# Seeing the strongly-correlated zero-bias anomaly in double quantum dot
measurements

Rachel Wortis  Department of Physics & Astronomy, Trent University,
Peterborough, Ontario, K9L0G2, Canada  Joshua Folk  Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, British Columbia,
V6T1Z4, Canada  Department of Physics and Astronomy, University of British
Columbia, Vancouver, British Columbia, V6T1Z1, Canada  Silvia Lüscher  Stewart
Blusson Quantum Matter Institute, University of British Columbia, Vancouver,
British Columbia, V6T1Z4, Canada  Department of Physics and Astronomy,
University of British Columbia, Vancouver, British Columbia, V6T1Z1, Canada
Sylvia Luyben  Department of Physics & Astronomy, Trent University,
Peterborough, Ontario, K9L0G2, Canada

###### Abstract

Experiments in doped transition metal oxides often show suppression in the
single-particle density of states at the Fermi level, but disorder-induced
zero-bias anomalies in strongly-correlated systems remain poorly understood.
Numerical studies of the Anderson-Hubbard model have identified a zero-bias
anomaly that is unique to strongly correlated materials, with a width
proportional to the inter-site hopping amplitude $t$.Chiesa et al. (2008) In
ensembles of two-site systems, a zero-bias anomaly with the same parameter
dependence also occurs, suggesting a similar physical origin.Wortis and
Atkinson (2010) We describe how this kinetic-energy-driven zero-bias anomaly
in ensembles of two-site systems may be seen in a mesoscopic realization based
on double quantum dots, where ensemble averaging is effected by scanning over
a range of gate voltages.

##  I Introduction

Semiconductor-based quantum dots and transition-metal oxides are both subjects
of current interest in condensed matter physics, yet with seemingly little
connection to each other. Transition-metal oxides are studied typically as
bulk materials in which atomic-scale Coulomb interactions profoundly shape the
electronic structure and induce phenomena that range from high temperature
superconductivity to colossal magnetoresistance. Quantum dots, on the other
hand, are typically built from semiconductors described by relatively simple
band structures, with atomic-scale interactions folded into Fermi liquid
parameters such as effective mass. Here we show that a simple mesoscopic
structure involving two coupled semiconductor-based quantum dots can provide a
window into the behavior of bulk strongly-correlated systems.

Specifically, we focus on the non-trivial role that disorder plays in altering
the electronic properties of transition-metal oxides. A well-known effect of
disorder in interacting systems is the appearance of a feature in the density
of states pinned to the Fermi level, referred to as a zero-bias anomaly. The
properties of such anomalies can be used as a probe of electronic structure.
Well established theoretical frameworks exist for understanding the energy
dependence of this feature in weakly correlated metals, which is known as the
Altshuler-Aronov zero-bias anomalyAltshuler and Aronov (1985), and of a
similar feature in insulators, known as the Efros-Shklovskii Coulomb gapEfros
and Shklovskii (1975). A crossover between these metallic and insulating
limits in the weakly correlated regime has been observed in experimentsButko
et al. (2000).

When similar measurements are made in transition metal oxides, zero-bias
anomalies are found, but they generally do not match either the Altshuler-
Aronov or the Efros-Shklovskii pictures.Sarma et al. (1998) Progress has been
made in constructing an alternative framework for understanding zero-bias
anomalies in strongly-correlated systems using the Anderson-Hubbard
model.Chiesa et al. (2008); Wortis and Atkinson (2010); Chen and Atkinson
(2010); Wortis and Atkinson (2011); Chen et al. (2011); Hong-Yi Chen (2012) To
use the disorder-driven zero-bias anomaly that is observed in measurements as
a probe of strong correlations in real materials, more robust contact between
theory and experiment is needed.

Theoretical work suggests that key physics responsible for the zero-bias
anomaly in strongly-correlated systems is captured in an ensemble of two-site
Anderson-Hubbard systems.Wortis and Atkinson (2010) Here, we build on this
insight to describe a link between the incompletely-understood zero-bias
anomaly in bulk disordered strongly-correlated systems, and measurements in
double quantum dots (DQDs) that have been commonplace for decades in the
mesoscopics community.van der Wiel et al. (2002) To researchers in bulk
strongly-correlated materials, the message of this work is that DQDs provide a
controlled environment in which to see the kinetic-energy-driven zero-bias
anomaly unique to disordered strongly-interacting systems. To researchers in
the semiconductor mesoscopics community, the message is that an unconventional
analysis of transport data through two parallel-coupled quantum dots displays
a zero-bias anomaly that is in fact a fingerprint of strong correlations.

Whereas transition metal oxides can generically be described within a many-
site version of the Anderson-Hubbard model, DQDs can be described by a two-
site Anderson-Hubbard model with each dot corresponding to a single site. In
order to connect bulk tunneling measurements with DQD measurements, we
consider DQDs in the parallel configuration in which both dots couple to both
the source and the drain such that the tunneling process may occur through
both dots simultaneously.Chan et al. (2002); Holleitner et al. (2003); Chan et
al. (2002); Chen et al. (2004); Hübel et al. (2008); Keller et al. (2014) This
is in contrast to the more conventional series coupling in which electrons
tunnel sequentially through the two dots. Disorder that would be present in a
bulk material is mimicked in the DQD system by averaging over gate-voltage-
induced energy level offsets. The resulting average conductance through the
DQD system contains, under some conditions, a zero-bias anomaly that has not
been recognized previously.

This paper begins with a brief presentation of the two-site Anderson-Hubbard
model and its realization in DQDs, followed by an overview of past work on the
zero-bias anomaly in the ensemble-averaged density of states. We include
discussion of the effect of nearest-neighbor interactions (electrostatic
coupling), a Hamiltonian term that has been explored in bulk systemsSong et
al. (2009); Hong-Yi Chen (2012); Wortis and Mulindwa (2014) but which was not
included in prior discussion of ensembles of two-site systemsWortis and
Atkinson (2010); Chen and Atkinson (2010); Wortis and Atkinson (2011) and is
relevant to DQDs. We then explore how the zero-bias anomaly is reflected in
DQD measurements, proposing an experimental approach to identify the kinetic-
energy-driven effect that is unique to strongly-correlated systems.

##  II The two-site Anderson-Hubbard model and a double quantum dot

We are considering systems consisting of two sites, each of which has a
variable site potential and between which tunneling may occur (Fig. 1a). A
two-site Anderson-Hubbard model is described by the following Hamiltonian:

$\displaystyle{\hat{H}}$ $\displaystyle=$
$\displaystyle-t\sum_{\sigma=\uparrow,\downarrow}\left({\hat{c}}_{1\sigma}^{{\dagger}}{\hat{c}}_{2\sigma}+{\hat{c}}_{2\sigma}^{{\dagger}}{\hat{c}}_{1\sigma}\right)$
(1)
$\displaystyle+\sum_{i=1,2}\left(\epsilon_{i}{\hat{n}}_{i}+U{\hat{n}}_{i\uparrow}{\hat{n}}_{i\downarrow}\right)$

${\hat{c}}_{i\sigma}$ and ${\hat{n}}_{i\sigma}$ are the annihilation and
number operators for lattice site $i$ and spin $\sigma$. $t$ is the hopping
amplitude and $U$ the strength of the on-site Coulomb repulsion. Disorder is
modelled by choosing $\epsilon_{i}$, the energy of the orbital at site $i$,
from a uniform distribution of width $\Delta$.

DQDs offer an approach to realize this Hamiltonian in experiment (Fig. 1b).
DQDs are commonly built in semiconductor-based two-dimensional electron gases
using metal gates to confine and control the electronsvan der Wiel et al.
(2002). The energy of an orbital on dot 1, $\epsilon_{1}$, can be tuned by
applying a voltage $V_{g1}$ to an adjacent gate. Likewise for dot 2, $V_{g2}$
tunes $\epsilon_{2}$. $U$ effectively represents the single-dot charging
energies $e^{2}/C$, where $C$ is the self capacitance of each dot, assumed to
be the same. In the measurements described here, site potentials
$\epsilon_{1}$ and $\epsilon_{2}$ are varied continuously over a range
$\Delta>U$, large enough to cause the ground-state occupation of each dot to
vary by $\pm 1$. The height of the tunnel barrier between the two dots, and
hence the hopping amplitude $t$, is typically controlled by an additional
gate.

