# Improving Classifier Robustness through Active Generation of Pairwise
Counterfactuals

Ananth Balashankar, Xuezhi Wang, Yao Qin, Ben Packer, Nithum Thain,
Jilin Chen, Ed H. Chi, Alex Beutel
Google Research
USA Corresponding author<EMAIL_ADDRESS>
###### Abstract

Counterfactual Data Augmentation (CDA) is a commonly used technique for
improving robustness in natural language classifiers. However, one fundamental
challenge is how to discover meaningful counterfactuals and efficiently label
them, with minimal human labeling cost. Most existing methods either
completely rely on human-annotated labels, an expensive process which limits
the scale of counterfactual data, or implicitly assume label invariance, which
may mislead the model with incorrect labels. In this paper, we present a novel
framework that utilizes counterfactual generative models to generate a large
number of diverse counterfactuals by actively sampling from regions of
uncertainty, and then automatically label them with a learned pairwise
classifier. Our key insight is that we can more correctly label the generated
counterfactuals by training a pairwise classifier that interpolates the
relationship between the original example and the counterfactual. We
demonstrate that with a small amount of human-annotated counterfactual data
(10%), we can generate a counterfactual augmentation dataset with learned
labels, that provides an 18-20% improvement in robustness and a 14-21%
reduction in errors on 6 out-of-domain datasets, comparable to that of a fully
human-annotated counterfactual dataset for both sentiment classification and
question paraphrase tasks.

##  1 Introduction

Counterfactual data augmentation (CDA) has been used to make models robust to
distribution shift and mitigate biases towards spuriously correlated
attributes. Often, counterfactuals are generated as labeled examples through
pre-specified templates [1, 2] or crowd-sourcing [3]. While natural text
templates codify a specific number of assumptions of how counterfactual
sentences and labels might vary, crowd-sourcing that can cover various types
of counterfactuals, can be expensive. On the other hand, many existing methods
[4, 5, 6, 7] simply rely on a label-invariance assumption: the label of the
generated counterfactual example is the same as the corresponding original
example. However, this simple label-invariance assumption does not always hold
[8, 9, 10] and thus greatly increases the risk of using incorrect labels for
counterfactuals during training. For example, for many NLP tasks a small
perturbation can easily change the ground-truth label [3, 11], e.g., changing
the input from This movie is great to This movie is supposed to be great for
sentiment classification, or changing the hypothesis from The lady has three
children to The lady has many children for natural language inference.
Therefore, in this work, we mainly focus on addressing this challenging
research problem:

_“How can we automatically explore diverse counterfactual examples and learn
their labels, given a counterfactual text generator?”_

Beyond costly human annotation or simplifying assumptions of label invariance,
researchers have explored to use a classifier $f$ that has learnt to predict
the label on the original dataset $(X,Y)$. Such a classifier has been used to
directly label generated examples (our “trust” baseline; [3]) or to weight
generated examples based on the model uncertainty (our weighted-trust
baseline; [12]). However, we see that using such simplistic labeling
assumptions for counterfactual data augmentation have limited benefits for
improving robustness (defined as the accuracy over a counterfactual test set
of interest).

In this paper we propose an alternative approach to this problem: we leverage
the sample efficiency of generative models and exploration capabilities of
active learning [13] to 1) first generate a large number of diverse
counterfactuals, and 2) then train an auxiliary classifier to automatically
annotate the generated counterfactual data based on the difference between the
original and counterfactual labels. Specifically, we propose to generate
counterfactual examples that lie in the region of uncertainty of the
classifier $f$, and learn a pairwise classifier $h$ to predict the
counterfactual label $y^{\prime}$. The pipeline of our method is shown in
Figure 1.

In particular, we utilize a very small set of human-annotated counterfactual
examples to train the pairwise counterfactual classifier $h$, which takes in
the pair of original and counterfactual sentences $(x,c_{s}(x))$ and the
original label $y$ as input. Then in the inference stage, the pairwise
counterfactual classifier $h$ is used to predict the labels to produce a large
counterfactual augmentation dataset used to fine-tune $f$ to improve
robustness.

Figure 1: Overview of proposed approach: We propose to generate diverse
counterfactuals through active sampling, and label them using a pairwise
counterfactual classifier at scale. We use the labeled counterfactuals as data
augmentation over the original classifier (not shown) to significantly improve
robustness.

By using active sampling over counterfactual generators and auxiliary pairwise
counterfactual classifiers, we show that we greatly reduce the number of
counterfactual examples for which we need human annotation, while providing
similar gains in robustness comparable to a fully human annotated
counterfactual dataset. We attribute this to two core components of our
method. First, the active-learning based sampling method helps diversify the
types of generated counterfactuals and enables them to capture different
robustness issues not previously captured by our classifier model. Second, the
proposed auxiliary pair-wise classifier can automatically annotate the
generated counterfactuals with more accurate labels and help efficiently scale
up the size of the counterfactual augmentation dataset. Our core contributions
in this work include:

  * •

We propose an active-learning based sampling method to generate diverse
counterfactuals and effectively improve robustness of classifiers for the
sentiment classification task on Stanford Sentiment Treebank (SST-2) dataset,
and the question paraphrase task on Quora Question Pair (QQP) dataset.

  * •

We propose a novel pairwise counterfactual classifier to automatically label
counterfactually generated examples based on a small set of annotated
counterfactuals, improving sample efficiency of counterfactual data
augmentation.

  * •

The generated augmented dataset, which uses just 10% of human-annotated
labels, produces an improvement in counterfactual robustness of 18-20%,
comparable to a fully human annotated dataset, and a reduction in errors by
14-21% on out-of-domain datasets that were not used during training: IMDB,
Amazon, SemEval (Twitter), and Yelp reviews.

##  2 Related Work

Our work is built on advances from various domains as outlined below:

Adversarial Text Generation Training against adversarial examples which
perturb inputs in the vicinity of the existing training data by making
geometric assumptions [10, 14] on a lower dimensionality of the data to
improve robustness has been extensively studied recently. Natural examples
which are syntactically and semantically similar to the original sentence, but
produce different model predictions have been produced [7]. Similarly,
defenses against adversarial attacks on self-attentive models have shown
improvement in robustness to label invariant examples [15]. In FairGAN [4],
they showed it is possible for a discriminator to achieve statistical parity
on the real dataset, while performing the auxiliary task of detecting real and
generated examples. Such controlled adversarial generative approaches [16]
have demonstrated the effectiveness of automating data augmentation in text-
based tasks. Generative models which optimize for fluency have passed human
annotation checks where the model generated text is almost indistinguishable
from human generated ones [17, 18]. We build on this body of work and utilize
a generative model [19] that captures template-based counterfactuals to
improve robustness. Generic adversarial notions of robustness however
applicable, fail to incorporate specific counterfactuals directly in their
training and orthogonal to our scope of study. Through carefully disentangling
specific attributes and the rest of the latent variables in text, we generate
counterfactuals across all possibilities, and utilize human-annotated
templates to label a small fraction of the generated examples to train a
pairwise counterfactual classifier.

Semi-Supervised and Self-Supervised Learning Labeling functions which provide
crude estimates of the label have been used in semi-supervised methods [20,
21, 22], and are further used to learn a generative model to generalize over
them. Further, utilizing unlabeled data [23] to improve adversarial robustness
leverages geometric smoothing-based techniques to bridge the sample complexity
gap between accuracy and robustness [24]. Thus, semi-supervised learning
approaches aim to generate examples where the discriminator is least confident
about [12]. Language models with very large number of parameters have also
shown to be few-shot learners with minimal supervision [25]. Similarly,
reinforcement learning based approaches with minimal labels have been proposed
to combine the objectives of accuracy and counterfactual robustness [26].
Generalization against counterfactual examples by making models not to rely on
salient features (easy examples) have been extensively studied by modeling
biases in corpora [27, 28, 29, 30, 31, 32]. While the goal in these works have
been building ensembles or end-to-end bias mitigation models, our goal is to
minimize the number of human labels required to achieve an equivalent
improvement in robustness. In this spirit of efficiently capturing the
patterns already prevalent in the original dataset, and learning only the new
ones introduced in the counterfactual templates, we learn the pairwise
counterfactual classifier on a small number of samples, and use it to capture
the label variations in the remaining counterfactual dataset.

