# Ultralow-loss integrated photonics enables bright, narrow-band, photon-pair
sources

Ruiyang Chen  Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China  International
Quantum Academy, Shenzhen 518048, China  Yi-Han Luo<EMAIL_ADDRESS>International Quantum Academy, Shenzhen 518048, China  Jinbao Long
International Quantum Academy, Shenzhen 518048, China  Baoqi Shi
International Quantum Academy, Shenzhen 518048, China  Department of Optics
and Optical Engineering, University of Science and Technology of China, Hefei,
Anhui 230026, China  Chen Shen  International Quantum Academy, Shenzhen
518048, China  Qaleido Photonics, Shenzhen 518048, China  Junqiu Liu
<EMAIL_ADDRESS>International Quantum Academy, Shenzhen 518048, China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China

Photon-pair sources are critical building blocks for photonic quantum systems.
Leveraging Kerr nonlinearity and cavity-enhanced spontaneous four-wave mixing,
chip-scale photon-pair sources can be created using microresonators built on
photonic integrated circuit. For practical applications, a high microresonator
quality factor $Q$ is mandatory to magnify photon-pair sources’ brightness and
reduce their linewidth. The former is proportional to $Q^{4}$, while the
latter is inversely proportional to $Q$. Here, we demonstrate an integrated,
microresonator-based, narrow-band photon-pair source. The integrated
microresonator, made of silicon nitride and fabricated using a standard CMOS
foundry process, features ultralow loss down to $3$ dB/m and intrinsic $Q$
factor exceeding $10^{7}$. The photon-pair source has brightness of
$1.17\times 10^{9}$ Hz/mW2/GHz and linewidth of $25.9$ MHz, both of which are
record values for silicon-photonics-based quantum light source. It further
enables a heralded single-photon source with heralded second-order correlation
$g^{(2)}_{\mathrm{h}}(0)=0.0037(5)$, as well as a energy-time entanglement
source with a raw visibility of $0.973(9)$. Our work evidences the global
potential of ultralow-loss integrated photonics to create novel quantum light
sources and circuits, catalyzing efficient, compact and robust interfaces to
quantum communication and networks.

Quantum science and technology have revolutionized our information society by
offering a new paradigm to generate, transmit and process information. Photons
– travelling at $3\times 10^{8}$ m/s and free from decoherence – are
irreplaceable carriers of quantum information Pan  _et al._ (2012); O’Brien
_et al._ (2009). Photons offer unrivalled coherence and immunity to
perturbation for quantum computation Wang  _et al._ (2019); Zhong  _et al._
(2020); Madsen  _et al._ (2022), and have been ubiquitously used in quantum
communication Kimble (2008); Bouwmeester  _et al._ (1997); Lo  _et al._
(2014); Chen  _et al._ (2021); Lu  _et al._ (2022). Currently, an emerging
trend is to realize quantum information processing using photonic integrated
circuit (PIC), which features small size, weight, and power consumption Wang
_et al._ (2020); Elshaari  _et al._ (2020); Pelucchi  _et al._ (2022). In
addition, PIC-based quantum chips can be manufactured with large volume and
low cost using established CMOS foundries. Indeed, in the last decade,
integrated photonics has enabled increasingly diverse applications on quantum
states manipulation Vigliar  _et al._ (2021); Bao  _et al._ (2023), quantum
key distribution Sibson  _et al._ (2017), quantum networks Zheng  _et al._
(2023), and quantum frequency conversion Holzgrafe  _et al._ (2020); Weaver
_et al._ (2024).

For photonic quantum systems, photon-pair sources are key building blocks.
Particularly, narrow-band photon-pair sources are essential for long-distance
quantum communication using quantum repeaters Gisin and Thew (2007); Lvovsky
_et al._ (2009). Here the narrow bandwidth is critical, as it must match the
natural linewidth of atomic transition to ensure efficient photon-atom
interface. Meanwhile, narrow-band photon-pair sources suffer minimum
distortion from chromatic dispersion when transmitting over fibers. Moreover,
a narrow band boosts the brightness (i.e. photon flux rate) of photon-pair
sources, an equally central parameter for practical applications.

To build narrow-band photon-pair sources, nonlinear optical microresonators
are particularly useful Vahala (2003); Gómez  _et al._ (2024). Compared to the
methods using cavity-enhanced spontaneous parametric down-conversion (SPDC)
with bulky crystals Ou and Lu (1999); Bao  _et al._ (2008) or using atomic
transition Balić  _et al._ (2005); Shu  _et al._ (2016), optical
microresonators are small, compact and robust. By leveraging the resonant
nature for intracavity power enhancement, optical microresonators can offer
exaggerated nonlinear effects, and have already been extensively used for
optical frequency comb generation Kippenberg  _et al._ (2018); Kues  _et al._
(2019) and wideband frequency translation Li  _et al._ (2016); Lu  _et al._
(2019a).

Previously, using microresonators on silicon PIC, photon-pair sources have
achieved brightness of $1.6\times 10^{8}$ Hz/mW2/GHz and bandwidth of $\sim 2$
GHz Ma  _et al._ (2017). The latter, due to the high optical loss (typically
$>1$ dB/cm) of silicon waveguides, is overwhelmingly larger than the natural
linewidth of atomic transition ($\sim 10$ MHz). The quest for even lower
optical loss in PIC has motivated the rising and quick maturing of silicon
nitride (Si3N4) integrated photonics. Integrated waveguides based on Si3N4
feature ultralow optical loss down to 1 dB/m Liu  _et al._ (2021) and
tailorable dispersion Okawachi  _et al._ (2014), and have become the leading
platform for nonlinear photonics Moss  _et al._ (2013); Gaeta  _et al._
(2019), narrow-linewidth lasers Jin  _et al._ (2021); Xiang  _et al._ (2022),
and linear networks Taballione  _et al._ (2019); Arrazola  _et al._ (2021)
etc. Indeed, Si3N4 has unleashed new capability for on-chip photonic quantum
information processing Ramelow  _et al._ (2015); Lu  _et al._ (2019b); Samara
_et al._ (2019, 2021); Wu  _et al._ (2021); Fan  _et al._ (2023); Wen  _et
al._ (2023).

Figure 1:  Principle and characterization of integrated Si3N4 microresonators.
a. The principle of cavity-enhanced SFWM in an optical microresonator. Two
pump photons annihilate, creating a pair of signal and idler photons aligned
with the resonance grid. Meanwhile, nonlinear scattering process can
incoherently transfer pump photons to other modes. b. Photo of the Si3N4 chip
containing tens of microresonators. Lensed fibers are used for fiber-chip
light coupling. c. Optical microscope image showing the layout of a 100-GHz-
FSR microresonator. The inset presents the SEM image of the Si3N4 waveguide’s
cross section, superposed with the simulated TE00 optical mode. d. A segment
of microresonator transmission spectrum, where the pump/signal/idler mode
locates in C35/C37/C33. e. Resonance profile in C33 with a Lorentzian fit. f.
Measured microresonator dispersion profile fitted with Eq. 1.