To observe the effects of interest here, the size of each individual dot must
be small enough that the orbital level spacing is large relative to the
thermal energy and to the hopping amplitude between the dots. Furthermore, we
assume the dots contain just 0, 1, or 2 electrons so higher orbitals may be
neglected. We remind the reader that $\Delta$ in this paper refers to the
disorder distribution width, not to orbital level spacing (as is common in DQD
literature) as the orbital spacing is assumed to be so large as never to play
a role.

In DQD experiments, the two dots are generally close enough that the presence
of an electron on dot 1 increases the potential at dot 2. In the tight-binding
model (1) this electrostatic coupling can be represented by adding a nearest-
neighbor interaction term $V{\hat{n}}_{1}{\hat{n}}_{2}$ in the Hamiltonian.

##  III The zero-bias anomaly in the ensemble-average density of states

Figure 1:  (Color online.) (a) Schematic diagram of a single two-site system.
(b) The DQD realization of the system in (a), with tunnel coupling $t$ between
sites and interdot electrostatic coupling $V$. The dots are connected to
source and drain leads in the parallel configuration of relevance to this
paper.

The density of states (DOS) of an ensemble of two-site Anderson-Hubbard
systems (Fig. 1a) has previously been shown to be suppressed at low
energy.Wortis and Atkinson (2010); Chen and Atkinson (2010); Wortis and
Atkinson (2011) In order to understand the structure and origin of this zero-
bias anomaly, we focus first on a single two-site system and then an ensemble
average, showing in each case the effect of inter-site hopping on the the
single-particle DOS. When the system is non-interacting ($U=V=0$), the DOS is
just a histogram of the single-particle states as a function of energy. For a
single two-site system, the DOS will consist of just two peaks of equal
weight. Without hopping (tunneling) between the dots, these peaks will be at
the site energies $\epsilon_{1}$ and $\epsilon_{2}$. With hopping they will be
at the energies of the bonding and anti-bonding states.

When interactions are present, the state of the system can no longer be
described in terms of single-particle states, and the single-particle DOS is
the density of single-particle _transitions_ – transitions from the many-body
ground state (with minimum grand potential $\Omega=E-\mu N$) to many-body
excited states for which the particle number is one more or one less than that
of the ground state. With multiple many-body states available, the single-
particle DOS of a single interacting two-site system consists of between two
and five sharp peaks (depending on the number of possible transitions from the
ground state) with varying heights (depending on the magnitude of the
corresponding matrix elements). Fig. 2(b) shows an example of the DOS, both
without and with hopping, for a specific two-site system with site energies as
shown in Fig. 2(a).

Figure 2:  (Color online.) (a) Diagram of one possible arrangement of Hubbard
orbitals relative to the chemical potential. (b) The DOS of the system shown
in (a) without hopping (black) and with hopping (red). (c) The ensemble-
average DOS with $\Delta=12$, $U=8$, $V=0$ and $t$ as indicated. (d) The
ensemble average DOS with $\Delta=12$, $U=8$, $t=0$ and $V$ as indicated. (e)
The ensemble-average DOS with $\Delta=12$, $U=8$, $V=2$ and $t$ as indicated.
In all cases, $\mu=V+U/2$, corresponding to half filling.

An average over infinitely many disorder configurations (an ensemble of
systems) will result in a smooth DOS, because, for each pair of site
potentials, the peaks appear at different energies. The case of purely on-site
interactions has been explored in detail elsewhere.Wortis and Atkinson (2010);
Chen and Atkinson (2010); Wortis and Atkinson (2011) Two central results from
those references are (i) without hopping there is no zero-bias anomaly, and
(ii) with hopping, and for sufficiently strong on-site interaction and
disorder, there is a zero-bias anomaly the width of which grows linearly with
$t$, as shown in Fig. 2(c).

The DOS suppression can ultimately be traced to the simple lowering of kinetic
energy when electrons can spread themselves over both sites.Wortis and
Atkinson (2010) This generic effect manifests in a unique way in systems in
which both disorder and interactions are strong, because in these systems the
difference in potential between neighboring sites can be roughly equal to $U$
and much greater than $t$. In order to understand how this leads to
suppression of the DOS, consider again Fig. 2(a), where the alignment of the
levels is such that $\epsilon_{2}\lesssim\mu\lesssim\epsilon_{1}+U$. For this
system, the $t=0$ ground state contains two particles, one on each site. When
hopping is added, the energy of this ground state is lowered by an amount
linear in $t$ due to the spreading of the second electron over both sites. To
linear order in $t$, the 1-particle and 3-particle excited states are not
shifted. Therefore, the energies of the transitions from the 2-particle ground
state to both 1-particle and 3-particle excited states are increased: In Fig.
2(b) the red peaks (with hopping) are farther from the origin than the black
peaks (without hopping) by an amount of order $t$. This shift of the
transitions away from zero energy demonstrates the key physics in one system.

The zero-bias anomaly is in the DOS of an ensemble of systems. One might
expect the shifts in transition energies in different systems to cancel each
other. Indeed, changes in peak position and peak height with $t$ do depend on
the specific site potentials and some cancelation does occur. Taken together,
however, the net effect is a systematic shift of weight away from the Fermi
level.Wortis and Atkinson (2010)

When nearest-neighbor interactions are included, there is a minimum in the DOS
at zero energy even when $t=0$. Fig. 2(d) shows the evolution of the $t=0$ DOS
as nearest-neighbor interactions are turned on, favoring charge
ordering.Wortis and Mulindwa (2014) The effect of nonzero $V$ is important for
comparison with DQD experiments in which electrostatic coupling will always be
present. However the resulting $V$-dependent ZBA is of less interest because
it is independent of the strength of the on-site interactions. In contrast,
the $t$-dependent ZBA is unique to the strongly-correlated regime. Taking both
effects together, Fig. 2(e) shows the evolution of the DOS for increasing $t$
at nonzero $V$. It is this last panel that is most relevant to experiments
because, in DQDs, $V$ will always be nonzero and $t$ is relatively easy to
control.

##  IV Experimental signatures of the zero-bias anomaly

We discuss here two ways in which the physics of the zero-bias anomaly is
reflected in the DQD transport measurement sketched in Fig. 1(b). We begin by
describing the connection between a suppression in the ensemble DOS near the
Fermi level and the stability diagrams often produced in DQD measurements.Chen
et al. (2004) We then discuss how the energy dependence of the zero-bias
anomaly can be measured, to identify its physical origin. In particular, our
focus is on distinguishing the unique effect of $t$, in a system with both
strong interactions and strong disorder, from other effects such as the charge
ordering associated with $V$.

###  IV.1 Seeing the reduction of the DOS near the Fermi level

Figure 3: (Color online.) Stability diagrams for a parallel-coupled double
quantum dot system with (a) no hopping, no electrostatic coupling; (b) no
hopping, $V=2$; and (e) $t=0.6$, $V=2$. The color scale indicates the
magnitude of the current as a function of $V_{g1}$ and $V_{g2}$. The ground-
state occupation of each dot is indicated in the two $t=0$ plots, panels (a)
and (b), while in panel (e) the numbers in parentheses indicate the overall
ground-state occupation. In all cases $\Delta=12$, $U=8$, $\mu=V+U/2$ and
$V_{sd}=0.2$. Transition spectrum diagrams for $(eV_{1},eV_{2})=(-3,-4.01)$
(c) $t=0$ and (f) $t\neq 0$; and for $(eV_{1},eV_{2})=(-3,-4.03)$ (d) $t=0$
and (g) $t\neq 0$. Diagrams showing $\mu_{s}$ on the left, $\mu_{d}$ on the
right and the available single-particle-addition transitions. The height of
the transition lines indicates energy, and the length indicates the weight of
the transition.

It is common practice to plot the current through a double quantum dot system
as a function of the gate voltages on the two dots, resulting in what is often
called a stability diagram. This is effectively a plot of the DOS near the
Fermi level for an ensemble of two-site systems. Examples are shown in Fig. 3
panels (a), (b), (e), and (f) for different settings of $t$ and $V$. Here we
address how these plots reflect the suppression of the ensemble-average DOS
near the Fermi level shown in Fig. 2.