Counterfactual Applications The counterfactual datasets we use throughout this
paper were intended to highlight the shortcomings of existing models at the
time. Improving robustness through training on the augmented data has been
extensively explored [33, 34]. Learning how counterfactuals differ have been
explored by comparing against gradient supervision [35] and the
generalizability between original and counterfactuals [3]. The generated
counterfactuals have also been used for explanations [36], highlighting biases
[1] and debiasing through statistical methods [37]. This rich set of contrast
sets [11], checklists [8], paraphrases [38, 39], adversarial schemes [40] and
lexical diagnostic datasets [41] form the foundation of our method, which re-
purposes them to build a counterfactual generative model and improve
counterfactual robustness.

Generative Learning Generative adversarial active learning has been proposed
with pool-based and synthesizing-based sampling strategies [42]. While the
pool-based strategy selects from an existing sample of generated examples, the
synthesizing-based sampling re-samples from the generator based on information
theoretic measures like mutual information [43, 44] or model uncertainty [45]
or informativeness [46, 13] over the initial sample. Further, recent work has
highlighted that using large language models for generation and annotation of
that generated data can be very useful [47, 48]. We build on this work, and
use the synthesizing-based Bayesian Generative Active Learning approach [44],
where we condition on a counterfactual with low uncertainty in the model, to
actively generate more counterfactual examples. We then iteratively sample the
examples with the highest classifier uncertainty and annotate manually. The
human annotations are then used to train the pairwise classifier $h$, which is
then used to scale the annotation process for all other generated
counterfactuals.

##  3 Methodology

### Problem Framing

Let $x$, $y$ be the input sentence and its associated label in the original
dataset, respectively. We assume $y\in\\{0,1\\}$ throughout the paper (i.e.,
we focus on binary classification tasks), but our framework can be extended to
multi-class tasks as well.

Our core challenge is what is the true label $y^{\prime}$ for a generated
counterfactual $x^{\prime}$? Although we can further obtain human annotations,
this can quickly become time consuming and budget intensive to do at scale. If
we make the simplified assumption of label invariance throughout the
counterfactual inputs $x\textquoteright$ generated, which is a common
assumption in adversarial literature [49, 6, 7], we could end up with an
incorrect counterfactual dataset which might hurt robustness and accuracy. Our
goal is thus, to generate a counterfactual augmentation dataset that produces
a comparable improvement in accuracy and robustness as that of human-annotated
counterfactuals with minimal supervision.

We frame this problem as how to learn when the labels flip, i.e., identifying
when the label of the counterfactual is different from the label of the
original sentence: $P(y\neq y\textquoteright)=\delta$, $(0<\delta<1)$, in the
counterfactual distribution $x\textquoteright\in X\textquoteright$. Given a
generation model $c$, we denote $c_{s}(x)$ as the generated counterfactual
over $x$ by changing an attribute $s$ in $x$. We also assume that a classifier
$f:X\rightarrow Y$ has been learnt on the original dataset $(X,Y)$ by
optimizing for accuracy $A$.

$\displaystyle A=$ $\displaystyle E_{(x,y)\in(X,Y)}\mathbb{I}(f(x)=y)$ (1)

In our paper, the objective is to use the counterfactual data to train a model
$f^{\prime}$ that improves robustness, i.e., to make sure the models we
trained generalize to unseen scenarios. We measure this by the counterfactual
accuracy $\tilde{A}$ of $f$ on multiple held-out counterfactual datasets
$(X^{\prime},Y^{\prime})$ split based on domains (OOD), patterns (e.g.
negation, insertion):

$\displaystyle\tilde{A}=$ $\displaystyle
E_{(x^{\prime},y^{\prime})\in(X^{\prime},Y^{\prime})}\mathbb{I}(f^{\prime}(x^{\prime})=y^{\prime})$
(2)

In the remainder of this section, we first explain how the counterfactuals are
generated using active learning. Then, we explain how we account for the
possibility that the label of the generated counterfactual $c_{s}(x)$ might
have flipped, using a pairwise classifier. Finally, we explain how when both
these components are combined, we can further improve robustness.

### Active Counterfactual Generation

To achieve the goal of improving counterfactual accuracy on held-out
counterfactual datasets (Eqn 2), we use a controlled generative model to
generate additional training counterfactual data $c_{s}(x)\in X_{t}^{\prime}$
(here the subscript $t$ denotes the training set) that modifies original input
$x\in X$ based on the attribute $s$. In natural language tasks, the attribute
$s$ cannot be directly inferred from the sentence $x$ and hence we rely on
templates to define the types of counterfactual (e.g., negation, insertion,
deletion) as commonly used in [8, 19] to infer the attribute $s$. Let $y\in
Y,y^{\prime}\in Y_{t}^{\prime}$ be the label for the original and
counterfactual sentences in our counterfactual training dataset. The training
objective of robustness is to minimize the error $\tilde{\mathcal{E}}_{t}$ of
the model $f$ aggregated by attribute $s$ on the training counterfactuals
$(X_{t}^{\prime},Y_{t}^{\prime})$, where $CE$ refers to the cross-entropy
loss, as follows:

$\displaystyle\tilde{\mathcal{E}}_{t}(s)$ $\displaystyle=E_{x\in
X,(c_{s}(x),y^{\prime})\in(X_{t}^{\prime},Y_{t}^{\prime})}CE(f(c_{s}(x)),y^{\prime})$
(3) $\displaystyle\tilde{\mathcal{E}}_{t}$ $\displaystyle=E_{s\in
S}\tilde{\mathcal{E}}_{t}(s)$ (4)

The counterfactual generator that optimizes the above cross-entropy loss then
generates several counterfactuals $c_{s}(x)$ by relying on instructions
provided in controlled generation methods [19] such as “negation”,
“restructure”. However, these counterfactuals are not necessarily diverse, and
fails to incorporate the classifier’s uncertainty to get the most informative
set of generated counterfactuals. To improve generalization across a diverse
set of counterfactual types, we fine-tune the generator to actively sample
counterfactuals the most informative set from the unlabeled dataset $x*\in
X_{t}^{\prime}$ (BALD, [43]) that synthesizes examples by maximizing the
acquisition function given by the Monte Carlo (MC) dropout approximation
method [45] using the class-wise probability scores of the pairwise classifier
$h$, where $H[y|x,.]$ represents the Shannon entropy of the corresponding
conditional probability:

$\displaystyle x^{*}=argmax_{x^{\prime}\in X_{t}^{\prime}}$
$\displaystyle[H[y^{\prime}|x^{\prime},X,Y]$ (5) $\displaystyle-$
$\displaystyle\mathbb{E}_{x\in X_{t}}H[y^{\prime}|x^{\prime},x,f(x)]]$

Since $y^{\prime}$ is not readily available for counterfactual generated
sentences $c_{s}(x)$ in our training dataset and gathering them for all
examples can be expensive, our goal is to minimize the number of human-
annotations of counterfactuals $y^{\prime}$ in the training dataset
$Y_{t}^{\prime}$, while achieving comparable improvement in robustness (Eqn
2). Hence, the training sentence and label set
$(X_{t}^{\prime},Y_{t}^{\prime})$ can be decomposed into two sets, one whose
labels are human-annotated: $(X_{a}^{\prime},Y_{a}^{\prime})$ and the other
with model generated labels: $(X_{g}^{\prime},Y_{g}^{\prime})$, such that
$X_{t}^{\prime}=X_{a}^{\prime}\cup
X_{g}^{\prime},Y_{t}^{\prime}=Y_{a}^{\prime}\cup Y_{g}^{\prime}$. Our goal is
to automatically discover informative counterfactuals $X^{\prime}_{g}$ and
learn their labels with access to a limited human-annotated counterfactual
data ($X^{\prime}_{a},Y^{\prime}_{a}$), where
$|Y^{\prime}_{a}|\ll|Y^{\prime}_{g}|$, while achieving counterfactual
robustness $\tilde{A}$ (Eqn 2) comparable to the scenario when all the
training labels are human-annotated.

