# Ghost imaging in scattering media

Wenlin Gong  Pengli Zhang  Xia Shen  Shensheng Han<EMAIL_ADDRESS>Key
Laboratory for Quantum Optics and Center for Cold Atom Physics, Shanghai
Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai
201800, China

###### Abstract

Ghost imaging with thermal light in scattering media is investigated. We
demonstrated both theoretically and experimentally for the first time that the
image with high quality can still be obtained in the scattering media by ghost
imaging. The scattering effect on the qualities of the images obtained when
the object is illuminated directly by the thermal light and ghost imaging is
analyzed theoretically. Its potential applications are also discussed.

###### pacs:

42.50.Ar, 42.68.Mj, 42.50.Dv, 42.62.Be, 42.30.Kq

##  I introduction

Multiple scattering has a great effect on the qualities of images and the
transmission of information. The information will be decayed and the images
suffer reduced resolution and contrast because of multiple scattering. For
example, the measurement of the laser radar Kamerman , satellite
communications Booker , the propagation and imaging of light in the atmosphere
Mckechnie , neutron imaging Raine  and the imaging and diagnosis in life and
medical science Maher . So the imaging in strong scattering media is always a
great problem and presents a key challenge for the research of better imaging
method and technique.

In clinic applications, the most common imaging modalities include ultrasound
imaging, X-ray computed tomography(CT), and magnetic resonance imaging (MRI)
Maher ; Wang . As the development of imaging technology, optical imaging is
becoming an increasing interesting method for the imaging in biological
tissue. By now most imaging methods are obtained by using the gating
techniques. Such as confocal imaging, spatial filtering, optical coherence
tomography (OCT), Mueller optical coherence tomography, Diffuse optical
tomography (DOT), Photoacoustic tomography (PAT), Ultrasound-Modulated optical
tomograph (UOT) and so on Wang ; Gibson ; Yodh ; Arridge1 ; Arridge2 ; Sachin
; Hebden ; Chance ; Laubscher ; Brian . Although the qualities of the images
have a great increase by these techniques, there is still lots of problems
which are difficult to be done. Because the imaging techniques in scattering
media discussed above mainly are only the first-order effect of light field,
detection and imaging are unseparated. When the information of the object is
distorted by multiple scattering, and the information of both multiple
scattering and the object is unknown, so we can not, in principle, obtain
exactly the images destroyed by the multiple scattering, which leads to be
impossible of the restoration of the qualities of images caused by multiple
scattering.

The first two-photon imaging experiment with entangled source was demonstrated
by Pittman _etal_. in 1995 Pittman , which shew that we could obtain a
nonlocal image by transmitting pairs of photons through a test and a reference
path. Since 2002, the theories and experiments demonstrated that the ghost
imaging could also be obtained with thermal light Cheng ; Gatti1 ; Gatti2 ;
Valencia ; Bennink ; Basano . And the fierce discussion on the essence of
ghost imaging at one time Scarcelli1 ; Gatti3 ; Scarcelli2 . Ghost imaging is
considered as the effect of second-order correlation of light field and is
caused by the undistinguishable relation of identical particles Zhang1 ;
Scarcelli1 ; Scarcelli2 . For the first time, detection and imaging are
separated by ghost imaging. The test path and the reference path are used to
detect the information from the object and imaging for the object,
respectively. Recently we find that the qualities of ghost images are
determined by both the reference path and test path Zhang . Because multiple
scattering only degrades the imaging quality in the test path, whereas there
is no multiple scattering in the reference path. By correlation measurement,
we may get a image with much better quality than the image obtained by
detection in a single path.

##  II theory and analysis

Figure 1: Schematic diagram for the light transmitting in a scattering media.

Schematic diagram for the light transmitting in a scattering media is shown in
Fig. 1. In the theory of linear systems Goodman , the light field $E(x)$ on
the plane $x$ is the convolution of the light field $E(x_{0})$ on the plane
$x_{0}$ and the impulse response function $h(x,x_{0})$.

$\displaystyle E(x)=\int{dx_{0}E(x_{0})}h(x,x_{0}).$ (1)

For light transmission in scattering media, the light field on the position
$x$ is the linear superposition the incident light and the scattering light.

$\displaystyle E(x)=\alpha\int{dx_{0}}E(x_{0})h_{in}(x,x_{0})$
$\displaystyle+\beta\int dx_{0}E(x_{0})h_{sca}(x,x_{0}),$ (2)
$\displaystyle\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1.$ (3)

where $h_{in}(x,x_{0})$ is the impulse response function with no scattering
media, and $h_{sca}(x,x_{0})$ is the impulse response function from the plane
$x_{0}$ to the plane $x$ because of the interactions of multiple scattering,
and $\alpha$, $\beta$ are the probability amplitudes of the incident light and
the scattering light, respectively. From Eqs. (1)-(3), we have

$\displaystyle h(x,x_{0})=\alpha h_{in}(x,x_{0})+\beta h_{sca}(x,x_{0}).$ (4)

The probability distribution function in scattering media is called point
scattering function. The impulse response function $h_{sca}(x,x_{0})$ has
close contact with the point scattering function which is Dirac delta function
when there is no scattering media. However, in the scattering media, it is a
spread function with a broadening length, and generally the point scattering
function has two forms: Lorentzian-shaped and Gaussian-shaped distribution
Hassanein ; Segre . In multiple scattering Mie theoryMie ; Toublanc ; Sharma ,
both of probability amplitudes $\alpha$, $\beta$ are depending on the diameter
size of the particle $D$, the wavelength of the incident light $\lambda$, the
concentration of suspended particles $w$ and the effective length of
scattering media $L$. According to the experiments and theories Jessica ;
Milun ; Mustafa ; Faulkner ; wells ; yura ; Hassanein ; Segre , we have

$\displaystyle\alpha=\alpha(D,\lambda,w,L)\propto\frac{{\lambda^{b_{\alpha}}}}{{D^{a_{\alpha}}w^{c_{\alpha}}L^{d_{\alpha}}}},$
(5a)
$\displaystyle\beta=\beta(D,\lambda,w,L)\propto\frac{{D^{a_{\beta}}w^{c_{\beta}}L^{d_{\beta}}}}{{\lambda^{b_{\beta}}}},$
(5b) $\displaystyle h_{sca}(x,x_{0})\propto\int
dx^{\prime}P(x^{\prime},x_{0})_{L_{A}}h(x,x^{\prime})_{(L+d)},$ (5c)
$\displaystyle P(x^{\prime},x_{0})_{L_{A}}=[\frac{2}{{\pi\Delta
x_{L_{A}}^{2}}}]^{1/4}$
$\displaystyle\times\exp\left\\{{-(\frac{{x^{\prime}-x_{0}}}{{\Delta
x_{L_{A}}}})^{2}}\right\\},$ (5d) $\displaystyle\Delta
x_{L_{A}}\propto\frac{{D^{a_{x}}w^{c_{x}}{L_{A}}^{d_{x}}}}{{\lambda^{b_{x}}}},$
(5e)
$\displaystyle\int{\left|{P(x^{\prime},x_{0})_{L_{A}}}\right|}^{2}dx^{\prime}=1.$
(5f)

where $P(x^{\prime},x_{0})_{L_{A}}$ is point scattering probability amplitude.
$\Delta x_{L_{A}}$ is broadening length because of the interactions of
multiple scattering, and it becomes wider with the increase of the scattering
length. With the increase of the broadening length, the frequency spectrum of
the optical transfer function becomes narrower, which is the main reason
leading to the degradation of the quality of information transmission and
images wells ; yura . We suppose point scattering function is Gaussian-shaped
distribution without considering the absorption of scattering media. All the
coefficients in Eq. (5) should be determined by specific experimental
conditions.

The scheme for ghost imaging with thermal light in the scattering media is
shown in Fig. 2. The light source $S$, first propagates through a beam
splitter, then is divided into a test and a reference path. In the test path,
the light propagates through a single lens of focal length $f_{1}$, the
scattering media and then to the detector $D_{t}$. In the reference path, the
light propagates through a single lens of focal length $f$ then to an array of
pixel detector $D_{r}$.

Figure 2: Scheme for ghost imaging with thermal light in the scattering media.

By optical coherence theory Cheng ; Glauber , we can obtain the correlation
function of intensity fluctuations between the detectors:

$\displaystyle\Delta G^{(2,2)}(x_{r},x_{t})=\left\langle{\Delta
I_{r}(x_{r})\Delta
I_{t}(x_{t})}\right\rangle=\left|{\Gamma(x_{r},x_{t})}\right|^{2},$ (6a)
$\displaystyle\Gamma(x_{r},x_{t})=\int{dx_{1}}\int{dx_{2}G^{(1,1)}(x_{1},x_{2})h_{r}^{*}(x_{r},x_{1})}$
$\displaystyle\times h_{t}(x_{t},x_{2}).$ (6b)

where $\Gamma(x_{r},x_{t})$ is the first-order cross-correlation function of
two different points from the test and reference paths.

Suppose the light source is fully spatially incoherent, then

$G^{(1,1)}(x_{1},x_{2})=I_{0}\delta(x_{1}-x_{2}).$ (7)

where $I_{0}$ is a constant, and $\delta(x)$ is the Dirac delta function.

