# Holographic superfluid solitons with backreaction

Zhongshan Xu<EMAIL_ADDRESS>School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Yiqiang Du
<EMAIL_ADDRESS>Institute for Theoretical Physics and Astrophysics
and Würzburg-Dresden Cluster of Excellence ct.qmat, Am Hubland 97074 Würzburg,
Germany  Department of Physics, Hanyang University, 222 Wangshimni-ro,
Sungdong-gu, Seoul, 04763, South Korea  Johanna Erdmenger
<EMAIL_ADDRESS>Institute for Theoretical Physics and
Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmat, Am Hubland
97074 Würzburg, Germany  René Meyer<EMAIL_ADDRESS>Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden
Cluster of Excellence ct.qmat, Am Hubland 97074 Würzburg, Germany  Yu Tian
<EMAIL_ADDRESS>School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Center for Theoretical Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China  Zhuo-
Yu Xian<EMAIL_ADDRESS>Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing 100190, China

###### Abstract

Solitons are important nonperturbative excitations in superfluids. For
holographic superfluids, we numerically construct “dark” solitons that have
the symmetry-restored phase at their core. A central point is that we include
the gravitational backreaction of the matter fields, which becomes important
at low temperatures. We study in detail the properties of these solitons under
variation of the backreaction strength via tuning the gravitational constant.
In particular, the depletion fraction of the particle number density at the
core of the solitons is carefully investigated. In agreement with the probe-
limit analysis, the depletion fraction shows the same qualitative behavior as
in Bogoliubov-de Gennes theory, even if the backreaction is included. We find
that the depletion decreases with increasing backreaction strength. Moreover,
the inclusion of backreaction enables us to obtain the effective energy
density of solitons within holography, which together with an evaluation of
the surface tension leads to a simple physical explanation for the snake
instability of dark solitons.

###### Contents

  1. I Introduction
  2. II Holographic Setup
  3. III Numerical Scheme and Results
  4. IV A Simple Mechanism for the Snake Instability
  5. V Conclusions and Discussions
  6. A Holographic stress-energy tensor
  7. B Particle number (charge) change with backreaction

##  I Introduction

Gauge/gravity duality Maldacena (1998); Witten (1998); Gubser  _et al._ (1998)
is a powerful tool to describe strongly coupled and correlated systems. Many
problems associated with strongly interacting condensed matter physics are
tractable in this setup Ammon and Erdmenger (2015). One of these problems is
unconventional superfluidity Herzog  _et al._ (2009); Hartnoll  _et al._
(2008a).

Superfluidity is a collective quantum phenomenon occurring in both bosonic and
fermionic systems at low temperatures. In particular, fermionic systems can
interpolate in a smooth way between the formation and condensation of loosely
bound Cooper pairs (BCS superconductivity), and Bose-Einstein condensation
(BEC) of preformed bosonic molecules. This is known as the BCS-BEC crossover
(see Ref. Randeria and Taylor (2014) for a review), which has been realized by
cooling fermionic gases to ultra-low temperatures and tuning the interactions
between the fermions with a controllable external magnetic field in the
laboratory. The qualitative essence of this crossover can be understood from
the phase diagram depicted in Fig. 1: the horizontal axis interpolates between
the BCS regime of a weakly attractive interaction between fermions and the BEC
regime of very strong attraction. Above the pairing onset temperature
$T^{\ast}$, the system is a normal Fermi liquid consisting of the unpaired
fermions on the BCS side. On the BEC side, the the strong attractive
interaction binds the fermions together to form bosonic molecules, which at
large enough temperatures form a Bose liquid. As the temperature on the BCS
side decreases, loosely bound Cooper pairs of fermions start to form at
$T<T^{\ast}$ and condense below a critical temperature $T_{c}$. On the BEC
side, Bose-Einstein condensation of the bosonic molecules occurs for
$T<T_{c}$. Between these two regimes and at $T<T_{c}$, there exists a strongly
coupled regime of unconventional superfluidity around the point of infinite
scattering length $\frac{1}{k_{F}a}=0$, which is known as the unitary Fermi
gas.

Figure 1: Qualitative phase diagram reproduced from Ref. Randeria and Taylor
(2014) of the BCS-BEC crossover as a function of temperature $T/E_{F}$ and
coupling $1/k_{F}a$, where $k_{F}$ is the Fermi momentum and $a$ is the
scattering length. $T^{\ast}$ is the pairing onset temperature, while $T_{c}$
is the critical temperature for the superfluidity.

Superfluids are quantum fluids which can sustain nonperturbative excitation
such as solitons and vortices Keränen  _et al._ (2010a, b). One kind of
soliton is the so-called dark soliton, which interpolates between the two
phases of the system111Dark solitons include gray solitons with definite
velocity and black solitons that are static Frantzeskakis (2010). We focus on
black solitons in this work.. The core of dark solitons is in the normal state
where the condensate vanishes, while asymptotically far away from the core,
the soliton is in the superfluid state. The core properties can be used to
provide insights into the microscopic structure of the superfluid Antezza  _et
al._ (2007). It was found that, at zero temperature, solitons display
distinctively different core properties in the BEC and BCS regimes. For the
BEC superfluid, solitons have vanishing core particle number density, because
bosonic molecules fully condense below the critical temperature, and there are
no non-condensed states available. But in the core of the BCS dark soliton,
there are still unpaired fermionic atoms or loosely bound Cooper pairs in
excited states, which lead to a non-vanishing core particle number density. At
finite temperature, even in the BEC superfluid there will be a small number of
uncondensed particles arising from thermal excitationKeränen  _et al._ (2011).
In order to quantify these properties, the depletion fraction of the particle
number density at the core is defined as
$\frac{\rho_{max}-\rho_{min}}{\rho_{max}}$. Here, $\rho_{min}$ is the density
at the core, while $\rho_{max}$ is particle number density far away from core.
The depletion fraction was investigated in Bogoliubov-de Gennes (BdG) theory
in Antezza  _et al._ (2007), where it was found that in the BEC regime the
core contains nearly no particles, i.e. the depletion is almost $100\%$, while
in the BCS regime the soliton core still contains some particles at very low
temperatures, with a depletion of about $30\%$. Furthermore, during the
crossover from the BCS regime to the BEC regime, the depletion correspondingly
undergoes a continuous variation from $30\%$ to $100\%$.

Holographic methods can be used to describe unconventional superconductivity
and superfluidity Hartnoll  _et al._ (2008a); Herzog  _et al._ (2009).
Holographic dark solitons in the probe limit, i.e, neglecting the backreaction
on the black-brane space-times forming the background, have been obtained in
Refs. Keränen  _et al._ (2009, 2010a, 2011). The depletion fraction for the
particle number density was calculated in the two different quantizations
possible in AdS/CFT, i.e, standard and alternative quantization. As explained
in more detail below, both quantizations are related by a double-trace
deformation. By comparing the holographic results with the depletion
characteristics described above, a condensate with standard (alternative)
quantization was found to show similar characteristics as in the BCS (BEC)
regime in Ref. Keränen  _et al._ (2011). For ease of comparison, we will below
refer to the condition of standard (alternative) quantization as BCS (BEC)
case. Moreover, the authors of Refs. Keränen  _et al._ (2010a, 2011)
conjectured that one may implement the BCS-BEC crossover in holographic
superfluid systems by varying the scaling dimension of the condensing
operator.

It was also argued in Ref. DeWolfe  _et al._ (2017) that by introducing
double-trace deformations of the charged scalar, it is possible to model the
interaction between fermionic constituents of the boundary system. By
continuously tuning the coupling constant (denoted as $\kappa$ below) of the
deformation, the salient features of the physics of the BCS-BEC crossover can
be captured. In fact, the two independent arguments in Refs. Keränen  _et al._
(2009, 2010a, 2011) and Ref. DeWolfe  _et al._ (2017) are related: according
to Refs. Witten ; Hartman and Rastelli (2008); Vecchi (2011); Erdmenger  _et
al._ (2013), perturbing the large $\mathcal{N}$ boundary theory by a relevant
double-trace deformation corresponds to imposing “mixed” boundary conditions
$\psi_{+}=\kappa\psi_{-}$ for a scalar field in the bulk, where $\kappa$ is
related to the coupling constant of the double-trace deformation. $\kappa=0$
($\kappa=-\infty$) is the boundary conditions for alternative (standard)
quantization. Adding the perturbation will trigger a renormalization group(RG)
flow from the original conformal field theory(CFT) in the UV to another
conformal fixed point in the IR. Under this flow, the conformal dimension of
$\mathcal{O}$ varies between $\Delta_{-}$ in the UV and $\Delta_{+}$ in the
IR, i.e, the two inequivalent boundary CFTs $\Delta=\Delta_{\pm}$ can be
recovered as two limits of the same deformed theory.

The argument of Refs. Keränen  _et al._ (2009, 2010a, 2011) leaves an
important caveat: while the experimental results are obtained at nearly zero
temperature, the holographic probe limit that ignores the backreaction of the
matter fields onto the metric is known to break down in the low-temperature
regime. In particular, the condensate in the alternative quantization diverges
near zero temperature Hartnoll  _et al._ (2008b), which is a sign of the
backreaction becoming important at low temperatures. In this work, we hence
study the behavior of dark solitons in a holographic superfluid system
including backreaction. The dark soliton configurations are constructed by
numerically solving the Einstein equations coupled to the matter fields. In
particular, we employ the DeTurck method for finding stationary solutions,
first introduced in AdS/CFT in Refs. Headrick  _et al._ (2010) and further
developed in Ref. Figueras  _et al._ (2011), This method can reformulate the
Einstein equations into a manifestly elliptic form by adding a covariant
gauge-fixing term to the Einstein equations, which gives a well-posed boundary
value problem.

Another drawback of the probe limit is that the boundary stress-energy tensor
of the condensate cannot be investigated, which conceals information about
important thermodynamic quantities such as the effective energy (mass) and
entropy of the soliton. Taking into account the backreaction allows us to
extract these quantities. Interestingly, it turns out that our result for the
effective energy (mass) of the dark soliton, together with the surface tension
of our holographic dark solitons, is consistent with previous expectations
Fedichev  _et al._ (1999); Cetoli  _et al._ (2013) for the physical mechanism
of a particular instability of dark solitons: the so-called snake instability.
The snake instability is an instability of solitons under transverse
perturbations, leading to the spontaneous formation of a snake-like bending of
the solitons. The snake instability was observed in different physical systems
Anderson  _et al._ (2001); Dutton (2001); Yefsah  _et al._ (2013), and has
attracted much theoretical attention Muryshev  _et al._ (1999); Busch and
Anglin (2000, 2001); Gangardt and Kamenev (2010); Cetoli  _et al._ (2013). In
holography, the authors of Refs. Guo  _et al._ (2020) identified the snake
instability of holographic superfluids in the probe limit via the appearance
of unstable quasinormal modes in the bulk, and observed the final decay of the
“snake” into vortex-antivortex pairs. The investigation of Ref. Guo  _et al._
(2020) was systematic but not as intuitive as effective arguments from
mechanics or hydrodynamics (see, e.g, Ref. Fedichev  _et al._ (1999); Cetoli
_et al._ (2013)). In this work we holographically confirm the explanation of
the snake instability of dark solitons Fedichev  _et al._ (1999); Cetoli  _et
al._ (2013) by calculating the negative effective mass responsible for the
self-acceleration effect Fedichev  _et al._ (1999), as well as the positive
surface tension responsible for the spontaneous generation of ripples on the
soliton Cetoli  _et al._ (2013).

This paper is organized as follows. In next section, we briefly introduce our
holographic superfluid model and analyze the Ansätze and boundary conditions
necessary for solving the equations of motion. In sec. III, our numerical
scheme and main numerical results are discussed. Sec. IV is devoted to the
thermodynamics of the holographic dark soliton and to holographically
confirming the mechanism of the snake instability. A summary and outlook are
included in Sec. V.