### Pairwise-Counterfactual (PC)

In order to generate labels for the counterfactuals, we construct a novel
auxiliary pairwise classifier $h$, which at inference time, takes in as input
both the original dataset $(x,y)\in(X,Y)$, and a corresponding counterfactual
$c_{s}(x)\in X_{g}^{\prime}$, to output $y^{\prime}\in Y_{g}^{\prime}$. This
classifier $h$ is trained on _pairs_ of input sentences $x,c_{s}(x)$ and the
original label $y$ to predict the human-annotated label $y^{\prime}\in
Y_{a}^{\prime}$.

Specifically, the classifier $h$ takes in the original input sentence $x$ and
its associated label $y$, as well as its corresponding counterfactual example
$c_{s}(x)$. The output of the classifier $h(x,c_{s}(x),y)$ is the predicted
label of the counterfactual example $c_{s}(x)$. In the training stage, the
classifier $h$ is optimized on the counterfactual examples with human-
annotated labels $(c_{s}(x),y^{\prime})\in(X_{a}^{\prime},Y_{a}^{\prime})$ via
minimizing the loss function:

$\ell_{h}=E_{\begin{subarray}{c}(x,y)\in(X,Y)\\\
(c_{s}(x),y^{\prime})\in(X_{a}^{\prime},Y_{a}^{\prime})\end{subarray}}CE(h(x,c_{s}(x),y),y^{\prime})$
(6)

With the well-trained classifier $h$, we can generate the labels for any
counterfactual example $c_{s}(x)\in X^{\prime}_{g}$ (the counterfactual set
without human annotation) according to:

$y^{\prime}=h(x,c_{s}(x),y):(x,y)\in(X,Y),c_{s}(x)\in X_{g}^{\prime}$ (7)

### Classifier-Aware Pairwise-Counterfactual (CAPC)

Additionally, since we know that $f$ is already optimized to predict the label
accurately on the original dataset, the auxiliary classifier $h$ could
potentially leverage $f$ in its pairwise prediction through transfer learning.
Specifically, if we decompose the counterfactual distribution
$(X^{\prime},Y^{\prime})$ as a mixture of samples from the original
distribution $(X,Y)$ and those that are independent of the original
distribution, we would benefit by training $h$ to identify samples from the
latter distribution. In addition, assuming the correspondence between $f(x)$
and $f(c_{s}(x))$ is easier to learn (e.g., with a lower model complexity), we
could also benefit from learning a classifier-aware function to better capture
this correspondence. Thus, we propose to augment the predictions of the
original classifier $f(x),f(c_{s}(x))$ as input to $h$ as follows:

$\displaystyle y^{\prime}\in Y_{g}^{\prime}=h(x,c_{s}(x),y,f(x),f(c_{s}(x))):$
(8) $\displaystyle(x,y)\in(X,Y),c_{s}(x)\in X_{g}^{\prime}$

Any uncertainty that $f$ has on the counterfactual samples $P(f(c_{s}(x))\neq
y^{\prime})$ can be mitigated by the auxiliary classifier $h$ by identifying
patterns in $c_{s}(x)$ when $f$ predicts incorrectly. As a simple example,
without any human annotation, the original model $f$ might make incorrect
assumptions on $c_{s}(x)$ that lead to incorrect predictions $f(c_{s}(x))\neq
y^{\prime}$, e.g., a sentiment analysis model might give “positive” sentiment
predictions due to the presence of qualifiers like “terrific”, “amazing”
(_this movie was amazing_) even when the counterfactual input $c_{s}(x)$
alters aspects of a sentence that changes the label (_this movie was supposed
to be amazing_). But, this can be corrected using Eqn 8 after $h$ has observed
some data over the correct correlation between $x,c_{s}(x),y,f(x),f(c_{s}(x))$
and $y^{\prime}$, especially if there exists a lower-complexity function
mapping between them - for instance, adding the phrase “supposed to be" may
alter the label of a review.

##  4 Evaluation

We evaluate on two NLP tasks, sentiment classification and question
paraphrase, using two datasets namely the Stanford Sentiment Treebank (SST-2)
[50] and the Quora Question Pair (QQP) [51, 52]. When the CAPC classifier is
used in conjunction with generated examples through active learning, we
correspondingly prefix the model name as p-CAPC (pool-based sampling with no
retraining as per Eqn 5) or s-CAPC (examples synthesized with re-generation of
counterfactuals optimizing Eqn 5).

### Counterfactual Generator: Polyjuice

We use a general purpose counterfactual text generator called Polyjuice [19],
which extends CheckList [8], that has shown promise by improving diversity,
fluency and grammatical correctness as evaluated by user studies. It covers a
wide variety of commonly used counterfactual types including patterns of
negation [3], adding or changing quantifiers [11], shuffle key phrases [38],
word or phrase swaps which do not alter POS tags [40] or parse trees [39],
along with insertions or deletion of constraints that do not alter the parse
tree [41]. Specifically, we use 8 types of counterfactuals - negation,
quantifier, lexical, resemantic, insert, delete, restructure, shuffle; in
Polyjuice to generate the augmented dataset. Other text generative models like
[5, 3, 6] that improve adversarial robustness or like [53, 54] that allow
controlled generation could be used as well.

### Experiment Setup

We test our methods on two popular text datasets. We briefly describe the two
datasets below, and discuss the different evaluations of counterfactual
robustness we perform over them.

Stanford Sentiment Treebank (SST-2): The sentiment analysis task in SST-2 [50]
assigns a binary sentiment (negative/positive) to a sentence mined from
RottenTomatoes movie reviews. The corresponding counterfactuals are generated
using the Polyjuice generator [19]. The original dataset contained 4,000
samples, while the counterfactual dataset had 2,000 samples with human labels
against which we evaluate. We show a sample of the dataset in the following:
Positive: A dog is embraced by the dog Negative: A dog is not embraced by the
dog

Quora Question Pair: In the QQP dataset [51, 52], given a pair of questions,
the task is to predict if they are semantically equivalent, hence marked as
duplicate. Here, again the second question is modified by Polyjuice [19] as
per the templates used for the SST-2 dataset including negation, insertion,
deletion, rephrasing, etc, out of which 1,911 samples were human annotated for
evaluation. The original dataset had 20,000 samples.
Duplicate: How can I help a friend experiencing serious depression?; How can I
help a friend who is in depression? Non-duplicate: How can I help a friend
experiencing serious depression?; How can I play with a friend who is in
depression?
Evaluation: In both datasets, we have a small number of counterfactual human
annotations available (SST-2: 2,000; QQP: 1,911) [19]. We divide these
examples into two sets, one for training and annotating using $h$, and another
held-out test dataset used to compute counterfactual robustness of $f$. The
former dataset is used for fine-tuning $f$ for counterfactual robustness,
while the latter is used only as a held-out test set. In the SST-2 dataset,
this means we split out 1,000 samples for training/annotation and 1,000 as the
test set, while in the QQP dataset, we use 1,000 samples for
training/annotation and the remaining 911 samples for testing counterfactual
robustness. However, our aim is to use a minimal subset of the 1,000 samples
available for training the base classifier directly. Instead, we use a smaller
training dataset (100) to train our pairwise classifier which in-turn can then
_artifically_ annotate the remaining (say 900) samples. The combination of
these (sum to 1000) will then be used to train the base classifier. Thus, in
all our experiments, the number of counterfactual samples available to the
base classifier to train on remains the same, although at different levels of
human labeling costs.

