# Multiple backscattering in trivial and non-trivial topological photonic
crystal edge states with controlled disorder

S. Arora  T. Bauer  Kavli Institute of Nanoscience, Delft University of
Technology, 2600 GA, Delft, The Netherlands  R. Barczyk  E. Verhagen  Center
for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
L. Kuipers<EMAIL_ADDRESS>Kavli Institute of Nanoscience, Delft
University of Technology, 2600 GA, Delft, The Netherlands

###### Abstract

We present an experimental investigation of multiple scattering in photonic-
crystal-based topological edge states with and without engineered random
disorder. We map the spatial distribution of light as it propagates along a
so-called bearded interface between two valley photonic crystals which
supports both trivial and non-trivial edge states. As the light slows down
and/or the disorder increases, we observe the photonic manifestation of
Anderson localization, illustrated by the appearance of localized high-
intensity field distributions. We extract the backscattering mean free path
(BMFP) as a function of frequency, and thereby group velocity, for a range of
geometrically engineered random disorders of different types. We observe that
the BMFP is an order of magnitude higher for the non-trivial edge state than
for the trivial. However, the BMFP for the non-trivial mode decreases rapidly
with increasing disorder. As the light slows down the BMFP for the trivial
state decreases as expected, but the BMFP for the topological state shows a
non-trivial relation to group velocity. While the topological mode is not
immune to backscattering at disorder that breaks the protecting crystalline
symmetry, it displays a larger robustness than the trivial mode for a specific
range of parameters in the same structure. Intriguingly, the topological
nature of the edge states appears to break the conventional relationship in
which slower light always leads to more backscattering.

##  I Introduction

Robust scattering-free photonic transport is a major driving force in device
miniaturization, on-chip photonic networks, and photonic quantum technology
[1, 2, 3, 4]. In principle, using slow light will benefit developments through
the concomitant increase in light-matter interactions [5, 6, 7]. However, the
much-wanted increased light-matter interaction itself has unwanted yet
fundamental side effects. No real-life photonic structure is perfect and as
the light slows down, imperfections increasingly scatter the propagating light
[8, 9]. Moreover, the fraction of light being backscattered will grow with
increased slowdown, inevitably resulting in Anderson localization for one-
dimensional systems like waveguides [10, 11]. The advent of crystalline
topological insulators in electronic as well as photonic and acoustic systems
spurred investigation into robust transmission channels that are inherently
protected from scattering [12, 13]. These systems offer the possibility of
topological transport without a magnetic field that breaks time-reversal
symmetry. Instead, the protection is directly linked to crystalline symmetry.
The realization of photonic analogs consisting of topologically non-trivial
photonic crystals (PhCs) [14, 15, 16, 17, 18, 19, 13, 20, 21, 22] with reduced
backscattering [23, 24, 25, 26, 27] offer passive implementation and ability
for on-chip, nanoscale integration. In the specific case of quantum valley
Hall effect emulating systems [28, 29, 30, 31, 32] that operate below the
light line, robustness against backscattering relies fully on geometric
symmetries.This immediately raises the question to what extent the topological
protection holds in the presence of random disorder, which by its very nature
destroys the underlying and required symmetry [33, 34].

To address this intriguing question, one must simultaneously consider photonic
slowdown and topological robustness. As the propagating light slows down,
light becomes more susceptible to backscattering due to a combination of an
increase in light-matter interaction and local density of states (proportional
to the slow down) that increases the overall scattering at imperfections, and
increases the fraction of scattered light into the backward direction also
proportional to the density of states and slowdown [9, 35, 36]. Recently, a
design called bearded interface has been introduced that offers both a
topologically trivial and non-trivial mode within the bandgap of the same
silicon-on-insulator device, i.e., with the same unavoidable fabrication-
induced disorder [37]. Moreover, this interface offers the prospect of
achieving two topologically distinct modes over an extended range of
achievable group velocities. Initial transmission measurements have found that
the transmission along straight and symmetry-preserving corners of the non-
trivial mode was indeed higher than for the trivial mode [37, 38, 39], with
losses increasing for slower light. A more recent study on scattering losses
due to intrinsic fabrication disorder found no appreciable difference between
the robustness of topologically trivial and non-trivial edge states [40] with
all losses being enhanced by the slowdown of the propagating light. By
observing indications of the onset of Anderson localization, the study
suggested the presence of multiple (back)scattering. In all cases, the
determination of the group velocity came from simulations. It is important to
note that, interestingly, due to small differences in the exact geometrical
parameters (relative hole sizes and corner rounding) of the realized photonic
crystals, the frequency for which the group velocity is lowest is not
equivalent in both investigations: in one it is located inside the non-trivial
frequency range [38] and in the other, inside the trivial frequency range
[40]. This highlights, that for probing the limits of topological protection,
investigations that use engineered disorder, combined with a local
determination of the group velocity would be highly valuable.

In this paper, we experimentally quantify the robustness of edge states
propagating along a bearded interface using a universal metric known as
backscattering mean free path (BMFP) [41, 42, 43]. We directly map the spatial
distribution of the complex electric field of the edge states with phase-
resolved near-field scanning optical microscopy [44, 45]. We first investigate
the bearded interface without induced disorder and systematically compare the
BMFP of the trivial and non-trivial modes for the same range of group
velocities. Subsequently, we study the effect of intentionally induced
positional or hole size disorders of various magnitudes. While both trivial
and non-trivial waveguides show scattering losses, we observe a different
qualitative and quantitative response for the topologically trivial and non-
trivial edge states, which depends on the type and magnitude of the disorder.
For the largest magnitudes of the disorder, we observe the photonic
manifestation of Anderson localization, illustrated by the appearance of
localized high-intensity states. Interestingly, we find that the onset of
Anderson localization is different for the distinct edge states. We also
observe that the conventional direct relation between the slowdown factor and
backscattering seemingly breaks down for the topologically non-trivial mode.

##  II Experimental Results

###  II.1 Mapping of the near-field edge state

We experimentally investigate edge states at the interface of topological PhCs
mimicking the quantum valley Hall effect. A non-vanishing Berry curvature at
the valleys is associated with an intrinsic magnetic moment, resulting in a
valley Chern number $C_{K,K^{\prime}}=\pm\frac{1}{2}$ [46, 47, 48, 49, 28, 50,
51, 52, 53]. This gives rise to photonic bulk bands of a topologically non-
trivial nature. For a unit cell with underlying $C_{3}$ symmetry, there are
several ways to realize valley-protected edge state interfaces, akin to edge
terminations in a honeycomb lattice structure such as graphene. Edge state
modes appear at the K and K′ points of the Brillouin zone (BZ) as a
consequence of broken lattice symmetry [54, 50, 51, 55, 56] by mirror
inversion of the PhC lattices at the interface where $y=0$. The two valley-
protected degenerate eigenmodes traverse the photonic band gap with a linear
dispersion. This is referred to as a zigzag interface [46, 47]. The bearded
interface combines parity inversion and glide symmetry [37] and the resulting
interface enforces a degeneracy at the BZ edge [57] resulting in two
eigenmodes in each valley with a slow-light region around the BZ edge. The
high-energy eigenmode is attributed to have a trivial nature as it can be
observed to ‘drop down’ from the conduction band when holes at the interface
are adiabatically reduced in size [37]. At the same time, the low-energy
eigenmode is attributed to be non-trivial. It is observed to exhibit higher
transmission through sharp bends in comparison to the high-energy eigenmode,
as is expected for a non-trivial edge state [38]. From this point on, the low-
energy mode is thus referred to as the non-trivial mode, whereas the high-
energy mode is called the trivial one. The continuous deformation of the
interface holes and the resulting glide plane symmetry gives rise to the two
eigenmodes bending towards the BZ edge. This results in a slow-light region
for both the trivial and non-trivial modes enabling access to different
backscattering regimes [58].

Figure 1: (a) Scanning electron micrograph (SEM) of the bearded interface in
the fabricated sample with the color-coded regions depicting the two mirror-
inverted lattices with glide symmetry (blue and green). The lattice
periodicity is $a=$510\text{\,}\mathrm{n}\mathrm{m}$$. (b) Schematic
representation of the experimental setup employed to study the near-field of
the bearded interface with amplitude and phase resolution. (c) The
experimentally measured dispersion diagram of the first Brillouin zone of the
edge state in a pristine PhC with two edge modes: at lower frequencies, non-
trivial and at higher frequencies, a trivial mode that is degenerate at
$1558\text{\,}\mathrm{n}\mathrm{m}$ or
$192.4\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. The blue and red dashed lines
indicate the numerically calculated dispersion curves of the non-trivial and
trivial modes, respectively. Figure 2: Real and reciprocal space of
electromagnetic field of an ensemble-averaged realization of a bearded
interface without the intentional disorder. (a) Near-field spatial-spectral
amplitude map normalized to the incoupling region collected as a function of
frequency and position along the interface. Light propagates the TPC from the
left. (b) Ensemble-averaged dispersion curve. The white dashed horizontal line
denotes where trivial and non-trivial modes become degenerate at the Brillouin
zone edge. (c) Calculated group index curve from the measured ensemble
averaged dispersion. The blue and red shaded regions depict the non-trivial
and trivial modes separated by the degeneracy point given by the white dashed
line at $0\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. The dark blue and dark red
regions indicate the slow light region for the non-trivial and trivial modes
with a group index of $n_{g}>15$, respectively.

