903$\pm$0.2459 0.7467$\pm$0.1314 2.3971$\pm$0.2494 0.8922$\pm$0.1599
100 90 0.9640$\pm$0.1457 0.7120$\pm$0.1282 0.9897$\pm$0.1465 0.8717$\pm$0.1335

(a) Comparing all methods

(b) Comparing FedIW and FedDA

Figure 1: Performances of methods with different covariate shifts. The
horizontal axis represents the parameters of the source domains
$c\in\\{1,1.5,2,2.5,3,3.5,4,4.5,5\\}$. (a) The vertical axis represents the
mean of the performance of each method. (b) The vertical axis represents the
ratio of the mean performance of FedIW to FedDA

From the results of Case 1 in Table 3, we can see that FedDA has the best mean
performance with the smallest standard error in each situation. The results of
Case 2 in Fig 1 indicate that FedDA significantly outperforms the other
methods for every value of $c$. Fig. 1(a) shows that the advantage of FedDA
becomes more and more strengthened when the covariate shifts between the
target and sources become larger. The curve in Fig. 1(b) shows that FedDA
outperforms FedIW to an increasing degree as the covariate shifts between the
target and sources become larger. FedDA significantly outperforms Naive
because FedDAE can well adapt to the FedCS setting, while the Naive method
does not. FedDA also outperforms FedIW because FedDAE has a smaller asymptotic
variance than FedIWE. Compared with Reference, FedDA performs better because
it can make full use of additional source samples. These results are
consistent with our theoretical analysis.

##  4 Real data Analysis

We verify the performance of our proposed method on a real dataset,
Parkinson’s remote monitoring data.[26] The dataset consists of a series of
biomedical voice measurement data from 42 patients with early Parkinson’s
disease recruited to participate in a 6-month remote monitoring symptom
progression test. These recordings were automatically recorded at the
patients’ home. Here, we regard each patient’s home as a small hospital, and
all the recordings of the patient are the data owned by this hospital. The
columns of the dataset contain subject number, subject age, subject gender,
time interval from baseline recruitment date, total UPDRS (Parkinson’s disease
score scale) score, and 16 biomedical voice measurements. Each row corresponds
to one of the 5875 voice records of these people. The primary purpose of the
dataset is to predict the total UPDRS score from the 16 voice measurements.

We choose one hospital as the target and the others as the sources. The
patient disease data of different source hospitals can not be fused. We use
two models to train the data: the weighted least squares with exponential
weight and the ridge regression. The hyperparameters are the flattening
parameter of the weight and the regularization parameter, respectively. The
loss function is square loss.

Experimental process: Use the target data without output values and the source
data with output values; Divide the original target data into the training set
and test set according to 7:3; Train the model on the training set of the
target data and the source data; Predict the disease scores of the test set of
the target data; Calculate the mean absolute error of the prediction as to the
performance of the method used; Repeat the above steps 100 times with
different random seeds.

Report the mean, standard error and worst-case performances of the 100
experimental results of the weighted least squares and the ridge regression in
Table 4 and Table 4, respectively. The results indicate that FedDA is better
than other methods in mean and worst-case with the smallest standard error.

Weighted least squares on real data  Methods Mean Standard error Worst Case
Naive 0.9003 0.0753 1.0862 FedIW 0.9519 0.0871 1.1573 FedDA 0.8578 0.0617
1.0034 Reference 0.8834 0.0657 1.0524

Ridge regression on real data  Methods Mean Standard error Worst Case Naive
0.8645 0.0587 1.0072 FedIW 0.8694 0.0588 1.0121 FedDA 0.8468 0.0577 0.9848
Reference 0.8578 0.0684 1.0312

FedDA performs better than Naive because FedDAE can adapt to the FedCS setting
between the hospitals. FedDA performs better than FedIW because FedDAE is
stabler than FedIWE asymptotically. FedDA performs better than Reference
because the samples of the source hospitals are more sufficient than that of
the target hospital, and FedDA can make full use of existing data information.

##  5 Conclusion

This paper explores a new problem setting that extend the MS-CS setting under
the framework of federated learning. The output values of the target data are
completely missing, while the output values of the source data are available,
and each source does not allow its data to leave its local area or be
disclosed. To estimate the optimal hyperparameter, we first propose the
federated importance weighting estimate of the target risk, which is
asymptotically unbiased and can adapt to covariate shift. We further propose
the federated domain adaptation estimate and show that it achieves the smaller
asymptotic variance among a class of asymptotically unbiased estimates of the
target risk. Then, we construct a weighted model by weighting all source
models for the target task whose error can be bounded. Finally, we propose the
federated covariate shift adaptation algorithm, a general and tractable
hyperparameter optimization process. The experimental results indicate that
our method is effective. However, our method does not take into account the
nature of the under-sampled situation, which deserves further study in our
future work.

## Acknowledgments

We acknowledge support from the National Natural Science Foundation of China
(Grant Nos. 71873128, 12171451) .

