1

Information Geometrically Generalized Covariate Shift Adaptation

Masanari Kimura1, Hideitsu Hino†,2,3
1SOKENDAI, Graduate University for Advanced Studies.
Shonan Village, Hayama, Kanagawa 240-0193 Japan
2The Institute of Statistical Mathematics.
10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
3Center for Advanced Intelligence Project, RIKEN.
1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
†corresponding author<EMAIL_ADDRESS>
Keywords: Information Geometry, Domain Adaptation, Covariate Shift

Abstract

Many machine learning methods assume that the training and test data follow
the same distribution. However, in the real world, this assumption is very
often violated. In particular, the phenomenon that the marginal distribution
of the data changes is called covariate shift, one of the most important
research topics in machine learning. We show that the well-known family of
covariate shift adaptation methods is unified in the framework of information
geometry. Furthermore, we show that parameter search for geometrically
generalized covariate shift adaptation method can be achieved efficiently.
Numerical experiments show that our generalization can achieve better
performance than the existing methods it encompasses.

##  1 Introduction

When considering supervised learning methods, it is often assumed that the
training and test data follow the same distribution (Bishop,, 1995; Duda et
al.,, 2006; Hastie et al.,, 2009; Vapnik,, 2013; Mohri et al.,, 2018).
However, this common assumption is violated in the real world in most cases
(Huang et al.,, 2007; Zadrozny,, 2004; Cortes et al.,, 2008; Quionero-Candela
et al.,, 2009; Jiang,, 2008).

Covariate shift (Shimodaira,, 2000) is a prevalent setting for supervised
learning in the real world, where the input distribution differs in the
training and test phases, but the conditional distribution of the output
variable given the input variable remains unchanged. Covariate shift is a
commonly observed phenomenon in real-world machine learning applications, such
as emotion recognition (Hassan et al.,, 2013; Jirayucharoensak et al.,, 2014),
3D pose estimation (Yamada et al.,, 2012), brain computer interfaces (Li et
al.,, 2010; Raza et al.,, 2016), spam filtering (Bickel et al.,, 2009), and
human activity recognition (Hachiya et al.,, 2012). In addition, there has
been recent discussion on the relationship between covariate shift and the
robustness of deep learning (Ioffe and Szegedy,, 2015; Arpit et al.,, 2016;
Santurkar et al.,, 2018; Nado et al.,, 2020; Huang and Yu,, 2020; Awais et
al.,, 2020).

Ordinary empirical risk minimization (ERM) (Vapnik,, 1998, 2013) may not
generalize well to the test data under covariate shift because of the
difference between the training and test distributions. However, importance
weighting for training examples has been shown to be effective in mitigating
the effect of covariate shift (Shimodaira,, 2000; Sugiyama and Müller, 2005b,
; Sugiyama and Müller, 2005a, ; Zadrozny,, 2004). The main idea of these
strategies is weighting the training loss terms according to their importance,
which is the ratio of the training input density to the test input density.
The importance weighting is widely adopted even in modern covariate shift
studies with deep neural networks (DNN) (Fang et al.,, 2020; Zhang et al.,,
2021).

In this paper, we consider the generalization of these methods in the
framework of information geometry (Amari,, 1985; Amari and Nagaoka,, 2007), a
tool that allows us to deal with probability distributions on Riemannian
manifolds. This generalization makes it possible to search for good weighting
without searching for a large number of parameters. Our contributions is
summarized as follows:

  * •

(Section 4.1 and 4.2) We generalize existing methods of covariate shift
adaptation in the framework of information geometry. By our information
geometrical formulation, geometric biases of conventional methods are
elucidated.

  * •

(Section 4.3) We show that our geometrically generalized covariate shift
adaptation method has a much larger solution space than existing methods
controlled by only two parameters. Efficient weighting is obtained by
searching for parameters using an information criterion or Bayesian
optimization.

  * •

(Section 5) Numerical experiments show that our generalization can achieve
better performance than the existing methods it encompasses.

##  2 Preliminaries

###  2.1 Problem formulation

First, we formulate the problem of supervised learning. We denote by
$\mathcal{X}\subset\mathbb{R}^{d}$ the input space. The output space is
denoted by $\mathcal{Y}\subset\mathcal{R}$ (regression) or
$\mathcal{Y}\subset\\{1,\dots,K\\}$ ($K$-class classification). We assume that
training examples $\\{(\bm{x}^{tr}_{i},y^{tr}_{i})\\}^{n_{tr}}_{i=1}$ are
independently and identically distributed (i.i.d.) according to some fixed but
unknown distribution $p_{tr}(\bm{x},y)$, which can be decomposed into the
marginal distribution and the conditional probability distribution, i.e.,
$p_{tr}(\bm{x},y)=p_{tr}(\bm{x})p_{tr}(y|\bm{x})$. We also denote the test
examples by $\\{(\bm{x}^{te}_{i},y^{te}_{i})\\}^{n_{te}}_{i=1}$ drawn from a
test distribution $p_{te}(\bm{x},y)=p_{te}(\bm{x})p_{te}(y|\bm{x})$.

Let $\mathcal{H}$ be a hypothesis class. The goal of supervised learning is to
obtain a hypothesis $h:\mathcal{X}\to\mathbb{R}\ (h\in\mathcal{H})$ with the
training examples that minimizes the expected loss over the test distribution:

$\mathcal{R}(h)\coloneqq\mathbb{E}_{(\bm{x}^{te},y^{te})\sim
p_{te}(\bm{x},y)}\Big{[}\ell(h(\bm{x}^{te}),y^{te})\Big{]},$ (1)

where $\ell:\mathbb{R}\times\mathcal{Y}\to\mathbb{R}$ is the loss function
that measures the discrepancy between the true output value $y$ and the
predicted value $\hat{y}\coloneqq h(\bm{x})$. In this paper, we assume that
$\ell$ is bounded from above, i.e., $\ell(y,y^{\prime})<\infty\ (\forall
y,y^{\prime}\in\mathcal{Y})$.

######  Definition 2.1 (Covariate shift assumption)

We consider that the two distributions $p_{tr}(\bm{x},y)$ and
$p_{te}(\bm{x},y)$ satisfy the covariate shift assumption if the following
three conditions hold: 1) $p_{tr}(\bm{x})\neq p_{te}(\bm{x})$, 2)
$\operatorname{supp}(p_{tr}(\bm{x}))\supset\operatorname{supp}(p_{te}(\bm{x}))$
and 3) $p_{tr}(y|\bm{x})=p_{te}(y|\bm{x})$.

Under the covariate shift assumption, the goal of covariate shift adaptation
is still to obtain a hypothesis $h$ that minimizes the expected loss (1) by
utilizing both labeled training examples
$\\{(\bm{x}^{tr}_{i},y^{tr}_{i})\\}^{n_{tr}}_{i=1}$ and unlabeled test
examples $\\{(\bm{x}^{te}_{i})\\}^{n^{te}}_{i=1}$.

###  2.2 Previous works

Ordinary empirical risk minimization (ERM) (Vapnik,, 1998, 2013), a standard
approach in supervised learning, may fail under the covariate shift because it
assumes that the training and test data follow the same distribution.
Importance weighting has been shown to be effective in mitigating the effect
of covariate shift (Shimodaira,, 2000; Sugiyama and Müller, 2005b, ; Sugiyama
et al.,, 2007; Zadrozny,, 2004):

$\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}w(\bm{x}^{tr}_{i})\ell(h(\bm{x}^{tr}_{i}),y^{tr}_{i}),$
(2)

where $w:\mathcal{X}\to\mathbb{R}_{\geq 0}$ is a certain weighting function.

######  Definition 2.2 (IWERM (Shimodaira,, 2000))

If we choose the density ratio $p_{te}(\bm{x})/p_{tr}(\bm{x})$ as the
weighting function, ERM according to

$\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\frac{p_{te}(\bm{x}^{tr}_{i})}{p_{tr}(\bm{x}^{tr}_{i})}\ell(h(\bm{x}^{tr}_{i}),y^{tr}_{i})$
(3)

has consistency.

This is called importance weighted ERM (IWERM). However, IWERM tends to
produce an estimator with high variance. We can reduce the variance by
flattening the importance weights, which is called adaptive IWERM (AIWERM):

######  Definition 2.3 (AIWERM (Shimodaira,, 2000))

Let $\lambda\in[0,1]$. If we choose
$(p_{te}(\bm{x})/p_{tr}(\bm{x}))^{\lambda}$ as the weighting function, we can
obtain the variance-reduced estimator:

$\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\Big{(}\frac{p_{te}(\bm{x}^{tr}_{i})}{p_{tr}(\bm{x}^{tr}_{i})}\Big{)}^{\lambda}\ell(h(\bm{x}^{tr}_{i}),y^{tr}_{i}).$
(4)

Relative IWERM (RIWERM), a stable version of AIWERM, has also been proposed:

###### Definition 2.4

(RIWERM (Yamada et al.,, 2011)) Let $\lambda\in[0,1]$. If we choose
$p_{te}(\bm{x})/\lambda p_{tr}(\bm{x})+(1-\lambda)p_{te}(\bm{x})$ as the
weighting function, we can directly estimate a flattened version of the
importance weight:

$\min_{h\in\mathcal{H}}\frac{1}{n_{tr}}\sum^{n_{tr}}_{i=1}\frac{p_{te}(\bm{x}^{tr}_{i})}{\lambda
p_{tr}(\bm{x}^{tr}_{i})+(1-\lambda)p_{te}(\bm{x}^{tr}_{i})}\ell(h(\bm{x}^{tr}_{i}),y^{tr}_{i}).$
(5)

All of the above methods are considered as different weighting methods for
each point of the training data. More generally, the method of covariate shift
adaptation can be essentially rephrased as a weighting strategy for training
data.

##  3 Statistical Model and Exponential Family

Information geometry (Amari,, 1985; Amari and Nagaoka,, 2007) is a powerful
framework that allows us to deal with statistical models on Riemannian
manifolds. For theoretical investigation, we need the notion of dual
connection and curvature tensor associated with Fisher metric, but these
details are deferred to the Appendix Appendix A: Statistical Manifolds and
Straight Line in Exponential Family and we here present minimum required
definitions and notations. We note that the assumption on the parametric
family is only required for the information geometric analysis in Section 4.2.
The algorithmic framework of the proposed method is independent of the
parametric model.

Since $p_{tr}(y|\bm{x})=p_{te}(y|\bm{x})=p(y|\bm{x})$ from the assumption of
Definition 2.1, what we are interested in is the model manifold
$(\mathcal{M},g(\bm{\theta}))$ to which the marginal distribution
$p(\bm{x};\bm{\theta})$ belongs:

$\mathcal{M}=\Big{\\{}p(\bm{x};\bm{\theta})\ ;\bm{\theta}\in\Theta\Big{\\}}.$
(6)

Here, $p_{tr}(\bm{x};\bm{\theta}),p_{te}(\bm{x};\bm{\theta})\in\mathcal{M}$.
We note that elements in $\mathcal{M}$ is specified by its parameter
$\bm{\theta}$ and we identify the parameter vector $\bm{\theta}$ to the
density function $p(\bm{x};\bm{\theta})$ and write
$p(\bm{x};\bm{\theta})\simeq\bm{\theta}$ if necessary. In this paper, we
assume that $\mathcal{M}$ is an exponential family and the probability density
function can be written as

$p(\bm{x};\bm{\theta})=\exp\Big{\\{}\theta^{i}T_{i}(\bm{x})+k(\bm{x})-\psi(\bm{\theta})\Big{\\}},$
(7)

where $\bm{x}$ is a random variable,
$\bm{\theta}=(\theta^{1},\dots,\theta^{p})$ is an $p$-dimensional vector
parameter to specify a distribution,
$\bm{T}(\bm{x})=(T_{1}(\bm{x}),\dots,T_{p}(\bm{x}))$ are sufficient statistics
of $\bm{x}$, $k(\bm{x})$ is a function of $\bm{x}$ and $\psi$ corresponds to
the normalization factor. In Eq. (7), and hereafter the Einstein summation
convention will be assumed, so that summation will be automatically taken over
indices repeated twice in the term, e.g.,
$\bm{a}^{i}\bm{b}_{i}=\sum_{i}\bm{a}^{i}\bm{b}_{i}$.

In the exponential family, the natural parameter $\bm{\theta}$ forms the
affine coordinate system, i.e.,

$\bm{\theta}(t)=(1-t)\bm{\theta}_{1}+t\bm{\theta}_{2}\ \
(\forall\bm{\theta}_{1},\bm{\theta}_{2}\in\Theta,\ \forall t\in[0,1])$ (8)

is a geodesic on $\mathcal{M}$. As a dual coordinate of $\bm{\theta}$, the
expectation parameter $\bm{\eta}$ is defined by the Legendre transformation

$\displaystyle\bm{\eta}=$
$\displaystyle\nabla\psi(\bm{\theta}),\quad\bm{\theta}=\nabla\varphi(\bm{\eta}),$
$\displaystyle\mbox{where}\;\;\varphi(\bm{\eta})=$
$\displaystyle\max_{\bm{\theta}^{\prime}}\Big{\\{}\bm{\theta}^{\prime}\cdot\bm{\eta}-\psi(\bm{\theta}^{\prime})\Big{\\}}.$

Existing weights for covariate shift adaptation are geometrically
characterized, then a generalized weight function is designed based on this
geometric formulation.

