# The central surface density of “dark halos” predicted by MOND

Mordehai Milgrom  The Weizmann Institute Center for Astrophysics, Rehovot,
76100, Israel

###### Abstract

Prompted by the recent claim, by Donato & al., of a quasi-universal central
surface density of galaxy dark matter halos, I look at what MOND has to say on
the subject. MOND, indeed, predicts a quasi-universal value of this quantity
for objects of all masses and of any internal structure, provided they are
mostly in the Newtonian regime; i.e., that their mean acceleration is at or
above $a_{\scriptscriptstyle 0}$. The predicted value is
$\gamma\Sigma_{\scriptscriptstyle M}$, with $\Sigma_{\scriptscriptstyle
M}\equiv a_{\scriptscriptstyle 0}/2\pi G=138(a_{\scriptscriptstyle
0}/1.2\times 10^{-8}{\rm cm~{}s^{-2}})M_{\scriptscriptstyle\odot}{\rm
pc}^{-2}$, and $\gamma$ a constant of order 1 that depends only on the form of
the MOND interpolating function. For the nominal value of
$a_{\scriptscriptstyle 0}$, $\log(\Sigma_{\scriptscriptstyle
M}/M_{\scriptscriptstyle\odot}{\rm pc}^{-2})=2.14$, which is consistent with
that found by Doanato & al. of $2.15\pm 0.2$.

MOND predicts, on the other hand, that this quasi-universal value is not
shared by objects with much lower mean accelerations. It permits halo central
surface densities that are arbitrarily small, if the mean acceleration inside
the object is small enough. However, for such low-surface-density objects,
MOND predicts a halo surface density that scales as the square root of the
baryonic one, and so the range of the former is much compressed relative to
the latter. This explains, in part, the finding of Donato & al. that the
universal value applies to low acceleration systems as well. Looking at
literature results for a number of the lowest surface-density disk galaxies
with rotation-curve analysis, I find that, indeed, their halo surface
densities are systematically lower then the above “universal” value.

The prediction of $\Sigma_{\scriptscriptstyle M}$ as an upper limit, and
accumulation value, of halo central surface densities, pertains, unlike most
other MOND predictions, to a pure “halo” property, not to a relation between
baryonic and “dark matter” properties.

galaxies: kinematics and dynamics; cosmology: dark matter, theory.

##  1 introduction

Donato & al. (2009) have recently looked at the distribution of the central
surface densities, $\Sigma_{c}$, of the dark matter halos (hereafter CHSD) of
galaxies of different types. They find that the distribution is rather narrow,
with a central value $\Sigma_{c}=10^{2.15\pm
0.2}M_{\scriptscriptstyle\odot}{\rm pc}^{-2}$. This finding agrees with
previous studies, in particular with that of Milgrom & Sanders 2005, who dealt
with the relevance to MOND, and with others (see references in Donato & al.
2009). $\Sigma_{c}$ is defined by Donato & al. as the product of the central
halo density, $\rho_{0}$, and the core radius, $r_{0}$, both derived by
fitting halo-plus-baryons models to various observations, such as rotation
curves, weak lensing results, or velocity dispersion data. In deducing
$\rho_{0}$ and $r_{0}$, the halo is sometimes assumed to have a density
distribution of the cored isothermal form; Donato & al. assumed a spherical
Burkert profile.

A surface density of special role, $\Sigma_{c}$, translates into an
acceleration of a special role $\Sigma_{c}G$, and this immediately evokes
MOND. One is thus naturally led to consider whether such a special value for
the CHSD is predicted by MOND.

Brada & Milgrom (1999) showed that MOND predicts an absolute acceleration
maximum, of order $a_{\scriptscriptstyle 0}$, that any phantom halo can
produce, anywhere in an object. Milgrom & Sanders (2005), in a precursor to
Donato & al. (2009), tested this MOND prediction by plotting, for a sample of
17 Ursa Major galaxies, the deduced $\rho_{0}$ and $r_{0}$ against each other.
(These were deduced for a cored isothermal sphere model, not a Burkert one,
with a variety of assumptions on the stellar $M/L$ values: maximum disc,
population synthesis values, best MOND fits to rotation curves, etc.) They
found, for their sample, that these parameters lie near a line of constant
$\Sigma_{c}=10^{2}M_{\scriptscriptstyle\odot}{\rm pc}^{-2}$ (their Fig.4), in
agreement with the value Donato & al. find. This was interpreted by Milgrom &
Sanders (2005) as indicating a maximum halo acceleration as suggested by Brada
& Milgrom (1999), because the sample used was devoid of truly low-surface
brightness galaxies, for which “halo” accelerations are supposedly lower.

Here I will show, as a new result, that MOND does indeed predict a quasi-
universal value for the CHSD of the imaginary, or phantom, dark matter (DM),
but only for baryonic systems that are, by and large, in the Newtonian regime,
having mean internal accelerations of order $a_{\scriptscriptstyle 0}$ or
larger. In contradistinction, MOND predicts that, in principle, we can have
galaxies with arbitrarily small values of $\Sigma_{c}$, if the baryonic
surface density is low enough. However, the predicted CHSD scales as the
square root of the baryonic surface density, and so will have a rather
contracted span in a given sample.

Of course, each of the objects in the sample studied can, and should, be used
to subject MOND to a detailed, individual test. Inasmuch as MOND passes theses
tests, as it seems to do quite well, we can deduce that there is an acceptable
halo model whose analog of $\Sigma_{c}$ agrees with the MOND prediction. If
other halo models do not agree with the MOND prediction, it only shows that
there is a range of acceptable halo parameters, within the uncertainties in
the model parameters or assumptions (assumed density law for the halo, stellar
$M/L$ values, etc.).

Individual tests are, collectively, more decisive than tests of general rules,
which they subsume. Nevertheless, deducing and testing such general rules,
such as the mass-rotational-speed relation (aka the baryonic Tully-Fisher
relation), or the MOND prediction underlying the Faber-Jackson relation, have
obvious merits of their own, as they focus attention on certain unifying
principles. In this light it is important to consider the prediction of a
quasi-universal CHSD in itself.

In section 2, I explain how the quasi-universal CHSD arises in MOND, for high-
acceleration systems. In section 3, I treat systems with low surface density;
in particular, I show from results in the literature that disk galaxies with
the lowest surface densities analyzed to date, do have $\Sigma_{c}$ values
that fall systematically below the quasi-universal value. The discussion
section 4 deals with the special significance of the prediction at hand, in
comparison with other MOND predictions.

##  2 The emergence of a quasi-universal “halo” central surface density in
high acceleration systems

I shall be using the formulation of MOND as modified gravity put forth by
Bekenstein & Milgrom (1984). In this theory the MOND gravitational potential,
$\phi$, is determined by a nonlinear generalization of the Poisson equation

$\vec{\nabla}\cdot[\mu(|\vec{\nabla}\phi|/a_{\scriptscriptstyle
0})\vec{\nabla}\phi]=4\pi G\rho,$ (1)

$\rho$ being the true (“baryonic”) matter density. Here $\mu(x)$ is the
interpolating function characterizing the theory, and $a_{\scriptscriptstyle
0}$ is the MOND acceleration constant, known from various analyses to be
$a_{\scriptscriptstyle 0}\approx 1.2\times 10^{-8}{\rm cm~{}s^{-2}}$ (see,
e.g., Stark, McGaugh, & Swaters 2009 who find that gas dominated galaxies
satisfy the mass-asymptotic-rotational-velocity relation predicted by MOND,
$M=a_{\scriptscriptstyle 0}^{-1}G^{-1}V^{4}_{\infty}$, with this value of
$a_{\scriptscriptstyle 0}$). Similar results will follow from the pristine,
algebraic formulation of MOND (Milgrom 1983). Also, if the halo properties are
derived from rotation-curve analysis, the same results will follow in modified
inertia theories, since these theories predict the algebraic relation between
the Newtonian and MOND accelerations for circular orbits. We do not know
exactly what these modified inertia theories say about gravitational lensing,
but we expect similar results from this as well. Regarding lensing, the
existing relativistic extension of the modified-Poisson theory, TeVeS (see
Bekenstein 2006 and Skordis 2009 for reviews), says that we can use the halo
as deduced from the modified Poisson theory to derive lensing in the standard
way, at least when we can assume approximate spherical symmetry. Weak-lensing
halo properties can thus be compared directly with the predictions of this
theory.

