# Self-training through Classifier Disagreement for Cross-Domain Opinion
Target Extraction

Kai Sun<EMAIL_ADDRESS>Beijing Advanced Innovation Center for Big
Data and Brain Computing, School of Computer Science and Engineering, Beihang
UniversityBeijingChina ,  Richong Zhang<EMAIL_ADDRESS>Beijing
Advanced Innovation Center for Big Data and Brain Computing, School of
Computer Science and Engineering, Beihang UniversityBeijingChina ,  Samuel
Mensah<EMAIL_ADDRESS>Department of Computer Science,
University of Sheffield,SheffieldUnited Kingdom ,  Nikolaos Aletras
<EMAIL_ADDRESS>Department of Computer Science,
University of Sheffield,SheffieldUnited Kingdom ,  Yongyi Mao
<EMAIL_ADDRESS>School of Electrical Engineering and Computer Science,
University of OttawaOttawaCanada and  Xudong Liu<EMAIL_ADDRESS>Beijing Advanced Innovation Center for Big Data and Brain Computing, School of
Computer Science and Engineering, Beihang UniversityBeijingChina

(2023; 20 February 2007; 12 March 2009; 5 June 2009)

###### Abstract.

Opinion target extraction (OTE) or aspect extraction (AE) is a fundamental
task in opinion mining that aims to extract the targets (or aspects) on which
opinions have been expressed. Recent work focus on cross-domain OTE, which is
typically encountered in real-world scenarios, where the testing and training
distributions differ. Most methods use domain adversarial neural networks that
aim to reduce the domain gap between the labelled source and unlabelled target
domains to improve target domain performance. However, this approach only
aligns feature distributions and does not account for class-wise feature
alignment, leading to suboptimal results. Semi-supervised learning (SSL) has
been explored as a solution, but is limited by the quality of pseudo-labels
generated by the model. Inspired by the theoretical foundations in domain
adaptation (Ben-David et al., 2010), we propose a new SSL approach that opts
for selecting target samples whose model output from a domain-specific teacher
and student network disagree on the unlabelled target data, in an effort to
boost the target domain performance. Extensive experiments on benchmark cross-
domain OTE datasets show that this approach is effective and performs
consistently well in settings with large domain shifts.

domain adaptation, self-training, opinion mining

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
ACM Web Conference 2023; May 1–5, 2023; Austin, TX, USA††booktitle:
Proceedings of the ACM Web Conference 2023 (WWW ’23), May 1–5, 2023, Austin,
TX, USA††price: 15.00††doi: 10.1145/3543507.3583325††isbn:
978-1-4503-9416-1/23/04††ccs: Computing methodologies Information
extraction††ccs: Computing methodologies Semi-supervised learning settings

##  1\. Introduction

The growth of e-commerce websites has allowed consumers to directly interact
with products, leading to an increase in user-generated content. In
particular, reviews of products generated by users has grown at an
astronomical rate with the increasingly accessibility and affordability of the
internet. These reviews, which are often expressed as text, contain sentiment
information or opinion words expressed on different aspects of products
(referred to as opinion targets or aspect terms). As a result, opinion words
along with its corresponding aspect terms have become an important resource to
improve recommender systems for Web resource discovery. A typical example is
the recommendation of a book in Goodreads (Thelwall and Kousha, 2017) based on
the opinions expressed on a specific section in the book.

This phenomenon has led to the increased research in opinion mining (Liu,
2015). This paper focuses on opinion target extraction (OTE) (or aspect
extraction (AE)), a fundamental step in opinion mining. AE aims to extract
from opinionated sentences the aspects on which opinions have been expressed.
Traditional approaches (Jakob and Gurevych, 2010; Yang and Eisenstein, 2015)
utilize hand-crafted features, which heavily rely on feature extraction. With
the advances in deep learning, recent approaches (Ma et al., 2019; Li et al.,
2020; Wei et al., 2020) are based on neural networks that are trained in a
supervised manner. However, as with any other supervised learning method,
these approaches perform poorly when there is a change in domain upon
deployment. Cross-domain OTE (Ding et al., 2017) has emerged as a solution by
using unsupervised domain adaptation (UDA) techniques (Bollegala et al., 2013;
Zhou et al., 2016; He et al., 2018) to reduce the domain shift between a
labelled source and unlabelled target domain.

One typical line of work aims to reduce domain shifts via domain adversarial
neural networks (DANN) (Ganin et al., 2016). Given a labelled source and
unlabelled target domain data, DANNs attempt to learn representations that are
discriminative on the source domain and invariant to the domain distribution.
However, DANNs align the feature distributions of the source and target data
inputs (i.e., aligning the marginal distribution), neglecting the feature
alignment at the class-level (Tan et al., 2019). As a consequence, the
resulting target features are non-discriminative with respect to the class
labels, which consequently leads to suboptimal target domain performance.

Semi-supervised learning (SSL) (Chapelle et al., 2009) has been explored to
learn target discriminative features by generating pseudo-labels from the
unlabelled target data. While SSL approaches have been heavily employed to
boost domain adaptation in vision tasks (Tarvainen and Valpola, 2017; Ke et
al., 2019; Xie et al., 2020), it has been lightly touched in cross-domain OTE
(Yu et al., 2021; Zhou et al., 2021). The state-of-the-art method Adaptive
Hybrid Framework (AHF) (Zhou et al., 2021) adapts a mean teacher (i.e.,
teacher and student networks) (Tarvainen and Valpola, 2017) into the task. The
teacher is modelled as a feedforward network while the student is a DANN
(i.e., developed by augmenting the feedforward network with a discriminator).
Here, knowledge on the target’s output of the teacher-student networks is
shared among the networks to learn the target-discriminative features.
Although AHF demonstrates the importance of SSL, the fundamental weakness of
the mean-teacher cannot be ignored. Specifically, Ke et al. (2019) provided
theoretical and empirical proof to show that the weights of the teacher
quickly converges to that of the student as training progresses, which
consequently leads to a performance bottleneck.

---
Figure 1. Illustrative example of source and target distributions induced by a
teacher and student network (Best viewed in color). Target samples that change
class due to adversarial learning by the student network are selected to self-
train the student.

These findings motivate us to decouple the student-teacher networks and
optimize the networks through independent paths to prevent the networks from
collapsing into each other (Ke et al., 2019). We propose a novel SSL approach,
which performs Self-training through Classifier Disagreement (SCD), to
effectively explore the outputs of the student and teacher networks on the
unlabelled target domain. SCD is inspired by the theory of domain adaptation
(Ben-David et al., 2010), which allows us to detect high-quality pseudo-
labelled target samples in the student feature space to self-train the student
for cross-domain OTE. As demonstrated in Fig. 1, SCD achieves this by
comparing the two target distributions induced separately by the student and
teacher networks. These high-quality pseudo-labelled target samples are those
that disagree (i.e., discrepancy in target predictions) with their
correspondence in the teacher feature space. We perform extensive experiments
and find that SCD not only achieves impressive performance but also performs
consistently well in large domain shifts on cross-domain OTE.

Our contribution can be summarized as follows:

  * •

We develop a novel SSL approach for cross-domain OTE, referred to as Self-
training through Classifier Disagreement (SCD) which leverages high-quality
pseudo-labelled target samples in the student feature space to improve target
performance in cross-domain OTE.

  * •

We demonstrate that SCD is favourable in large domain divergence - a key
direction in the domain adaptation research.

  * •

We perform extensive experiments and show that SCD achieves state-of-the-art
results in nine out of ten transfer pairs for the cross-domain OTE task.

##  2\. Related Work

There is a growing literature on OTE (Li et al., 2018; Xu et al., 2018; Li et
al., 2019a; Ma et al., 2019; Li et al., 2020; Wei et al., 2020) but they
mostly focus on single domain learning. However, in real-world scenarios, the
training distribution used by a classifier may differ from the test
distribution, which is a big challenge for single domain learning methods.

Cross-domain learning has been explored for the OTE task. Traditional methods
use hand-crafted domain-independent features and use Conditional Random Fields
(CRFs) (Jakob and Gurevych, 2010; Li et al., 2012; Chernyshevich, 2014). While
hand-crafted features are useful, they are manually engineered and require
human experts, making them time-consuming and expensive to obtain. So far,
some neural models have been proposed for cross-domain OTE (Ding et al., 2017;
Wang and Pan, 2018; Li et al., 2019b; Gong et al., 2020; Chen and Qian, 2021;
Yu et al., 2021; Zhou et al., 2021). The common paradigm in prior work is to
reduce the domain shift between the source and target domains. Among recent
work, Ding et al. (2017) proposed a hierarchical network trained with joint
training (Hier-Joint). This method uses domain-independent rules to generate
auxiliary labels and use a recurrent neural network to learn a domain-
invariant hidden representation for each word. However, the manually defined
rules have limited coverage. A similar method, namely, Recursive Neural
Structural Correspondence Network (RNSCN) (Wang and Pan, 2018) introduces an
opinion word extraction as an auxiliary task based on a critical assumption
that associative patterns exist between aspect terms and opinion words
irrespective of the domain. They use syntactic relations in dependency trees
as the pivot to bridge the domain gap for cross-domain OTE. However, the
external linguistic resources used are derived from traditional feature-
engineered NLP systems which may propagate errors. More recent methods,
including the Aspect Boundary Selective Adversarial Learning model (AD-SAL)
(Li et al., 2019b) uses an adversarial network with attention mechanisms to
learn domain-invariant features. Gong et al. (2020) proposed BERTE-UDA to
integrate BERT fine-tuned on domain information for the task. Chen and Qian
(2021) proposed a Semantic Bridge network (SemBridge) which constructs
syntactic and semantic bridges to transfer common knowledge across domains.

While significant progress has been made, the majority of the proposed models
neglect the feature distribution alignment at the class-level. Hence, their
performance cannot be guaranteed because they do not learn target
discriminative features. Recently, a Cross-Domain Review Generation model
based on BERT (BERTE-CDRG) (Yu et al., 2021) generated target domain data with
fine-grained annotations aiming to learn the target discriminative features.
Perhaps, AHF (Zhou et al., 2021) is the first to use SSL in the task. AHF
adapts a mean teacher in which the teacher and student networks are found to
be tightly coupled during training, leading to a performance bottleneck (Ke et
al., 2019). Elsewhere, researchers have delicately designed SSL approaches
that allow individual models to iteratively learn from each other, thus,
preventing these models from collapsing into each other (Qiao et al., 2018; Ke
et al., 2019; Chen et al., 2021). Such approaches have demonstrated
substantial improvements over the mean-teacher.