To understand the connection between the DOS and the stability diagram,
consider first the simplest case of two independent dots. The conductance
through both dots in parallel is shown in Fig. 3(a), with each dot coupled to
the source and drain with applied bias $V_{sd}\equiv\mu_{s}-\mu_{d}$. We will
focus here on a system in which the barrier to the drain is much lower than
that to the source, such that $\mu_{d}$ sets the ground state of each dot.
Current flows through a dot when a single-particle transition from the ground
state has an energy less than $V_{sd}$. These low energy transitions occur
only for parameters very close to a change in the ground state occupation, so
nonzero current flow appears in Fig. 3(a) only along lines marking the
boundaries between different ground states. Note that, in contrast to most
theoretical work, we adopt the experimental convention that the axes of the
stability diagram correspond to increasing occupancy up and to the right.

When the two parallel-coupled dots are nearby to each other and therefore not
fully independent, but still without tunneling, electrostatic coupling affects
the ground state occupancy of each dot and hence the stability diagram, Fig.
3(b). Fig. 3(c) shows an example of the transition spectrum (single-particle
DOS) corresponding to a set of parameters for which there is a transition with
energy less than $V_{sd}$, so current does flow. Indeed there are two such
transitions, from the 1-particle ground state $\ket{\uparrow 0}$ to two
2-particle excited states $\ket{\uparrow,\uparrow}$ and
$\ket{\uparrow\downarrow}$, which are degenerate when $t=0$ because there is
no singlet-triplet splitting. In contrast, Fig. 3(d) shows an example of the
transition spectrum for a system in which current does not flow: Here the same
transitions discussed above have an energy outside the window between source
and drain. As an example, Fig. 2(c) of Ref. Chan et al., 2002 shows a measured
stability diagram of the type shown in Fig. 3(b) with regions of fixed site
occupancy separated by straight lines where current flows. Note that, while in
a calculation each parameter may be tuned independently, in experiments they
are often interdependent. In particular, increasing $V_{g1}$ will also
slightly increase the potential on dot 2, resulting in lines of current in the
experimental diagram that are slanted rather than strictly horizontal and
vertical lines as in the theory diagram.

Now consider the case of non-zero $t$, when hopping is allowed between the two
dots. The transitions giving rise to current flow are between many-body states
of the DQD system, which here have non-integer weight on each dot. The current
through the system is proportional to the DOS of the DQD system

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle{\rho_{1}(\omega)+\rho_{2}(\omega)\over 2}$ (2)

where $\rho_{i}(\omega)$ is the local DOS on dot $i$, given by the imaginary
part of the local single-particle Green’s function.

$\displaystyle\rho_{i}(\omega)$ $\displaystyle=$
$\displaystyle-{1\over\pi}{\rm Im}\ G_{ii}(\omega)$ (3)

Fig. 3(e) and (f) show examples of the resulting stability diagram, without
and with electrostatic coupling respectively. Fig. 3(g) and (h) show examples
of the transition spectra (single-particle DOS) for two different parameter
settings that would and would not give rise to current flow, respectively. An
experimental example of the curved lines of current shown in Fig. 3(f) can be
seen in Fig. 1(b) of Ref. Chen et al., 2004.

The calculated zero-bias anomalies shown in Figs. 2(c)-(e) emerge from an
ensemble of two-site systems with site potentials chosen from a flat
distribution as described in Section II. The experimental equivalent, measured
in a DQD device for which the parameters may be tuned in situ, is to scan
gates $V_{g1}$ and $V_{g2}$ over an appropriate region of the stability
diagram. The integrated current over such a scan then reflects the ensemble-
average DOS for two-site systems, and a decline in the value of this integral
as the barrier height is lowered (and hence as $t$ grows) is consistent with
the development of a kinetic-energy-driven zero-bias anomaly.

The suppression of ensemble-average DOS at zero bias emerges from two
characteristics of the stability diagrams in Fig. 3: both the length and the
colour of the lines depend on $V$ and $t$. The most obvious effect giving rise
to the suppression is that the total length of the lines decreases from panel
(a), to (b), to (f). Because the width of the lines is fixed by $V_{sd}$,
shorter lines means smaller average DOS near the Fermi energy and therefore a
smaller integrated current, in the absence of other effects. Comparing panel
(b) with panel (a), we see the DOS suppression associated with inter-site
interactions, and by comparing panel (f) with panel (a) we see line-length
influence of DOS suppression associated with both inter-site interactions
($V$) and inter-site hopping ($t$).

However, line length is not the whole story. The zero-bias anomaly is also due
to changes in the magnitude of the contributions (and hence current, indicated
by colour) within each line. Comparing panels (c) and (g) in Fig. 3, one can
see that the weight of the transition that allows the current is much less
when hopping is present. This change in weight is indicated in the stability
diagram by the much darker color (reduced current) of the line separating the
1-particle and 2-particle ground state regions in panel (f) relative to the
corresponding line in panel (b). In contrast, the weight of the transition
facilitating current at the 2-particle to 3-particle boundary is increased,
but not by as much as the other is decreased. The difference in behavior at
the two boundaries arises from the inherent asymmetry of the
experiment–stronger coupling to the drain, with $\mu_{s}$ above
$\mu_{d}$–which thereby enables transitions associated with the addition, not
the removal, of a particle. The combined effect of the changes in the weight
(current) at the two boundaries when hopping is added is a further reduction
of the ensemble-average DOS near the Fermi level, beyond the reduction due to
reduced line length.

###  IV.2 Seeing the energy dependence of the zero-bias anomaly

The analysis above, illustrating the suppression of integrated current in the
DQD system due to finite $t$ and $V$, cannot distinguish between the kinetic-
energy-driven zero-bias anomaly and other sources of DOS suppression near the
Fermi level, such as zero-bias anomalies driven by alternative physical
mechanisms or broader reduction in the DOS. To make matters worse, measuring
the drop in current due to a change in $t$ would be very challenging from an
experimental point of view, as gates to change $t$ would typically also affect
coupling from the dots to source and drain. A more effective approach is to
measure the zero-bias anomaly by the rise in current at finite energy, that
is, with finite applied bias ($V_{sd}$).

Figure 4: (Color online.) The ensemble-average density of states as a function
of frequency near the Fermi level for (a) $U=8$ and (f) $U=1$. In all cases,
$\Delta=12$, $V=0.6$, $\mu=V+U/2$. (b)-(e) stability diagrams for $U=8$ and
(g)-(j) stability diagrams for $U=1$ with $t$ and $V_{sd}$ as indicated and
the same color scale as shown in Fig. 3.

Differential conductance is a common probe of the DOS in DQD structures, and
can readily be measured also as function of $V_{sd}$. To probe the ensemble-
average DOS, the differential conductance across the stability diagram can be
averaged over $V_{g1}$ and $V_{g2}$, as discussed above. Fig. 4 shows the
calculated results from such a gate average over DQD systems with two values
of on-site interactions: $U=1$ and $U=8$. All other parameters are the same
for both figures: disorder $\Delta=12$, nearest-neighbor interactions $V=0.6$,
and four values of hopping $t$. Panel (a) shows the case of strong
correlations, in the sense that $t<<U$ for all hopping values shown, while in
panel (f) $t$ approaches the value of $U$ because $U$ is much smaller.
Although implementing $U\ll\Delta$ as in panel (f) in a DQD system while
remaining in the single-electron regime is impossible, the small $U$ case is
relevant to bulk systems with screening, and the comparison highlights what is
unique about the strongly correlated regime.

Panels (a) and (f) show the ensemble-average DOS close to the Fermi level for
the large and small $U$ cases. The curves without hopping illustrate the DOS
suppression caused by nearest-neighbor interactions, resulting in linear
dependence on energy out to the energy $V$ in both panels (a) and (f). With
hopping, there is a clear difference in the response of the ensembles with
large and small $U$. In the strongly-correlated (large $U$) case a conspicuous
zero-bias anomaly opens as tunneling between the dots is increased, while in
the weakly-correlated case no such feature appears.

Why are the two limits so different? Fig. 4 panels (b) and (c) show the change
in the stability diagram in the strongly-correlated case when hopping is
added, while panels (g) and (h) show the corresponding comparison with weak
onsite interactions. Two energy ratios are relevant – the relative strength of
hopping versus interactions $t/U$ and the relative strength of interactions
versus disorder $U/\Delta$. The ratio $t/U$ is important because the kinetic-
energy-driven zero-bias anomaly arises from an alignment in energy of the
potential on one site with the potential on the other site plus $U$ as shown
in Fig. 2(a).Wortis and Atkinson (2010) As described in Section III, the
linear-$t$ behaviour is generated by a mixing of just two atomic orbitals
$\epsilon_{2}$ and $\epsilon_{1}+U$, but not all four orbitals. This is only
possible with $t/U<<1$.