Here, we report a microresonator-based photon-pair source using ultralow-loss
Si3N4 integrated photonics. The photon pairs are generated via cavity-enhanced
spontaneous four-wave mixing (SFWM) in the optical microresonator Helt  _et
al._ (2010); Luo  _et al._ (2015), as the principle shown in Fig. 1a. In the
presence of the microresonator’s resonance grid, all the photons exist from
the resonances of frequency $\omega(\mu)/2\pi$, where $\mu$ denotes the
resonance mode index. A continuous-wave (CW) laser of frequency
$\omega(\mu_{\mathrm{p}})/2\pi$ is used to pump the microresonator. Via SFWM,
two pump photons annihilate, creating a pair of signal and idler photons of
frequency $\omega(\mu_{\mathrm{s}})/2\pi$ and $\omega(\mu_{\mathrm{i}})/2\pi$.
The subscript p/s/i corresponds to the pump/signal/idler mode. In this
process, the phase matching condition is automatically satisfied given
$2\mu_{\mathrm{p}}=\mu_{\mathrm{s}}+\mu_{\mathrm{i}}$. The energy conservation
$2\omega(\mu_{\mathrm{p}})=\omega(\mu_{\mathrm{s}})+\omega(\mu_{\mathrm{i}})$
requires anomalous (negative) group-velocity dispersion (GVD). A weak GVD is
beneficial for a locally equidistant resonance grid. Additionally, strong
optical confinement in the waveguides enhances intra-cavity electric field
intensity, further facilitating SFWM.

Microresonator characterization. To satisfy these requirements, we design and
fabricate integrated Si3N4 microresonators using a deep-ultraviolet (DUV)
subtractive process on 6-inch wafers Ye  _et al._ (2023). Figure 1b shows a
photograph of one final Si3N4 chip of $5\times 5$ mm2 size containing tens of
microresonators. Figure 1c shows the microresonator’s layout under an optical
microscope. The Si3N4 microresonators are formed with waveguides of 810-nm
thickness and 2.4-$\mu$m width. The free spectral range (FSR) of the
microresonators is 100 GHz. Figure 1c inset shows the scanning electron
microscope (SEM) image of the waveguide’s cross-section, together with the
simulated optical field distribution of the fundamental transverse-electric
(TE00) mode. It is apparent that the waveguide geometry enables tight optical
confinement. The gap between the bus waveguide and the microring resonator is
300 nm for over-coupling Cai  _et al._ (2000). A pulley-style coupler where
the bus waveguide adiabatically approaches the microring is applied to ensure
high coupling ideality Pfeiffer  _et al._ (2017).

Figure 2:  Measurement of photon-pair brightness and coincidence-to-accidental
ratio. a. Schematic of the experiment. The Si3N4 microresonator is pumped with
a CW laser whose frequency locates in C35. The generated photons are detected
with SNSPDs and analyzed with a time-to-digital converter for brightness
measurement. The photon pair created at an earlier moment, labeled as
$|e_{\mathrm{i}}\rangle|e_{\mathrm{s}}\rangle$, is coherently superposed with
the photon pair $|l_{\mathrm{i}}\rangle|l_{\mathrm{s}}\rangle$ generated
later, resulting in a energy-time entanglement state
$|\Psi\rangle=(|e_{\mathrm{i}}\rangle|e_{\mathrm{s}}\rangle+|l_{\mathrm{i}}\rangle|l_{\mathrm{s}}\rangle)/\sqrt{2}$.
b. The measured two-photon correlation histogram normalized to its maximum.
The bin width is 0.2 ns. The histogram is obtained with on-chip pump power of
$P=5.2$ $\mu$W and integration time of 3600 seconds. The FWHM of the peak at
zero delay is 5.6 ns. The inset shows the zoom-in of background in the region
far from the zero delay. c. The single count rates of signal (C37, green) and
idler (C33, red) photons with different on-chip power, and their polynomial
fit curves. d. The coincident count rate with different on-chip power, and its
polynomial fit. The coincidence window width $\Delta t=40$ ns is chosen. In
panels (c, d), error bars are calculated as the standard deviation of ten
independent measurements, which are however smaller than the size of the data
points. e. The measured CAR with different on-chip power.

We characterize the Si3N4 microresonator using a vector spectrum analyzer in
the telecommunication band Luo  _et al._ (2024). Here we study the TE00 mode
that has anomalous GVD and lowest loss among all microresonator’s spatial
modes. A segment of the microresonator’s transmission spectrum is shown in
Fig. 1d, where the frequency axis is calibrated with a relative precision
better than 500 kHz. Each resonance of the 100-GHz-FSR microresonator locates
in one International Telecommunication Union (ITU) channel, which can be
separated using a dense wavelength-division multiplexer (DWDM). We plot and
fit these resonances to extract the central frequency $\omega/2\pi$, the
intrinsic loss $\kappa_{0}/2\pi$, and the external coupling strength
$\kappa_{\mathrm{ex}}/2\pi$ Li  _et al._ (2013). As an example, the resonance
profile in Channel 33 (C33) is shown in Fig. 1e, with marked
$\kappa_{0}/2\pi$, $\kappa_{\mathrm{ex}}/2\pi$ and the full width at half
minimum/maximum (FWHM). The intrinsic and loaded quality factors are
calculated as $Q_{0}=\omega/\kappa_{0}=1.02\times 10^{7}$ and
$Q=\omega/\kappa=4.97\times 10^{6}$, where
$\kappa=\kappa_{0}+\kappa_{\mathrm{ex}}$. The measured FWHM is larger than
$\kappa/2\pi$ due to the presence of mode split Li  _et al._ (2013) (see Note
1 in Supplementary Materials for more details).

With the measured $\omega/2\pi$ for each resonance, the microresonator’s
dispersion is fitted with

$D_{\mathrm{int}}(\mu)=\omega(\mu)-\omega(0)-D_{1}\mu=D_{2}\mu^{2}/2+\mathcal{O}(\mu^{3}),$
(1)

where $\omega(0)/2\pi$ is the pump resonance’s frequency, $D_{1}/2\pi$ is the
microresonator FSR, $D_{2}/2\pi$ is the GVD parameter. Higher-order dispersion
terms are irrelevant in this work and thus neglected. The $D_{\mathrm{int}}$
profile and the extracted parameters are shown in Fig. 1f. The measured
$D_{2}/2\pi=895.9$ kHz is sufficiently small compared to resonance linewidth
$\kappa/2\pi$, satisfying energy conservation.

Photon-pair generation. Experimentally we select the mode in Channel 35 (C35)
as the pump mode. The signal and idler photons are generated in pairs in
Channel 37 (C37) and C33, as illustrated in Fig. 1d. The photon-pair
generation rate (PGR) via cavity-enhanced SFWM is
$\alpha\propto\gamma^{2}P^{2}Q^{3}$, where $\gamma$ is the effective Kerr
coefficient and $P$ is the on-chip pump power Helt  _et al._ (2010); Luo  _et
al._ (2015). The photon-pair brightness is calculated as $\alpha/P^{2}$
divided by the photon linewidth ($\sim 0.6\kappa/2\pi$, discussed later), thus
is $\propto Q^{4}$. Consequently, a high-$Q$ microresonator is extremely
critical, as the photon-pair brightness is biquadratically magnified by the
high $Q$.

We note that the correct measurement of photon-pair brightness poses several
requirements. In the ideal case that only SFWM presents, the signal and idler
photons are created strictly in pairs, marked as shaded blue in Fig. 2a.
Considering the system efficiency $\eta_{\mathrm{s/i}}$, single count rate for
the signal/idler photon on the superconducting-nanowire single-photon
detectors (SNSPD) is $n_{\mathrm{s/i}}=\alpha\eta_{\mathrm{s/i}}$, resulting
in the coincidence count rate
$n_{\mathrm{cc}}=\alpha\eta_{\mathrm{i}}\eta_{\mathrm{s}}$. By measuring
$n_{\mathrm{s}}$, $n_{\mathrm{i}}$ and $n_{\mathrm{cc}}$, the PGR is
calculated as

$\alpha=n_{\mathrm{s}}n_{\mathrm{i}}/n_{\mathrm{cc}}$ (2)

However, in reality, nonlinear scattering process such as Raman scattering
Karpov  _et al._ (2016) occurs in microresonators. It can transfer the pump
photons to other states, as the dashed arrows in Fig. 1a. The scattering-noise
photons are unpaired and randomly mixed with the SFWM photon pairs, as shaded
grey in Fig. 2a. Experimentally, scattering-noise photons cannot be
distinguished from the SFWM photon pairs, as photons can be lost due to
$\eta_{\mathrm{s/i}}<1$, marked with dashed outline in Fig. 2a. Therefore, Eq.
2 fails and requires modification.