## References

  * [1] Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan.  A theory of learning from different domains.  Machine learning, 79(1):151–175, 2010.
  * [2] James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl.  Algorithms for hyper-parameter optimization.  In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 24. Curran Associates, Inc., 2011.
  * [3] John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman.  Learning bounds for domain adaptation.  Advances in neural information processing systems, 20, 2007.
  * [4] Keith Bonawitz, Vladimir Ivanov, Ben Kreuter, Antonio Marcedone, H Brendan McMahan, Sarvar Patel, Daniel Ramage, Aaron Segal, and Karn Seth.  Practical secure aggregation for privacy-preserving machine learning.  In proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, pages 1175–1191, 2017.
  * [5] Edwin V Bonilla, Kian Chai, and Christopher Williams.  Multi-task gaussian process prediction.  Advances in neural information processing systems, 20, 2007.
  * [6] Víctor Elvira, Luca Martino, David Luengo, and Mónica F Bugallo.  Efficient multiple importance sampling estimators.  IEEE Signal Processing Letters, 22(10):1757–1761, 2015.
  * [7] Matthias Feurer and Frank Hutter.  Hyperparameter optimization.  In Automated machine learning, pages 3–33. Springer, Cham, 2019\.
  * [8] Matthias Feurer, Benjamin Letham, Frank Hutter, and Eytan Bakshy.  Practical transfer learning for bayesian optimization.  arXiv preprint arXiv:1802.02219, 2018.
  * [9] Ran Gilad-Bachrach, Nathan Dowlin, Kim Laine, Kristin Lauter, Michael Naehrig, and John Wernsing.  Cryptonets: Applying neural networks to encrypted data with high throughput and accuracy.  In International conference on machine learning, pages 201–210. PMLR, 2016.
  * [10] Takafumi Kanamori, Shohei Hido, and Masashi Sugiyama.  A least-squares approach to direct importance estimation.  The Journal of Machine Learning Research, 10:1391–1445, 2009.
  * [11] Takafumi Kanamori, Taiji Suzuki, and Masashi Sugiyama.  Statistical analysis of kernel-based least-squares density-ratio estimation.  Machine Learning, 86(3):335–367, 2012.
  * [12] Christiane Lemieux.  Control Variates.  American Cancer Society, 2017.
  * [13] Yang Liu, Yan Kang, Chaoping Xing, Tianjian Chen, and Qiang Yang.  A secure federated transfer learning framework.  IEEE Intelligent Systems, 35(4):70–82, 2020.
  * [14] Mingsheng Long, Zhangjie Cao, Jianmin Wang, and Michael I Jordan.  Conditional adversarial domain adaptation.  Advances in neural information processing systems, 31, 2018.
  * [15] Payman Mohassel and Peter Rindal.  Aby3: A mixed protocol framework for machine learning.  In Proceedings of the 2018 ACM SIGSAC conference on computer and communications security, pages 35–52, 2018.
  * [16] Payman Mohassel and Yupeng Zhang.  Secureml: A system for scalable privacy-preserving machine learning.  In 2017 IEEE symposium on security and privacy (SP), pages 19–38. IEEE, 2017.
  * [17] Masahiro Nomura and Yuta Saito.  Efficient hyperparameter optimization under multi-source covariate shift.  In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pages 1376–1385, 2021.
  * [18] Sinno Jialin Pan and Qiang Yang.  A survey on transfer learning.  IEEE Transactions on knowledge and data engineering, 22(10):1345–1359, 2009.
  * [19] Xingchao Peng, Zijun Huang, Yizhe Zhu, and Kate Saenko.  Federated adversarial domain adaptation.  arXiv preprint arXiv:1911.02054, 2019.
  * [20] Xingchao Peng, Zijun Huang, Yizhe Zhu, and Kate Saenko.  Federated adversarial domain adaptation.  arXiv preprint arXiv:1911.02054, 2019.
  * [21] Valerio Perrone, Rodolphe Jenatton, Matthias W Seeger, and Cédric Archambeau.  Scalable hyperparameter transfer learning.  Advances in neural information processing systems, 31, 2018.
  * [22] Hidetoshi Shimodaira.  Improving predictive inference under covariate shift by weighting the log-likelihood function.  Journal of statistical planning and inference, 90(2):227–244, 2000\.
  * [23] Virginia Smith, Chao-Kai Chiang, Maziar Sanjabi, and Ameet S Talwalkar.  Federated multi-task learning.  Advances in neural information processing systems, 30, 2017.
  * [24] Kimberly A Smith-Jentsch, Eduardo Salas, and Michael T Brannick.  To transfer or not to transfer? investigating the combined effects of trainee characteristics, team leader support, and team climate.  Journal of applied psychology, 86(2):279, 2001.
  * [25] Masashi Sugiyama, Matthias Krauledat, and Klaus-Robert Müller.  Covariate shift adaptation by importance weighted cross validation.  Journal of Machine Learning Research, 8(5), 2007.
  * [26] Athanasios Tsanas, Max Little, Patrick McSharry, and Lorraine Ramig.  Accurate telemonitoring of parkinson’s disease progression by non-invasive speech tests.  Nature Precedings, pages 1–1, 2009.
  * [27] Makoto Yamada, Taiji Suzuki, Takafumi Kanamori, Hirotaka Hachiya, and Masashi Sugiyama.  Relative density-ratio estimation for robust distribution comparison.  Neural computation, 25(5):1324–1370, 2013.
  * [28] Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong.  Federated machine learning: Concept and applications.  ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1–19, 2019.
  * [29] Kaichao You, Ximei Wang, Mingsheng Long, and Michael Jordan.  Towards accurate model selection in deep unsupervised domain adaptation.  In International Conference on Machine Learning, pages 7124–7133. PMLR, 2019.
  * [30] Erheng Zhong, Wei Fan, Qiang Yang, Olivier Verscheure, and Jiangtao Ren.  Cross validation framework to choose amongst models and datasets for transfer learning.  In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 547–562. Springer, 2010.

## Appendix A

A1. Lemma1 Consider $\\{f_{n}(y)\\}$ is a sequence of functions defined on the
interval $[a,b]$. Suppose $f_{n}(y)$ converges uniformly to
$f(y):[a,b]\to\mathbb{R}$ on $[a,b]$ as $n\to\infty$, then
$\lim\limits_{n\to\infty}\underset{y\in[a,b]}{\min}f_{n}(y)=\underset{y\in[a,b]}{\min}f(y)$.

Proof: For any $\epsilon>0$, take $y_{0}\in[a,b]$ such that

$f(y_{0})\leq\underset{y\in[a,b]}{\min}f(y)+\epsilon.$ (11)

then since

$f(y_{0})=\lim\limits_{n\to\infty}f_{n}(y_{0})\geq\lim\limits_{n\to\infty}\underset{y\in[a,b]}{\min}f_{n}(y),$

let $\epsilon\to 0$ we get

$\underset{y\in[a,b]}{\min}f(y)\geq\lim\limits_{n\to\infty}\underset{y\in[a,b]}{\min}f_{n}(y).$
(12)

Take a positive constant $N$ such that $\forall n\geq N$,
$|f_{n}(y)-f(y)|\leq\epsilon$ for any $y\in[a,b]$. This gives that $\forall
n\geq N$,

$\displaystyle-\epsilon\leq f_{n}(y)-f(y)\leq\epsilon,$ (13)
$\displaystyle-\epsilon\leq f_{n}(y_{0})-f(y_{0})\leq\epsilon.$

From (13) we can obtain that

$\displaystyle-2\epsilon\leq f_{n}(y)-f(y)+f(y_{0})-f_{n}(y_{0})\leq
2\epsilon$ $\displaystyle\implies f(y)-f_{n}(y)+f_{n}(y_{0})\leq
f(y_{0})+2\epsilon.$

Then

$\displaystyle\underset{y\in[a,b]}{\min}f(y)$
$\displaystyle=\underset{y\in[a,b]}{\min}[f(y)-f_{n}(y)+f_{n}(y)-f_{n}(y_{0})+f_{n}(y_{0})]$
$\displaystyle\leq\underset{y\in[a,b]}{\min}[f(y_{0})+2\epsilon+f_{n}(y)-f_{n}(y_{0})]$
$\displaystyle=\underset{y\in[a,b]}{\min}f_{n}(y)+f(y_{0})-f_{n}(y_{0})+2\epsilon$
$\displaystyle\leq\underset{y\in[a,b]}{\min}f_{n}(y)+3\epsilon.$