The linear dependence on $t$ is visible in Fig. 2(c), specifically in the
distance between the sharp maxima in the DOS.Wortis and Atkinson (2010) The
influence of nearest-neighbor interaction $V$ obscures this linearity in Fig.
2(e) and Fig. 4(a).

The ratio $U/\Delta$ is also important because it determines over what
fraction of the ensemble the changes just discussed will manifest. In the
$U=1$ case shown here, there is very little change in the stability diagram at
most values of the site potentials. Other than at the center, both the
locations of the lines, indicating nonzero DOS contribution, and their colour,
indicating the magnitude of the contribution, are very similar in panels (g)
and (h). The $U=8$ case, in contrast, shows significant changes in both the
position and the colour of the lines between panels (b) and (c). Because
$U/\Delta$ is small in the $U=1$ case shown, when $t=0.2$ such that $t/U$ is
small, the center of the diagram will show similar changes to those seen in
the $U=8$ case, but there is no dramatic zero-bias anomaly because the
addition of hopping has little effect for most values of the site potentials.

A closer examination of changes in the stability diagram with bias can help to
understand the origin of the sharp edge in the zero-bias anomaly, for $U=8$.
Panels (c)-(e) and (h)-(j) of Fig. 4 show how the stability diagram evolves
with $V_{sd}$ for the strongly and weakly correlated cases respectively. In
the weakly correlated case ((b)-(e)), the only change is the width of the
region in which nonzero current flows. In the strongly correlated case
((g)-(j)), on the other hand, new transitions–specifically those to the two
triplet states–begin to contribute at higher values of $V_{sd}$, sharply
increasing the current flow for specific potential configurations. It is the
entrance of these new transitions that creates the sharp edge to the zero-bias
anomaly in the ensemble-average DOS. However, while the sharp edge is visually
striking, the connection with bulk systems is in the largest energy scale of
the anomaly rather than its detailed shape.

##  V Conclusion

In summary, we have pointed out two ways in which a characteristic transport
signature of disordered strongly-correlated bulk materials can be seen in DQD
measurements. Comparing stability diagrams at fixed $V_{sd}$, without and with
hopping, shows that hopping suppresses the DOS. While such diagrams have been
observed in DQD systems, they have not previously been connected with the bulk
zero-bias anomaly. Moreover, comparing stability diagrams at fixed hopping as
$V_{sd}$ is increased allows exploration of the energy dependence of the zero-
bias anomaly. Such an analysis, including averaging over gate voltages, has
not, to our knowledge, been performed. The presence of a zero-bias anomaly
that depends only on hopping, and for which the maximum energy scale grows
linearly with $t$, is a unique signature of strongly-correlated systems.
Observing this effect in DQD systems would establish a first solid point of
contact between theory and experiment in the study of the disorder-driven
zero-bias anomaly in strongly correlated systems.

##  VI Acknowledgements

R.W. gratefully acknowledges support from the Natural Sciences and Engineering
Research Council (NSERC) of Canada and thanks the Pacific Institute for
Theoretical Physics for hosting her. J.F. and S.L. acknowledge support from
NSERC, the Canada Foundation for Innovation, the Canadian Institute for
Advanced Research, and the Stewart Blusson Quantum Matter Institute.

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# From Factor Models to Deep Learning:
Machine Learning in Reshaping Empirical Asset Pricing

Junyi Ye∗ Bhaskar Goswami∗ Jingyi Gu∗ Ajim Uddin&Guiling Wang New Jersey
Institute of Technology {jy394, bg362, jg95, ajim.uddin<EMAIL_ADDRESS>
###### Abstract

This paper comprehensively reviews the application of machine learning (ML)
and AI in finance, specifically in the context of asset pricing. It starts by
summarizing the traditional asset pricing models and examining their
limitations in capturing the complexities of financial markets. It explores
how 1) ML models, including supervised, unsupervised, semi-supervised, and
reinforcement learning, provide versatile frameworks to address these
complexities, and 2) the incorporation of advanced ML algorithms into
traditional financial models enhances return prediction and portfolio
optimization. These methods can adapt to changing market dynamics by modeling
structural changes and incorporating heterogeneous data sources, such as text
and images. In addition, this paper explores challenges in applying ML in
asset pricing, addressing the growing demand for explainability in decision-
making and mitigating overfitting in complex models. This paper aims to
provide insights into novel methodologies showcasing the potential of ML to
reshape the future of quantitative finance.

**footnotetext: These authors contributed equally to this work.

##  1 Introduction

The finance sector, often recognized as the backbone of economic society,
plays an indispensable role in facilitating smooth economic operations and
growth. This sector faces unique challenges, stemming from the complexity of
financial markets, regulatory constraints, the need for precise decision-
making, big data, and the rapid pace of technological change. The integration
of Machine Learning (ML) and Artificial Intelligence (AI) into this domain,
particularly in asset pricing, is not merely an advancement but a necessity.

Empirical asset pricing models, which aim to elucidate the complex
relationships between financial assets and their expected returns, are crucial
for investors, fund managers, and policymakers. Traditional models like the
Capital Asset Pricing Model (CAPM) Sharpe (1964) and the Fama-French models
Fama and French (2015) have been foundational yet often struggle to capture
the multifaceted and nonlinear dynamics of financial markets.

In literature, the issues of less predictive accuracy, difficult variable
selection, and less flexible functional forms of traditional models are well
documented Welch and Goyal (2008); He and Krishnamurthy (2013). [Gu et al., Gu
et al. (2020), propose addressing these issues by introducing ML in asset
pricing. ML models offer better predictive power and the ability to model
complex non-linear relationships with much more flexibility. They also enable
the integration of non-traditional data sources, e.g., text, image, video, and
audio data, enriching the decision-making processes. In addition, fine-tuning
and parameter optimization strategies enable continuous learning based on new
information, facilitating real-time decision-making. As a result, we see a
surge in the application of ML in finance, especially for modeling
complexities in asset pricing.

In this paper, we offer a comprehensive review of empirical asset pricing
using machine learning (ML). We examine how ML-based approaches have
transformed traditional models, providing a renewed perspective on challenges
in asset pricing. Unlike some previous works that focused solely on either
Finance Giglio et al. (2022) or those detailing the taxonomy of financial-risk
tasks linked to machine learning methods (Mashrur et al. (2020)), our approach
involves an examination of recent research contributions from both finance and
computer science fields, offering a comprehensive view of the
interdisciplinary advancements. Moreover, we conduct a critical evaluation of
current challenges in adopting ML for empirical asset pricing, identifying
research gaps and providing insights into future research directions. As a
result, this review serves as a valuable resource for future researchers,
offering a starting point to understand the current advancements in this
interdisciplinary domain. Pioneering in its scope, our survey is the first to
provide a detailed emphasis on the ML algorithm development perspective
specifically for asset pricing, filling a critical void in existing
literature.

The organization of the rest of the paper is as follows: Section 2 addresses
the fundamental asset risk premia estimation problem and covers a spectrum
from traditional factor models to ML for various tasks. Section 3 explores
portfolio optimization from traditional approaches to supervised learning and
reinforcement learning strategies. Section 4 delves into recent advancements
in asset pricing techniques. Section 5 examines the challenges and anticipates
future trends.

##  2 Risk Assessment and Price Prediction

Asset pricing pertains to the valuation of various financial instruments, such
as equities, options, fixed-income securities, and cryptocurrencies. In
traditional finance, the fundamental problem of asset pricing boils down to
estimating asset risk premia defined as the conditional expected return over
the risk-free rate.

$y_{i,t}=\mathbb{E}[y_{i,t}]+\epsilon_{i,t},$ (1)

where $y_{i,t}$ is the individual assets excess return over the risk-free
rate, i.e., t-bill rate, with $\mathbb{E}[\epsilon_{i,t}]=0$. The conditional
expected return $\mathbb{E}[y_{i,t}]$ often modeled as an unknown function
$g(\cdot)$ of some $P$-dimensional predictors $x_{i,t}$ and parameterized by
$\theta$ as:

$y_{i,t}=g(x_{i,t-1};\theta)+\epsilon_{i,t}.$ (2)

Asset pricing models commonly incorporate asset-specific characteristics and
macroeconomic factors as predictors $X$ for estimating excess return $y$.