Considering that $\alpha\propto P^{2}$ and the generation rate of scattering-
noise photons is $\alpha_{\mathrm{sn}}\propto P$, $n_{\mathrm{s/i}}$ and
$n_{\mathrm{cc}}$ can be expressed as

$\displaystyle n_{\mathrm{s/i}}$
$\displaystyle=\eta_{\mathrm{s/i}}\left(aP^{2}+b_{\mathrm{s/i}}P\right),$ (3)
$\displaystyle n_{\mathrm{cc}}-n_{\mathrm{acc}}$
$\displaystyle=\eta_{\mathrm{i}}\eta_{\mathrm{s}}aP^{2}+\mathcal{O}(P^{3}),$
(4)

where $n_{\mathrm{acc}}$ is accidental coincidence count rate (discussed
later), $a$ is the PGR coefficient (i.e. $\alpha=aP^{2}$), and
$b_{\mathrm{s/i}}$ is the coefficient corresponding to the scattering-noise
photon generation. We note that, in the case that $\alpha$ and
$\alpha_{\mathrm{sn}}$ are comparable, $n_{\mathrm{acc}}$ contains higher-
order terms proportional to $P^{3}$ and $P^{4}$, which should be included in
the righthand of Eq. 4. Since the values of $n_{\mathrm{s/i}}$,
$n_{\mathrm{cc}}$ and $n_{\mathrm{acc}}$ can be experimentally measured at
different pump power $P$, $\eta_{\mathrm{s}}a$, $\eta_{\mathrm{i}}a$ and
$\eta_{\mathrm{i}}\eta_{\mathrm{s}}a$ can be extracted via polynomial fit,
allowing the calculation of $a$. More details on the fit formula can be found
in Note 2 in Supplementary Materials.

The schematic of the experimental setup is shown in Fig. 2a, with more details
found in Note 3 in Supplementary Materials. The photon-pair generation is
evidenced by the two-photon correlation histogram shown in Fig. 2b, describing
the distribution of the two-photon arrival time difference. The peak at the
zero delay proves that the signal and idler photons are generated in pairs.
The FWHM of the histogram peak is 5.6 ns. By summing up the bins within a
window centered at zero delay and taking a specific width $\Delta t$, the
coincidence count $n_{\mathrm{cc}}$ is obtained, while $n_{\mathrm{acc}}$ is
obtained with the same $\Delta t$ but far from the zero delay. We emphasize
that, to obtain the correct brightness, $\Delta t$ must be sufficiently large
to account all the coincidence events. Otherwise the brightness will be
overestimated. Here we take $\Delta t=40$ ns for brightness measurement. More
details can be found in Note 4 in Supplementary Materials.

The measured $n_{\mathrm{s/i}}$, $n_{\mathrm{cc}}$ and $n_{\mathrm{acc}}$
versus on-chip power $P$, together with their fit curves, are shown in Figs.
2d and 2e, respectively. The error bars are calculated as the standard
deviation of ten independent measurements. Experimentally we obtain
$a=3.04\times 10^{7}$ Hz/mW2. Detailed fit results are found in Note 4 in
Supplementary Materials. The generated photons’ linewidth is estimated as
$\delta\nu=25.9$ MHz. The brightness is thus calculated as $1.17\times 10^{9}$
Hz/mW2/GHz. More details on the linewidth estimation are elaborated in Note 5
in Supplementary Materials.

In addition to the brightness, the noise level is equally important. As shown
in Fig. 2b, for the region far outside the zero delay, the bin counts are not
exactly zero, mainly due to the SFWM multi-photon excitation and the
scattering noise. The coincidence-to-accidental ratio (CAR) is defined as
$(n_{\mathrm{cc}}/n_{\mathrm{acc}}-1)$. Figure 2c shows the measured CAR with
$\Delta t=5.6$ ns, i.e. the FWHM of the histogram peak in Fig. 2b. As the pump
power $P$ decreases, both the SFWM multi-photon excitation and the scattering
noise decrease, resulting in a higher CAR. Experimentally, with $P=5.2$
$\mu$W, we achieve $n_{\mathrm{cc}}/n_{\mathrm{acc}}-1=1438\pm 22$ with 3600
second integration. Meanwhile, the measured coincidence count rate is
$n_{\mathrm{cc}}=22$ Hz.

Figure 3:  Measurement of the second-order correlation. a. Schematic of the
experimental setup. The idler photons (C33, red) are directly detected with an
SNSPD as trigger (marked as path 1). The signal photons (C37, green) are
equally splitted and detected (marked as paths 2 and 3). The two-photon
correlation histogram between paths 2 and 3 is measured with and without the
path 1 clicking. b. Measured $g^{(2)}_{\mathrm{h}}(0)$ versus different idler
photon count rate. c. Measured $g^{(2)}_{\mathrm{nh}}(\tau)$. The bin width is
chosen as 0.5 ns. The maximum $g^{(2)}_{\mathrm{nh}}(\tau)=1.942(14)$ is
achieved at zero delay.  Figure 4:  Two-photon interference and energy-time
entanglement. a. Schematic of the experimental setup. To avoid single-photon
interference, a 14-meter-long fiber link is used to introduce 69 ns delay. A
portion of the pump laser is used to actively stabilize the phase difference
between the two branches via controlling the added phase $\phi$ caused by the
fiber stretcher. b. Two-photon correlation histogram. In the upper/lower
panel, the peak at zero delay appears/vanishes due to constructive/destructive
two-photon interference. c. Two-photon interference fringe. The sinusoidal fit
shows a raw visibility of $V=0.973(9)$ without background subtraction. Error
bars are much smaller than the point size and thus are invisible.

Second-order correlation. Photon-pair sources can be used as heralded single-
photon sources. Once the SNSPD clicks upon the arrival of an idler photon, it
heralds the existence of a signal photon. The photon anti-bunching can be
observed with the heralded signal photons. To characterize the single-photon
purity, a Hanbury Brown and Twiss (HBT) setup is utilized to measure the
heralded second-order correlation $g^{(2)}_{\mathrm{h}}(0)$. As shown in Fig.
3a, the DWDM’s C37 is equally splitted into two branches with marked path
indices 2 and 3. Together with C33 of path index 1, the
$g^{(2)}_{\mathrm{h}}(0)$ is calculated as
$g^{(2)}_{\mathrm{h}}(0)=n_{123}n_{1}/(n_{12}n_{13}$), where $n_{1}$ is the
single count rate, $n_{12}$, $n_{13}$, $n_{123}$ are the coincidence count
rates. Experimentally, we observe $g^{(2)}_{\mathrm{h}}(0)=0.0037(5)$ at idler
count rate $n_{\mathrm{s}}=13.3$ kHz, with $P=40$ $\mu$W and $\Delta t=5.6$
ns. Due to the SFWM multi-pair excitation, $g^{(2)}_{\mathrm{h}}(0)$ increases
with increasing $P$ (thus $n_{\mathrm{i}}$), as shown in Fig. 3b.

Next we characterize the spectral purity of the photon pairs by measuring the
non-heralded second-order correlation $g^{(2)}_{\mathrm{nh}}(\tau)$ Christ
_et al._ (2011). Such measurement does not involve idler photons. The
$g^{(2)}_{\mathrm{nh}}(\tau)$ histogram between paths 2 and 3 is shown in Fig.
3c. It is observed that $g^{(2)}_{\mathrm{nh}}(\tau)=1.942(14)$ is reached at
zero delay. It indicates that, a high multi-photon interference visibility
between two independent photon-pair sources requires a narrow coincidence
window Huang  _et al._ (2010). This is mainly due to the strong spectral
correlation of the photon pairs generated with the CW pump. The influence of
the spectral correlation can be eliminated by introducing a dual interference
scheme Liu  _et al._ (2023).