When interpreted by a Newtonist, the departure predicted by MOND, and
encapsulated in the difference between the MOND acceleration field
$\vec{\nabla}\phi$ and the Newtonian one, is explained by the presence of
“dark matter”, or “phantom matter” whose density is (Milgrom 1986)

$\rho_{p}={1\over 4\pi G}\Delta\phi-\rho.$ (2)

Using the field equation (1) we can write

$\rho_{p}=-{1\over 4\pi Ga_{\scriptscriptstyle
0}}(\mu^{\prime}/\mu)\vec{\nabla}|\vec{\nabla}\phi|\cdot\vec{\nabla}\phi+(\mu^{-1}-1)\rho,$
(3)

which can be cast in another form

$\rho_{p}=-{a_{\scriptscriptstyle 0}\over 4\pi G}{\bf
e}\cdot\vec{\nabla}\mathcal{U}(|\vec{\nabla}\phi|/a_{\scriptscriptstyle
0})+(\mu^{-1}-1)\rho,$ (4)

where $\mathcal{U}(x)=\int L(x)dx$, with $L=x\mu^{\prime}/\mu$ the logarithmic
derivative of $\mu$, and ${\bf e}$ is a unit vector in the direction of
$\vec{\nabla}\phi$. This form is particularly useful for calculating column
densities of $\rho_{p}$ along field lines, as we want to do here. This
relation is exact. Expression (4), with $\vec{\nabla}\phi$ replaced by $-{\bf
g}$, holds exactly in the more primitive, algebraic formulation, whereby the
MOND acceleration ${\bf g}$ is given by $\mu(|{\bf g}|/a_{\scriptscriptstyle
0}){\bf g}={\bf g}_{\scriptscriptstyle N}$; ${\bf g}_{\scriptscriptstyle N}$
being the Newtonian acceleration; ${\bf g}$ is not generally derivable from a
potential.

Consider first an arbitrary point mass, and integrate expression (4) along a
line through the point mass. This gives the central surface density of the
phantom matter halo surrounding the mass, $\Sigma(0)$. Inside the mass
$\mu\approx 1$ so the second term does not contribute. The integral is
performed in two segments: from $-\infty$ to the point mass (where ${\bf e}$
is opposite the direction of integration) and from the other side of the point
mass to $\infty$. The two combined give

$\Sigma(0)=\int_{-\infty}^{\infty}\rho_{p}dz=\Sigma_{\scriptscriptstyle
M}[\mathcal{U}(\infty)-\mathcal{U}(0)]=\Sigma_{\scriptscriptstyle
M}\int_{0}^{\infty}L(x)dx\equiv\lambda\Sigma_{\scriptscriptstyle M},$ (5)

where,

$\Sigma_{\scriptscriptstyle M}\equiv{a_{\scriptscriptstyle 0}\over 2\pi G}$
(6)

is the relevant surface density proxy for $a_{\scriptscriptstyle 0}$ in the
present context. In the deep MOND regime ($x\ll 1$) $L(x)\approx 1$ , and far
outside the MOND regime $L(x)\approx 0$; so $\lambda$ is of order 1, and
depends only on the interpolating function $\mu(x)$.

I am dealing all along with central column density
$\Sigma(0)=2\int_{0}^{\infty}\rho dr$ of the MOND phantom halo. For a Burkert
halo this column density is related to the quantity $\Sigma_{c}$, used by
Donato & al., by $\Sigma(0)=(\pi/2)\Sigma_{c}$. So, translating the column
density to the MOND analog of $\Sigma_{c}$, call it $\Sigma_{c}^{*}$,

$\Sigma_{c}^{*}=(2\lambda/\pi)\Sigma_{\scriptscriptstyle
M}\equiv\gamma\Sigma_{\scriptscriptstyle M}.$ (7)

We have

$\Sigma_{\scriptscriptstyle M}=138(a_{\scriptscriptstyle 0}/1.2\times
10^{-8}{\rm cm~{}s^{-2}})M_{\scriptscriptstyle\odot}{\rm pc}^{-2},$ (8)

or, for the nominal value of $a_{\scriptscriptstyle 0}$,
$\log(\Sigma_{\scriptscriptstyle M}/M_{\scriptscriptstyle\odot}{\rm
pc}^{-2})=2.14$, compared with the value
$\log(\Sigma_{c}/M_{\scriptscriptstyle\odot}{\rm pc}^{-2})=2.15\pm 0.2$ found
by Donato et al.111The predicted MOND “halo” of an isolated system is not well
described by a Burkert profile: The MOND “halo” density behaves asymptotically
as $r^{-2}$, not $r^{-3}$, and it is expected to have a depression around the
center not a decreasing density profile everywhere. Nevertheless, these
differences are expected to produce only differences by a factor of order 1 in
the resulting $\Sigma_{c}$. The very near equality of
$\Sigma_{\scriptscriptstyle M}$ and the central value found by Donato & al. is
thus somewhat fortuitous..

For the limiting form of $\mu(x)$–with $\mu(x)=x$, for $x\leq 1$, and
$\mu(x)=1$, for $x>1$–we have $\lambda=1$, and $\gamma=2/\pi$. For
$\mu(x)=x(1+x^{2})^{-1/2}$, we have $\lambda=\pi/2$, and $\gamma=1$. Values of
$\lambda$ for other forms of $\mu$ can be read off Fig. 3 of Milgrom & Sanders
(2008) (where they were deduced numerically, and appear for other purposes).
One sees that $1\lesssim\lambda\lesssim 3$, and so $0.7\lesssim\gamma\lesssim
2$ for the range of $\mu$ forms studied there222The coefficient $\lambda$
diverges if $1-\mu(x)$ behaves at large $x$ as $x^{-1}$ or slower. The
divergence does not occur in the MOND regime, but comes from the Newtonian
regime very near the point mass. Such a behavior of $\mu$ is, however excluded
strongly from solar system constraints, and I preclude it..

Equations (5)-(7) are our basic result, around which all else in the paper
revolves. They tell us that for the simple case of a point mass a universal
value of $\Sigma_{c}$ is indeed predicted by MOND; its value is
$\approx\Sigma_{\scriptscriptstyle M}$, which agrees very well with the value
found by Donato & al..

Consider now an extended mass, $M$. If the mass is well contained within its
MOND transition radius, $R_{\scriptscriptstyle M}=(MG/a_{\scriptscriptstyle
0})^{1/2}$, namely if the Newtonian accelerations, and hence the MOND
accelerations, are high everywhere within the mass, then the procedure we
followed for a point mass applied approximately, and we get again
$\Sigma_{c}^{*}\approx\gamma\Sigma_{\scriptscriptstyle M}$.

Here I have to pause, and comment on a subtlety in the use of eq.(4), and in
interpreting the results thereof. This I demonstrate with two examples. First
consider a mass of finite extent whose density does not increase towards its
center as $r^{-1}$ or faster. In this case, the Newtonian acceleration, and so
also the MOND acceleration, goes to zero at the center, even if these
accelerations are much higher than $a_{\scriptscriptstyle 0}$ in most of the
bulk. In other words, there are two MOND regimes: one within some small sphere
of radius $r_{1}$ around the center, and another beyond the MOND transition
radius, $R_{\scriptscriptstyle M}=(MG/a_{\scriptscriptstyle 0})^{1/2}$. The
small $r$ region contributes to $\Sigma(0)$ through the first term in eq.(4),
an amount $-\Sigma_{\scriptscriptstyle M}\int_{0}^{X_{0}}L(x)dx$, where
$X_{0}$ is the maximum (MOND) acceleration in units of $a_{\scriptscriptstyle
0}$. This contribution is $\approx-\lambda\Sigma_{\scriptscriptstyle M}$ for
$X_{0}\gg 1$. The outer region contributes a positive quantity of the same
magnitude. In addition, the inner region contributes through the second term
in eq.(4), and its total contribution is positive (the phantom density is
always positive in the spherical case). The inner region of phantom mass, even
if it contributes to $\Sigma(0)$, has only little mass, is dynamically
unimportant, at large, and should not be included when comparing with results
for global halo parameters. I shall thus ignore it, and take
$\lambda\approx\int_{0}^{X_{0}}L(x)dx$. When the baryonic surface density is
low, the central, low-acceleration region is expanded and encompasses the
whole mass. The contribution of the first term in eq.(4) then can, indeed, be
taken to vanish, and the contribution to $\Sigma(0)$ comes from the second
term.