###  2.1 Traditional Factor Models

Historically, factor models have stood out as pivotal frameworks for asset
pricing. The fundamental idea behind these factor-based models is that the
high-dimensional predictors $X$ can be replaced with low-dimensional factors
$F$. This replacement allows for the explanation of cross-sectional asset
returns through their sensitivity to the risk factor. In a factor model,
excess return can be modeled as:

$y_{i,t}=\alpha_{i,t-1}+\beta^{\prime}_{i,t-1}f_{t}+\epsilon_{i,t},$ (3)

where, the loading $\beta_{i,t-1}$ represent an asset $i$’s exposure to a
common risk factor $f_{t}$ and $\alpha_{i,t-1}$ denotes the mispricing
component, assuming $\alpha_{i,t-1}=0,\forall i$ and $t$.

Although some factors, such as industrial production growth, are known and
observable, the empirical asset pricing literature assumes that most factors
are unobservable or latent. Researchers employ two approaches to estimate
these latent factors. The first approach involves constructing characteristic-
sorted portfolios based on prior knowledge about the cross-section of returns,
and then considering the long-short portfolio returns as the observable
factors. The well-known Fama-French Five-Factor model Fama and French (2016),
encompassing market, size, value, profitability, and investment, serves as an
example of such factors. Over the years, the empirical asset pricing
literature has reported hundreds of factors that contribute to explaining
cross-sectional asset returns. Despite these efforts, empirical evidence
indicates that factor models developed using these techniques have failed to
produce $\alpha_{i,t-1}=0$. Researchers attribute this limitation to a partial
understanding of the variability in cross-sectional average returns Kelly et
al. (2019); Giglio et al. (2022). Constructing an all-encompassing model in
this approach requires complete knowledge of the cross-section of returns and
the inclusion of every influencing characteristic. However, our current
understanding remains partial at best.

The second approach is to estimate the latent factors from the panel of
realized returns using factor analysis techniques, e.g., PCA Connor and
Korajczyk (1986). This approach does not require ex-ante knowledge of the
cross-section of average returns. In a static PCA-based approach, the
estimation for factors occurs in two steps: firstly, it combines a large set
of predictors into linear combinations of latent factors $f$, and then
utilizes these latent factors to model excess returns through predictive
regression. The relationship between the latent factor $f_{t}$ and observable
characteristics $x_{t}$ is represented as $x_{t}=\beta f_{t}+u_{t}$. In order
to estimate time-varying factors and factor loadings, Kelly et al. (2019)
propose IPCA, as:

$y_{i,t}=x_{i,t-1}\beta_{i}f_{t}+\epsilon_{i,t}.$ (4)

Here, $\beta\in\mathbb{R}^{N\times K}$ and $f_{t}\in\mathbb{R}^{K\times T}$
are unknown parameters, and $\epsilon_{i,t}$ is a composite error, including
$\alpha_{i,t-1}$. A significant strand of traditional asset pricing literature
focuses on developing appropriate techniques to reduce the dimensionality of
predictors and estimate these latent factors Feng et al. (2020); Giglio and
Xiu (2021) (a detailed discussion is provided in Section 4.1).

These traditional approaches serve as the building blocks for estimating asset
risk premia and the various factors influencing asset returns. However, these
methods have multiple limitations. First, the ever-increasing factor universe
leads to a large number of free parameters in Equation 4, resulting in
inefficient estimations using traditional regression-based models. Second, the
factor models rely on a linear approximation of risk exposure based on
observable characteristics. Nevertheless, theoretical asset pricing models
suggest nonlinearity in return dynamics Bansal and Yaron (2004); He and
Krishnamurthy (2013). In the following section, we discuss how integrating
machine learning techniques into traditional models can overcome these
limitations and further augment their capabilities.

Figure 1: ML in Asset Pricing

###  2.2 AI-Augmented Prediction Models

Unlike traditional models, ML takes into account the temporal variability of
financial markets and their non-linear interactions. ML also provides built-in
solutions for factor selection and handling high-dimensional data. As a
result, we observe numerous successful applications of ML models in empirical
asset pricing, encompassing risk premia estimation Chen et al. (2023), feature
engineering Gu et al. (2020); Long et al. (2019), and dimension reduction
Kelly et al. (2019, 2023).

To simplify, most ML-based model tries to estimate the function $g(\cdot)$
identified in Equation 2. The objective is to find the optimal functional
representation for the cross-sectional return $y_{t}$ based on the information
(characteristics) available at previous time point $x_{i,t-1}$ as:

$\hat{g}(\cdot)=\arg\min_{g\in\mathcal{G}}\sum_{i=1}^{N}\sum_{t=1}^{T}(y_{i,t}-g(x_{i,t-1}))^{2}.$
(5)

Researchers have employed various specifications and models with flexible
structures to estimate the function $g(\cdot)$. These models range from simple
linear models, such as linear regression, to more complex ones like non-linear
decision trees, ensemble random forests, gradient-boosted regression trees,
and deep neural networks Gu et al. (2020); Leippold et al. (2022). The
application domain is also extensive, covering equities Gu et al. (2020); Chen
et al. (2023), cryptocurrencies, futures, and options Goudenege et al. (2020).

The functional form, as depicted in Equation 5, is primarily employed for
cross-sectional return prediction. Beyond this, ML applications extend to
predicting individual asset returns using time-series models and forecasting
price movements or rankings through the use of classification and regression
algorithms.

Figure 2: General Pipelines of Temporal Models (Left) and Spatio-Temporal
Models (Right).

Time-series models forecast asset values by analyzing historical price data to
identify temporal patterns and dependencies, such as trends, seasonality, and
cycles. For return prediction, common evaluation metrics include distance-
based measurements such as RMSE, MAE, and MAPE. Asset movement prediction,
driven by classification algorithms, focuses on the direction of price changes
(upward, stationary, and downward) by examining market trends and
correlations. In this context, evaluation metrics such as Accuracy, Precision,
Recall, F1 Score, and MCC are employed. Specifically, MCC is often emphasized
in financial data analysis due to its effectiveness in handling imbalanced
datasets, offering a more nuanced performance measure compared to Accuracy and
F1 Score. Meanwhile, asset ranking prediction uses regression models to
organize assets within a market or portfolio, aiding investors and fund
managers in identifying securities likely to outperform, thereby supporting
strategic investment decisions. Ranking predictions are typically evaluated
using metrics such as MAP, MRR, and NDCG. Details of these performance
measures are presented in Table 1.

In the following sections, we delve into recent advancements in ML models that
capture the temporal and spatial dependencies in financial data, highlighting
the innovative approaches and techniques that have emerged in prediction and
ranking.

####  2.2.1 Temporal Models

To address the inherent temporal dependencies in financial data, a number of
temporal models have been developed for individual assets. These models aim to
decode historical dynamics to predict future market trends accurately.
Temporal models operate by harnessing a sequence of data points from prior
time stamps, referred to as lagged observations, to project future values.
These inputs range from simple constructs, like lagged prices or returns, to
more intricate arrays comprising diverse financial indicators and market data.
Mathematically, the nexus between historical inputs and future predictions is
encapsulated as follows:

$y_{i,t},y_{i,t+1},\ldots=g_{t}(x_{i,t-1},x_{i,t-2},\ldots,x_{i,1}).$ (6)

Here, $y_{i,t},y_{i,t+1},\ldots$ signify the predicted values for asset $i$
(e.g., future prices or returns) at future time $t,t+1,\ldots$. The terms
$x_{i,t-1},x_{i,t-2},\ldots,x_{i,1}$ represent the historical input features
up to time $t-1$. Each $x_{i,t}$ is a vector that encapsulates one or multiple
attributes such as lagged prices, returns, or other pertinent financial
indicators at time $t$. The function $g_{t}$ is designed to capture the
intrinsic temporal patterns and dependencies within the data, effectively
mapping past observations to future values.