Let $\epsilon\to 0$ and $n\to\infty$, we get

$\underset{y\in[a,b]}{\min}f(y)\leq\lim\limits_{n\to\infty}\underset{y\in[a,b]}{\min}f_{n}(y).$
(14)

Combine (12) and (14),
$\lim\limits_{n\to\infty}\underset{y\in[a,b]}{\min}f_{n}(y)=\underset{y\in[a,b]}{\min}f(y).$

A2. Proof of Theorem 2.2
[(ii)]

For a given hyperparameter $\bm{\theta}\in\Theta$,
$h\left(\bm{x};\bm{\omega},\bm{\theta}\right)\in\mathcal{Y}$ and
$y\in\mathcal{Y}$ are both bounded, and $L$ satisfies the Lipschitz condition,
so that

$\mathbb{E}\Big{|}L\Big{(}h(\bm{x};\bm{\omega},\bm{\theta}),y\Big{)}\Big{|}<\infty.$

$(\bm{x}_{j,1}^{tr},y_{j,1}^{tr}),\cdots,(\bm{x}_{j,n_{j}^{tr}}^{tr},y_{j,n_{j}^{tr}}^{tr})$
are i.i.d.$\sim p_{j}(\bm{x},y)$, Since $h$ and $L$ are continues and
measurable, then for any given hyperparameter $\bm{\theta}\in\Theta$ and model
parameter $\omega\in\Omega$,

$L\Big{(}h(\bm{x}_{j,1}^{tr};\bm{\omega},\bm{\theta}),y_{j,1}^{tr}\Big{)},\cdots,L\Big{(}h(\bm{x}_{j,n_{j}^{tr}}^{tr};\bm{\omega},\bm{\theta}),y_{j,n_{j}^{tr}}^{tr}\Big{)}$

are i.i.d. According to the weak law of large numbers, when
$n_{j}^{tr}\to\infty$,

$\frac{1}{n_{j}^{tr}}\sum_{i=1}^{n_{j}^{tr}}L\Big{(}h(\bm{x}_{j,i}^{tr};\bm{\omega},\bm{\theta}),y_{j,i}^{tr}\Big{)}\overset{P}{\to}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}L\Big{(}h(\bm{x};\bm{\omega},\bm{\theta}),y\Big{)}.$

For a given hyperparameter $\bm{\theta}\in\Theta$. Let

$\displaystyle f_{n_{j}^{tr}}(\bm{\omega};D_{j}^{tr})$
$\displaystyle=\frac{1}{n_{j}^{tr}}\sum_{i=1}^{n_{j}^{tr}}L\Big{(}h(\bm{x}_{j,i}^{tr};\bm{\omega},\bm{\theta}),y_{j,i}^{tr}\Big{)},$
$\displaystyle f_{j}(\bm{\omega})$ $\displaystyle=\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}L\Big{(}h(\bm{x};\bm{\omega},\bm{\theta}),y\Big{)}.$

Then
$f_{n_{j}^{tr}}(\bm{\omega};D_{j}^{tr})\overset{P}{\to}f_{j}(\bm{\omega}),\forall\bm{\omega}\in\Omega$.
For any $\epsilon_{1}>0$ and $\delta_{1}>0$, let

$\tilde{G}_{j}=\cup\\{D_{j}^{tr}:|f_{n_{j}^{tr}}(\bm{\omega};D_{j}^{tr})-f_{j}(\bm{\omega})|>\epsilon_{1},D_{j}^{tr}\sim
p_{j}(\bm{x},y)\\},$

then $\exists N_{1}>0$, when $n_{j}^{tr}\geq N_{1}$,
$\mu(\tilde{G}_{j})<\delta_{1}$. So that when
$D_{j}^{tr}\subset\tilde{G}^{c}_{j}$, $f_{n_{j}^{tr}}(\bm{\omega};D_{j}^{tr})$
uniformly converges to $f_{j}(\bm{\omega})$ on $\Omega$. By Lemma1 in appendix
A1 we have

$\lim\limits_{n_{j}^{tr}\to\infty}\underset{\bm{\omega}\in\Omega}{\min}f_{n_{j}^{tr}}(\bm{\omega};D_{j}^{tr})=\underset{\bm{\omega}\in\Omega}{\min}f_{j}(\bm{\omega}).$

Since the minimum value of $L$ about $\bm{\omega}$ is unique, then for any
hyperparameter $\bm{\theta}\in\Theta$, when
$D_{j}^{tr}\subset\tilde{G}^{c}_{j}$ and $n_{j}^{tr}\to\infty$,

$\hat{\bm{\omega}}_{j}(\bm{\theta})=\underset{\bm{\omega}\in\Omega}{\arg\min}f_{n_{j}^{tr}}(\bm{\omega})\to\underset{\bm{\omega}\in\Omega}{\arg\min}f_{j}(\bm{\omega})={\bm{\omega}}_{j}(\bm{\theta}).$

Since $h$ is continuous for $\bm{\omega}$ and $L$ is continuous, then when
$D_{j}^{tr}\subset\tilde{G}^{c}_{j}$, for any
$(\bm{x}_{j,i}^{val},y_{j,i}^{val})\in D_{j}^{val}$,

$\lim_{n_{j}^{tr}\to\infty}L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}=L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}.$
(15)

According to Theorem 2 of Ref. kanamori2012statistical, we can obtain that
when $n_{j}^{de},n_{T}\to\infty$,
$\hat{r}_{j}(\cdot)\overset{P}{\to}r_{j}(\cdot)$, $j=1,2,\cdots,K.$ Here
$\hat{r}_{j}(\cdot)$ is learned on $D_{j}^{de}$ and $D_{T}$. For any
$\epsilon_{2}>0$ and $\delta_{2}>0$, let

$\hat{G}_{j}=\cup\\{D_{j}^{de}:|\hat{r}_{j}(\bm{x}_{j,i}^{val})-r_{j}(\bm{x}_{j,i}^{val})|>\epsilon_{2},D_{j}^{de}\sim
p_{j}(\bm{x},y)\\},$

then $\exists N_{2}>0$, when $\min(n_{j}^{de},n_{T})\geq N_{2}$,
$\mu(\hat{G}_{j})<\delta_{2}$. Let
$G_{j}=\tilde{G}_{j}^{c}\cap\hat{G}_{j}^{c}$, then
$G^{c}_{j}=\tilde{G}_{j}\cup\hat{G}_{j}$ and