##  II Holographic Setup

We work with the simplest holographic superfluid model which requires gravity
coupled to a Maxwell field $A_{\mu}$ and a massive charged scalar field $\Psi$
with charge $e$. The bulk action reads

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2\kappa_{4}^{2}}\left(R-2\Lambda\right)-\frac{1}{e^{2}}\left(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left|D\Psi\right|^{2}+m^{2}\left|\Psi\right|^{2}\right)\right],$
(1)

where we have rescaled the gauge field $A_{\mu}$ and the scalar $\Psi$ to
$\frac{A_{\mu}}{e}$ and $\frac{\Psi}{e}$ compared to the original form
Hartnoll  _et al._ (2008a); Herzog  _et al._ (2009). $L$ is the anti-de
Sitter(AdS) radius related to the cosmological constant as
$\Lambda=-\frac{3}{L^{2}}$, and $m$ is the mass of the charged scalar. The
covariant derivative is $D_{\mu}=\nabla_{\mu}-iA_{\mu}$, where $\nabla_{\mu}$
is the Christoffel covariant derivative with respect to the background metric
$g_{\mu\nu}$. $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
field strength. In the rest of the paper we set $L=1$. The equations of motion
derived from the action take the following form:

$\displaystyle R_{\mu\nu}-\Lambda
g_{\mu\nu}-\frac{2\kappa_{4}^{2}}{e^{2}}\left\\{\frac{1}{2}\left[D_{\mu}\Psi\left(D_{\nu}\Psi\right)^{\dagger}+D_{\nu}\Psi\left(D_{\mu}\Psi\right)^{\dagger}+g_{\mu\nu}m^{2}\left|\Psi\right|^{2}\right]\right.$
$\displaystyle\left.+\left(\frac{1}{2}F_{\mu\sigma}F_{\nu}^{\
\sigma}-\frac{g_{\mu\nu}}{8}F_{\rho\sigma}F^{\rho\sigma}\right)\right\\}=0,$
(2)

$D^{\mu}D_{\mu}\Psi-m^{2}\Psi=0,$ (3)

$\nabla_{\mu}F^{\mu\nu}=ig^{\mu\nu}\left[\Psi^{\dagger}\left(D_{\mu}\Psi\right)-\Psi\left(D_{\mu}\Psi\right)^{\dagger}\right].$
(4)

In the probe limit $\frac{2\kappa_{4}^{2}}{e^{2}}\ll 1$, the backreaction of
the terms involving the gauge field and the charged scalar on the background
geometry in Eq. (2) can be neglected. In this limit, one can first solve the
Einstein equations $R_{\mu\nu}-\Lambda g_{\mu\nu}=0$ for the fixed background
metric $g_{\mu\nu}$, and then solve the matter equations (3) and (4) on top of
that fixed background. Once $\frac{2\kappa_{4}^{2}}{e^{2}}$ is not small this
is no longer possible, and full coupled set of equations (2)–(4) have to be
solved. In this work we are interested in the effect of backreaction. In the
following we will set the charge of the scalar $e=1$ and define the
backreaction parameter $\epsilon\equiv 2\kappa_{4}^{2}$ as a measure of the
strength of backreaction.

In the absence of the charged scalar in Eq. (1), the solution of the Einstein
equations is the Reissner-Nordström-AdS black brane

$ds^{2}=\text{$\frac{1}{z^{2}}\left[-f\left(z\right)dt^{2}+\frac{dz^{2}}{f\left(z\right)}+dz^{2}+dy^{2}\right]$},$
(5) $\displaystyle f\left(z\right)$
$\displaystyle=1-\left(1+\frac{\epsilon\mu^{2}z_{+}^{2}}{4}\right)\left(\frac{z}{z_{+}}\right)^{3}+\frac{\epsilon\mu^{2}z_{+}^{2}}{4}\left(\frac{z}{z_{+}}\right)^{4},$
(6) $A=\mu\left[1-\left(\frac{z}{z_{+}}\right)\right]dt,$ (7)

where $\mu$ is the chemical potential and $z_{+}$ parametrizes the black-brane
temperature via

$T=\frac{1}{4\pi z_{+}}\left(3-\frac{\epsilon\mu^{2}z_{+}^{2}}{4}\right)\,.$
(8)

For numerical convenience Cremonini  _et al._ (2017), we make the following
radical coordinate transformation with $z_{h}\equiv\frac{1}{z_{+}}$:

$z=\frac{1-r^{2}}{z_{h}}.$ (9)

In order to construct the backreacted geometries, we use the following Ansatz
compatible with staticity and translation invariance in the second boundary
direction $y$:

$ds^{2}=\text{$\frac{z_{h}^{2}}{\left(1-r^{2}\right)^{2}}\left[-Q_{1}f\left(r\right)dt^{2}+\frac{4r^{2}Q_{2}dr^{2}}{z_{h}^{2}f\left(r\right)}+Q_{4}\left(dx-\frac{2r}{z_{h}}Q_{3}dr\right)^{2}+Q_{5}dy^{2}\right]$},$
(10) $\Psi=\left(\frac{1-r^{2}}{z_{h}}\right)Q_{6},$ (11) $A=\mu
r^{2}Q_{7}dt\,.$ (12)

Here ${\left\\{Q_{i}|i=1,2,\cdots,7\right\\}}$ are functions of $r$ and $x$ to
be determined by solving Eqs. (2)–(4). In the coordinate (9), the conformal
boundary is located at $r=1$, while the horizon is at $r=0$ where the
regularity of $Q_{i}$ must be imposed. Expanding the equations of motion near
the horizon as a power series in $r$ and requiring the leading order to vanish
yields the boundary conditions

$Q_{1}|_{r=0}=Q_{2}|_{r=0};\left(\partial_{r}Q_{j}\right)|_{r=0}=0,j=2,3,\cdots
7.$ (13)

In particular, the Dirichlet condition for $Q_{1}$ in Eq. (13) ensures that
the temperature of the black brane is still given by Eq. (8). At the conformal
boundary, we demand that the metric approaches AdS4, i.e.

$Q_{1}|_{r=1}=Q_{2}|_{r=1}=Q_{4}|_{r=1}=Q_{5}|_{r=1}=1;Q_{3}|_{r=1}=0.$ (14)

In the asymptotically AdS regime, the scalar field $\Psi$ behaves in the $z$
coordinate as

$\Psi=\psi_{-}z^{\Delta_{-}}+\psi_{+}z^{\Delta_{+}}+\ldots$ (15)

Here $\Delta_{\pm}=3/2\pm\sqrt{9/4+m^{2}L^{2}}$ is the conformal dimension of
the operator $\mathcal{O}$ dual to $\psi$. We choose $m^{2}L^{2}=-2$.
According to the AdS/CFT correspondence, in the standard quantization, one
identifies $\psi_{-}$ with the source of the operator
$\mathcal{O}_{\text{$2$}}$ having the conformal dimension $\Delta_{+}=2$,
while $\psi_{+}$ is regarded as the response (expectation value).
Alternatively, one can exchange the roles of $\psi_{-}$ and $\psi_{+}$ in the
dual field theory, i.e, $\psi$ is dual to operator $\mathcal{O}_{1}$ having
the conformal dimension $\Delta_{-}=1$ with source $\psi_{+}$ and expectation
value $\psi_{-}$. Since we want to investigate a spontaneously generated
condensate in the absence of sources, we will set $\psi_{-}=0$ (standard case)
or $\psi_{+}=0$ (alternative case) on the conformal boundary. Generally, the
mixed boundary condition $\psi_{+}=\kappa\psi_{-}$ can also be imposed, which
is known as double-trace boundary condition since it corresponds to a
deformation of the dual theory by adding a term $\sim\int
d^{3}x\mathcal{O}^{\dagger}\mathcal{O}$ to its boundary action Sbdry Witten ;
Hartman and Rastelli (2008); Vecchi (2011). Finally, the gauge field admits
the usual UV expansion

$A_{t}=\mu-\rho z+\ldots$ (16)

$\mu$ is interpreted as the chemical potential, and $\rho$ is the charge (or
particle number) density. Since the ground state of the system is conformal,
we can scale out one dimensionful quantity. Throughout this paper, we do so by
normalizing all dimensionful quantities with the chemical potential, which we
set to the fixed value $\mu=5.6$.222This value turns out to be numerically
convenient in terms of the convergence speed of our code.

##  III Numerical Scheme and Results

We employ the DeTurck method to numerically solve the Einstein equations, for
a recent review, see Ref. Dias  _et al._ (2016). This method consists of
adding the gauge-fixing term
$\frac{1}{2}\mathcal{L_{\xi}}g_{\mu\nu}=\nabla_{(\mu}\xi_{\nu)}$ to the
Einstein equations (2), which breaks all diffeomorphisms and yields elliptic
Einstein-DeTurck equations,

$\displaystyle R_{\mu\nu}-\Lambda
g_{\mu\nu}-\epsilon\left\\{\frac{1}{2}\left[D_{\mu}\Psi\left(D_{\nu}\Psi\right)^{\dagger}+D_{\nu}\Psi\left(D_{\mu}\Psi\right)^{\dagger}+g_{\mu\nu}m^{2}\left|\Psi\right|^{2}\right]\right.$
$\displaystyle\left.+\left(\frac{1}{2}F_{\mu\sigma}F_{\nu}^{\
\sigma}-\frac{g_{\mu\nu}}{8}F_{\rho\sigma}F^{\rho\sigma}\right)\right\\}-\nabla_{(\mu}\xi_{\nu)}=0.$
(17)

Here the DeTurck vector $\xi^{\nu}\equiv
g^{\rho\sigma}\left[\text{$\Gamma_{\rho\sigma}^{\nu}$$\left(g\right)$}-\text{$\overline{\Gamma}_{\rho\sigma}^{\nu}$$\left(\bar{g}\right)$}\right]$
is constructed from the difference of the Christoffel symbols of the metric
$g_{\mu\nu}$ (which we aim to solve for) and a reference metric
$\bar{g}_{\mu\nu}$. The reference metric has to have the same asymptotics and
symmetries as the metric $g_{\mu\nu}$ we are trying to solve for. In our
scheme, we take the standard Reissner-Nordström-AdS metric (5), corresponding
to $Q_{1}=Q_{2}=Q_{4}=Q_{5}=1$ and $Q_{3}=0$ in Eq. (10). We then find
solutions to the Einstein-DeTurck equation with the constraint condition
$\xi^{\mu}=0$, which ensures that our solution coincides with a solution to
Einstein equations (2).333We have checked that $|\xi^{2}|<10^{-10}$ when the
size of the grids reaches $30\times 150$.

The nonlinear partial differential equations(PDEs) (3), (4), and (17),
together with the boundary conditions (13)–(14) are then solved via the
Newton-Kantorovich method. To be specific, we first linearize the PDEs and
then discretize the linear partial differential equations into algebraic
equations via the standard pseudospectral procedure, where we represent
unknown functions as linear combinations of Chebyshev polynomials in the $z$
coordinate and a Fourier series in the $x$ coordinate.444Since a single
soliton has no periodicity in the _x_ direction, for the simplicity of the
spatial boundary conditions and efficiency of the numerics, we follow Ref. Lan
_et al._ (2017) and instead construct the double soliton (kink-antikink)
configuration, and then analyze a single soliton. Our integration domains
lives on a rectangular grid,
$\left(r,x\right)\in\left(0,1\right)\times\left(-\frac{L_{x}}{2},\frac{L_{x}}{2}\right)$.
The resulting linear system is solved by lower-upper decomposition or other
iterative techniques.

(a) Critical temperature (b) Normalized critical temperature

Figure 2: Absolute (a) and normalized (b) critical temperature for varying
backreaction parameter $\epsilon$. $T_{0}$ is the critical temperature in the
probe limit $\epsilon=0$. The critical temperature drops with increasing
backreaction for both the standard and alternative quantization, indicating a
suppression of the condensate mechanism.