##  3\. Preliminaries

Our method is inspired by the theory of domain adaptation proposed by Ben-
David et al. (2010), which provides an upper bound on the target error in
terms of the source error and the domain divergence. Suppose $h\in\mathcal{H}$
is a hypothesis, Ben-David et al. (2010) theorized that the target error
$\epsilon_{\mathcal{T}}(h)$ (which can also be viewed as the target
performance) is bounded by the source error $\epsilon_{\mathcal{S}}(h)$ (i.e.,
the source performance) and the symmetric difference hypothesis divergence
$\mathcal{H}\Delta\mathcal{H}$-divergence between the source $\mathcal{S}$ and
target $\mathcal{T}$ distributions, denoted as
$d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})$ (i.e., a measure
of the domain shift). Formally,

(1) $\displaystyle\forall
h\in\mathcal{H},\epsilon_{\mathcal{T}}(h)\leq\epsilon_{\mathcal{S}}(h)+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})+\beta$

where $\beta$ is the optimal joint error on the source and target domains
which should be small for domain adaptation. Note, $\beta$ is a constant which
is independent of $h$. To obtain a better estimate of
$\epsilon_{\mathcal{T}}(h)$, a learner can either reduce the source error
$\epsilon_{\mathcal{S}}(h)$ or/and the divergence
$d_{\mathcal{H}\Delta\mathcal{H}}(\mathcal{S},\mathcal{T})$, which can be
estimated from finite samples of the source and target domains (Ben-David et
al., 2010).

##  4\. Problem Statement

The OTE task is formulated as a sequence labeling problem. Given the $j$-th
input sentence $\mathbf{x}_{j}=\\{x_{ij}\\}_{i=1}^{n}$ with $n$ words, the
word $x_{ij}$ is represented as a feature vector. The goal is to predict the
label sequence $\mathbf{y}_{j}=\\{y_{ij}\\}_{i=1}^{n}$, with
$y_{ij}\in\mathcal{Y}=\\{\rm B,I,O\\}$, denoting the Beginning, Inside and
Outside of an opinion target or aspect term.

In this paper, we focus on the cross-domain setting which is typically tackled
through unsupervised domain adaptation (UDA). Particularly, UDA aims to
transfer knowledge from a labelled source domain to an unlabelled target
domain, whose data distribution has a considerable shift from that of the
source domain. Formally, suppose a labelled source domain dataset with
$N_{\mathcal{S}}$ sentence and label pairs
$D_{\mathcal{S}}=\\{(\mathbf{x}_{j}^{\mathcal{S}},\mathbf{y}_{j}^{\mathcal{S}})\\}_{j=1}^{N_{\mathcal{S}}}$,
and an unlabeled dataset in a target domain with $N_{\mathcal{T}}$ unlabelled
sentences
$D_{\mathcal{T}}=\\{(\mathbf{x}_{j}^{\mathcal{T}})\\}_{j=1}^{N_{\mathcal{T}}}$.
Our goal is to predict labels of testing samples in the target domain using a
model trained on $D_{\mathcal{S}}\cup D_{\mathcal{T}}$. 111Hereinafter,
subscripts or superscripts are omitted for clarity, and the term “aspect” will
be used instead of “opinion target” to avoid confusion with the target domain.

##  5\. Methodology

Our method is based on a teacher-student network structure. Teacher $A$ learns
on the source data $D_{\mathcal{S}}$; and Student $B$ learns on both the
source $D_{\mathcal{S}}$ and target domain data $D_{\mathcal{T}}$. Both
trained networks generate pseudo-labelled target samples on the unlabelled
target domain, which are then compared to detect high quality pseudo-labelled
target samples to self-train the student for cross-domain OTE.

###  5.1. Teacher Network

The teacher network $A=\\{A_{e},A_{l}\\}$ is a neural network, consisting of a
feature encoder $A_{e}$ and a label classifier $A_{l}$. In our work, $A_{e}$
is modelled using a BiLSTM (Hochreiter and Schmidhuber, 1997) or BERT (Devlin
et al., 2019) since they are both widely used approaches for sequence
labelling problems. $A_{l}$ on the other hand is modelled using a softmax
function. Although the CRF (Lafferty et al., 2001) is a typical choice to
model the label classifier for sequence labelling problems, the softmax offers
comparable performance in cross-domain OTE (Li et al., 2019b). Hence, given
the sentence $\mathbf{x}_{j}=\\{x_{ij}\\}_{i=1}^{n}$, $A_{e}$ extracts the
context features $\mathbf{f}^{A_{e}}_{j}=\\{f^{A_{e}}_{ij}\\}_{i=1}^{n}$. Now,
for each word-level feature $f^{A_{e}}_{ij}$, the label classifier $A_{l}$ is
applied to output the prediction probability $P(\hat{y}^{A_{l}}_{ij})$ over
the tag set $\mathcal{Y}$. As the teacher is trained over the source data
only, the classification loss by the teacher network is given by:

(2)
$\mathcal{L}^{A}_{y}=\frac{1}{N_{\mathcal{S}}}\sum_{j=1}^{N_{\mathcal{S}}}\sum_{i=1}^{n}\ell(P(\hat{y}^{A_{l}}_{ij}),y_{ij})$

where $P(\hat{y}_{ij}^{A_{l}})$ is the probability prediction for the word
$x_{ij}\in\mathbf{x}_{j}^{\mathcal{S}}$ and
$y_{ij}\in\mathbf{y}_{j}^{\mathcal{S}}$ is the ground-truth of $x_{ij}$.
$\ell$ is the cross-entropy loss function.

Now suppose $\mathbf{F}^{{A}_{e}}_{\mathcal{S}}$ and
$\mathbf{F}^{{A}_{e}}_{\mathcal{T}}$ are fixed representations of the
respective source and target domain data produced by the trained teacher
$A_{e}$. The upper bound on the target error $\epsilon_{\mathcal{T}}({A}_{l})$
of the label classifier $A_{l}$ can be expressed as:

(3)
$\epsilon_{\mathcal{T}}({A}_{l})\leq\epsilon_{\mathcal{S}}({A}_{l})+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{A}_{e}}_{\mathcal{S}},\mathbf{F}^{{A}_{e}}_{\mathcal{T}})+\beta$

It is easy to see that the teacher network simply reduces the source error
$\epsilon_{\mathcal{S}}({A}_{l})$ by (2) while the domain shift
$d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{A}_{e}}_{\mathcal{S}},\mathbf{F}^{{A}_{e}}_{\mathcal{T}})$
remains large since the network does not have an appropriate component to
reduce the domain shift. This leads to a suboptimal estimate for the bound of
the target errors $\epsilon_{\mathcal{T}}({A}_{l})$.

###  5.2. Student Network

As we have seen in the previous section, the teacher applies domain-specific
knowledge (i.e., the source domain) for inference, which may underperform on
the target domain due to difference in the data distribution. Ideally, the
network should have the ability to perform in different domains. We introduce
the student network as a solution.

The student network is analogous to a student who learns several subjects
simultaneously in order to perform well in those subjects. This is different
from teachers who are normally experts in a single subject. This implies that
the student network not only desires to be as excellent as the domain-specific
teacher on the source data but also aims to perform well on the target data.
To this end, the student network is developed by augmenting a teacher network
with a discriminator (or domain classifier), following DANN (Ganin et al.,
2016). Accordingly, the student network $B=\\{B_{e},B_{l},B_{d}\\}$ consists
of a feature encoder $B_{e}$; label classifier $B_{l}$; and domain classifier
$B_{d}$, which determines if the sample comes from the source or target
domain. $B_{e}$ extracts the context features $\mathbf{f}^{B_{e}}_{j}$ from
the sentence $\mathbf{x}_{j}\in D_{\mathcal{S}}\cup D_{\mathcal{T}}$ and feeds
to $B_{l}$ to learn discriminative features on the source domain, following a
similar classification loss with Eqn. (2). Formally, the classification loss
is defined as:

(4)
$\mathcal{L}^{B}_{y}=\frac{1}{N_{\mathcal{S}}}\sum_{j=1}^{N_{\mathcal{S}}}\sum_{i=1}^{n}\ell(P(\hat{y}^{B_{l}}_{ij}),y_{ij})$

where $P(\hat{y}^{B_{l}}_{ij})$ is the probability prediction for the word
$x_{ij}\in\mathbf{x}_{j}^{\mathcal{S}}$ and
$y_{ij}\in\mathbf{y}_{j}^{\mathcal{S}}$ is the ground-truth. At the same time,
$\mathbf{f}^{B_{e}}_{j}$ is fed to a domain classifier $B_{d}$ to learn
domain-invariant features through a gradient reversal layer (GRL) (Ganin et
al., 2016). Formally, the GRL $R_{\lambda}(\cdot)$ acts as an identity
function in the forward pass, i.e.,
$R_{\lambda}(\mathbf{f}^{B_{e}}_{j})=\mathbf{f}^{B_{e}}_{j}$, and
backpropagates the negation of the gradient in the backward pass, i.e.,
$\partial
R_{\lambda}(\mathbf{f}^{B_{e}}_{j})/\partial\mathbf{f}^{B_{e}}_{j}=-\lambda
I$. Consequently, $B_{e}$ maximizes the domain classification loss
$\mathcal{L}_{d}^{B}$ through the GRL while $B_{d}$ minimizes
$\mathcal{L}_{d}^{B}$ to make $\mathbf{f}^{B_{e}}_{j}$ domain-invariant. The
domain classification loss $\mathcal{L}_{d}^{B}$ is defined as follows:

(5) $\displaystyle\begin{split}\mathcal{L}_{d}^{B}=\sum_{j=1}^{N}d_{j}{\rm
log}(P(\hat{d}^{B_{d}}_{j}))+(1-d_{j}){\rm
log}(1-P(\hat{d}^{B_{d}}_{j}))\end{split}$

where $d_{j}=1$ indicates that the $j$-th sentence comes from the source
domain, otherwise $d_{j}=0$; $P(\hat{d}^{B_{d}}_{j})$ is the domain
probability prediction of the sentence-level feature $\mathbf{x}_{j}$;
$N=N_{\mathcal{S}}+N_{\mathcal{T}}$.