We fabricate the bearded interface in a silicon-on-insulator platform (pseudo-
colored in SEM image Fig. 1(a)). The unit cell with lattice constant
$a=$510\text{\,}\mathrm{n}\mathrm{m}$$ consists of equilateral triangular
holes with side lengths $L_{1}=0.6a$ and $L_{2}=0.4a$. An interface is created
by transforming the triangular holes at the interface of one lattice from
$L_{1}$ to $L_{2}$. We measure the complex in-plane optical fields propagating
along the length of the PhC spanning $180$ unit cells as a function of laser
excitation wavelength
$\lambda=[$1480\text{\,}\mathrm{n}\mathrm{m}$-$1640\text{\,}\mathrm{n}\mathrm{m}$]$
using phase-sensitive near-field scanning optical microscopy [44]. The
employed heterodyne detection scheme allows us to decompose each spatial field
map into its Fourier components $\mathcal{F}(k_{x},k_{y})$. Thus, we obtain
the mode dispersion. Figure 1(b) shows the dispersion of a bearded interface
without any engineered disorder as an example. The region above
$194.5\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ represents the start of top
bulk bands. Rather than a linear dispersion as supported by the zigzag
interface [31], we observe both predicted modes each with a steep linear slope
around $k_{x}=2\pi/3a$, and also a slow light region close to the Brillouin
zone edge ($k_{x}=\pi/a$). The red (trivial) and blue (non-trivial) dashed
outline indicates an excellent correspondence with numerically simulated
dispersion curves for the two modes.

###  II.2 Visualizing real and reciprocal space of the edge state

We measure the electric field of light that enters the crystal through a feed
waveguide and propagates along the interface until it leaves at the end-facet,
which serves as a termination of the PhC. An exemplary measured electric field
amplitude of a bearded interface without engineered disorder is shown in
Supplementary Fig. S1. The end-facet in addition to fabrication imperfections
such as surface and side wall roughness may cause backscattering and create
intrinsic loss channels [31, 59]. The procedure that we follow for all
spatial-spectral maps in this work includes obtaining an ensemble average of
five realizations of bearded interface PhCs with the same magnitude of
disorder, summed over modes with the same group velocity [60]. Due to slight
variations in nanofabrication, the dispersion relation of individual PhCs may
exhibit slight shifts in the degeneracy point $\omega_{D}$ (DP) of the trivial
and non-trivial modes. We therefore shift all measurements to ensure that the
DPs coincide before averaging. As a consequence, the vertical axis of Fig. 2
indicates modified units of frequency in terahertz denoting the frequency
difference with respect to that of the DP ($\omega-\omega_{D}$). The spatial-
spectral amplitude map displayed in Fig. 2(a) shows the ensemble-averaged
electric field amplitude as a function of $\omega-\omega_{D}$ and interface
length (in units of lattice constant $a$). To account for the excitation-
dependent incoupling efficiencies, each horizontal line in the spatial-
spectral map is normalized to the mean amplitude of the first four unit cells
after the feed waveguide. The corresponding ensemble-averaged dispersion is
shown in Fig. 2(b). The white dashed line denotes the DP of the trivial and
non-trivial mode in the ensemble dispersion at the boundary of the Brillouin
zone edge. By fitting a hexic polynomial function (polynomial of degree six)
to the measured ensemble-averaged dispersion curve we determine the group
velocity as a function of frequency. The extracted group index for the
ensemble $n_{g}=c/v_{g}$, where $c$ is the speed of light in vacuum and
$v_{g}$ is the group velocity of the measured edge state, is shown in Fig.
2(c). More information on this procedure is provided in Sec. III.

For qualitative scrutiny of the ensemble-averaged spatial-spectral map (Fig.
2(a)), we separate the edge mode of the bearded interface into two regimes:
fast dispersive ($n_{g}\leq 15$) and slow diffusive ($n_{g}>15$). Below
$-0.66\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ (blue dashed line), the non-
trivial mode has a group index of $n_{g}\leq 15$. Here, the non-trivial mode
retains a nearly uniform amplitude distribution over the entire interface
length consistent with unity transmission [31], and the corresponding
dispersion shows a positive steep slope within $k_{x}<\pi/a$. Amplitude
oscillations (with a spatial frequency of roughly $2\pi/3a$ and lower) in the
spatial-spectral maps arise due to the interference of the backward and
forward propagating Bloch modes. As the non-trivial mode enters the diffusive
regime ($n_{g}>15$) for larger frequencies, the mode nominally slows down
further and the amplitude starts to rapidly decay with distance. In reciprocal
space (Fig. 2(b)) more spatial frequencies appear around the DP. We observe
that the non-trivial mode reaches a maximum group index of $n_{g}=50$ at
$\omega-\omega_{D}=-0.17$ (shown in Fig. 2(c)). Importantly, this maximum
group index is found to occur not at the DP but at a lower frequency, i.e., in
the frequency range of the non-trivial edge state (see Supplementary Sec. III.
A and ref. [40]). As we cross over to the trivial region, we observe a gradual
decrease of $n_{g}$ as the frequency is increased away from the BZ edge with a
group index $n_{g}=30$ at the BZ edge. The influence of unavoidable
fabrication-induced disorder on the trivial mode is evident. A clear
indication of light undergoing backscattering along the waveguide is visible
in the frequency region around the DP. The local high-intensity features are
characteristic of Anderson localized modes [61, 62]. Above the red dashed
line, where the trivial mode has $n_{g}\leq 15$, the dispersion is roughly
linear. However, multiple interference patterns along the interface are
observed in the spatial-spectral maps with a distinct decay, in sharp contrast
to the homogeneous distribution of the non-trivial mode with the same $n_{g}$
range. Thus, from qualitative scrutiny, it is apparent that the trivial edge
state suffers more backscattering than the non-trivial edge state.

Figure 3: Spatial-spectral maps of the measured electrical field amplitude
(normalized to the input waveguide) for various degrees of intentionally
induced disorder averaged over five realizations of the disorder. The two
columns represent position- and size-induced disorder, respectively, as a
function of interface length in units of lattice constant
$a=$510\text{\,}\mathrm{n}\mathrm{m}$$. The white dashed line denotes DP of
trivial and non-trivial mode. The red and blue dashed lines (with a white
background for better visibility) denote the frequency away from DP where the
trivial and non-trivial mode has a group index of $n_{g}=15$, respectively.

###  II.3 Introducing random intentional disorder

To experimentally unravel the degree of protection that a trivial and non-
trivial mode may offer against multiple scattering, we intentionally introduce
disorder in the bearded interface PhC design. We engineer two types of
intentional disorder: (a) random displacement of the holes and (b) hole size
variations. We introduced several magnitudes of disorder for each type of
disorder and fabricated five different PhC interface realizations per
magnitude. The triangular holes are randomly displaced and deformed in
accordance with a normal distribution with standard deviation $\sigma$ as a
measure of position and size disorder. The value of $\sigma$ is scaled with
the lattice constant $a$ and the hole diameters $L_{1}$ and $L_{2}$ for
position disorder and size disorder, respectively. Figure 3 shows the
ensemble-averaged spatial-spectral amplitude maps for position and size
disorders with increasing disorder magnitude
$\sigma=[0.0025,0.005,0.01,0.02]$. To account for variations in in-coupling
efficiency for different excitation frequencies, each horizontal line on the
spatial-spectral maps is also normalized to the mean intensity at the feed
waveguide. Qualitatively, the following observations can be made for the non-
trivial (bottom) and trivial (top) modes as we increase the magnitude of the
engineered disorder. Every ensemble-averaged spatial-spectral map in Fig. 3
reveals that the light in the fast non-trivial mode region (the limit denoted
by the blue dashed line where $n_{g}=15$) exhibits an almost homogeneous
spatial profile. Interference of the forward and backward modes due to back
reflection at the end is evidenced by fast amplitude oscillations. As the
frequency is increased into the slow non-trivial region close to the DP, the
intensity along the interface rapidly decays to a negligible value. The
propagation losses become prevalent and the penetration of the light along the
interface decreases. Similar behavior is evident in the slow trivial region
below the red dashed line where $n_{g}=15$. However, the fast trivial region
above the red dashed line looks different than that of the non-trivial mode,
showing multiple interference patterns along the interface. In the vicinity of
the DP, we observe high-intensity resonances that feature spectral and spatial
dependence. These sharp features of intensities are fingerprints of localized
states and they become more pronounced as the magnitude of the disorder
increases. In general, we observe a rather sharp transition in propagation
behavior for the non-trivial region as the mode slows down as the frequency
increases, from an edge state that propagates along the entire interface to a
state that is hardly able to penetrate the structure. In contrast, the
transition in propagation behavior from fast to slow light in the trivial mode
is more gradual with frequency. We scale the magnitude of both types of
engineered disorder to the lattice constant in a linear fashion. If we use a
similar scaling as was used in previous works for both two-dimensional and
three-dimensional photonic crystals [63, 64, 33] to compare the effects of
different types of disorder, it suggests that the influence of position
disorder is more prominent on edge state propagation than size disorder, as
evidenced by the slower onset of speckle-like interference patterns in trivial
and non-trivial regions.