Under the paraxial and small angle approximation, and when the effective
apertures of the lenses in the optical system are large enough, the impulse
response function of the reference system is

$h_{r}(x_{r},x_{1})\propto\exp\left\\{{\frac{{j\pi}}{{\lambda
f}}(1-\frac{z}{f})x_{1}^{2}-\frac{{2j\pi}}{{\lambda f}}x_{r}x_{1}}\right\\}.$
(8)

And $f_{2},z_{2}+L_{2},z_{3}$ obey the Lens Law

$\frac{1}{{z_{2}+L_{2}}}+\frac{1}{{z_{3}}}=\frac{1}{{f_{2}}}.$ (9)

when

$\displaystyle\frac{{f_{1}}}{f}=\frac{{z_{2}+L_{2}}}{{z_{3}}}.$ (10)

The impulse response function of the test system is

$\displaystyle
h(x_{t},x_{2})\propto\int{dx^{\prime}}[\alpha_{1}\exp\left\\{{-\frac{{2j\pi}}{{\lambda
f_{1}}}x^{\prime}x_{2}}\right\\}+\beta_{1}\int dx_{2}^{\prime}$
$\displaystyle\times
P(x^{\prime},x_{2}^{\prime})_{L_{1A}}\exp\left\\{{-\frac{{2j\pi}}{{\lambda
f_{1}}}x_{2}^{\prime}x_{2}}\right\\}]t(x^{\prime})C(x^{\prime})$
$\displaystyle\times\exp\left\\{{\frac{{j\pi}}{{\lambda
f_{1}}}(1-\frac{{z_{1}+L_{1}}}{{f_{1}}})x_{2}^{2}}\right\\},$ (11a)

$\displaystyle
C(x)=[\alpha_{2}\delta(x+\frac{{f_{1}}}{f}x_{t})\exp\left\\{{\frac{{j\pi}}{{\lambda(L_{2}+z_{2})}}x^{2}}\right\\}+\beta_{2}$
$\displaystyle\times
P(-\frac{{f_{1}}}{f}x_{t},x)_{L_{2A}}\exp\left\\{{\frac{{j\pi}}{{\lambda(L_{2}+z_{2})}}(\frac{{f_{1}}}{f}x_{t})^{2}}\right\\}].$
(11b)

Substituting Eqs. (7) and (9)-(11b) into Eq. (6b), we can get the intensity
distribution in the test path

$\displaystyle
I(x_{t})\propto\int{dx^{\prime}}\int{dx^{\prime\prime}}t(x^{\prime})t^{*}(x^{\prime\prime})\\{\left|{\alpha_{1}}\right|^{2}\delta(x^{\prime}-x^{\prime\prime})$
$\displaystyle+2P(x^{\prime},x^{\prime\prime})_{L_{1A}}{\mathop{\rm
Re}\nolimits}[\alpha_{1}^{*}\beta_{1}]+\left|{\beta_{1}}\right|^{2}\int{dx_{2}^{\prime}}$
$\displaystyle\times
P(x^{\prime},x_{2}^{\prime})_{L_{1A}}P(x^{\prime\prime},x_{2}^{\prime})_{L_{1A}}\\}C(x^{\prime})C^{*}(x^{\prime\prime}).$
(12)

Eq. (12) describes the images in the scattering media for the direct
illumination of the thermal light . When

$\displaystyle\frac{{1-\frac{z}{f}}}{f}=\frac{{1-\frac{{z_{1}+L_{1}}}{{f_{1}}}}}{{f_{1}}}.$
(13)

substituting Eqs. (7)-(11b) and (13) into Eq. (6), we can obtain the
correlation function of intensity fluctuations

$\displaystyle\Delta
G^{(2)}(x_{r},x_{t})\propto\left|{\alpha_{1}\alpha_{2}C_{1}+\alpha_{1}\beta_{2}C_{2}}\right.$
$\displaystyle\left.{+\beta_{1}\alpha_{2}C_{3}+\beta_{1}\beta_{2}C_{4}}\right|^{2},$
(14a) $\displaystyle C_{1}=\delta(x_{r}+x_{t})t(-\frac{{f_{1}}}{f}x_{t}),$
(14b) $\displaystyle
C_{2}=t(\frac{{f_{1}}}{f}x_{r})P(-\frac{{f_{1}}}{f}x_{t},\frac{{f_{1}}}{f}x_{r})_{L_{2A}},$
(14c) $\displaystyle
C_{3}=t(-\frac{{f_{1}}}{f}x_{t})P(-\frac{{f_{1}}}{f}x_{t},\frac{{f_{1}}}{f}x_{r})_{L_{1A}},$
(14d) $\displaystyle
C_{4}=\int{dx^{\prime}t(x^{\prime})}P(x^{\prime},\frac{{f_{1}}}{f}x_{r})_{L_{1A}}P(-\frac{{f_{1}}}{f}x_{t},x^{\prime})_{L_{2A}}.$
(14e)

The ghost imaging in the scattering media is described by the Eqs.
(14a)-(14e). And it is a image with high quality for $C_{1}$ and $C_{2}$,
which implies we can still get a image with high quality by ghost imaging in
the scattering media.

From Eqs. (14a)-(14e), if the test detector is an array of pixel detector, and
$x_{t}=-x_{r}$, after some calculation, then

$\displaystyle\Delta
G^{(2)}(x_{r},-x_{r})\propto\left|{\left\\{{\alpha_{1}\alpha_{2}+\alpha_{1}\beta_{2}\left[{\frac{2}{{\pi\Delta
x_{L_{2A}}^{2}}}}\right]^{1/4}}\right.}\right.$
$\displaystyle\left.{+\beta_{1}\alpha_{2}\left.{\left[{\frac{2}{{\pi\Delta
x_{L_{1A}}^{2}}}}\right]^{1/4}}\right\\}t(\frac{{f_{1}}}{f}x_{r})+\beta_{1}\beta_{2}C_{4}}\right|^{2}.$
(15)

if the test detector is a bucket detector, then

$\displaystyle\Delta
G^{(2)}(x_{r})\propto\int{dx_{t}}\left|{\alpha_{1}\alpha_{2}C_{1}+\alpha_{1}\beta_{2}C_{2}}\right.$
$\displaystyle\left.{+\beta_{1}\alpha_{2}C_{3}+\beta_{1}\beta_{2}C_{4}}\right|^{2}.$
(16)

Eqs. (15) and (16) represent ghost imaging when the test detector is an array
of pixel detector or a bucket detector, respectively. From Eqs. (15), if
$\beta_{1}=0$(namely $L_{1}=0$) or $\beta_{2}=0$(namely $L_{2}=0$), it is
clear to find that we can still obtain a image with high quality by ghost
imaging in the scattering media. if $\beta_{1}\neq 0$ and $\beta_{2}\neq 0$
and as the increase of $\beta_{1}$ and $\beta_{2}$, the qualities of ghost
images will reduce, and the term including $C_{4}$ is the main reason leading
to the degradation of the qualities of the images.

For $L_{1}=0$, there is only multiple scattering between the object plane and
the test detector, then we have $\left|\alpha_{1}\right|=1,\beta_{1}=0$. By
Eq. (12), the intensity distribution in the test path is

$\displaystyle
I_{t}(x_{t})\propto(\left|{\alpha_{2}}\right|^{2}+2C_{5t})\left|{t(-\frac{{f_{1}}}{f}x_{t})}\right|^{2}+C_{6t}\left|{\beta_{2}}\right|^{2},$
(17a) $\displaystyle C_{5t}=[\frac{2}{{\pi\Delta
x_{L_{2A}}^{2}}}]^{1/4}{\mathop{\rm Re}\nolimits}[\alpha_{2}^{*}\beta_{2}],$
(17b) $\displaystyle
C_{6t}=\int{dx^{\prime}}\left|{t(x^{\prime})}\right|^{2}\left|{P(-\frac{{f_{1}}}{f}x_{t},x^{\prime})_{L_{2A}}}\right|^{2}$
$\displaystyle=[\frac{2}{{\pi\Delta
x_{L_{2A}}^{2}}}]^{1/2}\int{dx^{\prime}}\left|{t(x^{\prime})}\right|^{2}$
$\displaystyle\times\exp\left\\{{-\frac{2}{{\Delta
x_{L_{2A}}^{2}}}(x^{\prime}+\frac{{f_{1}}}{f}x_{t})^{2}}\right\\}.$ (17c)

the last term in Eq. (17a) is the main reason leading to the decrease of the
quality of the image when there is multiple scattering between the object
plane and the detector. With the increase of $\beta_{2}$ (and the decrease of
the probability amplitude $\alpha_{2}$), the quality of the image will be
further degraded.

Form Eq. (15), when the test detector is an array of pixel detector, after
some calculation, then

$\displaystyle\Delta
G^{(2)}(x_{r},-x_{r})\propto\left|{\alpha_{2}+\beta_{2}\left[{\frac{2}{{\pi\Delta
x_{L_{2A}}^{2}}}}\right]^{1/4}}\right|^{2}$
$\displaystyle\times\left|{t(\frac{{f_{1}}}{f}x_{r})}\right|^{2}.$ (18)

If the test detector is a bucket detector, By Eq. (16), then

$\displaystyle\Delta
G^{(2)}(x_{r})\propto(\left|\alpha_{2}\right|^{2}+2C_{5}+C_{6}\left|\beta_{2}\right|^{2})\left|{t(\frac{{f_{1}}}{f}x_{r})}\right|^{2},$
(19a) $\displaystyle C_{5}=[\frac{2}{{\pi\Delta
x_{L_{2A}}^{2}}}]^{1/4}{\mathop{\rm Re}\nolimits}[\alpha_{2}^{*}\beta_{2}],$
(19b) $\displaystyle
C_{6}=\int{dx_{t}\left|{P(-\frac{{f_{1}}}{f}x_{t},\frac{{f_{1}}}{f}x_{r})_{L_{2A}}}\right|^{2}}\sim
1.$ (19c)

from Eqs. (18)-(19c), we find that whether the test detector is an array of
pixel detector or a bucket detector, the qualities of ghost images can be
obtained even though there is multiple scattering between the object plane and
the test detector.

For $L_{2}=0$, there is only multiple scattering between the source and the
object plane, then we can gain $\left|\alpha_{2}\right|=1,\beta_{2}=0$. By Eq.
(12), the intensity distribution in the test path is

$\displaystyle
I_{t}(x_{t})\propto(\left|{\alpha_{1}}\right|^{2}+2C_{7t}+C_{8t}\left|{\beta_{1}}\right|^{2})\left|{t(-\frac{{f_{1}}}{f}x_{t})}\right|^{2},$
(20a) $\displaystyle C_{7t}=[\frac{2}{{\pi\Delta
x_{L_{1A}}^{2}}}]^{1/4}{\mathop{\rm Re}\nolimits}[\alpha_{1}^{*}\beta_{1}],$
(20b) $\displaystyle
C_{8t}=\int{dx_{2}^{\prime}}\left|{P(-\frac{{f_{1}}}{f}x_{t},x_{2}^{\prime})_{L_{1A}}}\right|^{2}\sim
1.$ (20c)

From Eqs. (20a)-(20c), we find that the multiple scattering between the source
and the object plane has no effect on the quality of the image.