Two-photon interference visibility. Photon-pair sources can be regarded as
narrow-band, energy-time entanglement light sources, which are advantageous
for long-distance distribution due to their robustness against decoherence in
optical fibers Marcikic  _et al._ (2004); Inagaki  _et al._ (2013); Yu  _et
al._ (2020). As shown in Fig. 2a, the photon pair created at an earlier
moment, labeled as $|e_{\mathrm{i}}\rangle|e_{\mathrm{s}}\rangle$, is
coherently superposed with the photon pair
$|l_{\mathrm{i}}\rangle|l_{\mathrm{s}}\rangle$ generated later, where $e$/$l$
denotes the earlier/later generated photon and the subscript s/i marks the
signal/idler photon. Thus energy-time entangled photons
$|\Psi\rangle=(|e_{\mathrm{i}}\rangle|e_{\mathrm{s}}\rangle+|l_{\mathrm{i}}\rangle|l_{\mathrm{s}}\rangle)/\sqrt{2}$
are created. To reveal the quantum interference, the two photons are projected
along the state $|\phi\rangle=(|e\rangle+e^{i\phi}|l\rangle)/\sqrt{2}$
individually. The probability of measuring the state $|\Psi\rangle$ in
$|\phi_{\mathrm{i}}\rangle|\phi_{\mathrm{s}}\rangle$ is calculated as
$p=(1+\cos 2\phi)/4$.

We use a folded Franson interferometer Franson (1989); Ou  _et al._ (1990) to
interfere two temporally separated photons. The setup is shown in Fig. 4a. The
photons are equally splitted into two branches. In the lower branch, a
14-meter-long fiber link, corresponding to 69 ns delay, is introduced to avoid
single-photon interference. A fiber stretcher is used to control the added
phase $\phi$. The two branches are then recombined and interfere. To suppress
high-frequency phase fluctuation in the fiber due to ambient temperature
fluctuation, we place the interferometer in a heat-insulated container.
Additionally, a portion of the pump laser is used for active phase locking
with a PID of 10 kHz bandwidth. The phase $\phi$ is varied by tuning the
locking voltage and the PID’s locking point.

The two-photon interference is evidenced by the two-photon correlation
histogram shown in Fig. 4b. The central peak is post-selected, as it
corresponds to the measurement of $|\Psi\rangle$. For $\phi=0$, the central
peak reaches maximum that is fourfold to the sidebands. When $\phi=\pi/2$, the
central peak vanishes due to destructive interference. Detailed analysis is
found in Note 6 in Supplementary Materials.

The measured two-photon interference is shown in Fig. 4c, where $\Delta t=5.6$
ns is identical to that of CAR measurement. The coincidence count
$n_{\mathrm{cc}}$ oscillates with a period of $\pi$. A sinusoidal fit using
$n_{\mathrm{cc}}=0.5N[1+V\cos(\phi)]$ is applied to extract a raw visibility
$V=0.973(9)$ at $P=97$ $\mu$W pump power, well above 0.707 and thus violating
Clauser–Horne–Shimony–Holt (CHSH) inequality Clauser  _et al._ (1969). Due to
the scattering noise, the background of the histogram in Fig. 4b is much
larger than that of a crystal-based photon-pair source Zhong  _et al._ (2018).
The background can severely deteriorate the interference visibility. In our
experiment, the background is calculated by averaging the bin values far from
the peaks. By subtracting the averaged background, we obtain a visibility of
$V=0.995(9)$.

Conclusion and discussion. In summary, we have studied integrated, high-$Q$,
Kerr-nonlinear microresonators for bright, narrow-band, photon-pair
generation. Using an over-coupled microresonator on ultralow-loss Si3N4 PIC
and fabricated with a standard CMOS-foundry process, we realize a photon-pair
source with brightness exceeding $1.17\times 10^{9}$ Hz/mW2/GHz and bandwidth
below 25.9 MHz, both of which are record values for silicon-photonics-based
quantum light sources. Moreover we achieve a CAR of
$n_{\mathrm{cc}}/n_{\mathrm{acc}}-1=1438\pm 22$, a heralded second-order
correlation of $g^{(2)}_{\mathrm{h}}(0)=0.0037(5)$, and a raw two-photon
interference visibility of $V=0.973(9)$. A comparison of integrated quantum
light sources using microresonators is presented in Note 7 in Supplementary
Materials.

Since the optical loss of our Si3N4 waveguides is far above the material limit
of absorption loss Liu  _et al._ (2021); Gao  _et al._ (2022), there is major
space to further improve the microresonator $Q$ factors by optimizing the
material growth and fabrication process. Consequently, the brightness
($\propto Q^{4}$) can well exceed the best performance of recently developed
highly nonlinear AlGaAs-on-insulator microresonators Steiner  _et al._ (2021);
Chang  _et al._ (2020). Combined with the wide transparency window of Si3N4
and broadband dispersion engineering, photon-pair generation from the visible
to mid-infrared can be envisaged, paving the way to interfacing integrated
photonics with a variety of quantum devices.

Acknowledgments: We thank Yuan Cao and Hui-Nan Wu for the fruitful discussion
on Franson interferometer phase locking, Yun-Ru Fan for the suggestion on the
energy-time entanglement measurement, and Shuyi Li for the suggestion on
second-order correlation measurement. J. Liu acknowledges support from the
National Natural Science Foundation of China (Grant No.12261131503),
Innovation Program for Quantum Science and Technology (2023ZD0301500),
Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation (HZQB-
KCZYB2020050), and from the Guangdong Provincial Key Laboratory
(2019B121203002). Y.-H L. acknowledges support from the China Postdoctoral
Science Foundation (Grant No. 2022M721482). The silicon nitride chips were
fabricated by Qaleido Photonics. R. C. and Y.-H. L. contributed equally to
this work.

Author contributions: Y.-H. L. and J. Liu conceived the experiment. R. C. and
Y.-H. L. built the experimental setup, assisted with J. Long. S. C. fabricated
the Si3N4 chip device. B. S. characterized the Si3N4 chip device. R. C., Y.-H.
L. and J. Liu analyzed the data and prepared the manuscript with input from
others. J. Liu supervised the project.

Conflict of interest: The authors declare no conflicts of interest.

Data Availability Statement: The code and data used to produce the plots
within this work will be released on the repository Zenodo upon publication of
this preprint.