At a critical temperature $T=T_{c}$, an instability of the black brane to form
charged scalar hair occurs. This instability corresponds to a phase transition
in the boundary theory. We begin by a discussion of the critical temperature
$T_{c}$ and its dependence on the backreaction as given by the backreaction
parameter $\epsilon=2\kappa_{4}^{2}$. To determine where our solutions become
unstable, we have to determine the critical temperature for a given
$\epsilon$. We do this by perturbing the Reissner-Nordström-AdS (5) with the
scalar field $\psi=\phi\left(r\right)e^{-i\omega_{p}t}$. At the onset of the
instability, the frequency of the unstable mode crosses zero, $\omega_{p}=0$.
The critical temperature $T_{c}$ itself is therefore found by looking for a
static normalizable solution to the scalar equation of motion with
$\omega_{p}=0$. The results are shown in Fig. 2. We find that in alternative
quantization, $T_{c}/\mu$ is always higher than in the standard case, since
the alternative boundary conditions allow for more modes to become unstable in
the IR, i.e. in the deep interior of the space, in the sense that they violate
the Breitenlohner-Freedman bound there Faulkner  _et al._ (2011). Furthermore,
in AdS/CFT the mass of the scalar field in the bulk determines the dimension
of the the operator in standard ($\Delta_{+}$)/alternative ($\Delta_{-}$)
quantization via
$\Delta_{\pm}=\frac{d}{2}\pm\sqrt{\frac{d^{2}}{4}+m^{2}L^{2}}$. From the field
theoretic point of view, the dimension of the operator in alternative
quantization is always smaller than in the standard case,
$\Delta_{-}<\Delta_{+}$.555Equality is reached at the Breitenlohner-Freedman
bound $m^{2}L^{2}=-d^{2}/4$. The operator with smaller scaling dimension is
more relevant in the RG sense, and its effects will be more pronounced already
at larger values of the renormalization scale. This operator hence will
condense at higher temperatures, explaining the larger $T_{c}$ in the
alternative case.

In relation to this, we find that for for a value of the backreaction
parameter in the range $\epsilon\in[0,1]$, the critical temperature for the
alternative (BEC) case has only a milder dependence on $\epsilon$ compared to
the standard (BCS) case. Within this range, the backreaction is relatively
weak, and we can expand the blackening factor $f(z)$ in a power series in
$\epsilon$ near the probe limit $\epsilon=0$ Hartnoll  _et al._ (2008a,
b),666A possible hyperscaling violating factor has been ruled out by the
construction of the IR fixed point in Faulkner  _et al._ (2011). with the
first correction being

$\displaystyle
f\left(z\right)=1-\left(\frac{z}{z_{+}}\right)^{3}+\epsilon\left(\frac{3\mu}{8\pi
T}\right)^{2}\left(\frac{z}{z_{+}}\right)^{3}\left(\frac{z}{z_{+}}-1\right)+\dots.$
(18)

Since $\epsilon$ is multiplied by $(\mu/T)^{2}$, just below the critical
temperature the effect of backreaction is suppressed for higher $T_{c}/\mu$.
Since $T_{c}/\mu$ is at least a factor 4 larger in the alternative case, the
backreaction-induced change on $f(z)$ in alternative quantization is reduced
by a factor of 16 as compared to the standard case. This implies that in the
alternative case, the equations of motions are far less sensitive to changes
in the backreaction parameter. In particular the change in $T_{c}/\mu$ to
order $\epsilon$ is also going to be suppressed compared to the standard case.
In summary, the insensitivity of $T_{c}/\mu$ on the backreaction parameter in
the alternative case is due to the enhancement of $T_{c}/\mu$ in the probe
limit as compared to the standard case.

Figure 3: Profiles of $\Psi/\Psi_{h}$ just below the critical temperature,
where $\Psi_{h}$ is the value of $\Psi$ on the horizon. We show the cases of
$\epsilon=0$ (solid line), $\epsilon=1$ (dashed line) and $\epsilon=10$ (dot-
dashed line). The blue (red) line refers to the profile in the standard
(alternative) quantization. Figure 4: $T_{c}/\mu$ in the standard quantization
(blue) and alternative quantization (red).

The previous argument is corroborated by the change in the scalar field
profile $\Psi$ as shown for the scalar in Fig. 3, which is also found to be
much weaker in alternative quantization compared to the standard case. Also
the boundary condition of $\Psi$ on the horizon,

$\displaystyle\left.\frac{\partial_{\xi}\Psi}{\Psi}\right|_{\xi=1}=-\frac{8}{\epsilon(3\mu/4\pi
T)^{2}-12}+\dots,\quad\xi=\frac{z}{z_{+}},$ (19)

depends more mildly on $\epsilon$ in alternative quantization. Thus we see
that in all cases $T_{c}/\mu$ in the probe limit controls the first correction
in the backreaction parameter $\epsilon$, and hence in alternative
quantization depends more weakly on the backreaction. However, in a wider
range of $\epsilon$, the dependence on $\epsilon$ can only be found
numerically, as shown in Fig. 4.

(a) $Q_{1}$ (b) $Q_{3}$ (c) $Q_{6}$ (d) $Q_{7}$

Figure 5: The profile of (a) the function $Q_{1}$ setting the $tt$ metric
component, (b) the function $Q_{3}$ setting, together with $Q_{4}$, the $rr$
metric component, (c) the function $Q_{6}$ setting the charged scalar profile,
and (d) the function $Q_{7}$ setting the $t$ component of the gauge field, at
$\epsilon=0.25,T/T_{c}=0.5$ in the standard quantization.

(a) $Q_{1}$ (b) $Q_{3}$ (c) $Q_{6}$ (d) $Q_{7}$

Figure 6: The profile of (a) the function $Q_{1}$ setting the $tt$ metric
component, (b) the function $Q_{3}$ setting, together with $Q_{4}$, the $rr$
metric component, (c) the function $Q_{6}$ setting the charged scalar profile,
and (d) the function $Q_{7}$ setting the $t$ component of the gauge field, at
$\epsilon=0.25,T/T_{c}=0.5$ in the alternative quantization.

Based on the above results for the critical temperature, we now construct the
solution with backreaction for $T<T_{c}$. These are hairy charged black holes
dual to the superfluid phase. The seed configurations of the matter field can
be chosen as $\psi_{\pm}\sim\tanh(x-L_{x}/4)\tanh(-x-L_{x}/4)$. As a result,
the components of metric and the configuration of the matter fields are shown
in Figs. 5 and 6. We can see that larger fluctuations of the spacetime metric
appear only near the core of the soliton.

(a) Condensate (order parameter) (b) Particle number density

Figure 7:  The condensate (a) and the particle number density (b) as a
function of $x$ at $\epsilon=0.25,T/T_{c}=0.5$. $\varrho_{+}$ and
$\varrho_{-}$ are particle number densities normalized with their equilibrium
values at $x\rightarrow\pm\infty$. Red solid lines correspond to the
alternative (BEC) case, while blue dashed lines correspond to the standard
(BCS) case. The charge depletion is considerably larger in the BEC case
compared to the BCS case.

From the asymptotic form of the matter fields, we then read off the
expectation value of the charged condensate and the particle number density in
the dual field theory. These are shown in Fig. 7. As found in Refs. Keränen
_et al._ (2010a, 2011) in the probe limit for the BEC superfluid, the soliton
shows a larger depletion fraction which is, however, smaller than
$100\%$.777One possible explanation for this finding is that the temperature
here is not low enough. In fact, the solution with backreaction at lower
temperature is still very unreliable and difficult to obtain in numerics. We
expect that the depletion will be close to $100\%$ at lower temperature, where
backreaction has to be included. The dependence of the depletion factor on
temperature for different $\epsilon$ is plotted in Fig. 8. Contrary to
expectations, for the BEC soliton, the depletion in the core is considerably
smaller than $100\%$, and even lower than that in the probe limit at low
temperature. As Fig. 9 shows, the depletion decreases as the backreaction
increases. This behavior is observed in both the BCS and BEC regime of our
holographic model. We give a qualitative interpretation of this behavior in
terms of the balance between uncondensed and condensed charge in the boundary
theory as the homogeneous and normal state is reached under the limit of large
backreaction in Appendix B. We think that the underlying reason for this
behavior is the nature of the condensate zero-temperature IR fixed point,
which may be uncharged. In the near future we plan to analyze this fixed point
using analytic methods along the lines of Ref. Charmousis  _et al._ (2010),
and also construct other fixed points which show increasing depletion with
decreasing temperature.

(a) Standard (BCS) case (b) Alternative (BEC) case

Figure 8: The depletion of particle number density as a function of $T/T_{c}$
for different $\epsilon$ in the (a) standard (BCS) and (b) alternative (BEC)
case. In both cases, the depletion factor decreases with increasing
backreaction, i.e, more charge is present at the soliton core.

##  IV A Simple Mechanism for the Snake Instability

(a) Standard (BCS) case (b) Alternative (BEC) case

Figure 9:  The depletion of particle number density as a function of
$\epsilon$ at $T/T_{c}=0.5$. The inset panel shows the change in the particle
number density far away from the core of the soliton and at the core. In both
cases, increasing backreaction reduces the charge depletion.

In this section we turn to the discussion of the thermodynamics of our
holographic dark soliton solution, as well as its instabilities. Since the
boundary chemical potential and temperature are fixed, our system is in the
grand canonical ensemble characterized by the grand potential

$\text{$\Omega=E-TS-\mu N$}\,.$ (20)

Here $N$ is the total particle number obtained by integrating the charge
density $\rho$ over space. The internal energy is found to be

$E=\int_{\Sigma_{t}}d^{2}x\sqrt{\eta}\left[T_{\mu\nu}\left(\partial_{t}\right)^{\mu}\right]t^{\nu}.$
(21)

Here $\eta_{\mu\nu}$ is the induced metric on the surface $\Sigma_{t}$ at
$z=0$ and $t=$ const, with unit normal vector $t^{\nu}$, and $T_{\mu\nu}$ is
the holographic stress-energy tensor, see Appendix A for a detailed
calculation. $S$ is the usual black hole entropy, given by

$S=\frac{A_{h}}{4G}=\frac{4\pi
z_{h}^{2}}{\epsilon}\int\sqrt{Q_{4}\left(0,x\right)Q_{5}\left(0,x\right)}d^{2}x.$
(22)

(a) Total energy density (b) Subtracted energy density

Figure 10: Total (a) and subtracted (b) energy density at $\epsilon=0.25$,
$T=0.5T_{c}$. The subtracted energy density, which can be interpreted as the
effective mass density of the soliton in the condensate phase, is found to be
negative.

Since the soliton extends in a noncompact spatial direction, in what follows
we will consider densities of the above thermodynamic quantities. In
particular, the grand potential $\Omega$ and the energy $E$ are replaced by
their respective densities $\omega$ and $\varepsilon$. Far away from the
soliton center, these local quantities will approach their homogeneous
equilibrium values. Therefore, the soliton core is characterized by the
difference between these local densities and their equilibrium values.

The energy density difference is displayed in Fig. 10b, where we see an
obvious energy depletion around the soliton core.888In this section, we only
show the explicit results for the soliton in the BCS superfluid, while the
results for the BEC case are similar. Upon integration along $x$, this
depletion yields a negative effective energy difference $\Delta E=-7.220$ (in
units of the chemical potential $\mu$). This energy difference $\Delta E$ can
be seen to set the effective mass of the soliton, which is negative as
expected for a dark soliton Fedichev  _et al._ (1999).

(a) Total grand potential density (b) Subtracted grand potential density

Figure 11: Total (a) and subtracted (b) grand potential density at
$\epsilon=0.25$, $T=0.5T_{c}$. After integration over the width of the
soliton, the latter becomes the surface tension of the soliton, which is found
to be positive.

The grand potential density difference is plotted in Fig. 11b, from which we
see that there is a grand potential cost for the soliton with respect to the
homogeneous background. Upon integration along $x$, the grand potential cost
of the soliton yields the surface tension coefficient of the soliton. The
surface tension coefficient $\sigma$ of a domain wall such as the soliton is
defined as the external work $W$ necessary to enlarge the surface by a unit
area while keeping temperature and chemical potential fixed. Under these
conditions, the external work is just the increase of the grand potential due
to the enlargement of the domain wall surface, $W=\Omega-\Omega_{0}$, with
$\Omega_{0}$ being the grand potential of the corresponding homogeneous system
without the domain wall. For the case displayed in Fig. 11b, we numerically
determine the surface tension in units of the chemical potential to be
$\sigma=6.615$. As a consistency check for our numerics, we also plot the
pressure anisotropy $B\equiv p_{x}-p$ in Fig. 12a and check the thermodynamic
relation $\omega=-p$, where $p$ is the average pressure, which should hold far
away from the soliton center.