Suppose $\mathbf{F}^{{B}_{e}}_{\mathcal{S}}$ and
$\mathbf{F}^{{B}_{e}}_{\mathcal{T}}$ are fixed representations of the
respective source and target domain data produced by the trained student
encoder $B_{e}$. The upper bound on the student label classifier $B_{l}$ can
be expressed as:

(6)
$\epsilon_{\mathcal{T}}({B}_{l})\leq\epsilon_{\mathcal{S}}({B}_{l})+\frac{1}{2}d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{B}_{e}}_{\mathcal{S}},\mathbf{F}^{{B}_{e}}_{\mathcal{T}})+\beta$

The source error $\epsilon_{\mathcal{S}}({B}_{l})$ is comparable with
$\epsilon_{\mathcal{S}}({A}_{l})$ since the student and teacher are trained on
the source data using the same network pipeline (comparing (2) and (4), and
also empirically demonstrated in Table 5). But the student network has been
shown to reduce the domain divergence with a theoretical guarantee via the GRL
(Ganin et al., 2016). This means
$d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{B}_{e}}_{\mathcal{S}},\mathbf{F}^{{B}_{e}}_{\mathcal{T}})$
is relatively small, i.e.,
$d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{B}_{e}}_{\mathcal{S}},\mathbf{F}^{{B}_{e}}_{\mathcal{T}})\leq
d_{\mathcal{H}\Delta\mathcal{H}}(\mathbf{F}^{{A}_{e}}_{\mathcal{S}},\mathbf{F}^{{A}_{e}}_{\mathcal{T}})$,
and therefore leads to a better estimate of $\epsilon_{\mathcal{T}}({B}_{l})$.
In other words, the student performs better than the domain-specific teacher
on the target data due to the mitigation of the domain shift.

###  5.3. Self-training through Classifier Disagreement

$D_{\mathcal{T}}$Student Encoder $B_{e}$Student Classifier
$B_{l}$$\bar{\mathcal{L}}^{B}_{y}$Teacher Encoder $A_{e}$Teacher Classifier
$A_{l}$$D_{d}$$D_{a}$Labeled SourceLabeled TargetUnlabel TargetClass 1Class
2$D_{d}$$D_{a}$$D_{\mathcal{S}}$TrainPredictCompare
---
Figure 2. Overview of our SSL Approach. Both Teacher and Student networks have
been earlier trained by Eqn. (2), (4) and (5). The Student network alone is
further self-trained through classifier disagreement on the target domain.
This figure is best viewed in color.

The student network improves target performance by aligning the source and
target data distributions. It just so happens that it simply aligns the data
distribution without considering the alignment at the class-level (Tan et al.,
2019), leading to suboptimal performance. Such a situation occurs due to the
lack of labelled target data to learn target discriminative features. The
fundamental challenge is that we do not have access to labelled target data.

To this end, we introduce a strikingly simple approach to collect high-quality
pseudo-labelled target samples to improve the class-level alignment of the
student network. Fig 2 shows an overview of our approach, which we refer to as
Self-training through Classifier Disagreement (SCD). Suppose the trained
student and teacher networks (i.e., trained by Eqn. (2), (4) and (5)) assign
pseudo-labels to the unlabelled target data. Eqns (3) and (6) indicate that
the increase in target performance by the student can be explained by the
target samples that have shifted toward the domain-invariant feature space
(i.e., the student feature space). Our goal is to self-train the student
network by leveraging the target samples responsible for the performance
improvement in the target domain.

This strategy is only beneficial if the domain shift is large since this will
lead to a large set of high-quality pseudo-labelled target samples. Otherwise,
both networks will have comparable performance on the unlabelled target domain
and the performance gain is minimal. To extend the approach to problems with
close similarity between domains, we split the self-training learning problem
by paying attention to: 1) $D_{d}$, the target samples in the student feature
space that disagree with their counterpart in the teacher feature space; and
2) $D_{a}$, the target samples in the student feature space that agree with
their counterpart in the teacher feature space.

Formally, let us suppose the student and teacher networks are already trained
(i.e., without self-training). As we aim to self-train the Student network, we
can rewrite the classification loss expressed in (4) as
$\mathcal{L}^{B(0)}_{y}$ to represent the initial classification loss of the
Student network. Now, let us suppose the teacher and student networks assign
the pseudo-labels
$\bar{\mathbf{y}}_{j}^{A_{l}}=\\{\bar{y}_{ij}^{A_{l}}\\}_{i=1}^{n}$ and
$\bar{\mathbf{y}}_{j}^{B_{l}}=\\{\bar{y}_{ij}^{B_{l}}\\}_{i=1}^{n}$ for each
sentence $\mathbf{x}^{\mathcal{T}}_{j}\in D_{\mathcal{T}}$, respectively.
Self-training is formulated as training the student network on the set
$D_{\mathcal{S}}\cup D_{d}\cup D_{a}$, where the sets $D_{d}$ and $D_{a}$ are
defined as follows:

(7)
$\displaystyle\begin{split}&D_{d}:=\\{(\mathbf{x}_{j}^{\mathcal{T}},\bar{\mathbf{y}}_{j}^{B_{l}})|\exists
x_{ij}\in\mathbf{x}_{j}^{\mathcal{T}}\,\,{\rm
s.t.}\,\,\bar{y}_{ij}^{{B_{l}}}\neq\bar{y}_{ij}^{{A_{l}}}\\}\\\
&D_{a}:=\\{(\mathbf{x}_{j}^{\mathcal{T}},\bar{\mathbf{y}}_{j}^{B_{l}})|\forall
x_{ij}\in\mathbf{x}_{j}^{\mathcal{T}}\,\,{\rm
s.t.}\,\,\bar{y}_{ij}^{{B_{l}}}=\bar{y}_{ij}^{{A_{l}}}\\}\\\ \end{split}$

Here, $\bar{y}_{ij}^{A_{l}}\in\bar{\mathbf{y}}_{j}^{A_{l}}$ is the teacher
network’s pseudo-label assignment on $x_{ij}\in\mathbf{x}_{j}^{\mathcal{T}}$.
Let $r$ index the self-training round. Then the self-training loss for the
student network at a specific self-training round $r$ can be formulated as
follows:

(8)
$\displaystyle\begin{split}\bar{\mathcal{L}}^{B(r)}_{y}={\mathcal{L}}^{B(r)}_{y}+\frac{1}{|D_{d}^{(r)}|}\sum_{(\mathbf{x}_{j}^{\mathcal{T}},\bar{\mathbf{y}}_{j}^{B_{l}})\in
D_{d}^{(r)}}\sum_{x_{ij}\in\mathbf{x}_{j}^{\mathcal{T}}}\ell(P(\hat{y}^{B_{l}}_{ij}),\bar{y}_{ij}^{B_{l}})\\\
+\eta\frac{1}{|D_{a}^{(r)}|}\sum_{(\mathbf{x}_{j}^{\mathcal{T}},\bar{\mathbf{y}}_{j}^{B_{l}})\in
D_{a}^{(r)}}\sum_{x_{ij}\in\mathbf{x}_{j}^{\mathcal{T}}}\ell(P(\hat{y}^{B_{l}}_{ij}),\bar{y}_{ij}^{B_{l}})\end{split}$

where $r\geq 1$, $\eta\in[0,1]$ is a variable to control the weight of the
loss on $D_{a}^{(r)}$. Since the similarity between source and target domains
can only be measured empirically, $\eta$ is treated as a hyper-parameter to be
tuned. $\eta$ is expected to be large when the source and target domains are
similar, otherwise small. Notice that when $\eta=1$,
$\bar{\mathcal{L}}^{B(r)}_{y}$ becomes a special case of the pseudo-labelling
loss function expressed in Eq. 15 of (Lee et al., 2013) with $\alpha(t)=1$,
which we refer to as a standard pseudo-labelling method.

The total loss function $\mathcal{L}$ for SCD can now be formulated as

(9) $\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{d}^{B}+\mathcal{L}^{B(0)}_{y}+\sum_{r\geq
1}\bar{\mathcal{L}}^{{B}(r)}_{y}$

In each self-training round, $D^{(r)}_{pl}=D_{d}^{(r)}\cup D_{a}^{(r)}$ is
generated using the current trained student network. The self-training stops
when $D^{(r)}_{pl}$ is approximately equal in successive rounds.