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# Meta-Learning with
MAML on Trees

Jezabel R. Garcia  MediaTek Research, Cambourne Business Park, Cambridge CB23
6DW, United Kingdom  Federica Freddi  MediaTek Research, Cambourne Business
Park, Cambridge CB23 6DW, United Kingdom  Feng-Ting Liao  MediaTek Research,
Cambourne Business Park, Cambridge CB23 6DW, United Kingdom  Jamie McGowan
MediaTek Research, Cambourne Business Park, Cambridge CB23 6DW, United Kingdom
Tim Nieradzik  MediaTek Research, Cambourne Business Park, Cambridge CB23 6DW,
United Kingdom
Da-shan Shiu  MediaTek Research, Cambourne Business Park, Cambridge CB23 6DW,
United Kingdom  Ye Tian  MediaTek Research, Cambourne Business Park, Cambridge
CB23 6DW, United Kingdom  Alberto Bernacchia  MediaTek Research, Cambourne
Business Park, Cambridge CB23 6DW, United Kingdom

###### Abstract

In meta-learning, the knowledge learned from previous tasks is transferred to
new ones, but this transfer only works if tasks are related. Sharing
information between unrelated tasks might hurt performance, and it is unclear
how to transfer knowledge across tasks with a hierarchical structure. Our
research extends a model agnostic meta-learning model, MAML, by exploiting
hierarchical task relationships. Our algorithm, TreeMAML, adapts the model to
each task with a few gradient steps, but the adaptation follows the
hierarchical tree structure: in each step, gradients are pooled across tasks
clusters, and subsequent steps follow down the tree. We also implement a
clustering algorithm that generates the tasks tree without previous knowledge
of the task structure, allowing us to make use of implicit relationships
between the tasks. We show that the new algorithm, which we term TreeMAML,
performs better than MAML when the task structure is hierarchical for
synthetic experiments. To study the performance of the method in real-world
data, we apply this method to Natural Language Understanding, we use our
algorithm to finetune Language Models taking advantage of the language
phylogenetic tree. We show that TreeMAML improves the state of the art results
for cross-lingual Natural Language Inference. This result is useful, since
most languages in the world are under-resourced and the improvement on cross-
lingual transfer allows the internationalization of NLP models. This results
open the window to use this algorithm in other real-world hierarchical
datasets.

##  1 Introduction

Deep learning models require a large amount of data in order to perform well
when trained from scratch. When data is scarce for a given task, we can
_transfer_ the knowledge gained in a source task to quickly learn a target
task, if the two tasks are related. _Multi-task learning_ studies how to learn
multiple tasks simultaneously with a single model, by taking advantage of task
relationships Ruder (2017); Zhang and Yang (2018). However, in Multi-task
learning models, a set of tasks is fixed in advance and they do not generalize
to new tasks. Instead, _Meta-learning_ is inspired by the human ability to
learn how to quickly learn new tasks by using the knowledge of previously
learned ones.

Meta-learning has been widely used in multiple domains, especially in recent
years since the advent of Deep Learning Hospedales et al. (2020). A successful
model for meta-learning, MAML Finn et al. (2017), does not diversify task
relationships according to their similarity and it is unclear how to modify it
for that purpose. Furthermore, there is still a lack of methods for sharing
information across tasks that have a hierarchical structure, and the goal of
our work is to fill this gap.

The use of MAML-like algorithms in NLP has just recently been proved
successful for Natural Language Inference (NLI) and Question Answering (QA)
Nooralahzadeh et al. (2020). These results represent a practical meta-learning
solution to the fundamental problem of applying NLP models to under-resourced
languages where data annotation is scarce. This work, combined with the fact
that languages can be organized hierarchically using their phylogenetic tree
Dunn et al. (2011), motivated us to develop a hierarchical meta-learning
algorithm, that we call TreeMAML.

In this work, we make the following contributions:

  * •

We propose a novel modification of MAML to account for a hierarchy of tasks.
The algorithm uses the tree structure of data during adaptation, by pooling
gradients across tasks at each adaptation step and subsequent steps follow
down the tree (see Figure 1a).

  * •

We modify the hierarchical clustering from Menon et al. (2019) to allow
asymmetric tree structure. We apply this clustering algorithm to learn dynamic
trees that exploit the similarity between tasks.

  * •

We introduce new benchmarks for testing a hierarchy of tasks in meta-learning
using a multidimensional linear regression task. We compare our algorithm to
MAML and a baseline model, where we train on all tasks without any meta-
learning algorithm applied.

  * •

We apply TreeMAML to few-shot NLI, using the XNLI dataset Conneau et al.
(2018), obtaining accuracies higher than previous state-of-the-art.

##  2 Related work

The problem of quantifying and exploiting task relationships has a long
history in Multi-task learning and is usually approached by parameter sharing,
see Ruder (2017); Zhang and Yang (2018) for reviews. However, Multi-task
Learning is fundamentally different from Meta-learning as it does not consider
the problem of generalizing to new tasks Hospedales et al. (2020). Recent work
includes Zamir et al. (2018), who studies a large number of computer vision
tasks and quantifies the transfer between all pairs of tasks. Achille et al.
(2019) proposes a novel measure of task representation by assigning an
importance score to each model parameter in each task. The score is based on
each task’s loss function gradients with respect to each model parameter. This
work suggests that gradients can be used as a measure of task similarity and
we use this insight in our proposed algorithm.

In Meta-learning, a few papers have been recently published on learning and
using task relationships. The work of Yao et al. (2019) applies hierarchical
clustering to task representations learned by an autoencoder and uses those
clusters to adapt the parameters to each task. The model of Liu et al. (2019)
maps the classes of each task into the edges of a graph, it meta-learns
relationships between classes and how to allocate new classes by using a graph
neural network with attention. However, these algorithms are not model-
agnostic; they have a fixed backbone and loss function and are thus difficult
to apply to new problems. Instead, we design our algorithm as a
straightforward generalization of Model-agnostic meta-learning (MAML, Finn et
al. (2017)) and it can be applied to any loss function and backbone.

A couple of studies looked into modifying MAML to account for task
similarities. The work of Jerfel et al. (2019) finds a different initial
condition for each cluster of tasks and applies the algorithm to the problem
of continual learning. The work of Katoch et al. (2020) defines parameter
updates for a task by aggregating gradients from other tasks according to
their similarity. However, in contrast with our algorithm, both of these
models are not hierarchical, tasks are clustered on one level only and cannot
be represented by a tree structure.

Recently, MAML has been applied to cross-lingual meta-learning Gu et al.
(2018); Dou et al. (2019). In particular, the implementation by Nooralahzadeh
et al. (2020), called XMAML, obtained good results on NLI and QA tasks. As in
the previously mentioned computer vision studies, some of these NLP algorithms
looked into the relationships among languages to select the support languages
used in their meta-learning algorithm, but they do not use the hierarchical
structure of the languages.

##  3 The meta-learning problem

We follow the notation of Hospedales et al. (2020). We assume the existence of
a distribution over tasks $\tau$ and, for each task, a distribution over data
points $\mathcal{D}$ and a loss function $\mathcal{L}$. The loss function of
the meta-learning problem, $\mathcal{L}^{meta}$, is defined as an average
across both distributions of tasks and data points:

$\mathcal{L}^{meta}\left(\bm{\omega}\right)=\mathop{\mathbb{E}}_{\tau}\mathop{\mathbb{E}}_{\mathcal{D}|\tau}\mathcal{L}_{\tau}\left(\bm{\theta}_{\tau}(\bm{\omega});\mathcal{D}\right)$
(1)

The goal of meta-learning is to minimize the loss function with respect to a
vector of meta-parameters $\bm{\omega}$. The vector of parameters
$\bm{\theta}$ is task-specific and depends on the meta-parameters
$\bm{\omega}$. Different meta-learning algorithms correspond to a different
choice of $\bm{\theta}_{\tau}(\bm{\omega})$. We describe below the choice of
MAML that will also be followed by TreeMAML.

During meta-training, the loss is evaluated on a sample of $m$ tasks and
$n_{v}$ validation data points for each task.

$\mathcal{L}^{meta}\left(\bm{\omega}\right)=\frac{1}{mn_{v}}\sum_{i=1}^{m}\sum_{j=1}^{n_{v}}\mathcal{L}_{\tau_{i}}\left(\bm{\theta}_{\tau_{i}}(\bm{\omega});\mathcal{D}_{ij}\right)$
(2)

For each task $i$, the parameters $\bm{\theta}_{\tau_{i}}$ are learned by a
set of $n_{t}$ training data points, distinct from the validation data. During
meta-testing, a new (target) task is given and the parameters $\bm{\theta}$
are learned by a set of $n_{r}$ target data points. In this work, we also use
a batch of training data points to adapt $\bm{\theta}$ at test time. No
training data is used to compute the model’s final performance, which is
computed on separate test data of the target task.