In another example, consider two arbitrary point masses along the line of
sight. Integrating the phantom density in eq.(4) along the line of sight now
gives $\Sigma(0)=2\lambda\Sigma_{\scriptscriptstyle M}$. (We now have to
integrate over four segments over which ${\bf e}$ changes sign: from $-\infty$
to the first mass, from there to the zero-field point somewhere between the
masses, from there to the second mass, and from there to infinity). This value
is exact and independent of the distance between the masses. How is this
consistent then with our deduction that
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$ for all systems well
within their transition radius? When the two masses are well separated, by
more then their joint transition radius, there is an extended halo surrounding
each of the masses, each halo with its own
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$, and the two column
densities add up. When the two masses are near each other, well within their
joint $R_{\scriptscriptstyle M}$, there will be a common halo of phantom
matter residing roughly beyond $R_{\scriptscriptstyle M}$, and this indeed has
$\Sigma(0)\approx\lambda\Sigma_{\scriptscriptstyle M}$ [arising from
integrating eq(4) in the outer two segments]. In addition, there is a small
region around the point of zero field between the two masses, which
contributes the same amount to the central column density, but which contains
little mass, is dynamically unimportant in the present context, and should be
eliminated from the result that is to be compared with the observations.

Keeping these caveats in mind, the reasoning leading to eq.(5) can be applied
not only to spherical systems. For example, for a disk galaxy with a high
central surface (baryonic) density,
$\Sigma_{b}(0)\gg\Sigma_{\scriptscriptstyle M}$, we can use this equation to
calculate the column density either along the symmetry axis, or along a
diameter in the plane of the disc (in both cases the field is always parallel
or antiparallel to the line of integration). If we ignore the small region of
phantom matter near the very center (or if we add a small matter cusp that
prevents the acceleration from vanishing at the center) we again get
$\Sigma(0)=\lambda\Sigma_{\scriptscriptstyle M}$.

Take now, more generally the extent of our mass to be $R$, and its mean
density $\rho$, and define $\Sigma_{b}=\rho R$, the baryonic equivalent of
$\Sigma_{c}$. The second term in eq.(4) can be estimated to contribute to
$\Sigma(0)$

$\approx 2\rho R[\mu^{-1}(g/a_{\scriptscriptstyle 0})-1],$ (9)

where $g$ is the MOND mean acceleration inside the mass, and is given by

$(g/a_{\scriptscriptstyle 0})\mu(g/a_{\scriptscriptstyle 0})\approx{4\pi\over
3}{\rho RG\over a_{\scriptscriptstyle 0}}={2\over
3}{\Sigma_{b}\over\Sigma_{\scriptscriptstyle M}}.$ (10)

The first term in eq.(4) is taken to contribute
$\approx\Sigma_{\scriptscriptstyle M}\int_{0}^{X_{0}}L(x)dx$, where
$X_{0}=g/a_{\scriptscriptstyle 0}$. Thus, we can write

$\Sigma_{c}^{*}=(2/\pi)\Sigma(0)\approx\Sigma_{\scriptscriptstyle
M}\\{(6/\pi)X_{0}[1-\mu(X_{0})]+\int_{0}^{X_{0}}L(x)dx\\}.$ (11)

For $X_{0}\gg 1$ this gives
$\Sigma_{c}^{*}\approx\gamma\Sigma_{\scriptscriptstyle M}$, again.

##  3 Low surface density systems

MOND does permit arbitrarily low values of $\Sigma_{c}$ for phantom halos in
low acceleration systems. When $X_{0}\ll 1$, namely, when the maximum (MOND)
acceleration in the system is much smaller than $a_{\scriptscriptstyle 0}$, we
get from eq.(11), to lowest order in $X_{0}$,

$\Sigma_{c}^{*}\approx(6/\pi+1)\Sigma_{\scriptscriptstyle M}X_{0}\approx
2.4\left({\Sigma_{b}\over\Sigma_{\scriptscriptstyle
M}}\right)^{1/2}\Sigma_{\scriptscriptstyle M}.$ (12)

Such low acceleration systems are characterized by low baryonic surface
densities $\Sigma_{b}/\Sigma_{\scriptscriptstyle M}\ll 1$. Note, however, that
the departure from the universal $\Sigma_{c}^{*}$ sets in at rather low
baryonic surface densities, since $\Sigma_{c}^{*}/\Sigma_{\scriptscriptstyle
M}$ scales as the square root of $\Sigma_{b}/\Sigma_{\scriptscriptstyle M}$.
The lowest acceleration disc galaxies studied to date have $X_{0}$ values only
down to 0.1-0.2; and we see from eq.(12) that even for values of $X_{0}$ as
low as 1/5 we get $\Sigma_{c}^{*}\approx 0.6\Sigma_{\scriptscriptstyle M}$.
Clearly, however, MOND does predicts that, for extremely low baryonic surface
density galaxies, the CHSD falls increasingly below the quasi-universal value.

To superficially check this expectation, I looked (rather randomly) in the
literature for derived halo parameters for the lowest acceleration disk
galaxies with rotation-curve analysis. Three such galaxies were analyzed in
light of MOND by Milgrom & Sanders (2007), showing rather satisfactory
agreement. These were also analyzed earlier in terms of cored isothermal
halos: For KK98 250 and KK98 251, I find in Begum & Chengalur (2005) best-fit
parameters that give $\Sigma_{c}=56$, and $66M_{\scriptscriptstyle\odot}/{\rm
pc}^{2}$, respectively. For NGC 3741, Begum & al. (2005) find parameters that
yield $\Sigma_{c}=56M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$. All three
values fall substantially below the nominal quasi-universal value of
$\Sigma_{c}=140M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$, and are consistent,
within the uncertainties, with our rough estimate (12), having $X_{0}$ values
of between 0.1 and 0.3. The first two galaxies were not included in the Donato
& al. analysis; but NGC 3741 was included, based on the analysis of Gentile &
al. (2007) (assuming a Burkert, not a cored isothermal halo), whose results
give $\Sigma_{c}=74$. This value is higher than the result of Begum & al.
(2005) (though consistent within the uncertainties), but still only about half
the quasi-universal value.

Another low acceleration galaxy that is worth analyzing in detail (and is not
included in the Donato & al. sample) is the dwarf Andromeda IV. Its rotation
curve is given in Chengalur & al. (2007). To my knowledge, its photometry and
HI distribution are not yet available publicly for rotation curve analysis.
However, according to Chengalur et &. (2007) it is heavily dominated by gas
with $M_{\scriptscriptstyle gas}/L\approx 18$, and it shows a very strong mass
discrepancy with $M_{\scriptscriptstyle dyn}/M_{\scriptscriptstyle gas}\approx
14$ at the last measured point. In deriving a cored isothermal halo parameters
we can thus approximately ignore the baryons and fit the rotation curve with
the halo alone. Doing this, I find, tentatively, $\Sigma_{c}\sim
45M_{\scriptscriptstyle\odot}/{\rm pc}^{2}$, about three times lower than
$\Sigma_{\scriptscriptstyle M}$. Since in this case $X_{0}\sim 0.1-0.15$, this
is also in agreement with the estimate of eq.(12).