Model | $\mathbb{S}\rightarrow\mathbb{R}$ | $\mathbb{L}\rightarrow\mathbb{R}$ | $\mathbb{D}\rightarrow\mathbb{R}$ | $\mathbb{R}\rightarrow\mathbb{S}$ | $\mathbb{L}\rightarrow\mathbb{S}$ | $\mathbb{D}\rightarrow\mathbb{S}$ | $\mathbb{R}\rightarrow\mathbb{L}$ | $\mathbb{S}\rightarrow\mathbb{L}$ | $\mathbb{R}\rightarrow\mathbb{D}$ | $\mathbb{S}\rightarrow\mathbb{D}$ | AVG
---|---|---|---|---|---|---|---|---|---|---|---
CRF | 17.00 | 17.00 | 2.50 | 8.80 | 8.60 | 4.50 | 10.90 | 11.60 | 9.00 | 9.70 | 9.96
FEMA | 37.60 | 35.00 | 20.70 | 10.80 | 14.80 | 8.80 | 26.60 | 15.00 | 22.90 | 18.70 | 21.09
Hier-Joint | 52.00 | 46.70 | 50.40 | 19.80 | 23.40 | 23.50 | 31.70 | 30.00 | 32.00 | 33.40 | 34.29
RNSCN | 48.89 | 52.19 | 50.39 | 30.41 | 31.21 | 35.50 | 47.23 | 34.03 | 46.16 | 32.41 | 40.84
AD-SAL | 52.05 | 56.12 | 51.55 | 39.02 | 38.26 | 36.11 | 45.05 | 35.99 | 43.76 | 41.21 | 43.91
AHF | 54.98 | 58.67 | 61.11 | 40.33 | 47.17 | 45.78 | 56.58 | 36.62 | 48.24 | 44.16 | 49.36$\pm$3.23
SCD | 59.52 | 71.40 | 61.85 | 48.30 | 48.67 | 52.58 | 59.68 | 42.40 | 54.45 | 54.01 | 55.28$\pm$1.07
BERTB | 54.29 | 46.74 | 44.63 | 22.31 | 30.66 | 33.33 | 37.02 | 36.88 | 32.03 | 38.06 | 37.60
BERTE | 57.56 | 50.42 | 45.71 | 26.50 | 25.96 | 30.40 | 44.18 | 41.78 | 35.98 | 35.13 | 39.36
BERTE-UDA | 59.07 | 55.24 | 56.40 | 34.21 | 30.68 | 38.25 | 54.00 | 44.25 | 42.40 | 40.83 | 45.53
BERTE-CDRG | 59.17 | 68.62 | 58.85 | 47.61 | 54.29 | 42.20 | 55.56 | 41.77 | 35.43 | 36.53 | 50.00
BERT-SCD | 64.10 | 67.61 | 64.75 | 55.83 | 51.33 | 58.92 | 55.64 | 49.76 | 49.62 | 53.29 | 57.08$\pm$1.17
Table 1. Comparison of F1 performance. Best performance is in bold format. Model | $\mathbb{S}\rightarrow\mathbb{R}$ | $\mathbb{L}\rightarrow\mathbb{R}$ | $\mathbb{D}\rightarrow\mathbb{R}$ | $\mathbb{R}\rightarrow\mathbb{S}$ | $\mathbb{L}\rightarrow\mathbb{S}$ | $\mathbb{D}\rightarrow\mathbb{S}$ | $\mathbb{R}\rightarrow\mathbb{L}$ | $\mathbb{S}\rightarrow\mathbb{L}$ | $\mathbb{R}\rightarrow\mathbb{D}$ | $\mathbb{S}\rightarrow\mathbb{D}$ | AVG
---|---|---|---|---|---|---|---|---|---|---|---
SCD | 59.52 | 71.40 | 61.85 | 48.30 | 48.67 | 52.58 | 59.68 | 42.40 | 54.45 | 54.01 | 55.28$\pm$1.07
SCD($\eta=0.0$) | 59.18 | 71.40 | 61.85 | 48.22 | 48.52 | 52.25 | 57.81 | 40.13 | 52.78 | 45.95 | 53.80$\pm$1.91
SCD($\eta=1.0$) | 57.76 | 67.49 | 59.06 | 47.83 | 46.13 | 51.03 | 55.62 | 42.40 | 53.80 | 54.01 | 53.51$\pm$0.96
Student | 55.39 | 63.69 | 56.52 | 47.19 | 45.48 | 50.69 | 52.66 | 41.22 | 52.39 | 44.28 | 50.95$\pm$1.23
Teacher | 52.10 | 57.46 | 48.02 | 24.88 | 28.48 | 33.09 | 48.08 | 40.92 | 50.75 | 45.35 | 42.87$\pm$1.10
Student(MMD) | 0.041 | 0.040 | 0.046 | 0.035 | 0.094 | 0.080 | 0.054 | 0.042 | 0.045 | 0.043 | 0.052$\pm$0.009
Teacher(MMD) | 0.215 | 0.197 | 0.415 | 0.364 | 0.170 | 0.263 | 0.198 | 0.134 | 0.158 | 0.106 | 0.222$\pm$0.023
Table 2. Ablation Study: F1 Performance of different ablated models (top).
Student(MMD) (or Teacher(MMD)) is an estimate of the discrepancy between the
learned source and target distributions by the Student (or Teacher).

##  6\. Experiments and Results

###  6.1. Experimental Setup

####  6.1.1. Comparison Methods

We evaluate SCD as well as our BERT-based version BERT-SCD in this section.
Comparison methods include, CRF (Jakob and Gurevych, 2010), FEMA (Yang and
Eisenstein, 2015), Hier-Joint (Ding et al., 2017), RNSCN (Wang and Pan, 2018),
AD-SAL (Li et al., 2019b), AHF (Zhou et al., 2021) as well as the BERT-based
models BERTE-UDA (Gong et al., 2020) and BERTE-CDRG (Yu et al., 2021). Two
strong single-domain OTE models BERTB and BERTE (Gong et al., 2020), which are
trained only on the source-domain to investigate the capacity of BERT without
domain adaptation. SemBridge (Chen and Qian, 2021) is excluded in our
comparison since its dataset setup is different from that used in compared
works.

####  6.1.2. Datasets

We use benchmark datasets from four domains following previous work (Wang and
Pan, 2018; Li et al., 2019b). The Laptop dataset consists of reviews in the
laptop domain taken from the SemEval ABSA challenge 2014 (Pontiki et al.,
2014). The Restaurant dataset is the set of all restaurant reviews in SemEval
ABSA challenge 2014, 2015 and 2016 (Pontiki et al., 2014, 2015, 2016). The
Device dataset, originally provided by (Hu and Liu, 2004) contains reviews in
the device domain. The Service dataset, introduced by (Toprak et al., 2010)
contains reviews related to the web service domain. We use the preprocessed
data provided by (Li et al., 2019b). Dataset statistics are shown in Table 3.

Dataset | Domain | Sentence | Train | Test
---|---|---|---|---
$\mathbb{L}$ | Laptop | 1869 | 1458 | 411
$\mathbb{R}$ | Restaurant | 3900 | 2481 | 1419
$\mathbb{D}$ | Device | 1437 | 954 | 483
$\mathbb{S}$ | Service | 2153 | 1433 | 720
Table 3. Statistics of the datasets.

####  6.1.3. Evaluation Protocol

We follow prior work (Li et al., 2019b; Gong et al., 2020) and evaluate on 10
transfer pairs $D_{\mathcal{S}}\rightarrow D_{\mathcal{T}}$ from the datasets.
We use the test set of the source domain as a development set to tune our
models. The test set of the target domain is used for evaluation purposes. We
evaluate an exact match,222Exact Match: the predicted label sequence should
exactly match the gold label sequence and compute the Micro-F1 score. Reported
results are the average over $5$ runs.

####  6.1.4. Implementation Details

Following Zhou et al. (2021), we use $100$-dim fixed pretrained Word2Vec
emebeddings (Mikolov et al., 2013) or BERT-Mini embeddings for word
features.333We use BERT-Mini implementation from https://github.com/google-
research/bert We use Adam with $1e^{-3}$ learning rate, 100 epochs for both
Teacher and Student networks, and 50 epochs during self-training, word
embedding dropout rate in $[0.3,0.5,0.7]$, BiLSTM dimensions in
$[100,200,300]$, adaption rate $\lambda\in[1.0,0.7,0.5,0.3]$, batch size in
$[32,64,128]$ and $\eta\in[0.0,0.1,\ldots,\\\ 0.9,1.0,1e^{-2},1e^{-3}]$. Each
batch contains half labeled source and half unlabelled target data. All
sentences are padded to a max length $n$. During self-training, we adopt
repeated sampling on the labeled source data with the same size as the pseudo
labeled target data in each epoch.

###  6.2. Main Results

Table 1 summarizes our main results. We find that neural methods, including
RNSCN and Hier-Joint surpass hand-crafted feature methods FEMA and CRF,
highlighting the importance of leveraging neural networks for the task. We
also find that adversarial methods such as AD-SAL and AHF outperforms both
Hier-Joint and RNSCN, indicating that adversarial learning is effective in
mitigating the domain shift to yield performance. However, by learning target
discriminative features, the SOTA method AHF achieves a better performance
over AD-SAL by about 5.45 F1 on average. We see similar performance on the
SOTA BERT-based model BERTE-CDRG that consider learning target discriminative
features . Specifically, BERTE-CDRG outperforms the previous SOTA BERTE-UDA by
about 4.47 F1 on average. This clearly shows the importance of learning target
discriminative features. However, AHF considers the a mean teacher while
BERTE-CDRG considers a generation model to learn these target discriminative
features. In contrast, we consider to learn an adversarial model (i.e.,
Student) based on self-training through classifier disagreement. Our results
suggest the effectiveness of our approach where we outperform AHF and BERTE-
CDRG by an average F1 of 5.92 and 7.08. In particular, we obtain SOTA results
on nine out of 10 transfer pairs with relative stability when compared to AHF.

###  6.3. Ablation Study

We study the contribution of model components. Table 2 presents our results.
The upper portion of the table shows the performance of different ablated
models. The lower portion is the Maximum Mean Discrepancy (MMD) (Gretton et
al., 2012), which measures the distance between source and target domain
distributions.444MMD from https://github.com/easezyc/deep-transfer-learning/

First, we note that the Teacher and Student networks have comparable
performance on the source domain (see results in Table 5). This means the
performance of the Student over Teacher is due to the divergence (measured by
MMD). Since Student(MMD) is lower than Teacher(MMD) for all transfer pairs, it
is not surprising to see the Student network outperforming the Teacher
network. Conversely, SCD($\eta=1.0$) is simply standard pseudo-labelling.
Although it improves performance, we find that SCD($\eta=0.0$) offers
comparable performance for the average F1 by focusing on learning only on
pseudo-labelled samples with prediction disagreement with the Teacher network.
Interestingly, we find that on pairs such as $\mathbb{S}\rightarrow\mathbb{L}$
and $\mathbb{S}\rightarrow\mathbb{D}$, Teacher(MMD) is already low. Although
Student(MMD) becomes smaller due to adversarial learning, SCD($\eta=0.0$)
cannot leverage sufficient pseudo-labelled samples to achieve satisfactory
performance. This is because Student can only shift few samples to the domain
invariant-distribution to bring about a prediction disagreement. But we see
the benefit of prediction disagreement on pairs such as
$\mathbb{D}\rightarrow\mathbb{R}$, where Teacher(MMD) is large and
corresponding Student(MMD) is low, improving the Student network from 56.52 to
61.85 (i.e., performance on SCD($\eta=0.0$)).

These results indicate that the pseudo-labelled samples help to learn the
discriminative features, achieving better performance as compared to recent
works.