The classifier $f$ is first trained on the original classifier and then fine-
tuned on the counterfactual dataset. We also perform 10 random initializations
of the model $f$ and $h$ and a 10-fold cross-validation split on the
training/annotation data, thus report the mean and standard error bounds
$\sigma/\sqrt{n}$ over $n=1000$ runs for each model-based annotation and
training for counterfactual robustness. We used the standard hyperparameters
provided in RoBERTa for training $f$ on $(X,Y)$ and the hyperparameters for
fine-tuning $f$ on $(X_{t}^{\prime},Y_{t}^{\prime})$ include learning rate of
$5e^{-5}$, batch size of $16$ and a sequence length of 120 for 20 epochs. The
pairwise counterfactual classifier’s hyperparameters were chosen after a grid
search to have a learning rate of $5e^{-4}$, batch size of $32$ for 50 epochs,
sequence length of $240$ including the original label and classifier
predictions with special marker characters. While the base classifier $f$ is
trained on contextual embeddings of the sentence(s), $h$ is trained by further
augmenting the original and counterfactual sentence embeddings as input to
RoBERTa followed by the base classifier’s predictions separated by special
delimiters [DEL]. A similar 10-fold cross-validation split is used to finetune
the parameters of the classifier $h$.

Out-of-Distribution (OOD): To test the methodology on out-of-domain datasets,
we test on sentiment analysis tasks in 6 reviews datasets - IMDB movie (3
including contrast sets) reviews, Amazon, SemEval, and Yelp reviews [55]. The
IMDB reviews (1,700) were collected by [3] through careful human elicitation
to produce label varying counterfactuals of existing IMDB reviews. In the Yelp
reviews [56], the task is to predict the ratings of 115,907 reviews on a scale
of 1-5, and in the Amazon reviews [57], we evaluate on the 57,947 reviews in
the clothing product category. Each of these datasets was not used for
training either the base classifier or the pairwise classifier, and the
training relies solely on the SST-2 dataset. So, we can measure the
generalizability of the pairwise classifier based data augmentation
methodology.

### Baselines

We now briefly describe five different baselines used to generate the labels
of counterfactual augmented data ($Y_{g}^{\prime}$), given access to a small
number of annotated labels $Y_{a}^{\prime}$.

  * •

No-cda: $f$ without any counterfactual data used for robustness.

  * •

Label-invariant (invariant) : the labels of the counterfactual examples are
assumed to be the same as the original sentence: $y^{\prime}=y$ (except for
the counterfactuals generated for the negation type, where it is the
opposite).

  * •

Trust: we trust the classifier $f$ to annotate the counterfactual labels
$y^{\prime}=f(c_{s}(x))$ \- a form of semi-supervision based on the existing
base classifier.

  * •

Weighted-trust (w-trust): the label of the counterfactual example is computed
via the maximum score weighted by the confidence score of the classifier $f$
on the pair for a label $l:$ $p_{l}(x)$ such that
$y^{\prime}=\arg\max_{l}p_{l}(x)\cdot p_{l}(c_{s}(x))$.

  * •

Random: In order to understand the importance of the counterfactual sentences
used in the pairwise classifier, we also evaluate against a classifier which
takes two randomly paired sentences from the original dataset as input and
predicts the second label given the label of one sentence.

  * •

Training: we only use those counterfactual examples with human-annotated
labels ($X^{\prime}_{a},Y^{\prime}_{a})$ and drop all other counterfactual
examples.

For all these baselines as well as our proposed methods, we use the RoBERTa
[58] fine-tuned model as the choice of classifier $f$, and a corresponding
pairwise fine-tuning task using RoBERTa 111huggingface.co/roberta-large-mnli,
textattack/roberta-base-SST-2, ji-xin/roberta_base-QQP-two_stage for the
auxiliary pairwise counterfactual classifier $h$.

##  5 Results

### Improving Counterfactual Robustness

To demonstrate the effectiveness of our proposed method: actively synthesized
classifier-aware pairwise-counterfactual (s-CAPC), we perform counterfactual
data augmentation using 10$\%$ counterfactual examples with human-annotated
labels as well as 90$\%$ counterfactual examples (a total of 1,000 samples),
whose labels are predicted using each method. The error rate on the hold-out
counterfactual examples (referred as robustness) as well as on the original
test set are shown in Figure 2.

Figure 2: (a) Robustness: (first row) Training on 10% of human-annotated
counterfactuals, and annotating the rest using the auxiliary classifier, we
achieve a comparable improvement in robustness (lower error rate) for both
Stanford Sentiment and Quora Question Pair datasets; (b) Accuracy:  This
improvement in robustness does not sacrifice the accuracy on the original
held-out dataset.

We can clearly see that (1) the error rate of our proposed method: s-CAPC
significantly outperforms other baselines on models’ robustness. (2) Comparing
PC and CAPC, we can see that CAPC performs slightly better than PC. This
indicates that the prediction of the original classifier $f(x),f(c_{s}(x))$
does provide additional information to help with labels prediction. (3) In
addition, we also compare our methods with the extreme case that all the
counterfactual examples (100$\%$) are provided human-annotated labels, denoted
as (human-labels). Surprisingly, our methods, which only use 10$\%$ human-
annotated labels and predict the labels for the other 90$\%$ counterfactual
data, achieve comparable performance in improving models’ robustness. This
sufficiently supports that our proposed methods can effectively predict the
labels for counterfactual examples. (4) Looking at the error rate on the hold-
out original test set, all the methods share a similar performance on SST-2
and our methods are better than other baselines and comparable to human-labels
on QQP.

### How much human-annotated data do we need?

To understand the impact of the training data provided to the auxiliary
classifier $h$, we increased the % of data $Y_{a}^{\prime}$ provided to the
classifier. While this increases costs of annotation, it is important to
understand the headroom improvement in counterfactual robustness one would get
had they opted for complete human-annotation. Figure 3 shows that across both
datasets, the improvement in accuracy and robustness in providing more human
annotations to train $h:$ CAPC and subsequently training the model $f:$
RoBERTa-{SST-2, QQP} is not significant and hence further demonstrates that,
with just 10% of the augmentation dataset, we can already achieve an
improvement comparable to a fully human annotated dataset. This further
confirms our method can achieve high sample efficiency in improving models’
robustness.

Figure 3: Impact of training size: As the number of samples $|Y_{a}^{\prime}|$
increases more than $10\%$ in the s-CAPC model , there is not much headroom in
counterfactual accuracy, and does not significantly impact the accuracy on the
held-out original test dataset on both SST-2 and QQP datasets (overlapping
error bounds).

### Generalization across Counterfactual Types

We evaluate the generalization of our pairwise counterfactual classifier $h$
by ablating one counterfactual type (e.g negation, quantifier, etc) at a time
during training $h$, but still annotate them to generate the augmented
training data for $f$. The results are shown in Table 1 (rows 2-9). We see
that for the SST-2 task, our approach outperforms existing baselines on
counterfactual robustness. This further indicates the importance of learning a
counterfactual classifier which captures patterns of label invariance that
generalizes across counterfactual templates. Finally, we evaluate if our
generated augmentation dataset can be used to improve unseen counterfactual
types - ablated while training both $h$ and $f$. While this is not the goal of
our paper, it is useful to understand what types of counterfactuals are
captured by our generator and if any overlap between the types of
counterfactuals is leveraged. Table 1 (row 10) shows that our approach is
comparable with baselines (rows 2-5 in Table 1) when a specific counterfactual
type is ablated completely from the data augmentation pipeline. This is
consistent with existing work [59, 60] and further highlights the need to
incorporate diverse types of counterfactuals to perform data augmentation.