##  4 Geometrical Generalization of Covariate Shift Adaptation

###  4.1 Information Geometrically Generalized IWERM

In order to derive a generalized covariate shift adaptation method, we prepare
the following function.

######  Definition 4.1 ($f$-interpolation (Kimura and Hino,, 2021))

For any $a,b,\in\mathbb{R}$, some $\lambda\in[0,1]$ and some
$\alpha\in\mathbb{R}$, we define $f$-interpolation as

$m_{f}^{(\lambda,\alpha)}(a,b)=f^{-1}_{\alpha}\Big{\\{}(1-\lambda)f_{\alpha}(a)+\lambda
f_{\alpha}(b)\Big{\\}},$ (9)

where

$f_{\alpha}(a)=\begin{cases}a^{\frac{1-\alpha}{2}}&(\alpha\neq 1)\\\ \log
a&(\alpha=1)\end{cases}$ (10)

is the function that defines the $f$-mean (Hardy et al.,, 1952).

We can easily see that this family includes various known weighted means
including the $e$-mixture and $m$-mixture for $\alpha=\pm 1$ in the literature
of information geometry (Amari,, 2016):

$\displaystyle m_{f}^{(\lambda,1)}(a,b)=$ $\displaystyle\exp\\{(1-\lambda)\log
a+\lambda\log b\\},$ $\displaystyle m_{f}^{(\lambda,-1)}(a,b)=$
$\displaystyle(1-\lambda)a+\lambda b,$ $\displaystyle
m_{f}^{(\lambda,0)}(a,b)=$
$\displaystyle\Big{(}(1-\lambda)\sqrt{a}+\lambda\sqrt{b}\Big{)}^{2},$
$\displaystyle m_{f}^{(\lambda,3)}(a,b)=$
$\displaystyle\frac{1}{(1-\lambda)\frac{1}{a}+\lambda\frac{1}{b}}.$

Also, for any $\bm{u},\bm{v}\in\mathbb{R}^{d}\ (d>0)$, we write

$\displaystyle\bm{m}=m_{f}^{(\lambda,\alpha)}(\bm{u},\bm{v}),\mbox{where}\quad\bm{m}_{i}=m_{f}^{(\lambda,\alpha)}(\bm{u}_{i},\bm{v}_{i}).$

Using this function, we generalize the existing methods of covariate shift
adaptation.

######  Lemma 4.1 ($f$-representation of AIWERM)

The marginal positive measures generated by the weighting of AIWERM can be
expressed by using the $f$-interpolation function as

$p_{A}^{(\lambda)}(\bm{x})=m_{f}^{(\lambda,1)}(p_{tr}(\bm{x}),p_{te}(\bm{x})).$
(11)

###### Proof 1

From Eq .(4), we consider its expectation as

$\displaystyle\hat{h}=$
$\displaystyle\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\Big{(}\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\Big{)}^{\lambda}\ell(h(\bm{x}),y)p_{tr}(\bm{x},y)d\bm{x}dy$
$\displaystyle=$
$\displaystyle\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}),y)p^{(\lambda)}_{A}(\bm{x})p_{tr}(y|\bm{x})d\bm{x}dy.$

Here,

$\displaystyle p^{(\lambda)}_{A}(\bm{x})$
$\displaystyle=\Big{(}\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\Big{)}^{\lambda}p_{tr}(\bm{x})$
$\displaystyle\log p^{(\lambda)}_{A}(\bm{x})$ $\displaystyle=\alpha(\log
p_{te}(\bm{x})-\log p_{tr}(\bm{x}))+\log p_{tr}(\bm{x})$ $\displaystyle=$
$\displaystyle(1-\lambda)\log p_{tr}(\bm{x})+\lambda\log p_{te}(\bm{x})$
$\displaystyle p^{(\lambda)}_{A}(\bm{x})$
$\displaystyle=\exp\\{(1-\lambda)\log p_{tr}(\bm{x})+\lambda\log
p_{te}(\bm{x})\\}$ $\displaystyle=$ $\displaystyle
m_{f}^{(\lambda,1)}(p_{tr}(\bm{x}),p_{te}(\bm{x})).$

######  Lemma 4.2 ($f$-representation of RIWERM)

The marginal positive measures generated by the weighting of RIWERM can be
expressed by using the $f$-interpolation function as

$p^{(\lambda)}_{R}(\bm{x})=m^{(\lambda,3)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x})).$
(12)

###### Proof 2

From Eq. (5),

$\displaystyle p^{(\lambda)}_{R}(\bm{x})$
$\displaystyle=\frac{p_{te}(\bm{x})p_{tr}(\bm{x})}{\lambda
p_{tr}(\bm{x})+(1-\lambda)p_{te}(\bm{x})}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda\frac{1}{p_{te}(\bm{x})}+(1-\lambda)\frac{1}{p_{tr}(\bm{x})}}=m^{(\lambda,3)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x})).$

From the above discussion, the following generalized method of covariate shift
adaptation is derived using the $f$-representation.

######  Theorem 4.1 (Geometrically generalized IWERM)

For $\lambda\in[0,1]$ and $\alpha\in\mathbb{R}$, AIWERM and RIWERM is
generalized as

$\hat{h}=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}w^{(\lambda,\alpha)}(\bm{x})\ell(h(\bm{x}),y)p_{tr}(\bm{x},y)d\bm{x}dy,$
(13)

where

$w^{(\lambda,\alpha)}(\bm{x})=\frac{m^{(\lambda,\alpha)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}.$
(14)

See the Appendix Appendix B: Proofs of main results for the proof. From
Definition 4.1, we can confirm that

$\displaystyle m_{f}^{(0,\alpha)}(p_{tr}(\bm{x}),p_{te}(\bm{x}))$
$\displaystyle=p_{tr}(\bm{x}),\;\;\mbox{and}$ $\displaystyle
m_{f}^{(1,\alpha)}(p_{tr}(\bm{x}),p_{te}(\bm{x}))$
$\displaystyle=p_{te}(\bm{x}),$

for all $\alpha\in\mathbb{R}$, and this means that we can obtain the set of
all curves that connect $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$.

We note that Zhang et al., (2021) proposed a method based on basis expansion
to estimate a flexible importance weight. It is similar to our proposal in the
sense that improves the degree of freedom for designing the importance weight.
However, our method considers the parametric form of weight, which enables us
to achieve information geometric insight.

In many studies of covariate shift problems using the density ratio weighting
including Yamada et al., (2011), the direct estimation of the density ratio is
often employed (Sugiyama et al.,, 2012). Our proposed weight function in (14)
is also represented as density ratio:

$\displaystyle w^{(\lambda,\alpha)}(\bm{x})=$
$\displaystyle\frac{\left[(1-\lambda)p_{tr}(\bm{x})^{\frac{1-\alpha}{2}}+\lambda
p_{te}(\bm{x})^{\frac{1-\alpha}{2}}\right]^{\frac{2}{1-\alpha}}}{p_{tr}(\bm{x})}$
$\displaystyle=$
$\displaystyle\left[1-\lambda+\lambda\left(\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}\right)^{\frac{1-\alpha}{2}}\right]^{\frac{2}{1-\alpha}},\quad(\alpha\neq
1).$

It is then also possible to apply the direct estimation of the density ratio
using, for example kernel expansion. In our implementation, we simply used the
given $p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$ separately because they are
explicitly known by the construction of the training and the test datasets as
explained in Section 5.1. In the practical application of the proposed method
in which the generative processes of the covariates of training and test data
are unknown, direct density estimation would be a promising approach.

###  4.2 Geometric Bias

AIWERM and RIWERM connects two distributions $p_{tr}$ and $p_{te}$ in
different ways. Statistical bias and variance of IWERM, AIWERM, and RIWERM are
discussed in the respective papers. In this subsection, we study the geometric
bias of these methods to have a deeper understanding of these methods from the
geometric viewpoint.

The proposed generalization of IWERM is independent from a specific
parametrization of density functions. In this subsection, for theoretical
treatment, the exponential model manifold which contains
$p_{tr}(\bm{x};\bm{\theta})$ and $p_{te}(\bm{x};\bm{\theta})$ are considered,
hence geodesics can be described by a linear combination of parameters as
explained in Appendix Appendix A: Statistical Manifolds and Straight Line in
Exponential Family. With this assumption, specifying $\lambda$ and $\alpha$ is
equivalent to selecting a point on the geodesic connecting $p_{tr}$ and
$p_{te}$.

######  Definition 4.2 ($\alpha$-divergence (Amari,, 1985))

Let $\alpha$ be a real parameter. The $\alpha$-divergence between two
probability vectors $\bm{p}$ and $\bm{q}$ is defined as

$D_{\alpha}[\bm{p}:\bm{q}]=\frac{4}{1-\alpha^{2}}\Big{(}1-\sum_{i}p_{i}^{\frac{1-\alpha}{2}}q_{i}^{\frac{1+\alpha}{2}}\Big{)}.$
(15)

###### Definition 4.3

($\alpha$-representation (Amari,, 2009)) For some positive measure
$m_{i}^{\frac{1-\alpha}{2}}$, the coordinate system $\bm{\theta}=(\theta^{i})$
derived from the $\alpha$-divergence is
$\theta^{i}=m_{i}^{\frac{1-\alpha}{2}}=f_{\alpha}(m_{i})$ and denote by
$\theta^{i}$ the $\alpha$-representation of a positive measure
$m_{i}^{\frac{1-\alpha}{2}}$.

######  Definition 4.4 ($\alpha$-geodesic (Amari,, 2016))

The $\alpha$-geodesic connecting two probability vectors $p(\bm{x})$ and
$q(\bm{x})$ is defined as

$\displaystyle r_{i}(\lambda)=$ $\displaystyle
c(t)f^{-1}_{\alpha}\Big{\\{}(1-\lambda)f_{\alpha}(p(x_{i}))+\lambda
f_{\alpha}(q(x_{i}))\Big{\\}},$ $\displaystyle c(\lambda)=$
$\displaystyle\left(\sum^{p}_{i=1}r_{i}(\lambda)\right)^{-1}.$ (16)

Let $\psi_{\alpha}(\bm{\theta})=\frac{1-\alpha}{2}\sum_{i=1}m_{i}$, the dual
coordinate system $\bm{\eta}$ is given by
$\bm{\eta}=\nabla\psi_{\alpha}(\bm{\theta})$ as

$\eta_{i}=(\theta^{i})^{\frac{1+\alpha}{1-\alpha}}=f_{-\alpha}(m_{i}),$ (17)

which is the $-\alpha$-representation of $m_{i}$.

From Definitions 4.1 and 4.4, we see that $f$-interpoloation is the
unnormalized version of the $\alpha$-geodesic. We write
$\tilde{m}^{(\lambda,\alpha)}_{f}$ for a suitably normalized
$f$-interpolation. The important properties of $\alpha$-geodesics are

  * •

the $\alpha$-geodesic is a geodesic in the $\alpha$-coordinate system derived
from $\alpha$-divergence,

  * •

the $-\alpha$-geodesic is linear in the $-\alpha$-representation.

Let $\gamma_{c}$ be the geodesic connecting two distributions parameterized by
$\bm{\theta}_{tr}$ and $\bm{\theta}_{te}$. Now, we define two types of
geometric biases to characterize the dispersion of $\bm{\theta}_{tr}$ from
$\bm{\theta}_{te}$ with respect to the direction along the $\alpha$-geodesic
and to the direction orthogonal to the $\alpha$-geodesic.

######  Definition 4.5 (Geodesic bias and curvature bias)

If we write the unit vector along the $\alpha$-geodesic direction as $e_{1}$
and any unit vector in the orthogonal direction to $e_{1}$ as $e_{2}$, the
bias relative to the test distribution due to weighting can be decomposed as
follows:

  * •

geodesic bias: $b_{g}=(1-\lambda)e_{1}$,

  * •

curvature bias: $b_{c}=(1-\lambda)tr_{g}(\mathrm{Ric})e_{2}$,

where $tr_{g}$ is the trace operation on the metric tensor $g$ and
$\mathrm{Ric}$ is the Ricci curvature of the curve connecting the two points
generated by the weighting:

$\mathrm{Ric}=R_{ikj}d\bm{\theta}^{i}\otimes d\bm{\theta}^{j}.$ (18)

Here, $R_{ikj}$ is the Riemannian curvature tensor.