From Eq. (15), when the test detector is an array of pixel detector, after
some calculation, then

$\displaystyle\Delta
G^{(2)}(x_{r},-x_{r})\propto\left|{\alpha_{1}+\beta_{1}\left[{\frac{2}{{\pi\Delta
x_{L_{1A}}^{2}}}}\right]^{1/4}}\right|^{2}$
$\displaystyle\times\left|{t(\frac{{f_{1}}}{f}x_{r})}\right|^{2}.$ (21)

which reveals that we can still obtain a image with high quality when the test
detector is an array of pixel detector even if there is multiple scattering
between the source and the object plane.

If the test detector is a bucket detector, By Eq. (16), then

$\displaystyle\Delta
G^{(2)}(x_{r})\propto(\left|\alpha_{1}\right|^{2}+2C_{7})\left|{t(\frac{{f_{1}}}{f}x_{r})}\right|^{2}+C_{8}\left|\beta_{1}\right|^{2},$
(22a) $\displaystyle C_{7t}=[\frac{2}{{\pi\Delta
x_{L_{1A}}^{2}}}]^{1/4}{\mathop{\rm Re}\nolimits}[\alpha_{1}^{*}\beta_{1}],$
(22b) $\displaystyle
C_{8}=\int{dx_{t}}\left|{t(-\frac{{f_{1}}}{f}x_{t})P(-\frac{{f_{1}}}{f}x_{t},\frac{{f_{1}}}{f}x_{r})_{L_{1A}}}\right|^{2}$
$\displaystyle=\left[{\frac{2}{{\pi\Delta
x_{L_{1A}}^{2}}}}\right]^{1/2}\int{dx_{t}}\left|{t(-\frac{{f_{1}}}{f}x_{t})}\right|^{2}$
$\displaystyle\times\exp\left\\{{-\frac{{2f_{1}^{2}}}{{\Delta
x_{L_{1A}}^{2}f^{2}}}(x_{t}+x_{r})^{2}}\right\\}.$ (22c)

where $C_{8}$ is the main factor leading to the decrease of the qualities of
ghost images. $f_{1}>f$ is helpful to improve the quality of the image.

The second-order degree of coherence $g^{(2)}(x_{1},x_{2})$ at the positions
$x_{1}$ and $x_{2}$ can be defined as follows

$\displaystyle g^{(2)}(x_{1},x_{2})=\frac{{\left\langle{I_{1}\cdot
I_{2}}\right\rangle}}{{\left\langle{I_{1}}\right\rangle\left\langle{I_{2}}\right\rangle}}=1+\frac{{\left\langle{\Delta
I_{1}\cdot\Delta
I_{2}}\right\rangle}}{{\left\langle{I_{1}}\right\rangle\left\langle{I_{2}}\right\rangle}}.$
(23)

for ghost imaging, The second-order degree of coherence
$g^{(2)}(x_{t1},x_{t2})$ obtained only by the correlation measurement in the
test path reveals the characteristic of the source, whereas
$g^{(2)}(x_{r},x_{t})$ describes the correlation between two paths. And the
degradation of $g^{(2)}(x_{r},x_{t})$ will lead to the decay of intensity
fluctuations, which has a great effect on the visibility of ghost images.

Figs. 3-7 present numerical results of imaging a single slit in scattering
media based on Eqs. (12) and (15)-(16) (in which we take $\lambda$=650nm,
$f_{1}$=400mm, $f$=250mm, the single slit width a=0.2mm). From Fig. 3(a), as
the increase of $L_{2}$, the qualities of the images will decrease obviously
when the object is illuminated directly by the thermal light. The quality of
the image also degrade rapidly with the increase of the broadening length
$\triangle x_{2}$ and the decrease of the probability amplitude $\alpha_{2}$
when the object is fixed in the position of $L_{1}$=0mm, $L_{2}$=100mm (Fig.
3(b), (c)), which accord with the results described by the Eqs (17a)-(17c).

Figure 3: Factors which have the effect on the qualities of the images when
the object is illuminated directly by the thermal light. (a). Images of a
single slit at different positions in the scattering media; (b). Images of a
single slit for different probability amplitude $\alpha_{2}$ when $L_{1}$=0mm,
$L_{2}$=100mm and the broadening length $\triangle x_{2}$=5.0; and (c). Images
of a single slit for different probability amplitude $\triangle x_{2}$ when
$L_{1}$=0mm, $L_{2}$=100mm and the broadening length $\alpha_{2}$=0.05.

Fig. 4 shows the numerical results when the position of the object in the
scattering media is shifted. By Fig. 4(a), we can find that if the test
detector is an array of pixel detector, multiple scattering has no effect on
the qualities of ghost images when there is only multiple scattering either
between the object plane and the test detector or between the object plane and
the source. As the object is fixed closing to the middle of scattering media,
the qualities of ghost images will decay. However, if the test detector is a
bucket detector, the qualities of ghost images will decrease sharply with the
degradation of $L_{1}$, which is opposite absolutely to imaging when the
object is illuminated directly by the thermal light (Fig. 3(a)).

Figure 4: Relationship between the qualities of ghost images and the position
of the object in scattering media. (a). the test detector is an array of pixel
detector; and (b). the test detector is a bucket detector.

In Fig. 5, We give the comparison of the qualities of images between imaging
by the direct illumination of the thermal light and ghost imaging when the
object is fixed in the middle of scattering media. It is easy to find that we
can only obtain the same image with low quality as imaging by the direct
illumination of the thermal light when the test detector is a bucket detector
by ghost imaging. But a image with high quality can still be gained if the
test detector is an array of pixel detector.

Figure 5: Comparison between imaging by the direct illumination of the thermal
light and ghost imaging when $L_{1}$=50mm, $L_{2}$=50mm, the broadening length
$\triangle x_{1}$=2.0, $\triangle x_{2}$=2.0 and $\alpha_{1}$=0.1,
$\alpha_{2}$=0.1.

Results shown in Fig. 6 reveals that the qualities of ghost images will reduce
obviously as the degradation of the probability amplitude $\alpha_{1}$ and
$\alpha_{2}$. And the object is still fixed in the middle of scattering media.

Figure 6: Dependence of probability amplitude $\alpha_{2}$ on the qualities of
ghost images when $L_{1}$=50mm, $L_{2}$=50mm and the broadening length
$\triangle x_{1}$=2.0, $\triangle x_{2}$=2.0.

In order to improve the qualities of ghost images further, we investigate a
new method to the effect on the qualities of ghost images by using a circular
aperture gating to change the transverse coherent length near the scattering
media in Fig. 7, which is different from the method to change the transverse
size of the source. But when the space interval $\Delta s$ is greater than the
characteristic scale of the object, the diffraction will emerge.

Figure 7: The effect of the space interval $\Delta s$ on the qualities of
ghost images when a circular aperture gating with different space interval
$\Delta s$ is fixed on the plane $x_{0}$. $L_{1}$=50mm, $L_{2}$=50mm, the
broadening length $\triangle x_{1}$=2.0, $\triangle x_{2}$=2.0 and
$\alpha_{1}$=0.1, $\alpha_{2}$=0.1.

##  III experiment

In the experiments, we prepare a suspension liquid which is composed by
emulsion polymerization particles with particle diameter $D$=3.26$\mu$m and
the solution $NaCl$ with the density $\rho$=1.19 $g/cm^{3}$. The vessel used
to put the suspension liquid is designed as 40mm$\times$40mm$\times$20mm. The
liquid can be considered as strong multiple scattering media. And we take
$\lambda$=650nm, $f_{1}$=400mm, $f$=250mm, $z$=211mm, $z_{1}$=300mm,
$z_{2}$=390mm, $z_{3}$=243.8mm. The minimum characteristic scale of the object
(‘zhong’ ring) is 60 $\mu$m and the diameter of the ring is 1.6mm. And the
detectors in both paths are arrays of pixel detectors.

Images shown in Fig. 8 (1) and (2) were the experimental results of the object
(‘zhong’ ring) by direct illumination and ghost imaging when the object was
fixed at different positions in the scattering media, respectively. Form Fig.
8, when the object is illuminated directly by the thermal light, the qualities
of images will reduce as the increase of the length $L_{2}$ of scattering
media. However, when there is only strong multiple scattering between the
object plane and the source or between the object plane and the detector
$D_{t}$, we can both obtain ghost images with high qualities. If the object is
fixed in the middle of the scattering media, the visibility of ghost images
will reduce, but the resolution doesn’t degrade. All discussed above accord
with the theoretical results described by the Eqs. (1).

Figure 8: Images of the aperture (‘zhong’ ring) when the object is fixed in
different position of the scattering media. (a). $L_{1}$=0mm, $L_{2}$=40mm;
(b). $L_{1}$=20mm, $L_{2}$=20mm; and (c). $L_{1}$=40mm, $L_{2}$=0mm. (1). when
the object was illuminated directly by thermal light; (2). ghost imaging.

In Fig. 9, we demonstrated experimentally that the effect of the concentration
of scattering media or the coherent length located at the object plane on
ghost imaging when the object is fixed in the middle of scattering media. The
visibility of ghost images will reduce with the increase of the concentration
of scattering media. When the coherent length located at the object plane
becomes long, the visibility of ghost images will improve. However, the
resolution will degrade.

Figure 9: Effect of the concentration of scattering media and the coherent
length located at the object plane on ghost imaging. (a). $L_{c}$=40.6$\mu$m,
6 drops; (b). $L_{c}$=40.6$\mu$m, 3 drops; and (c). $L_{c}$=192.5$\mu$m, 6
drops. (1). when the object was illuminated directly by thermal light; (2).
ghost imaging.

Carves shown in Figs. 10 and 11 are the effect of concentration of emulsion
polymerization particles on $g^{(2)}$ when $L_{1}$=40mm and $L_{2}$=0mm and
when $L_{1}$=20mm and $L_{2}$=20mm without the object, respectively. We can
find that $g^{(2)}(x_{t1},x_{t2})$ and $g^{(2)}(x_{r},x_{t})$ decay sharply
with the increase of concentration of emulsion polymerization particles. When
there is two drops emulsion polymerization particles put into the solution
$NaCl$, the cross-correlation coefficient will be lower than 1.10.