Sliced Error by Counterfactual Type $\%$
---
Model | negation | quantifier | lexical | resemantic | insert | delete | restructure | shuffle
s-CAPC-no-ablation | 2.20 | 1.81 | 1.94 | 1.40 | 2.01 | 1.75 | 2.01 | 2.12
no-cda | 19.12 | 18.10 | 21.40 | 20.65 | 17.54 | 20.99 | 18.32 | 17.42
invariant | 14.62 | 4.82 | 4.32 | 3.10 | 7.72 | 7.83 | 6.48 | 9.24
trust | 12.96 | 4.15 | 4.73 | 3.00 | 4.95 | 12.49 | 3.74 | 9.02
w-trust | 5.09 | 3.55 | 8.91 | 10.60 | 7.72 | 5.57 | 10.51 | 10.60
random | 4.74 | 4.04 | 6.92 | 2.22 | 7.42 | 5.55 | 5.72 | 4.96
training | 4.53 | 3.53 | 6.32 | 2.62 | 7.24 | 5.32 | 5.83 | 4.83
Slice error when counterfactual type is ablated from training $h$
PC | 4.50 | 5.35 | 2.73 | 3.20 | 2.12 | 2.13 | 5.30 | 5.10
CAPC | 4.04 | 2.20 | 4.76 | 2.10 | 4.56 | 4.67 | 3.56 | 4.50
p-CAPC | 3.12 | 2.01 | 2.21 | 1.78 | 2.66 | 2.65 | 2.08 | 2.48
s-CAPC | 2.54 | 1.84 | 2.19 | 1.46 | 2.07 | 1.86 | 2.02 | 2.14
Sliced error when counterfactual type is ablated from training $h$ and $f$
s-CAPC | 11.17 | 13.02 | 7.55 | 13.33 | 4.98 | 5.76 | 10.77 | 9.01

Table 1: Generalization of Counterfactual Types: Comparison of error rates (%)
sliced by different counterfactual sentence types shows that our approach
s-CAPC continues to perform well even when those types are held out during
training $h$. However, when we ablate the counterfactual type both while
training $f$ and $h$, our approaches performs comparably to the baselines
sliced error rates. This shows that $h$ does not just memorize the templates,
but training on diverse counterfactual types continues to be important for
robustness.

Test error rate % |
---|---
Model | IMDB | Yelp | Amazon | SemEval | IMDB-cont | IMDB-CAD
no-CDA | 9.2 | 15.7 | 20.0 | 15.2 | 7.8 | 13.5
invariant | 11.3 | 15.9 | 21.5 | 15.4 | 8.0 | 13.8
trust | 9.3 | 15.8 | 20.5 | 15.5 | 8.1 | 13.8
w-trust | 9.2 | 15.5 | 20.2 | 15.5 | 8.0 | 13.7
random | 10.4 | 16.3 | 23.8 | 17.2 | 9.5 | 14.3
PC | 8.0 | 14.3 | 18.1 | 14.2 | 7.4 | 12.9
CAPC | 7.2 | 13.1 | 17.2 | 13.6 | 6.0 | 10.3
p-CAPC | 9.2 | 13.7 | 15.9 | 13.6 | 5.9 | 10.1
s-CAPC | 7.2 | 10.4 | 12.9 | 11.9 | 5.5 | 9.9
domain-trained | 6.7 | 10.0 | 11.7 | 10.8 | 5.4 | 9.5

Table 2: Out-of-domain reviews: Using data augmentation with SST-2
counterfactuals from the Polyjuice generator and classified using s-CAPC
performs comparable to a model trained on within-domain data.

### Checklist Evaluation

To further validate that the generated labels by our auxiliary model can be
used for other tasks, we evaluate it against the labels in CheckList [8] which
capture other types of counterfactuals. We measure the _Absolute Failure Gap_
: $|\epsilon-\epsilon_{a}|$ computed as the difference between the true error
rate $\epsilon$ and the error rate as reported by using our augmented dataset
$\epsilon_{a}$ while evaluating the models and tasks in the CheckList dataset.
In Figure 4, we see that even when the training data provided to the auxiliary
classifier is synthetically made explicitly label-invariant (90%), evaluating
against counterfactuals with minimal label-invariance (10%), our model
generalizes with a lower failure gap than other augmentation approaches.
However, on the original Checklist dataset there is no significant improvement
in failure gap compared to reporting the failure gap just on the training data
alone.

Figure 4:  Checklist Evaluation - (a) Out of distribution data: Our methods
perform well over different label-invariant distributions with 90%
counterfactual label flips ($y\neq y^{\prime}$) in the Checklist dataset even
when the training distribution has only 10% counterfactual label flips; (b)
Model Comparison: However, on the original Checklist dataset [8], we achieve a
comparable failure gap with the golden error rate to other model-based
annotations

### Out-of-Domain Reviews

To validate that the counterfactuals we augment through our pairwise
classifier’s annotations have generalizability to 6 out-of-domain datasets, we
evaluate the reduction in error rates of the base RoBERTa model when they are
trained on the pairwise classifier’s data augmentation in Table 2. In the IMDB
reviews dataset [61], we see an improvement in error rates from 9.2% without
data augmentation to 7.2% through CAPC and s-CAPC. This out-of-domain error
rate is comparable to the error rate obtained by the model trained by [3]
after incorporating samples from the counterfactuals drawn from the same
distribution as part of the training (6.7%). In the Yelp reviews too
222https://www.yelp.com/dataset/, we see a reduction from 15.7% to 10.4%
whereas other baseline approaches lead to an increase in error rates. In the
Amazon reviews, the s-CAPC approach (12.9%) outperforms the baselines and is
comparable to the augmentation from the training split from the Amazon reviews
(11.7%). Similar improvements can be seen on the SemEval [62] and IMDB
contrast sets (IMDB-cont, IMDB-CAD) [11, 3]. Each of these improvements has to
be viewed with the context that it was achieved in a more sample efficient
manner (1,000 counterfactuals generated from the original SST-2 dataset by
Polyjuice) as compared to the in-distribution training approach, where the
training data has 3,400 samples from their own respective datasets. This
further confirms that training on augmented counterfactuals using a generator
and pairwise classifier approach is comparable to human-annotated samples from
other domains, while providing us the ability to scale both in terms of domain
generalization as well as labeling efficiency.

##  6 Limitations

In this work, we have demonstrated new methods to safely use more diverse
counterfactuals and their value, but in taking on this broader goal, we
discover a number of further steps that could take the work further forward.
One of the limitations of our paper is that the set of counterfactuals we
improved robustness over is limited and restricted to perturbations in the
English language. Our analysis indicates the value of using more diverse
counterfactual types that require a case-by-case contextual understanding. We
show that adding more counterfactual types can be done in a sample efficient
manner by using a generator trained to produce counterfactuals. However, this
still suffers from the limitation that to extend to more counterfactuals and
languages, a classifier which labels them by training on a small set of human
annotations is required. Further, we do not investigate the quality of the
counterfactuals annotated, and we do not study the performance using more
nuanced counterfactuals with low levels of inter-rater agreement. Since we use
an auxiliary classifier to label the generated counterfactuals, the risk of
label drift remains a clear challenge and we do not control for this label
drift based on the certainty of these labels from the auxiliary classifier.
Further, a natural drift in concepts based on active exploration might render
invalid sentences that are not grammatically or semantically correct, and new
methods would be needed to filter based on these text patterns.

As in other generative models, the risk of perpetuating or amplifying biases
in the generated text data continues to be important and while we believe
counterfactual generation and augmentation can help address such biases, there
is also uncertainty in using more flexible, generated counterfactuals. For
example, it is quite possible that one of the generated counterfactuals relies
on an identify term in the generated sentence, and attributes a negative
sentiment spuriously based on prevalent stereotypes in the text corpus. For
this reason, we refer the reader to incorporating bias mitigation strategies
like [63] in addition to improving counterfactual robustness.