With the results for these thermodynamic variables, we can confirm the
following explanation for the so-called snake instability of the dark soliton
Cetoli  _et al._ (2013): The soliton moves through the condensate as a heavy,
i.e, nonrelativistic, particle Fedichev  _et al._ (1999). Since it has a
negative effective mass $M_{eff}=\Delta E<0$, its energy
$E_{s}=\frac{M_{e}ff}{2}\dot{q}^{2}$ decreases with increasing velocity
$\dot{q}$. As shown in Refs. Fedichev  _et al._ (1999), for a homogeneous
solitonic configuration, the velocity grows exponentially. This is the so-
called self-acceleration instability Fedichev  _et al._ (1999) of the dark
soliton. The self-acceleration originates from the dissipative interaction of
the soliton with the surrounding condensate. As discussed in Refs. Fedichev
_et al._ (1999), for a homogeneous soliton configuration the self-acceleration
terminates once the soliton velocity reaches the speed of sound, at which
point the soliton decays into sound waves that dissipate away in the
condensate.

On the other hand, the soliton has a positive surface tension coefficient
$\sigma$. As shown in the hydrodynamic approximation in Refs. Cetoli  _et al._
(2013), the combined effect of the negative effective mass together with the
positive surface tension leads to a growing transverse bending mode with a
finite wave vector. The same instability was also found in the holographic
quasinormal mode spectrum in Refs. Guo  _et al._ (2020). Once this transverse
bending mode starts to grow, the self-acceleration instability enhances the
local bending, leading to the formation of a snake-like structure.

(a) Pressure anisotropy (b) Validity of the thermodynamic relation $\omega=-p$

Figure 12: Pressure anisotropy (a) and thermodynamical relation (b) at
$\epsilon=0.25$, $T=0.5T_{c}$. Panel (b) in particular shows that the system
is in a thermalized state away from the soliton core.

##  V Conclusions and Discussions

We investigated the implications of including the gravitational backreaction
onto solitons in holographic superfluid systems. We numerically solved
Einstein equations coupled with the relevant matter fields. As compared to the
probe limit, and contrary to our original expectations, increasing the
backreaction decreases the depletion of the particle number density in the
soliton core. We gave a qualitative interpretation of this in terms of the
balance of the ratios of condensate and noncondensate over total charge in the
dual field theory as a homogeneous state is reached at strong backreaction.
Finally, we computed the holographic stress-energy tensor of the system and
confirmed a simple holographic explanation for the snake instability of the
dark soliton.

In this work, we restricted ourselves to the asymptotic regimes of BEC and BCS
superfluidity. In particular we did not investigate the Robin boundary
conditions necessary to model the actual unitary regime, in which the strongly
coupled unitary fermion system is expected to live. We plan to investigate the
behavior of the soliton in the crossover regime in future work. Such an
investigation may in particular provide a better description of the
intermediate unitary fermion regime, for which the Bogoliubov-de Gennes theory
provides only a broad approximation Cetoli  _et al._ (2013). Furthermore, our
qualitative interpretation of the reduction of the depletion factor implicitly
relies on the assumption of a noncondensate homogeneous infrared fixed point
at strong backreaction or low temperatures. In our holographic superfluid
strong backreaction implies a smaller critical temperature, and hence the
limit of strong backreaction is equivalent to the limit of low temperatures.
Only if the homogeneous infrared fixed point is noncondensate, i.e, the
condensing operator is irrelevant at the infrared fixed point, can the system
return to the noncondensate state in the low-temperature limit. We will
investigate the possible infrared fixed points in our system along the lines
of Refs. Charmousis  _et al._ (2010) to support our qualitative interpretation
in future work. In the thin-domain-wall limit of the soliton, an analytic
treatment of the domain wall in terms of a brane with junction conditions is
also conceivable.

###### Acknowledgements.

We would like to thank Xin Li, Hong Liu, Jie Ren, Ya-Wen Sun, Xiaoning Wu and
Hongbao Zhang for useful discussions. In particular, Zhongshan Xu is grateful
to Jie Ren for his explanations concerning the DeTurck method. This work is
partially supported by the NSFC with Grant No. 11975235 and 11847229. Y. D. is
supported by the China Scholarship Council. Y. D., J. E and R. M acknowledge
financial support through the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), project-id 258499086 - SFB 1170 “ToCoTronics”, and
through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology
in Quantum Matter - ct.qmat (EXC 2147, project-id 39085490). Y. T. is also
supported by the “Strategic Priority Research Program of the Chinese Academy
of Sciences” with Grant No.XDB23030000. Z.-Y. X also acknowledges the support
from the National Postdoctoral Program for Innovative Talents BX20180318,
funded by the China Postdoctoral Science Foundation.

##  Appendix A Holographic stress-energy tensor

In order to compute the holographic stress energy tensor, following the
process of Refs. Dias  _et al._ (2014); Tallarita  _et al._ (2019), we need to
find the asymptotic expansion of the metric at the conformal boundary. The
expansion is obtained by solving Eq. (17) with boundary conditions order by
order in $\left(1-r\right)$ and in addition imposing $\xi^{\mu}=0$,

$Q_{i}\left(r,x\right)=1-\frac{\epsilon
Q_{6}^{\left(0\right)}\left(x\right)}{z_{h}^{2}}\left(1-r\right)^{2}+q_{i}\left(x\right)\left(1-r\right)^{3}+O\left[\left(1-r\right)^{4}\right],i=1,4,5,$
(23a) $Q_{2}\left(r,x\right)=1+\frac{8\epsilon
Q_{6}^{\left(0\right)}\left(x\right)Q_{6}^{\left(1\right)}\left(x\right)}{3z_{h}^{2}}\left(1-r\right)^{3}+O\left[\left(1-r\right)^{4}\right],$
(23b) $Q_{3}\left(r,x\right)=\frac{2\epsilon
Q_{6}^{\left(0\right)}\left(x\right)\partial_{x}Q_{6}^{\left(0\right)}\left(x\right)}{z_{h}^{3}}\left(1-r\right)^{3}+O\left[\left(1-r\right)^{4}\right],$
(23c)
$Q_{6}\left(r,x\right)=Q_{6}^{\left(0\right)}\left(x\right)+Q_{6}^{\left(1\right)}\left(x\right)\left(1-r\right)+\cdots,$
(23d)
$Q_{7}\left(r,x\right)=Q_{7}^{\left(0\right)}\left(x\right)+Q_{7}^{\left(1\right)}\left(x\right)\left(1-r\right)+\cdots.$
(23e)

Here $q_{1}\left(x\right),q_{4}\left(x\right),q_{5}\left(x\right)$ satisfy
Eqs. (25) related to the tracelessness and conservation of the boundary
stress-energy tensor,

$T_{i}^{i}=0,\ \ \ \partial_{i}T^{ij}=0.$ (24)

Using these relations, one can explicitly show the following conditions:

$q_{1}\left(x\right)+q_{4}\left(x\right)+q_{5}\left(x\right)=-\frac{\epsilon
Q_{6}^{\left(0\right)}\left(x\right)\left(-3Q_{6}^{\left(0\right)}\left(x\right)+8Q_{6}^{\left(1\right)}\left(x\right)\right)}{z_{h}^{2}},$
(25a)

$\partial_{x}q_{4}\left(r,x\right)=-\frac{2\epsilon\left[\left(-3Q_{6}^{\left(0\right)}\left(x\right)+8Q_{6}^{\left(1\right)}\left(x\right)\right)\partial_{x}Q_{6}^{\left(0\right)}\left(x\right)+2Q_{6}^{\left(0\right)}\left(x\right)\partial_{x}Q_{6}^{\left(1\right)}\left(x\right)\right]}{3z_{h}^{2}}.$
(25b)

Having obtained the asymptotic behavior of the metric functions, one then
changes to Fefferman-Graham coordinates $\left(z,v\right)$ by an expansion of
the series (26a) and demands $g_{zz}=\frac{1}{z^{2}}$ and $g_{zv}=0$ to
determine the two functions
$\left\\{a_{k}\left(v\right),b_{k}\left(v\right)\right\\}$ order by order in
$z$. Here we provide the first few terms necessary for the computation of
holographic stress-energy tensor:

$\begin{cases}r=1-\frac{z_{h}}{2}z+\sum_{k=2}^{\infty}a_{k}\left(v\right)z^{k},\\\
x=v+\sum_{k=1}^{\infty}b_{k}\left(v\right)z^{k},\end{cases}$ (26a)
$a_{2}\left(v\right)=-\frac{z_{h}^{2}}{8},\ \ \
a_{3}\left(v\right)=-\frac{z_{h}^{3}}{16},$ (26b)
$a_{4}\left(v\right)=\frac{z_{h}^{2}\left(24\epsilon\mu^{2}+51z_{h}^{2}+32\epsilon
Q_{6}^{\left(0\right)}\left(v\right)Q_{6}^{\left(1\right)}\left(v\right)\right)}{1152},$
(26c) $b_{1}\left(v\right)=b_{2}\left(v\right)=b_{3}\left(v\right)=0,$ (26d)
$b_{4}\left(v\right)=-\frac{\epsilon
Q_{6}^{\left(0\right)}\left(v\right)\partial_{v}Q_{6}^{\left(0\right)}\left(v\right)}{16}.$
(26e)

Finally, the holographic stress-energy tensor is computed by using Eq. (27a)
in the standard quantization and Eq. (27b) in the alternative quantization
Balasubramanian and Kraus (1999); Skenderis (2001),

$T_{\mu\nu}=\frac{1}{\kappa_{4}^{2}}\lim_{z\rightarrow
0}\frac{1}{z}\left(K_{\mu\nu}-\gamma_{\mu\nu}K-2\gamma_{\mu\nu}-\frac{\epsilon}{2}\left|\Psi\right|^{2}\gamma_{\mu\nu}\right),$
(27a) $T_{\mu\nu}=\frac{1}{\kappa_{4}^{2}}\lim_{z\rightarrow
0}\frac{1}{z}\left[K_{\mu\nu}-\gamma_{\mu\nu}K-2\gamma_{\mu\nu}+\frac{\epsilon}{2}\left(-\Psi^{\dagger}n^{\sigma}D_{\sigma}\Psi-C.C.+\left|\Psi\right|^{2}\right)\gamma_{\mu\nu}\right].$
(27b)

Here $K_{\mu\nu}$ is the extrinsic curvature associated with an inward-
pointing unit normal vector $n^{\sigma}$ on the constant $z=\epsilon$ surface
near the boundary. $\gamma_{\mu\nu}$ is the induced metric on the cutoff
surface. The last term in Eq. (27) cancels the divergences due to the presence
of the scalar field Guo  _et al._ (2016).

##  Appendix B Particle number (charge) change with backreaction

(a) Total particle number density. (b) Condensed particle number density. (c)
Uncondensed particle number density. (d) Soliton contribution.

Figure 13: Change of the total charge density (a), condensed charge density
(b), uncondensed charge density (c) and soliton contribution (d) with
backreaction at $T=0.5T_{c}$ in the standard (BCS) case.

Here we give a qualitative interpretation of the depletion that decreases with
increasing backreaction, in terms of the balance between uncondensed and
condensed charge in the boundary theory as the homogeneous and normal state is
reached under the limit of large backreaction. For our static but
inhomogeneous charged scalar and gauge field configuration, by integrating the
$t$ component of the Maxwell equations (4) over the holographic $r$
coordinate, one obtains the total particle number density in terms of three
contributions,

$\sqrt{-g}F^{tr}|_{r=1}=\sqrt{-g}F^{tr}|_{r=0}+\int_{0}^{1}(-\sqrt{-g}J^{t})dr+\int_{0}^{1}\partial_{x}\sqrt{-g}F^{xt}dr,$
(28)

$J^{\nu}=ig^{\mu\nu}\left[\Psi^{\dagger}\left(D_{\mu}\Psi\right)-\Psi\left(D_{\mu}\Psi\right)^{\dagger}\right].$
(29)

On the right-hand side of Eq. (28), the first term yields the uncondensed
particle number density, given by the electric flux evaluated at the horizon.
The second term is the condensed particle number density. The third term is a
contribution arising from the inhomogeneous configuration of the soliton.