##  III Quantifying backscattering

To quantify the robustness to backscattering of the non-trivial mode in
comparison to the trivial mode in the bearded interface, we use the
backscattering mean free path (BMFP or $\xi$) as a figure of merit [65, 66].
It is defined as the absolute displacement along the bearded interface after
which light suffers (multiple) scattering events in the absence of other loss
channels such as absorption [67] or out-of-plane losses.

Figure 4: (a) Normalized ensemble-averaged electromagnetic field intensity
map, averaged over 5 realizations of bearded interface without intentionally
induced disorder of the non-trivial mode with integrated over fast light
region. Inset: Ensemble-averaged field intensity profile fitted with a linear
slope to obtain the backscattering mean free path.

We extract the backscattering mean free path along the interface for an
exemplary waveguide as shown in the inset of Fig. 4 using the following
relation [68]:

$-\frac{x}{\xi(\omega)}=\left\langle\ln\left[\frac{I(\omega)}{I_{0}(\omega)}\right]\right\rangle$
(1)

where $x$ signifies the distance from the feed waveguide, $\frac{I}{I_{0}}$ is
the normalized electric field intensity, and the brackets indicate the
statistical ensemble average over different PhCs with the same type and
magnitude of disorder. The frequency ($\omega$) dependence of $\xi$ is stated
to differentiate between the contributions of disorder on the non-trivial and
trivial modes.

Figure 5: Measured backscattering mean free path as a function of
intentionally induced disorder. The triangles indicate the measured BMFP from
each ensemble averaged disorder realization for the fast non-trivial (red) and
fast trivial (blue) modes. The left and right side of the graph represents the
fabricated size and position-induced disorder respectively.  Figure 6:
Qualitative analysis of the slow light region. a) Zoom-in of Fig. 2(a)
denoting the spatial-spectral map confined to the slow light region of the two
modes. b) Group index curve for the frequencies in the range $[-1,1]$, with
the red and blue dashed line denoting the frequencies where $n_{g}=15$ for the
trivial and non-trivial region, respectively. c) Backscattering mean free path
as a function of frequency away from DP. The red and blue points denote the
measured BMFP, whereas the black curve denotes the non-trivial and trivial
BMFP for a sliding window of width
$0.17\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. It is clear that the minimum
BMFP neither occurs at the DP nor the region of the maximum group index. d)
Backscattering mean free path as a function of $n_{g}$. The blue and red
circles denote the non-trivial and trivial mode for a sliding window of width
$0.17\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, respectively. The increasing
darkness of the filled circles indicates the relative frequency away from the
DP where the darkest red and darkest blue denote the slow light region,
whereas light red and light blue denote fast light of trivial and non-trivial
mode, respectively.

The experimentally determined backscattering mean free path $\xi$ as a
function of induced disorder is plotted in Fig. 5. The red and blue triangles
indicate the BMFP of fast dispersive regimes ($n_{g}\leq 15$) for trivial and
non-trivial modes in the ensemble with an equal group index range,
respectively (see Supplementary Table I for BMFP values and mean group indices
of the trivial and non-trivial regions). For the PhC ensemble with no
intentionally engineered disorder ($\sigma_{0}$), the non-trivial mode is more
robust to backscattering by an order of magnitude compared to the trivial
mode. As the magnitude of the disorder increases, the BMFP for the non-trivial
mode drops by almost an order of magnitude for the largest disorder. The
decrease in BMFP is strongest for increases in positional disorder.
Surprisingly, the trivial mode, doesn’t suffer a similar reduction and
exhibits a BMFP that remains roughly constant with increasing disorder [33].
Importantly, we observe backscattering for all degrees of engineered disorder,
and thus symmetry-induced protection partly breaks down due to the induced
disorder. However, the non-trivial mode suffers less backscattering than the
trivial mode of the waveguide without intentional disorder. This suggests that
a certain degree of robustness of the non-trivial mode remains in effect for
the studied types and magnitudes of disorder. Nevertheless, the rapid decrease
in BMFP does suggest a rapid deterioration of the symmetry-induced protection
[34]. At the maximum value of induced disorder, for size and position ($2\%$),
the magnitude of the BMFP for the non-trivial and trivial mode is almost the
same, suggesting that for the largest disorder topological protection is
destroyed to a certain degree. It is important to note that for the fast non-
trivial and trivial mode without the engineered disorder, the deviation in the
mean group indices of the two modes is within $1.05\%$ (where $n_{g}\leq 15$).
See Table S1 for BMFP values and mean group indices of the trivial and non-
trivial regions of the ensemble. An upper limit of the deviation in the mean
group indices of the trivial and non-trivial regions is within $13.39\%$ and
it occurs for size disorder $0.5\%$ and does not directly dictate the trend in
BMFP. The difference in BMFP observed in Fig. 5 of the two fast modes can
therefore not be explained by differences in group velocity. The anomalous
value of a low BMFP for both non-trivial and trivial mode at size disorder
$\sigma_{0.25\%}$ is attributed to limited statistics. Our observation is in
agreement with the work in [33], where the initial introduction of disorder
results in a quick reduction of BMFP, and thereafter the decay is slower with
an increasing amount of disorder.

###  III.1 Group index contribution to backscattering

A complete understanding of the edge mode’s robustness also requires a
detailed analysis of the extracted BMFP with respect to light slowing down as
defined by the group index $n_{g}$. In Fig. 6, we show how the two modes’ BMFP
depend on $n_{g}$. For conventional waveguides, $\xi$ is directly linked to
the group index as $\xi\propto n_{g}^{-2}$ for ordinary light propagation [9,
69]. While the concept of a waveguide mode’s group index falters in the slow-
light regime due to multiple scattering and the diffusive nature of light [58,
66], the nominal $n_{g}$ is typically still useful to describe the average
behavior of the light propagation [6, 70]. From Fig. 6(a) we qualitatively
investigate the slow diffusive behavior of the two distinct modes for the case
without intentional disorder using the ensemble-averaged spatial-spectral mode
profile. Around the DP (white dashed line in Fig. 6(a) and black dashed line
in Fig. 6(b)), the propagation losses are significant as evidenced by the
small penetration along the interface. The non-trivial mode above the blue
dashed line where $n_{g}>15$ suffers a discrete drop in intensity as a
function of frequency, nonetheless maintaining a homogeneous distribution
along the interface (See Supplementary Fig. S6 for the central spatial-
spectral map). In contrast, a gradual decay is observed in the trivial mode as
we move from the fast to the slow light regime ($n_{g}>15$). Figure 6(c) and
Fig. 6(d) show the experimentally determined BMFP as a function of
$\omega-\omega_{D}$ and $n_{g}$ for the ensemble without engineered disorder.
It is important to note that the occasional occurrence of localization events
for a single frequency along the length of the waveguide can result in
meaningless values of the BMFP. To reduce the effect of such occurrences on
our investigation of the role of slowdown on the BMFP, we use a sliding
averaging window with a width of $0.17\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
to obtain a smooth curve denoted by the black circles in Fig. 6(c). For BMFP
as a function of $n_{g}$ in Fig. 6(d), the smoothed values in the non-trivial
region are denoted as blue circles whereas the smoothed values in the trivial
region are depicted in red. The increase in shade of the circles indicates the
frequency concerning $\omega_{D}$ and therefore also indicates a transition
from the fast to the slow light region, with the darkest shade of red and blue
denoting the value at the DP. For a narrow range of $10<n_{g}<15$, the fast
non-trivial mode has a higher BMFP than the fast trivial mode. It is
interesting to note that the lowest BMFP ($\xi=26.71a$) does not occur at the
frequency of largest group index in the non-trivial region below the DP, but
occurs when the edge mode lies in the trivial region with $n_{g}=22$. For the
non-trivial mode, a behavior is observed that is different from the behavior
of conventional slow light in photonic crystal waveguides. The slowest non-
trivial mode exhibits $n_{g}=54$. At this maximal slowdown, we observe modes
propagating through the entire structure with a relatively high BMFP
suggesting a degree of robustness against backscattering larger than that of
much faster light in the trivial mode. In the slow light non-trivial region,
the electric field intensity retains a high BMFP for a large range of $n_{g}$,
whereas the trivial mode’s BMFP is sensitive to variations in the modes’ group
velocity around the fast-slow transition (vertical grey dashed line in Fig.
6d). However, as the frequency of the non-trivial mode approaches the DP and
its group velocity increases, the BMFP actually decreases in complete contrast
to the dependence on backscattering in conventional waveguides. We suggest
that the effect of a smaller change in wavevector as the DP is approached
apparently has a stronger influence in increased backscattering than the
reduction in light-matter interactions resulting from the decreased group
index.