$\displaystyle\mu(G^{c}_{j})=1-\mu(G_{j})<$ $\displaystyle
1-\max(\mu(\tilde{G}_{j}^{c}),\mu(\hat{G}_{j}^{c}))$ (16) $\displaystyle=$
$\displaystyle 1-\max(1-\mu(\tilde{G}_{j}),1-\mu(\hat{G}_{j}))$
$\displaystyle=$ $\displaystyle\min(\mu(\tilde{G}_{j}),\mu(\hat{G}_{j}))$
$\displaystyle=$ $\displaystyle\min(\delta_{1},\delta_{2}).$

For any given hyperparameter $\bm{\theta}\in\Theta$, from (15) we obtain that
for any $\epsilon_{3}>0$, $\exists N_{3}>0$, when $D_{j}^{tr}\subset G_{j}$
and $n_{j}^{tr}\geq N_{3}$,

$\Big{|}L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}-L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}\Big{|}<\epsilon_{3}.$

Then, for any $(\bm{x},y)\in D_{j}^{val}$, when $D_{j}^{de}\subset G_{j}$ and
$D_{j}^{tr}\subset G_{j}$, $\min(n_{j}^{de},n_{T})>N_{2}$ and
$n_{j}^{tr}>N_{3}$,

$\displaystyle\Big{|}\hat{r}_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}$
(17) $\displaystyle=$ $\displaystyle\Big{|}\hat{r}_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}$
$\displaystyle+r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\hat{r}_{j}(\bm{x})-r_{j}(\bm{x})\Big{|}\cdot\Big{|}L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}$
$\displaystyle+\Big{|}r_{j}(\bm{x})\Big{|}\cdot\Big{|}L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\epsilon_{2}\Big{|}L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}+C\cdot\epsilon_{3}.$

Let $\epsilon_{2}\to 0$ and $\epsilon_{3}\to 0$, then

$\Big{|}\hat{r}_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}\to
0.$

Form (16), we can obtain that for any $\delta_{4}>0$ and $\epsilon_{4}>0$,
$\exists N_{4}>0$, when $n_{j}^{de},n_{j}^{tr},n_{T}\geq N_{4}$,

$\displaystyle\mu\Big{(}\cup\Big{\\{}D:$
$\displaystyle\Big{|}\hat{r}_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}-r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{|}$
$\displaystyle>\epsilon_{4},D_{j}^{de}\sim p_{j}(\bm{x},y),D_{j}^{tr}\sim
p_{j}(\bm{x},y),D=D_{j}^{de}\cup
D_{j}^{tr}\Big{\\}}\Big{)}=\mu(G_{j}^{c})<\delta_{4}.$

Thus, for any given hyperparameter $\bm{\theta}\in\Theta$ and $(\bm{x},y)\in
D_{j}^{val}$, when $n_{j}^{de},n_{j}^{tr},n_{T}\to\infty$,

$\hat{r}_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\overset{P}{\to}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}.$
(18)

So we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{IW}(\bm{\theta};D_{j})=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{r}_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
(19) $\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)},$
$\displaystyle\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{IW}(\bm{\theta};D_{j})=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\mathbb{E}r_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle=$
$\displaystyle\mathbb{E}\frac{p_{T}(\bm{x}_{j,1}^{val})}{p_{j}(\bm{x}_{j,1}^{val})}L\Big{(}h\Big{(}\bm{x}_{j,1}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,1}^{val}\Big{)}$
$\displaystyle=$
$\displaystyle\mathbb{E}\int_{\mathcal{X}}\frac{p_{T}(\bm{x})}{p_{j}(\bm{x})}L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,1}^{val}\Big{)}p_{j}(\bm{x})d\bm{x}$
$\displaystyle=$
$\displaystyle\mathbb{E}\int_{\mathcal{X}}p_{T}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,1}^{val}\Big{)}d\bm{x}$
$\displaystyle\overset{(*)}{=}$
$\displaystyle\mathbb{E}\int_{\mathcal{X}}L\Big{(}h\Big{(}\bm{t};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},u\Big{)}p_{T}(\bm{t})d\bm{t}$
$\displaystyle=$
$\displaystyle\mathbb{E}L\Big{(}h\Big{(}\bm{t};\bm{\omega}(\bm{\theta}),\bm{\theta}\Big{)},u\Big{)}$
$\displaystyle=$ $\displaystyle f_{T}(\bm{\theta}).$

$(*)$ in (19) refers to Ref. 2000Improving and 2007Covariate. Since
$\mathcal{Y}$ is bounded, $L$ satisfies the Lipschitz condition and
$r(\cdot)\in[0,C]$ for a constant $C$, then
$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{IW}(\bm{\theta};D_{j})$
is bounded. Thus, by the bounded convergence theorem, we have

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\hat{f}_{IW}(\bm{\theta};D_{j})=\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{IW}(\bm{\theta};D_{j})=f_{T}(\bm{\theta}).$

The variance of $\hat{f}_{IW}$ is

$\displaystyle\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}=$
$\displaystyle\mathbb{V}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}$
$\displaystyle-\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}\Big{\\}}^{2}.$

By (18) and (19), according to the operation rules of limit and the bounded
convergence theorem, we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}.$

And by the same token, we can derive that

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle=$
$\displaystyle\Big{\\{}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle=$ $\displaystyle\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{[}\hat{r}_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle=$ $\displaystyle\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle=$ $\displaystyle f^{2}_{T}(\bm{\theta}).$

Thus,

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},\atop
n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}=$
$\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}-f^{2}_{T}(\bm{\theta}).$

∎

A2. Proof of Corollary 2.3

[(ii)]

Since $\hat{r}_{j}(\cdot)\to r_{j}(\cdot)$ a.s. when $n_{j}^{de}\to\infty$ and
$n_{T}\to\infty$, by (LABEL:A11) and the asymptotic unbiasedness of the sample
variance and sample covariance, we can obtain that

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\widehat{\operatorname{Var}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}=\operatorname{Var}_{\bm{x}\sim
p_{j}(\bm{x})}\Big{[}r_{j}(\bm{x})\Big{]},$
$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\widehat{\operatorname{Cov}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\cdot
L\Big{(}h\big{(}\bm{x}_{j}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\big{)},y_{j}^{val}\Big{)},\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}$
$\displaystyle=$ $\displaystyle\operatorname{Cov}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\big{)},y\Big{)},r_{j}(\bm{x})\Big{]}.$