$\underbrace{\int d^{2}x\sqrt{-g}F^{tr}|_{r=1}}\limits_{N}=\underbrace{\int
d^{2}x\sqrt{-g}F^{tr}|_{r=0}}\limits_{N_{b}}+\underbrace{\int
d^{2}x\int_{0}^{1}(-\sqrt{-g}J^{t})dr}\limits_{N_{\psi}}.$ (30)

(a) Total particle number density. (b) Condensed particle number density. (c)
Uncondensed particle number density. (d) Soliton contribution.

Figure 14: Change of the total charge density (a), condensed charge density
(b), uncondensed charge density (c) and soliton contribution (d) with
backreaction at $T=0.5T_{c}$ in the alternative (BEC) case.

(a) Standard (BCS) case (b) Alternative (BEC) case

Figure 15: The condensate as a function of $\epsilon$ at $T/T_{c}=0.5$.

A relation between the total charge $N$, uncondensed charge $N_{b}$ and
condensed charge $N_{\psi}$ can be obtained by integrating (28) over the
spatial part.999The spatial domain of all integrations above is
$\left(-\frac{L_{x}}{2},\frac{L_{x}}{2}\right)\times\left(-\frac{L_{y}}{2},\frac{L_{y}}{2}\right)$.
Due to the translation symmetry along the $y$ direction, we normalize these
charges with regard to $L_{y}$. After integration, the soliton contribution in
Eq. (28) becomes a total derivative and hence vanishes if we integrate over a
symmetric interval around $x=0$. The remaining terms in Eq. (28) yield the
relation (30) in which we denote the three terms in order as $N$, $N_{b}$, and
$N_{\psi}$. The ratios of uncondensed and condensed charge over the total
charge then have to add up to one,

$\frac{N_{b}}{N}+\frac{N_{\psi}}{N}=1.$ (31)

(a) Standard (BCS) case (b) Alternative (BEC) case

Figure 16: The ratio of two kinds of particle number as a function of
$\epsilon$ at $T/T_{c}=0.5$.

As can be seen from Figs. 13 and 14, the backreaction suppresses the condensed
particle density and promotes the uncondensed particle number density. On the
other hand, from Fig. 15, the condensate (order parameter) far away from the
soliton core also decreases strongly with increasing backreaction. It seems
that the system returns to a homogeneous noncondensate state in the limit of
large backreaction. This is, in particular, obvious for the BEC case [Fig.
15b], but the trend is also obvious for the BCS case [Fig. 15a]. Furthermore,
this interpretation of the data is consistent with the general expectation
that backreaction inhibits the formation of the condensate in a holographic
superconductor Wang and Liu (2016). The suppression of the condensate implies
that the ratio of condensate to total charge monotonically decreases with
increasing backreaction. This can be seen from Fig. 16. Since the ratios of
tthe condensate and non-condensate charge over total charge is bound to add up
to one by charge conservation, the ratio of noncondensate to total charge
hence must increase with increasing backreaction. Taking these two facts
together, we deduce that the depletion at the soliton core must decrease and
vanish as the system approaches a homogeneous and normal state.

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$\displaystyle\leq
2\sqrt{\varepsilon\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}+4\varepsilon^{\frac{3}{4}}\|h(x^{*};\cdot)\|_{\infty},$

where the first inequality follows from the fact that $\chi^{2}$-divergence
satisfies Assumption 3 with probability at least $1-\delta$. The second and
third inequalities follow from Lemma 9.3 for two pairs
$(\mathbb{P},\hat{\mathbb{Q}})$ and $(\mathbb{P}^{*},\hat{\mathbb{Q}})$. And
the fourth inequality follows from:

$\displaystyle\text{Var}_{\hat{\mathbb{Q}}}[h]-\text{Var}_{\mathbb{P}^{*}}[h]$
$\displaystyle\leq\left|\mathbb{E}_{\hat{\mathbb{Q}}}[h^{2}]-\mathbb{E}_{\mathbb{P}^{*}}[h^{2}]\right|+2\|h\|_{\infty}\left|\mathbb{E}_{\hat{\mathbb{Q}}}[h]-\mathbb{E}_{\mathbb{P}^{*}}[h]\right|$
(38)
$\displaystyle\leq\sqrt{2\chi^{2}(\mathbb{P}^{*},\hat{\mathbb{Q}})\text{Var}_{\mathbb{P}^{*}}[h^{2}]}+2\|h\|_{\infty}\sqrt{2\chi^{2}(\mathbb{P}^{*},\hat{\mathbb{Q}})\text{Var}_{\mathbb{P}^{*}}[h]}.\quad\hfill\square$

###  11.3 Improved Results from Theorem 3.3 for General $f$-divergence

The result in Theorem 3.3 can be improved for general $f$-divergence from
$\|h(x^{*};\cdot)\|_{\infty}$ to
$\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]}$ in terms of the
dependence of $\varepsilon$, without requiring (10) as long as some mild
conditions hold for the cost function $h(x;\cdot)$ and sample size $n$. This
is achieved by applying a general duality property of $f$-divergence DRO to
bound $\hat{Z}^{P-DRO}(x^{*})-Z(x^{*})$. The result, which is presented in
Theorem 11.4 below, holds for any $f$-divergence DRO regardless of the
distribution center $\hat{\mathbb{Q}}$.

###### Theorem 11.4

When the sample size $n$ is large enough, for general metrics $d_{f}$ in
$f$-divergence and $\|h(x^{*};\cdot)\|_{\infty}<\infty$, we have:

$\sup_{\mathbb{P}:d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]+C(f,\varepsilon)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]\varepsilon},$
(39)
$\inf_{\mathbb{P}:d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]\geq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-C(f,\varepsilon)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]\varepsilon},$
(40)

where $C(f,\varepsilon)$ only depends on the metric $d_{f}$ and $\varepsilon$
and is bounded by some numerical constant in classical $f$-divergences (See
Examples 11.5 and 11.6 below).

With this, the result in case $(b)$ in Theorem 3.3 can be improved to:

$\displaystyle\mathcal{E}(x^{\texttt{P-DRO}})$
$\displaystyle\leq\sup_{\mathbb{P}:d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]+C(f,\varepsilon)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon}$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\inf_{\mathbb{P}:d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]+C(f,\varepsilon)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon}$
$\displaystyle\leq
2C(f,\varepsilon)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon},$

where the second and fourth inequalities follow from the result in Theorem
11.4. The first and third inequalities follow from
$\mathbb{P}[d_{f}(\mathbb{P}^{*},\hat{\mathbb{Q}})\leq\varepsilon]\geq
1-\delta$ such that
$\inf_{d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]\leq\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]\leq\sup_{\mathbb{P}:d_{f}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]$
with probability at least $1-\delta$. After that, we can use the same argument
as before to bound $\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}$.

Proof of Theorem 11.4. We first show (39). We have:

$\displaystyle\sup_{d_{f}(\mathbb{P}^{*},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]$
$\displaystyle\leq\min_{\lambda\geq
0,\mu}\left\\{\lambda\mathbb{E}_{\hat{\mathbb{Q}}}\left[f^{*}\left(\frac{h(x^{*};\xi)-\mu}{\lambda}\right)\right]+\lambda\varepsilon+\mu\right\\}$
(41)
$\displaystyle\leq\hat{\lambda}\mathbb{E}_{\hat{\mathbb{Q}}}\left[f^{*}\left(\frac{h(x;\xi)-\hat{\mu}}{\hat{\lambda}}\right)\right]+\sqrt{\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon}{f^{\prime\prime}(1)}}+\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]+\left(\frac{1}{\sqrt{f^{{}^{\prime\prime}}(1)}}+\frac{\sqrt{f^{{}^{\prime\prime}}(1)}(f^{*})^{{}^{\prime\prime}}(0)C_{0}(\varepsilon)}{2}\right)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon},$

where the first inequality above is based on weak duality, i.e., Theorem 1 in
Ben-Tal et al. (2013) (Note that although strong duality holds generally in
this problem, we only need weak duality in our proof). The second inequality
above is given by
$\hat{\lambda}=\sqrt{\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}{f^{\prime\prime}(1)\varepsilon}},\hat{\mu}=\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]$
as the feasible dual solution, and the third inequality follows from plugging
in the values of $\hat{\lambda}$ and $\hat{\mu}$, and then taking the Taylor
expansion up to the second order for $f^{*}$ with a Maclaurin remainder
$C_{0}(\varepsilon)$ upper bounded by some constant and $C_{0}(0)=1$:

$\displaystyle\hat{\lambda}\mathbb{E}_{\hat{\mathbb{Q}}}\left[f^{*}\left(\frac{h(x^{*};\xi)-\hat{\mu}}{\hat{\lambda}}\right)\right]$
$\displaystyle\leq\hat{\lambda}\mathbb{E}_{\hat{\mathbb{Q}}}\left[f^{*}(0)+(f^{*})^{\prime}(0)\left(\frac{h(x^{*};\xi)-\hat{\mu}}{\hat{\lambda}}\right)+\frac{(f^{*})^{{}^{\prime\prime}}(0)C_{0}(\varepsilon)}{2}\left(\frac{h(x^{*};\xi)-\hat{\mu}}{\lambda}\right)^{2}\right],$
$\displaystyle=\hat{\lambda}\mathbb{E}_{\hat{\mathbb{Q}}}\left[\frac{(f^{*})^{{}^{\prime\prime}}(0)C_{0}(\varepsilon)}{2}\left(\frac{h(x^{*};\xi)-\hat{\mu}}{\hat{\lambda}}\right)^{2}\right]=\frac{(f^{*})^{{}^{\prime\prime}}(0)C_{0}(\varepsilon)\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}{2\hat{\lambda}},$

where the first equality follows from $f^{*}(0)=0$ and
$\hat{\mu}=\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]$. Then (39) follows
from (41) if we denote
$C(f,\varepsilon)=\frac{1}{\sqrt{f^{{}^{\prime\prime}}(1)}}+\frac{\sqrt{f^{{}^{\prime\prime}}(1)}(f^{*})^{{}^{\prime\prime}}(0)C_{0}(\varepsilon)}{2}$.
For (40), we only need to consider $-h(x;\cdot)$ and plug in the result of
(39). $\hfill\square$

We now show several common divergences satisfying (10) and give some concrete
values for $C(f,\varepsilon)$ above. To avoid redundancy, we only consider
(39) and ignore (40).

######  Example 11.5 (KL divergence)

We take $f(t)=t\log t-(t-1)$. Then $f^{*}(t)=e^{t}-1$ with
$f^{{}^{\prime\prime}}(1)=1$. We use the inequality $e^{t}-1\leq t+t^{2}$ when
$t\in(-1,1)$, i.e., we need
$\left|\frac{h(x^{*};\xi)-\hat{\mu}}{\hat{\lambda}}\right|\leq 1$, which
implies when
$\sqrt{\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}{\varepsilon}}=\hat{\lambda}\geq
2\|h(x^{*};\cdot)\|_{\infty}$, i.e.,
$\varepsilon\leq\frac{\text{Var}_{\hat{\mathbb{P}}}[h(x^{*};\xi)]}{4\|h(x^{*};\cdot)\|_{\infty}^{2}}$.
Therefore, if
$\varepsilon\leq\frac{\text{Var}_{\hat{\mathbb{P}}}[h(x^{*};\xi)]}{4\|h(x^{*};\cdot)\|_{\infty}^{2}}$,
we have:

$\sup_{\mathbb{P}:KL(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]+3\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon}.$

######  Example 11.6 ($H^{2}$-distance)

We take $f(t)=(\sqrt{t}-1)^{2}$ and $f^{{}^{\prime\prime}}(1)=\frac{1}{2}$.
Then for $t<1$, $f^{*}(t)=\frac{t}{1-t}=\frac{1}{1-t}-1\leq t+2t^{2}$ when
$t\in[-\frac{1}{2},\frac{1}{2}]$. Therefore, if
$\varepsilon\leq\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}{2\|h(x^{*};\cdot)\|_{\infty}^{2}}$,
we have:

$\sup_{\mathbb{P}:H^{2}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{*};\xi)]\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]+(2+\sqrt{2})\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\varepsilon}.$

Therefore, following from the same argument as for case $(c)$ of Theorem 3.3,
if
$\Delta(\delta,\Theta)\leq\varepsilon\leq\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}{c_{0}2\|h(x^{*};\cdot)\|_{\infty}^{2}}$,
$\mathcal{E}(x^{\texttt{P-DRO}})$ can be improved by
$c_{1}\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}+c_{2}\varepsilon^{\frac{3}{4}}\|h(x^{*};\cdot)\|_{\infty}$
for KL divergence and Hellinger distance with probability at least $1-\delta$.