###  3.1 TreeMAML

MAML aims at finding the optimal initial condition $\omega$ from which a
suitable parameter set can be found, separately for each task, after $K$
gradient steps Finn et al. (2017). For task $i$, we define the single gradient
step with learning rate $\alpha$ as

$U_{i}(\bm{\omega})=\bm{\omega}-\frac{\alpha}{n_{t}}\sum_{j=1}^{n_{t}}\nabla\mathcal{L}(\bm{\omega};\mathcal{D}_{ij})$
(3)

Then, MAML with $K$ gradient steps corresponds to $K$ iterations of this step.

$\bm{\theta}_{\tau_{i}}(\bm{\omega})=U_{i}(U_{i}(...U_{i}(\bm{\omega})))\;\;\;\;\;\;\;\;\;\;\;\;\mbox{($K$
times)}$ (4)

This update is usually referred to as _inner loop_ and is performed separately
for each task, while optimization of the loss 2 is referred to as _outer
loop_.

We propose to modify MAML in order to account for a hierarchical structure of
tasks. The idea is illustrated in Figure 1.

Figure 1: Illustration of the MAML(a) and TreeMAML(b) algorithms. Both
algorithms are designed to quickly adapt to new tasks with a small number of
training samples. MAML achieves this by introducing a gradient step in the
direction of the single task. TreeMAML follows a similar approach, but it
exploits the relationship between tasks by introducing the hierarchical
aggregation of the gradients.

At each gradient step $k$, we assume that tasks are aggregated into $C_{k}$
clusters and the parameters for each task are updated according to the average
gradient across tasks within the corresponding cluster (in Fig.1b, we use
$K=3$ steps and $C_{1}=2$, $C_{2}=4$, $C_{3}=8$). We denote by
$\mathcal{T}_{c}$ the set of tasks in cluster $c$. Then, the gradient update
for the parameters of each task belonging to cluster $c$ is equal to

$U_{c}(\bm{\omega})=\bm{\omega}-\frac{\alpha}{n_{t}\left|\mathcal{T}_{c}\right|}\sum_{i\in\mathcal{T}_{c}}\sum_{j=1}^{n_{t}}\nabla\mathcal{L}(\bm{\omega};\mathcal{D}^{(i)}_{j})$
(5)

Furthermore, we denote by $c_{i}^{k}$ the cluster to which task $i$ belongs at
step $k$. Then, TreeMAML with $k$ gradient steps corresponds to $K$ iterations
of this step.

$\bm{\theta}_{\tau_{i}}(\bm{\omega})=U_{c_{i}^{K}}(U_{c_{i}^{K-1}}(...U_{c_{i}^{1}}(\bm{\omega})))$
(6)

The intuition is the following: if each task has scarce data, gradient updates
for single tasks are noisy and adding up gradients across similar tasks
increases the signal. Note that we recover MAML if $C_{k}$ is equal to the
total number of tasks $m$ at all steps. On the other hand, if $C_{k}=1$, then
the inner loop would take a step with a gradient averaged across all tasks.

Because at one specific step the weight updates are equal for all tasks within
a cluster, it is possible to define the steps of the inner loop update per
cluster $c$ instead of per task $\bm{\theta}_{\tau_{i}}$. Given a cluster $c$
and its parent cluster $p_{c}$ in the tree, the update at step $k$ is given by

$\bm{\theta}_{c,k}=\bm{\theta}_{p_{c},k-1}-\frac{\alpha}{n_{t}\left|\mathcal{T}_{c}\right|}\sum_{i\in\mathcal{T}_{c}}\sum_{j=1}^{n_{t}}\nabla\mathcal{L}(\bm{\theta}_{p_{c},k-1};\mathcal{D}_{ij})$
(7)

where $\bm{\theta}^{c}_{k}$ is the parameter value for cluster $c$ at step
$k$. In terms of the notation used in expression 6, we have the equivalence
$\bm{\theta}_{\tau_{i}}(\bm{\omega})=\bm{\theta}_{c_{i},K}$, which depends on
the initial condition $\bm{\omega}$. The full procedure is described in
Algorithm 1

We consider two versions of the algorithm, depending on how we obtain the tree
structure similar to Srivastava and Salakhutdinov (2013):

  * •

Fixed TreeMAML. The tree is fixed by the knowledge of the tree structure of
tasks when this structure is available. In that case, the values of $C_{k}$
are determined by such tree.

  * •

Learned TreeMAML. The tree is unknown _a priori_ and is learned using a
hierarchical clustering algorithm. In that case, the values of $C_{k}$ are
determined at each step by the clustering algorithm.

In the latter case, we cluster tasks based on the gradients of each task loss,
consistent with recent work Achille et al. (2019). After each step $k$ at
cluster $c_{i}$, the clustering algorithm takes as input the gradient vectors
of the children tasks $i$

$\mathbf{g}_{ik}=\frac{1}{n_{t}}\sum_{j=1}^{n_{t}}\nabla\mathcal{L}(\bm{\theta}_{c_{i},k};\mathcal{D}_{ij})$
(8)

and these gradients are further allocated into clusters according to their
similarity. The clustering algorithm is described in subsection 3.2.

Similar to MAML, adaptation to a new task is performed by computing
$\bm{\theta}^{(i)}(\bm{\omega})$ on a batch of data of the target task. In
order to exploit task relationships, we first reconstruct the tree structure
by using a batch of training data and then we introduce the new task.

Algorithm 1 TreeMAML

0: distribution over tasks $p(\tau)$; distribution over data for each task
$p(\mathcal{D}|\tau)$;

0: number of inner steps $K$; number of training tasks $m$; learning rates
$\alpha,\beta$;

0: number of clusters $C_{k}$ for each step $k$; loss function
$\mathcal{L}_{\tau}(\bm{\omega},\mathcal{D})$ for each task

randomly initialize $\bm{\omega}$

while not done do

sample batch of $i=1:m$ tasks $\\{\tau_{i}\\}\sim p(\tau)$

for all tasks $i=1:m$ initialize a single cluster $c_{i}=1$

initialize $\bm{\theta}_{1,0}=\bm{\omega}$

for steps $k=1:K$ do

for tasks $i=1:m$ do

sample batch of $j=1:n_{v}$ data points $\\{\mathcal{D}_{ij}\\}\sim
p(\mathcal{D}|\tau_{i})$

evaluate gradient
$\mathbf{g}_{ik}=\frac{1}{n_{t}}\sum_{j=1}^{n_{t}}\nabla\mathcal{L}_{\tau_{i}}(\bm{\theta}_{c_{i},k-1};\mathcal{D}_{ij})$

end for

regroup tasks into $C_{k}$ clusters $\mathcal{T}_{c}=\\{i:c_{i}=c\\}$

according to similarity of $\\{\mathbf{g}_{ik}\\}$ and parent clusters
$\\{p_{c}\\}$

update
$\theta_{c,k}=\theta_{p_{c},k-1}-\frac{\alpha}{|\mathcal{T}_{c}|}\sum_{i\in\mathcal{T}_{c}}\mathbf{g}_{ik}$
for all clusters $c=1:C_{k}$

end for

update
$\bm{\omega}\leftarrow\bm{\omega}-\beta\frac{1}{mn_{v}}\sum_{i=1}^{m}\sum_{j=1}^{n_{v}}\nabla_{\bm{\omega}}\mathcal{L}_{\tau_{i}}\left(\bm{\theta}_{c_{i},K}(\bm{\omega});\mathcal{D}_{ij}\right)$

end while

###  3.2 Clustering Algorithm

We employ a hierarchical clustering algorithm to cluster the gradients of our
model parameters in the learned TreeMAML case. We specifically opt for an
online clustering algorithm to maximise computational efficiency at test time
and scalability. When a new task is evaluated, we reuse the tree structure
generated for a training batch and add the new task. This process saves us
from computing a new task hierarchy from scratch for every new task. Moreover,
with offline hierarchical clustering, all the data needs to be available to
the clustering algorithm simultaneously, which becomes a problem when dealing
with larger batch sizes. Therefore online clustering favours scalability.