Because $Var_{\bm{x}\sim p_{j}(\bm{x})}\left[r_{j}(\bm{x})\right]\neq 0$,

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\frac{\widehat{\operatorname{Cov}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\cdot
L\Big{(}h\big{(}\bm{x}_{j}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\big{)},y_{j}^{val}\Big{)},\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}}{\widehat{\operatorname{Var}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}}$
$\displaystyle=$
$\displaystyle\frac{\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\widehat{\operatorname{Cov}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\cdot
L\Big{(}h\big{(}\bm{x}_{j}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\big{)},y_{j}^{val}\Big{)},\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}}{\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\widehat{\operatorname{Var}}\Big{[}\hat{r}_{j}(\bm{x}_{j}^{val})\Big{]}}$
$\displaystyle=$ $\displaystyle\frac{\operatorname{Cov}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\big{)},y\Big{)},r_{j}(\bm{x})\Big{]}}{Var_{\bm{x}\sim
p_{j}(\bm{x})}\Big{[}r_{j}(\bm{x})\Big{]}}.$

Thus, we obtain that
$\lim_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{\eta}_{j}(\bm{\theta})=\eta_{j}(\bm{\theta}).$

Then, we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j})$
$\displaystyle=$ $\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},\atop
n_{T}\to\infty}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}\hat{r}_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(x_{j,i}^{val})-1\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\lim\limits_{n_{j}^{de},n_{j}^{tr},\atop
n_{T}\to\infty}\Big{[}\hat{r}_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(x_{j,i}^{val})-1\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}r_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(x_{j,i}^{val})-1\Big{)}\Big{]}.$

Since

$\mathbb{E}r_{j}(\bm{x})=\int_{\mathcal{X}}r_{j}(\bm{x})p_{j}(\bm{x})d\bm{x}=\int_{\mathcal{X}}\frac{p_{T}(\bm{x})}{p_{j}(\bm{x})}p_{j}(\bm{x})d\bm{x}=\int_{\mathcal{X}}p_{T}(\bm{x})d\bm{x}=1,$

we can derive that

$\displaystyle\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j})$
(20) $\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\mathbb{E}\Big{[}r_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(x_{j,i}^{val})-1\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}r_{j}(\bm{x}_{j,1}^{val})L\Big{(}h\Big{(}\bm{x}_{j,1}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,1}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(x_{j,1}^{val})-1\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}r_{j}(\bm{x}_{j,1}^{val})L\Big{(}h\Big{(}\bm{x}_{j,1}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,1}^{val}\Big{)}\Big{]}+\mathbb{E}\Big{[}\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(x_{j,1}^{val})-1\Big{)}\Big{]}$
$\displaystyle\overset{(**)}{=}$ $\displaystyle
f_{T}(\bm{\theta})+\eta_{j}(\bm{\theta})\cdot\mathbb{E}\left[r_{j}(x_{j,1}^{val})-1\right]$
$\displaystyle=$ $\displaystyle f_{T}(\bm{\theta}).$

$(**)$ in (20) is derived from (19). Similar to the proof of Theorem 2.2 and
by the bounded convergence theorem, we have

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\hat{f}_{CV}(\bm{\theta};D_{j})=\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j})=f_{T}(\bm{\theta}).$

According to the operation rules of limit and the bounded convergence theorem,
we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}$
(21) $\displaystyle=$
$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}^{2}-\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{\\{}\mathbb{E}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}\Big{\\}}^{2}$
$\displaystyle=$
$\displaystyle\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}^{2}-\Big{\\{}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}\Big{\\}}^{2}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}^{2}-f_{T}^{2}(\bm{\theta})$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{\\{}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}r_{j}(\bm{x}_{j,i}^{val})L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle-f_{T}^{2}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x})-1\Big{)}\Big{]}^{2}$
$\displaystyle\quad\quad\quad-f_{T}^{2}(\bm{\theta})\Big{\\}}$

To derive the last equation of (21), let $Z,Z_{1},\cdots,Z_{n}$ be i.i.d.
random variables,

$\displaystyle\mathbb{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}Z_{i}\Big{]}^{2}-(\mathbb{E}Z)^{2}=\frac{1}{n^{2}}\mathbb{E}\Big{[}\sum_{i=1}^{n}Z_{i}^{2}+2\sum_{i<j}Z_{i}Z_{j}\Big{]}-(\mathbb{E}Z)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\mathbb{E}Z^{2}+\frac{2}{n^{2}}\frac{n(n-1)}{2}(\mathbb{E}Z)^{2}-(\mathbb{E}Z)^{2}=\frac{1}{n}[\mathbb{E}Z^{2}-(\mathbb{E}Z)^{2}].$

By the same token, we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2}-f^{2}_{T}(\bm{\theta})\Big{\\}}.$

The control variate method[12] implies that

$\displaystyle\mathbb{E}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(x)-1\Big{)}\Big{]}^{2}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\Big{[}r_{j}(\bm{x})L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}\Big{]}^{2},$

so we have

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}\leq\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{IW}(\bm{\theta};D_{j})\Big{]}.$

∎

A3. Proof of Corollary 2.6

[(ii)]

For a given hyperparameter $\bm{\theta}\in\Theta$ and any set of weights for
sources $\bm{\lambda}=\\{\lambda_{1},\cdots,\lambda_{K}\\}$,

$\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}\cdot\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j}).$

Similar to the proof of Theorem 2.2, we can obtain that

$\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\mathbb{E}\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}\cdot\mathbb{E}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\hat{f}_{CV}(\bm{\theta};D_{j})=\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}\cdot
f_{T}(\bm{\theta})=f_{T}(\bm{\theta}).$

We can see that
$\hat{f}_{FedDA}\left(\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\right)$ is a special
case of $\hat{f}_{\bm{\lambda}}\left(\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\right)$
where $\lambda_{j}=\hat{\lambda}_{j}(\bm{\theta}),j=1,2,\cdots,K$, so

$\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{E}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=f_{T}(\bm{\theta}).$

Because $K$ source datasets are independent and each source dataset is i.i.d.,
we have

$\mathbb{V}\Big{[}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}=\sum_{j=1}^{K}\mathbb{V}\Big{[}\lambda_{j}n_{j}^{val}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}=\sum_{j=1}^{K}\lambda_{j}^{2}(n_{j}^{val})^{2}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}.$

According to the expression of
$\operatorname{Div}_{j}\left(\bm{\theta}\right)$ and (21), we can obtain that

$\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}=\frac{1}{n_{j}^{val}}\operatorname{Div}_{j}\left(\bm{\theta}\right),$

Then, by the operation rules of limit

$\displaystyle\lim\limits_{\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\sum_{j=1}^{K}\lambda_{j}^{2}(n_{j}^{val})^{2}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\sum_{j=1}^{K}\lambda_{j}^{2}n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right).$

To investigate the asymptotic variance of FedDAE, we first study the limit of
$\hat{\lambda}_{j}(\bm{\theta})$, $j=1,2,\cdots,K$. According to the operation
rules of limit, we have