For more detail on the geometric concepts, see textbooks on Riemannian
manifolds (Jost,, 2017). This definition of geometric biases is consistent
with the fact that IWERM, which corresponds to $\lambda=1$, leads to an
unbiased estimator of the risk in the test dataset.

###### Proposition 4.2

For AIWERM, the geometric bias $b_{A}(\lambda)$ is computed as

$b_{A}(\lambda)=(1-\lambda)e_{1}.$ (19)

Figure 1: Geometry of covariate shift adaptation methods. In the
$\bm{\theta}$-coordinate system, the dashed line corresponds to AIWERM and the
dotted line corresponds to RIWERM. We write unit vector along the
$\alpha$-geodesic direction as $e_{1}$ and any unit vector in the orthogonal
direction to $e_{1}$ as $e_{2}$. Here, $\lambda=0$ and $\lambda=1$ correspond
to $\theta_{tr}$ (ERM) and $\theta_{te}$ (IWERM), respectively, and $\alpha=1$
and $\alpha=3$ correspond to the AIWERM and RIWERM curves in the figure.
Figure 2: Visualization of grid search for $\alpha$ and $\lambda$ on LIBSVM
dataset. Algorithm 1 Bayesian optimization for IGIWERM

0: acquisition function $a(\lambda,\alpha|D)$, target function
$L(h;\lambda,\alpha)$, initial points $D_{init}$ compose of a set of
parameters $\Xi=\\{(\lambda,\alpha)\\}$ and corresponding values of the target
function

0: $(\lambda^{\ast},\alpha^{\ast})$ that minimizes
$\min_{h\in\mathcal{H}}L(h;\lambda,\alpha)$

Initialize $D=D_{init}$

while Not converge do

$\hat{\lambda},\hat{\alpha}=\mathop{\rm
arg~{}min}\limits_{\lambda,\alpha}a(\lambda,\alpha|D),\;\;\Xi=\Xi\cup\\{(\hat{\lambda},\hat{\alpha})\\}$

$\hat{e}=L(h;\hat{\lambda},\hat{\alpha}),\;\;D=D\cup\\{(\hat{\lambda},\hat{\alpha},\hat{e})\\}$

end while

$(\lambda^{\ast},\alpha^{\ast})=\mathop{\rm
arg~{}min}\limits_{(\lambda,\alpha)\in\Xi}\left\\{\min_{h\in\mathcal{H}}L(h;\lambda,\alpha)\right\\}$

###### Proposition 4.3

For RIWERM, the geometric bias $b_{R}(\lambda)$ is computed as

$b_{R}(\lambda)=(1-\lambda)\Big{\\{}e_{1}+tr_{g}\Big{(}-4\Lambda_{ikj}d\bm{\theta}^{i}\otimes
d\bm{\theta}^{j}\Big{)}e_{2}\Big{\\}}.$ (20)

Here, $\Lambda$ is a tensor that depends on the connection.

These propositions are proved by straightforward calculation as detailed in
Appendix Appendix B: Proofs of main results

Figure 3: Left: generated data from $y=x^{2}+\varepsilon$. We see that $p_{tr}(x)$ and $p_{te}(x)$ are different. Right: results of fitting by ERM, IWERM, and IGIWERM. Figure 4: Bayesian optimization for IGIWERM. The coordinates of the purple circles are the parameters explored by Bayesian optimization, and the size of the purple circles indicates the goodness of the parameters (inverse of the MSE). Table 1: Mean squared errors of covariate shift adaptation methods in regression problems over $10$ trials. Here, IGIWERM (bopt) is the Bayesian optimization based, and IGIWERM (IC) is the information criterion based strategy. Weighting strategy | MSE
---|---
ERM | $160.19(\pm 4.25)$
IWERM | $33.76(\pm 3.82)$
AIWERM | $31.14(\pm 2.97)$
RIWERM | $30.03(\pm 2.74)$
IGIWERM (bopt) | ${\bf 28.89(\pm 2.42)}$
IGIWERM (IC) | ${\bf 28.38(\pm 2.12)}$

Figure 1 shows the curves on the manifolds created by AIWERM and RIWERM. Both
of them satisfy

  * •

for $\lambda=0$, it is equivalent to unweighted ERM,

  * •

for $\lambda=1$, it is equivalent to IWERM.

Note that the curvature bias $b_{c}$ vanishes for all $\lambda\in[0,1]$ in
AIWERM, while RIWERM does not guarantee the vanishing of the curvature bias
for $\lambda\in(0,1)$.

Intuitively, the geometric bias reveals in which direction the two parameters
are misaligned. IWERM, which corresponds to AIWERM and RIWERM with
$\lambda=1$, is optimal when the sample size is large enough, but in real
problems with limited sample size, it is often desirable to adopt a point
between $\bm{\theta}_{tr}$ and $\bm{\theta}_{te}$. AIWERM and RIWERM consider
distinct curves and specify a point on them by the parameter $\lambda$. Our
geometric analysis revealed that these curves are included in the set of
curves represented by dual $f$-representation of the parameter coordinate
system, and the geometric biases of these particular cases (AIWERM and RIWERM)
are identified. The results presented in this subsection do not claim
superiority of a particular method and are of importance in their own right as
a geometric analysis of the covariate shift method.

###  4.3 Optimization of the generalized IWERM

The existing covariate shift adaptation methods described above can be
regarded as having determined a good “weighting direction” in some sense in
advance and then the “weighting magnitude” is adjusted according to the
parameter $\lambda$. This approach is very convenient in terms of
computational efficiency since the only optimized parameter is
$\lambda\in[0,1]$.

However, geometrically, these methods only consider certain curves on the
manifold as candidate solutions, as can be seen from Figure 1, which means
that the solution space is very small.

Our information geometrical IWERM (IGIWERM) can handle all curves
$\gamma_{\alpha}(\lambda)$ in $\Pi_{(p_{tr},p_{te})}$ that connect
$p_{tr}(\bm{x})$ and $p_{te}(\bm{x})$, by adding only one parameter. For
example, by setting $\alpha\in[1,3]$, shaded area in Figure 1 can be used as
the solution space. The problem of how to determine $\lambda$ and $\alpha$
remains.

####  4.1 Information criterion

When the predictive model is of a simple parametric form, information
criterion derived in (Shimodaira,, 2000) is available (see appendix of
(Shimodaira,, 2000) for the proof.):

######  Theorem 4.4 (Information criterion for IGIWERM)

Let the information criterion for IWERM be

$IC_{GW}\coloneqq-2L_{1}(\hat{\bm{\theta}})+2tr(J_{w}H_{w}^{-1}),$ (21)

where $L_{1}(\bm{\theta})=\sum^{n_{tr}}_{i=1}dr(\bm{x}^{tr}_{i})\log
p(y^{tr}_{i}|\bm{x}^{tr}_{i},\bm{\theta}),\;dr(\bm{x})=\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x)}}$
and

$\displaystyle J_{w}$
$\displaystyle=-\mathbb{E}_{p_{tr}}\Biggl{[}dr(\bm{x})\frac{\partial\log
p(y|\bm{x},\bm{\theta})}{\partial\bm{\theta}}\Biggr{|}_{\bm{\theta}^{*}_{w}}\times\frac{\partial\Big{(}\frac{m_{f}^{\lambda,\alpha}(p_{tr}(x),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}\log
p(y|\bm{x},\bm{\theta})\Big{)}}{\partial\bm{\theta}^{\prime}}\Biggr{|}_{\bm{\theta}^{*}_{w}}\Biggr{]}$
$\displaystyle H_{w}$
$\displaystyle=\mathbb{E}_{p_{tr}}\Biggl{[}\frac{\partial^{2}\Big{(}\frac{m_{f}^{\lambda,\alpha}(p_{tr}(x),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}\log
p(y|\bm{x},\bm{\theta})\Big{)}}{\partial\bm{\theta}\partial\bm{\theta}^{\prime}}\Biggr{]}.$

Here, $\bm{\theta}^{*}_{w}$ is the minimizer of the weighted empirical risk.
The matrices $J_{w}$ and $H_{w}$ may be replaced by their consistent
estimates. Then, $IC_{GW}/2n$ is an unbiased estimator of the expected loss up
to $O(n^{-1})$ term:

$\mathbb{E}_{p_{tr}}\Big{[}IC_{GW}/2n\Big{]}=\mathbb{E}_{p_{tr}}\Big{[}\ell_{1}(\hat{\bm{\theta}}_{w})\Big{]}+o(n^{-1}).$
(22)

Table 2: Mean misclassification rates averaged over $10$ trails on LIBSVM
benchmark datasets. The numbers in the brackets are the standard deviations.
For the methods with (optimal), the optimal parameters for the test data are
obtained by linear search.

Dataset | #features | #data | unweighted | IWERM | AIWERM (optimal) | RIWERM (optimal) | ours
---|---|---|---|---|---|---|---
australian | $14$ | $690$ | $33.46(\pm 23.65)$ | $22.13(\pm 3.37)$ | $21.98(\pm 3.36)$ | $21.73(\pm 3.82)$ | ${\bf 18.85(\pm 3.99)}$
breast-cancer | $10$ | $683$ | $38.28(\pm 10.98)$ | $41.23(\pm 15.39)$ | $36.41(\pm 9.68)$ | $36.13(\pm 10.81)$ | ${\bf 31.65(\pm 8.49)}$
heart | $13$ | $270$ | $45.17(\pm 6.98)$ | $39.94(\pm 8.55)$ | $39.76(\pm 8.49)$ | $39.76(\pm 8.92)$ | ${\bf 35.37(\pm 6.84)}$
diabetes | $8$ | $768$ | $33.19(\pm 5.69)$ | $37.22(\pm 6.63)$ | $33.11(\pm 6.45)$ | $33.38(\pm 5.74)$ | ${\bf 32.83(\pm 5.62)}$
madelon | $500$ | $2,000$ | $47.78(\pm 1.53)$ | $47.28(\pm 2.20)$ | $47.10(\pm 2.13)$ | $47.12(\pm 1.65)$ | ${\bf 46.56(\pm 2.12)}$

####  4.2 Bayesian optimization

This information criterion does not work for complicated nonparametric models.
As a method that can be applied in general situations, we consider using
Bayesian optimization Snoek et al., (2012); Frazier, (2018) to find the
optimal weighting by IGIWERM. Bayesian optimization assumes that the target
function is drawn from a prior distribution over functions, typically a
Gaussian process, updating a posterior as we observe the target function value
in new places. We use the validation loss as the target function:

$L(h;\lambda,\alpha)=\frac{1}{n_{val}}\sum^{n_{val}}_{i=1}\frac{p_{te}(\bm{x}_{i}^{val})}{p_{tr}(\bm{x}_{i}^{val})}\ell(h_{\lambda,\alpha}(\bm{x}^{val}_{i}),y^{val}_{i}).$
(23)

where $n_{val}$ is the validation sample size and $h_{\lambda,\alpha}$ is
given by IWERM with $\lambda$ and $\alpha$. This validation procedure is based
on the importance weighted cross validation used in (Sugiyama et al.,, 2007).
In Bayesian optimization, an acquisition function $a(\lambda,\alpha|D)$ is
used for measuring goodness of candidate point $(\lambda,\alpha)$ based on
current dataset $D$. As the acquisition function, we adopt the expected
improvement Mockus et al., (1978); Jones et al., (1998). In this strategy, we
choose the next query point which has the highest expected improvement over
the current minimum target value. See Appendix Appendix D: Optimization of the
generalized IWERM for more detail. The overall picture is summarized in
Algorithm 1.

###  4.4 Learning guarantee

Generalization bounds of weighted maximum likelihood estimator for the target
domain are derived in Cortes et al., (2010), and our weight function (14) is
compatible with their bound. The weight defined by Eq. (14) is bounded when
$\alpha\neq 1$ and achieves a standard rate $O(n_{tr}^{-1/2})$. When
$\alpha=1$, the weight is unbounded and its rate is $O(n_{tr}^{-3/8})$.
Details are shown in Appendix Appendix C: Learning guarantee.

##  5 Numerical Experiments

In this section, we present experimental results of domain adaptation problems
under covariate shift using both synthetic and real data111Source code to
reproduce the results is available from https://github.com/nocotan/IGIWERM.
Since the main purpose of the experiments is to see the effect of our
generalization of the importance weighted ERM and comparison to the proposed
and conventional IWERM methods, in all experiments, we assume that $p_{tr}$
and $p_{te}$ are known as detailed in Section 5.1.

###  5.1 Induction of Covariate Shift

Since each dataset is composed of data points generated from independent and
identical distributions, we need to artificially induce covariate shifts. We
induce the covariate shift as follows Cortes et al., (2008):

  1. 1.

As a preprocessing step, we perform Z-score standardization on all input data.

  2. 2.