Acknowledging the possible misinterpretation when comparing Fig. 6(a) and Fig.
6(c), we address the apparent contradiction between the spatial-spectral map
and the measured BMFP. When quantifying the BMFP (shown in Fig. 6(c)) we find
that the minimum BMFP occurs for frequencies just below the red dashed line
around $0.39\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$. A cursory glance at the
corresponding spatial-spectral map (Fig. 6(a)) seems to show a large
penetration up to $\approx 55a$ for these frequencies. Conversely, frequencies
around $-0.08\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, directly below the DP,
seemingly exhibit the smallest penetration in the spatial-spectral map but
show a comparatively high BMFP of $\xi=70.14a$. However, the cursory, visual
interpretation of a small penetration in the non-trivial region actually
neglects a much longer intensity tail all the way to the end termination of
the PhC. In contrast, the trivial mode around
$\omega-\omega_{D}=$0.39\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$ experiences
a drastic reduction in intensity right after the first high-intensity region,
even leading to a negligible intensity of light reaching the end termination.
This is evident from the spatial-spectral map in logarithmic intensity scale
overlaid on the spatial-spectral amplitude map in Supplementary Fig. S6. These
findings emphasize the significance of employing appropriate normalization and
background correction methods when assessing the magnitude of and mechanisms
behind backscattering.

In conclusion, a direct experimental evaluation of backscattering of fast and
slow light propagating along a bearded interface between two topological
photonic crystals emulating the quantum valley Hall effect has been performed.
The spectral and spatial maps of the full interface enable us to completely
visualize and probe the appearance of localized events caused by fabrication
disorders. Induced disorder in terms of position and size variations affects
the electric field distribution in different ways. For the slowest light we
find an indication of Anderson localization at increasing magnitude of
disorder illustrated by the localized areas of high light intensity for both
the non-trivial and trivial edge states. However, the sensitivity to disorder
is found to be different for the two modes for the investigated range of
disorder, with the non-trivial mode exhibiting more robustness than the
trivial ones. This study sheds light on the potential relevance of topological
crystals for on-chip photonic devices for practical applications. We find that
all edge states undergo backscattering. Having said that, our observations do
indicate that for the same slowdown topologically non-trivial edge states can
exhibit less backscattering that their trivial counterparts. However, the
difference is less than an order of magnitude. Both types of edge states will
exhibit multiple backscattering as the light is slowed down significantly or
when the amount of disorder increases, ultimately leading to Anderson
localization.

##  IV Appendix: Methods

Measurement technique Numerical simulations were performed using MIT Photonic-
Bands. To match the calculated edge state to the measured dispersion relation,
the refractive index of silicon was taken as $n=3.48$. To account for the
corner roundness arising from fabrication, a fillet of
$60\text{\,}\mathrm{n}\mathrm{m}$ radius was added to the triangular holes of
lattice constant $517\text{\,}\mathrm{n}\mathrm{m}$. The unit cell consisted
of equilateral triangles, with a larger triangle side length $d_{1}=0.6a$ and
a smaller triangle side length $d_{2}=0.3a$.

Device fabrication. The PhC slab was fabricated on a silicon-on-insulator
(SOI) platform with a $220\text{\,}\mathrm{n}\mathrm{m}$ thick silicon layer
on a $3\text{\,}\mu\mathrm{m}$ buried oxide layer. A positive electron-beam
resist AR-P 6200.09 of thickness $240\text{\,}\mathrm{n}\mathrm{m}$ was spin-
coated. The PhC design was patterned using e-beam lithography on a Raith
Voyager with $100\text{\,}\mathrm{k}\mathrm{V}$ beam exposure. The e-beam
resist was developed in pentyl acetate/O-Xylene/isopropanol, and the chip
underwent reactive-ion etching in HBr and Ar. In the next step, a photo-
lithography resist S1813 was patterned using an EVG 620 NT mask aligner to
define a wet-etching window around the PhC, not covering the ridge waveguides
to prevent buckling. After development with MF321, the buried-oxide layer was
removed in 7:1 buffered hydrofluoric acid for
$18\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{s}$. The PhC was then
cleaned in piranha solution and carefully removed from the solution, before
being mounted in the near-field optical microscopy setup. The PhCs support an
incoupling single TE-mode waveguide into the crystal to allow better index-
matching for efficient coupling. [46]

Two different VPC domain walls were fabricated to facilitate transmission and
back-scattering comparisons. The straight edge VPC has dimensions $215a\times
71a$, where $a=$510\text{\,}\mathrm{n}\mathrm{m}$$ with the designed band gap
falling within the excitation wavelength range
$1480-$1640\text{\,}\mathrm{n}\mathrm{m}$$. A total of 45 PhCs were measured,
with 9 configurations of disorder.

Near-field optical microscopy setup. The employed aperture-based near-field
optical microscopy setup has been discussed in detail elsewhere [44]. In
short, an aperture-based probe picks up the decaying evanescent fields on top
of the VPC. It consists of a tapered optical fiber coated with
$140\text{\,}\mathrm{n}\mathrm{m}$ aluminum and an aperture of ca.
$170\text{\,}\mathrm{n}\mathrm{m}$ is created via focused ion beam milling.
Scanning the probe over the silicon membrane at a relative height of ca.
$20\text{\,}\mathrm{n}\mathrm{m}$ is controlled via shear force. The amplitude
and phase information are determined using a heterodyne detection scheme, with
the coherent reference light beam shifted by $\Delta
f=$40\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ in frequency.

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# UNO-DST: Leveraging Unlabelled Data in Zero-Shot Dialogue State Tracking

Chuang Li12, Yan Zhang2, Min-Yen Kan2, Haizhou Li23
1NUS Graduate School for Integrative Sciences and Engineering
2National University of Singapore
3Chinese University of Hong Kong, Shenzhen
<EMAIL_ADDRESS>
{eleyanz, kanmy<EMAIL_ADDRESS>
###### Abstract

Previous zero-shot dialogue state tracking (DST) methods only apply transfer
learning, but ignore unlabelled data in the target domain. We transform zero-
shot DST into few-shot DST by utilising such unlabelled data via joint and
self-training methods. Our method incorporates auxiliary tasks that generate
slot types as inverse prompts for main tasks, creating slot values during
joint training. Cycle consistency between these two tasks enables the
generation and selection of quality samples in unknown target domains for
subsequent fine-tuning. This approach also facilitates automatic label
creation, thereby optimizing the training and fine-tuning of DST models. We
demonstrate this method’s effectiveness on large language models in zero-shot
scenarios, improving average joint goal accuracy by $8\%$ across all domains
in MultiWOZ111Code and data are available at
https://github.com/lichuangnus/UNO-DST.

##  1 Introduction

Dialogue state tracking (DST) is a crucial task in understanding users’
intentions by extracting the dialogue states from the dialogue history
Balaraman et al. (2021), where a single dialogue state is a combination of a
slot type (e.g.,<hotel-name>) and a slot value (e.g.,<Hilton hotel>), as in
Figure 1. Dialogue states are a set of those combinations (e.g.,<hotel-name:
Hilton hotel>) retrieved by DST models, given dialogue history and slot types.
Traditional methods train and evaluate DST models with manually-labelled
dialogue states in each domain, which can be costly and time consuming Wu et
al. (2020b); Hosseini-Asl et al. (2020). Recently, DST under zero and few-shot
settings draw increased attention Lin et al. (2021b); Hudeček et al. (2021).
Compared with few-shot methods, zero-shot approaches are more challenging, due
to unseen slot types and data scarcity in unknown target domains.

Figure 1: Examples of zero-shot methods in DST.

In both zero and few-shot settings, the majority of existing methods convert
the DST problem into other common problem settings in natural language
processing (NLP): for example, Question Answering (QA; Lin et al., 2021a; Li
et al., 2021), prompt learning Lee et al. (2021), summarization Shin et al.
(2022) and instruction learning Gupta et al. (2022). For example in Figure 1,
a given slot type (<hotel-name>) can be transformed into a QA setting by
queries like “What is the hotel name in the context?” and the slot values can
be predicted by a QA model accordingly.

Transfer learning methods also convert DST tasks to generation ones, more
suited for pre-trained language models (PLMs; Devlin et al., 2019). However,
such methods cannot fully leverage the capability of PLMs in generation and
selection. Two main difficulties emerge: 1) the performance of the chosen NLP
tasks can be unpredictable for unseen slot types in a new domain due to domain
divergence; and 2) existing models are only trained in the known domains,
without utilizing any unlabeled data in the new target domain.