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{[}\hat{r}_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{T}\to\infty}\Big{[}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\hat{\bm{\omega}}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\hat{\eta}_{j}(\bm{\theta})\cdot\Big{(}\hat{r}_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{)}\Big{]}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\Big{[}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{(}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{)}\Big{]}^{2}.$

Since $\eta_{j}(\bm{\theta})$ is bounded, we can use strong law of large
numbers and the operation rules of limit to obtain that

$\displaystyle\lim\limits_{n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)$
$\displaystyle=$
$\displaystyle\lim\limits_{n_{j}^{val}\to\infty}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{[}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\lim\limits_{n_{j}^{val}\to\infty}\Big{[}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{(}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{)}\Big{]}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\Big{\\{}\lim\limits_{n_{j}^{val}\to\infty}\frac{1}{n_{j}^{val}}\sum_{i=1}^{n_{j}^{val}}\Big{(}r_{j}(\bm{x}_{j,i}^{val})\cdot
L\Big{(}h\Big{(}\bm{x}_{j,i}^{val};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y_{j,i}^{val}\Big{)}$
$\displaystyle\quad\quad+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x}_{j,i}^{val})-1\Big{)}\Big{)}\Big{\\}}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x})-1\Big{)}\Big{]}^{2}$
$\displaystyle-\Big{\\{}\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x})-1\Big{)}\Big{]}\Big{\\}}^{2}$
$\displaystyle=$ $\displaystyle\mathbb{E}_{(\bm{x},y)\sim
p_{j}(\bm{x},y)}\Big{[}r_{j}(\bm{x})\cdot
L\Big{(}h\Big{(}\bm{x};\bm{\omega}_{j}(\bm{\theta}),\bm{\theta}\Big{)},y\Big{)}+\eta_{j}(\bm{\theta})\cdot\Big{(}r_{j}(\bm{x})-1\Big{)}\Big{]}^{2}-f^{2}_{T}(\bm{\theta})$
$\displaystyle=$
$\displaystyle\operatorname{Div}_{j}\left(\bm{\theta}\right).$

According to the operation rules of limit, we can derive that

$\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\hat{\lambda}_{j}(\bm{\theta})$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\Bigg{(}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}$
$\displaystyle=$ $\displaystyle\Bigg{(}\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\widehat{\operatorname{Div}}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}$
$\displaystyle=$
$\displaystyle\Bigg{(}\operatorname{Div}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}$
$\displaystyle=$ $\displaystyle\lambda_{j}(\bm{\theta}).$

And by the same token, we have

$\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},n_{j}^{val},n_{T}\to\infty}\sum_{j=1}^{K}\hat{\lambda}_{j}(\bm{\theta})^{2}(n_{j}^{val})^{2}\mathbb{V}\Big{[}\hat{f}_{CV}(\bm{\theta};D_{j})\Big{]}$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\sum_{j=1}^{K}\lambda_{j}^{2}(\bm{\theta})n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right)$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\sum_{j=1}^{K}\Bigg{(}\operatorname{Div}_{j}\left(\bm{\theta}\right)\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-2}n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right)$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-2}$
$\displaystyle=$ $\displaystyle\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}.$

Then, for any given set of weights of sources
$\bm{\lambda}=\\{\lambda_{1},\cdots,\lambda_{K}\\}$, we can use the Cauchy-
Schwarz inequality to obtain that,

$\displaystyle
1=\Bigg{(}\sum_{j=1}^{K}\lambda_{j}n_{j}^{val}\Bigg{)}^{2}\leq\Bigg{(}\sum_{j=1}^{K}\lambda_{j}^{2}n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right)\Bigg{)}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}$
$\displaystyle\implies\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}\leq\Bigg{(}\sum_{j=1}^{K}\lambda_{j}^{2}n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right)\Bigg{)}$
$\displaystyle\implies\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\Bigg{(}\sum_{j=1}^{K}\frac{n_{j}^{val}}{\operatorname{Div}_{j}\left(\bm{\theta}\right)}\Bigg{)}^{-1}\leq\lim\limits_{\forall
j,n_{j}^{val}\to\infty}\Bigg{(}\sum_{j=1}^{K}\lambda_{j}^{2}n_{j}^{val}\operatorname{Div}_{j}\left(\bm{\theta}\right)\Bigg{)}$
$\displaystyle\implies\lim\limits_{\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{\bm{\lambda}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}.$

∎

A4. Proof of Corollary 2.7

Let $\tilde{\bm{\lambda}}$ be a set of weights for sources with
$\tilde{\lambda}_{j}=1/n^{val},j=1,2,\cdots,K$, it is easily to prove that
$\tilde{\bm{\lambda}}$ satisfies $\tilde{\lambda}_{j}\geq 0,j=1,2,\cdots,K$
and $\sum_{j=1}^{K}\tilde{\lambda}_{j}n_{j}^{val}=1$. Then, by Corollary 2.6
we can derive that

$\lim\limits_{\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{\tilde{\bm{\lambda}}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]},$

where

$\hat{f}_{\tilde{\bm{\lambda}}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}=\sum_{j=1}^{K}\tilde{\lambda}_{j}n_{j}^{val}\hat{f}_{CV}(\bm{\theta};D_{j})=\frac{1}{n^{val}}\sum_{j=1}^{K}n_{j}^{val}\cdot\hat{f}_{CV}(\bm{\theta};D_{j}).$

According to (4) and the inequality in Corollary 2.3, we have

$\lim\limits_{\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{\tilde{\bm{\lambda}}}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedIW}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}.$

Thus, we obtain

$\lim\limits_{\forall j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedDA}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}\leq\lim\limits_{\forall
j,n_{j}^{de},n_{j}^{tr},\atop
n_{j}^{val},n_{T}\to\infty}\mathbb{V}\Big{[}\hat{f}_{FedIW}\Big{(}\bm{\theta};\\{D_{j}\\}_{j=1}^{K}\Big{)}\Big{]}.$

∎


[a,b]Axel Schmidt

# Charged-averaged elastic lepton-proton scattering cross section results from
OLYMPUS

###### Abstract

Measurements of the proton’s form factor ratio made with polarization transfer
show a striking discrepancy relative to the ratio extracted from unpolarized
elastic electron-proton scattering cross sections. One hypothesis is that the
discrepancy is caused by hard two-photon exchange (TPE), a typically neglected
radiative correction that may bias the two approaches differently. This
hypothesis has been challenging to confirm. Theoretical estimates of TPE are
model-dependent, and recent experimental determinations of TPE lacked the
kinematic reach to be conclusive. The possible impact of TPE remains a cloud
over our knowledge of the proton’s form factors. Recently, the OLYMPUS
experiment published new elastic scattering cross sections that are
insensitive to the effects of TPE: specifically the average of electron-proton
and positron-proton cross sections. The OLYMPUS experiment, conducted at DESY,
Hamburg, measured elastic $e^{-}p$ and $e^{+}p$ scattering by detecting the
scattered lepton and recoiling proton in coincidence in a large-acceptance,
toroidal magnetic spectrometer. OLYMPUS was designed to measure the
$e^{+}p/e^{-}p$ cross section raio to isolate the effects of TPE. By
exploiting the over determined kinematics of the reaction, the absolute
efficiency of spectrometer could be verified, allowing cross sections to be
extracted from the data. These results can help refine our knowledge of the
proton’s form factors, especially in the squared momentum-transfer region of
1–2 GeV2, where some previous measurements are in tension.