Then, an example $(\bm{x},y)$ is assigned to the training dataset with
probability $\exp(v)/(1+\exp(v))$ and to the test dataset with probability
$1/(1+\exp(v))$, where $v=16\bm{w}^{T}\bm{x}/\sigma$ with $\sigma$ being the
standard deviation of $\bm{w}^{T}\bm{x}$ determined by using the given
dataset, and $\bm{w}\in\mathbb{R}^{d}$ is a given projection vector. Here, the
projection vector $\bm{w}$ is given randomly for each experimental process.

By this construction of the training and test datasets, $p_{tr}$ and $p_{te}$
are explicitly determined as

$\displaystyle p_{tr}(\bm{x})$
$\displaystyle=\frac{\exp(16\bm{w}^{T}\bm{x}/\sigma)}{1+\exp(16\bm{w}^{T}\bm{x}/\sigma)},$
$\displaystyle p_{te}(\bm{x})$
$\displaystyle=\frac{1}{1+\exp(16\bm{w}^{T}\bm{x}/\sigma)},$

when the projection vector $\bm{w}\in\mathbb{R}^{d}$ is given. Although
density ratio estimation could be employed in our experiments, we assume that
the distribution is known in order to compare the performance of the proposed
method without relying on the accuracy of the density or density ratio
estimation.

###  5.2 Illustrative Example in Regression

Here, we predict the response $y\in\mathbb{R}$ using ordinary linear
regression: $y=\beta_{0}+\beta_{1}x+\varepsilon,\
\varepsilon\sim\mathcal{N}(0,\sigma^{2})$, where $\mathcal{N}(a,b)$ denotes
the normal distribution with mean $a$ and variance $b$. In the numerical
example below, we assume the true $p(y|x)$ given as $y=x^{2}+\varepsilon,\
\varepsilon\sim\mathcal{N}(0,5).$ The $p_{tr}(x)$ and $p_{te}(x)$ of the
covariate $x$ are $x^{tr}\sim\mathcal{N}(0,5),\
x^{te}\sim\mathcal{N}(-5,0.5).$ The training sample size is $n_{tr}=1000$ and
the test sample size is $n_{te}=300$. The left-hand side of Fig. 3 shows the
data to be generated. We can see that $p_{tr}(x)\neq p_{te}(x)$.

The right panel of Fig. 3 shows the results of fitting by unweighted ERM,
IWERM, and IGIWERM. Here, the parameters of IGIWERM are explored by using
Algorithm 1, as shown in Fig. 4. The coordinates of the purple circles are the
parameters explored by Bayesian optimization, and the radius of the purple
circles is proportional to the goodness $r(\bm{\beta})$ of the parameters
(inverse of the MSE):
$r(\bm{\beta})=\left(\frac{1}{n}\sum^{n}_{i=1}(y_{i}-h(x_{i},\bm{\beta}))^{2}\right)^{-1}$.
By choosing the size $r(\beta)$ of the plot for each point in this manner, the
better-evaluated parameters can be plotted in larger circles. From this
figure, it can be seen that our generalized weighting is not restricted to
lying just on two curves corresponding to AIWERM and RIWERM.

For the normal linear regression, the information criterion (21) is calculated
from

$\displaystyle
IC_{GW}(\lambda,\alpha)=\frac{1}{2}\sum^{n_{tr}}_{i=1}dr(\bm{x}_{i}^{tr})\Biggl{\\{}\frac{\hat{\epsilon}^{2}_{1}}{\hat{\sigma}^{2}+\log(2\pi\hat{\sigma}^{2})}\Biggr{\\}}+$
$\displaystyle\sum^{n_{tr}}_{i=1}dr(\bm{x}_{i}^{tr})\Biggl{\\{}\frac{\hat{\epsilon}^{2}_{i}}{\hat{\sigma}^{2}}\hat{h}_{i}+\frac{m_{f}^{\lambda,\alpha}(p_{tr}(\bm{x}^{tr}),p_{te}(\bm{x}^{tr}))}{2\hat{c}_{w}p_{tr}(\bm{x}^{tr})}\Big{(}\frac{\hat{\epsilon}^{2}_{i}}{\hat{\sigma}^{2}}-1\Big{)}^{2}\Biggr{\\}}.$

Here,
$\hat{c}_{w}=\sum^{n}_{i=1}\frac{m_{f}^{\lambda,\alpha}(p_{tr}(x_{i}^{tr}),p_{te}(x_{i}^{tr}))}{p_{tr}(x_{i}^{tr})}$,
$\hat{\sigma}^{2}=\sum^{n}_{i=1}\frac{m_{f}^{\lambda,\alpha}(p_{tr}(x_{i}^{tr}),p_{te}(x_{i}^{tr}))}{p_{tr}(x_{i}^{tr})}\hat{\epsilon}_{i}^{2}/\hat{c}_{w}$
and $\hat{\epsilon}_{i}$ is the residual. Table 1 shows that the IGIWERM
outperforms existing methods.

###  5.3 Experiments on binary classification problem

We show the results of our experiments on the LIBSVM
dataset222https://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/.

In the experiments, we randomly generate a mapping vector $\bm{w}$ for each
trial and perform $10$ trials for each dataset. We use SVM with Radial Basis
Function (RBF) kernel as the base classifier. In this experiment, the
parameters $\lambda$ of AIWERM and RIWERM are chosen optimally by linear
search using the test data. The experimental results on benchmark datasets are
summarized in Table 2. The table shows that the proposed IGIWERM outperforms
the conventional methods even when the parameters of those methods are
optimized by using the test dataset. More experimental results on other
datasets with various models are reported in Appendix Appendix E: Additional
experimental results.

###  5.4 Computational Cost

Here, we investigate the computational cost of our IGIWERM. The experimental
setup is the same as in Section 5.2. The mean and standard deviation of the
computation time obtained in the 10 trials are shown in Table 3. From this
table, we can see that our IGIWERM takes constant times longer to compute than
the vanilla ERM.

Table 3: Computational cost of ERM and IGIWERM. Method | Computation time [sec]
---|---
ERM | $1.130(\pm 0.238)$
IGIWERM | $9.887(\pm 0.845)$

##  6 Conclusion and Discussion

We generalized existing methods of covariate shift adaptation in the
geometrical framework. By our information geometrical formulation, geometric
biases of conventional methods are elucidated. Unlike the dominant approaches
restricted to a specific curve on a manifold in the literature, our
generalization has a much larger solution space with only two parameters. Our
experiments highlighted the advantage of our method over previous approaches,
suggesting that our generalization can achieve better performance than the
existing methods. A drawback of our proposed method is its relatively high
computational cost for optimizing parameters $\alpha$ and $\lambda$. We used
Bayesian optimization for efficient parameter search, and further
sophisticated approaches would be explored in our future work.

As mentioned in the introduction, the importance weighting is used with deep
neural network models (Fang et al.,, 2020), in which the importance weight in
the feature representation obtained by DNN is considered. It is also worth
mentioning that Sakai and Shimizu, (2019) used RIWERM in the study of
covariate shift on the learning from positive and unlabeled data. Our
generalization will be applicable to their methods to improve the performance
under a small sample regime. In particular, in a standard approach for
optimizing the implicit weight function $w(\bm{x})$, it is common to add a
regularization term $(w(\bm{x})-p_{tr}(\bm{x})/p_{te}(\bm{x}))^{2}$ to the
optimization objective. The use of the derived geodesic and curvature biases
to regularize the optimal weight function will be investigated in connection
with the modern weight learning approach using deep neural network models.
Finally, the relation between geometric bias and statistical bias should be
explored.

### Acknowledgement

Part of this work is supported by JST CREST JPMJCR1761, JPMJCR2015, JSPS
KAKENHI 17H01793, JP22H03653 and NEDO (JPNP18002). Finally, we express our
special thanks to the anonymous reviewers whose valuable comments helped to
improve the manuscript.

## Appendix A: Statistical Manifolds and Straight Line in Exponential Family

Let $M$ be a $d$-dimensional differentiable manifold with a Riemannian metric
$g$. For each $\bm{x}\in M$, $T_{\bm{x}}M$ is its tangent space.

###### Definition 6.1

Let $g_{\bm{x}}$ an inner product

$g_{\bm{x}}:T_{\bm{x}}(M)\times
T_{\bm{x}}(M)\to\mathbb{R}\quad\forall\bm{x}\in M.$ (24)

When, for any $X,Y\in M$, the map $\bm{x}\to
g_{\bm{x}}(X_{\bm{x}},Y_{\bm{x}})$ is differentiable with respect to
$\bm{x}\in M$, $g_{\bm{x}}$ is denoted as the Riemannian metric.

The correspondence $X:M\ni\bm{x}\mapsto X_{\bm{x}}\in T_{\bm{x}}M$ is called a
vector field on $M$. For $\bm{x}\in M$, let coordinate expression of
$X_{\bm{x}}$ be $X_{\bm{x}}=(v^{1}(\bm{x}),\dots,v^{d}(\bm{x}))$. Then,
$v^{i}(\bm{x})$ defines a real-valued function $v^{i}$ on $M$ and $X$ is
expressed as $X=(v^{1},\dots,v^{d})$. When a function on $M$ is $k$ times
continuously differentiable, it is called the class $C^{k}$, and the set of
all functions of class $C^{k}$ on $M$ is denoted as $C^{k}(M)$. A vector field
$X$ is called class $C^{k}$ when all of $v^{i},\;i=1,\dots,d$ are class
$C^{k}$. The set of all class $C^{\infty}$ vector fields is denoted as
$\mathfrak{X}(M)$. A tangent space $T_{\bm{x}}(M)$ is a vector space spanned
by differentials $\frac{\partial}{\partial x^{i}}$, namely,

$T_{\bm{x}}(M)=\left\\{a^{i}\left(\frac{\partial}{\partial
x^{i}}\right)_{\bm{x}}\middle|\forall\;a_{i}\in\mathbb{R}\right\\}.$ (25)

Following the notational convention of differential geometry, we use
$\partial_{i}=\frac{\partial}{\partial x^{i}}$ and the Einstein summation
convention. The vector field on a manifold $M$ is then written as

$\mathfrak{X}(M)=\left\\{v^{i}\partial_{i}\middle|\;v^{i}\in
C^{\infty}(M)\right\\}.$ (26)

For $X\in\mathfrak{X}(M)$ and $f\in C^{\infty}(M)$, $fX\in\mathfrak{X}(M)$ is
defined by $(fX)_{\bm{x}}=f(\bm{x})X_{\bm{x}},\;(\bm{x}\in M)$. Differential
of a function $f$ with respect to a vector field $X$ is denoted as $Xf\in
C^{\infty}(M)$ and defined by $(Xf)(\bm{x})=X_{\bm{x}}(f),\;(\bm{x}\in M)$.
When two vector fields are expressed as $X=v^{i}\partial_{i}$ and
$Y=u^{i}\partial_{i}$, we have

$X(Yf)-Y(Xf)=(v^{j}\partial_{j}u^{i}-u^{j}\partial_{j}v^{i})\partial_{i}f.$
(27)

The commutator product of $X$ and $Y$ is defined as
$[X,Y]\in\mathfrak{X}(M),\;[X,Y]f=(XY-YX)f$, and

$[X,Y]=(v^{j}\partial_{j}u^{i}-u^{j}\partial_{j}v^{i})\partial_{i}.$ (28)

###### Definition 6.2

Consider a map $\nabla:\mathfrak{X}(M)\times\mathfrak{X}(D)\to\mathfrak{X}(M)$
which assigns a pair of vectors $(X,Y)\in\mathfrak{X}(M)\times\mathfrak{X}(M)$
to a vector $\nabla_{Y}X\in\mathfrak{X}(M)$. $\nabla_{Y}X$ is called a
covariant derivative of $X$ with respect to $Y$, and $\nabla$ is called an
affine connection when the following conditions hold for any
$X,Y,Z\in\mathfrak{X}(M)$ and $f\in C^{\infty}(M)$:

  * •

$\nabla_{Y+Z}X=\nabla_{Y}X+\nabla_{Z}X$

  * •

$\nabla_{fX}X=f\nabla_{Y}X$

  * •

$\nabla_{Z}(X+Y)=\nabla_{Z}X+\nabla_{Z}Y$

  * •

$\nabla_{Y}(fX)=(Yf)X+f\nabla_{Y}X$

###### Definition 6.3

Let $\nabla$ be an affine connection on $M$, and define a map

$\displaystyle T:\mathfrak{X}(M)\times\mathfrak{X}(M)\to$
$\displaystyle\mathfrak{X}(M)$ $\displaystyle(X,Y)\mapsto$ $\displaystyle
T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y].$ (29)

The map $T$ is called the torsion tensor field of $\nabla$. When $T=0$ for all
$X,Y\in\mathfrak{X}(M)$, the connection $\nabla$ is called torsion-free.