This work proposes UNO-DST222“Uno”, Spanish for “one”, embodies our proposed
strategy in this paper: transitioning from zero to one and subsequently from
one to all., a method to leverage the unlabelled data for zero-shot DST in
target domain. Inspired by the popularity of multi-task learning and self-
supervised learning Zhang and Yang (2021); Tsai et al. (2021), UNO-DST employs
a two-step training framework invoking both joint and self-training (Figure
2). Aside from the main task of generating slot values, we design an auxiliary
task of generating slot types. We then jointly train both tasks using the
labelled training data in known source domains. For the self-training period,
we implement the concept of cycle consistency within our two tasks Zhu et al.
(2017). That is, a text output from the main task serves as input to the
auxiliary task, and the resultant text produced by the auxiliary task should
match the original input text. This process forms a full cycle, ensuring
consistent generation and selection of dialogue states from the unlabelled
data, which will be further used for fine-tuning the model. In this way, we
convert zero-shot problems into few-shot ones. Importantly, our framework is
model-agnostic and can be applied to different baseline models.

Our main contributions are as follows:

  * •

To the best of our knowledge, we are the first zero-shot DST work to use
unlabelled training data in an unknown target domain under a two-step training
strategy;

  * •

We introduce an auxiliary task to facilitate the training of the main task,
the selection of fine-tuning samples, and the generation of unseen or new slot
types;

  * •

Our results achieve a new state-of-the-art. Our analysis also identifies the
lower and upper bounds of each zero-shot DST method.

  * •

We demonstrate our methods with PLMs and large language models, showing its
effectiveness on two popular DST datasets.

##  2 Related Works

Figure 2: Overview of UNO-DST which consists of two periods: 1) joint training
for both task A (slot value prediction) and B (slot type prediction), 2) self-
training in the unknown target domain. Step 1: Generation of slot values and
types from task A and B; Step 2: Selection of good samples using cycle
consistency between two tasks; Step 3: Fine-turning the PLM with selected
samples.

Existing DST methods are generally classified as either 1) full-data or 2)
low-resource DST. However, regardless of which method is adopted, unlabeled
training data in the target domain is unutilized.

Full-data DST are commonly trained with fully annotated multi-domain
conversations Wu et al. (2020a); Hosseini-Asl et al. (2020). SOTA models focus
on DST tasks with well-annotated datasets Mrkšić et al. (2017); Ren et al.
(2018). However, the annotation work for data in a new domain can be costly.
Hence there is interest in transferring the knowledge of a model from a known
domain into an unknown domain and conducting DST tasks in a low-resource
setting.

Low-resource DST uses zero- or few-shot learning in the unknown target domain.
Here, the state-of-the-art use a single NLP task to transfer knowledge from
the source domains to the unknown target domain Lin et al. (2021b); Shin et
al. (2022). While transfer learning tasks achieve good results, each method is
task-dependent. Thus, task-independent strategies have been proposed Wang et
al. (2022); Yang et al. (2023).

Multi-task Learning involving simultaneous training of a model on diverse
tasks, to boost performance on trained downstream tasks. It also holds promise
for enhancements on new tasks Raffel et al. (2020); Zhang and Yang (2021).
However, existing methods typically neglect to assess the consistency across
multiple tasks after joint training, while our approach leverages the cycle
consistency for selection Zhu et al. (2017); Wang et al. (2023).

##  3 Methodology

In Figure 2, we show an overview of UNO-DST with joint training and self-
training periods. Our method includes two tasks: a main task for slot value
prediction (§3.1) and an auxiliary task for slot type prediction (§3.2). In
the joint training period, both tasks are jointly trained in the known source
domains (§3.3). In the self-training period, we introduce three steps to
generate dialogue states, select good samples, and fine-tune the PLM (§3.4).
Lastly, we elaborate on the transferability of our strategy with a lower and
upper bound (§3.5).

###  3.1 Task Definition

The main task for DST is predicting the <slot-type:slot-value> pairs with
given dialogue history and slot types from a pre-defined slot type list, as
shown in Figure 1. For each domain, there are seen slot types which appear in
other domains (e.g.,“hotel-name” and “restaurant-name”) or unseen slot types
which are unique in the specific domain (e.g.,“hotel-stars”). The ratio of the
occurrences of these unseen slot types represents the difficulty of zero-shot
DST for each domain Wang et al. (2022).

We denote the dialogue history in a $t$-turn conversations as
$C_{t}=\\{c_{1},c_{2}...c_{t}\\}$ and slot types $S$ in domain $h$ as
$S_{h}=\\{s_{1},s_{2}...s_{n}\\}$. For each conversation turn, the main goal
is to predict slot values $v^{\prime}$. Therefore, the input for the $PLM$ is
combined of dialogue history and slot types, with the output being slot
values, as shown in Eq. 1.

$v^{\prime}_{i}=PLM(C_{t},s_{i})$ (1)

Compared with methods that select slot values from a constant ontology list
using classification models Shi et al. (2017), we enhance the capability of
text-to-text PLMs for text generation Heck et al. (2020). For the case when
there are no slot values related to a given slot type, we train the model to
output a “none” value, indicating that there are no dialogue states from the
current conversation turn.

To better utilise the capability of PLMs in different tasks, we utilize
different prompt functions “$P(.)$” to generate the prompt in correct format.
For example, given a slot $s$ and context $c$, the QA prompt $p$ for the DST
can be $p^{main}=$“What is the value of slot $s$ in context $c$?”. We
formulate the way of using prompts for the DST main task as:

$v^{\prime}_{i}=PLM(p_{i}^{main})=PLM(P(s_{i},C_{t}))$ (2)

###  3.2 Auxiliary task

As joint training can improve the accuracy of PLMs, we design an auxiliary
task to facilitate the training of the main task Zhang and Yang (2021); Su et
al. (2022); Yang et al. (2023). We propose an auxiliary task to help the model
better understand the semantic and context information from the dialogue
history in the joint training period and serve as a regulator to check the
main task predictions obtained during the self-training period.

We design the auxiliary task as the inverse (converse) prompt of the main
task. In opposition to the main task, the auxiliary task thus takes the slot
values $v$ as input and generates the slot types $s^{\prime}$ as outputs,
which forms a cycle-consistent loop as a foil to the main task. To make it
easier for PLMs, we convert the slot values $v$ and dialogue history $C_{t}$
into a masked dialogue history $C_{t}^{m}$ for the model to make better masked
predictions, as in Eq. 3. The inverse QA prompt $p^{aux}$ is generated as
“What is the masked slot type in context $C_{t}^{m}$?” from inverse prompt
function “$IP(.)$” (Figure 2). We implement the auxiliary task during both the
joint training and self-training periods to facilitate slot values generation
and selection.

$s^{\prime}_{i}=PLM(p_{i}^{aux})=PLM(IP(v_{i},C_{t}^{m}))$ (3)

###  3.3 Joint training with auxiliary tasks

We conduct a simple version of joint training with only two tasks: the main
DST task and the auxiliary task. The training samples for the main tasks are
created using a prompt of dialogue history and slot type, while the samples
for the auxiliary DST tasks are created using an inverse prompt of masked
dialogue history.

As the auxiliary task is an inverse of the main task, the model is trained for
the same knowledge in a cycle-consistent way. By predicting the masked slot
type from the masked dialogue history, the model is familiar with the context
and different slot types. With our specially-designed auxiliary task, the
generation model reuses the existing data for another round of training
without the need to increase the amount of training data or model parameters.
We formulate the loss function for the main task $L_{m}$ and auxiliary task
$L_{a}$ as:

$L_{m}=-\sum^{n}_{i}{\log p(v^{\prime}_{i}|C_{t},s_{i})}$ (4)
$L_{a}=-\sum^{n}_{i}{\log p(s^{\prime}_{i}|C_{t}^{m},v_{i})}$ (5)

The final loss is a simple average of both. To keep the process simple, we do
not add hyperparameters to the existing model framework. Importantly, as the
auxiliary task samples are generated using the main task’s inverse prompt, the
task ratio mirrors the natural distribution of both tasks throughout the joint
training period.

###  3.4 Self-training with auxiliary tasks

Compared with other zero-shot DST models, the key novelty of our strategy is
in using the unlabelled training data in the unknown target domain for self-
training. Self-training aims to generate pseudo labels and select data samples
that further fine-tune the models. In the self-training period, we divide the
strategy into three steps: termed generation, selection and fine-tuning
(Algorithm 1).

Step 1 Generation tests both tasks using the unlabelled training data in the
unknown target domain to generate predicted slot values $v^{\prime}$ and slot
types $s^{\prime}$. Auxiliary tasks in self-training are created by value
masking, as shown in Figure 2. For training samples with slot values that do
not directly copy from the original context (such as “yes/no” for “hotel-
parking”), masking the slot value in the original context does not work. Such
samples are omitted in creating the masked dialogue history in auxiliary
tasks.

Step 2 Selection tests the cycle consistency between main tasks and auxiliary
tasks by comparing the predicted slot types $s^{\prime}$ with the original
slot types $s$ in each dialogue turn. A simplified selection process is shown
in Figure 2. In experiments, only conversation turns with all correct slot
types are selected as good samples, similar to joint goal accuracy settings,
aiming to reduce the selection error rate.