###  6.4. Sensitivity of Hyperparameter $\eta$

We now study the sensitivity of our model for the hyperparameter $\eta$. At
$\eta=0$, we pay attention to the learning of pseudo-labelled samples by the
student network that disagree with those produced by the Teacher network. At
$\eta=1$, we are simply performing the standard pseudo-labelling. We study the
sensitivity of $\eta$, particularly on pairs that have a high or low MMD on
the Teacher network. That is, the respective $\mathbb{D}\rightarrow\mathbb{R}$
and $\mathbb{S}\rightarrow\mathbb{D}$ pairs. With low MMD, the source and
target domains are similar, but diverges with high MMD. The idea is to
understand how the domain divergence affects $\eta$.

Figure 6.4 shows the results on this experiment where we report the F1
performance for different values of $\eta$ on the pairs. We find that on
$\mathbb{S}\rightarrow\mathbb{D}$, the learning problem moves toward standard
pseudo-labelling since the best performance is achieved at $\eta=1.0$.
However, on $\mathbb{D}\rightarrow\mathbb{R}$ the best performance is achieved
at $\eta=0$. These results suggest the importance of attention placed on the
learning of these pseudo-labelled samples. Particularly, we observe that when
the domain divergence is high it is beneficial to learn on pseudo-labelled
samples that disagree with the Teacher network. On the other hand, when the
source and target domains are similar, pseudo-labelling seems sufficient for
the problem. This model behaviour guides in the selection of $\eta$.

& $0$$0.2$$0.4$$0.6$$0.8$$1$$46$$48$$50$$52$$54$Value of $\eta$ (S-¿D)Macro F1

Figure 3. F1 Performance of SCD for different $\eta$ values on
$\mathbb{D}\rightarrow\mathbb{R}$ (left) and $\mathbb{S}\rightarrow\mathbb{D}$
(right) .

(a) Teacher | (b) Student | (c) SCD
---|---|---
Figure 4. The t-SNE visualization of features learned by the (a) Teacher, (b)
Student, and (c) SCD for the transfer pair $\mathbb{D}\rightarrow\mathbb{R}$
(light shade: target, dark shade: source)

###  6.5. Quality of Pseudo-Labels

We perform additional experiments to study the quality of pseudo-labels
generated by our method. Figure 6.4 shows the experiments, where we report the
F1 performance for different models for the pairs under study;
$\mathbb{D}\rightarrow\mathbb{R}$ (left) and $\mathbb{S}\rightarrow\mathbb{D}$
(right). Since SCD($\eta=0.0$) and SCD have equivalent performance on
$\mathbb{D}\rightarrow\mathbb{R}$ and SCD($\eta=1.0$) and SCD have equivalent
performance on $\mathbb{S}\rightarrow\mathbb{D}$, we omit the curves of SCD to
clearly show the benefit of pseudo-labelled samples under different
strategies. Other compared methods include AHF.

On $\mathbb{D}\rightarrow\mathbb{R}$, we find that both SCD($\eta=0.0$) and
SCD($\eta=1.0$) improves steeply but becomes unstable after the fifth and
eight epochs respectively. However, the improvement of SCD($\eta=0.0$) over
SCD($\eta=1.0$) is highly notable. This observation points us to the fact,
with high Teacher(MMD), prediction disagreement offers high quality pseudo-
labelled samples particularly in the early rounds of training to improve
performance. However, when Teacher(MMD) is low such as on the
$\mathbb{S}\rightarrow\mathbb{D}$, we are not able to take advantage of
pseudo-labelled samples with prediction disagreement. Hence, the standard
pseudo-labelling can outperform prediction disagreement as seen in the figure.
AHF on the other hand underperforms, indicating that our SSL approach is
effective as compared to the mean teacher.

& $0$$10$$20$$30$$40$$50$$40$$45$$50$$55$Number of epochs (S-¿D)Macro
F1AHFSCD($\eta$=1.0)SCD($\eta$=0.0)

Figure 5. F1 performance of different models for training epochs, aiming to
evaluate the quality of pseudo labels.

###  6.6. Feature Visualization

Fig. 4 depicts the t-SNE (Van der Maaten and Hinton, 2008) visualization of
features learned using the Teacher, Student and SCD models on the transfer
pair $\mathbb{D}\rightarrow\mathbb{R}$ (1000 instances sampled randomly in
each domain). As there are three class labels, namely BIO labels, an ideal
model should clearly align the source and target data into three clusters. For
the Teacher network, we can observe that the distribution of source samples is
relatively far from the distribution of the target samples. Through domain
adaptation, the Student network improves the alignment of the source and
target samples. However, by learning target discrimnative features through
SCD, we gradually observe three clusters forming. The results indicate that
SCD improves the class-level alignment.

Case |  Sentence |  Teacher |  Student |  SCD
---|---|---|---|---
1 |  But the space is small and lovely, and the service is helpful. |  space, service |  space, service |  space, service
2 |  Although small, it has beautiful ambience, excellent food and catfish is delicious. |  ambience, food |  ambience, food |  ambience, food, catfish
3 |  The pasta primavera was outstanding as well, lots of fresh veggies |  NULL |  pasta primavera |  pasta primavera, veggies
4 |  I would definitely go back, if only for some of those exotic martinis on the blackboard. |  NULL |  NULL |  martinis
Table 4. Case study on $\mathbb{D}\rightarrow\mathbb{R}$. Gold aspect terms are boldfaced. “NULL” indicates that no aspect term has been extracted. Model | $\mathbb{S}\rightarrow\mathbb{R}$ | $\mathbb{L}\rightarrow\mathbb{R}$ | $\mathbb{D}\rightarrow\mathbb{R}$ | $\mathbb{R}\rightarrow\mathbb{S}$ | $\mathbb{L}\rightarrow\mathbb{S}$ | $\mathbb{D}\rightarrow\mathbb{S}$ | $\mathbb{R}\rightarrow\mathbb{L}$ | $\mathbb{S}\rightarrow\mathbb{L}$ | $\mathbb{R}\rightarrow\mathbb{D}$ | $\mathbb{S}\rightarrow\mathbb{D}$ | AVG
---|---|---|---|---|---|---|---|---|---|---|---
Teacher | 69.63 | 76.35 | 66.73 | 82.67 | 75.43 | 67.59 | 80.00 | 69.20 | 79.74 | 69.16 | 73.65$\pm$0.60
Student | 67.48 | 77.88 | 65.72 | 82.63 | 74.44 | 67.53 | 80.17 | 70.03 | 80.64 | 69.41 | 73.59$\pm$0.62
Table 5. F1 performance of Teacher and Student on the test set of the source
domain.

###  6.7. Case Study

To test the effectiveness of our approach, some case examples from the
transfer pair with the largest domain divergence
($\mathbb{D}\rightarrow\mathbb{R}$) are selected for demonstration. Table 4
shows the aspect term extraction results on these case examples.

In the first case, we find that the Teacher, Student and SCD are all capable
of identifying the aspect terms “service” and “space”. As these aspect terms
appear in both Device and Restaurant domains, domain adaptation is not
necessary to extract the aspect terms. It is therefore not surprising to
observe that all models identify the aspect terms in the Restaurant domain.

In the second case example, the aspect terms “ambience”, “food” and “catfish”
are found in the Restaurant domain and not the Device domain. However, the
Teacher was able to extract the aspect terms “ambience” and “food”.
Introspecting further, we found that 81% of aspect terms extracted by the
Teacher in the Restaurant domain are accompanied with opinion words (e.g.,
“great”) that are also present in the Device domain. Hence, the Teacher was
able to learn the correspondences between opinion words and aspect terms in
the Device domain and use that knowledge to locate “ambience” and “food” in
the Restaurant domain. However, both Teacher and Student networks fail to
extract the aspect term “catfish”. This highlights the importance of learning
target discriminative features, as there is no correspondence between the word
“delicious” and an aspect term to be learned in the Device domain but only in
the Restaurant domain. SCD solves this problem by collecting high quality
pseudo-labelled samples in the Restaurant domain. As a result, SCD is able to
extract the aspect term “catfish”.

In the third case example, we found that the Teacher network failed to
identify the aspect terms “pasta primavera” and “veggies” as they do not exist
in the Device domain. However, by reducing the domain shift between the two
domains, the Student network is able to extract “pasta primavera” but not
“veggies”. Upon investigation, we found that the opinion word “fresh” which
expresses an opinion on “veggies” frequently appears 83 times in the
Restaurant dataset and 0 times in the Device dataset. Ideally, by learning
target discriminative features, we can learn correspondences that exist
between “fresh” and aspect terms. Such knowledge as learned by SCD offers
supervisory training signals, enabling SCD to detect the aspect term
“veggies”.

Finally, in the fourth case example, both the Teacher and Student networks
completely failed to detect the aspect term “martinis”. While it is no
surprise that the Teacher network fails (i.e., “martinis” is not seen during
training), the failure of the Student network highlights the limitations of
simply reducing the domain shift and suggests the importance of learning
target discriminative features for successful cross-domain OTE.

###  6.8. Performance Comparison on Source Domain

We argued that the difference between the target errors (or F1 performance) of
the teacher and student networks can be explained by the
$\mathcal{H}\Delta\mathcal{H}$ divergence when the source errors of these
networks are approximately equal. According to Ben-David et al. (2010), the
source error as well as the divergence can be estimated from finite samples of
the source and target domains, under the assumption of the uniform convergence
theory (Vapnik and Chervonenkis, 2015). Table 5 therefore reports the F1
performance on the source test set. We discover that for each transfer pair,
the F1 performance is approximately equal, comparing the Teacher and Student.
This suggest that adversarial learning performed by the Student to reduce the
domain shift has little to no effect on the classification on the source data.
Most importantly, the results suggest that the difference between the Teacher
and Student on the target data is due to the target samples shifted to the
domain-invariant space within the Student feature space.

##  7\. Conclusion

We have proposed a Self-training through Classifier Disagreement for cross-
domain OTE. We demonstrated that by simultaneously training a Teacher and a
Student network, we can benefit from the information that comes from their
predictions on the unlabelled target domain. Specifically, by leveraging
pseudo-labelled samples that disagree between the Teacher and Student
networks, the Student network is significantly improved, even in large domain
divergences. This model behaviour however leads to the potential limitation.
In cases of small domain shifts, the model tends to favor pseudo-labeling (Lee
et al., 2013), an SSL approach that risks confirmation bias (Tarvainen and
Valpola, 2017) (i.e., prediction errors are fit by the network). Nevertheless,
small domain shifts have little to no interest in cross-domain learning since
the source and target domains can be considered to be similar. In the future,
we will consider data augmentation strategies to mitigate confirmation bias
brought by pseudo-labelling in such situations (Arazo et al., 2020). We
believe our model is generic and can be applied to other cross-domain tasks
such as cross-domain named entity recognition.