###  11.4 Proof of Theorem 3.4

In case (a) where $d$ is an IPM, we have:

$\displaystyle\mathcal{E}(x^{\texttt{P-ERM}})$
$\displaystyle\leq|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-ERM}};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-ERM}};\xi)]|+|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]|$
$\displaystyle\leq
2\sup_{x\in\mathcal{X}}\bigg{|}\mathbb{E}_{\mathbb{P}^{*}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]\bigg{|}$
$\displaystyle\leq
2\sup_{x\in\mathcal{X}}\mathcal{V}_{d}(x)d(\mathbb{P}^{*},\hat{\mathbb{Q}}).$

where the second inequality follows from the uniform bound $\forall
x\in\mathcal{X}$, and the last inequality follows from the fact that $d$ is an
IPM such that $d(\mathbb{P},\mathbb{Q})=\sup_{\mathcal{V}_{d}(f)\leq
1}|\mathbb{E}_{\mathbb{P}}[f]-\mathbb{E}_{\mathbb{Q}}[f]|.\hfill\square$

In case (b) where $d$ satisfies the inequality (10), similarly, we have:

$\displaystyle\mathcal{E}(x^{\texttt{P-ERM}})$
$\displaystyle\leq|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-ERM}};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-ERM}};\xi)]|+|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]|$
$\displaystyle{\leq}2\sup_{x\in\mathcal{X}}\bigg{|}\mathbb{E}_{\mathbb{P}^{*}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]\bigg{|}$
$\displaystyle{\leq}4Md_{TV}(\mathbb{P}^{*},\hat{\mathbb{Q}})\leq
4C_{d}M\sqrt{d(\mathbb{P}^{*},\hat{\mathbb{Q}})}.$

In the special case (c) where $d$ is (modified) $\chi^{2}$-divergence,
following from the previous decomposition and Lemma 9.3, we have:

$\displaystyle\mathcal{E}(x^{\texttt{P-ERM}})$
$\displaystyle\leq|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-ERM}};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-ERM}};\xi)]|+|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]|$
(42)
$\displaystyle\leq\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\left(\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{\texttt{P-ERM}};\xi)]}+\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}\right)$
$\displaystyle\leq
2\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}+\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\sqrt{\left|\mathbb{E}_{\mathbb{P}^{*}}[h^{2}(x^{\texttt{P-ERM}};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h^{2}(x^{*};\xi)]\right|}$
$\displaystyle\leq
2\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}+\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\sqrt{4M^{2}d_{TV}(\hat{\mathbb{Q}},\mathbb{P}^{*})}$
$\displaystyle\leq
2\sqrt{2\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*})}\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}+2M(\chi^{2}(\hat{\mathbb{Q}},\mathbb{P}^{*}))^{\frac{3}{4}},$

where the fourth inequality in (42) follows from:

$\displaystyle\mathbb{E}_{\mathbb{P}^{*}}[h^{2}(x^{\texttt{P-ERM}};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h^{2}(x^{*};\xi)]$
$\displaystyle\leq\mathbb{E}_{\mathbb{P}^{*}}\left[(h(x^{\texttt{P-ERM}};\xi)+h(x^{*};\xi))((h(x^{\texttt{P-ERM}};\xi)-h(x^{*};\xi))\right]$
$\displaystyle\leq
2M\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-ERM}};\xi)-h(x^{*};\xi)]\leq
4M^{2}d_{TV}(\hat{\mathbb{Q}},\mathbb{P}^{*}).$

And the fifth inequality in (42) follows from Lemma 9.2.

For each case above, we then apply Assumption 3 to obtain the
result.$\hfill\square$

##  12 Further Details and Proofs for Section 4

We denote
$x^{\texttt{P-DRO}_{m}}\in\argmin_{x\in\mathcal{X}}\max_{d(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]$.
In general, we want to investigate the required sample size $m$ such that
$\mathcal{E}(x^{\texttt{P-DRO}_{m}})\approx\mathcal{E}(x^{\texttt{P-DRO}})$ in
Theorem 3.3. The idea is to understand when the Monte Carlo sampling error is
dominated by the generalization error in each P-DRO case.

Across statements in Section 4, we ignore the polynomial dependence on
$\log(1/\delta)$ for the required Monte Carlo size when we express the
required sample sizes for different generalization error bounds. That is to
say, when we write “the required sample size
$\cdots\geq\sqrt{\text{Comp}_{m}(\mathcal{H})}$”, we mean “the required sample
size $\cdots\geq\sqrt{\text{Comp}_{m}(\mathcal{H})+c_{1}\log(1/\delta)}$” for
some constant $c_{1}$. In other words, we ignore the dependence of
$\log(1/\delta)$ and the required sample sizes are at most polynomial in this
ignored term.

###  12.1 Proof of Theorem 4.1

Comparing the result with Theorem 3.3, we only need to show that
$\mathbb{P}^{*}\in\hat{\mathcal{A}}$ with probability at least $1-\delta$.
Then the other parts follow directly from the case (a) in Theorem 3.3.
Following the triangle inequality, we have:

$\displaystyle W_{1}(\mathbb{P}^{*},\hat{\mathbb{Q}}_{m})$ $\displaystyle\leq
W_{1}(\mathbb{P}^{*},\hat{\mathbb{Q}})+W_{1}(\hat{\mathbb{Q}},\hat{\mathbb{Q}}_{m})$
$\displaystyle\leq\frac{\varepsilon}{2}+\left(\frac{C}{m}\right)^{\frac{1}{D_{\xi}}}\log(1/\delta)\leq\varepsilon,$

where the second inequality follows from Lemma 9.1 and third inequality
follows from $\frac{\varepsilon}{2}\geq\Delta(\delta,\Theta)$ in Assumption 3
and
$\left(\frac{C}{m}\right)^{\frac{1}{D_{\xi}}}\log(1/\delta)\leq\frac{\varepsilon}{2}$,
i.e., $m\geq C(\frac{2\log(1/\delta)}{\varepsilon})^{D_{\xi}}$ for some
constant $C$. Then the subsequent steps are analogous to the proof of case (a)
in Theorem 3.3, only replacing $\hat{\mathbb{Q}}$ with $\hat{\mathbb{Q}}_{m}$.
$\hfill\square$

###  12.2 Proofs of Theorem 4.2

We present results and proofs with respect to $\chi^{2}$-divergence and
1-Wasserstein distance separately, with
$m\approx\text{Comp}(\mathcal{H})n^{\alpha}$ and $\alpha$ being independent of
$D_{\xi}$. Theorem 12.1 and Theorem 12.2 are more general results from which
Theorem 4.2 follows.

######  Theorem 12.1 (Generalization bounds for $\chi^{2}$ P-DRO with Monte
Carlo errors)

Suppose Assumption 3 holds and the cost function $h(x;\xi)\in[0,M],\forall
x,\xi$ with $\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]>0$. The size of the
ambiguity set $\varepsilon\geq\Delta(\delta,\Theta)$. If the Monte Carlo size
$m\geq
C_{0}\left(\frac{LM}{\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]}\varepsilon}\right)^{2}{\text{Comp}_{m}(\mathcal{H})}$
for some numerical constant $C_{0}$, when $d$ is $\chi^{2}$-divergence, with
probability at least $1-\delta$, we have:

$\mathcal{E}(x^{\texttt{P-DRO}_{m}})\leq\begin{cases}2\mathcal{E}_{\chi^{2}}+C_{1}\sqrt{\frac{\varepsilon}{L}}M,&~{}\text{if}~{}\text{Var}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-DRO}_{m}};\xi)]\leq
2\varepsilon M^{2}\\\
2\mathcal{E}_{\chi^{2}},&~{}\text{otherwise}{}\end{cases},$

where $L\geq 1$ and $\mathcal{E}_{\chi^{2}}$ is the generalization error upper
bound in the case $(c)$ of Theorem 3.3.

Note that this result depends on another term $L$ due to “incomplete” exact
variance regularization of $\chi^{2}$-divergence. However, when
$\text{Var}_{\mathbb{P}^{*}}[h(x;\xi)]$ is sufficiently large, as long as the
required Monte Carlo size $m\geq
C_{0}\left(\frac{M}{\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)}\varepsilon}\right)^{2}{\text{Comp}_{m}(\mathcal{H})}$,
$\mathcal{E}(x^{\texttt{P-DRO}_{m}})\leq 2\mathcal{E}_{\chi^{2}}$. On the
other hand, even if the variance is not large enough, as long as
$\sqrt{\frac{\varepsilon}{L}}M\leq\mathcal{E}_{\chi^{2}}\leq\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]\varepsilon}$,
i.e., $L\geq\frac{M^{2}}{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]}$ and
$m\geq\left(\frac{M}{\sqrt{\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\cdot)]}\varepsilon}\right)^{6}{\text{Comp}_{m}(\mathcal{H})}$
for some numerical constant $C_{0}$, we still have
$\mathcal{E}(x^{\texttt{P-DRO}_{m}})\leq 3\mathcal{E}_{\chi^{2}}$, which is
the required sample size in Theorem 4.2.

We obtain a dimension-free required Monte Carlo sample size for the
1-Wasserstein case too.

######  Theorem 12.2 (Generalization bounds for 1-Wasserstein P-DRO with Monte
Carlo errors)

Suppose Assumption 3 holds and the random quantity $h(x;\xi)$ is sub-Gaussian
with parameter $M$ when $\xi$ follows any distribution
$\mathbb{Q}\in\mathcal{P}_{\Theta},\forall x\in\mathcal{X}$. Besides, $\Xi$ is
unbounded and $h(x;\xi)$ is Lipschitz continuous and convex w.r.t. $\xi$.
Furthermore, there exists $\xi_{0}\in\Xi$ such that
$\limsup_{\|\tilde{\xi}-\xi_{0}\|\to\infty}\frac{h(x;\xi)-h(x;\xi_{0})}{\|\tilde{\xi}-\xi_{0}\|}=\|h(x;\cdot)\|_{\text{Lip}},\forall
x\in\mathcal{X}$. The size of the ambiguity set
$\varepsilon\geq\Delta(\delta,\Theta)$.

If the Monte Carlo size $m\geq
C_{0}\left(\frac{M}{\|h(x^{*};\cdot)\|_{\text{Lip}}\varepsilon}\right)^{2}\text{Comp}_{m}(\mathcal{H})$
for some numerical constant $C_{0}$, when $d$ is 1-Wasserstein distance, then
with probability at least $1-\delta$, we have:
$\mathcal{E}(x^{\texttt{P-DRO}_{m}})\leq
4\|h(x^{*};\cdot)\|_{\text{Lip}}\varepsilon.$

Before presenting the proofs, we introduce two uniform concentration
inequalities for the empirical mean and variance over $\mathcal{H}$.

######  Definition 12.3 (Sub-Gaussian Random Variable)

A random variable $g(\xi)$ over $\mathbb{R}$ is called sub-Gaussian with
parameter $\sigma$ when $\xi\sim\mathbb{Q}$ if
$\mathbb{E}_{\mathbb{Q}}[g(\xi)]<\infty$ and
$\mathbb{E}_{\mathbb{Q}}[\exp(t(g(\xi)-\mathbb{E}_{\mathbb{Q}}[g(\xi)]))]\leq\exp(\sigma^{2}t^{2}/2),\forall
t\in\mathbb{R}$.

######  Lemma 12.4 (Uniform Hoeffding (Sub-Gaussian) Inequality)

Suppose the random quantity $h(x;\xi)$ is sub-Gaussian with parameter $M$ when
$\xi\sim\hat{\mathbb{Q}},\forall x\in\mathcal{X}$, then with probability at
least $1-\delta$, we have:

$\displaystyle\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]$
$\displaystyle\leq C_{1}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}},$ (43)

where $C_{1}$ is some numerical constant independent of the function
complexity and sample size.