For an affine connection, the Christoffel symbol $\Gamma^{k}_{ij}\in
C^{\infty}(M)$ is defined by

$\nabla_{\partial_{i}}\partial_{j}=\Gamma^{k}_{ij}\partial_{k}.$ (30)

With this formula, the connection and the Christoffel symbol are often
identified. The affine connection $\nabla$ is torsion-free when and only when
$\Gamma^{k}_{ij}=\Gamma^{k}_{ji}$.

Suppose a manifold $M$ is equipped with a Riemannian metric $g$. When

$Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z)$ (31)

holds for all $X,Y,Z\in\mathfrak{X}(M)$, the connection $\nabla$ is called a
metric connection. In general, an affine connection is not a metric
connection, but there uniquely exists an affine connection $\nabla^{\ast}$
which satisfies

$Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\nabla^{\ast}_{X}Z).$ (32)

The connection $\nabla^{\ast}$ is called the dual connection of $\nabla$.

Given a Riemannian metric $g$, another reperesentation of the Christoffel
symbol is given by

$\Gamma_{ij,k}=g\left(\nabla_{\partial_{i}}\partial_{j},\partial_{k}\right).$
(33)

###### Definition 6.4

When an affine connection $\nabla$ is torsion-free and a metric connection
with respect to the Riemannian metric $g$, it is called a Levi-Civita
connection with respect to the metric $g$.

In general, when a $(0,3)$-tensor $\bar{T}$ is given in addition to an affine
connection $\nabla$ and a Riemannian metric $g$, an alternative connection
$\tilde{\nabla}$ is defined as

$g(\tilde{\nabla}_{Y}X,Z)=g(\nabla_{Y}X,Z)+\bar{T}(X,Y,Z).$ (34)

Let $\Omega$ be a set for which probability measure is defined, and define a
$d$-dimensional statistical model

$S=\\{p(\cdot;\bm{\xi})|\bm{\xi}\in\Xi\\},$ (35)

where the parameter space $\Xi$ is isomorphic to $\mathbb{R}^{d}$. As a
Riemannian metric associated with the statistical model $S$, we consider the
Fisher metric defined as

$g_{ij}(\bm{\xi})=\mathbb{E}_{\bm{\xi}}[(\partial_{i}l_{\bm{\xi}})(\partial_{j}l_{\bm{\xi}})],$
(36)

where $\mathbb{E}_{\bm{\xi}}[\cdot]$ is expectation with respect to a
probability density $p(\cdot;\bm{\xi})$ and $l_{\bm{\xi}}(x)=\log
p(x;\bm{\xi})\;(x\in\Omega)$ is the log-likelihood. Now, consider a
$(0,3)$-tensor $\bar{T}$ on $S$ defined by

$(\bar{T})_{ijk}(\bm{\xi})=\sum_{x\in\Omega}(\partial_{i}l_{\bm{\xi}}(x))(\partial_{j}l_{\bm{\xi}}(x))(\partial_{k}l_{\bm{\xi}}(x))p(x;\bm{\xi}),$
(37)

and based on the Levi-Civita connection $\nabla$ associated with the Fisher
metric $g$ on $S$, we define a affine connection $\nabla^{(\alpha)}$ by

$g(\nabla^{(\alpha)}_{Y}X,Z)=g(\nabla_{Y}X,Z)-\frac{\alpha}{2}\bar{T}(X,Y,Z),\quad(X,Y,Z\in\mathfrak{X}(S)).$
(38)

This connection is called the $\alpha$-connection. The Christoffel symbols
associated with connections $\nabla$ and $\nabla^{(\alpha)}$ are

$\displaystyle\Gamma_{ij,k}=$
$\displaystyle\mathbb{E}_{\bm{\xi}}\left[\left\\{\partial_{i}\partial_{j}l_{\bm{\xi}}+\frac{1}{2}(\partial_{i}l_{\bm{\xi}})(\partial_{j}l_{\bm{\xi}})\right\\}(\partial_{k}l_{\bm{\xi}})\right],$
$\displaystyle\Gamma_{ij,k}^{(\alpha)}=$
$\displaystyle\mathbb{E}_{\bm{\xi}}\left[\left\\{\partial_{i}\partial_{j}l_{\bm{\xi}}+\frac{1-\alpha}{2}(\partial_{i}l_{\bm{\xi}})(\partial_{j}l_{\bm{\xi}})\right\\}(\partial_{k}l_{\bm{\xi}})\right].$

From $\Gamma_{ij,k}^{(\alpha)}=\Gamma_{ji,k}^{(\alpha)}$, the
$\alpha$-connection is torsion-free. Note that the dual connection of
$\nabla^{(\alpha)}$ is $\nabla^{(-\alpha)}$, and it also holds that

$\nabla^{(\alpha)}=\frac{1+\alpha}{2}\nabla^{\ast}+\frac{1-\alpha}{2}\nabla.$
(39)

###### Definition 6.5

For an affine connection $\nabla$ of a manifold $M$, a map

$\displaystyle R:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)\to$
$\displaystyle\mathfrak{X}(D)$ $\displaystyle(X,Y,Z)\mapsto$ $\displaystyle
R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$

is called the curvature tensor field of the connection $\nabla$.

The curvature tensor is expressed with coordinate and the Christoffel symbol
as

$R(\partial_{i},\partial_{j})\partial_{k}=(\partial_{i}\Gamma_{jk}^{l}-\partial_{j}\Gamma^{l}_{ik})\partial_{l}+(\Gamma^{l}_{jk}\Gamma^{m}_{il}-\Gamma^{l}_{ik}\Gamma^{m}_{jl})\partial_{m}.$
(40)

###### Definition 6.6

When both the torsion and curvature are zero, the connection $\nabla$ is said
to be flat.

Let $\gamma$ be a map from a close interval $I$ to a manifold $M$. The map
$\gamma$ is parameterized by a real-valued parameter $t\in I$ as $\gamma(t)$
and called a curve on $M$. When the value of $\gamma$ at two endpoints of $I$
is fixed, the shortest path between these two points is defined by using the
variational principle. The pararell shift of $\frac{d\gamma}{dt}$ along with
$\gamma$ is expressed as

$\nabla_{\frac{d}{dt}}^{\gamma}\frac{d\gamma}{dt}=\left(\frac{d^{2}\gamma_{k}}{dt^{2}}+(\Gamma^{k}_{ij}\circ\gamma)\frac{d\gamma_{i}}{dt}\frac{d\gamma_{j}}{dt}\right)\partial_{k}.$
(41)

###### Definition 6.7

An equation

$\nabla_{\frac{d}{dt}}^{\gamma}\frac{d\gamma}{dt}=\bm{0}$ (42)

is called the geodesic equation, and the curve satisfying this equation is
called a geodesic.

Note that if $\Gamma_{ij,k}=0\;\forall i,j,k$, the geodesic equation is of the
form $\frac{d^{2}\gamma_{k}}{dt^{2}}=0$, hence the geodesic is a straight
line.

###### Definition 6.8

Let $S$ be a $d$-dimensional statistical model. When each element of the model
in $S$ is represented by

$p(x;\bm{\theta})=\exp\left(k(x)+\theta^{i}F_{i}(x)-\psi(\bm{\theta})\right),$
(43)

by using functions $k,F_{1},\dots,F_{d}:\Omega\to\mathbb{R}$ and
$\psi:\Theta\to\mathbb{R}$, the statistical model $S$ is called an exponential
family, and $\bm{\theta}$ is called the natural parameter of the model.

Note that in a general statistical model $S$, $\xi$, and $\Xi$ are often used
as its parameter and the parameter space, while for an exponential family,
$\theta$ and $\Theta$ are often used to represent its parameter and the
parameter space. Consider an exponential family with $\alpha$ connection
$\nabla^{(\alpha)}$. The Christoffel symbols are

$\Gamma^{(\alpha)}_{ij,k}=\mathbb{E}_{\bm{\theta}}\left[\left\\{\partial_{i}\partial_{j}l_{\bm{\theta}}+\frac{1-\alpha}{2}(\partial_{i}l_{\bm{\theta}})(\partial_{j}l_{\bm{\theta}})\right\\}(\partial_{k}l_{\bm{\theta}})\right],$
(44)

and

$\partial_{i}l_{\bm{\theta}}=F_{i}(x)-(\partial_{i}\psi)(\bm{\theta}),\quad(\partial_{i}\partial_{j}\psi)(\bm{\theta}).$
(45)

So, when $\alpha=1$, we have

$\Gamma^{(1)}_{ij,k}=\mathbb{E}_{\bm{\theta}}[(-(\partial_{i}\partial_{j}\psi)(\bm{\theta}))(\partial_{k}l_{\bm{\theta}})]=0,$
(46)

namely, the exponential family is flat with the Fisher metric and $\alpha=1$
connection. This implies that in exponential family, for the
$\alpha=1$-connection $\nabla^{(1)}$ associated with the Fisher metric, the
geodesic between two points correspond to natural parameters $\bm{\theta}_{1}$
and $\bm{\theta}_{2}$ is of the form $t\bm{\theta}_{1}+(1-t)\bm{\theta}_{2}$.

## Appendix B: Proofs of main results

######  Proof 3 (Derivation of the information geometrically generalized
IWERM)

Let $h_{A}$ be a hypothesis generated by AIWERM. From Lemma 4.1, we can write

$\displaystyle\hat{h}_{A}$
$\displaystyle=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}),y)p^{(\lambda)}_{A}(\bm{x})p_{tr}(y|\bm{x})d\bm{x}dy$
$\displaystyle=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}),y)m_{f}^{(\lambda,1)}(p_{tr}(\bm{x}),p_{te}(\bm{x}))p_{tr}(y|\bm{x})d\bm{x}dy$
$\displaystyle=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}),y)\frac{m_{f}^{(\lambda,1)}(p_{tr}(\bm{x}),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}p_{tr}(\bm{x},y)d\bm{x}dy.$
(47)

From Lemma 4.2, we also have

$\displaystyle\hat{h}_{R}=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}\ell(h(\bm{x}),y)\frac{m_{f}^{(\lambda,3)}(p_{tr}(\bm{x}),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}p_{tr}(\bm{x},y)d\bm{x}dy.$
(48)

Then, we consider

$\hat{h}=\min_{h\in\mathcal{H}}\int_{\mathcal{X}\times\mathcal{Y}}w^{(\lambda,\alpha)}(\bm{x})\ell(h(\bm{x}),y)p_{tr}(\bm{x},y)d\bm{x}dy,$
(49)

where

$w^{(\lambda,\alpha)}(\bm{x})=\frac{m^{(\lambda,\alpha)}_{f}(p_{tr}(\bm{x}),p_{te}(\bm{x}))}{p_{tr}(\bm{x})}.$
(50)

We can see that AIWERM is a special case when $\alpha=1$ and RIWERM is a
special case when $\alpha=3$.

######  Proof 4 (Proofs of Propositions 4.2 and 4.3)

Let

$\bm{\theta}^{(\lambda,\alpha)}=m^{(\lambda,\alpha)}_{f}(\bm{\theta}_{tr},\bm{\theta}_{te}),$
(51)

and let $R^{(\alpha)}$ be the Riemann curvature tensor defined in Definition
6.5 with respect to the $\alpha$-connection $\nabla^{(\alpha)}$.