## Acknowledgements

This work was supported in part by the National Key R&D Program of China under
Grant 2021ZD0110700, in part by the Fundamental Research Funds for the Central
Universities, in part by the State Key Laboratory of Software Development
Environment. In addition, SM and NA received support from the Leverhulme Trust
under Grant Number: RPG#2020#148.

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Finally, we fit the theoretical prediction Eq. (61) to the measured shear
elongation orientations. To this end, define in accordance with Eqs. (61)–(62)
a fit function
$\alpha_{\text{fit}}(\phi)=\phi+\frac{\pi}{4}\text{sgn}(\cos\phi)+\delta\alpha_{\text{fit}}(\phi)$
with

$\delta\alpha_{\text{fit}}(\phi)=\frac{1}{2}\arg\left[(1+ip_{1})\cos\phi+(p_{2}+ip_{3})\sin\phi\right]-\frac{1}{2}\arg(\cos\phi).$
(65)

Weighted estimates and standard errors of the three dimensionless fitting
parameters$p_{1}=2\nu^{o}/(1+\nu)$, $p_{2}=2\gamma_{1}/(1+\nu)$ and
$p_{3}=(\gamma_{2}-1)/(1+\nu)$ (assuming $\nu>-1$) are determined using
MATLAB’s `nlmfit` function (Tab. 3). The curve $\alpha_{\text{fit}}(\phi)$
corresponding to the best fit is shown as a red solid line in Fig. 4b of the
main text.

In a standard passive linearly elastic solid with $B,\mu>0$ and all other
moduli zero, one expects $p_{1}=p_{2}=p_{3}=0$, which seems – despite a fairly
large standard error – not sufficient to fit strain elongation axis
orientation (main text Fig. 4b, Tab. 3). The fact that $p_{3}$ is notably
different from zero suggests that stress-strain couplings mediated through
interactions with the surrounding fluid [$\Gamma,\Lambda$, see Eq. (63d)] are
contributing to the effective mechanical response of the crystal. In addition,
we note that for a coarse-grained hexagonal network with ideal linearly
elastic odd interactions one finds $p_{1}\sim\nu^{o}=0$ [12]. It is therefore
not possible to directly infer unambiguous information about odd bulk and
shear moduli $A$ and $K^{o}$, respectively, from fits to the shear elongation
axis alone. A complementary fitting approach that overcomes this limitation is
described in the following section.

####  3.11.5 Estimating elastic moduli from shear strain fields

The fits from the previous section are consistent with the presence of odd
moduli in the elastic response of a living chiral crystal, but do not provide
unambiguous information about the relative magnitude and signs of such moduli.
To determine the latter, we now explicitly fit theoretical predictions of the
spatial dependence of strain components to measurements. Importantly, these
fits specify magnitude and signs of odd moduli, which in turn can be compared
against predictions based on embryo-embryo interactions within the crystal
(see Sec. 2.3.3).

Using the coordinate system in which $\textbf{b}=|\mathbf{b}|\mathbf{e}_{x}$
and centering the dislocation at the coordinate origin as before, the spatial
dependence of strain components in a general isotropic linearly elastic solid
is given by [13]

$u^{\alpha}=\frac{|\mathbf{b}|}{2\pi r^{2}}\tilde{u}^{\alpha}$ (66)

with

$\displaystyle\tilde{u}^{0}(x,y)$ $\displaystyle=-2\gamma_{1}x-(1-\nu)y$ (67a)
$\displaystyle\tilde{u}^{1}(x,y)$ $\displaystyle=(1+\gamma_{2})x-2\nu^{o}y$
(67b) $\displaystyle\tilde{u}^{2}(x,y)$
$\displaystyle=-[2\nu^{o}x+(\gamma_{2}-1)y]\frac{x^{2}-y^{2}}{x^{2}+y^{2}}-[(1+\nu)x+2\gamma_{1}y]\frac{2xy}{x^{2}+y^{2}}$
(67c) $\displaystyle\tilde{u}^{3}(x,y)$
$\displaystyle=-[2\nu^{o}x+(\gamma_{2}-1)y]\frac{2xy}{x^{2}+y^{2}}-[(1+\nu)x+2\gamma_{1}y]\frac{y^{2}-x^{2}}{x^{2}+y^{2}}.$
(67d)

Fit Parameter | Fit value | Standard Error
---|---|---
$\nu$ | 0.049 | 0.0097
$\nu^{o}$ | 0.053 | 0.0039
$\gamma_{1}$ | 0.010 | 0.0046
$\gamma_{2}$ | 0.74 | 0.0088
Table 4: Mean and standard errors for fits of the shear strain components
$u^{2}$ and $u^{3}$ using theoretical predictions [13] Eq. (66) with
$\tilde{u}^{2}$ and $\tilde{u}^{3}$ given in Eqs. (67c) and (67d). The
corresponding spatial shear strain profiles together with experimental
measurements are depicted in Fig. S14a. Predictions for divergent and rotary
strain components using these fitting parameters in Eq. (66) with
$\tilde{u}^{0}$ and $\tilde{u}^{1}$ are shown in Fig. S14b (same data as shown
in main text Fig. 4c).

The goal is then to determine $\nu,\,\nu^{o},\,\gamma_{1}$ and $\gamma_{2}$ by
fitting measured spatial profiles of strain around the dislocation. To make
this analysis comparable to the fitting of shear orientations described the
previous section, we determine $\nu,\,\nu^{o},\,\gamma_{1}$ and $\gamma_{2}$
by fitting spatial profiles of the shear strain components $u^{2}$ and $u^{3}$
(Eq. (66) with $\tilde{u}^{2}$ and $\tilde{u}^{3}$). As before, we restrict
the fitting to a specific subset of embryo positions around the dislocation
(black dots in Fig. S12) to mitigate spurious effects from other nearby
dislocations and from domain boundaries (see Sec. 3.11.2 for details). The
shear component fits are implemented using MATLAB’s `lsqcurvefit` function.
The mean and the standard error of each fit parameter, computed from the
collection of single-frame fits, are depicted in Tab. 4. The strain components
$u^{2}$ and $u^{3}$ corresponding to this best fit are plotted for comparison
with measurements in Fig. S14a.

Figure S14: Spatial strain component analysis near a dislocation. a, Shear
components measured (top, see Sec. 3.11.1–3.11.3) and fitted (bottom, see Sec.
3.11.5) using Eq. (66) with $\tilde{u}^{2}$ (Shear 1) and $\tilde{u}^{3}$
(Shear 2). Each disk represents an average strain measurement or value of the
fitting function centered at the average position of lattice sites near the
dislocation (depicted black in Fig. S12). The resulting fit parameters are
given in Tab. 4. b, Comparison of measured divergent and rotary strain
components (top) with predictions (bottom) from Eq. (66) with $\tilde{u}^{0}$
(Divergence) and $\tilde{u}^{1}$ (Curl) using the fitting paramers determined
in a (same data as shown in main text Fig. 4c). Scale bar, 0.4 mm. c, Spatial
profiles of strain components on a continuous domain as computed from Eq. (66)
with $\tilde{u}^{\alpha}$ given in Eqs. (67a)–(67d) using parameters given in
Tab. 4 found from fitting data in a (top). Scales and color bars same as in
panels a and b.

As a consistency check for this alternative fitting approach, we compute the
parameters $p_{1}\,,p_{2}$ and $p_{3}$ found from strain elongation
orientation fits (see Sec. 3.11.4) using the effective material parameters
from Tab. 4, where we find $p_{1}\approx 0.1$, $p_{2}\approx 0.02$ and
$p_{3}\approx-0.25$. These values deviate, but are still within the error
margins of the fit values listed Tab. 3.

The fit results in Tab. 4 lead to nontrivial predictions for divergent and
rotary strain components surrounding a dislocation (Eq. (66) with
$\tilde{u}^{0}$ and $\tilde{u}^{1}$), which so far had not been included into
our analysis. Importantly, these predictions can be compared against
independent experimental measurements of the corresponding strain components
in the crystal, where we find good quantitative agreement (see main text Fig.
4c, reproduced for convenience in Fig. S14b).

Modulus | Estimate | Standard Error
---|---|---
$A/\mu$ | 7.7 | 0.61
$K^{o}/\mu$ | 7.1 | 0.59
$\Gamma/\mu$ | 0.32 | 0.082
$\Lambda/\mu$ | -1.0 | 0.097
Table 5: Relative values of elastic moduli determined from an inversion of
Eqs. (63) for given fit parameters $\nu,\,\nu^{o},\,\gamma_{1}$ and
$\gamma_{2}$ from Tab. 4. The signs of the odd bulk and shear moduli $A>0$ and
$K^{o}>0$ ($\mu>0$ required for stability) are consistent with predictions
based on the handedness of transverse embryo interactions (see Sec. 2.3.3) and
indicate broken Maxwell-Betti reciprocity [13].

Finally, we explicitly determine elastic moduli by inverting Eqs. (63) for
given fit parameters $\nu$, $\nu^{o}$, $\gamma_{1}$ and $\gamma_{2}$ from Tab.
4. From dimensionless strain measurements, we can only expect to find relative
values of elastic moduli, where we choose the passive shear modulus $\mu$ to
make all other moduli dimensionless. To make the inversion result of Eqs. (63)
unique, we additionally impose $B=2\mu$, which holds for a linearly elastic
hexagonal spring network with nearest neighbor interactions [12, 13]. Using
MATLAB’s `solve` function, we can then determine from the values Tab. 4 and
Eqs. (63) a unique solution for the parameters $A/\mu$ and $K^{o}/\mu$ (odd
bulk and shear moduli), as well as for $\Gamma/\mu$ and $\Lambda/\mu$
(equilibrium and nonequilibrium moduli coupling rotations to torque and
pressure). The final parameter values and standard errors of these moduli are
given in Tab. 5.