Why then do Donato & al. suggest that the quasi-universal value of
$\Sigma_{c}$ applies to all galaxies, including the very low-acceleration
ones? This is based mostly on the analysis of dwarf spheroidal satellites of
the Galaxy. Their analysis includes only one well studied low-acceleration
disk, the above mentioned, NGC 3741–for which, as we saw, the actual
$\Sigma_{c}$ could be lower–and one somewhat higher acceleration galaxy, DDO
47. As regards the dwarf spheroidal Milky-Way satellites, MOND would indeed
predict lower values of $\Sigma_{c}$ than adopted by Donato & al.. However, as
Donato & al. emphasize themselves, the analysis of these systems is beset by
uncertainties in the model assumptions (e.g., assumptions on orbital
anisotropies), leading to non-unique results. Angus (2008) has analyzed these
dwarf spheroidals in MOND, and found that, with two exceptions perhaps, they
can be well explained by MOND, assuming appropriate orbit anisotropy
distributions. This would mean, as I stressed above, that there are acceptable
“halo” models that are consistent with the predictions of MOND. The disparate
values adopted by Donato & al. only demonstrate the non-uniqueness of the
halo-parameter determination.

##  4 Discussion

I have shown that the acceleration constant of MOND $a_{\scriptscriptstyle 0}$
defines a special surface density parameter $\Sigma_{\scriptscriptstyle
M}=a_{\scriptscriptstyle 0}/2\pi G$. This serves as a quasi-universal central
surface density of phantom halos around objects of all masses and structures,
provided they are themselves in the Newtonian regime (i.e., with bulk
accelerations of order $a_{\scriptscriptstyle 0}$ or higher).

This is a particularly interesting prediction of MOND, because most of the
other salient MOND predictions relate properties of the true matter (baryons)
to those of the putative dark matter halo. This is the case for the mass-
velocity (baryonic Tully-Fisher) relation, the Faber-Jackson relation, the
transition from baryon dominance to DM dominance at a fixed acceleration, the
full prediction of rotation curves, the necessity of a disk component of DM,
in disk galaxies, in addition to a spheroidal halo, etc. (see Milgrom 2008 for
a more detailed list, and explanations). Here, however, we have a prediction
that speaks of a property of halos themselves, without regard to the true mass
that engenders them, apart from the requirement that the baryons be well
concentrated. $\Sigma_{\scriptscriptstyle M}$ may also be viewed as an upper
limit, and accumulation value, for “halo” central surface densities,
irrespective of baryonic properties. It is clear then, that the
$a_{\scriptscriptstyle 0}$ that appears in this prediction need have nothing
to do with the $a_{\scriptscriptstyle 0}$ that appears in other relations, in
the framework of the DM doctrine. We could have a sample of halos all
satisfying the present prediction, and add to them baryons arbitrarily, so as
not to satisfy, e.g., the baryonic-mass-velocity relation
$MGa_{\scriptscriptstyle 0}=V^{4}$, which also revolves around some
acceleration constant. The fact that the $a_{\scriptscriptstyle 0}$ emerging
from the phenomenology here is the same as that appearing in the other
phenomenological relation should be viewed as another triumph of MOND.

There are two other MOND predictions that speak of properties of the halo
alone. The first is that the density profile of the “halo” of any isolated
object behaves asymptotically as $r^{-2}$ (asymptotic flatness of rotation
curves). The other such prediction is the maximally allowed acceleration (of
order $a_{\scriptscriptstyle 0}$) that a halo can produce (Brada & Milgrom
1999). This is simply a reflection of the MOND tenet that the phantom mass
cannot be present where accelerations are higher than roughly
$a_{\scriptscriptstyle 0}$. The prediction I discuss here can be understood,
qualitatively, as a result of the above two: On the asymptotic, $r^{-2}$, tail
of the “halo”, the acceleration it produces is $g_{h}\approx 4\pi G\rho r$.
Going inward, the maximum-acceleration prediction tells us that this behavior
can continue only down to a radius where $g_{h}\sim a_{\scriptscriptstyle 0}$.
Below this radius the halo density profile must become shallow and produce a
core. This gives $\rho_{0}r_{0}\sim a_{\scriptscriptstyle 0}/4\pi G$. It is
this that underlies our more quantitative result here.

However, there is nothing in MOND to forbid the halo density profile from
becoming shallow within a radius much larger then that where $g_{h}\sim
a_{\scriptscriptstyle 0}$. This can happen at arbitrarily large radii,
producing arbitrarily small values of $\Sigma_{c}^{*}$, as indeed our detailed
analysis shows.

## Acknowledgements

This research was supported by a center of excellence grant from the Israel
Science Foundation.

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This paper has been withdrawn by the authors.


# Estimating Individual Advertising Effect in E-Commerce

Hao Liu1,3111Contact Author Yunze Li2 Qinyu Cao2 Guang Qiu2&Jiming Chen1,3
1College of Control Science and Engineering, Zhejiang University
2Alibaba Group
3Alibaba-Zhejiang University Joint Institute of Frontier Technologies
{vectorliu<EMAIL_ADDRESS>{yunze.lyz, qingyu.cqy, guang.qiug}@alibaba-
inc.com

###### Abstract

Online advertising has been the major monetization approach for Internet
companies. Advertisers invest budgets to bid for real-time impressions to gain
direct and indirect returns. Existing works have been concentrating on
optimizing direct returns brought by advertising traffic. However, indirect
returns induced by advertising traffic such as influencing the online organic
traffic and offline mouth-to-mouth marketing provide extra significant
motivation to advertisers. Modeling and quantization of causal effects between
the overall advertising return and budget enable the advertisers to spend
their money more judiciously. In this paper, we model the overall return as
individual advertising effect in causal inference with multiple treatments and
bound the expected estimation error with learnable factual loss and distance
of treatment-specific context distributions. Accordingly, a representation and
hypothesis network is used to minimize the loss bound. We apply the learned
causal effect in the online bidding engine of an industry-level sponsored
search system. Online experiments show that the causal inference based bidding
outperforms the existing online bidding algorithm.

##  1 Introduction

The last two decades have seen the prosperity of e-commerce. Taking Taobao as
an example, as the biggest e-commerce marketplace in China Edquid (2016),
Taobao search service covers over $300$ million consumers each day, bringing
daily $10$ billion search queries and subsequent page views (PVs), providing
advertisers sufficient opportunities to promote their commodities online
222https://zhitongche.taobao.com.

In sponsored search advertising, advertisers bid for keywords associated with
their commodities (ADs 333A commodity is also called an AD in Taobao, which
represents both the commodity and the associated advertisement.) and pay the
platform when consumers land their commodity/store homepage by clicking the
advertisement (Pay-Per-Click, PPC). The payment equals the minimum bid price
required to keep the advertising slot in the real-time competition Wilkens et
al. (2017). The returns of advertising can be summarized in two aspects. The
_direct returns_ of advertising are the impressions, clicks and conversions
occurred upon the advertising PVs. Meanwhile, advertising also yields even
more valuable _indirect returns_ by connecting with wider online audience,
thus impressing more audience via unobserved social interactions.
Additionally, in e-commerce platform like Taobao, there is a ranking index
called “sales volume” for organic search traffics, which reflects the
purchasing popularity of the commodity among its peers and has been an
important shopping guideline for consumers. In this way advertisers can
accumulate sales volume via advertising PVs to gain more exposures in gigantic
organic search PVs.

The direct and indirect returns motivate advertisers to invest advertising
budget to prosper their online business. Despite its significance, however, to
the best of our knowledge, existing work mostly focus on optimizing direct
returns Zhang (2016); Zhu et al. (2017). This might be caused by the fact that
direct returns are seamlessly observable in the closed-loop e-commercial
platform like Taobao. Meanwhile, there are so many factors leading to the
_overall returns_ , making it intractable to quantify the indirect returns
attributed to advertising. Nonetheless, the ability of inferring the overall
advertising effect including both direct and indirect returns provide
advertisers the opportunities to allocate their advertising budget more wisely
in the product life cycle.