Figure 10: The effect of concentration of emulsion polymerization particles on
$g^{(2)}$ when $L_{1}$=40mm and $L_{2}$=0mm without the object.

Figure 11: The effect of concentration of emulsion polymerization particles on
$g^{(2)}$ when $L_{1}$=20mm and $L_{2}$=20mm without the object.

When there are five drops emulsion polymerization particles put into the
solution $NaCl$, the intensity distribution on the detector $D_{r}$ is
heterogenous because there is no multiple scattering. However, we can find the
homogeneous intensity distribution on the detector $D_{t}$ (in Fig. 12).

Figure 12: Images from a single frame speckle distribution when $L_{1}$=20mm
and $L_{2}$=20mm without the object and there are five drops emulsion
polymerization particles put into the solution $NaCl$, (a). from the CCD
camera $D_{t}$, (b). from the CCD camera $D_{r}$.

##  IV discussion and inclusion

When the object is illuminated directly by thermal light, multiple scattering
between the object plane and the detector $D_{t}$ is the main reason leading
to the degradation of the qualities of images (Fig. 3). In scattering media,
different from images obtained by the first-order correlation of light field,
ghost imaging causes the separation of detection and imaging. Even if the
information from the object is distorted by multiple scattering, there is no
effect on ghost imaging. Because multiple scattering destroys the correlation
of the light field, the visibility of ghost images is degraded but the
resolutions doesn’t degrade. However, the visibility of ghost images can be
enhanced by means of the decrease of the resolution Shen ; Cai1 ; Cai2 ;
Gatti4 . The experimental result in Fig. 9 (c) also demonstrated that.
Transmission information of the light on the object plane (namely
$\alpha_{1}$) has a great effect on the visibility of ghost images. Because
multiple scattering leads to the decay of the correlation between the test and
reference pathes with the decrease of Transmission information of the light
(Figs 8 and 9). The higher the Transmission information is, the better
visibility the ghost image has. When there is no multiple scattering between
the object plane and the source, then ballistic component of the light on the
object plane is equal to 1, so we can always obtain a ghost image with high
quality whether there is multiple scattering between the object plane and the
detector $D_{t}$. Otherwise, because the ballistic component of the light is
impossible to be 0, we can always see the shapes of the object (Figs. 3(b) and
6). The degradation of the qualities of the images is provoked by the decay of
visibility in scattering media, which is different absolutely from the low
resolution caused by diffraction-limited. When the object is fixed in the
middle of the scattering media, as shown in Fig. 11, even if the second-order
degree of coherence $g^{(2)}$ is very low, we can obtain a ghost image with
much better quality than the image gained when the object is illuminated
directly by thermal light. By the characteristics of ghost imaging, the
qualities of ghost images does not depend on the intensity of the light but on
the intensity fluctuations. As the multiple scattering increased, the
intensity fluctuations will reduce (Figs 10 and 11), which leads to the
degradation of the visibility of the image. The expression in the Eq. (15) can
explain basically the experimental results and the fact discussed above, which
shows that the results described by the Eq. (15) are reasonable for the
description of ghost imaging in scattering media. However, when the test
detector is a bucket detector, the qualities of ghost images depend on the
multiple scattering between the object plane and the source, which is opposite
to imaging by the direct illumination of the thermal light (Fig. 3(a)). By the
results in Fig. 8 (a) and (c), maybe we can also obtain a ghost image with
high visibility by means of eliminating the multiple scattering between the
object plane and the source or between the object plane and the detector
$D_{t}$. One is like the technique of DOT. Firstly we obtain a improved image
by iterative recovery method of DOT from a single frame of image collected by
the detector $D_{t}$, then we will get a ghost image with improved visibility
by the correlation measurement of intensity fluctuations between the detector
$D_{r}$ and the improved images. The other is that we get the intensity
distribution on the object plane by the measurement to transmission and
reflection coefficients, then do the correlation measurement of the intensity
fluctuations between the detector $D_{t}$ and intensity distribution obtained
by numerical simulation. The method discussed in Fig. 7 can improve the
visibility of ghost images to some extent. Generally speaking, by ghost
imaging method, we can always obtain a image with much better quality on the
base of the image obtained by a novel conventional imaging technique with
thermal light. For entangled source, because of the entanglement
characteristic of the two photons, maybe we can also improve the visibility of
ghost images. Based on the effect of photon bunching, and the new source with
high $g^{(2)}$ can be obtained, the visibility of ghost images may also be
enhanced obviously.

In medical science, in order to avoid ionizing radiation of X-ray, optical
photons provides nonionizing and safe radiation for medical applications.
Recently there has been increasing interest in the field of the imaging, test
and diagnosis of the biological tissues with the infrared and the near
infrared light Wang ; Sachin ; Brian . Because the near infrared light around
700-nm wavelength can penetrate several centimeters into biological tissue
Brian . But several factors still limit the imaging quality. Because most
biological tissues are characterized by strong optical scattering and hence
are referred to as scattering media or turbid media. The images suffer reduced
resolution and contrast due to multiple scattering, which leads to a low
efficiency and accuracy of diagnosis and a difficulty of analysis in medical
science. So the diffusion-like behavior of light in biological tissue presents
a key challenge for optical imaging. The ghost imaging discussed here may
solve the problems about the low quality of the imaging in biological tissues.

In conclusion, for the first time, detection and imaging of the object
information are separated by ghost imaging, which provides a new way for
imaging in scattering media. we have demonstrated experimentally and
theoretically for the first time that the images with high qualities in a
scattering media can still be obtained for ghost imaging when the test
detector is an array of pixel detector and there is only multiple scattering
between the object plane and the source or between the object plane and the
detector. When the object is fixed in the middle of scattering media, we can
also gain ghost images with much better qualities than the images obtained
when the object is illuminated directly by the thermal light. The probability
amplitudes of the incident light $\alpha_{1}$ and $\alpha_{2}$ have a great
effect on the visibility of ghost images. These results will be very useful
for the imaging and diagnosis in medical science.

The work was partly supported by the Hi-Tech Research and Development Program
of China, Project No. 2006AA12Z115, and Shanghai Fundamental Research Project,
Project No. 06JC14069.

## References

  * (1) G. W. Kamerman, Proc. SPIE, 4377, 126-131 (2001).
  * (2) H. G. Booker, and D. C. Miller, Joural of Atmospheric and Terrestrial Physics, 42, 3 (1980).
  * (3) T. S. Mckechnie, J. Opt. Soc. Am. A, 8,2 (1991).
  * (4) Raine D. A. , and Brenizer J. S. Materials evalution, 55, 1174 (1997).
  * (5) K. P. Maher, J. F . Malone, and Brain Chance, Contemporary Physics, 38, 131 (1997).
  * (6) Lihong V. Wang and Hsin-i Wu, Biomedical optics: principle and imaging, (2007).
  * (7) A. P. Gibson, J. C. Hebden and S. R. Arridge, Phys. Med. Biol. 50 R1-R43 (2005).
  * (8) A. Yodh, and B. Chance, Phys. Today, 48, 34 (1995).
  * (9) S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
  * (10) S. R. Arridge, Topical Review, R41-93 (1999).
  * (11) Sachin V. patwardhan, and Joseph P. Culver, J. Biomed. Opt. 13(1), 011009 (2008).
  * (12) J. C. Hebden, S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 42, 825 (1997).
  * (13) B. Chance and R. R. Alfano, Proc. SPIE, 2389 (1995).
  * (14) M. Laubscher, M. Ducros, B. Karamata, Opt. Exp. 10, 9 (2002)
  * (15) Brian C. Wilson, Eva M. Sevick, Michael S. Patterson, and Britton Chance, Proceedings of the IEEE. 80, 6 (1992).
  * (16) T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A. 52. R3429 (1995).
  * (17) J. Cheng and S. Han, Phys. Rev. Lett. 92. 093903 (2004).
  * (18) A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A. 70. 013802 (2004).
  * (19) A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. Lett. 93. 093602 (2004).
  * (20) A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94. 063601 (2005).
  * (21) R. S. Bennink, S. J. Bentley, and R. W. Boyd, Phys. Rev. Lett. 89. 113601 (2002).
  * (22) L. Basano and P. Ottonello, Appl. Phys.Lett. 89, 091109 (2006).
  * (23) Minghui Zhang, Qing Wei, and Shensheng Han, Phys. Lett. A. 366, 569-574 (2007).
  * (24) G. Scarcelli, V. Berardi and Y. Shih, Phys. Rev. Lett. 96, 063602 (2006).
  * (25) A. Gatti, M. Bondani, L. A.Lugiato, M. G. A. Pairs, and C. Fabre, Phys. Rev. Lett. 98, 039301 (2007).
  * (26) G. Scarcelli, V. Berardi and Y. Shih, Phys. Rev. Lett. 98, 039302 (2007).
  * (27) Pengli Zhang, Wenlin Gong, Xia Shen, and Shensheng Han, arXiv. Quantum-ph/0804.0575 (2008).
  * (28) R. J. Glauber, Phys. Rev. 130, 2529 (1963).
  * (29) J. W. Goodman, Introduction to Fourier optics (1976).
  * (30) G. Mie, Ann. Phys. 330, 377 (1908).
  * (31) D. Toublanc, Appl. Opt. 35, 3270 (1996).
  * (32) S. K. Sharma, and A. K. Roy, J. Quant. Spectrosc. Radiat. Transfer. 64, 327 (2000).
  * (33) Jessica C. Ramella-Roman, Scott A. Prahl, and Steven L. Jacques, Opt. Exp. 13, 25 (2005).
  * (34) Milun J. Rakovic and George W. kattawar, Appl. opt. 37, 15 (1999).
  * (35) A. A. M. Mustafa and Daphne F. Jackson, Phys. Med. Biol. 26, 461 (1981).
  * (36) J. S. Faulkner and Eva A. Horvath, Phys. Rev. B. 44, 8467 (1991).
  * (37) W. H. Wells, J. Opt. Soc. Am. 59, 6 (1969).
  * (38) H. T. Yura, Appl. opt. 10, 1 (1971).
  * (39) Rene Hassanein, Eberhard Lehmann, and Peter Vontobel, Nucl. Ins. Meth. Phys. Res. A. 542, 353 (2005).
  * (40) P. N. Segre and P. N. Pusey, Phys. Rev. Lett. 77, 4 (1996).
  * (41) Shen Xia, Bai Yan-Feng, and Han Shen-Sheng, Chin. Phys. Lett. 25, 11 (2007).
  * (42) Y. Cai and S. Zhu, Opt. Lett. 29, 2716 (2004).
  * (43) Y. Cai and S. Zhu, Opt. Lett. 30, 388 (2005).
  * (44) A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, J. Mod. Opt. 53, 739 (2006).