While we show generalization across label variance in templates, we cannot
guarantee that by learning solely on label invariant counterfactuals, a
classifier can generalize over label modifying counterfactuals or obtain the
same levels of sample efficiency on harder classification tasks. While
generators like Polyjuice [19] have been evaluated for fluency, diversity,
etc., there is a need to evaluate them within the context of a task and its
labels. However, the gains in robustness shown in Figure 3 and Table 2 further
illustrate the need for dataset generation in an efficient manner. As future
work, one can also look towards an efficient crowdsourcing strategy that
minimizes the gain provided by the pairwise classifiers as each sample in the
annotated dataset provides a unique and diverse counterfactual.

##  7 Conclusion

Counterfactual Data Augmentation approaches have been extensively used to
train for counterfactual robustness. As the types of counterfactuals - both
label-invariant and label-modifying, over which to evaluate natural language
models increase, there is a need to adopt a methodology that can scale with
increasing types of counterfactuals. We overcome a significant challenge in
doing so, by learning an auxiliary pairwise counterfactual classifier that
leverages the patterns of counterfactuals produced by vairous generative
models. Using only a small amount of human annotated counterfactual samples,
we demonstrate that our method can produce a dataset that improves
counterfactual robustness comparable to a fully human-annotated dataset.

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Mukherjee et al

*

# Resolved simulations of jet-ISM interaction: Implications for gas dynamics
and star formation

Dipanjan Mukherjee  Geoffrey V. Bicknell  Alexander Y. Wagner  Inter-
University Centre for Astronomy and Astrophysics (IUCAA), Pune University,
Ganeshkhind, Pune - 411007, India  Research School of Astronomy and
Astrophysics, Australian National University, Canberra, ACT 2611, Australia
Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai,
Tsukuba, Ibaraki, 305-8577<EMAIL_ADDRESS>
(26 April 2016; 6 June 2016; 6 June 2016)

###### Abstract

Relativistic jets can interact with the ambient gas distribution of the host
galaxy, before breaking out to larger scales. In the past decade several
studies have simulated jet-driven outflows to understand how they affect the
nearby environment, and over what spatial and temporal scales such
interactions occur. The simulations are able to capture the interaction of the
jets with the turbulent clumpy interstellar medium and the resultant
energetics of the gas. In this review we summarise the results of such recent
studies and discuss their implications on the evolution of the dynamics of the
gas distribution and the star formation rate.

###### keywords:

galaxies: jets, galaxies: evolution, hydrodynamics, methods: numerical

††articletype: Article Type

D. Mukherjee, G.V. Bicknell, and A.Y. Wagner, (2021), Resolved simulations
jet-ISM interaction: Implications for gas dynamics and star formation, Astron.
Nachr., 01.

##  1 Introduction

Feedback from active galactic nuclei (AGN) in the form of outflows is an
important driver of galaxy evolution (Silk Rees, 1998; Fabian, 2012). Large
scale relativistic jets powered by the central black hole are one such means
by which the AGN can transfer energy to its environment, affecting the stellar
mass assembly of galaxies (Fabian, 2012). Although relativistic jets have
primarily been considered in the context of large scale heating of the circum-
galactic environment to prevent cooling flows (Gaspari ., 2011; Fabian, 2012;
Yang Reynolds, 2016), studies have established that jets can also strongly
interact with the gas in the host galaxy itself, before evolving to larger
structures.

Understanding the impact of compact jets evolving through their host is also
of interest for the studies of a class of radio galaxies, called the Giga-
Hertz Peaked Spectrum (GPS) or Compact Steep Spectrum (CSS) sources (O’Dea,
1998; O’Dea Saikia, 2021). These sources are characterised by a turnover in
their radio spectrum, resulting either from synchrotron self-absorption inside
the radio emitting plasma itself (Kellermann ., 1969) or free-free absorption
by the ionised dense gas, external to the jet (Bicknell ., 1997, 2018). Which
of the above processes dominate is still a question of debate, with both
giving equally acceptable interpretation of observations (O’Dea Saikia, 2021).
However, recently, several radio galaxies with steeply inverted optically-
thick spectral indices $>2.5$ have been discovered (Gopal-Krishna ., 2014;
Callingham ., 2015; Mhaskey ., 2019, 2020). This supports a free-free process
as the absorbing mechanism and implies the presence of dense gas surrounding
the jets. Additionally, high rotation measures and lower polarisation in such
sources (O’Dea, 1998; O’Dea Saikia, 2021), along with spatially resolved
observations of shocked emission from the local gas (Zovaro, Sharp ., 2019;
Zovaro, Nesvadba ., 2019), further support the scenario of such sources
harbouring jets that are evolving through a gas-rich environment (see also
Patil ., 2020). Thus the physics of jet-ISM interactions is an important topic
to understand in the context of young evolving jets.

The inability of large scale cosmological simulations to resolve the
interaction of AGN driven outflows with individual dense molecular clouds due
to restricted spatial resolutions, calls for zoomed-in simulations of isolated
galaxies. That such an interaction between a fast supersonic jet and a thermal
cloud may occur, has long been theorised (Blandford Königl, 1979), and further
promoted by the ‘dentist drill’ scenario of evolving young jets (Scheuer,
1982; Bicknell ., 1997). With the advent of better computational resources,
non-relativistic simulations of jet-cloud interactions have been performed by
a number of authors (such as Bicknell ., 2003; Saxton ., 2005; Sutherland
Bicknell, 2007; Gaibler ., 2011, 2012; Dugan ., 2017; Cielo ., 2018). In
recent times, development of robust numerical techniques to solve for the
relativistic hydrodynamic flows have resulted in several papers exploring how
relativistic jets break out of a dense ISM of its host (e.g. Wagner Bicknell,
2011; Mukherjee ., 2016). We review below some of the results of these works
and their implications on the evolution of the host galaxy.

##  2 Resolved simulations of Jet-ISM interaction

Interaction of a relativistic jet with an ISM with fractal clouds were
investigated in a series of papers, starting with Wagner Bicknell (2011) and
Wagner . (2012). A relativistic jet was propagated through a lognormally
distributed static cloud distribution placed into an otherwise homogeneous
atmosphere. These were followed up by simulations where the fractal ISM with a
turbulent velocity dispersion was first allowed to settle in a galactic
potential followed by a jet launched from the centre (Mukherjee ., 2016, 2017;
Bicknell ., 2018; Mukherjee, Wagner ., 2018; Mukherjee, Bicknell ., 2018).
Some of the key results of these studies can be summarised as follows:

  * •

The flood and channel phase:

Figure 1: 3D Volume rendering of the jet tracer and dense clouds of simulation
E of Bicknell . (2018) at $t\sim 1$ Myr. The primary axis of the jet with the
highest concentration of the tracer is shown in magenta in the centre. The
head of the jet stream shows bifurcations into filaments due to interaction
with clouds. This is better represented in the image of the density in the
$y_{Z}$ plane shown in the inset in the top right corner. Surrounding the
central jet axis is a diffuse filamentary cloud of jet plasma, depicted in
orange, forming the so-called ‘flood and channel’ phase. This results from the
jet interacting with the clouds as it breaks out from central clouds shown in
yellow-green colors. The active interaction of the jet with clouds in the iSM
is also evident from the cross-section of the density in the $Y-Z$ plane.