##  1 Introduction

The proton’s electromagnetic form factors, $G_{E}$ and $G_{M}$, show a
significant discrepancy depending on the technique used to measure them. When
determined from unpolarized elastic electron scattering cross sections, the
form factors generally exhibit scaling, i.e., the ratio $\mu_{p}G_{E}/G_{M}$
is approximately constant as a function of momentum transfer, $Q^{2}$. By
contrast, measurements of polarization transfer or other equivalent
polarization observables indicate a sharply decreasing ratio, as shown in Fig.
2. The leading hypothesis is that the effects of hard two-photon exchange
(TPE), a typically neglected radiative correction, might be affecting the two
techniques differently [1, 2]. Several recent experiments have attemped to
measure hard TPE directly by comparing positron-proton and electron-proton
elastic scattering cross sections, but the results have not been conclusive
[3, 4, 5, 6]. Accounting for hard TPE remains a problem for understanding the
proton form factors at high $Q^{2}$. For a recent review, see Ref. [7].

Figure 1:  The protons form factor ratio, $\mu_{p}G_{E}/G_{M}$, shows a
striking discrepancy between unpolarized Rosenbluth separations and direct
measurements using polarization observables. The unpolarized global fit comes
from Ref. [8]. The polarized data are from Refs. [9, 10, 11, 12, 13, 14, 15].

One of these recent experiments, OLYMPUS, has recently published absolute
cross sections for both electron and positron elastic scattering [16]. The
average of the two—the lepton charge averaged cross section—has the convenient
feature of being insensitive to two-photon exchange effects at lowest order.
This can observable can help constrain proton form factors by removing the
uncertainty associated with two photon exchange corrections.

##  2 The OLYMPUS Experiment

Figure 2:  Left: illustration of the OLYMPUS detector, described in detail in
Ref. [17]. Right: the OLYMPUS measurement of $R_{2\gamma}$ [6] showed a slight
increase with $Q^{2}$ as expected from the two-photon exchange hypothesis, but
was slightly lower than theoretical expectations [18, 8].

The OLYMPUS Experiment collected data at DESY, in Hamburg, Germany in 2012.
Alternating beams of electrons and positrons in the DORIS storage ring were
passed through a windowless unpolarized hydrogen target that was internal to
the ring vacuum [19]. The OLYMPUS spectrometer [17] consisted of two
instrumented sectors of an 8-coil toroid magnet [20] surrounding the target,
shown in Fig. 2 left. Drift chambers were used for charged particle tracking
and panels time-of-flight (ToF) scintillators were used for triggering,
timing, and particle identification. Scattered leptons and recoiling protons
were detected in coincidence and over-determined kinematics were used to
cleanly separate elastic scattering events from inelastic background. The
relative luminosity between electron and positron running modes was determined
from the rate of multi-interaction events in a pair of lead fluoride
calorimeters positioned downstream at the angle of symmetric Møller/Bhabha
scattering [21].

OLYMPUS’s primary result was a measurement of the ratio of positron-proton to
electron-proton elastic scattering cross sections, i.e.,
$R_{2\gamma}\equiv\sigma_{e^{+}p}/\sigma_{e^{-}p}$, shown in Fig. 2 right [6].
The results show a definitive indication of TPE: $R_{2\gamma}$ has a positive
slope with increasing $Q^{2}$ (and decreasing $\epsilon$). However the results
are generally below theoretical predictions [18, 8]. It is not clear whether
this slight disagreement is due to additional effects not accounted for in the
predictions or to the overall normalization of the experiment (estimated to be
better than $\pm 0.5\%$). In any case, the OLYMPUS results are completely
consistent with the magnitude of the form factor discrepancy in the $Q^{2}$
range covered by the experiment [22].

##  3 Charge-Averaged Cross Section Analysis

While the ratio $\sigma_{e^{+}p}/\sigma_{e^{-}p}$ amplifies the effects of
TPE, the lepton charge-averaged cross section,
$(\sigma_{e^{+}p}+\sigma_{e^{-}p})/2$, cancels the effects of TPE at lowest
order. This combination therefore provides a more accurate measure of the Born
cross section, removing the need to apply a correction for hard TPE.

Determining absolute cross sections was difficult for OLYMPUS, which was
designed to measure cross section ratios in which the effects of acceptance,
efficiency, and absolute luminosity normalization largely cancel.
Nevertheless, the over-determined kinematics in elastic scattering allowed
several cross checks to be performed in order to quantify the uncertainty
coming from each effect.

Figure 3:  The charge-averaged cross section as measured by OLYMPUS [16],
compared to other recent proton form-factor predictions [23, 24, 25, 26]. A
normalization uncertainty of $\pm 7.5\%$ is not shown.

The efficiency of each sub-detector element—each scintillator panel and each
drift chamber cell—was determined from data and then modelled in a Geant4
simulation. The efficiency for reconstructing elastic scattering events was
determined from simulation. To cross check that there was no additional
tracking inefficiency in real data relative to simulated data, elastic events
were identified using the lepton (or proton) track information alone, to
determine the probability for the successful reconstruction of the opposite
side proton (or lepton). There was no indication of any additional tracking
inefficiency, at the percent level.

OLYMPUS had no mechanism to deterimine the absolute luminosity, which is a
combination of the well-measured beam current and the crudely known density of
the hydrogen gas target. The best measure of the absolute luminosity came from
the forward Møller/Bhabha calorimeters, but even this was estimated to have an
accuracy of $\pm 7\%$. The OLYMPUS results, therefore, are primarily a measure
of the shape of the cross section, rather than one of absolute scale.

##  4 Results

The results [16] are shown in Fig. 3, in comparison to other recent global
fits of proton cross section data [23, 24, 25, 26]. The results are in good
qualitative agreement with these global fits. However, the data do not show
the cusp predicted by the Bernauer et al. fit [8]. It will be interesting to
see how this cusp changes when the Bernauer fit is updated to include the new
OLYMPUS data. In general, the OLYMPUS results can help constrain future global
fits with reduced ambiguity from uncertain corrections for hard TPE.