# Carbon-Efficient Neural Architecture Search

Yiyang Zhao  Worcester Polytechnic Institute<EMAIL_ADDRESS>and  Tian Guo
Worcester Polytechnic Institute<EMAIL_ADDRESS>
(2023)

###### Abstract.

This work presents a novel approach to neural architecture search (NAS) that
aims to reduce energy costs and increase carbon efficiency during the model
design process. The proposed framework, called carbon-efficient NAS (CE-NAS),
consists of NAS evaluation algorithms with different energy requirements, a
multi-objective optimizer, and a heuristic GPU allocation strategy. CE-NAS
dynamically balances energy-efficient sampling and energy-consuming evaluation
tasks based on current carbon emissions. Using a recent NAS benchmark dataset
and two carbon traces, our trace-driven simulations demonstrate that CE-NAS
achieves better carbon and search efficiency than the three baselines.

Sustainability, carbon aware, neural architecture search

††copyright: acmcopyright††journalyear: 2023††copyright:
acmlicensed††conference: 2nd Workshop on Sustainable Computer Systems; July 9,
2023; Boston, MA, USA††booktitle: 2nd Workshop on Sustainable Computer Systems
(HotCarbon ’23), July 9, 2023, Boston, MA, USA††price: 15.00††doi:
10.1145/3604930.3605708††isbn: 979-8-4007-0242-6/23/07††ccs: Social and
professional topics Sustainability††ccs: Computing methodologies Neural
networks††ccs: Computing methodologies Search methodologies

##  1\. Introduction

Deep Learning (DL) has become an increasingly important field in computer
science, with applications ranging from healthcare to transportation to energy
management. However, DL training is notoriously energy-intensive and
significantly contributes to today’s carbon emissions (Strubell et al., 2019;
Zoph et al., 2018). The main culprit comes down to the iterative nature of
training, which requires evaluating and updating model parameters based on a
large amount of data.

Neural architecture search (NAS) has emerged as a means to automate the design
of DL models. At the high level, NAS often involves leveraging search
algorithms to explore a massive architecture design space, ranging from
hundreds of millions to trillions of candidates (Zoph et al., 2018; Real et
al., 2019; Wang et al., 2019b; Zhao et al., 2021, 2022; Liu et al., 2019), by
training and evaluating a subset of architectures. In searching for the best
architecture for different application domains, many NAS works have reported
using thousands of GPU-hours (Zoph et al., 2018; Real et al., 2019; Wang et
al., 2019b, a; Zhao et al., 2022).

The environmental impact of NAS, if left untamed, can be substantial. While
recent works have significantly improved the search efficiency of NAS (Zoph et
al., 2018; Real et al., 2019; Wang et al., 2019b, a; Zhao et al., 2021), e.g.,
reducing the GPU-hours to tens of hours without sacrificing the architecture
quality, there still lacks _conscious efforts in reducing carbon emissions_.
As noted in a recent vision paper by Bashir et al. (Bashir et al., 2021),
energy efficiency can help reduce carbon emissions but is not equivalent to
carbon efficiency. This paper aims to bridge the gap between carbon and energy
efficiency with a new NAS framework designed to be carbon-aware from the
outset.

The proposed framework, termed CE-NAS, will tackle the high carbon emission
problem from two main aspects. First, CE-NAS will regulate the model design
process by _deciding when to use different NAS evaluation strategies based on
the varying carbon intensity_. To elaborate, given a fixed amount of GPU
resources, CE-NAS will allocate more resources to energy-efficient NAS
evaluation strategies, e.g., one-shot NAS (Pham et al., 2018; Bender et al.,
2018; Liu et al., 2019; Zhao et al., 2021; Cai et al., 2019, 2020), during
periods of high carbon intensity. Conversely, during periods of low carbon
intensity, CE-NAS will shift the focus to running energy-intensive but more
effective NAS evaluation strategies, e.g., vanilla NAS (Zoph et al., 2018;
Real et al., 2019; Wang et al., 2019b, a). Second, the CE-NAS framework will
support energy and carbon-efficient DL model design via multi-objective
optimization. Specifically, we will leverage a recent learning-based multi-
objective optimizer LaMOO (Zhao et al., 2022) and integrate it to CE-NAS to
achieve search efficiency.

Based on these two design guidelines, we sketch out the basis of the proposed
CE-NAS framework in Figure 1 and implement a trace-driven simulator to
investigate the promise of CE-NAS in improving carbon and search efficiencies.
Using carbon emission traces from electricityMap (Map, [n. d.]) and a new NAS
dataset called HW-NASBench (Li et al., 2021), we show that CE-NAS has the
least relative carbon emissions and only marginally lower search efficiency
compared to vanilla LaMOO (Zhao et al., 2022). Based on our investigation, we
believe there are many fruitful directions in the context of CE-NAS which we
outline in §5. We hope this discussion will serve as the blueprint and a
baseline for building a carbon-efficient NAS framework.

Table 1. Comparison of energy-efficient NAS evaluation methods. _Eval. cost_
refers to the cost of obtaining the evaluation results. _Init. cost_ describes
additional dataset preparation and the time required for training the model
(e.g., supernet or predictor). _Accuracy_ measures the rank correlation
between the evaluation method and the actual rank. Predictor-based methods
require _Extra data_ as a training set to construct the prediction model.

Method | Eval. cost | Init. cost | Accuracy | Extra data
---|---|---|---|---
One-shot (Liu et al., 2019; Pham et al., 2018; Zhao et al., 2021; Xu et al., 2020; Cai et al., 2020) | Low | Low | Intermediate | No
Predictor (Liu et al., 2018; Domhan et al., 2015; Wang et al., 2019b) | Low | High$\dagger$ | High$\dagger$ | Yes
Low-fidelity (Zoph et al., 2018; Real et al., 2019; Liu et al., 2019; Wang et al., 2019b; Klein et al., 2017) | High | None | Intermediate$\ddagger$ | No
Gradient Proxy (Xu et al., 2021) | Low | Low | Intermediate | No

  * $\dagger$

It depends on the size of extra data.

  * $\ddagger$

It depends on the extent of the fidelity.

Figure 1. An overview of CE-NAS. The sampling and evaluation tasks will be
dispatched with different GPU resources based on carbon emission intensity
during the neural architecture search.

##  2\. NAS and its Carbon Impact

Neural architecture search (NAS) is a technique for automating the design of
neural network architectures. NAS aims to find an optimal network architecture
that performs well on a specific task without human intervention. NAS-designed
neural architectures achieved state-of-the-art performances on many AI
applications, such as image classification, object detection, and image
segmentation (Zoph et al., 2018; Wang et al., 2020b; Ghiasi et al., 2019; Real
et al., 2019; Wang et al., 2019b).

However, NAS typically requires significant computational resources (e.g.,
GPUs) to find the optimal architecture, with most of these resources being
used for architecture evaluation. For example, Zoph et al. (Zoph et al., 2018)
used 800 GPUs for 28 days, equivalent to 22,400 GPU hours, to obtain the final
architectures. Strubell et al. (Strubell et al., 2019) found that a single NAS
solution can emit as much carbon as five cars during its lifetime. These
findings highlight the need for energy and carbon-efficient NAS methods to
reduce the environmental impact of AI research.

Existing works on energy-efficient NAS often focus on improving the evaluation
phase, e.g., via weight-sharing (Pham et al., 2018; Liu et al., 2019; Xu et
al., 2020; Chen et al., 2019; Cai et al., 2020), performance predictor (Liu et
al., 2018; Cai et al., 2020; Wang et al., 2019b; Zela et al., 2022), low-
fidelity NAS evaluation (Zoph et al., 2018; Real et al., 2019; Wang et al.,
2019b, a; Liu et al., 2019; Klein et al., 2017), and gradient proxy NAS (Xu et
al., 2021). A comparison of these methods can be found in Table 1. Weight-
sharing leverages the accuracy estimated by a _supernet_ as a proxy for the
true architecture performance, while gradient proxy NAS uses the gradient as a
proxy. These proxy-based methods, although incurring smaller search costs in
terms of energy, can have lower search efficiency because their estimated
architecture accuracy may have poor rank correlation (Zhao et al., 2021).
Performance predictors provide a more accurate performance prediction than
weight-sharing and gradient proxy NAS. Still, their accuracy heavily relies on
the volume and quality of the training data, which can be very expensive to
create (Ying et al., 2019; Zela et al., 2022). Low-fidelity evaluation still
requires training each searched architecture, leading to limited energy
savings.

In short, utilizing existing energy-efficient NAS methods requires careful
consideration of the search quality and efficiency trade-offs; however,
naively applying these methods may not even lead to energy savings, not to
mention lower carbon emissions. In this work, we achieve the goals of search
efficiency, search quality, and carbon efficiency by leveraging a generic
multi-objective optimizer (Zhao et al., 2022), a mix of energy-efficient
(Bender et al., 2018; Pham et al., 2018; Liu et al., 2019; Zhao et al., 2021)
and energy-consuming (Zoph et al., 2018; Real et al., 2019; Wang et al.,
2019b, a) evaluation methods, and a carbon-aware GPU resource allocation
strategy.

##  3\. Research Roadmap

In this section, we present an overview of the proposed CE-NAS framework
(Figure 1) and sketch the basis for each component. We hope this discussion
will serve as the blueprint and a baseline for designing a carbon-efficient
NAS framework.