The jets pass through a ‘flood and channel’ evolutionary phase (Sutherland
Bicknell, 2007; Wagner Bicknell, 2011) as they drill through the clouds before
breakout. During such a phase, the jet streams get deflected and broken into
smaller channels by the clouds. This spreads the collimated jet into
filamentary low density streams of non-thermal plasma, percolating through the
gaps of the thermal clouds and engulfing them. A representation of this is
depicted in Fig. 1, which shows the volume rendering of the jet tracer111The
’jet-tracer’ is a passive scalar advected with the fluid flow. It denotes the
location of the jet plasma injected into the computational domain. of
simulation E from Bicknell . (2018). Here a jet of power $P_{\rm
jet}=10^{45}\mbox{ erg s}^{-1}$ is ploughing through a cloud distribution
extended up to $\sim 2$ kpc from the centre. The main jet is a thin collimated
structure (in magenta), which shows some bifurcation at its head due to active
interaction with clouds dragged by the jet flow. However, it is surrounded by
filaments of low density ($\sim 10^{-3}$ of the main jet) jet tracer material.

The extended distribution of the low density jet plasma implies that the jet
can potentially drive shocks into the otherwise cold ISM, up to distances a
few kpc away from the main jet axis. Observing such tenuous extended jet
plasma may be difficult, as their surface brightness is nearly $\sim 3-4$
orders of magnitude lower than the peak value at the jet-head (Bicknell .,
2018). However, signatures of shocks from such material will be evident from
multi-phase observations of the thermal gas. This has been recently
demonstrated in spatially resolved observations of jet-ISM interactions in
some galaxies Zovaro, Sharp . (2019); Zovaro, Nesvadba . (2019), where the
extent of the shocked gas was found to be several times larger ($\sim 1-2$
kpc), than the apparent size of the radio jet ($\lesssim 100$ pc).

  * •

Impact of resolving cloud structures: An important aspect of these results was
the strong dependence of the mean outflow velocities of the clouds on the
cloud size and ISM filling factor (Wagner ., 2012). An ISM with a lower
filling factor or smaller cloud sizes results in larger mean outflow
velocities and hence more gas removal, similar to the results from large scale
cosmological simulations. However, larger cloud sizes or a higher filling
factor of the ISM may result in lower outflow speeds. This is more drastic in
the simulations with an external gravitational field (Mukherjee ., 2016, 2017)
where the net mass-weighted mean radial velocity is negative for most
simulations. This follows from the fact that for the same power ($P_{w}$) of
an incident wind, the approximate velocity of the cloud of density $\rho_{c}$
will be: $v_{c}\sim\left(P_{w}/\rho_{c}\right)^{1/2}$. Thus, low density,
unresolved cloud structures will be more easily swept out of the galaxy, as
often found in large scale simulations (Springel, 2005), whereas resolved
simulations with larger density contrasts show more resistance to outflows.

  * •

Galactic fountain: The jets can transfer a significant fraction of their
energy and momentum to the clouds, implying strong coupling with the ISM. The
simulations of Wagner Bicknell (2011) and Wagner . (2012) found the mechanical
advantage (ratio of cloud momentum to that of the jet) to be greater than
unity. A significant fraction of the jet’s energy ($\lesssim 30\%$) can be
imparted to the clouds as kinetic energy. However, this does not imply that
the AGN driven outflows can clear the gas from the galaxy and hence
substantially reduce the star forming fuel, as envisaged earlier in large
scale simulations (Springel, 2005).

Although the jets can cause outflows of diffuse ionised gas exceeding speeds
of $\sim 1000\mbox{ km s}^{-1}$, the net mass weighted radial velocity of the
entire ISM is not higher than the escape velocity (Mukherjee ., 2016, 2017).
Considering a purely ballistic trajectory of the clouds, only a small fraction
of the dense gas ($\lesssim 1-10\%$) would reach a height of $\sim 10$ kpc.
This proposes a ‘galactic fountain’ scenario, where local outflows triggered
by the jet remain within the galaxy’s potential and will eventually fall back
in. Thus, although the jets may not be very efficient in clearing out the gas
in the galaxy, they do increase the kinetic energy budget of the dense gas,
which will induce turbulence (as also shown in Mukherjee, Bicknell ., 2018).

##  3 Impact on star formation of the host

How AGN driven outflows may influence the stellar mass assembly of the host
galaxy is an important question. In this context, relativistic jets have
primarily been considered for their role in the large scale heating of the
circum-galactic medium and offsetting cooling flows (Gaspari ., 2012; Fabian,
2012; Yang Reynolds, 2016). However, there has been growing evidence both from
theoretical simulations, as detailed above, as well as observations (both
direct and indirect) that jets can both enhance and quench the star formation
rate (SFR) of their host. We summarise below the current understanding on how
these processes may be mediated, and the future work needed to unravel them
better.

###  3.1 Induced SFR: positive feedback

Jets aligned with star forming regions have often been considered to be
evidence of jet induced star formation e.g. 4C 41.17 (Bicknell ., 2000;
Nesvadba ., 2020), Minkowskis object (Salomé ., 2015; Lacy ., 2017), 3C 285
(Salomé ., 2015). Shocks from relativistic jets can be more efficient in
compressing the ambient gas than non-relativistic outflows. For example, the
advance speed of a relativistic shock extending into a medium is (see eq. (19)
- eq. (22) of Mukherjee ., 2020, for the detailed derivation):

$v_{c}\approx\gamma_{j}v_{j}\left(\frac{\rho_{j}}{\rho_{c}}\right)^{1/2}\left[1+\frac{\Gamma
p_{j}}{(\Gamma-1)\rho_{j}c^{2}}\right]^{1/2}\propto\gamma_{j}v_{c,\mbox{\small
NR}}.$ (1)

Here ($\rho_{j},v_{j}$) are the density and velocity of the jet or wind and
($\rho_{c},v_{c,\mbox{\small NR}}$) are the pre-shock (up-stream) density in
the cloud and the velocity of the shock in the cloud. The corresponding non-
relativistic velocity is given by $v_{c,\mbox{\small
NR}}=v_{j}\left(\frac{\rho_{j}}{\rho_{c}}\right)^{1/2}$ (Fragile ., 2004).
Thus a relativistic outflow can create shock speeds higher by the Lorentz
factor of the jet. Also, if the jets are collimated, they can retain their
momentum up to much larger distances, as opposed to adiabatically expanding
energy driven winds, whose advance speeds are likely to decrease with radius
as $V\propto t^{-(2-\beta)/(5-\beta)}$ (see Appendix A of Mukherjee, Wagner .,
2018).

This leads to an overall enhancement of the density of the gas in the
simulations of jet-ISM interactions (Sutherland Bicknell, 2007; Wagner .,
2016; Mukherjee ., 2016, 2017). Such strong compression are expected to
enhance gravitational collapse and hence aid the formation of stars. This has
been investigated by several authors, as compression from direct interactions
of jets with the ISM (Fragile ., 2004; Gaibler ., 2012; Fragile ., 2017;
Bourne Sijacki, 2017; Mukherjee, Wagner ., 2018) or by indirect compression of
the extended gas disk due to the over-pressured lobes driven by the jets
(Bieri ., 2016).

###  3.2 Turbulence induced negative feedback

However, an often important point missed in these analyses is the impact of
turbulence induced by the jet. Resolved simulations of Mukherjee, Wagner .
(2018) have indeed shown that the turbulent velocity dispersion increases by
$\sim 2-4$ times its initial value (depending on jet power). This is mediated
both by direct impact of the jet as well as shocks at the outer edges of the
disk due to the backflow of the jet material, forming a high pressure bubble
that engulfs the disk. Several observational studies (Nesvadba ., 2010, 2011;
Alatalo ., 2011) have found the star formation rate to be significantly
reduced in host galaxies with a radio jet of modest power. The outflows
launched in such systems are not expected to significantly deplete the dense
gas.

Figure 2:  Distribution of the $\mbox{SFR}_{\rm ff}$ given in Eq. (2) as a
function of the virial parameter ($\alpha_{\rm vir}$) and mach number
($\mathcal{M}$). The $\mbox{SFR}_{\rm ff}$ of the three different cases in
Table 1 are marked as: i) Case A in filled yellow square, ii) Case B in open
yellow square and iii) Case C in open cyan circle.