E-commercial advertisers are eager to know the growth of advertising returns
if they invest more budget via a specified advertising channel. Specifically,
in the PPC advertising, the cost is equivalent to the number of clicks
occurred in advertising PVs given a relatively stable payment per click. And
the overall advertising returns can be observed as the number of total clicks
of the advertising AD accumulated in all the online channels. Therefore, we
want to infer the _individual advertising effect (IAE)_ via predicting the
incremental number of all-channel clicks in a period under the intervention of
advertising clicks 444Numerical analysis also shows that the Pearson’s
coefficient between the number of advertising clicks and all-channel clicks is
approximately $25\%$ larger than that between advertising impressions and all-
channel clicks. We hide the detailed coefficient value due to commercial
secrets..

The problem of inferring IAE resembles the estimation of _individual treatment
effect (ITE)_ in the field of _causal inference_ or learning from
_observational data_ Rubin (2005). In causal inference, we only have
observational data which contains the past actions, their outcomes and
possibly more context. However, we do not know the mechanism which gave rise
to the action. In the scenario of advertising, the context might correspond to
the features representing the current status of the AD, while the action and
outcome are the number of advertising clicks and all-channel clicks
(containing all direct and direct returns), respectively. The key difference
between IAE and ITE is that actions of the latter are binary or categorical,
but those of the former might be continuous and transitive. Furthermore, for
any specified context, there exists only one exact action (acquire specific
number of advertising clicks) in the data. We can never know exactly the
potential advertising outcome if it applies a different action in exactly the
same context. Besides, the observed advertising outcomes can be influenced by
a lot of factors including online sources such as in or out of Taobao
recommendation/organic search, and offline mouth-to-mouth marketing by the
audience etc, which is similar to the _confounding_ factor in classic causal
inference. Since the effect of advertising is accumulating in the whole-time
horizon, we assume that the context together with the action contain all the
necessary information to determine the outcome, i.e., the “no-hidden
confounding” assumption holds in the analysis.

In this paper we model the causal effect between advertising cost and returns
via a formal definition of IAE. We propose a representation network and a
hypothesis network combined to predict the individual advertising effect
referring Shalit et al. (2017). Different from binary or categorical
treatments, advertising treatments are continuous and transitive. Relying on
this property, we derive a rigorous theoretical upper bound of the expected
IAE estimation error by way of a learnable factual regression loss and the
distance of context distribution among different treatments. Then the network
is trained to minimize the derived theoretical upper bound. Furthermore, we
derive a time-varying factor called _leverage rate_ ($lvr$) based on IAE to
reflect the AD-level potential to lever the overall advertising returns. The
learned $lvr$ is used in the online bidding engine to achieve better overall
advertising performance in Taobao sponsored search. The contributions of this
paper can be summarized as follows:

  1. 1.

We model the problem of predicting the overall advertising return in the
framework of causal inference. In this framework, the formal definition of
individual advertising effect is given.

  2. 2.

We derive a general theoretical upper bound on the expected IAE estimation
error in advertising scenarios with multiple continuous and transitive
treatments. Subsequently, a representation and hypothesis network is learned
to predict IAE.

  3. 3.

IAE-induced $lvr$ is integrated in the online bidding engine, which yields
better overall advertising returns compared with the existing bidding engine
in Taobao sponsored search.

##  2 Related Work

When considering only direct advertising returns, estimating individual
advertising effect has been investigated in both ex ante and ex post way. The
key of the ex ante estimation lies in three separate models of predicting the
winning rate of specific bids, click-through-rate and conversion-rate of the
advertising PV Zhang (2016); Zhu et al. (2017). Meanwhile, attribution
modeling Dalessandro et al. (2012); Diemert et al. (2017) corresponds to the
ex post estimation of advertising effect, i.e. attributing the later
conversion to the previous customer-commodity/store contact. All these works
ignore the indirect returns brought by the advertising PVs.

Causal inference has already been used in complex real-world ad-placement
systems Bottou et al. (2013). In the scenario of estimating individual
advertising effect, given a context (AD), a naive way might be direct least
square regression to fit the advertising effect, either taking the number of
advertising clicks as a feature or separately fit each action. However, such
estimation might be biased due to the fact that different contexts should have
priority of choosing specific actions in the dataset. To alleviate the bias,
propensity score, which characterizes the probability vector of choosing
specific actions, Austin (2011) is used to evaluate the similarity of two
contexts. Therefore, counterfactual samples can be constructed by comparing
the propensity scores via various approaches such as nearest neighbor matching
Lopez et al. (2017). Besides propensity score, various methods such as random
forestsWager and Athey (2017); Athey and Imbens (2016) and expensive random
control trials Taddy et al. (2016); Peysakhovich and Lada (2016) are also used
to tackle the binary treatment causal inference.

Recently deep representation is also used to encode the contexts. Atan _et
al._ proposed an auto encoder-decoder network to represent the raw context, to
ensure that the propensity score vector of the mapped contexts is similar,
therefore removing the selection bias Atan et al. (2018). Johansson _et al._
also designed a deep representation network to embed the original contexts, to
guarantee that the distribution of contexts after the representation is
similar between two different treatments, as well as the small regression loss
Johansson et al. (2016). In its later version, a theoretical error bound on
the expected ITE is given to yield a more rigorous estimation algorithm Shalit
et al. (2017). Deep models prove to be advanced but when faced with the
multiple treatments, theoretical error bound is non-trivial. In this paper we
design a similar network structure as Shalit et al. (2017) but derive our
theoretical upper bound considering the continuous action space in the
advertising scenario.

Back to the bidding application, perhaps the most relevant work is lift-based
bidding proposed by Xu _et al_. By predicting the ex ante and ex post click-
through-rate of an advertising impression, the bid price is adjusted to be
proportional to the lift Xu et al. (2016). However, we point out that the
observed outcome might also ignore the abundant indirect returns.

##  3 Individual Advertising Effect Formalization

We adopt the Rubin-Neyman potential outcomes modeling framework Rubin (2005)
in causal inference but tailor it for the e-commerce advertising scenario. Let
$\mathcal{X}\subset\mathbb{R}^{d}$ be the set of contexts,
$\mathcal{T}=\\{T_{1},...,T_{n}\\}$ the $n$-action (also known as treatment or
intervention) set, $\mathcal{Y}\subset\mathbb{R}$ the observed _overall_
performance index. For each context $x\in\mathcal{X}$, there is a treatment
assignment $T\in\mathcal{T}$ and with $n$ potential outcomes,
$Y_{T_{1}},Y_{T_{2}},...,Y_{T_{n}}\in\mathcal{Y}$. The samples we have can be
denoted as $\\{x_{i},t_{i},y_{i}\\}_{i=1}^{N}$, where $y_{i}=Y_{t_{i}}$. We do
not observe any of the other potential outcomes (i.e., $Y_{T}$ for $T\neq
t_{i}$).

In e-commercial advertising, context $x$ can be features representing the
status of an AD in the beginning of the day. Treatment $T$ refers to the
number of clicks acquired from the advertising PVs during the day, while
potential outcome $y$ might be observed as the overall whole-site clicks
obtained by the same AD until the end of the day. Apparently, the potential
outcome can be influenced by a lot of factors including online channels such
as in or out of Taobao recommendation/organic search, and offline mouth-to-
mouth marketing by extroverted audience etc. Specifically, let
$T_{i}=i-1,i=1,...,n$, where $n-1$ can be interpreted as the context-specific
largest possible advertising clicks in a day. Note that we restrict the
advertising effects $y$ to be happened in the same day, but ignores the
persisting effects in the far future. This naturally coincides with the
advertising logic that advertisers are accustomed to adjusting the budget of
an AD day by day. We can also alleviate the influence of persisting dependency
by following the “strong ignorability” assumption in causal inference.

######  Assumption 1.

(_Strong Ignorability_) $Y_{T_{1}},Y_{T_{2}},...,Y_{T_{n}}\perp T|x$, which
means that, given a context $x$, the potential outcome is independent of the
treatment assignment.