This result is extracted from Theorem 6 in Maurer and Pontil (2009). And a
special case in Lemma 12.4 is when $0\leq h(x;\xi)\leq M,\forall x,\xi$.

######  Lemma 12.5 (Uniform Variance Concentration Inequality)

When $0\leq h(x;\xi)\leq M,\forall x,\xi$, with probability at least
$1-\delta$, we have:

$\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}\geq\sqrt{1-\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\frac{2M^{2}}{m}-C_{2}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}},$
(44)

where $C_{2}$ is some numerical constant independent of the function
complexity and sample size.

###### Proof 12.6

Proof of Lemma 12.5. Consider the variance concentration inequality (extracted
from Lemma A.1 in Duchi and Namkoong (2019)), $\forall x\in\mathcal{X}$, when
$m\geq 3$, with probability at least $1-\delta$:

$\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}\geq\sqrt{1-\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\frac{2M^{2}}{m}-M\sqrt{\frac{2\log(1/\delta)}{m}}.$
(45)
$\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}\leq\sqrt{1+\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}+M\sqrt{\frac{2\log(1/\delta)}{m}}.$
(46)

From Definition 10.1 with $\ell:=N_{\infty}(\mathcal{H},\frac{1}{m},m)$, we
consider functions $h(x_{1};\cdot),\ldots,h(x_{\ell};\cdot)$ such that
$\mathcal{H}_{m}(\bm{\xi})$ is contained in the union of balls
$D_{k}:=\\{(h(x;\xi_{1}),\ldots,h(x;\xi_{m}))|x\in\mathcal{X},\sup_{\bm{\xi}\in\Xi^{m}}\sup_{i\in[m]}|h(x;\xi_{i})-h(x_{k};\xi_{i})|\leq
1/m\\},k\in[\ell]$. Then we apply the union bound to (45) such that with
probability at least $1-\delta$:

$\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x_{i};\xi)]}\geq\sqrt{1-\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x_{i};\xi)]}-\frac{2M^{2}}{m}-M\sqrt{\frac{2\log(N_{\infty}(\mathcal{H},\frac{1}{m},m)/\delta)}{m}},\forall
i\in[\ell].$

Besides, for any other $h(x;\cdot)$ with
$x\not\in\\{x_{1},\ldots,x_{\ell}\\}$, we can always find one $h(x_{k};\cdot)$
with $k\in[\ell]$ such that
$\sup_{\bm{\xi}\in\Xi^{m}}\sup_{i\in[m]}|h(x;\xi_{i})-h(x_{k};\xi_{i})|\leq\frac{1}{m}$
by the definition of the covering number $\ell$.

Therefore, we have:

$\displaystyle\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\cdot)]}$
$\displaystyle=\sqrt{\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h^{2}(x;\cdot)]-(\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\cdot)])^{2}}$
$\displaystyle\geq\sqrt{\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h^{2}(x_{k};\cdot)]-(\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x_{k};\cdot)])^{2}-4M|\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x_{k};\cdot)-h(x;\cdot)]|}$
$\displaystyle\geq\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x_{k};\cdot)]}-2\sqrt{M\frac{1}{m}}$
$\displaystyle\geq\sqrt{1-\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x_{k};\cdot)]}-\frac{2M^{2}}{m}-M\sqrt{\frac{2\log(N_{\infty}(\mathcal{H},\frac{1}{m},m)/\delta)}{m}}-2\sqrt{\frac{M}{m}}$
$\displaystyle\geq\sqrt{1-\frac{1}{m}}\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\cdot)]}-\frac{2M^{2}}{m}-M\sqrt{\frac{2\log(N_{\infty}(\mathcal{H},\frac{1}{m},m)/\delta)}{m}}-4\sqrt{\frac{M}{m}},$

where the first inequality follows from $h(x;\cdot)$ being “close” to some
$h(x_{k};\cdot)$ from the covering number argument. And the second inequality
follows from the ball size being $\frac{1}{m}$ and
$\sqrt{a-b}\geq\sqrt{a}-\sqrt{b}$ when $\sqrt{a-b}\geq 0$. And the third
inequality follows from the union bound above. The fourth inequality follows
from
$|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x_{k};\cdot)-h(x;\cdot)]|\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x_{k};\xi)-h(x;\xi)]\leq\frac{1}{m}$,
i.e.:

$\displaystyle\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x_{k};\cdot)]}$
$\displaystyle\geq\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\cdot)]-2M|\mathbb{E}_{\hat{\mathbb{Q}}}[h(x_{k};\cdot)-h(x;\cdot)]|}$
$\displaystyle\geq\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\cdot)}-\sqrt{\frac{2M}{m}},$

Then denote
$\text{Comp}_{m}(\mathcal{H})=2\log(N_{\infty}(\mathcal{H},\frac{1}{m},m)/\delta)$,
we obtain (44). $\hfill\square$

####  12.2.1 Proof of Theorem 12.1

We denote
$\hat{Z}^{P-DRO}(x):=\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]$
and the discrete approximation
$\hat{Z}_{m}^{P-DRO}(x):=\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]$
here. The key is to show that
$\sup_{x\in\mathcal{X}}|\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)|$ is small
so that we can borrow results from Theorem 3.3. In the beginning, we present
the error decomposition between the empirical variance and the true variance
under $\mathbb{P}^{*}$. That is, with probability at least $1-\delta$:

$\displaystyle\left|\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]-\text{Var}_{\mathbb{P}^{*}}[h(x;\xi)]\right|$
$\displaystyle=\left|\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]-\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]\right|+\left|\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\text{Var}_{\mathbb{P}^{*}}[h(x;\xi)]\right|$
(47) $\displaystyle\leq
M^{2}\left(C_{1}\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}+3\sqrt{2\varepsilon}\right),\forall
x\in\mathcal{X},$

where the first term in the inequality follows from the uniform Hoeffding
inequality. And the second term in the inequality follows from the choice of
$\varepsilon$ such that
$\chi^{2}(\mathbb{P}^{*},\hat{\mathbb{Q}})\leq\varepsilon$ and
$\|h\|_{\infty}\leq M$ in (38).

The main proof is divided into the following three steps.

Step 1: Variance Regularization. Following from Lemma 9.3, we have:

$\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]+\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]},$
(48)
$\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\leq\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]\leq\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]+\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]},$
(49)

We now choose the required size $m$ so that the equality condition of RHS
holds in (49).

Note that for any $\varepsilon$, the objective value
$\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]$
is equivalent to the optimal objective value of the following optimization
problem:

$\max_{p\in\mathbb{R}^{m}_{+}}\sum_{i=1}^{m}p_{i}h(x;\xi_{i}),~{}\text{s.t.}:~{}\sum_{i=1}^{m}\left(p_{i}-\frac{1}{m}\right)^{2}\leq\frac{2\varepsilon}{m},\sum_{i=1}^{m}p_{i}=1.$

The optimal objective value above is the same as RHS of (49) if
$\sqrt{2\varepsilon}\frac{h(x;\xi)-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}{\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}}\geq-1$.
Since $h(x;\xi)\in[0,M],\forall x,\xi$, this condition holds if:

$\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\geq 2\varepsilon M^{2},\forall
x\in\mathcal{X}.$ (50)

In general, we obtain the following variance-dependent lower bound of
$\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]$:

$\sup_{\mathbb{P}:\chi^{2}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]\geq\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]+\sqrt{\Delta\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]},$
(51)

as long as $\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\geq\Delta M^{2}$.

For any integer $L\geq 1$ and the output $\hat{\mathbb{Q}}$, we partition the
decision space $\mathcal{X}$ into the following regions
$\mathcal{X}_{1}\cup\ldots\mathcal{X}_{L+1}\cup\mathcal{X}_{L+2}$, where
$\mathcal{X}_{L+2}=\\{x\in\mathcal{X}:\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]\geq
2\varepsilon M^{2}\\}$, and:

$\mathcal{X}_{\ell}=\left\\{x\in\mathcal{X}:\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]\in\left[\frac{\ell-1}{L}2\varepsilon
M^{2},\frac{\ell}{L}2\varepsilon
M^{2}\right)\right\\},\ell\in\\{1,\ldots,L+1\\}.$

We first choose the Monte Carlo size such that:

$MC_{1}\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}=:\Delta_{\mathbb{E}}\leq\frac{2\varepsilon
M}{L},$ (52)

so that with probability at least $1-\delta$, following from (47):

$\bigg{|}\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\bigg{|}\leq\frac{2\varepsilon
M^{2}}{L},\forall x\in\mathcal{X}.$ (53)

Step 2: Monte Carlo Error Decomposition. We partition the problem of bounding
$\hat{Z}(x)-\hat{Z}_{m}(x)$ into the following different regimes of the
decision space.

(2.1) $\forall x\in\mathcal{X}_{L+2}$, due to (53), (50) holds. Therefore by
(48) and (51), we obtain:

$\displaystyle\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]+\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]-\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}$
(54)
$\displaystyle=(\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)])+\sqrt{2\varepsilon}(\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}).$

For the first term of RHS in (54), we choose the Monte Carlo size $m$ such
that
$C_{1}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}:=\Delta_{\mathbb{E}}\leq\frac{2\varepsilon
M}{L}$ in (43) in Lemma 12.4. Then, plugging (43) and (44) into (54), with
probability at least $1-\delta$, $\forall x\in\mathcal{X}_{L+2}$, we have:

$\displaystyle\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)$
$\displaystyle\leq\Delta_{\mathbb{E}}+\sqrt{2\varepsilon}\left(C_{2}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}+\left(1-\sqrt{1-\frac{1}{m}}\right)\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}+\frac{2M^{2}}{m}\right)$
$\displaystyle\leq\Delta_{\mathbb{E}}+C_{2}M\sqrt{\frac{2\varepsilon\text{Comp}_{m}(\mathcal{H})}{m}}+\frac{\sqrt{2\varepsilon}(2M^{2}+\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)])}{m}$
$\displaystyle\leq
C_{3}\Delta_{\mathbb{E}}+\frac{3\sqrt{2\varepsilon}M^{2}}{m}\leq
C_{3}^{\prime}\Delta_{E},$

where $C_{3},C_{3}^{\prime}$ are numerical constants independent of the
function complexity and sample size. The second inequality follows from the
fact that TV distance is an IPM.

(2.2) $\forall x\in\mathcal{X}_{i},i\in\\{1,\ldots,L+1\\}$, we have
$\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\geq\max\left\\{\frac{i-2}{L}2\varepsilon
M^{2},0\right\\}$.