The evolution of these models has been significantly influenced by the rise of
deep learning, replacing traditional models like ARIMA and VAR. Starting from
the simple feedforward model the focus soon shifted to Recurrent Neural
Networks (RNN) and especially Long-Short Term Memory (LSTM) Selvin et al.
(2017). Recently, the trend in time-series forecasting has shifted from RNNs
towards all-MLP (Multilayer Perceptron) models such as N-BEATS Oreshkin et al.
(2019), which are dominating the time series benchmark leaderboards. Notably,
TS-Mixer Ye et al. (2023) has outperformed both RNNs and N-BEATS in predicting
S&P 500 returns, showcasing the potential of sophisticated designs, such as
residual connections and Mixer blocks in MLP architectures, jointly learning
the correlations among features and temporal dynamics.

Another significant effort involves transitioning from single-scale to multi-
scale (time, frequency, resolution) approaches. This shift, incorporating
different types of scale techniques (Fourier transform, wavelets,
downsampling), is crucial for dissecting the multifaceted behavior of stocks,
ranging from rapid intraday fluctuations to long-term trend formations. MTDNN
Liu et al. (2020a) utilizes wavelet-based and downsampling techniques to
analyze stock patterns at varying scales, from fine-grained to broader trends.
Transformer-based approach Ding et al. (2020) with its Multi-Scale Gaussian
Prior excels in capturing long-term, short-term as well as hierarchical
dependencies of financial time series.

Task |  Evaluation Metric |  Formula
---|---|---
Return Prediction |  RMSE ($\downarrow$) |  $\sqrt{\frac{1}{T}\frac{1}{N}\sum_{(i,t)\in\Omega}(y_{i,t}-\hat{y}_{i,t})^{2}}$
|  MAE ($\downarrow$) |  $\frac{1}{T}\frac{1}{N}\sum_{(i,t)\in\Omega}|y_{i,t}-\hat{y}_{i,t}|$
|  MAPE ($\downarrow$) |  $\frac{1}{T}\frac{1}{N}\sum_{(i,t)\in\Omega}\left|\frac{y_{i,t}-\hat{y}_{i,t}}{y_{i,t}}\right|\times 100\%$
Movement Prediction |  Accuracy ($\uparrow$) |  $\frac{\sum_{i=1}^{N}I(\hat{D}_{i}=D_{i})}{N}$
|  Precision ($\uparrow$) |  $\frac{\text{TP}}{\text{TP+FP}}$
|  Recall ($\uparrow$) |  $\frac{\text{TP}}{\text{TP+FN}}$
|  F1 Score ($\uparrow$) |  $\frac{2\cdot\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}}$
|  MCC ($\uparrow$) |  $\frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}$
Ranking Prediction |  MAP ($\uparrow$) |  $\frac{1}{T}\frac{1}{K}\sum_{(k,t)}P(k)_{t}\cdot\text{rel}(k)_{t}$
|  MRR ($\uparrow$) |  $\frac{1}{T}\sum_{t=1}^{T}\frac{1}{rank_{t}}$
|  NDCG ($\uparrow$) |  $\frac{\text{DCG}}{\text{IDCG}},\text{ where }\text{DCG}=\sum_{k=1}^{K}\frac{\text{rel}(k)}{\log_{2}(k+1)}$
Portfolio Optimization |  Accumulated Return ($\uparrow$) |  $\prod\limits_{t=1}^{T}\sum_{i}^{N}w_{i,t-1}y_{i,t}$
|  Volatility ($\downarrow$) |  $\sigma(PR_{t})$
|  Maximum Drawdown ($\uparrow$) |  $\max_{0\leq a\leq b\leq T}1-\frac{\prod\limits_{t=1}^{b}\sum_{i}^{N}w_{i,t-1}y_{i,t}}{\prod\limits_{t=1}^{a}\sum_{i}^{N}w_{i,t-1}y_{i,t}}$
|  Sharpe Ratio ($\uparrow$) |  $\frac{\sum_{t=1}^{T}(PR_{t})/T}{\sigma(PR_{t})}$
|  Calmar Ratio ($\uparrow$) |  $\frac{\sum_{t=1}^{T}(PR_{t})/T}{\text{MDD}}$
|  Sortino Ratio ($\uparrow$) |  $\frac{\sum_{t=1}^{T}(PR_{t})/T}{\text{DD}}$
Table 1: Evaluation Metrics Used in Empirical Asset Pricing222Note: RMSE: Root
Mean Square Error, MAE: Mean Absolute Error, MAPE: Mean Absolute Percentage
Error, MCC: Mathews Correlation Coefficient, MAP: Mean Average Precision, MRR:
Mean Reciprocal Rank, NDCG: Normalized Discounted Cumulative Gain, $T$: Number
of timestamps, $N$: Number of assets, $D$: Price Movement Direction, TP: True
Positive, TN: True Negative, FP: False Positive, FN: False Negative, $K$:
Maximum top-ranked assets evaluated, $P(k)$: Precision at Rank $k$,
$\text{rel}(k)$: 1 if rank $k$ asset is suitable, 0 otherwise, $rank$:
Position of first relevant asset, DCG: Discounted Cumulative Gain, IDCG: Ideal
Discounted Cumulative Gain, $\sigma(\cdot)$: Standard Deviation, PR: Portfolio
Return, DD: Downside Deviation, MDD: Maximum Drawdown.. $\uparrow$ means the
larger the better while $\downarrow$ means the smaller the better.

####  2.2.2 Spatio-Temporal Models

The financial market exhibits interconnectivity, where fluctuations in the
price of one asset can influence the pricing dynamics of its affiliated
counterparts. Temporal models adeptly capture chronological data but often
neglect spatial interconnections, which are pivotal in the financial market.
Incorporating these spatial connections can significantly enhance prediction
accuracy and reduce noise in asset pricing. Many studies have introduced Graph
Neural Networks (GNNs) to model the intricate relationships between assets,
capturing spatial dynamics in the process. GNNs are particularly adept at
learning sparse spatial dependencies, a trait that aligns well with the
complex and nuanced nature of financial markets.

The general form of spatio-temporal is denoted as:

$y_{t},y_{t+1},\ldots=g_{t}[g_{s}(X_{t-1}),g_{s}(X_{t-2}),\ldots,g_{s}(X_{1})].$
(7)

Here, $X_{t},X_{t-1},\ldots,X_{1}$ represent the feature matrices of the asset
pool (e.g., a collection of assets) at past time stamps, and
$y_{t},y_{t+1},\ldots$ are the predicted values for the future time steps.
Most existing spatio-temporal models first capture spatial correlations with
$g_{s}$, such as GNN, among assets and then their temporal evolutions with
temporal models $g_{t}$.

To model spatial dynamics, researchers adopt diverse relational frameworks for
constructing graphs, capturing the unique similarities among entities in the
financial market. Chen et al. (2018) explored shareholding ratios to map
corporate interconnections, employing graph embedding techniques such as
DeepWalk, LINE, node2vec, and Graph Convolutional Network (GCN). Meanwhile,
HATS Kim et al. (2019) applied Graph Attention Networks (GAT) to analyze
graphs based on ownership and organizational structures. Addressing the
limitations of traditional GNNs, which are restricted by pre-defined firm
relations AD-GAT Cheng and Li (2021) introduces an attribute-mattered
aggregator for capturing attribute-sensitive momentum spillovers. RSR Feng et
al. (2019) employed Temporal Graph Convolution, they use sector similarity and
supply-chain network to analyze relational dynamics and rank stocks by their
revenue potential. Similarly, STHAN-SR Sawhney et al. (2021) enhances stock
selection for quantitative trading through a Spatio-Temporal Hypergraph
Attention Network. This model utilizes a neural hypergraph structure, with
hyperedges representing industry and corporate connections. Uddin et al.
(2021) utilized attention mechanisms to develop evolving networks, linking
firms through Pearson Correlation, and learned from positive and negative
graphs separately. Besides GNNs, DTML Yoo et al. (2021) uses a Transformer
based model to encode asymmetric, and dynamic stock correlations.

These spatial models integrate temporal recurrent neural networks such as LSTM
and GRU and evolve into spatio-temporal models. This integration results in a
more sophisticated and accurate representation of market dynamics, effectively
capturing both spatial connections and temporal evolutions within the
financial landscape.