We define the relative curvature tensor as

$R^{(\alpha,\beta)}(X,Y,Z)=\Big{[}\nabla^{(\alpha)}_{X},\nabla^{(\beta)}_{Y}\Big{]}Z-\nabla^{(\alpha)}_{[X,Y]}Z$
(52)

and the difference tensor as

$K(X,Y)=\nabla^{*}_{X}Y-\nabla_{X}Y.$ (53)

For any $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$, we have

$\displaystyle\nabla^{(\alpha)}_{X}\nabla^{(\beta)}_{Y}Z$
$\displaystyle=\Big{(}\frac{1+\alpha}{2}\nabla^{*}_{X}+\frac{1-\alpha}{2}\nabla_{X}\Big{)}\Big{(}\frac{1+\beta}{2}\nabla^{*}_{Y}+\frac{1-\beta}{2}\nabla_{Y}\Big{)}Z$
$\displaystyle=\frac{(1+\alpha)(1+\beta)}{4}\nabla^{*}_{X}\nabla^{*}_{Y}Z+\frac{(1+\alpha)(1-\beta)}{4}\nabla^{*}_{X}\nabla_{Y}Z$
$\displaystyle\
\quad+\frac{(1-\alpha)(1+\beta)}{4}\nabla_{X}\nabla^{*}_{Y}Z+\frac{(1-\alpha)(1-\beta)}{4}\nabla_{X}\nabla_{Y}Z.$
(54) $\displaystyle\nabla^{(\beta)}_{Y}\nabla^{(\alpha)}_{X}$
$\displaystyle=\Big{(}\frac{1+\beta}{2}\nabla^{*}_{Y}+\frac{1-\beta}{2}\nabla_{Y}\Big{)}\Big{(}\frac{1+\alpha}{2}\nabla^{*}_{X}+\frac{1-\alpha}{2}\nabla_{X}\Big{)}Z$
$\displaystyle=\frac{(1+\alpha)(1+\beta)}{4}\nabla^{*}_{Y}\nabla^{*}_{X}Z+\frac{(1-\alpha)(1+\beta)}{4}\nabla^{*}_{Y}\nabla_{X}Z$
$\displaystyle\
\quad+\frac{(1+\alpha)(1-\beta)}{4}\nabla_{Y}\nabla^{*}_{X}Z+\frac{(1-\alpha)(1-\beta)}{4}\nabla_{Y}\nabla_{X}Z.$
(55) $\displaystyle\nabla^{(\alpha)}_{[X,Y]}Z$
$\displaystyle=\frac{1+\alpha}{2}\nabla^{*}_{[X,Y]}Z+\frac{1-\alpha}{2}\nabla_{[X,Y]}Z.$
(56)

Then

$\displaystyle R^{(\alpha,\beta)}(X,Y,Z)$
$\displaystyle=\nabla^{(\alpha)}_{X}\nabla^{(\beta)}_{Y}Z-\nabla^{(\beta)}_{X}\nabla^{(\alpha)}_{Y}Z-\nabla^{(\alpha)}_{[X,Y]}Z$
$\displaystyle=\frac{(1+\alpha)(1+\beta)}{4}(\nabla^{*}_{X}\nabla^{*}_{Y}-\nabla^{*}_{Y}\nabla^{*}_{X})Z$
$\displaystyle\
\quad+\frac{(1+\alpha)(1-\beta)}{4}(\nabla^{*}_{X}\nabla_{Y}-\nabla_{Y}\nabla^{*}_{X})Z$
$\displaystyle\
\quad+\frac{(1-\alpha)(1+\beta)}{4}(\nabla_{X}\nabla^{*}_{Y}-\nabla^{*}_{Y}\nabla_{X})Z$
$\displaystyle\
\quad+\frac{(1-\alpha)(1-\beta)}{4}(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X})Z$
$\displaystyle\
\quad-\frac{1+\alpha}{2}\nabla^{*}_{[X,Y]}Z-\frac{1-\alpha}{2}\nabla_{[X,Y]}Z$
$\displaystyle=\frac{(1+\alpha)(1+\beta)}{4}\Big{\\{}R^{*}(X,Y,Z)+\nabla^{*}_{[X,Y]}Z\Big{\\}}$
$\displaystyle\
\quad+\frac{(1+\alpha)(1-\beta)}{4}\Big{\\{}R^{(1,-1)}(X,Y,Z)+\nabla^{*}_{[X,Y]}Z\Big{\\}}$
$\displaystyle\
\quad+\frac{(1-\alpha)(1+\beta)}{4}\Big{\\{}R^{(-1,1)}(X,Y,Z)+\nabla_{[X,Y]}Z\Big{\\}}$
$\displaystyle\
\quad+\frac{(1-\alpha)(1-\beta)}{4}\Big{\\{}R^{(-1,-1)}(X,Y,Z)+\nabla^{*}_{[X,Y]}Z\Big{\\}}$
$\displaystyle\
\quad-\frac{1+\alpha}{2}\nabla^{*}_{[X,Y]}Z-\frac{1-\alpha}{2}\nabla_{[X,Y]}Z$
(57) $\displaystyle 4R^{(\alpha,\beta)}$
$\displaystyle=(1+\alpha)(1+\beta)R^{*}+(1-\alpha)(1-\beta)R$ $\displaystyle\
\quad+(1+\alpha)(1-\beta)R^{(1,-1)}+(1-\alpha)(1+\beta)R^{(-1,1)}.$ (58)

We also have

$\displaystyle K^{(\alpha,\beta)}(X,Y)$
$\displaystyle=\nabla^{(\beta)}_{X}Y-\nabla^{(\alpha)}_{X}Y$
$\displaystyle=\Big{\\{}\frac{1+\beta}{2}\nabla^{*}_{X}Y+\frac{1-\beta}{2}\nabla_{X}Y\Big{\\}}-\Big{\\{}\frac{1+\alpha}{2}\nabla^{*}_{X}Y+\frac{1-\alpha}{2}\nabla_{X}Y\Big{\\}}$
$\displaystyle=\frac{\beta-\alpha}{2}\nabla^{*}_{X}Y-\frac{\beta-\alpha}{2}\nabla_{X}Y=\frac{\beta-\alpha}{2}K(X,Y)$
(59) $\displaystyle K^{(\alpha,\beta)}\Big{(}X,K^{(\alpha,\beta)}(Y,Z)\Big{)}$
$\displaystyle=\frac{\beta-\alpha}{2}K\Big{(}X,K^{(\alpha,\beta)}(Y,Z)\Big{)}=\frac{(\beta-\alpha)^{2}}{4}K\Big{(}X,K(Y,Z)\Big{)}.$
(60)

Combining them, the following relations hold:

$\displaystyle K^{(\beta,\alpha)}\Big{(}X,K^{(\beta,\alpha)}(Y,Z)\Big{)}$
$\displaystyle=K^{(\beta,\alpha)}\Big{(}X,\nabla^{(\alpha)}_{Y}Z-\nabla^{(\beta)}_{Y}Z\Big{)}$
(61)
$\displaystyle=K^{(\beta,\alpha)}\Big{(}X,\nabla^{(\alpha)}_{Y}Z\Big{)}-K^{(\beta,\alpha)}\Big{(}X,\nabla^{(\beta)}_{Y}Z\Big{)}$
$\displaystyle=\nabla^{(\alpha)}_{X}\nabla^{(\alpha)}_{Y}Z-\nabla^{(\beta)}_{X}\nabla^{(\alpha)}_{Y}Z-\nabla^{(\alpha)}_{X}\nabla^{(\beta)}_{Y}Z+\nabla^{(\beta)}_{X}\nabla^{(\beta)}_{Y}Z$
(62) $\displaystyle\frac{(\alpha-\beta)^{2}}{4}K\Big{(}X,K(Y,Z)\Big{)}$
$\displaystyle=\nabla^{(\alpha)}_{X}\nabla^{(\alpha)}_{Y}Z-\nabla^{(\beta)}_{X}\nabla^{(\alpha)}_{Y}Z-\nabla^{(\alpha)}_{X}\nabla^{(\beta)}_{Y}Z+\nabla^{(\beta)}_{X}\nabla^{(\beta)}_{Y}Z.$
(63)

Swapping $X$ and $Y$, we have

$\displaystyle\
\frac{(\alpha-\beta)^{2}}{4}\Biggl{\\{}K\Big{(}X,K(Y,Z)\Big{)}-K\Big{(}Y,K(X,Z)\Big{)}\Biggr{\\}}$
$\displaystyle\
=R^{(\alpha)}(X,Y,Z)+R^{(\beta)}(X,Y,Z)-\Big{\\{}\Big{[}\nabla^{(\alpha)}_{X},\nabla^{(\beta)}_{Y}\Big{]}Z-\nabla^{(\alpha)}_{[X,Y]}Z\Big{\\}}$
$\displaystyle\
\quad-\Big{\\{}\Big{[}\nabla^{(\beta)}_{X},\nabla^{(\alpha)}_{Y}\Big{]}Z-\nabla^{(\beta)}_{[X,Y]}Z\Big{\\}}$
$\displaystyle=R^{(\alpha)}(X,Y,Z)+R^{(\beta)}(X,Y,Z)-R^{(\alpha,\beta)}(X,Y,Z)-R^{(\beta,\alpha)}(X,Y,Z).$
(64)

Making $\alpha=\beta$, we have

$\displaystyle 4R^{(\alpha)}$
$\displaystyle=(1+\alpha)^{2}R^{*}+(1-\alpha)^{2}R+(1-\alpha^{2})R^{(1,-1)}+(1-\alpha^{2})R^{(-1,1)}$
$\displaystyle=(1+\alpha^{2})R^{*}+(1-\alpha)^{2}R+(1-\alpha^{2})\Big{(}R^{(1,-1)}+R^{(-1,1)}\Big{)}.$
(65)

Making $\alpha=1$ and $\beta=-1$, we also have

$\displaystyle R^{(1,-1)}(X,Y,Z)+R^{(-1,1)}(X,Y,Z)$
$\displaystyle=R^{*}(X,Y,Z)+R(X,Y,Z)$ $\displaystyle\
\quad-\Biggl{\\{}K\Big{(}X,K(Y,Z)\Big{)}-K\Big{(}Y,K(X,Z)\Big{)}\Biggr{\\}}.$
(66)

From Eq. (65) and (66), we obtain

$\displaystyle 4R^{(\alpha)}$
$\displaystyle=(1+\alpha)^{2}R^{*}(X,Y,Z)+(1-\alpha)^{2}R(X,Y,Z)$
$\displaystyle\
\quad+(1-\alpha^{2})R^{*}(X,Y,Z)+(1-\alpha^{2})\Biggl{\\{}K\Big{(}X,K(Y,Z)\Big{)}-K\Big{(}Y,K(X,Z)\Big{)}\Biggr{\\}}$
$\displaystyle=2(1+\alpha)R^{*}(X,Y,Z)+2(1-\alpha)R(X,Y,Z)$ $\displaystyle\
\quad+(1-\alpha^{2})\Biggl{\\{}K\Big{(}Y,K(X,Z)\Big{)}-K\Big{(}X,K(Y,Z)\Big{)}\Biggl{\\}}$
(67)

Since the exponential family is dually flat, that is $R=0$ and $R^{*}=0$, and
the Riemann curvature tensor with respect to $\nabla^{(\alpha)}$ is

$\displaystyle R^{(\alpha)}(X,Y,Z)$
$\displaystyle=\frac{1-\alpha^{2}}{4}\Lambda,$ (68) $\displaystyle\Lambda$
$\displaystyle=\Big{(}K(Y,K(X,Z))-K(X,K(Y,Z))\Big{)}.$ (69)

Then, the geometric bias vector of $\bm{\theta}^{(\lambda,\alpha)}$ is

$\displaystyle
b(\alpha,\lambda)=(1-\lambda)\Big{\\{}e_{1}+tr_{g}\Big{(}\frac{1-\alpha^{2}}{2}\Lambda_{ikj}d\bm{\theta}^{i}\otimes
d\bm{\theta}^{j}\Big{)}e_{2}\Big{\\}},$ (70)

where $tr_{g}$ is the trace operation on the metric tensor $g$, and
$\Lambda_{ikj}$ is the element of $\Lambda$ in Eq. (69). Since AIWERM and
RIWERM are two special cases for $\alpha=1$ and $\alpha=3$, we have

$\displaystyle b(1,\lambda)$ $\displaystyle=(1-\lambda)e_{1},$ (71)
$\displaystyle b(3,\lambda)$
$\displaystyle=(1-\lambda)\Big{\\{}e_{1}+tr_{g}\Big{(}-4\Lambda_{ikj}d\bm{\theta}^{i}\otimes
d\bm{\theta}^{j}\Big{)}e_{2}\Big{\\}}.$ (72)

## Appendix C: Learning guarantee

Generalization bounds of weighted maximum likelihood estimator for the target
domain are derived in Cortes et al., (2010), and our weight function (14) is
compatible with their bound.

Then, the gap between the expected (with respect to test distribution) loss
$\mathcal{R}(h)$ and empirical risk $L(h;\lambda,\alpha)$ is bounded as

$\displaystyle|\mathcal{R}(h)-L(h;\lambda,\alpha)|\leq\left|\mathbb{E}_{p_{tr}}\left[\left\\{\frac{p_{te}(\bm{x})}{p_{tr}(\bm{x})}-w^{(\lambda,\alpha)}(\bm{x})\right\\}\right]\right|\ell(h(\bm{x},y(\bm{x})))$
$\displaystyle+$ $\displaystyle
2^{5/4}\max\left(\sqrt{\mathbb{E}_{p_{tr}}(w^{(\lambda,\alpha)}(\bm{x}))^{2}\ell^{2}(h(\bm{x},y(\bm{x})))},\sqrt{\mathbb{E}_{\hat{p}_{tr}}(w^{(\lambda,\alpha)}(\bm{x}))^{2}\ell^{2}(h(\bm{x},y(\bm{x}))}\right)$
$\displaystyle\times$
$\displaystyle\left(\frac{p\log\frac{2n_{tr}e}{p}+\log\frac{4}{\delta}}{n_{tr}}\right)^{\frac{3}{8}}.$
(73)

In the above inequality, $p$ is the pseudo-dimension of the function class
$\\{w^{\lambda,\alpha}(\bm{x})\ell(h(\bm{x}),y(\bm{x}))|h\in\mathcal{H}\\}$
where $y(\bm{x})$ is the ground truth function of connecting $\bm{x}$ and $y$
as $y=y(\bm{x})$. The first term of the r.h.s. of the above inequality is the
bias introduced by using $w^{\lambda,\alpha}$ instead of a standard density
ratio, and the second term reflects the variance. It is worth mentioning that
the term
$\mathbb{E}_{p_{tr}}(w^{(\lambda,\alpha)}(\bm{x}))^{2}\ell^{2}(h(\bm{x},y(\bm{x})))$
is further bounded by
$d_{2}(p_{te}||p_{tr})=\int_{x\in\mathcal{X}}\frac{p_{te}^{2}(\bm{x})}{p_{tr}(\bm{x})}d\bm{x}$.