(2.2.1) If $i\geq 2$, following (48) and (51) as well as the definition of
$\mathcal{X}_{i}$, we have:

$\displaystyle\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)$
$\displaystyle\leq\left(\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\right)+\sqrt{\frac{i}{L}2\varepsilon\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\sqrt{\frac{i-2}{L}2\varepsilon\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}$
(55)
$\displaystyle\leq\Delta_{\mathbb{E}}+\sqrt{\frac{2(i-2)\varepsilon}{L}}\left(\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}-\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}\right)+\sqrt{\frac{4\varepsilon}{L}\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}$
$\displaystyle\leq\Delta_{\mathbb{E}}+\sqrt{\frac{2(i-2)\varepsilon}{L}}\frac{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}{\sqrt{\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}+\sqrt{\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]}}+\sqrt{\frac{4\varepsilon}{L}\frac{2i}{L}M^{2}}$
$\displaystyle\leq\Delta_{\mathbb{E}}+\sqrt{\frac{2(i-2)\varepsilon}{L}}\frac{2\varepsilon
M^{2}/L}{2\sqrt{2(i-2)\varepsilon
M^{2}/L}}+C_{4}^{\prime}\sqrt{\frac{\varepsilon}{L}}M\leq\Delta_{\mathbb{E}}+C_{4}\frac{\varepsilon}{L}M+C_{4}^{\prime}\sqrt{\frac{\varepsilon}{L}}M.$

(2.2.2) If $i=1$, following from the definition of $\mathcal{X}_{1}$ and (48),
we have:

$\displaystyle\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)$
$\displaystyle\leq\left(\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\right)+\sqrt{\frac{2\varepsilon}{L}\text{Var}_{\hat{\mathbb{Q}}}[h(x;\xi)]}$
(56) $\displaystyle\leq\Delta_{\mathbb{E}}+\frac{2\varepsilon M}{L}.$

In general, combining cases in (2.2), with probability at least $1-\delta$, if
$\frac{\varepsilon}{L}\leq\sqrt{\frac{\varepsilon}{L}}\leq 1$, then:

$\hat{Z}^{P-DRO}(x)-\hat{Z}_{m}^{P-DRO}(x)\leq
C_{0}\sqrt{\frac{\varepsilon}{L}}M,\forall
x\in\mathcal{X}\backslash\mathcal{X}_{L+2}.$ (57)

Step 3: Generalization Error Decomposition. Plugging the solution
$x^{\texttt{P-DRO}_{m}}$ into (57), we have:

$\displaystyle~{}Z(x^{\texttt{P-DRO}_{m}})-\hat{Z}_{m}^{P-DRO}(x^{\texttt{P-DRO}_{m}})$
(58)
$\displaystyle\leq\hat{Z}^{P-DRO}(x^{\texttt{P-DRO}_{m}})-\hat{Z}_{m}^{P-DRO}(x^{\texttt{P-DRO}_{m}})\leq\begin{cases}\Delta_{\mathbb{E}}+C_{0}\sqrt{\frac{\varepsilon}{L}}M,~{}&\text{
if }x^{\texttt{P-DRO}_{m}}\not\in\mathcal{X}_{L+2}\\\
C_{0}^{\prime}\Delta_{\mathbb{E}},~{}&\text{ otherwise}\end{cases}.$

for some constant $C_{0}$ when $L$ is large. The first inequality follows from
Assumption 3 and Theorem 3.3, with probability at least $1-\delta$,
$\mathbb{P}^{*}\in\hat{\mathcal{A}}$ when
$\varepsilon\geq\Delta(\delta,\Theta)$. Besides,

$\hat{Z}_{m}^{P-DRO}(x^{*})-Z(x^{*})\leq(\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)])+\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]}.$
(59)

The first term of RHS in (59) can be further bounded by Bernstein inequality.
That is, with probability at least $1-\delta$:

$\displaystyle\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]$
$\displaystyle\leq(\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)])+(\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)])$
(60)
$\displaystyle\leq\sqrt{\frac{2\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]\log(1/\delta)}{m}}+\frac{\|h(x^{*};\cdot)\|_{\infty}\log(1/\delta)}{3m}+\sqrt{2\varepsilon\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}$
$\displaystyle\leq\|h(x^{*};\cdot)\|_{\infty}\left(\sqrt{\frac{2\log(1/\delta)}{m}}+\frac{\log(1/\delta)}{3m}\right)+\sqrt{2\varepsilon\text{Var}_{\mathbb{P}^{*}}[h(x^{*};\xi)]}.$

Following (46), with probability at least $1-\delta$, the second term of RHS
in (59) can be bounded by:

$\displaystyle\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]}$
$\displaystyle\leq\sqrt{2\varepsilon\text{Var}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)]}+2\|h(x^{*};\cdot)\|_{\infty}\sqrt{\frac{\varepsilon\log(1/\delta)}{m}}.$
(61)

Then following the same decomposition procedure,

$\displaystyle Z(x^{\texttt{P-DRO}_{m}})-Z(x^{*})$
$\displaystyle\leq(Z(x^{\texttt{P-DRO}_{m}})-\hat{Z}_{m}^{P-DRO}(x^{\texttt{P-DRO}_{m}}))+(\hat{Z}_{m}^{P-DRO}(x^{*})-Z(x^{*}))$
$\displaystyle\leq
2\mathcal{E}_{\chi}^{2}+C_{1}\sqrt{\frac{\varepsilon}{L}}M\mathbb{I}_{\\{x^{\texttt{P-DRO}_{m}}\in\mathcal{X}_{L+2}\\}},$

where the second inequality follows from the observation that the error
$Z(x^{\texttt{P-DRO}_{m}})-\hat{Z}_{m}(x^{\texttt{P-DRO}_{m}})$ in (58) is
bounded by $\hat{Z}_{m}(x^{*})-Z(x^{*})$ in (59) when the required sample size
$m$ is chosen as the way in Theorem 12.1. Therefore, we can attain the given
generalization bound.$\hfill\square$

####  12.2.2 Proof of Theorem 12.2

Before presenting the proof, we introduce the following lemma demonstrating
the regularization effects of the 1-Wasserstein DRO model:

######  Lemma 12.7 (Extracted from Theorem 6.3 of Esfahani and Kuhn (2018))

Suppose $h(x;\xi)$ is Lipschitz continuous and convex w.r.t. $\xi$ and $\Xi$
is unbounded. There exists $\xi_{0}\in\Xi$ such that
$\limsup_{\|\tilde{\xi}-\xi_{0}\|\to\infty}\frac{h(x;\xi)-h(x;\xi_{0})}{\|\tilde{\xi}-\xi_{0}\|}=\|h(x;\cdot)\|_{\text{Lip}}$,
Then for any $\hat{\mathbb{P}}$ we have:

$\sup_{W_{1}(\mathbb{P},\hat{\mathbb{P}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]=\mathbb{E}_{\hat{\mathbb{P}}}[h(x;\xi)]+\varepsilon\|h(x;\cdot)\|_{\text{Lip}}.$

Proof of Theorem 12.2. On one hand, with probability at least $1-\delta$, we
have:

$\displaystyle\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-DRO}_{m}};\xi)]$
$\displaystyle\leq\sup_{\mathbb{P}:W_{1}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{\texttt{P-DRO}_{m}};\xi)]$
(62)
$\displaystyle=\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-DRO}_{m}};\xi)]+\varepsilon\|h(x^{\texttt{P-DRO}_{m}};\cdot)\|_{\text{Lip}}$
$\displaystyle=\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{\texttt{P-DRO}_{m}};\xi)]+\left(\sup_{\mathbb{P}:W_{1}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{\texttt{P-DRO}_{m}};\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{\texttt{P-DRO}_{m}};\xi)]\right),$

where the first inequality follows from
$W_{1}(\mathbb{P}^{*},\hat{\mathbb{Q}})\leq\varepsilon$ under the ambiguity
size $\varepsilon\geq\Delta(\delta,\Theta)$. The first and second equalities
follow from Lemma 12.7 when taking the distribution center $\hat{\mathbb{P}}$
to be $\hat{\mathbb{Q}}$ and $\hat{\mathbb{Q}}_{m}$ respectively.

On the other hand, we have:

$\displaystyle\sup_{\mathbb{P}:W_{1}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{{\mathbb{P}}}[h(x^{*};\xi)]$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]+\varepsilon\|h(x^{*};\cdot)\|_{\text{Lip}}$
$\displaystyle\leq\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x^{*};\xi)]+(\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}}[h(x^{*};\xi)])+2\varepsilon\|h(x^{*};\cdot)\|_{\text{Lip}}.$

Therefore, we have:

$\displaystyle\mathcal{E}(x^{\texttt{P-DRO}_{m}})$
$\displaystyle\leq\left(\mathbb{E}_{\mathbb{P}^{*}}[h(x^{\texttt{P-DRO}_{m}};\xi)]-\sup_{\mathbb{P}:W_{1}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x^{\texttt{P-DRO}_{m}};\xi)]\right)+\left(\sup_{\mathbb{P}:W_{1}(\mathbb{P},\hat{\mathbb{Q}}_{m})\leq\varepsilon}\mathbb{E}_{{\mathbb{P}}}[h(x^{*};\xi)]-\mathbb{E}_{\mathbb{P}^{*}}[h(x^{*};\xi)]\right)$
$\displaystyle\leq
2\|h(x^{*};\cdot)\|_{\text{Lip}}\varepsilon+2\sup_{x\in\mathcal{X}}\bigg{|}\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}_{m}}[h(x;\xi)]\bigg{|}$
$\displaystyle\leq
2\|h(x^{*};\cdot)\|_{\text{Lip}}\varepsilon+2C_{1}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}.$

Then the required sample size $m$ is chosen such that the second term
$C_{1}M\sqrt{\frac{\text{Comp}_{m}(\mathcal{H})}{m}}$ above is smaller than
$\|h(x^{*};\cdot)\|_{\text{Lip}}\varepsilon$. $\hfill\square$

###  12.3 Generalization Results for $p$-Wasserstein Distance

In this part, we extend the result of bounding the Monte Carlo sampling error
to the case when $d$ is taken as $p$-Wasserstein distance with $p\in(1,2]$.

######  Definition 12.8 ($p$-Wasserstein Distance)

$p$-Wasserstein distance $(1\leq p<\infty)$ between two distributions
$\mathbb{P}$ and $\mathbb{Q}$ supported on $\Xi$ is defined as:

$W_{p}(\mathbb{P},\mathbb{Q})=\inf_{\Pi\in\mathcal{M}{(\Xi\times\Xi)}}\left\\{\left(\int_{\Xi\times\Xi}\|x-y\|^{p}\Pi(dx,dy)\right)^{\frac{1}{p}}:\Pi_{x}=\mathbb{P},\Pi_{y}=\mathbb{Q}\right\\},$

where $\Pi_{x}$ and $\Pi_{y}$ are the marginal distributions of $\Pi$.

We present the following lemma establishing the regularization effect of
$p$-Wasserstein distance.

######  Lemma 12.9 (Extracted from Lemma 1 in Gao et al. (2022))

Suppose some mild conditions hold for $\mathcal{H}$ (i.e. the same conditions
of Assumption 1 and 2 from Gao et al. (2022), only replacing $\mathcal{F}$
there with $\mathcal{H}$). Consider $p$-Wasserstein distance with $p\in(1,2]$,
for any distribution $\hat{\mathbb{Q}}$, there exists $C>0$ such that:

$\bigg{|}\sup_{\mathbb{P}:W_{p}(\mathbb{P},\hat{\mathbb{Q}})\leq\varepsilon}\mathbb{E}_{\mathbb{P}}[h(x;\xi)]-\mathbb{E}_{\hat{\mathbb{Q}}}[h(x;\xi)]-\varepsilon\mathcal{V}_{\hat{\mathbb{Q}},q}(h(x;\cdot))\bigg{|}\leq
C\varepsilon^{p}.$

where $\mathcal{V}_{\hat{\mathbb{Q}},q}(h(x;\cdot))$ is the $L_{q}$ norm of
the vectorized random variable $\nabla_{\xi}h(x;\xi)$ under the measure
$\hat{\mathbb{Q}}$ with $\frac{1}{p}+\frac{1}{q}=1$.

We abbreviate $h:=h(x;\cdot),h^{*}:=h(x^{*};\cdot)$ in the following.

###### Corollary 12.10

Suppose Assumption 3 in the main body and Assumption 1 and 2 in Gao et al.
(2022) hold. The size of the ambiguity set
$\varepsilon\geq\Delta(\delta,\Theta)$, when $d$ is $p$-Wasserstein distance
with $p\in(1,2]$, if the Monte Carlo size satisfies:

$m\geq\max\left\\{\left(\frac{C_{1}+C_{2}\tilde{M}\sqrt{\text{Comp}_{m}(\partial(\mathcal{H}))}}{\mathcal{V}_{\mathbb{P}^{*},q}(h^{*})}\right)^{q},C_{0}\left(\frac{M}{\varepsilon\mathcal{V}_{\mathbb{P}^{*},q}(h^{*})}\right)^{2}\text{Comp}_{m}(\mathcal{H})\right\\},$

where
$\tilde{M}:=\sup_{x\in\mathcal{X},\xi\in\Xi}\left\|\nabla_{\xi}h(x;\xi)\right\|_{2}$
and
$\partial(\mathcal{H})=\left\\{\left\|\nabla_{\xi}h(x;\xi)\right\|_{2}^{q}:x\in\mathcal{X}\right\\}$
for some constants $C_{0},C_{1},C_{2}$, Then with probability at least
$1-\delta$, we have $\mathcal{E}(x^{\texttt{P-DRO}_{m}})\leq
4\varepsilon\mathcal{V}_{\mathbb{P}^{*},q}(h(x^{*};\cdot))+C\varepsilon^{p}$.