##  3 Portfolio Optimization

Portfolio optimization is a crucial component of asset pricing. Investors
construct an investment portfolio comprising diverse assets including stocks,
foreign currency, futures, bonds, cryptocurrencies, and other securities. The
Modern Portfolio Theory (MPT)Markowitz (1952) revolutionized the landscape of
portfolio construction and risk management. MPT argues that investors, being
inherently risk-averse, can strategically design portfolios to either maximize
expected returns within a specified level of risk or, conversely, to minimize
risk for a given level of expected return. MPT emphasizes the significance of
diversification, suggesting that by spreading investment across various assets
without positive correlation, investors can achieve a more optimal risk-return
trade-off. The utility maximizing portfolio solution can be defined as:

$w^{*}=\frac{1}{\gamma}\Sigma^{-1}\mu,$ (8)

where, $\mu\in\mathbb{R}^{N}$ and $\Sigma\in\mathbb{R}^{N\times N}$ represent
the mean and variance of return on $N$ assets, and $\gamma$ represent
investros risk aversion. An investor can develop an optimal portfolio of
$w^{*}\in\mathbb{R}^{N}$ in two ways. First, by using models to estimate the
return distribution and treating the estimations as known parameters $\mu$ and
$\Sigma$, then selecting weights $w$ that optimize the utility. Second, by
utilizing models to optimize the portfolio through parameterizing the weights.
Here, we discuss how ML facilitates both of these methods.

###  3.1 Supervised Learning Techniques

Supervised Learning models can facilitate portfolio optimization by estimating
$\mu$ and $\Sigma$ of assets. As discussed in Section 2, ML can predict
returns, rank, and movements. Investors can choose appropriate assets based on
these predictions and develop a long-short portfolio. Firstly in the process
of choosing assets: (1) Given a return prediction model or rank prediction
model, investors can take a long position on assets with the highest predicted
future returns or highest ranking, and a short position on assets with the
lowest predicted future returns or lowest ranking Uddin et al. (2021); Lin et
al. (2021); Duan et al. (2022). (2) Given a movement prediction model, the
portfolio can be developed by taking a long (short) position in an asset with
a predicted upward (down) direction Yoo et al. (2021). Subsequently, during
the portfolio construction process, investors establish position at the
beginning of period $t-1$ and liquidate/rebalance the portfolio at $t$.
Portfolio are typically allocated with equal weights $w_{i,t}=\frac{1}{N}$ or
weighted weights $w_{i,t}=\frac{v_{i,t}}{\sum_{i=1}^{N}{v_{i,t}}}$.

###  3.2 Reinforcement Learning Methods

The portfolio weight $w_{i,t}$ among $N$ asset at time $t$ can also be
directly determined with the aim of maximizing return and minimizing risk. The
most prominent approach to this end is Reinforcement learning (RL). Leveraging
its nature of sequence decision-making RL utilizes neural networks to
approximate the action policy function and estimate the reward function. In
the RL environment as shown in Figure 3, the state $s_{t}$ at the beginning of
each period is defined as the pair of historical prices $x_{t}$ and previous
portfolio weight $w_{t-1}$, observable in the environment; deep neural
networks are employed to determine the portfolio weight vector $w_{t}$ as the
action output $a_{t}$; the agent, representing the portfolio manager, performs
trading actions and obtains periodic logarithmic returns as the reward
$r_{t}$. The RL agent aims to find an optimal policy to maximize the action-
value function $Q^{\pi}(s,a)$:

$\displaystyle
Q^{\pi}(s,a)=\mathbb{E}[\sum_{i=t}^{\infty}\gamma^{i}r_{t+i}|s_{t}=x_{t},a_{t}=w_{t-1}]$
(9)

where $\gamma_{i}$ is the discount factor. A notable contribution in this
domain is EIIE Jiang et al. (2017), which proposes a financial-model-free RL
approach with deep learning solutions for the portfolio optimization problem.
It assumes _Zero market impact_ , positing that trading activities do not
significantly affect market prices due to relatively small investment capital.
EIIE is trained in an Online Stochastic Batch Learning (OSML) scheme and
conducts three distinct model species: CNN, RNN, and LSTM. This framework is
implemented across various asset classes, including stocks and
cryptocurrencies.

Figure 3: General Framework for RL in Portfolio Optimization

Following EIIE, a number of recent studies utilize deep RL for portfolio
optimization under more realistic trading scenarios. In Wang et al. (2019); Xu
et al. (2021), attention mechanisms are employed to model interrelationships
among assets. In these studies, short sales are incorporated under simple
assumptions, such as investors can borrow without limitations and will hold
equal long-short positions. Some other works focus on optimizing the reward
function with Imitative Recurrent Deterministic Policy Gradient Liu et al.
(2020b), and contrastive learning Lien et al. (2023). Trading execution and
trading costs are also considered. The Hierarchical RL framework developed in
Wang et al. (2021b) enables the placement of small-sized limit orders at
desired prices and quantities during the trading period to minimize trading
costs. Smart Trader Yang et al. (2022) employs a Geometric Brownian Motion
process for portfolio weight and optimal trading point decisions to achieve
excess intraday profit. Deep RL also contributes significantly to algorithmic
trading. For example, iRDPG applied Imitative Learning and Behavior Cloning to
create a trading bot that consistently takes intraday greedy actions, with the
aim of developing a profitable policy Liu et al. (2020b). Meta Trader
initially acquires multiple diverse trading policies and subsequently learns
to choose the most suitable policy based on market conditions Niu et al.
(2022).

Additionally, a notable research avenue focuses on the generalized RL
framework. The openly accessible FinRL library Liu et al. (2022) provides
diverse market environments with data APIs and automatic pipelines. It enables
users to select popular benchmarks based on their requirements and design
novel strategies. Margin Trader Gu et al. (2023) integrates margin accounts
and constraints to enhance the realism of the trading environment for both
long and short positions.

##  4 Innovations in Asset Pricing Techniques

In recent years, the impact of ML on finance has been most prominently
recognized in its role in price prediction and portfolio optimization.
However, the scope of ML’s contributions extends far beyond these domains.
Notably, ML has made significant strides in the development of advanced
techniques and processes for asset pricing. These innovations not only enhance
the accuracy of predictions but also simplify the application of traditional
models. This section explores the multifaceted contributions of ML,
specifically in advancing techniques that facilitate asset pricing studies.

###  4.1 Dimensionality Reduction

In asset pricing, researchers are often faced with a ‘factor zoo’ – a term
referring to the extensive array of factors (hundreds, sometimes) proposed in
the literature for predicting asset returns. This not only complicates the
models but also raises concerns about overfitting and interpretability. This
is where dimensionality reduction techniques become invaluable. The primary
motivation for employing dimensionality reduction in financial data analysis
is to distill this multitude of factors into a more manageable, representative
set. In traditional finance, Partial Least Squares (PLS) Gu et al. (2020) and
Principal Component Analysis (PCA) Giglio and Xiu (2021), along with its
advanced variants, are widely used to achieve these objectives. Factor
selection models, like LASSO, is another popular alternative. For example,
Feng et al. Feng et al. (2020) proposed a two-step LASSO approach to identify
the most influential factors among the 150 previously identified factors for
explaining cross-sectional returns.

In recent years, autocoder-based models have gained significant traction for
dimension reduction in financial data. In Gu et al. (2021); Uddin and Yu
(2020), the authors use auto-encoder to learn latent asset pricing factors by
jointly modeling both asset pricing characteristics and excess return.
Building on this, Variational Autoencoders (VAEs) advance the concept by
introducing a probabilistic twist to the encoding process, effectively
distilling meaningful factors from noisy market data. FactorVAE Duan et al.
(2022) addresses the challenge of low signal-to-noise ratios in financial
datasets by fusing dynamic factor models (DFM) with a VAE framework, thereby
harnessing factors as latent variables to enhance asset pricing models.
Concurrently, DiffusionVAE (D-Va) Koa et al. (2023) combines a hierarchical
VAE structure with diffusion techniques to efficiently process stochastic
stock data, ensuring precise, denoised predictions.

###  4.2 Imputing Missing Data

Financial data, especially in asset pricing, often involves missing values.
These values may be random or non-random and can occur for multiple reasons,
including poor data curation or intentional non-reporting. In dealing with
missing values, historically, researchers and practitioners have either
removed observations with missing values or imputed the missing values with
zeros or with class mean Bryzgalova et al. (2022). However, both approaches
have major drawbacks, as they may either remove important information or alter
the distribution of the data.