Strong ignorability assumption also ensures that there is a positive
probability of choosing any action in each context $x$.

######  Definition 1.

(Individual Advertising Effect, IAE) The IAE of context $x$ from treatment
$T_{i}$ to $T_{j}$ can be defined as:

$\alpha_{i,j}(x)=m_{j}(x)-m_{i}(x),\forall i,j=1,...,n,$ (1)

where $m_{i}(x)=\mathbb{E}[Y_{T_{i}}|x],\forall i,j=1,...,n$.

For $n$ treatments, we can obtain an antisymmetric matrix
$\alpha(x)=[\alpha_{i,j}]\subset\mathbb{R}^{n\times n}$ which corresponds to
the IAE in context $x$. Matrix $\alpha(x)$ has the following properties:

  1. $\bullet$

_Antisymmetric:_ $\alpha_{i,j}=-\alpha_{j,i}$;

  2. $\bullet$

_Monotonicity:_ $\alpha_{i,j}\leq\alpha_{i,k},\forall j\leq k,i=1,...,n$;
$\alpha_{i,j}\geq\alpha_{k,j},\forall i\leq k,j=1,...,n$;

  3. $\bullet$

_Zero-Diagonal:_ $\alpha_{i,i}=0,\forall i=1,..,n$;

  4. $\bullet$

_Transitivity:_ $\alpha_{i,j}+\alpha_{j,k}=\alpha_{i,k},\forall
i,j,k=1,...,n$.

In causal inference, IAE is analogous to individual treatment effect. However,
different from classical causal inference with only two interventions of
either treatment or non-treatment, multiple actions are available in IAE.
Since the action and outcome are accumulated in a day, there is some kind of
ambiguity of advertising effect with clicks assigned to different time slots.
Therefore, we take the expectation in the right-hand side of Eqn. (1) to
eliminate the ambiguity. In this sense, IAE is the average advertising effect
and should be useful among different ADs.

To learn IAE, we further define a representation function
$\Phi:\mathcal{X}\to\mathcal{R}$ where $\mathcal{R}$ is the representation
space. Let $h:\mathcal{R}\times\mathcal{T}\to\mathcal{Y}$ be a hypothesis
function which yields the outcome. Putting it together, we denote
$f(x,T)=h(\Phi(x),T)$.

######  Definition 2.

Given a hypothesis $f$, the IAE estimation for context $x$ is:

$\hat{\alpha}_{i,j}^{f}(x)=f(x,T_{j})-f(x,T_{i}).$ (2)

With a little abuse of notation, we will omit the superscript $f$ and write it
as $\hat{\alpha}_{i,j}(x)$ without confusion.

######  Definition 3.

The IAE estimation error of treatment pairs $(T_{i},T_{j})$ satisfies that:

$\tau_{i,j}(x)=\hat{\alpha}_{i,j}(x)-\alpha_{i,j}(x).$ (3)

######  Definition 4.

The expected _Precision in Estimation of Heterogeneous Effect (PEHE)_ Shalit
et al. (2017) loss of $f$ is:

$\epsilon_{\text{PEHE}}(f)=\frac{1}{n(n-1)}\sum\limits_{i=1}^{n}{\sum\limits_{j=1,j\neq
i}^{n}{\int_{\mathcal{X}}{\tau_{i,j}^{2}(x)p(x)dx}}}.$ (4)

For completeness, we also give the notion of _Integral Probability Metric_
(IPM), which is a class of distance metrics between probability distributions
Shalit et al. (2017). For two probability density functions $p,q$ defined over
$\mathcal{S}\subset\mathbb{R}^{d}$, and for a function family $G$ as
$g:\mathcal{S}\to\mathbb{R}$, it holds that

$\text{IPM}_{G}(p,q):=\sup\limits_{g\in
G}|\int_{\mathcal{S}}{g(s)(p(s)-q(s))ds}|.$

##  4 Learning to Infer Individual Advertising Effect

The key of learning IAE lies in minimizing the PEHE loss in Eqn. (4). The idea
is similar as Shalit et al. (2017). Firstly, we map the original context into
the representation space $\mathcal{R}$ via $\Phi$. In the new space, we denote
the probability density function of representation space $\mathcal{R}$ given
treatment $T$ as $p_{\Phi}^{T}$. To remove the selection bias in the original
space, the idea is that the distance between
$\\{p_{\Phi}^{T_{i}}\\}_{i=1}^{n}$ should be as small as possible, which is
guaranteed by the representation network. Given the similar context
distribution in the representation space, the hypothesis network should try to
minimize the regression loss of fitting the advertising return. The neural
network architecture is displayed in Fig. 1.

Figure 1: Neural network architecture for IAE estimation. $L$ is a loss
function and $\Phi$ is a representation of the original context $x$. $h$
represents the hypothesis and $f$ denotes the complete function.

The architecture resembles that in Shalit et al. (2017). However, the key
differences lies in two aspects. Firstly, the hypothesis network in that of
Shalit et al. (2017) are separate for treatment/non-treatment. In the
advertising scenario, the treatments can be seen as continuous actions and
should be generalizable, therefore different treatments share the same
hypothesis network. Secondly, IPM for binary treatments are straightforward
while it is not obvious for multiple treatments. We simplify the IPM term
based on the transitive property of treatment effects. In the following part
we will first give the theoretical upper error bound of PEHE error and
elaborate the detailed IPM we use, followed by the description of the IAE
estimation algorithm.

###  4.1 Loss Error bound

Before analyzing our main result, we first give a lemma considering the binary
treatment case.

######  Lemma 1.

Shalit et al. (2017) Let $\Phi:\mathcal{X}\to\mathcal{R}$ be a one-to-one
representation function with inverse $\Psi$. Let
$h:\mathcal{R}\times\\{T_{i},T_{j}\\}\to\mathcal{Y}$ be an hypothesis. Assume
there exists a constant $B_{\Phi}$ such that for $T\in\\{T_{i},T_{j}\\}$, the
per-unit expected loss functions $\ell_{h,\Phi}(\Psi(r),T)$ obey
$\frac{1}{B_{\Phi}}\cdot\ell_{h,\Phi}(\Psi(r),T)\in G$, where
$\ell_{h,\Phi}(x,T)=\int_{\mathcal{Y}}{L(h(\Phi(x),T),Y_{T})p(Y_{T}|x)dY_{T}}$.
Assuming that the loss $L$ is the squared loss, we have that

$\displaystyle\int_{\mathcal{X}}{\tau^{2}_{i,j}(x)p(x)dx}\leq$ $\displaystyle
2(\epsilon_{F}^{T=T_{i}}(h,\Phi)+\epsilon_{F}^{T=T_{j}}(h,\Phi)+B_{\Phi}\text{IPM}_{G}(p_{\Phi}^{T=T_{i}},p_{\Phi}^{T=T_{j}})),$

where we omit the term of minus variance in the righthand side. And
$\epsilon_{F}^{T}(h,\Phi)=\int_{\mathcal{X}}{\ell_{h,\Phi}(x,T)p^{T}(x)dx}$,
which represents the learnable factual loss of treatment $T$.

In the binary case, Shalit _et al._ separate the regression loss of
treatment/non-treatment for the categorical treatments. In our scenario, the
treatments are continuous advertising clicks and should be generalizable
across different treatments. In this case we define the regression loss with
respect to treatments $\\{T_{i},T_{j}\\}$ as

$\epsilon_{i,j}(h,\Phi)=\epsilon_{F}^{T=T_{i}}(h,\Phi)+\epsilon_{F}^{T=T_{j}}(h,\Phi).$
(5)

Then we are ready to decompose the PEHE error.