Step 3 fine-tunes the model $PLM(.)$ with selected samples and predicted slot
values $v^{\prime}$. This completes the conversion of zero-shot DST into few-
shot DST, helping the model adapt to unknown domains without increasing data
annotation and model parameters. For models that are difficult to fine-tune,
we propose other solutions (§ 7).

###  3.5 Lower and upper bound for zero-shot DST models

Even though there are many studies working on zero-shot DST, to the best of
our knowledge there are no common methods to identify the threshold below
where results are unreasonably low (lower bound) or the peak performance that
each model can potentially achieve (upper bound). Here, we discuss our
proposed algorithm with respect to the lower and upper bounds, aiming to limit
the research that is unsuitable for zero-shot DST by the lower bound and
benchmark our method against oracular results for each model as an upper
bound.

Lower bound. PLMs generate either a “slot value” representing specific
information, or “none” if no information is relevant to the dialogue history,
for all slot types in our main tasks. Notably, most dialogues do not associate
slot types with specific information, resulting in “none” being the prevalent
prediction. For example, slot type “hotel-area” is linked to “none” for
dialogue in Figure 2. Therefore, we set the lower bound as the outcome when
models predict “none” for all slot types across all dialogue turns.

Upper bound. According to our self-training methods (§3.4), zero-shot DST can
always be converted into a few-shot DST by selecting good samples with self-
generated slot values for fine-tuning, but cycle consistency cannot ensure
$100\%$ correct data selection. Oracular performance comes when we select only
the correct samples from all the self-generated slot values and use them to
fine-tune the model. We define such performance as the upper bound for the
zero-shot DST model.

##  4 Experiments and Datasets

Dataset. We train and test our model on both MultiWOZ 2.1 Budzianowski et al.
(2018) and the Schema-Guided Dialogue (SGD; Rastogi et al., 2020). MultiWOZ
and SGD have dialogues distributed in both training and testing distributions
over 7, 13 domains, representing 7K, 16K training examples in English,
respectively. We use standard means for data pre-processing Budzianowski et
al. (2018) for data pre-processing and follow the MultiWOZ leave-one-out
settings for zero-shot training and testing in both datasets Wu et al. (2019);
Rastogi et al. (2020).

| MultiWOZ | SGD
---|---|---
| JGA | JGA | AGA
Benchmarks | 25.8 | 27.6 | 58.0
T5DST | 32.4 | NA | NA
SD-T5 | 35.6 | NA | NA
TransferQA | 35.8 | 21.3 | 60.8
UNO-DST-JT |  36.6 (+0.8) |  36.9 (+15) |  75.9 (+15)
UNO-DST-ST |  40.8 (+5.0) |  47.4 (+26) |  81.8 (+21)
Table 1: Average zero-shot JGA and AGA results on MultiWOZ and SGD. JT/ST stands for joint/self-training and red figures calculate the performance increase of UNO-DST (60M) over TransferQA (770M). MultiWOZ | T5DST$*$ | SD-T5 | UNO-DST† | UNO-DST‡
---|---|---|---|---
Domains |  |  | JT | ST | JT | ST
Attraction | 30.45 | 33.92 | 33.50 | 36.05 | 32.86 | 33.09
Hotel | 19.38 | 19.85 | 21.04 | 23.00 | 22.91 | 25.66
Restaurant | 20.42 | 20.75 | 22.36 | 24.03 | 29.47 | 30.99
Taxi | 66.32 | 66.25 | 65.23 | 65.03 | 66.00 | 65.48
Train | 25.60 | 36.96 | 38.72 | 47.95 | 31.68 | 48.90
Average | 32.44 | 35.55 | 36.17 | 39.21 | 36.58 | 40.82
Table 2: Zero-shot JGA results for domains in MultiWOZ. Bold indicates the
best results, $*$ shows results of our replicated T5DST model, while † and ‡
give “t5-small” and “t5-QA” as our model checkpoints. JT/ST stands for
joint/self-training.

Evaluation metrics. The primary metric for DST evaluation is Joint Goal
Accuracy (JGA), which compares the set of generated predicted values with the
set of ground truth after each turn of conversation and returns the fraction
of correct matches Henderson et al. (2014). Average Goal Accuracy (AGA)
calculates the JGA only for active slot types, which is used in the SGD
baseline Rastogi et al. (2020).

Baselines and experiment setup. We use T5 Raffel et al. (2020) as our baseline
model. For a fair analysis, we also compare our results with previous DST
benchmarks: TRADE Wu et al. (2019) and the SGD baseline Rastogi et al. (2020),
and current SOTA models: T5DST Lin et al. (2021b), TransferQA Lin et al.
(2021a) and SD-T5 Wang et al. (2022). We adopt the cross-domain settings for
both datasets, experimenting on two checkpoints,
“t5-small”333https://huggingface.co/t5-small and
“t5-QA”444https://github.com/facebookresearch/Zero-Shot-DST. We detail our
experimental setup and parameters in § A.1.

SGD | TransferQA | UNO-DST-JT | UNO-DST-ST
---|---|---|---
Domains | JGA | AGA | JGA | AGA | JGA | AGA
Flights | 3.6 | 42.9 | 26.4 | 75.1 | 25.3 | 72.7
RideSharing | 31.2 | 61.7 | 33.3 | 64.3 | 73.5 | 89.8
Homes | 31.7 | 80.6 | 16.8 | 77.6 | 17.9 | 76.3
Events | 15.6 | 56.8 | 11.5 | 58.0 | 23.1 | 71.6
Movies | 24.0 | 56.2 | 35.5 | 86.7 | 52.6 | 86.7
Services | 37.2 | 75.6 | 75.1 | 92.1 | 77.2 | 92.4
Travel | 14.0 | 24.2 | 55.2 | 76.7 | 56.4 | 77.8
Weather | 40.3 | 59.4 | 93.8 | 98.0 | 94.3 | 98.5
Hotels | 13.5 | 60.1 | 44.8 | 85.6 | 75.9 | 94.6
RentalCars | 10.8 | 73.8 | 7.5 | 72.9 | 5.4 | 79.4
Restaurants | 16.3 | 68.9 | 31.8 | 74.7 | 35.9 | 78.5
Media | 30.2 | 67.5 | 37.0 | 69.7 | 60.0 | 89.2
Music | 8.9 | 62.4 | 11.6 | 54.9 | 19.1 | 55.5
Average | 21.3 | 60.8 | 36.9 | 75.9 | 47.4 | 81.8
Table 3: Zero-shot JGA and AGA results for domains in SGD. Bold shows the best
results and JT/ST stands for joint/self-training.

##  5 Results

Table 1 analyzes the zero-shot results of our UNO-DST and the baseline models,
including the state-of-the-art (SOTA) TransferQA model, across both datasets.
Our model surpasses all baselines for both joint and self-training phases,
inclusive of TransferQA using “t5-large” (a magnitude larger in model size).
Detailed outcomes for each domain and specific training period (either joint
or self-training) across each dataset are presented in Tables 2 and 3.

Joint training results. For the joint training period in the MultiWOZ dataset
(Table 2), we are using the same model and prompt as T5DST. UNO-DST shows an
increase of more than $4\%$ for JGA across all checkpoints. For SGD (Table 3),
our joint training period increases the previous baseline by even larger
margins of $15.6\%$ in JGA ($15.1\%$ in AGA). The joint training period is
critical as it prepares a model for self-training. Table 2 shows that using
different model checkpoints is also critical even when using the same model
architecture and parameter size. The model performs best when we follow the
prompt format in each pre-training checkpoint.

Self-training results. For the MultiWOZ dataset (Table 2), self-training
further improves average JGA by $3.09\%$ after joint training. Compared with
the baseline, the best performance increases by $8.38\%$ in JGA. In SGD (Table
3), self-training improves the average JGA and AGA in 12 out of 13 domains by
an average of $10.5\%$ and $5.9\%$ compared with the joint-training alone, and
over $26\%$ and $21\%$ compared to the baseline.

The success of self-training proves the possibility of using pseudo labels
generated from zero-shot DST models to bootstrap performance. However,
carefully selecting good samples to fine-tune the model is challenging because
not all the domains benefit from the self-training process. For example, the
result for the “Taxi” domain in MultiWOZ and “Flights” domain in SGD decreased
after self-training. We examine the rationale behind the gains obtained
through self-training, which is associated with the upper bound results in
each domain and further discussed in § 6.

As shown in Table 4, as we lengthen self-training from a single round to
multiple rounds, our framework’s performance continues to improve. However,
the performance gap between results from different rounds shows diminishing
returns, signalling a plateau. The best result with UNO-DST comes when adding
more variation is insignificant and so we stop the training when the margin is
below $0.1$ in JGA. Future work is required to systematically study this
strategy over multi-round self-training.