###  3.1. CE-NAS Overview

As observed in (Bashir et al., 2021), grid carbon emissions vary
geographically and temporally based on the mix of active generators.
Consequently, different carbon emissions arise even when consuming the same
electricity at different locations or times. Operating the NAS framework
without considering costs across every carbon period will lead to carbon waste
when utilizing carbon-consuming but effective NAS methods. Conversely,
employing carbon-saving yet sample-inefficient NAS methods may deteriorate
search performance.

To address this issue, we propose a carbon-aware adaptive NAS search strategy
that balances energy consumption during high-carbon and low-carbon periods.
Our strategy decouples the two parts of a NAS search process—evaluation
(energy-consuming) and sampling (energy-saving)—and handles them independently
across different carbon periods. The basic idea involves leveraging the
energy-efficient one-shot NAS (Bender et al., 2018; Pham et al., 2018; Liu et
al., 2019; Zhao et al., 2021) to effectively estimate the accuracy of
architectures in the sampling process during periods of high carbon intensity.
Meanwhile, we will run the expensive evaluation part primarily during low-
carbon periods. In the following sections, we will provide a detailed
explanation of the carbon-aware NAS strategy.

###  3.2. Search Initialization

Similar to other optimization problems (Daulton et al., 2020; Wang et al.,
2020a; Zhao et al., 2022; Real et al., 2019), the first step in our proposed
carbon-efficient NAS framework involves initializing the search process by
randomly selecting several architectures, a, from the search space, $\Omega$,
and evaluating their accuracy, $E(\textbf{a})$, carbon emissions,
$C(\textbf{a})$, and inference energy, $I(\textbf{a})$. The resulting samples
are then added to the observed samples set, $\mathcal{P}$.

Here, we define two types of methods for evaluating the accuracy of
architectures. One is actual training, which trains the architecture $a$ from
scratch until convergence and evaluates it to obtain its true accuracy,
$E(a)$. Another method is called _one-shot evaluation_ (Bender et al., 2018;
Pham et al., 2018; Liu et al., 2019), which leverages a trained _supernet_ to
estimate the accuracy of the architecture, denoted as $E^{{}^{\prime}}(a)$.
Note that obtaining $E^{{}^{\prime}}(a)$ is energy-efficient; however, due to
the co-adaption among operations (Zhao et al., 2021), $E^{{}^{\prime}}(a)$ is
often not as accurate as $E(a)$. We train all the sampled architectures in the
initialization stage to obtain their true accuracy for further search.

###  3.3. Energy-Efficient Architecture Sampling

To search for architectures with high inference accuracy and low inference
energy, we formulate the search problem as a multi-objective optimization
(MOO).

#### Primer.

Mathematically, in multi-objective optimization we optimize $M$ objectives
${\bm{f}}({\bm{x}})=[f_{1}({\bm{x}}),f_{2}({\bm{x}}),\ldots,f_{M}({\bm{x}})]\in{\textnormal{r}}^{M}$:

(1) $\displaystyle\min\;$ $\displaystyle
f_{1}({\bm{x}}),f_{2}({\bm{x}}),...,f_{M}({\bm{x}})$
$\displaystyle\mathrm{s.t.}\;$ $\displaystyle{\bm{x}}\in\Omega,$

where $f_{i}({\bm{x}})$ denotes the function of objective $i$. Modern MOO
methods aim to find the problem’s entire _Pareto frontier_ , the set of
solutions that are not _dominated_ by any other feasible solutions. Here we
define _dominance_ ${\bm{y}}\prec_{\bm{f}}{\bm{x}}$ as $f_{i}({\bm{x}})\leq
f_{i}({\bm{y}})$ for all functions $f_{i}$, and there exists at least one $i$
s.t. $f_{i}({\bm{x}})<f_{i}({\bm{y}})$, $1\leq i\leq M$. If the condition
holds, a solution ${\bm{x}}$ is always better than solution ${\bm{y}}$.

#### Multi-objective search space partition.

We leverage the recently proposed multi-objective optimizer called LaMOO (Zhao
et al., 2022) that learns to partition the search space from observed samples
to focus on promising regions likely to contain the Pareto frontier. LaMOO is
a general optimizer; we can extend it to NAS as follows.

We utilize LaMOO (Zhao et al., 2022) to partition the search space, $\Omega$,
into several sub-search spaces. This partitioning will be based on the
architectures and their true accuracy ($E(\textbf{a})$) and inference energy
($I(\textbf{a})$) as observed in the sample set, $\mathcal{P}$. Specifically,
LaMOO recursively divides the search space into promising and non-promising
regions. Each partitioned region can then be mapped to a node in a search
tree. Using Monte-Carlo Tree Search (MCTS), LaMOO selects the most promising
sub-space (i.e., tree node) for further exploration based on their UCB values
(Auer et al., 2002). This optimal sub-space is denoted as $\Omega_{best}$.

Next, we will construct and train a supernet (Bender et al., 2018; Zhao et
al., 2021), $\mathcal{S}_{best}$, for $\Omega_{best}$. We then use a NAS
search algorithm to identify new architectures that will be used to refine the
search space. In this work, we employ the state-of-the-art multi-objective
Bayesian optimization algorithm qNEHVI (Daulton et al., 2021). This algorithm
will generate new architectures, denoted as $\bm{a_{n}}$, from
$\Omega_{best}$, and estimate their approximate accuracy,
$E^{{}^{\prime}}(\bm{a_{n}})$, using $\mathcal{S}_{best}$. At the same time,
these architectures $\bm{a_{n}}$ are added to a ready-to-train set,
$\mathcal{R}$, consisting of architecture candidates for further training.

Currently, to avoid unnecessary training and energy consumption, we define the
maximum capacity of $\mathcal{R}$ as $Cap(\mathcal{R})$. When the capacity
reaches, i.e., when there are more architectures to train than we have
resources for, the sampling process blocks until spaces free up in
$\mathcal{R}$. The accuracy of architectures, either estimated by
$\mathcal{S}_{best}$ or obtained from training, will be fed back into the
search engine as shown in Figure 1 to repeat the process described above.

As mentioned in §3.2, obtaining estimated accuracy through supernet is energy-
efficient because these architectures can be evaluated without the time-
consuming training. Therefore, during high carbon emission periods, CE-NAS
will try to perform this process to save energy and produce as little carbon
as possible, as shown in the middle left part of Figure 1.

###  3.4. Energy-Consuming Architecture Evaluation

If we perform the entire NAS only using the process described in §3.3, CE-NAS
essentially is performing one-shot NAS within the sub-space
$\mathcal{S}_{best}$. However, it is possible to improve LaMOO’s space
partition with more observed samples, as Zhao et al. showed (Zhao et al.,
2022). This section describes the process to evolve $\mathcal{S}_{best}$
during low carbon emission periods.

At the high level, we will pick new architectures to train to convergence and
use them to refine the search space partition. That is, the architecture
$\bm{a}$, with its true accuracy, $E(\bm{a})$, will be added to the observed
sample set $\mathcal{P}$ to help identify a more advantageous sub-space,
$\Omega_{best}$, for the architecture sampling process. In this work, we sort
the architectures in the ready-to-train set $\mathcal{R}$ by their _dominance
number_. The dominance number $o({\bm{a}})$ of an architecture $\bm{a}$ is
defined as the number of samples that dominate $\bm{a}$ in search space
$\Omega$:

(2)
$o({\bm{a}}):=\sum_{{\bm{a}}_{i}\in\Omega}\mathbb{I}[{\bm{a}}_{i}\prec_{f}{\bm{a}},\
{\bm{a}}\neq{\bm{a}}_{i}],$

where $\mathbb{I}[\cdot]$ is the indicator function. With the decreasing of
the $o({\bm{a}})$, $\bm{a}$ would be approaching the Pareto frontier;
$o(\bm{a})$ = $0$ when the sample architecture $\bm{a}$ locates in the Pareto
frontier. The use of dominance number allows us to rank an architecture by
considering both the estimated accuracy $E^{{}^{\prime}}(\bm{a})$ and its
inference energy cost $I(\bm{a})$ at the same time. CE-NAS will first train
the architectures with lower dominance number values when GPU resources are
available. Once an architecture is trained, it is removed from $\mathcal{R}$.

This process is depicted in the middle right part of Figure 1. Note that this
process includes actual time-consuming DL training, which is energy-intensive.
Hence, CE-NAS will try to prioritize this process during periods of low carbon
intensity.

###  3.5. GPU Allocation Strategy

The carbon impact of the above two processes in a NAS search is materialized
through the use of GPU resources. A key decision CE-NAS needs to make is _how_
to allocate GPUs among these two interdependent processes. Assigning too many
GPUs to the architecture sampling could impact the search efficiency, i.e.,
the searched architectures are far from the Pareto frontier; assigning too
many GPUs to the architecture evaluation could significantly increase energy
consumption. CE-NAS must consider these trade-offs under varying carbon
intensity and re-evaluate the GPU allocation strategy.

Below we describe a heuristic strategy that automatically allocates GPU
resources between the sampling and evaluation processes given the carbon
emissions $C_{t}$ at time $t$. This allocation is based on the energy
characteristics of the processes: architecture sampling is often energy-
efficient because it does not involve actual training of architectures, while
architecture evaluation is often energy-consuming because it does. We assume
that the maximum and minimum carbon intensities $C_{max}$ and $C_{min}$ for a
future time window are known. $G_{t}$ denotes the total number of available
GPUs. $\lambda_{e}$ and $\lambda_{s}$ represent the ratio of GPU numbers
allocated to the evaluation and sampling processes at a given moment, and
$\lambda_{e}+\lambda_{s}=1$. We calculate $\lambda_{s}$ as
$\frac{C_{cur}-C_{min}}{C_{max}-C_{min}}$, where $C_{cur}$ is the current
carbon intensity. The GPU allocations for the sampling and evaluation
processes are, therefore, $G_{t}*\lambda_{s}$ and $G_{t}*\lambda_{e}$. This
simple heuristic allocation allows CE-NAS to prioritize more energy-efficient
sampling tasks during periods of higher carbon intensity, whereas, during low-
carbon periods, CE-NAS will allocate more resources for energy-intensive
evaluation tasks.

##  4\. Preliminary Results

We prototype the CE-NAS framework described in §3. This section presents a
preliminary analysis of CE-NAS for its carbon and search efficiency based on
trace-driven simulations. Specifically, we evaluate LaMOO’s performance in
partitioning the search space for NAS on HW-NASBench (Li et al., 2021).