######  Lemma 2.

In the continuous and transitive treatments scenario, it satisfies that

$\epsilon_{\text{PEHE}}(f)\leq\sum\limits_{i=1}^{n-1}{\int_{\mathcal{X}}{\tau_{i,i+1}^{2}(x)p(x)dx}}.$
(6)

###### Proof.

It follows that:

$\displaystyle\tau_{i,j}(x)$
$\displaystyle=\hat{\alpha}_{i,j}(x)-\alpha_{i,j}(x)$
$\displaystyle=f(x,T_{j})-f(x,T_{i})-m_{j}(x)+m_{i}(x)$
$\displaystyle=f(x,T_{j})-f(x,T_{j-1})+f(x,T_{j-1})-f(x,T_{j-2})$
$\displaystyle+...+f(x,T_{i+1})-f(x,T_{i})$ $\displaystyle-
m_{j}(x)+m_{j-1}(x)-...-m_{i+1}(x)+m_{i}(x)$
$\displaystyle=\hat{\alpha}_{j-1,j}(x)+...+\hat{\alpha}_{i,i+1}(x)$
$\displaystyle-\alpha_{j-1,j}(x)-...-\alpha_{i,i+1}(x)$
$\displaystyle=\tau_{i,i+1}(x)+\tau_{i+1,i+2}(x)+...+\tau_{j-1,j}(x)$

Therefore, according to the Cauchy–Schwarz inequality, we have that

$\tau_{i,j}^{2}(x)\leq(j-i)[\tau_{i,i+1}^{2}(x)+...+\tau_{j-1,j}^{2}(x)],\forall
j>i.$

Apparently, $\tau_{i,j}(x)=-\tau_{j,i}(x)$. Then the PEHE loss can then be
written as:

$\displaystyle\epsilon_{\text{PEHE}}(f)$
$\displaystyle\leq\frac{2}{n(n-1)}\sum\limits_{i=1}^{n}{\sum\limits_{j>i}^{n}{\int_{\mathcal{X}}{\tau_{i,j}^{2}(x)p(x)dx}}}$
$\displaystyle=\sum\limits_{i=1}^{n-1}{\int_{\mathcal{X}}{\tau_{i,i+1}^{2}(x)p(x)dx}}.$

∎

Combining Lemma 1, Lemma 2 and Eqn. (5), we obtain a learnable upper bound for
PEHE.

######  Theorem 1.

Under the assumptions in Lemma 1 and Lemma 2, we have that:

$\epsilon_{\text{PEHE}}(f)\leq
2\sum\limits_{i=1}^{n-1}{[\epsilon_{i,i+1}(h,\Phi)+B_{\Phi}\text{IPM}_{G}(p_{\Phi}^{T=T_{i}},p_{\Phi}^{T=T_{i+1}})]}.$
(7)

Theorem 1 directly points out an algorithm for learning IAE. Relying on
continuous and transitive treatments, it generalizes the binary treatments to
the multiple treatments advertising scenario. Note that we can compare
arbitrary pairwise treatment effect via customized definition of PEHE loss.

###  4.2 Algorithm implementation

In the advertising data, the context $x$ mainly embeds the status of an AD
with its features in major traffic channels. In our case, we choose the
following features to characterize an AD before applying treatment $T$:

  1. $\bullet$

IDs: including the commodity ID, the shop ID, the category ID and other one-
hot encoding features such as weekdays/weekends etc;

  2. $\bullet$

PV sources of the AD in the last day: we aggregate the advertising effects
from the major online PV sources of the last day, mainly including sponsored
search, recommendations, organic search, etc. Moreover, we also include the
time when the AD was created and shelved;

  3. $\bullet$

PV sources of the AD in the last week: the exponential decaying average
advertising effects of last week, including major online PV sources as the
above one;

  4. $\bullet$

Shop features: including the number of impressions and clicks of the shop in
the last day. Moreover, the count of total clicks of the shop, the total
number of ADs and also the ADs in sponsored search advertising campaign of the
shop are also included;

  5. $\bullet$

Competition of the last day: the average ranking of the AD and shop in the
bidding process of the last day, with the logarithm of the two values
included.

We believe the above features cover most of the data we can fetch online for
an AD in Taobao platform.

We learn to infer IAE by minimizing the upper bound of the nominal PEHE loss,
using the following objective:

$\displaystyle\min\limits_{h,\Phi}$
$\displaystyle{\frac{2}{N}\sum\limits_{i=1}^{N}{w_{i}\cdot
L(h(\Phi(x_{i}),t_{i}),y_{i})}+\lambda\cdot\mathscr{R}(h)}$ (8)
$\displaystyle-\frac{\mu_{1}}{N}\cdot\sum\limits_{i=1}^{N}{L(h(\Phi(x_{i}),t_{i}),y_{i})\mathbf{1}_{t_{i}=T_{1}}}$
$\displaystyle-\frac{\mu_{n}}{N}\cdot\sum\limits_{i=1}^{N}{L(h(\Phi(x_{i}),t_{i}),y_{i})\mathbf{1}_{t_{i}=T_{n}}}$
$\displaystyle+\beta\cdot\sum\limits_{i=1}^{n-1}{\text{IPM}_{G}(p_{\Phi}^{T=T_{i}},p_{\Phi}^{T=T_{i+1}})},$
with
$\displaystyle\mu_{j}=\frac{N_{j}}{N},w_{i}=\mu_{t_{i}},N_{j}=\sum\limits_{i=1}^{N}{\mathbf{1}_{T_{j}=t_{i}}},j=1,...,n,$
and $\displaystyle\mathscr{R}\ \text{is a model complexity term}.$

Note that $w_{j},j=1,...,n$ corresponds to the proportion of units applying
treatment $T_{j}$ in the whole population, which is approximated by the sample
population. And $\mathbf{1}_{\text{condition}}$ is an indicator function which
yields $1$ when the condition is true; otherwise $0$. We use the same
$B_{\Phi}=\beta$ across all the IPM distance since they have the same
importance in the PEHE definition. We train the models by using stochastic
gradient descent to minimize (8) with $\ell_{2}$-regularization and
$1$-Lipschitz function family $G$. Samples belonging to different treatments
share a common representation and hypothesis as shown in Fig. 1. The details
of the training process is shown in Algorithm 1. The algorithm structure is
much like that in Shalit et al. (2017) with differences on objective function
and gradients. We put it here for completeness.

Algorithm 1 Learning Individual Advertising Effect

0: Samples $\\{(x_{i},t_{i},y_{i})\\}_{i=1}^{N}$, loss function $L(\cdot)$,
regularization factor $\lambda$ and $\beta$, representation network
$\Phi_{\mathbf{W}}$ with initial weights $\mathbf{W}$, hypothesis network
$h_{\mathbf{V}}$ with initial weights, function family $G$ for IPM distance,
regularization norm $\mathscr{R}$.

0: Representation and hypothesis network.

Compute $N_{j}=\sum\limits_{i=1}^{N}{\mathbf{1}_{T_{j}=t_{i}}}$

Compute $\mu_{j}=\frac{N_{j}}{N},w_{i}=\mu_{t_{i}}$

while not converged do

Sample mini-batch ${i_{1},i_{2},...,i_{l}}\subset\\{1,2,...,N\\}$

Calculate the gradient of the IPM sum term:

$g_{1}=\nabla_{\mathbf{W}}{\sum\limits_{i=1}^{n-1}{{\text{IPM}_{G}(p_{\Phi}^{T=T_{i}},p_{\Phi}^{T=T_{i+1}})}}}$

Calculate the gradients of the empirical loss:

$g_{2}=\nabla_{\mathbf{V}}{L}$, $g_{3}=\nabla_{\mathbf{W}}{L}$

where $L=\frac{2}{m}\sum_{j}{w_{i_{j}}\cdot
L(h(\Phi(x_{i_{j}}),t_{i_{j}}),y_{i_{j}})}-\frac{\mu_{1}}{m}\cdot\sum\limits_{i=1}^{N}{L(h(\Phi(x_{i_{j}}),t_{i_{j}}),y_{i_{j}})\mathbf{1}_{t_{i_{j}}=T_{1}}}-\frac{\mu_{n}}{m}\cdot\sum\limits_{i=1}^{N}{L(h(\Phi(x_{i_{j}}),t_{i_{j}}),y_{i_{j}})\mathbf{1}_{t_{i_{j}}=T_{n}}}$