According to the fit results Tab. 5, strain measurements around a dislocation
suggest that odd bulk and shear moduli $A$ and $K^{o}$, respectively,
contribute to the effective material properties of a living chiral crystal.
Their relative values compared to the shear modulus $\mu$ are larger than
estimated from the coarse-graining of the microscopic model [see Sec. 2.3.3,
Eqs. (34) and (35)]. However, the latter was based on interactions among an
isolated pair of embryos, which neglects interaction modulations that are most
likely present within clusters. Importantly, the signs determined for $A$ and
$K^{o}$ agree between the coarse-graining and the strain measurement
approaches. Ultimately, these signs result from the handedness of the
microscopic transverse interactions and therefore represent a macroscopic
signature of the well-defined handedness of embryo rotations within the
cystal. Furthermore, $A,K^{o}>0$ (together with $\Lambda<0$) implies broken
Maxwell-Betti reciprocity [13] and suggests that strain cycles observed during
the emergence of chiral displacement waves (main text Fig. 4d–h, Sec. 3.12.2,
Fig. S17a,b) do work on the surrounding [see Eq. (26)], highlighting once
again the nonequilibrium nature of living chiral crystals. In addition, we
find the equilibrium ($\Gamma$) and nonequlibrium ($\Lambda$) moduli coupling
rotations to stress (see Sec. 2.3.1) are nonzero, which is most likely due to
interactions of the crystal with the surrounding fluid.

###  3.12 Phase space analysis of strain cycles

In this section, we describe the systematic statistical analysis of the living
chiral crystal dynamics in strain space. In particular, we find closed cycles
in strain space that provide information about contributions to entropy
production [30] and, in the context of effective odd elastic properties, can
be related to mechanical work that is done on the environment [12].

####  3.12.1 Phase space currents and partial entropy production

The entropy production rate $\dot{S}$ of an overdamped stochastic system that
follows Langevin dynamics can be determined from [30]

$\dot{S}=k_{\text{B}}\int
d\textbf{x}\,\frac{\mathbf{j}(\textbf{x})\cdot\mathbf{D}^{-1}(\textbf{x})\cdot\mathbf{j}(\textbf{x})}{\rho(\textbf{x})},$
(68)

where $\rho(\textbf{x})$ and $\mathbf{j}(\textbf{x})$ denote the probability
density and corresponding probability current of a particular system
configuration x, respectively. $\mathbf{D}(\textbf{x})$ is an effective
diffusion matrix with inverse $\mathbf{D}^{-1}(\textbf{x})$. In the following,
we describe how $\rho(\textbf{x})$, $\mathbf{j}(\textbf{x})$ and
$\mathbf{D}(\textbf{x})$ can be approximated from experimental data of the
strain component dynamics to estimate the system’s partial entropy production
rate shown in main text Fig. 4f,h.

For the spatially resolved analysis shown in Fig. S15b,d, we tiled the cluster
domain into square regions of $200\,\mu$m (approximately one embryo diameter).
The center of each tile is located at some position $\hat{\mathbf{r}}$. In
each of these squares, we spatially average the strain components
$u^{\alpha}(\mathbf{r},t)$ ($\alpha=0,1,2,3$, see Sec. 3.9) to determine local
strain component values $\hat{u}^{\alpha}(\hat{\mathbf{r}},t)$. We then
introduce a strain component pair vector
$\hat{\bm{u}}(\hat{\mathbf{r}},t)=[\hat{u}^{\alpha}(\hat{\mathbf{r}},t),\hat{u}^{\beta}(\hat{\mathbf{r}},t)]^{\top}$.
Inspired by the curl-divergence and shear 1-shear 2 cycles that were suggested
as signatures of odd elastic oscillations [12], we considered for our analysis
the corresponding pairs $\alpha=1,\beta=0$ (Fig. 4f in the main text) and
$\alpha=2,\beta=3$ (Fig. 4h in the main text).

To estimate partial entropy production rates, we used time series of strain
component pairs
$\\{\hat{\bm{u}}_{1},\hat{\bm{u}}_{2},...,\hat{\bm{u}}_{N}\\}$, where
$\hat{\bm{u}}_{k}:=\hat{\bm{u}}(\hat{\mathbf{r}},k\Delta t)$ with
$k=1,2,...,N$ denotes pairs of strain components at $N=200$ successive time
points in $\Delta t=10$ s intervals. The continuous probability density
$\rho(\textbf{x})$ and the associated current $\mathbf{j}(\textbf{x})$ were
then estimated as [31]:

$\displaystyle\hat{\rho}(\textbf{x})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}K(\textbf{x},\hat{\bm{u}}_{i},\Sigma),$
(69a) $\displaystyle\hat{\mathbf{j}}(\textbf{x})$
$\displaystyle=\frac{\hat{\rho}(\textbf{x})}{2\Delta
t}\frac{\sum_{i=2}^{N-1}K(\textbf{x},\hat{\bm{u}}_{i},\Sigma)(\hat{\bm{u}}_{i+1}-\hat{\bm{u}}_{i-1})}{\sum_{i=2}^{N-1}K(\textbf{x},\hat{\bm{u}}_{i},\Sigma)},$
(69b)

where system configurations $\textbf{x}=(u^{\alpha},u^{\beta})^{\top}$ are
defined in the space of a strain component pair. In Eqs. (69),
$K(\bm{x},\bm{\mu},\Sigma)=\exp[-(\bm{x}-\bm{\mu})^{\top}\Sigma^{-1}(\bm{x}-\bm{\mu})/2]/[2\pi\det(\Sigma)]^{-1/2}$
is the bivariate Gaussian with a bandwidth $\Sigma$ determined using the so-
called “rule of thumb” [32, 30]. The latter aims to heuristically minimize the
asymptotic mean squared error of the estimated fields $\hat{\rho}(\textbf{x})$
and $\hat{\mathbf{j}}(\textbf{x})$ under the assumption of a standard normal
distribution as reference distribution. In two dimensions, it suggests the
bandwidth $\Sigma$ is a diagonal $2\times 2$ matrix with components
$\Sigma_{\alpha\beta}=\frac{1}{2}N^{-1/6}\sigma_{\alpha}\delta_{\alpha\beta}$
(no summation), where $N$ denotes the total number of time points and
$\sigma_{\alpha}$ is the standard deviation of the local strain component
values
$\\{\hat{u}_{1}^{\alpha},\hat{u}_{2}^{\alpha},\dots,\hat{u}_{N}^{\alpha}\\}$
defined above.

Figure S15: Partial entropy production rate during cluster oscillations. a,
Spatial map the local partial entropy production rate in the curl-divergence
space before the emergence of strain waves (left) and in the presence of
strain waves (right). Details of the analysis are described in Sec. 3.12.1. b,
Spatially integrated entropy production rate in the curl-divergence space.
Error bars indicate standard deviation over 100 bootstraps (see Sec. 3.12.1).
The time point of maximum entropy production rate $t_{2}$ is defined as 0 min.
c, d, Same analysis as a and b in the shear strain component space. Scale bar,
1 mm.

Examples of the probability densities and fluxes estimated at some position in
the cluster using Eqs. (69) are shown in the insets of Fig. 4f and h (main
text). Similarly, the effective diffusion matrix is estimated as [31]

$\bar{\mathbf{D}}(\textbf{x})=\frac{\hat{\rho}(\bm{u})}{\Delta
t}\frac{\sum_{i=1}^{N-1}K(\textbf{x},\hat{\bm{u}}_{i},\Sigma)(\hat{\bm{u}}_{i+1}-\hat{\bm{u}}_{i})\otimes(\hat{\bm{u}}_{i+1}-\hat{\bm{u}}_{i})}{\sum_{i=1}^{N-1}K(\textbf{x},\hat{\bm{u}}_{i},\Sigma)},$
(70)

where $\otimes$ denotes a dyadic product. To ensure an inverse diffusion
matrix required in Eq. (68) is well-defined, we finally use
$\bar{\mathbf{D}}(\textbf{x})$ given in Eq. (70) to define a constant,
weighted mean diffusion matrix $\hat{\mathbf{D}}$ as

$\hat{\mathbf{D}}=\int
d\textbf{x}\,\bar{\mathbf{D}}(\textbf{x})\hat{\rho}(\textbf{x})\mathbbm{1}_{\hat{\rho}(\textbf{x})>c}.$
(71)

Here, $\mathbbm{1}_{\hat{\rho}(\textbf{x})>c}$ is the indicator function
defined analog to Eq. (49) and $c$ is chosen such that only sufficiently well
populated regions of the phase space with $\hat{\rho}d\textbf{x}>0.01$ are
included in the analysis. The estimates Eqs. (69) and Eq. (71) are then used
in Eq. (68) to approximate a local partial entropy production rate
$\hat{\dot{S}}(\hat{\mathbf{r}})$. Fig. S15a and c each show two exemplary
spatial maps of the partial entropy production rate before (left) and during
(right) the presence of displacement waves.

Finally, we consider the spatially integrated entropy production rate for the
whole cluster

$\hat{\dot{S}}_{\text{tot}}=\sum_{\hat{\mathbf{r}}}\hat{\dot{S}}(\hat{\mathbf{r}}).$
(72)

In Fig. S15b,d shows that the integrated entropy production rate calculated in
both strain component spaces peaks when the displacement waves are most
prominent.

To quantify the robustness of the spatially integrated entropy production
rate, we performed the same analysis on bootstrapped [33, 34] time series of
the strain component pairs. In particular, we randomly sampled $N-2$ elements
from the time series of strain component pair vectors
$\\{\hat{\bm{u}}_{2},\hat{\bm{u}}_{3},...,\hat{\bm{u}}_{N-1}\\}$. With each
sampled vector $\hat{\bm{u}}_{k}$, we additionally stored $\hat{\bm{u}}_{k-1}$
and $\hat{\bm{u}}_{k+1}$ to be able to compute the flux and diffusion matrix
given in Eqs. (69b) and (71). Entropy production rates were finally determined
from this bootstrapped configuration space information. Repeating this
procedure 100 times yields a standard deviation that is depicted by error bars
in Fig. S15b,d.

Figure S16: Dynamics of total defect number and partial entropy production
rate. The increase and decrease of the spatially integrated partial entropy
production rate (see Sec. 3.12.1 for details of the analysis) is accompanied
by a corresponding increase and decrease of the total number of defects.