We follow the online top-down (OTD) approach set out by Menon et al. (2019)
and adapt this to approximate non-binary tree structures. Our clustering
algorithm is shown in Algorithm 2. Specifically, we introduce two
modifications to the original OTD algorithm:

  * •

_Maximum Tree Depth Parameter $D$_: This is equivalent to the number of inner
steps to take in the TreeMAML since the tree is a representation of the inner
loop where each layer in the tree represents a single inner step.

  * •

_Non-binary Tree Approximation_ : We introduce a hyperparameter $\xi$ which
represents how far the similarity of a new task needs to be to the average
cluster similarity in order to be considered a child of that same cluster.
This is not an absolute value of distance, but it is a multiplicative factor
of the standard deviation of the intracluster similarities. Introducing this
factor allows clusters at any level to have more than two children.

Algorithm 2 Online top down (OTD) - Non-binary

0: origin cluster node $C$ with a given set of children
$A=\\{x_{1},x_{2},..x_{N}\\}$

0: new task $x$; maximum depth allowed $D$; similarity metric, $\omega()$

0: standard deviation multiplicative hyperparameter $\xi$;

if $|A|=0$ then

new task becomes a new child $A$ = $\\{x\\}$

else if $|A|=1$ then

add new task to set of children $A\leftarrow A\cup\\{x\\}$

else if $\omega(A\cup\\{x\\})>\omega(A)$ then

identify most similar child
$x_{*}=\operatorname*{arg\,min}_{x_{i}}(\omega(\\{x_{i},x\\}))$

if reached maximum depth $\mathrm{C}_{\mathrm{depth}}+1=D$ then

add new task to set of children $A\leftarrow A\cup\\{x\\}$

else

recursively perform OTD to create new node $C^{\prime}=\textrm{OTD}(x_{*},x)$

add new node to set of children
$A\leftarrow(A\setminus\\{x_{*}\\})\cup{C^{\prime}}$

end if

else if $\omega(A\cup\\{x\\})<\omega(A)-\xi\sigma_{T}$ then

current node and new task become children to new cluster
$A\leftarrow\\{C,x\\}$

else

add new task to set of children $A\leftarrow A\cup\\{x\\}$

end if

##  4 Synthetic experiment

In this section, we introduce a toy experiment illustrates the behaviour of
fixed and learnt TreeMAML in a simple scenario. We consider a multidimensional
linear regression problem $y=\sum_{i=1}^{64}P_{i}x_{i}+\eta$ where the tasks
are randomly sampled from a set of 4 defined clusters of multidimensional
parameters $P$. These clusters of multi-dimensional parameters are selected to
simulate hierarchically structured data. $\eta$ is randomly generated Gaussian
noise. Even in this case, the parameter clusters are arranged hierarchically
such that $C_{1}=2$, $C_{2}=4$.

The data points for the tasks are sampled uniformly $x_{i}\sim U[-5.0,5.0]$
for all training and testing tasks where. The models are then trained and
tested on a set of tasks with $K=4,8,16,32,64$ and $128$ data points.

###  4.1 Fixed TreeMAML

In these experiments, we assume knowledge about the structure of the
underlying tasks and we use this to aggregate the gradients.

Figure 2:  Results of the multidimensional (N=64) linear regression task for
Fixed TreeMAML, MAML and baseline for varying number of task data points
$K=4,8,16,32,64$ and $128$.

TreeMAML outperforms MAML, especially when the number of tasks data points K
is low.

###  4.2 Learned TreeMAML

In this section, we assume no prior knowledge of the underlying structure of
the data. Therefore, the data hierarchy is learnt per-batch using the modified
OTD algorithm described in section 3.2. In the clustering algorithm, we set
the maximum depth to 2 and we use the cosine similarity metric. For this
setting, the TreeMAML algorithm will perform three inner steps, where the last
one is task-specific. Therefore, in order to make a fair comparison, in this
experiment MAML is also set to perform three inner steps.

Table 1 shows that TreeMAML outperforms the Baseline and MAML across all
numbers of data points. What is more, learned TreeMAML performs better than
the fixed tree for a larger number of data points. This is an expected effect
since, as the number of data points increases, the gradients used to cluster
the tasks will be less affected by the noise and become more accurate, leading
to better clustering.

Model | K=5 | k=10 | k=20
---|---|---|---
Baseline | $1.293\pm 0.074$ | $1.055\pm 0.047$ | $1.139\pm 0.061$
MAML | $1.025\pm 0.068$ | $0.950\pm 0.048$ | $0.785\pm 0.028$
Fixed TreeMAML (ours) | $0.621\pm 0.038$ | $0.535\pm 0.024$ | $0.483\pm 0.016$
Learned TreeMAML (ours) | $0.758\pm 0.047$ | $0.510\pm 0.024$ | $0.495\pm 0.018$
Table 1:  Loss (MSE) $\pm 95\%$ confidence intervals on multidimensional
linear regression task, averaged over 400 meta-testing tasks. The results are
presented for varying numbers of K data points and a noise level of 0.01

##  5 Cross-Lingual NLI

Languages can be embraced in a forest of phylogenetic trees Dunn et al.
(2011), for example, the Indo-European and generic Austric-Aisatic trees
(Figure 3). TreeMAML exploits this hierarchical structure to generalize the
performance of models across languages, including under-resourced languages,
useing all the available languages in the tree.

We adapt a high-resource language model, Multi-BERT Devlin et al. (2018), to a
NLI task. In particular, we consider the problem of Few-Shot NLI using the
XNLI data set Conneau et al. (2018).

This dataset consists of a crowd-sourced collection of 5,000 test and 2,500
dev sentence-label pairs from the MultiNLI corpus. They are annotated with
textual entailment and translated into 15 languages: English (en), French
(fr), Spanish (es), German (de), Greek(el), Bulgarian (bg), Russian (ru),
Turkish, Arabic, Vietnamese (vi), Thai (th), Chinese (zh), Hindi (hi), Swahili
and Urdu (ur). Twelve of these languages are part of the same phylogenetic
tree, and we focus our study on those languages (see Figure 3). We separately
set as target language each language of the tree and we used the eleven
remaining languages as auxiliary languages for meta-training.

Each sentence has also an associated topic, or genre, among a collection of
$10$ possible genres (Face-To-Face, Telephone, Government, 9/11, Letters,
Oxford University Press (OUP), Slate, Verbatim, and Government, Fiction). We
define each combination of a language and a genre as a task, and we consider
the problem of few-shot meta-learning using three shots for each task during
meta-training. We add the new target task to the original distribution of
tasks, we apply the TreeMAML algorithm and evaluate the model on the target
language test set.

Figure 3:  Simplified version of the phylogenetic language tree. The tree
include 12 of the 15 languages of XNLI data set and have depth three (Three
levels of hierarchy)

We use the TreeMAML algorithm to fine-tune the top layer of Multi-BERT (layer
12), with four inner steps. We compare our results with MAML, using the same
number of inner steps, with the baseline Multi-BERT, and with XMAML
Nooralahzadeh et al. (2020). An important difference of our approach is that,
while XMAML uses only two auxiliary languages to fine tune Multi-BERT to a
target language, we use all other languages as auxiliary languages.