##  5 Summary

Data from the OLYMPUS Experiment were analyzed to extract absolute cross
sections for both electron-proton and positron-proton elastic scattering. The
average of the two is unaffected by hard TPE, giving robust access to the Born
cross section and the proton’s electromagnetic form factors. These results
will aid future global fits to constrain $G_{E}$ and $G_{M}$ in the momentum
transfer range of $0.6<Q^{2}<2.0$ GeV${}^{2}/c^{2}$.

## References

  * [1] P.A.M. Guichon and M. Vanderhaeghen, _How to reconcile the Rosenbluth and the polarization transfer method in the measurement of the proton form-factors_ , _Phys. Rev. Lett._ 91 (2003) 142303 [hep-ph/0306007].
  * [2] P.G. Blunden, W. Melnitchouk and J.A. Tjon, _Two photon exchange and elastic electron proton scattering_ , _Phys. Rev. Lett._ 91 (2003) 142304 [nucl-th/0306076].
  * [3] I.A. Rachek et al., _Measurement of the two-photon exchange contribution to the elastic $e^{\pm}p$ scattering cross sections at the VEPP-3 storage ring_, _Phys. Rev. Lett._ 114 (2015) 062005 [1411.7372].
  * [4] CLAS collaboration, _Towards a resolution of the proton form factor problem: new electron and positron scattering data_ , _Phys. Rev. Lett._ 114 (2015) 062003 [nucl-ex/1411.6908].
  * [5] CLAS collaboration, _Measurement of two-photon exchange effect by comparing elastic $e^{\pm}p$ cross sections_, _Phys. Rev. C_ 95 (2017) 065201 [1603.00315].
  * [6] OLYMPUS collaboration, _Hard Two-Photon Contribution to Elastic Lepton-Proton Scattering: Determined by the OLYMPUS Experiment_ , _Phys. Rev. Lett._ 118 (2017) 092501 [1611.04685].
  * [7] A. Afanasev, P.G. Blunden, D. Hasell and B.A. Raue, _Two-photon exchange in elastic electron–proton scattering_ , _Prog. Part. Nucl. Phys._ 95 (2017) 245 [1703.03874].
  * [8] A1 collaboration, _Electric and magnetic form factors of the proton_ , _Phys. Rev. C_ 90 (2014) 015206 [1307.6227].
  * [9] O. Gayou et al., _Measurements of the Elastic Electromagnetic Form Factor Ratio $\mu_{p}G_{E_{p}}/G_{M_{p}}$ via Polarization Transfer_, _Phys. Rev. C_ 64 (2001) 038202.
  * [10] Resonance Spin Structure collaboration, _Proton $G_{E}/G_{M}$ from beam-target asymmetry_, _Phys. Rev. C_ 74 (2006) 035201 [nucl-ex/0606015].
  * [11] C. Crawford et al., _Measurement of the Proton’s Electric to Magnetic Form Factor Ratio from ${}^{1}{\rm H}(\vec{e},e{\textasciiacute}p)$_, _Phys. Rev. Lett._ 98 (2007) 052301 [nucl-ex/0609007].
  * [12] X. Zhan et al., _High-Precision Measurement of the Proton Elastic Form Factor Ratio $\mu_{p}G_{E}/G_{M}$ at low $Q^{2}$_, _Phys. Lett. B_ 705 (2011) 59 [1102.0318].
  * [13] V. Punjabi et al., _Proton elastic form-factor ratios to Q**2 = 3.5-GeV**2 by polarization transfer_ , _Phys. Rev. C_ 71 (2005) 055202 [nucl-ex/0501018].
  * [14] A.J.R. Puckett et al., _Final Analysis of Proton Form Factor Ratio Data at $\mathbf{Q^{2}=4.0}$, 4.8 and 5.6 GeV2_, _Phys. Rev. C_ 85 (2012) 045203 [1102.5737].
  * [15] A.J.R. Puckett et al., _Polarization Transfer Observables in Elastic Electron Proton Scattering at $Q^{2}=$2.5, 5.2, 6.8, and 8.5 GeV2_, _Phys. Rev. C_ 96 (2017) 055203 [1707.08587].
  * [16] OLYMPUS collaboration, _Measurement of the Charge-Averaged Elastic Lepton-Proton Scattering Cross Sectionby the OLYMPUS Experiment_ , _Phys. Rev. Lett._ 126 (2021) 162501 [2008.05349].
  * [17] OLYMPUS collaboration, _The OLYMPUS Experiment_ , _Nucl. Instrum. Meth. A_ 741 (2014) 1 [1312.1730].
  * [18] P.G. Blunden and W. Melnitchouk, _Dispersive approach to two-photon exchange in elastic electron-proton scattering_ , _Phys. Rev. C_ 95 (2017) 065209 [1703.06181].
  * [19] J.C. Bernauer, V. Carassiti, G. Ciullo, B.S. Henderson, E. Ihloff, J. Kelsey et al., _The OLYMPUS Internal Hydrogen Target_ , _Nucl. Instrum. Meth. A_ 755 (2014) 20 [1404.0579].
  * [20] J.C. Bernauer et al., _Measurement and tricubic interpolation of the magnetic field for the OLYMPUS experiment_ , _Nucl. Instrum. Meth. A_ 823 (2016) 9 [1603.06510].
  * [21] A. Schmidt, C. O’Connor, J.C. Bernauer and R. Milner, _A novel technique for determining luminosity in electron-scattering/positron-scattering experiments from multi-interaction events_ , _Nucl. Instrum. Meth. A_ 877 (2018) 112 [1708.04616].
  * [22] A. Schmidt, _How much two-photon exchange is needed to resolve the proton form factor discrepancy?_ , _J. Phys. G_ 47 (2020) 055109 [1907.07318].
  * [23] J.J. Kelly, _Simple parametrization of nucleon form factors_ , _Phys. Rev. C_ 70 (2004) 068202.
  * [24] J. Arrington, _Implications of the discrepancy between proton form-factor measurements_ , _Phys. Rev. C_ 69 (2004) 022201 [nucl-ex/0309011].
  * [25] J. Arrington, W. Melnitchouk and J. Tjon, _Global analysis of proton elastic form factor data with two-photon exchange corrections_ , _Phys. Rev. C_ 76 (2007) 035205 [0707.1861].
  * [26] Z. Ye, J. Arrington, R.J. Hill and G. Lee, _Proton and Neutron Electromagnetic Form Factors and Uncertainties_ , _Phys. Lett. B_ 777 (2018) 8 [1707.09063].