## Appendix D: Optimization of the generalized IWERM

In the expected improvement strategy, the $t+1^{th}$ point
$(\lambda,\alpha)_{t+1}$ is selected according to the following equation.

$(\lambda,\alpha)_{t+1}=\mathop{\rm
arg~{}min}\limits_{(\lambda,\alpha)}\mathbb{E}\Big{[}\max\Big{(}0,h_{t+1}(\lambda,\alpha)-L(\lambda^{\dagger},\alpha^{\dagger})\Big{)}\Big{|}D_{t}\Big{]},$

where $L(\lambda^{\dagger},\alpha^{\dagger})$ is the maximum value of
empirical risk that has been encountered so far, $h_{t+1}(\lambda,\alpha)$ is
the posterior mean of the surrogate at the $t+1^{th}$ step and
$D_{t}=\\{(\lambda,\alpha)_{i},L(\lambda_{i},\alpha_{i})\\}^{t}_{i=1}$. This
equation for Gaussian process surrogate is an analytical expression:

$\displaystyle a_{EI}(\lambda,\alpha)$
$\displaystyle=(\mu_{t}(\lambda,\alpha)-L(\lambda^{\dagger},\alpha^{\dagger}))\Phi(Z)+\sigma_{t}(\lambda,\alpha)\phi(Z),$
$\displaystyle Z$
$\displaystyle=\frac{\mu_{t}(\lambda,\alpha)-L(\lambda^{\dagger},\alpha^{\dagger})}{\sigma_{t}(\lambda,\alpha)},$

where $\Phi(\cdot)$ and $\phi(\cdot)$ are normal cumulative and density
functions, respectively, and $\mu_{t}$ and $\sigma_{t}$ are mean and standard
deviation of $\\{(\lambda,\alpha)_{i}\\}^{t}_{i=1}$.

## Appendix E: Additional experimental results

###  6.1 Experimental results on LIBSVM dataset

We show that for the LIBSVM dataset, IGIWERM is effective even for multiple
models. Table 4 shows the results for each model. We use the scikit-learn
Pedregosa et al., (2011) implementation of the models, and the hyperparameters
of each model are the default values of this library. Figure 5 also shows the
relationship between the two parameters of IGIWERM and the errors that can be
achieved. For this visualization, we explore the parameter pairs by grid
search and evaluate their performance at that time. From this figure, it is
seen that the best performance is often achieved when $\alpha\neq 1$ and
$\alpha\neq 3$, showing the sub-optimality of conventional methods.

###  6.2 Experimental results on regression problems

In this section, we present experimental results for the regression problem.
All the datasets used in this experiment are available in the scikit-learn
Pedregosa et al., (2011) dataset collection. We use SVR with Radial Basis
Function (RBF) kernel as the base regressor. Table 5 shows the results of this
experiment. In this experiment, we use MSE as a metric, and this table shows
that IGIWERM is superior to existing covariate shift adaptation methods.
Figure 6 shows the relationship between the two parameters of IGIWERM and the
mean squared errors that can be achieved. This figure shows that, as in the
case of binary classification, the optimal parameters do not necessarily match
those of existing methods.

###  6.3 Experimental results on multi-class classification

We also introduce the additional experimental results for the multi-class
classification problem. All the datasets used in this experiment are available
in the scikit-learn Pedregosa et al., (2011) dataset collection. We also use
the scikit-learn Pedregosa et al., (2011) implementation of the models, and
the hyperparameters of each model are the default values of this library. We
note that the number of training sample for covtype is so large hence the
results with SVM for this dataset are omitted. Table 6 shows the experimental
results, and we see that our proposed generalization outperforms existing
methods. Figure 7 shows the relationship between the two parameters of IGIWERM
and the errors that can be achieved. This figure also shows the sub-optimality
of conventional methods.

###  6.4 Visualization of covariate shift

In Section 5, we induce the covariate shift by the method of Cortes et al
Cortes et al., (2008). Figure 8 shows a plot by PCA of each dataset splitted
into the training set and test set. This figure shows that we are able to
induce a covariate shift by partitioning the dataset.

Table 4: Mean misclassification rates averaged over 10 trails on LIBSVM
benchmark datasets. The numbers in the brackets are the standard deviations.
For the methods with (optimal), the optimal parameters for the test data are
obtained by the linear search. The lowest misclassification rates among five
methods are shown with bold.

Dataset | model | unweighted | IWERM | AIWERM (optimal) | RIWERM (optimal) | ours
---|---|---|---|---|---|---
australian | Logistic Regression | $22.76(\pm 9.53)$ | $22.75(\pm 7.73)$ | $22.19(\pm 9.35)$ | $22.39(\pm 7.76)$ | ${\bf 21.47(\pm 8.91)}$
| SVM | $33.46(\pm 23.65)$ | $22.13(\pm 3.37)$ | $21.98(\pm 3.36)$ | $21.73(\pm 3.82)$ | ${\bf 18.85(\pm 3.99)}$
| AdaBoost | $10.20(\pm 5.09)$ | $16.35(\pm 10.67)$ | $11.23(\pm 3.49)$ | $15.47(\pm 2.72)$ | ${\bf 9.15(\pm 3.93)}$
| Naive Bayes | $17.04(\pm 6.02)$ | $14.92(\pm 5.96)$ | $16.33(\pm 6.07)$ | $15.49(\pm 6.07)$ | ${\bf 14.74(\pm 5.84)}$
| Random Forest | $8.89(\pm 3.72)$ | $8.64(\pm 3.20)$ | ${\bf 8.29(\pm 3.21)}$ | $8.38(\pm 3.47)$ | $8.61(\pm 3.73)$
breast-cancer | Logistic Regression | $32.32(\pm 3.08)$ | $32.87(\pm 3.56)$ | $32.32(\pm 3.08)$ | $32.32(\pm 3.08)$ | ${\bf 32.26(\pm 3.04)}$
| SVM | $38.28(\pm 10.98)$ | $41.23(\pm 15.39)$ | $36.41(\pm 9.68)$ | $36.13(\pm 10.81)$ | ${\bf 31.65(\pm 8.49)}$
| AdaBoost | $5.09(\pm 1.65)$ | $5.63(\pm 1.38)$ | $6.01(\pm 1.13)$ | $5.95(\pm 1.50)$ | ${\bf 4.84(\pm 1.50)}$
| Naive Bayes | $11.27(\pm 4.09)$ | $19.65(\pm 15.14)$ | $10.36(\pm 5.14)$ | $18.00(\pm 14.24)$ | ${\bf 10.03(\pm 3.53)}$
| Random Forest | $3.32(\pm 1.23)$ | $3.19(\pm 1.10)$ | $3.19(\pm 1.13)$ | $3.18(\pm 1.04)$ | ${\bf 3.13(\pm 1.04)}$
heart | Logistic Regression | $39.68(\pm 7.90)$ | $40.55(\pm 9.10)$ | $39.96(\pm 7.40)$ | $39.94(\pm 6.93)$ | ${\bf 36.56(\pm 8.29)}$
| SVM | $45.17(\pm 6.98)$ | $39.94(\pm 8.55)$ | $39.76(\pm 8.49)$ | $39.74(\pm 8.92)$ | ${\bf 35.37(\pm 6.84)}$
| AdaBoost | $30.87(\pm 12.04)$ | $29.24(\pm 6.47)$ | $29.37(\pm 12.19)$ | $31.27(\pm 8.37)$ | ${\bf 26.96(\pm 13.12)}$
| Naive Bayes | $22.79(\pm 6.17)$ | $24.78(\pm 7.99)$ | $22.87(\pm 6.02)$ | $24.58(\pm 7.98)$ | ${\bf 21.97(\pm 6.41)}$
| Random Forest | $20.87(\pm 6.64)$ | $20.98(\pm 6.62$ | $20.96(\pm 6.67)$ | $21.96(\pm 6.70)$ | ${\bf 19.95(\pm 6.61)}$
diabetes | Logistic Regression | $37.62(\pm 4.35)$ | $40.22(\pm 4.10)$ | $38.38(\pm 3.85)$ | $40.11(\pm 3.74)$ | ${\bf 36.86(\pm 4.81)}$
| SVM | $33.19(\pm 5.69)$ | $37.22(\pm 6.63)$ | $33.11(\pm 6.45)$ | $33.38(\pm 5.74)$ | ${\bf 32.83(\pm 5.62)}$
| AdaBoost | $37.69(\pm 4.28)$ | $40.13(\pm 5.28)$ | $40.76(\pm 4.31)$ | $41.26(\pm 5.16)$ | ${\bf 33.45(\pm 4.35)}$
| Naive Bayes | $39.29(\pm 3.98)$ | $39.21(\pm 3.18)$ | $39.26(\pm 2.97)$ | $39.35(\pm 2.85)$ | ${\bf 38.10(\pm 4.02)}$
| Random Forest | $30.09(\pm 3.03)$ | $30.90(\pm 3.52)$ | $31.07(\pm 3.10)$ | $30.51(\pm 3.67)$ | ${\bf 29.46(\pm 2.99)}$
madelon | Logistic Regression | $47.31(\pm 1.80)$ | $47.80(\pm 1.57)$ | $47.16(\pm 1.68)$ | $46.81(\pm 1.56)$ | ${\bf 46.31(\pm 1.69)}$
| SVM | $47.78(\pm 1.53)$ | $47.28(\pm 2.20)$ | $47.10(\pm 2.13)$ | $47.12(\pm 1.65)$ | ${\bf 46.56(\pm 2.12)}$
| AdaBoost | $42.92(\pm 1.40)$ | $42.91(\pm 1.68)$ | $43.36(\pm 1.81)$ | $42.90(\pm 1.40)$ | ${\bf 40.64(\pm 7.32)}$
| Naive Bayes | $41.90(\pm 1.05)$ | $41.43(\pm 8.76)$ | $41.79(\pm 8.05)$ | $41.62(\pm 7.43)$ | ${\bf 41.03(\pm 8.32)}$
| Random Forest | $35.90(\pm 0.83)$ | $35.42(\pm 1.75)$ | $35.13(\pm 1.30)$ | ${\bf 34.79(\pm 1.75)}$ | $35.56(\pm 2.03)$

Figure 5: Visualization of grid search for $\alpha$ and $\lambda$ on LIBSVM
dataset. For the sake of clarity, we apply a moving average. Table 5: Mean
squared errors averaged over 10 trails on scikit-learn Pedregosa et al.,
(2011) regression benchmark datasets. The numbers in the brackets are the
standard deviations. For the methods with (optimal), the optimal parameters
for the test data are obtained by the linear search. The lowest mean squared
errors are shown with bold.

Dataset | #features | #data | unweighted | IWERM | AIWERM (optimal) | RIWERM (optimal) | ours
---|---|---|---|---|---|---|---
boston | $13$ | $506$ | $83.22(\pm 5.72)$ | $69.87(\pm 2.31)$ | $69.68(\pm 1.46)$ | $69.96(\pm 1.84)$ | ${\bf 68.36(\pm 1.20)}$
diabetes | $10$ | $442$ | $0.049(\pm 0.007)$ | $0.0501(\pm 0.009)$ | $0.049(\pm 0.008)$ | $0.049(\pm 0.009)$ | ${\bf 0.048(\pm 0.007)}$
california housing | $8$ | $20,640$ | $1.432(\pm 0.095)$ | $1.3214(\pm 0.345)$ | $1.260(\pm 0.125)$ | $1.261(\pm 0.086)$ | ${\bf 1.232(\pm 0.095)}$

Figure 6: Visualization of grid search for $\alpha$ and $\lambda$ on scikit-
learn regression dataset. For the sake of visualization, we apply a moving
average. Table 6: Mean misclassification rates averaged over 10 trails on
scikit-learn Pedregosa et al., (2011) multi-class classification benchmark
datasets. The numbers in the brackets are the standard deviations. For the
methods with (optimal), the optimal parameters for the test data are obtained
by the linear search. The lowest misclassification rates among five methods
are shown with bold.