###  5.1 Fixed TreeMAML

In the case of fixed TreeMAML, we use the phylogenetic tree in Figure 3. Fine-
tuning of Multi-BERT for the target language benefits not only from proximal
(auxiliary) languages, but also from all other languages in the tree that
share roots with the target language. For example, if the target language is
German, the fine-tuning in fixed TreeMAML would use for the first step of the
gradient update all the remaining languages. In the second step, the auxiliary
languages would be all the training languages of the Indo-European branch. The
third and last steps uses only English. The accuracy of TreeMAML is
consistently higher than the one of the Baseline or MAML and an average of
$\sim 3\%$ better than the one achieved by XMAML, see Table 2.

Note that we used a relatively simple version of the phylogenetic tree. A more
detailed version could be used for testing under-resourced languages, or to
emphasize the dependencies inside the tree. For example, a Bavarian testing
data set would fall inside the German branch, or we could add depth to the
tree by adding Germanic sub-branches, such as high-german, anglo-frisian and
low-franconian.

###  5.2 Learned TreeMAML

While fixed TreeMAML uses previous knowledge to construct the tree, learned
TreeMAML allows learning the relation between languages and genres,
potentially reflecting a priory unknown relationships in the XNLI corpus, but
also potentially fitting some noise. Note that learned TreeMAML has one
additional parameter, the maximum tree depth, as explained in 3.2.

The relationships between languages and genres is learned at each step of
gradient descent, for each batch of data. Therefore, the tree for one batch
can be different from the tree for the next batch. This difference is due to
the fact that the clustering algorithm only cares about the similarity of the
gradients, and this similarity does not need to be always the same between two
languages. It may depend on the particular words used in the sentences, or in
the tasks genres. For example, for a particular batch, some sentences from the
same genre in English and French could have closer gradients than other
sentences in Germanic languages with a different genre.

The clustering process happens at both training and testing time, which means
that learned TreeMAML may improve the accuracy by improving the training, but
also by producing the best hierarchy for the target task at testing time. This
may be particularly useful for under-resourced languages where non-obvious
dependencies between the task in the target languages and the tasks in other
languages can be exploited to improve the test accuracy.

en fr es de el bg ru vi th zh hi ur avg two languages Nooralahzadeh et al.
(2020) Multi-BERT (Baseline) $81.94$ $75.39$ $75.79$ $73.25$ $69.54$ $71.60$
$70.84$ $73.23$ $61.18$ $73.93$ $64.37$ $63.71$ $71.23$ XMAML $82.71$ $75.97$
$76.51$ $74.07$ $70.66$ $72.77$ $72.12$ $73.87$ $62.5$ $74.85$ $65.75$ $64.59$
$72.20$ all languages (ours) Multi-BERT (Baseline) $83.56$ $76.22$ $76.89$
$73.11$ $72.89$ $72.89$ $71.33$ $74.67$ $57.56$ $74.89$ $63.11$ $63.33$
$71.70$ MAML $83.11$ $78.22$ $77.11$ $73.56$ $69.33$ $71.78$ $71.33$ $74.22$
$57.33$ $75.11$ $63.33$ $63.78$ $71.52$ Fixed TreeMAML $84.67$ $79.78$ $78.22$
$76.89$ $72.00$ $74.22$ $73.33$ $74.44$ $59.56$ $79.11$ $66.00$ $66.89$
$73.76$ Learned TreeMAML $84.22$ $77.33$ $79.78$ $78.00$ $71.56$ $73.78$
$74.00$ $74.89$ $59.78$ $76.44$ $65.11$ $65.56$ $73.37$

Table 2: The top part of the table shows the results of training with two
auxiliary languages Nooralahzadeh et al. (2020). The lower part of the table
shows the performance when using all languages (except the target) as
auxiliary languages. The difference in the amount of data used for training
may account for a part of the difference in performance between XMAML and
TreeMAML, and may also explain why our Baseline outperforms XMAML for some
languages. The results are reported for each of our experiment by averaging
the performance over three different runs. The standard deviation is for all
our experiments below 1%.

As shown in Table 2, fixed/learned TreeMAML outperforms other methods in
almost all languages. These results show that using the languages hierarchical
structure helps achieving better cross-lingual transfer and higher accuracy in
the XNLI task.

In the case of Greek (el), TreeMAML outperforms XMAML, but the baseline Multi-
BERT obtains a slightly higher accuracy. This result could be due to the
simplified structure of the tree that we use, which does not adequately
reflect the actual distance in between languages from the Indo-European
family. Besides Greek, Thai (th) is the only language for which TreeMAML does
not get higher accuracy. This is mainly due to oversimplified tree used. We
used a generic ”Asian” language tree, but Chinese, Vietnamese and Thai belong
to three separate language families.

Learned TreeMAML performs very similar to fixed TreeMAML in most experiments,
achieving higher values for some of the languages. We believe that the
difference depends on how well our clustering algorithm performs in each case.
For some languages, learned TreeMAML is just learning the same tree structure
that we use in fixed TreeMAML, and both algorithms produce almost the same
results. In other cases, the clustering algorithm assigns tasks to the wrong
branch, making learned TreeMAML perform worse. For some other languages,
learned TreeMAML performs better, possibly because it finds other useful
relationships, for example tasks belonging to the same genre in different
languages.

##  6 Discussion and Conclusion

This paper presents a method to exploit the data hierarchy in the meta-
learning framework, TreeMAML. This algorithm can use a priory knowledge of the
data set (fixed TreeMAML), or learn the hierarchical structure using our
modification of the OTD clustering algorithm (learned TreeMAML).

To illustrate the performance and the benefits of this approach, we applied
the TreeMAML algorithm to a multidimensional linear regression problem, where
the parameters of the tasks are sampled from a distribution that can be
described by hierarchical clustering. Both, fixed and learned TreeMAML
outperform the baseline and MAML by a significant margin on this synthetic
task, halving the MSE of the other methods ( Table 1). In the case of few shot
learning when the number of points of the test task is small, the learned
algorithm outperform the fixed tree by exploding similitude between tasks not
described by the fixed tree.

Since languages follow a hierarchical phylogenetic tree, we hypothesized that
we could use TreeMAML to meta-train models for cross-lingual understanding. We
applied TreeMAML to the cross-lingual XNLI problem and show an improvement in
accuracy $\sim 3\%$ with respect to the state of the art obtained by XMAML
Nooralahzadeh et al. (2020) (Table 2). The improvement with respect to XMAML
suggests that using all available languages results in increased performance.
Furthermore, the improvement with respect to MAML suggests that using the tree
structure of those languages also improves performance.

How much each auxiliary language contributes to the target language’s
performance may depend on its relative position in the language tree. These
results are especially encouraging for meta-training of cross-lingual
understanding tasks for under-resourced languages. Future work may include an
improved algorithm that takes into account not only the position of a language
in the tree, but also the distances between languages in a branch by, for
example, introducing weighted averaging of the gradients.

In our NLI experiments, learned TreeMAML is in most cases as good or even
better than fixed TreeMAML. One possible explanation is that clustering learns
the tree for each batch of data at each gradient step, which allows it to pick
up NLI-relevant similarities that are not described by a phylogenetic tree, as
the genre of the task in the XNLI data set, structural similarities and
lexical similarity, which can be the result of language contact, as for
example lexical borrowings. This may help with cross-lingual understanding
tasks for uncommon languages for which the exact position in the tree may be
unclear or not enough data may be available.

As discussed in section 5.2, the lack of improvement in accuracy in the Greek
language and the low performance in Thai can be rooted in the experimental
design’s naive assumptions about: which languages to include in the
experiment, and the correctness of the language tree. The results in which
TreeMAMl performs worse than the other algorithms speak in favour of the
robustness of this algorithm to properly learn cross-lingual relationships and
exploit them to perform natural language understanding tasks. Therefore, the
use of learned TreeMAML could help with the internationalization of the NLP
models.

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