Round | Att. | Hotel | Res. | Taxi | Train | Avg |  Gain $\uparrow$
---|---|---|---|---|---|---|---
0 | 32.86 | 22.91 | 29.47 | 66.00 | 31.68 | 36.58 |
1 | 33.09 | 25.66 | 30.99 | 65.48 | 48.90 | 40.82 | (+4.24)
2 | 35.53 | 27.22 | 31.44 | 64.71 | 54.60 | 42.70 | (+1.88)
3 | 36.62 | 27.09 | 31.14 | 65.48 | 53.31 | 42.73 | (+0.03)
Table 4: JGA for multiple rounds of self-training on MultiWOZ. Absolute gains
indicated in red. Figure 3: Gains by joint and self-training stages of UNO-DST
on the “t5-QA” checkpoint. We show the results of upper-bound (oracle) in each
domain for relative comparison.

##  6 Discussion

Lower and upper bound. We calculate the lower and upper bound for zero-shot
DST (cf § 3.5) for MultiWOZ. The lower bound is an average of $27.96\%$ for
JGA across all domains, a threshold below which the results are unreasonably
low Budzianowski et al. (2018). Of all the baselines in this experiment, only
results from TRADE are below the lower bound (Table 1) and based on that, we
claim that TRADE Wu et al. (2019) model is unsuitable for zero-shot DST.

For the upper bound, oracular calculation, we select only the $100\%$ correct
samples from the zero-shot predictions and use them for fine-tuning. In Figure
3, we visualise the gains of joint training and self-training alongside our
upper bound. While efficacy differs from domain to domain, an important
observation is that when the margin between the upper bound (blue columns) and
joint training (red columns) is large, the model has a larger gain from self-
training, as in “Train” domain. In contrast, for “Taxi” domain, the influence
of self-training is weak (cf § 5). Utilizing upper bounds calculations enables
us to swiftly evaluate whether a domain or model is apt for the self-training
period. Said another way, a larger margin between joint training and the upper
bound yields a larger potential improvement that the model can achieve with
fine-tuning.

All Generated Slot Types in Train Domain
---
$people^{5},day^{5},destination^{5},departure^{5},leave^{5},$
$arrive^{5},price^{5},type^{5},time^{5},area^{5},name^{5}$
Valid New Slot Types
${price}^{15},day^{1},parking^{3},name^{5}$
Dialogue Example [PMUL1359] $(price^{5},name^{5})$
System:“Okay, tr6572 departs at 05:29.”
User:“What is the price?”
Dialogue Example [PMUL3027] $(parking^{3})$
System:“I have 2 Turkish restaurants in the centre?”
User:“Do they offer free parking?”
Dialogue Example [PMUL1118] $(day^{1})$
User:“I am in Cambridge for the week and want to
know what museums you guys have there.”
Table 5: Newly-generated slot types with examples. The superscript on each
slot type indicates the domain information from: (1:attraction, 2:hotel,
3:restaurant, 4:taxi, 5:train)

(a) Correction of the wrong prediction.

(b) Confirmation of the correct prediction.

Figure 4: Case studies for conversations with ChatGPT for zero-shot DST.

New slot type generation. All the existing zero-shot DST methods require given
slot types in generating the slot values for both source and target domains
and our model also follows the same experiment settings (cf § 3.1). However,
we show that our model can also self-generate reasonable slot types either in
or beyond the 30 given slot types with our designed auxiliary task. In order
to self-generate new slot types, we do a case study on the MultiWoZ “Train”
domain and perform random word masking for all the dialogue history, inputting
those randomly masked dialogue histories to the auxiliary task for slot type
predictions, as shown in Figure 5(b). In Table 5, we show some valid new slot
types generated by our auxiliary tasks with dialogue history. For example,
“asking for the ticket price” in “Train” domain and “asking for parking
information” in “Restaurant” domain are reasonable new slot types, which can
also be included in the given slot types list.

Besides generating new slot types, we propose a future zero-shot DST without
any given slot types, where the self-training is conducted based on self-
generated slot types in the target domain. Due to the limited scope of this
paper, we discuss the details in Appendix § A.3. Even though our current
auxiliary task can predict all $6$ given slot types in “Train” domain,
additional steps for selection and merging are required to maintain the
quality and efficiency of self-generated slot types Hudeček et al. (2021).

##  7 UNO-DST with ChatGPT

While earlier sections analyse the effectiveness of our methods on PLMs like
“T5”, this section focuses on the potential application of UNO-DST with
current generation large language models (LLMs), such as Brown et al., 2020
and Touvron et al., 2023. Specifically, we examine UNO-DST for zero-shot DST
using OpenAI’s ChatGPT555https://chatgpt.openai.com as the backbone LLM, an
LLM which has been adopted as a language tool for information extraction with
strong capabilities, even without specific training or fine-tuning.

ChatGPT can be interacted with either an open-accessed web interface or
through its API. Our study will test the efficacy of our self-training
strategy in UNO-DST, on both these versions of ChatGPT, including the web
interface and the API666 ChatGPT API model: gpt-3.5-turbo-0301. Our objective
is to explore the potential application of our methods to LLMs featuring
conversational approaches (§7.1) and in-context learning (ICL; Hu et al., 2022
§7.2).

###  7.1 Conversational approaches

Implementation. We skip the joint training for ChatGPT and use conversations
as an inference approach. As shown in Figure 4, we implement main and
auxiliary tasks with conversations asking for slot values or types. The
selection and fine-tuning steps in the self-training strategy have been
converted into the correction or confirmation step, where we provide the slot
value and type predictions from the previous two questions to ChatGPT and ask
whether it needs to revise the slot value to the main task. We consider the
revised response from ChatGPT as the final answer to our main task. We
manually examine all the responses generated and give examples of their
performance.

Results and discussion. We show two cases of predictions made by ChatGPT using
the same dialogue history in Figure 4(a) and 4(b). In Figure 4(a), ChatGPT
first made a wrong main task prediction, followed by a correct but not
consistent auxiliary task prediction. When we provide all historical
information to ChatGPT and ask for a revised main task prediction, it realised
that the slot value from the first prediction is not consistent with the
original slot type and it self-corrected its wrong prediction. In Figure 4(b),
we show another case of correct predictions which happens for the majority of
the conversations. When ChatGPT is able to make correct predictions for both
the main and auxiliary tasks, it confirms the correct predictions for the
final question based on cycle consistency.

We illustrate how the cycle consistency between the main and auxiliary tasks
aids ChatGPT in rectifying incorrect answers or confirming correct answers
provided by the LLM Zhu et al. (2017). The strategy is applicable to the free
accessible web interface of ChatGPT and is easy to implement. However, as
conversations are difficult to quantify and evaluate, we only show the results
in a qualitative way and encourage future research to explore all the
potential implementations in LLMs Wang et al. (2023).

###  7.2 In-context Learning

Implementation. Following the settings in § 7.1, we skip the joint training
period and apply our strategy to ChatGPT by ICL, providing examples and
instructions in the prompt context. Specifically, we compare the ICL results
using prompts from two different types of examples: 1) examples from source
domain and 2) examples generated and selected by cycle consistency in target
domain. To illustrate, the ICL prompt is originally built by DST examples from
the source domains and DST test turn from the target domain. After selecting
good samples with the strategy discussed in § 7.1, the ICL prompt can be
updated with examples in the target domain. We conduct small-scale experiments
$3$ times with the “train” domain in the MultiWOZ dataset by randomly sampling
$100$ turns of conversations and inference with ICL prompts using examples
generated from the source or target domains, evaluated by JGA.

Results and discussion. The resulting average JGA for the original source
domain ICL prompt is $34.92\%$ while the JGA for the selected target domain
ICL prompt is $54.18\%$. Our self-training strategy works very well, serving
the LLM to select valuable in-domain examples for the ICL prompt, improving
the zero-shot DST performance by a large $19.25\%$ margin. By manually
examining the generated dialogue states, the ICL prompt modified with our
strategy performs better, especially for conversations with longer dialogue
turns and more slot types. Our self-training strategy demonstrates its
capability in generating and selecting dialogue state samples in LLMs which
can further improve the performance of zero-shot DST using an ICL prompt.
However, due to the scope of this paper, we only test the application of our
UNO-DST on ChatGPT with a small corpus of dialogues for case studies (§
Limitations).

In summary, this section extends the applicability of the UNO-DST strategy to
LLMs like ChatGPT, assessing its efficacy in both conversational approaches
and ICL. Conversational methods offer a straightforward mechanism for
rectifying in-discussion errors, whereas ICL, leveraging APIs or LLM
inferences, facilitates handling larger data corpora. Besides testing the
cycle consistency strategy in well-suited DST tasks, additional work is
required to extend it to other LLMs or more general NLP problems.