HW-NASBench was selected due to its inclusion of information on our two search
targets: inference energy and accuracy. To assess the search performance and
carbon cost of CE-NAS, we have CE-NAS search for optimal architectures on HW-
NASBench and compare the searched results to three different NAS search
methods. CE-NAS delivers the most effective search results within the same
carbon budget.

Figure 2. Carbon traces from electricityMap. Trace 1 is based on the US-CAL-
CISO data from 2021, specifically covering the period from 0:00, January 1,
2021, to 16:00, January 2, 2021. Trace 2 is also based on the US-CAL-CISO data
from 2021, covering the period from 17:00, January 2, 2021, to 9:00, January
4, 2021.

(a) Hypervolume

(b) Accuracy

(c) Inference energy

Figure 3.  Comparisons of architecture qualities between LaMOO-selected region
and the entire search space of HW-Nasbench. We ran LaMOO 10 times. For each
run, we randomly sampled 50 architectures from the LaMOO-selected space and
the whole search space.

(a) With carbon trace 1

(b) With carbon trace 2

Figure 4. Search progress over time. CE-NAS has the lowest relative carbon
emission while achieving the second best $HV_{\mathrm{log\\_diff}}$.

###  4.1. Setup

We conduct our experiments using CE-NAS and other baselines based on the two
carbon traces depicted in Figure 2. We initiate the process with ten samples
in the set $\mathcal{P}$ and set the maximum capacity of $\mathcal{R}$ to be
300. Each method is simulated ten times for consistency, and all search
processes in the simulation are executed on an Nvidia GeForce RTX 3090.

_Carbon Traces._ We used two carbon traces obtained from ElectricityMap (Map,
[n. d.]), a third-party carbon information service. Both carbon traces span 40
hours and consist of the per-hour average carbon intensity. We chose these two
traces because they exhibit varying carbon intensity, as shown in Figure 2,
which allowed us to evaluate both the search over time performance and CE-
NAS’s adaptiveness to carbon intensity.

_NAS Dataset._ A number of popular open-source NAS datasets, such as
NasBench101 (Ying et al., 2019), NasBench201 (Dong and Yang, 2020), and
NasBench301 (Zela et al., 2022) exist. However, none contain information on
architecture inference energy, one of our search objectives. We chose the new
NAS dataset called _HW-NASBench_ (Li et al., 2021) due to its inclusion of
information on our two search targets: inference energy and accuracy.
Specifically, HW-NASBench contains inference performance of all networks in
the NasBench201’s search space (Dong and Yang, 2020) on six hardware devices,
including commercial edge devices. In short, we use a combination of
architecture information, including inference accuracy, training time,
evaluation time, and energy cost in the edge GPU obtained from HW-NASBench and
NasBench201 (Dong and Yang, 2020).

(a) $CO_{2}$ cost: 5000g

(b) $CO_{2}$ cost: 10000g

(c) $CO_{2}$ cost: 20000g

(d) $CO_{2}$ cost: 30000g

Figure 5.  Search efficiency under carbon emission constraints. These results
are obtained using carbon trace 1, and we ran each method ten times.

(a) $CO_{2}$ cost: 5000g

(b) $CO_{2}$ cost: 10000g

(c) $CO_{2}$ cost: 20000g

(d) $CO_{2}$ cost: 30000g

Figure 6.  Search efficiency under carbon emission constraints. These results
are obtained using carbon trace 2, and we ran each method ten times.

_Metrics._ We use two main metrics to evaluate the carbon and search
efficiency of CE-NAS. First, we use _relative carbon emission_ to quantify the
amount of CO2 each NAS method is responsible for. The relative carbon emission
is calculated by summing the average carbon intensity (in gCO2/KwH) over the
search process. We assume that all NAS methods use the same type of GPU whose
power consumption remains the same throughout the search process. Second, we
use the metric _hypervolume_ (HV) to measure the ”goodness” of searched
samples. HV is a commonly used multi-objective optimization quality indicator
(Daulton et al., 2020, 2021; Zhao et al., 2022) that considers all dimensions
of the search objective. Given a reference point $R\in{\textnormal{r}}^{M}$,
the HV of a finite approximate Pareto set $\mathcal{P}$ is the M-dimensional
Lebesgue measure $\lambda_{M}$ of the space dominated by $\mathcal{P}$ and
bounded from below by $R$. That is,
$HV(\mathcal{P},R)=\lambda_{M}(\cup_{i=1}^{|\mathcal{P}|}[R,y_{i}])$, where
$[R,y_{i}]$ denotes the hyper-rectangle bounded by the reference point $R$ and
$y_{i}$. A higher hypervolume denotes better multi-objective results.

_Baselines._ We chose three types of baselines according to different GPU
allocation strategies and NAS evaluation algorithms. During the search
process, all search methods employ the state-of-the-art multi-objective
optimizer, LaMOO (Zhao et al., 2022). Specifically, _one-shot LaMOO_ is a
method that utilizes one-shot evaluations throughout the search process. The
_vanilla LaMOO_ relies on actual training for architecture evaluation
throughout the search. The _random GPU allocations_ is a strawman strategy
that randomly allocates GPUs between the energy-efficient sampling stage and
the more energy-consuming evaluation stage without considering the carbon
intensity.

###  4.2. Effectiveness of LaMOO for NAS

We conducted ten runs of LaMOO (i.e., search space split) with a random search
on the HW-NASBench dataset (Li et al., 2021). In addition, we performed random
sampling for both the LaMOO-selected region and the entire search space,
conducting 50 trials for each. The distribution of accuracy and edge GPU
energy consumption of the architectures in both the LaMOO selected region and
the entire search space can be seen in Figure 3.

Specifically, our results show that the architectures in the region selected
by LaMOO have higher average accuracy and lower average edge GPU energy
consumption compared to those in the entire search space. On average, the
accuracy of the architectures in the LaMOO selected region is 72.12, while the
accuracy in the entire search space is 68.28. The average edge GPU energy for
the LaMOO selected region is 16.59 mJ, as opposed to 22.84 mJ for the entire
space.

Furthermore, as illustrated in Figure 3(a), we observe that searching within
the LaMOO-selected region yielded a tighter distribution, and the median
hypervolume demonstrated an improvement compared to searching across the
entire search space. These results suggest the efficacy of using LaMOO to
partition the search space for NAS.

###  4.3. Carbon and Search Efficiency

In this section, we evaluate the search performance and carbon costs of our
CE-NAS framework, comparing it to three other baselines on the HW-NASBench
dataset (Li et al., 2021). We use the _log hypervolume_ difference, the same
as in (Daulton et al., 2020, 2021; Zhao et al., 2022), as our evaluation
criterion for HW-NASBench, since the hypervolume difference may be minimal
over the search process. Therefore, using log hypervolume allows us to
visualize the sample efficiency of different search methods. We define
$HV_{\mathrm{log\\_diff}}:=\log(HV_{\mathrm{max}}-HV_{\mathrm{cur}})$ where
$HV_{\mathrm{max}}$ represents the maximum hypervolume calculated from all
points in the search space, and $HV_{\mathrm{cur}}$ denotes the hypervolume of
the current samples, which are obtained by the algorithm within a specified
budget. The $HV_{\mathrm{max}}$ in this problem is 4150.7236. For our
simulation, we use the training and evaluation time costs for the
architectures derived from NasBench201 (Dong and Yang, 2020), and inference
energy costs measured on the NVIDIA Edge GPU Jetson TX2 from HW-NASBench (Li
et al., 2021). We ran the simulation 10 times with each method.

As depicted in Figure 4, as the search time progresses, vanilla LaMOO
demonstrates the highest performance in terms of $HV_{\mathrm{log\\_diff}}$.
Vanilla LaMOO’s superior performance can be attributed to its approach of
_training all sampled architectures_ to obtain their true accuracy, which
effectively steers the search direction. However, when considering the
relative carbon emission, vanilla LaMOO consumes 2.22X-3.48X carbon compared
to other approaches. This is expected because vanilla LaMOO is an energy-
consuming approach and is not designed to be aware of carbon emissions
associated with joules.

We show that CE-NAS’s search efficiency is only marginally lower than that of
vanilla LaMOO while having the least relative carbon emission under both
carbon traces. Note that we are plotting the $HV_{\mathrm{log\\_diff}}$ in the
Y-axis of Figure 4; the actual $HV$ values achieved by CE-NAS and Vanilla
LaMOO are about 4100 and 4117, differing only by 0.034%, even though the two
lines look far apart. This result suggests that only relying on energy-
efficient approaches (e.g., one-shot LaMOO in this case) is insufficient to
achieve carbon efficiency. For both traces, one-shot LaMOO has 1.17X-1.48X
carbon compared to CE-NAS.

Moreover, we observe that CE-NAS’s carbon efficiency is correlated to the
time-varying carbon intensity. When the coefficient of variation of carbon
intensity is higher, CE-NAS has more opportunity to explore the GPU allocation
trade-offs between energy-efficient sampling and energy-consuming evaluation
without impacting search quality. The relative carbon emission difference
between CE-NAS and _random GPU allocations_ represents how well CE-NAS makes
such trade-offs. Currently, we are using a heuristic approach, and it is
possible to devise more sophisticated strategies to further reduce relative
carbon emissions. For example, if the strategy could determine the GPU
resources based on the queued architectures and the current carbon intensity,
it can better shift the workload to periods of low carbon emission.

Finally, Figure 5 and 6 compare CE-NAS performance with baselines under
different carbon budgets. We show that CE-NAS outperforms all baselines in
terms of search efficiency. This is because when there is a carbon budget,
energy-consuming approaches (e.g., vanilla LaMOO) would exhaust the budget and
end the search earlier, as opposed to operating with an unlimited carbon
budget. This result suggests CE-NAS’s ability to dynamically adjust the search
process based on carbon budgets while still producing reasonable search
efficiency.