The advancement of recommender systems offers a new avenue for tackling
financial missing data. In Uddin et al. (2022a), a coupled matrix
factorization approach is proposed to impute missing analyst earnings
forecasts. In their work, the authors augment missing analyst data with firm
characteristics data and then use the imputed values for firm return
prediction. In Beckmeyer and Wiedemann (2022), the authors use a transformer
to impute missing firm characteristics data. Additionally, tensor imputation
shows significant promise for imputing spatio-temporal data in finance Uddin
et al. (2022b); Zhou et al. (2023). Despite these efforts, research in this
area is still limited. Advanced DL methods can efficiently impute missing
values by incorporating non-linearity and spatio-temporal interactions in
financial data, thereby offering an exciting direction for future research.

###  4.3 Incorporating Alternate Data

The fusion of alternate data streams—text, images, and speech—with traditional
financial information is revolutionizing asset pricing. This surge in
popularity is significantly driven by the advancements in transformers and
LLMs. By harnessing the power of Computer Vision (CV) and Natural Language
Processing (NLP), this multimodal approach augments traditional financial
analyses, providing a richer, more nuanced view of market dynamics. The result
is the creation of pricing models that are not only more accurate but also
significantly more sophisticated.

In financial technology, incorporating image analysis enhances data processing
significantly. Traders often rely on visual cues from financial charts, a
concept advanced by Cohen et al. (2020), who reimagined time-series analysis
as an image classification task using models like CNNs. Building on this, Zeng
et al. (2021) transformed price data of multiple assets into 2D images,
utilizing video prediction algorithms for market dynamics analysis.

Textual data, encompassing stock descriptions and social media content, has
emerged as a critical component in investment recommendation and market
analysis frameworks. Advanced methodologies, such as the Time-Aware Graph
Relational Attention Network (TRAN) Ying et al. (2020) demonstrate the
efficacy of integrating textual content with financial datasets to augment
stock profiling and recommendation systems. In particular, models like
Prediction-Explanation Network (PEN) Li et al. (2023) underscores the
paramount importance of synergizing social media inputs, inter-stock
relational dynamics, and traditional financial indicators, placing a
pronounced emphasis on the sentiments analysis for improving predictive
accuracy in stock movements.

Furthermore, the pioneering VolTAGE model Sawhney et al. (2020) represents a
significant stride in the realm of multimodal data integration for stock
volatility prediction. This model ingeniously incorporates vocal cues from
corporate executives’ earnings calls, analyzing them in conjunction with
traditional financial data within a Graph Convolutional Network (GCN). The
deployment of an inter-modal, multi-utterance attention mechanism within
VolTAGE is particularly noteworthy, exemplifying a comprehensive approach to
data synthesis.

###  4.4 Denoising and Non-IID Adaptation

Recent trends in financial data prediction have highlighted the integration of
cutting-edge AI techniques.Among these, contrastive learning and Mixture of
Experts (MoE) stand out for their effectiveness.

Contrastive learning, initially finding significant success in computer vision
tasks, has been adapted to refine models’ predictive capabilities in financial
domains by contrasting positive (similar) and negative (dissimilar) sample
pairs. This approach is exemplified in techniques such as Copula-based
Contrastive Predictive Coding (Co-CPC) Wang et al. (2021a) and the Contrastive
Multi-Granularity Learning Framework (CMLF) Hou et al. (2021). Co-CPC excels
in filtering out noise and enhancing stock representations by contrasting data
from consecutive time points, while CMLF uses its dual contrastive approach to
tackle the intricacies of data granularity and temporal relationships, thereby
improving prediction accuracy.

Currently, MoE is a leading-edge architecture in Large Language Models (LLMs)
that significantly enhances predictive modeling. Researchers utilize it to
address complexities often overlooked by the conventional i.i.d. (independent
and identically distributed) assumption in financial data. MoE leverages a set
of specialized sub-models or ‘experts’, each adept at interpreting specific
data patterns or market conditions, countering the traditional one-size-fits-
all approach. A sophisticated gating network, termed ‘router’, orchestrates
the outputs of these experts, ensuring that the most relevant insights are
prioritized based on the current data, thus dynamically adapting to market
heterogeneity. Advanced implementations of MoE, such as the Temporal Routing
Adaptor (TRA) Lin et al. (2021) and the Pattern Adaptive Specialist Network
(PASN) Huang et al. (2022), underscore its effectiveness. TRA optimizes the
assignment of data to predictors, facilitating accurate recognition of
temporal patterns, while PASN’s Pattern Adaptive Training fosters autonomous
adaptation to new and evolving market dynamics. Collectively, these
enhancements fortify the robustness and precision of financial market
predictions, surpassing the limitations of traditional i.i.d.-based predictive
models.

##  5 Challenges & Future Direction

Recent developments have made strides in addressing complex issues within
asset pricing, yet a substantial array of challenges demands further
attention. These include overarching challenges like overfitting and model
interpretability, along with domain-specific issues such as managing noisy
data and complying with regulatory standards. In this section, we critically
examine these challenges, offering insights into potential research
initiatives for effective resolution.

  * •

Data Availability and Quality: Financial data are often proprietary, with
limited access for academic researchers. Some data are provided by third-party
vendors (e.g., WRDS, IBES) at a significant cost and with notable time delays.
Consequently, existing research predominantly focuses on indexes and the
equity market based on historical data, neglecting other markets like bonds
and derivatives. Additionally, unlike other domains, there is a lack of
unified test data for evaluating asset pricing models. Establishing a
comprehensive dataset with unified assets or synthetic data that reflects the
complexity of the financial market is essential for robust and equitable model
evaluation.

  * •

Market Dynamics and Structural Changes: Financial data demonstrates time-
varying dynamics and structural changes due to economic uncertainty,
geopolitical events, and investor behavioral assumptions. Additionally, the
no-arbitrage theory suggests that if a model can identify any mispricing, it
will evaporate quickly, rendering the model absolute. As a result, it is
challenging for deep learning models to adapt to these changes. Although
advancements in NLP facilitate the incorporation of news and sentiment into
asset pricing models, the real-time integration of this information into the
model is still in its infancy. Modeling structural changes, with online
learning and meta-learning can be a step forward in modeling this ever-
evolving financial landscape.

  * •

Model Complexity and Overfitting Risks: Another issue with the time-varying
dynamics of financial data is that complex models, e.g., Transformers and
GNNs, often perform well on training data but poorly on unseen data. One
interesting future research direction involves using algorithms such as Meta-
learning, one-shot learning, and ensemble learning, along with early stopping
and dropout techniques, to develop more generalized asset pricing models.

  * •

Regulatory Compliance: The financial industry is heavily regulated, with
strict requirements demanding transparency, accountability, and adherence to
established guidelines. Ensuring that AI algorithms comply with regulations
such as GDPR, MiFID II, and Basel III entails documentation, robust
validation, and ongoing monitoring. Collaborative efforts between financial
institutions, AI developers, and regulatory authorities can facilitate the
implementation of ethical AI principles and foster transparency in the AI-
driven financial landscape.

  * •

Model Interpretability and Fairness: Deep learning models are infamous for
their ’black box’ nature and algorithmic bias. In contrast, both regulators
and investors are more concerned about the explainability and fairness of a
model than predictive accuracy. Understanding the causal inference and
economic significance of a model allows them to make more informed decisions
and design better policy alternatives. The rise of explainable AI represents a
significant leap in this direction. An exciting avenue for future research
could involve identifying the balance between interpretability and
performance, offering greater transparency to financial models.

Apart from the aforementioned challenges, the problem of missing values,
nonlinear data structure, and heterogeneous data sources persist in finance.
Despite some efforts to address these challenges, they present compelling
opportunities for future research.

##  6 Conclusion

In this paper, we aim to encapsulate the ever-growing literature on financial
AI. Our motivation is to offer researchers and practitioners an insight into
ML as a versatile tool for modeling the intricate workings of financial
markets. While much of the current research in empirical asset pricing using
ML focuses on enhancing prediction performance, this review demonstrates that
such applications represent only a fraction of ML’s potential in finance. In
less explored areas such as model explainability, optimization, data
augmentation, and compliance monitoring, AI can make valuable contributions in
the near future.

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