The theory of turbulence regulated star formation (Krumholz McKee, 2005;
Padoan Nordlund, 2011; Hennebelle Chabrier, 2011; Federrath Klessen, 2012)
indicates that besides the density, the two key parameters that affect the SFR
are: a) the virial parameter ($\alpha_{\rm vir}=2E_{\rm kin}/E_{\rm grav}$),
which is the ratio of twice the kinetic energy and gravitational potential
energy, b) the rms mach number ($\mathcal{M}$) of the turbulent gas. Using the
above, one can compute the star formation rate per freefall time (hereafter
$\mbox{SFR}_{\rm ff}$, see Krumholz McKee, 2005). The total star formation
rate of the cloud will then be $\mbox{SFR}=(M/t_{\rm ff})\mbox{SFR}_{\rm ff}$.

The $\mbox{SFR}_{\rm ff}$ for a multi-freefall model of turbulence regulated
star formation is given by (Federrath Klessen, 2012)

$\displaystyle\mbox{SFR}_{\rm
ff}=\frac{\epsilon}{2\phi_{t}}\exp\left(\frac{3}{8}\sigma_{s}^{2}\right)\left[1+\mbox{erf}\left(\frac{\sigma_{s}^{2}-s_{\rm
crit}}{\sqrt{2\sigma_{s}^{2}}}\right)\right],$ (2)
$\displaystyle\sigma_{s}^{2}=\ln(1+b^{2}\mathcal{M}^{2}),\,s_{\rm\small
crit}=\ln\left[\left(\frac{\pi^{2}}{5}\right)\phi_{x}^{2}\alpha_{\rm
vir}\mathcal{M}^{2}\right].$

The density of the cloud is assumed to be represented by a lognormal
probability distribution function (PDF), expressed in terms of
$s=\ln(\rho/\rho_{0})$, $\rho_{0}$ being the mean density. $\sigma_{s}$
denotes the dispersion of the density PDF. We refer the reader to Krumholz
McKee (2005) and Federrath Klessen (2012) for detailed derivation of Eq. (2).
In Fig. 2, we have plotted Eq. (2) for different values of $\alpha_{\rm vir}$
and $\mathcal{M}$. We have assumed the best fit values of the various
parameters, as found in Federrath Klessen (2012): $\phi_{x}=0.19$, $b=0.4$,
$1/\phi_{t}=0.49$ and $\epsilon=0.5$

Table 1: $\mbox{SFR}_{\rm ff}$ for different states of a spherical gas cloud of size $L=20\mbox{ pc}$ and mass $M=10^{5}M_{\odot}$ Case |  $\sigma_{V}$ ($\mbox{ km s}^{-1}$) | $\alpha_{\rm vir}$ |  $T$ (K) | $\mathcal{M}$ | $\mbox{SFR}_{\rm ff}$
---|---|---|---|---|---
A | 5 | 1 | 10 | 20 | 0.77
B | 61 | 150 | 100 | 24.4 | 0.03
C | 61 | 150 | 500 | 11 | 0.006

Using Eq. (2), one can demonstrate quantitatively the impact of increasing the
turbulent velocity dispersion on the star formation rate. Let us a consider a
cloud of mass $M\sim 10^{5}M_{\odot}$, with a velocity dispersion
$\sigma_{V}\simeq 5\mbox{ km s}^{-1}$ and size $\sim 20\mbox{ pc}$, typical of
nearby molecular clouds in the Milky Way (Miville-Deschênes ., 2017; Solomon
., 1987). Let us approximate the cloud to be a homogeneous sphere for
analytical ease, although far from a realistic description of typical
filamentary molecular cloud. The virial parameter for such a cloud will be
(Federrath Klessen, 2012)

$\displaystyle\alpha_{\rm vir}$ $\displaystyle\approx
0.96\times\left(\frac{\sigma_{V}}{5\mbox{ km
s}^{-1}}\right)^{2}\left(\frac{L}{20\mbox{
pc}}\right)\left(\frac{M}{10^{5}M_{\odot}}\right)^{-1}$ (3)

Assuming a temperature of 10 K and mean molecular weight of $\mu=2.3$ (Pineda
., 2008; Federrath Klessen, 2012), gives a sound speed of $c_{s}\simeq
0.25\mbox{ km s}^{-1}(T/10\mbox{K})^{1/2}$ and a mach number
$\mathcal{M}\simeq 20$. The above values result in a $\mbox{SFR}_{\rm ff}\sim
0.77$ (Case A of Table 1), placing the cloud in the lower left corner of Fig.
2, denoted by the filled yellow square.

Let us now consider that the above cloud lies in an ISM impacted by a jet,
which induces internal turbulence with a line of sight dispersion of
$\sigma_{1D}\sim 250\mbox{ km s}^{-1}$, similar to observed values in 3C 326
(Nesvadba ., 2011) and IC 5063 (Morganti ., 2015). This would imply an
internal 3D dispersion of $\sigma_{3D}=\sqrt{3}\sigma_{1D}\simeq 433\mbox{ km
s}^{-1}$. The observed values are for gas at kpc scales. Assuming the ISM to
be an extended turbulent distribution, the velocity dispersion will
approximately scale with length as (Larson, 1981; Solomon ., 1987; Federrath
Klessen, 2012)

$\sigma_{V,20\mbox{ pc}}=250\mbox{ km s}^{-1}\left(\frac{20\mbox{ pc}}{1\mbox{
kpc}}\right)^{1/2}\simeq 61\mbox{ km s}^{-1}$ (4)

This would imply a virial parameter (from Eq. (3)) of $\alpha_{\rm vir}\approx
150$. The shocks induced by the jet will also heat up the gas, increasing its
sound speed. The temperature of such a turbulent gas can be a few hundred
Kelvins, as inferred from excitation temperature of molecular gas (e.g. $T\sim
100$ K by Dasyra ., 2016), or the presence of shocked H2 gas in several
systems (e.g. in 4C 31.04 Zovaro, Sharp ., 2019). We have listed in Table 1
the $\mbox{SFR}_{\rm ff}$ for two values of the temperature of the jet
disturbed gas, and also demarcated their positions in Fig. 2. As can be seen,
the resultant $\mbox{SFR}_{\rm ff}$ is reduced by more than an order of
magnitude.

The above analysis, though very simplistic and approximate, gives some
indications that the SFR in the jet disturbed ISM can indeed be suppressed by
turbulence, as is predicted in several systems (Nesvadba ., 2010, 2011;
Alatalo ., 2011, 2015). Which of the above feedback mechanisms (positive vs
negative) dominates in a system is a complex problem and requires applying the
above prescription of turbulence regulated star formation to simulations of
jet-ISM interaction (as done in Mandal ., 2021).

##  4 Summary

Relativistic jets from the central supermassive black holes can significantly
affect the gas distribution of the host galaxy. Although jets can potentially
couple strongly with the gas, the resultant velocities may not be sufficient
to unbind a significant fraction of the gas. However, the jet driven outflows
can potentially affect the star formation rate (SFR) of the host in two ways.
They can either compress gas by direct interaction, enhancing the SFR as a
positive feedback. Alternatively, the turbulence induced by the jets can
significantly lower the star formation rate due to support from the turbulent
velocity dispersion and increased temperature of the gas (and hence lower Mach
number). To further test these implications, more resolved simulations of jet-
ISM interactions, with better subgrid models of star formation, is required in
future.

## Acknowledgments

The results discussed here spans the works from several publications in the
past few years and ongoing works. The authors would like to thank Ralph
Sutherland, Joseph Silk, Nicole Nesvadba, Raffaella Morganti, Christoph
Federrath, Ankush Mandal and M. Meenakshi for insightful interactions; and the
anonymous referee for helpful suggestions.

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## Author Biography

Dipanjan Mukherjee Assistant professor at the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India. His interest lies in
simulating astrophysical flows, with a focus on AGN feedback, relativistic
jets and related high energy phenomena.