##  5\. Conclusion and Future Directions

In this work, we described the design of a carbon-efficient NAS framework CE-
NAS by leveraging the temporal variations in carbon intensity. To search for
energy-efficient architectures, CE-NAS integrates a state-of-the-art multi-
objective optimizer, LaMOO (Zhao et al., 2022), with the one-shot and vanilla
NAS algorithms. These two NAS evaluation strategies have different energy
requirements, which CE-NAS leverages to schedule when to use each based on
average carbon intensity. Our trace-driven simulations show that CE-NAS is a
promising approach for reducing relative carbon emission while maintaining
search efficiency.

Based on our investigation, we believe there are many fruitful directions in
the context of CE-NAS. For example, one can train an agent, e.g., use deep
reinforcement learning, to automatically output different GPU allocation
strategies based on historical carbon traces. This can replace our current
heuristic GPU allocation strategy and will likely lead to better carbon and
search efficiency. Another direction is to develop models that are capable of
accurately predicting carbon intensity, similar to a recent work (Maji et al.,
2022). With such predictive models, CE-NAS can better schedule the NAS tasks
to a dynamic set of GPUs that can span across geographic locations without
adversely impacting the total search time.

###### Acknowledgements.

This work was supported in part by NSF Grants #2105564 and #2236987, and a
VMWare grant. We also thank electricityMap for its carbon intensity dataset.

## References

  * (1)
  * Auer et al. (2002) Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. 2002\.  Finite-time analysis of the multiarmed bandit problem.  _Machine learning_ 47, 2 (2002), 235–256.
  * Bashir et al. (2021) Noman Bashir, Tian Guo, Mohammad Hajiesmaili, David Irwin, Prashant Shenoy, Ramesh Sitaraman, Abel Souza, and Adam Wierman. 2021\.  Enabling sustainable clouds: The case for virtualizing the energy system. In _Proceedings of the ACM Symposium on Cloud Computing_. 350–358.
  * Bender et al. (2018) Gabriel Bender, Pieter-Jan Kindermans, Barret Zoph, Vijay Vasudevan, and Quoc Le. 2018\.  Understanding and Simplifying One-Shot Architecture Search. In _Proceedings of the 35th International Conference on Machine Learning_.
  * Cai et al. (2020) Han Cai, Chuang Gan, Tianzhe Wang, Zhekai Zhang, and Song Han. 2020.  Once for All: Train One Network and Specialize it for Efficient Deployment. In _International Conference on Learning Representations_.  https://arxiv.org/pdf/1908.09791.pdf
  * Cai et al. (2019) Han Cai, Ligeng Zhu, and Song Han. 2019.  ProxylessNAS: Direct Neural Architecture Search on Target Task and Hardware. In _International Conference on Learning Representations_.  https://arxiv.org/pdf/1812.00332.pdf
  * Chen et al. (2019) Xin Chen, Lingxi Xie, Jun Wu, and Qi Tian. 2019\.  Progressive DARTS: Bridging the Optimization Gap for NAS in the Wild.  _CoRR_ abs/1912.10952 (2019).  arXiv:1912.10952 http://arxiv.org/abs/1912.10952
  * Daulton et al. (2020) Samuel Daulton, Maximilian Balandat, and Eytan Bakshy. 2020\.  Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization.  _arXiv preprint arXiv:2006.05078_ (2020).
  * Daulton et al. (2021) Samuel Daulton, Maximilian Balandat, and Eytan Bakshy. 2021\.  Parallel bayesian optimization of multiple noisy objectives with expected hypervolume improvement.  _Advances in Neural Information Processing Systems_ 34 (2021), 2187–2200.
  * Domhan et al. (2015) Tobias Domhan, Jost Tobias Springenberg, and Frank Hutter. 2015\.  Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In _Twenty-fourth international joint conference on artificial intelligence_.
  * Dong and Yang (2020) Xuanyi Dong and Yi Yang. 2020.  NAS-Bench-201: Extending the Scope of Reproducible Neural Architecture Search. In _International Conference on Learning Representations (ICLR)_.  https://openreview.net/forum?id=HJxyZkBKDr
  * Ghiasi et al. (2019) Golnaz Ghiasi, Tsung-Yi Lin, Ruoming Pang, and Quoc V. Le. 2019\.  NAS-FPN: Learning Scalable Feature Pyramid Architecture for Object Detection.  _CoRR_ abs/1904.07392 (2019).  arXiv:1904.07392 http://arxiv.org/abs/1904.07392
  * Klein et al. (2017) Aaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter. 2017.  Fast bayesian optimization of machine learning hyperparameters on large datasets. In _Artificial intelligence and statistics_. PMLR, 528–536.
  * Li et al. (2021) Chaojian Li, Zhongzhi Yu, Yonggan Fu, Yongan Zhang, Yang Zhao, Haoran You, Qixuan Yu, Yue Wang, Cong Hao, and Yingyan Lin. 2021\.  {HW}-{NAS}-Bench: Hardware-Aware Neural Architecture Search Benchmark. In _International Conference on Learning Representations_.  https://openreview.net/forum?id=_0kaDkv3dVf
  * Liu et al. (2018) Chenxi Liu, Barret Zoph, Maxim Neumann, Jonathon Shlens, Wei Hua, Li-Jia Li, Li Fei-Fei, Alan L. Yuille, Jonathan Huang, and Kevin Murphy. 2018\.  Progressive Neural Architecture Search. In _European Conference on Computer Vision(ECCV)_.
  * Liu et al. (2019) Hanxiao Liu, Karen Simonyan, and Yiming Yang. 2019\.  DARTS: Differentiable Architecture Search. In _International Conference on Learning Representations(ICLR)_.
  * Maji et al. (2022) Diptyaroop Maji, Prashant Shenoy, and Ramesh K. Sitaraman. 2022\.  CarbonCast: Multi-Day Forecasting of Grid Carbon Intensity. In _Proceedings of the 9th ACM International Conference on Systems for Energy-Efficient Buildings, Cities, and Transportation_ (Boston, Massachusetts) _(BuildSys ’22)_. Association for Computing Machinery, New York, NY, USA, 198–207.  https://doi.org/10.1145/3563357.3564079
  * Map ([n. d.]) Electricity Map. [n. d.].  Electricity Map.  https://app.electricitymaps.com/map
  * Pham et al. (2018) Hieu Pham, Melody Y. Guan, Barret Zoph, Quoc V. Le, and Jeff Dean. 2018.  Efficient Neural Architecture Search via Parameter Sharing. In _International Conference on Machine Learning(ICML)_.
  * Real et al. (2019) Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V. Le. 2019.  Regularized Evolution for Image Classifier Architecture Search. In _Association for the Advancement of Artificial Intelligence(AAAI)_.
  * Strubell et al. (2019) Emma Strubell, Ananya Ganesh, and Andrew McCallum. 2019\.  Energy and policy considerations for deep learning in NLP.  _arXiv preprint arXiv:1906.02243_ (2019).
  * Wang et al. (2020a) Linnan Wang, Rodrigo Fonseca, and Yuandong Tian. 2020a.  Learning search space partition for black-box optimization using monte carlo tree search.  _Advances in Neural Information Processing Systems_ 33 (2020), 19511–19522.
  * Wang et al. (2019a) Linnan Wang, Saining Xie, Teng Li, Rodrigo Fonseca, and Yuandong Tian. 2019a.  Sample-Efficient Neural Architecture Search by Learning Action Space.  _CoRR_ abs/1906.06832 (2019).  arXiv:1906.06832 http://arxiv.org/abs/1906.06832
  * Wang et al. (2019b) Linnan Wang, Yiyang Zhao, Yuu Jinnai, Yuandong Tian, and Rodrigo Fonseca. 2019b.  AlphaX: eXploring Neural Architectures with Deep Neural Networks and Monte Carlo Tree Search.  _CoRR_ abs/1903.11059 (2019).  arXiv:1903.11059 http://arxiv.org/abs/1903.11059
  * Wang et al. (2020b) Ning Wang, Yang Gao, Hao Chen, Peng Wang, Zhi Tian, Chunhua Shen, and Yanning Zhang. 2020b.  NAS-FCOS: Fast Neural Architecture Search for Object Detection. In _IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_.
  * Xu et al. (2021) Jingjing Xu, Liang Zhao, Junyang Lin, Rundong Gao, Xu Sun, and Hongxia Yang. 2021\.  KNAS: green neural architecture search. In _International Conference on Machine Learning_. PMLR, 11613–11625.
  * Xu et al. (2020) Yuhui Xu, Lingxi Xie, Xiaopeng Zhang, Xin Chen, Guo-Jun Qi, Qi Tian, and Hongkai Xiong. 2020.  {PC}-{DARTS}: Partial Channel Connections for Memory-Efficient Architecture Search. In _International Conference on Learning Representations_.  https://openreview.net/forum?id=BJlS634tPr
  * Ying et al. (2019) Chris Ying, Aaron Klein, Eric Christiansen, Esteban Real, Kevin Murphy, and Frank Hutter. 2019\.  NAS-Bench-101: Towards Reproducible Neural Architecture Search. In _Proceedings of the 36th International Conference on Machine Learning_.
  * Zela et al. (2022) Arber Zela, Julien Niklas Siems, Lucas Zimmer, Jovita Lukasik, Margret Keuper, and Frank Hutter. 2022\.  Surrogate NAS Benchmarks: Going Beyond the Limited Search Spaces of Tabular NAS Benchmarks. In _International Conference on Learning Representations_.  https://openreview.net/forum?id=OnpFa95RVqs
  * Zhao et al. (2021) Yiyang Zhao, Linnan Wang, Yuandong Tian, Rodrigo Fonseca, and Tian Guo. 2021.  Few-Shot Neural Architecture Search. In _Proceedings of the 38th International Conference on Machine Learning_ _(Proceedings of Machine Learning Research, Vol. 139)_. PMLR, 12707–12718.  http://proceedings.mlr.press/v139/zhao21d.html
  * Zhao et al. (2022) Yiyang Zhao, Linnan Wang, Kevin Yang, Tianjun Zhang, Tian Guo, and Yuandong Tian. 2022\.  Multi-objective Optimization by Learning Space Partition. In _International Conference on Learning Representations_.  https://openreview.net/forum?id=FlwzVjfMryn
  * Zoph et al. (2018) Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc Le. 2018.  Learning Transferable Architectures for Scalable Image Recognition. In _Conference on Computer Vision and Pattern Recognition (CVPR)_.