Combining the dynamic analysis of lattice defects (Sec. 3.11.2) with the
quantification of partial entropy production (Fig. S15 and Sec. 3.12.1), we
can study the correlation of entropy production with the crystalline order of
the cluster. Specifically, we plot the total number of defects over time
together with the spatially integrated entropy production in Fig. S16, which
reveals a striking correlation between increasing (and decreasing) defect
numbers with the increase (and decrease) in the entropy production rate.
Qualitatively, we suspect that the presence of defects provides additional
space for embryos to fluctuate around their mean positions within the crystal
and thereby facilitates the emergence of substantial strain waves. Once the
defects are expelled through embryo rearrangements from the finite crystal
domain (Fig. S15), this extra space – and consequently strain waves – cease to
exists.

####  3.12.2 Strain cycle handedness

The handedness of the phase space currents in strain space computed in Sec.
3.12.1 corresponds to the handedness of strain cycles observed during cluster
oscillations. To quantify cycle handedness, we define a counter-clockwise
reference unit current field as
$\mathbf{j}_{\text{ref}}(\mathbf{x})=\\{-\sin\phi_{\alpha\beta},\cos\phi_{\alpha\beta}\\}$,
where $\phi_{\alpha\beta}\in[0,2\pi]$ represents the azimuthal angle in the
strain spaces spanned by $\mathbf{x}=(u^{1},u^{0})^{\top}$
($\alpha=1,\,\beta=0$) and $\mathbf{x}=(u^{2},u^{3})^{\top}$
($\alpha=2,\,\beta=3$, see insets in Fig. S17a,b). The strain cycle handedness
is then at each point in the embryo cluster empirically defined as

$H_{\alpha\beta}=\frac{1}{\max(|\hat{\mathbf{j}}|)}\int\hat{\mathbf{j}}\cdot\mathbf{j}_{\text{ref}}\,d\mathbf{x},$
(73)

such that $H_{\alpha\beta}=1$ and $H_{\alpha\beta}=-1$ indicate perfectly
counter-clockwise and clockwise cycles, respectively. Strain cycles handedness
maps, computed at the time of maximum partial entropy production of strain
wave oscillations ($t_{2}$ in Fig. S15b,d), are shown in Fig. S17a and b.
These maps reveal a predominant counter-clockwise cycle handedness
($H_{\alpha\beta}>0$). Comparing the handedness distribution with the local
partial entropy production rate map in Fig. S15, we found that handedness
amplitudes $|H_{\alpha\beta}|$ are larger in regions with high partial entropy
production rate. Finally, we analyzed the map of local correlation of the
handedness in either of the two strain spaces. Specifically, Fig. S17c depicts
the handedness product $H_{10}H_{23}$, which is largely positive and therefore
indicates a strong correlation between counter-clockwise oriented strain
cycles in both strain component spaces $\mathbf{x}=(u^{1},u^{0})^{\top}$ and
$\mathbf{x}=(u^{2},u^{3})^{\top}$.

Figure S17: Handedness maps of strain cycles during cluster oscillations. a–b,
Handedness maps of strain cycles computed via Eq. (73) during cluster
oscillations at the time of maximum partial entropy production (denoted
$t_{2}$ in Fig. S15b,d). In both strain spaces
$\mathbf{x}=(u^{1},u^{0})^{\top}$ (curl–divergence, a) and
$\mathbf{x}=(u^{2},u^{3})^{\top}$ (shear 1–shear 2, b), cycles are oriented
mostly counter-clockwise. c, Correlation between the handedness maps in a and
b, confirming the visual impression that the cycle handedness in both strain
spaces is strongly correlated. Scale bar, 1 mm.

##  4 Table of symbols

$\mathbf{v}$ | Fluid velocity
---|---
$\eta$ | Viscosity
$v_{s}$ | Sedimentation velocity of inactive embryos
Re | Reynolds number
$\mathbf{r}=(x,y,z)^{\top}$ |  Position vector with magnitude $r=|\mathbf{r}|$
$a$, $b$, $c$, $d$ | Stokeslet, force-dipole, force-quadrupole, source-dipole singularity amplitude
$\mathbf{p}$ | Embryo AP body-axis orientation
$\alpha$, $\beta$, $\gamma$ | Coefficients to describe flows near the surface of a microswimmer
$A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ | Mode coefficients in solution of the Hele-Shaw equation
$\mathbf{r}_{i}$ |  Centroid position of a disk/particle $i$
$t_{d}$ |  Developmental time, $t_{d}=0$: onset of cluster formation
$h$ | Distance of flow singularities from the fluid-air interface
$\mathbf{g}$/$\theta_{g}$ | Gravity/Orientation of gravity relative to fluid surface normal
$\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ |  Relative coordinate between disk $i$ and $j$
$\hat{\mathbf{r}}_{ij}$, $\hat{\mathbf{r}}_{ij}^{\perp}$ | Normalized relative coordinate and orthogonal normalized relative coordinate
$\omega_{0}$ | Single embryo angular spinning frequency
$R$ | Apparent top-view radius of surface-bound embryos (in the disk model)
$\omega_{i}$ |  Angular spinning frequency of disk/particle $i$
$\mathbf{F}_{\text{rep}}$, $f_{\text{rep}}$ | Effective repulsion force and effective repulsion force strength
$F_{\text{nf}}(|\mathbf{r}_{i}-\mathbf{r}_{j}|)$ | Effective lubrication forces in the disk model
$T_{\text{nf}}(|\mathbf{r}_{i}-\mathbf{r}_{j}|)$ | Effective lubrication torques in the disk model
$f_{0}$, $\tau_{0}$ | Strength of lubrication forces and torques in the disk model
$d_{c}$ | Cut-off length of lubrication interactions
$d_{ij}$ |  Shortest disk surface distance between disks $i$ and $j$
$N_{cl}$ | Total number of embryos or disks bound in a cluster
$\mathbf{e}_{x},\,\mathbf{e}_{y},\,\mathbf{e}_{z}$ | 3D Cartesian basis
$\mathbf{e}_{r},\,\mathbf{e}_{\theta},\,\mathbf{e}_{\phi}$ |  3D spherical coordinates basis, $x=r\sin\theta\cos\phi,y=r\sin\theta\sin\phi,z=r\cos\theta$
$\mathbf{e}_{\rho},\,\mathbf{e}_{\varphi}$ |  2D cylindrical coordinate basis, $\bar{x}=\rho\cos\varphi,\,\bar{y}=\rho\sin\varphi$.
$l$ | Average nearest neighbor centroid distance in bound pairs and triplets
$\ell_{\text{maj}}$, $\ell_{\text{min}}$ | Semi-major and semi-minor embryo body axis
$e=\ell_{\text{min}}/\ell_{\text{maj}}$ | Body axis ratio
$\chi=(1-e^{2})/(1+e^{2})$ | Microswimmer body shape anisotropy
$L=(\ell_{\text{maj}}+\ell_{\text{min}})/2$ |  Characteristic embryo size, $L=150\,\mu$m
$F_{\text{st}},F_{g}$ | Stokeslet force amplitude, Negatively buoyant weight force
$\mathbf{r}_{0}$, $\mathbf{r}_{0}^{\prime}$ | Position of flow singularity and of image flow singularity
$\mathbf{E}$, $\mathbf{E}^{\prime}$ | Symmetric strain rate tensor/Image above free fluid surface
$\mathbf{F}$, $\mathbf{F}^{\prime}$ | Stokeslet force and image Stokeslet force/Image above free fluid surface
$\mathbf{T}$, $\mathbf{T}^{\prime}$ | Rotlet torque and image rotlet torque/Image above free fluid surface
$\bar{\mathbf{v}}(\mathbf{r};\mathbf{p},\mathbf{r}_{0})$ | Singularity construction below free, nondeforming fluid surfaces
$\mathbf{u}$, $u_{ij}=\partial_{i}u_{j}$ | Displacement field and displacement gradient tensor
$u^{\alpha}$ |  2D strain components: dilation ($\alpha=0$), rotation ($\alpha=1$), shear ($\alpha=2,3$)
$\sigma^{\alpha}$ |  2D stress components: pressure ($\alpha=0$), torque ($\alpha=1$), shear ($\alpha=2,3$)
$B$, $\mu$ | Bulk modulus, Shear modulus
$A$, $K^{o}$ | Odd bulk modulus, Odd shear modulus
$\Lambda$, $\Gamma$ | Elastic moduli coupling rotations to stresses
$\nu$, $\nu^{o}$ | Poisson’s ratio, Odd ratio
$\gamma_{1}$, $\gamma_{2}$ | Effective material parameters of an isotropic elastic solid
$\hat{F}_{\text{nf}}$, $\hat{T}_{\text{nf}}$ | Stokeslet and rotlet fitting parameters for flows surrounding bound pairs and triplets
$\bar{a}$ |  Living chiral crystal lattice constant ($\approx 220\,\mu$m)
$\psi_{6}(\mathbf{r}_{i})$ |  Local hexagonal bond orientation parameter of embryo $i$
$g(r)$ | Radial pair distance distribution function
$\gamma_{L}(\tau)$ |  Dynamic Lindemann parameter as a function of lag time $\tau$
$\mathbf{b}$ | Burgers vector
$\phi_{\mathbf{b}}$ | Azimuthal Burgers vector orientation in cluster frame
$u_{s}$, $\alpha_{s}$ | Magnitude and orientation of the shear elongation axis
$\mathbf{x}=(u^{\alpha},u^{\beta})^{\top}$ | Strain space coordinate
$\rho(\textbf{x})$, $\mathbf{j}(\textbf{x})$, $\mathbf{D}(\textbf{x})$ | Probability density, probability current and diffusion matrix in strain space
$\hat{\rho}(\textbf{x})$, $\hat{\mathbf{j}}(\textbf{x})$, $\hat{\mathbf{D}}(\textbf{x})$ | Corresponding functions estimated from data
$H_{\alpha\beta}\ $ |  Strain cycle handedness averaged in the strain space $\mathbf{x}=(u^{\alpha},u^{\beta})^{\top}$
$\hat{\dot{S}}(\mathbf{r})$, $\hat{\dot{S}}_{\text{tot}}$ | Local and spatially integrated partial entropy production rate estimated from data

##  5 Extended Data Figures

Extended Data Figure 1: Uncropped embryo morphology images shown in Fig. 1b in
the main text.

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