Calculate step size scalar or matrix $\eta$ with Adam Kingma and Ba (2014)

$[\mathbf{W},\mathbf{V}]\leftarrow[\mathbf{W}-\eta(\beta
g_{1}+g_{3}),\mathbf{V}-\eta(g_{2}+2\lambda\mathbf{V})]$

Check convergence condition

end while

##  5 Causal Inference Based Bidding

Real-time bidding has been investigated thoroughly in recent years Zhu et al.
(2017); Perlich et al. (2012); Zhang et al. (2014). For e-commercial
advertisers in pursuit of conversions, the optimal bidding algorithms share a
common value-based form as:

$bid=\gamma*cvr*ip,$ (9)

where $cvr$, $ip$ are the predicted conversion rate and item price,
respectively. $\gamma$ is a used to regulate the _return-on-investment (ROI)_
/budget of the advertiser. Larger $\gamma$ leads to lower ROI but obtains more
impressions. In Taobao sponsored search practice, $\gamma$ is interpreted as
the inverse of the expected ROI of the advertiser, which can be estimated from
the historical auction log and advertiser’s keyword-level bid settings.
Apparently, the above bidding only considers the promotion value induced by
the advertising PVs, i.e., the direct returns. To bid for the overall returns,
we define an AD-level _leverage rate_ ($lvr$) based on IAE.

######  Definition 5.

The _leverage rate_ for context $x$ with advertising clicks changing from $s$
to $t$ is:

$\sigma_{s,t}(x)=\frac{\hat{\alpha}_{s,t}(x)}{t-s}.$ (10)

$lvr$ reflects the average number of all-channel clicks obtained per
advertising click invested. Apparently, in the perspective of an AD, $lvr$ is
changing as time evolves, which might be influenced by the shifting marketing
environment. In the slowly drifting bidding environment, the number of
advertising clicks an AD can obtain might also gradually change, which gives
an opportunity for the AD to take full advantage of the change. In this sense,
$lvr$ can also be seen as the partial derivative of the overall clicks with
respect to the advertising clicks. Specifically, for an AD in context $x$, we
define the _nominal $lvr$ by taking $s$ and $t$ to be the most recent daily
advertising clicks obtained by the same AD_. For example, let $t$ be the
number of advertising clicks obtained by the same AD yesterday while $s$
corresponds to that obtained the most recent day other than yesterday,
satisfying $s\neq t$. Then the AD-level nominal $lvr$ for today is $\sigma$.
We incorporate the nominal $lvr$ in the bidding equation as:

$bid=\sigma*\gamma*cvr*ip.$ (11)

In light of $lvr$, the bidding takes into account the average overall return
including both clicks from advertising PV itself and the indirect clicks
caused by the advertising effect. In the bidding process, the new bidding
formula as in Eqn. (11) should allocate more budget to those with the
potential to leverage more all-channel clicks. The overall returns of
advertising should be improved given the same budget.

##  6 Online Experiments

Different from classical machine learning tasks, to evaluate causal effect is
intractable due to the missing counterfactual outcomes in reality. Existing
binary-treatment work relies on synthetic or simulated dataset such as IHDP
and Jobs Yao et al. (2018) to evaluate. For multiple treatment scenario, to
the best of our knowledge, there only exists a multiple intervention breast
cancer dataset Yoon et al. (2017), which is not public however. To this end,
we turn to evaluate the causal effect estimation by applying it to the online
bidding engine in Taobao sponsored search, to compare the bidding performance
with the existing online bidding algorithm, which has already been proved to
be a strong baseline in practice.

In Taobao sponsored search system, we randomly choose a set of ADs
$\mathcal{A}$ to carry out the experiment. The nominal $lvr$ distribution in
$\mathcal{A}$ is similar with that in the whole set. We apply the
$lvr$-induced bidding for ADs in $\mathcal{A}$ and keep the remaining as the
control group. For AD $a_{i}\in\mathcal{A}$, the bidding equation is:

$\displaystyle bid_{i}$
$\displaystyle=\kappa*\frac{\sigma_{i}}{\bar{\sigma}}*\gamma*cvr*ip,\forall
a_{i}\in\mathcal{A},\bar{\sigma}=\frac{\sum_{i}{\sigma_{i}}}{|\mathcal{A}|}$

where $\sigma_{i}$ corresponds to the nominal $lvr$ of AD $a_{i}$ and
$|\mathcal{A}|$ is the cardinality of $\mathcal{A}$. Furthermore, $\kappa$ is
utilized to adjust the bidding formula to ensure that the total advertising
cost of all the ADs in $\mathcal{A}$ is approximately the same with that
applying the existing bidding equation. $\kappa$ is updated daily by an
offline replay system and applied online in the new day. The replay system
evaluates the cost with respect to bidding given the daily auction log. Note
that the same advertising cost implies that the number of advertising clicks
is also nearly the same.

We compare the number of all-channel clicks obtained in the $lvr$-bidding
group, with those of the control group as the baseline. Specifically, we
separately show the number of clicks obtained in Taobao organic search engine,
which is the major source of free clicks. For commercial secrets, we hide the
absolute value but show the relative incremental ratio based on the control
group. We display the relative incremental ratio of number of advertising
clicks (Ad)/all-channel clicks (All)/organic search clicks (Search) of the
$lvr$-bidding group in Fig. 2.

Figure 2: $lvr$-bidding performance. The red dashed lines in the upside,
middle, and downside plots show the number of advertising clicks (Ad), all-
channel clicks excluding advertising clicks (All), and organic search clicks
(Search) obtained by ADs in $\mathcal{A}$, respectively. Meanwhile, the green
solid line in the middle (All/Ad)/downside (Search/Ad) figure displays the
ratio of all-channel clicks (excluding advertising clicks)/organic search
clicks over advertising clicks. For commercial secrets, all the values
displayed are the relative ratio divided by the mean of the corresponding
values before treatment goes online, i.e., from Dec. 14, 2018 to Dec. 22, 2018
(included). The middle and downside figures are equipped with two y-axes, with
left for the red line and right for the green line.

The $lvr$-bidding goes online on Dec. 23, 2018 and offline at the end of Dec.
28, 2018 (included). It can be seen that as the $lvr$-bidding goes online,
although the number of advertising clicks goes down $2$ percent, the number of
all-channel clicks obtained by the same ADs sees an increase of $2$ percent,
while the number of organic search clicks increases $4$ percent, which is
significant improvement in a giant system like Taobao sponsored search.
Furthermore, the ratio of all-channel clicks/organic search clicks over
advertising clicks also experiences nearly the same increase pattern. The
contradicted variation between advertising and performance here means that
advertisers actually improve their marketing performance even with less
investments via allocating budgets based on causal effects. As the
$lvr$-bidding goes offline, the performance resembles that before the
experiment, which shows a relative stable marketing environment during the
whole period.

Causal inference-based bidding leads to a clever budget allocation paradigm
based on potential return. Such a paradigm is applicable in a lot of
e-commerce scenarios such as new product promotion and optimizing the budget
allocation in the product life cycle based on the evolving $lvr$.

##  7 Conclusions

Online advertising has been the major promotion approach for e-commercial
advertisers. However, a reasonable evaluation of both direct and indirect
returns induced by advertising has been ignored for a long time. In this
paper, we model the relation between advertising investment and return as a
causal inference problem with multiple treatments available. Relying on the
continuous and transitive treatments, we derive a theoretical upper bound for
the expected estimation error of the individual treatment effect. Then a deep
representation and hypothesis network is designed to balance the selection
bias in causal inference and learn the individual treatment effect. We
evaluate the effectiveness of the estimation algorithm by applying it to an
industry-level bidding engine, which shows that the causal inference-based
bidding outperforms the existing online bidding algorithm. We believe that the
proposed bidding paradigm provides new source of performance growth in
e-commercial advertising.

In the future, we might infer the causal effect of the more fine-grained PV-
level context. Furthermore, the causal effect might be used in the platform
level to influence the ranking of ADs for more efficient overall performance.

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