Dataset | model | unweighted | IWERM | AIWERM (optimal) | RIWERM (optimal) | ours
---|---|---|---|---|---|---
digits | Logistic Regression | $6.92(\pm 2.25)$ | $7.10(\pm 1.76)$ | $6.90(\pm 1.81)$ | $6.89(\pm 1.80)$ | ${\bf 6.79(\pm 1.82)}$
| SVM | $4.21(\pm 2.04)$ | $3.94(\pm 1.14)$ | $4.20(\pm 1.22)$ | $4.02(\pm 1.18)$ | ${\bf 3.88(\pm 1.11)}$
| AdaBoost | $67.98(\pm 12.35)$ | $71.77(\pm 6.25)$ | $71.26(\pm 8.21)$ | $70.47(\pm 6.46)$ | ${\bf 65.20(\pm 8.20)}$
| Naive Bayes | $19.07(\pm 2.45)$ | $18.68(\pm 2.61)$ | $19.31(\pm 2.64)$ | $18.78(\pm 2.68)$ | ${\bf 18.58(\pm 2.66)}$
| Random Forest | $6.85(\pm 2.40)$ | $6.30(\pm 1.80)$ | $6.23(\pm 1.53)$ | $6.27(\pm 1.64)$ | ${\bf 6.17(\pm 1.90)}$
iris | Logistic Regression | $54.69(\pm 20.51)$ | $36.49(\pm 22.39)$ | $45.78(\pm 18.09)$ | $35.37(\pm 23.19)$ | ${\bf 28.89(\pm 20.54)}$
| SVM | $55.16(\pm 22.60)$ | $36.65(\pm 22.56)$ | $33.21(\pm 20.25)$ | $30.46(\pm 21.14)$ | ${\bf 29.04(\pm 20.02)}$
| AdaBoost | $27.19(\pm 22.56)$ | $26.00(\pm 22.98)$ | $26.00(\pm 22.98)$ | $26.00(\pm 22.98)$ | ${\bf 19.62(\pm 21.36)}$
| Naive Bayes | $35.88(\pm 23.24)$ | $35.97(\pm 26.79)$ | $37.99(\pm 23.55)$ | $33.84(\pm 25.64)$ | ${\bf 27.52(\pm 21.00)}$
| Random Forest | $26.16(\pm 22.86)$ | $32.17(\pm 21.21)$ | $26.00(\pm 22.98)$ | $28.47(\pm 22.63)$ | ${\bf 22.77(\pm 21.74)}$
covtype | Logistic Regression | $45.36(\pm 13.08)$ | $32.04(\pm 5.833)$ | $30.62(\pm 4.64)$ | $25.20(\pm 8.99)$ | ${\bf 19.99(\pm 5.56)}$
| SVM | $-$ | $-$ | $-$ | $-$ | $-$
| AdaBoost | $47.53(\pm 14.51)$ | $25.55(\pm 12.27)$ | $25.47(\pm 14.14)$ | $27.86(\pm 10.37)$ | ${\bf 18.96(\pm 7.29)}$
| Naive Bayes | $41.13(\pm 15.65)$ | $30.66(\pm 15.09)$ | $28.48(\pm 15.66)$ | $27.20(\pm 15.79)$ | ${\bf 19.64(\pm 15.62)}$
| Random Forest | $23.51(\pm 3.31)$ | $18.18(\pm 2.01)$ | $17.28(\pm 2.05)$ | $17.13(\pm 2.26)$ | ${\bf 16.42(\pm 2.08)}$

Figure 7: Visualization of grid search for $\alpha$ and $\lambda$ on scikit-
learn multi-class classification dataset. For the sake of visualization, we
apply a moving average. Figure 8: Plot of covariate shifts using the method of
Cortes et al Cortes et al., (2008). Each dataset is included in LIBSVM and
mapped to two dimensions by PCA.

## References

  * Amari, (1985) Amari, S. (1985).  Differential-geometrical methods in statistics.  Lecture Notes on Statistics, 28:1.
  * Amari, (2009) Amari, S. (2009).  $\alpha$ -divergence is unique, belonging to both $f$-Divergence and bregman divergence classes.  IEEE Trans. Inf. Theory, 55(11):4925–4931.
  * Amari, (2016) Amari, S. (2016).  Information Geometry and Its Applications.  Springer.
  * Amari and Nagaoka, (2007) Amari, S. and Nagaoka, H. (2007).  Methods of Information Geometry.  American Mathematical Soc.
  * Arpit et al., (2016) Arpit, D., Zhou, Y., Kota, B. U., and Govindaraju, V. (2016).  Normalization propagation: A parametric technique for removing internal covariate shift in deep networks.
  * Awais et al., (2020) Awais, M., Iqbal, M. T. B., and Bae, S.-H. (2020).  Revisiting internal covariate shift for batch normalization.  IEEE Trans Neural Netw Learn Syst, PP.
  * Bickel et al., (2009) Bickel, S., Brückner, M., and Scheffer, T. (2009).  Discriminative learning under covariate shift.  J. Mach. Learn. Res., 10(9).
  * Bishop, (1995) Bishop, C. (1995).  Neural Networks for Pattern Recognition.  Clarendon Press.
  * Cortes et al., (2010) Cortes, C., Mansour, Y., and Mohri, M. (2010).  Learning bounds for importance weighting.  In Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010, NIPS 2010.
  * Cortes et al., (2008) Cortes, C., Mohri, M., Riley, M., and Rostamizadeh, A. (2008).  Sample selection bias correction theory.  In Algorithmic Learning Theory, pages 38–53. Springer Berlin Heidelberg.
  * Duda et al., (2006) Duda, R. O., Hart, P. E., and Others (2006).  Pattern classification.  John Wiley & Sons.
  * Fang et al., (2020) Fang, T., Lu, N., Niu, G., and Sugiyama, M. (2020).  Rethinking importance weighting for deep learning under distribution shift.  In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual.
  * Frazier, (2018) Frazier, P. I. (2018).  A tutorial on Bayesian optimization.
  * Hachiya et al., (2012) Hachiya, H., Sugiyama, M., and Ueda, N. (2012).  Importance-weighted least-squares probabilistic classifier for covariate shift adaptation with application to human activity recognition.  Neurocomputing, 80:93–101.
  * Hardy et al., (1952) Hardy, G. H., Littlewood, J. E., and Pólya, G. (1952).  Inequalities. By GH Hardy, JE Littlewood, G. Pólya.. University Press.
  * Hassan et al., (2013) Hassan, A., Damper, R., and Niranjan, M. (2013).  On acoustic emotion recognition: Compensating for covariate shift.  IEEE Trans. Audio Speech Lang. Processing, 21(7):1458–1468.
  * Hastie et al., (2009) Hastie, T., Tibshirani, R., and Friedman, J. (2009).  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition.  Springer Science & Business Media.
  * Huang et al., (2007) Huang, J., Gretton, A., Borgwardt, K., Schölkopf, B., and Smola, A. J. (2007).  Correcting sample selection bias by unlabeled data.  In Schölkopf, B., Platt, J. C., and Hoffman, T., editors, Advances in Neural Information Processing Systems 19, pages 601–608. MIT Press.
  * Huang and Yu, (2020) Huang, Y. and Yu, Y. (2020).  An internal covariate shift bounding algorithm for deep neural networks by unitizing layers’ outputs.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8465–8473. openaccess.thecvf.com.
  * Ioffe and Szegedy, (2015) Ioffe, S. and Szegedy, C. (2015).  Batch normalization: Accelerating deep network training by reducing internal covariate shift.
  * Jiang, (2008) Jiang, J. (2008).  A literature survey on domain adaptation of statistical classifiers.  URL: http://sifaka. cs. uiuc. edu/jiang4/domainadaptation/survey, 3:1–12.
  * Jirayucharoensak et al., (2014) Jirayucharoensak, S., Pan-Ngum, S., and Israsena, P. (2014).  EEG-based emotion recognition using deep learning network with principal component based covariate shift adaptation.  ScientificWorldJournal, 2014:627892.
  * Jones et al., (1998) Jones, D. R., Schonlau, M., and Welch, W. J. (1998).  Efficient global optimization of expensive black-box functions.  J. Global Optimiz., 13(4):455–492.
  * Jost, (2017) Jost, J. (2017).  Riemannian Geometry and Geometric Analysis.  Springer, Cham.
  * Kimura and Hino, (2021) Kimura, M. and Hino, H. (2021).  $\alpha$-geodesical skew divergence.  Entropy, 23(5).
  * Li et al., (2010) Li, Y., Kambara, H., Koike, Y., and Sugiyama, M. (2010).  Application of covariate shift adaptation techniques in Brain–Computer interfaces.  IEEE Transactions on Biomedical Engineering, 57(6):1318–1324.
  * Mockus et al., (1978) Mockus, J., Tiesis, V., and Zilinskas, A. (1978).  The application of Bayesian methods for seeking the extremum.  Towards global optimization, 2(117-129):2.
  * Mohri et al., (2018) Mohri, M., Rostamizadeh, A., and Talwalkar, A. (2018).  Foundations of Machine Learning, second edition.  MIT Press.
  * Nado et al., (2020) Nado, Z., Padhy, S., Sculley, D., D’Amour, A., Lakshminarayanan, B., and Snoek, J. (2020).  Evaluating Prediction-Time batch normalization for robustness under covariate shift.
  * Pedregosa et al., (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E. (2011).  Scikit-learn: Machine learning in Python.  Journal of Machine Learning Research, 12:2825–2830.
  * Quionero-Candela et al., (2009) Quionero-Candela, J., Sugiyama, M., Schwaighofer, A., and Lawrence, N. D. (2009).  Dataset Shift in Machine Learning.  The MIT Press.
  * Raza et al., (2016) Raza, H., Cecotti, H., Li, Y., and Prasad, G. (2016).  Adaptive learning with covariate shift-detection for motor imagery-based brain–computer interface.  Soft Computing, 20(8):3085–3096.
  * Sakai and Shimizu, (2019) Sakai, T. and Shimizu, N. (2019).  Covariate shift adaptation on learning from positive and unlabeled data.  In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, pages 4838–4845.
  * Santurkar et al., (2018) Santurkar, S., Tsipras, D., Ilyas, A., and Madry, A. (2018).  How does batch normalization help optimization?(no, it is not about internal covariate shift).  arXiv preprint arXiv:1805. 11604.
  * Shimodaira, (2000) Shimodaira, H. (2000).  Improving predictive inference under covariate shift by weighting the log-likelihood function.  J. Stat. Plan. Inference, 90(2):227–244.
  * Snoek et al., (2012) Snoek, J., Larochelle, H., and Adams, R. P. (2012).  Practical Bayesian optimization of machine learning algorithms.  In Pereira, F., Burges, C. J. C., Bottou, L., and Weinberger, K. Q., editors, Advances in Neural Information Processing Systems, volume 25, pages 2951–2959. Curran Associates, Inc.
  * Sugiyama et al., (2007) Sugiyama, M., Krauledat, M., and Müller, K.-R. (2007).  Covariate shift adaptation by importance weighted cross validation.  J. Mach. Learn. Res., 8(35):985–1005.
  * (38) Sugiyama, M. and Müller, K.-R. (2005a).  Generalization error estimation under covariate shift.  In Workshop on Information-Based Induction Sciences, pages 21–26. researchgate.net.
  * (39) Sugiyama, M. and Müller, K.-R. (2005b).  Input-dependent estimation of generalization error under covariate shift.  Statistics and Decisions-International Journal Stochastic Methods and Models, 23(4):249–280.
  * Sugiyama et al., (2012) Sugiyama, M., Suzuki, T., and Kanamori, T. (2012).  Density ratio estimation in machine learning.  Cambridge University Press.
  * Vapnik, (2013) Vapnik, V. (2013).  The Nature of Statistical Learning Theory.  Springer Science & Business Media.
  * Vapnik, (1998) Vapnik, V. N. (1998).  Statistical Learning Theory.  Wiley-Interscience.
  * Yamada et al., (2012) Yamada, M., Sigal, L., and Raptis, M. (2012).  No bias left behind: Covariate shift adaptation for discriminative 3D pose estimation.  In Computer Vision – ECCV 2012, pages 674–687. Springer Berlin Heidelberg.
  * Yamada et al., (2011) Yamada, M., Suzuki, T., Kanamori, T., Hachiya, H., and Sugiyama, M. (2011).  Relative Density-Ratio estimation for robust distribution comparison.  In Shawe-Taylor, J., Zemel, R., Bartlett, P., Pereira, F., and Weinberger, K. Q., editors, Advances in Neural Information Processing Systems, volume 24, pages 594–602. Curran Associates, Inc.
  * Zadrozny, (2004) Zadrozny, B. (2004).  Learning and evaluating classifiers under sample selection bias.  In Proceedings of the twenty-first international conference on Machine learning, ICML ’04, page 114, New York, NY, USA. Association for Computing Machinery.
  * Zhang et al., (2021) Zhang, T., Yamane, I., Lu, N., and Sugiyama, M. (2021).  A one-step approach to covariate shift adaptation.  SN Comput. Sci., 2(4):319.