##  8 Conclusion

We propose a novel approach to convert the zero-shot DST into a few-shot
setting by generating and selecting quality dialogue states from unlabeled
data in the target domain through joint and self-training periods. We
introduce and demonstrate how our proposed auxiliary task, which generates
slot types as the inverse prompt for the main task which generates slot
values, serves the whole model for 1) better accuracy of the main task in
joint training 2) quality data selection in the self-training period 3) new
slot types generation beyond the given slot type list and 4) upgrading to LLMs
without pre-training.

Our proposed strategies of UNO-DST are task-independent, which can be extended
to other prompt formats and generalised to LLMs. We look forward to future
works that engage additional auxiliary tasks which target new datasets and
apply zero-shot DST, even where no slot types are given.

## Limitations

This work has two main limitations: 1) due to the limitation of computational
resources, we only conduct experiments on small PLMs, which is “t5-small” and
simple NLP tasks, which is “QA”. Our future works will include more NLP tasks
with different PLMs to systematically test the performance of our proposed
models. 2) Our reported self-training results are only for a single round of
self-training because we could not find a way to continuously increase the
performance of self-training. Our future plan seeks to improve and examine the
best criteria for self-training using an early-stopping strategy. 3) The
experimental settings for ChatGPT can be improved in three aspects: a) a
larger data corpus can be applied with better instruction prompts in order to
limit ChatGPT in generating more accurate values, b) an open-source LLM
Touvron et al. (2023) can be applied to better evaluate and replicate the
results of the experiment, and c) a full self-training strategy that includes
generation, selection, and fine-tuning can be tested in order to demonstrate
the best performance with LLMs.

## Ethical Concerns

Our self- and joint-training tunes models to attenuate and amplify signals
from the original dataset. While this strategy does work well in our
experiments, if the dataset’s signal is weak to start with, our methods may
incorrectly amplify errors or biases. Application of our techniques in
practical settings should be evaluated before deployment.

This work experimented with publicly available datasets which require no
additional annotation from human annotators.

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##  Appendix A Appendix

###  A.1 Experimental Setup

We select QA as the main task in our framework for its popularity and test it
through different checkpoints holding parameters fixed. We adopt the open-
source t5-small Raffel et al. (2020) with 60M parameters as our baseline and
train using AdamW with a learning rate of 0.0001, batch size 8 for 1 epoch
(zero-shot setting) and 3 epochs (fine-tuning setting). We initialize the
model using “t5-small”777https://huggingface.co/t5-small and
“t5-QA”888https://github.com/facebookresearch/Zero-Shot-
DST/tree/main/TransferQA checkpoints. For “t5-small”, the same question prompt
is used as in T5DST Lin et al. (2021b). While for “t5-QA” which is pre-trained
following the steps in TransferQA, a similar prompt format is chosen Lin et
al. (2021a). We train on a single GeForce RTX3000 GPU.

During the training period, we adopt the cross-domain setting for MultiWOZ 2.1
all domains Wu et al. (2020a) and SGD seen domains Rastogi et al. (2020). For
example, under this setting, we use four of the five domains in MultiWOZ as
the source domains for the training of the model. The remaining domain is used
as the target domain for testing performance Wu et al. (2019). The unlabelled
training data in the target domain will be used for self-training and testing
data in target domain is only used for final testing.

###  A.2 Self-training Algorithm

Algorithm 1 Self-training with auxiliary task

1:Training dataset $K_{u}$ in unknown domains, slot type $s\in S$, dialogue
history $C_{t}$ and PLM $PLM(.)$ after joint training

2:The fine-tuned model $PLM(.)$

3:repeat

4:Step 1

5:for batch $k_{u}\in K_{u}$ do

6: for slot-type, context $s,C_{t}\in k_{u}$ do

7: $p^{main}=P(s,C_{t})$ –>prompting

8: $v^{\prime}\leftarrow PLM(p^{main})$ –>main task

9: $C_{t}^{m}$ = $M(C_{t},v^{\prime})$ –>value masking

10: $p^{aux}=IP(C_{t}^{m})$ –>inverse prompting

11: $s^{\prime}\leftarrow PLM(p^{aux})$ –>auxiliary task

12: end for

13:end for

14:Step 2

15:$G=[\ \ ]$

16:for batch $k_{u}\in Ku$ do

17: if $set(s)=set(s^{\prime})$ then

18: $G.append(k_{u})$

19: end if

20:end for

21:Step 3 fine-tuning $PLM(.)$ with $G$

In this section, we explain the detail for our self-training strategy in
Algorithm 1. The self-training is conducted using PLM after the joint training
and based on the unlabelled training data in the target domain. There are a
total of $3$ steps for self-training which are 1) generation of slot values
and types from the main and auxiliary task, 2) selection of good dialogue
state samples using cycle consistency between two tasks and 3) Fine-tuning the
PLM with selected samples.

###  A.3 Slot type discussion

####  A.3.1 Future zero-shot DST

(a) Future zero-shot DST

(b) Slot type prediction with randomly masked dialogue history

(c) Building slot type corpus with merging and selection

Figure 5: Zero-shot DST without pre-defined slot types

As discussed in the previous section, the auxiliary task can facilitate the
generation of new slot types beyond the pre-defined 30 slots in MultiWOZ.
Besides adding more slot types to the given slot type list, we believe that it
is possible for our proposed model to conduct DST tasks in an unknown target
domain without any given slot types and we describe the proposal of future
zero-shot DST in Figure 5

In order to eliminate the use of pre-defined slot types, we add a slot type
generation period between joint and self-training, which identifies and select
domain-relevant slot type corpus. Similar to the process proposed by Hudeček
et al. (2021), we can first use our auxiliary task to generate potential slot
types based on random masked dialogue history, as shown in Figure 5(b). The
generated text may contain domain-irrelevant or similar slot types and we
propose a weak selection and merging of task-relevant and similar slot types
for slot type corpus Hudeček et al. (2021). Secondly, those generated slot-
type corpus can be used for self-training in the unknown target domain, as
discussed in § 3.4.

During our testing, our auxiliary task can generate predictions including all
6 given slot types in the “train” domain, as well as valid slot types in other
domains, which demonstrates the potential of future zero-shot methods without
pre-defined slot types. We look forward to future works for zero-shot DST
without any labelled data in slot types and values in unknown target domains.

####  A.3.2 Unseen slot type prediction

For each domain, there are seen slot types which appear in other domains
(e.g.,“hotel-name” and “restaurant-name”) or unseen slot types which are
unique in the specific domain (e.g.,“hotel-stars”). The ratio of the
occurrences of these slot types represents the difficulty of zero-shot DST for
each domain Wang et al. (2022). As shown in table 6, the original setting for
the MultiWoz dataset has 30 given slot types. However, not all of them appear
in every domain. For some certain domains, like the “hotel” domain, there are
4 unique slot types which do not appear in other domains, including “stars”,
“internet”, “stay” and “parking”. In Figure 6, we show the slot accuracy for
the hotel domain. It shows that generally, the unseen slot types will perform
worse than the seen slot types Wang et al. (2022). Prediction for those slot
types in zero-shot cross-domain settings can be challenging as there is no
further information from the other source domains. In addition, half of the
unseen slot types in the “hotel” domain are related to “yes/no” slot values,
whereas in our joint training settings in § 3.3, we skip the masking of those
“yes/no” values from the context and the model is less trained compared with
other slot types. We hope that future works should improve on “yes” or “no”
value prediction.

All Given Slot Types in MultiWOZ 2.1
---
$area^{123},arriveby^{45},day^{235},departure^{45},$
$destination^{45},food^{3},internet^{2},leave^{45},$
$name^{123},people^{235},parking^{2},price^{23},stars^{2},$
$stay^{2},time^{3},type^{12}$
Seen Slot Types in Hotel Domain
$area^{123},day^{235},name^{123},people^{235},price^{23},type^{12}$
Unseen Slot Types in Hotel Domain
$internet^{2},parking^{2},stars^{2},stay^{2}$
Table 6: Seen and unseen slot types in hotel domain. The superscript on each
slot type indicates the domain information from: (1:attraction, 2:hotel,
3:restaurant, 4:taxi, 5:train) Figure 6: Slot Accuracy for seen and unseen
slot types in hotel domain. (Grey: Seen slot types, Red: Unseen slot types)

####  A.3.3 Auxiliary Task Design

In our model, in order to better facilitate the main task with cycle
consistency, we design the auxiliary task as an inverse prompt of the main
task and in order to better leverage the capability PLMs, we convert the
auxiliary prompt as mask language prediction. We conclude our design of the
auxiliary task into 3 criteria: 1) The auxiliary task and the main task should
have a similar format so that they can share the prompt function $P$. 2) The
auxiliary task needs to facilitate the main task in one or many different
ways. In UNO-DST, the auxiliary task serves the main task in two different
ways, which are generation and selection. 3) The auxiliary task should be
easier than the main task so that it can better serve the selection in self-
training. Our work shows the initiative of adding auxiliary tasks and we hope
that future work can propose different tasks, such as targeting “none” or
“yes/no” value predictions. Besides cycle consistency, the self-consistency
between multiple auxiliary tasks should also help with the main task in
generation and selection Wang et al. (2023).