summaries: This paper presents conditions that are necessary for counterfactual invariant predictions under certain assumptions. They derive these from causal graphs and show the connection to worst-case domain generalization. Experiments on natural language datasets show that the proposed conditions do lead to a decrease in worst-domain error.

summaries: In this paper, the authors study counterfactual invariance in machine learning models---intervening on “non causal” parts of the input should not change model predictions---and its relation to how models generalize out-of-domain. The authors show that counterfactually invariant predictors rely more on “causal” features as opposed to spurious correlations, thus generalizing to unseen distributions where these spurious correlations may not hold. The authors further propose two regularization schemes (one if the underlying data generation mechanism is represented by a causal model and another if it is an anticausal model) using which we can train counterfactually invariant predictors without access to counterfactual examples---as opposed to most prior work in which researchers often seek to either automatically construct or crowdsource counterfactual examples. They show that these strategies are heavily dependent on the underlying causal mechanisms, and that applying the regularization scheme designed for a causal model could lead to worse performance if the underlying data generation mechanism was anticausal and vice versa. The authors offer empirical evidence on text classification tasks (review helpfulness classification and natural language inference) to support their theoretical findings. I have some concerns about the paper (as discussed below) but overall I believe this paper builds on an exciting line of work and offers good theoretical and empirical contributions.

summaries: The paper casts robustness to spurious correlations as a model's dependence on input data that is not stable under stress testing. Stress testing here corresponds to changing part of the input data that a practitioner believes should not change the prediction. The authors term this counterfactual invariance and consider two different causal structures that could've produced the data. Identifying distribution-level implications (as opposed to unit-level) of these causal structures, the authors propose regularization schemes to enforce when learning predictors to ensure that counterfactual-invariance holds (under certain assumptions). The effectiveness of these schemes is shown both theoretically and empirically.

summaries: In this work, the authors focus on how to build models that are robust to spurious correlations. The focus of the paper is language datasets. The authors work with two types of DAGs -- a) association of the label with the invariant feature is anti-causal, and b) association of the label with invariant feature is causal. The authors propose a notion of invariance, which they call counterfactual invariance, that they require the predictor to satisfy. Under this notion, the prediction $f$ has to remain invariant under all the potential outcomes that $X$ can have under variations to the spurious feature $Z$. Since the notion of counterfactual invariance is hard to enforce as counterfactuals are not observed, the authors come up with necessary conditions that are then used as proxies to enforce counterfactual invariance. The necessary conditions are different in the two directions causal and anti-causal. Further, the authors analyze the relationship between best counterfactual invariant predictor on train domain and target domain (under appropriate notion of changes allowed in target domain). Finally, the authors establish conditions for min-max optimality of the predictor that is learned.

abstractText: Informally, a ‘spurious correlation’ is the dependence of a model on some aspect of the input data that an analyst thinks shouldn’t matter. In machine learning, these have a know-it-when-you-see-it character; e.g., changing the gender of a sentence’s subject changes a sentiment predictor’s output. To check for spurious correlations, we can ‘stress test’ models by perturbing irrelevant parts of input data and seeing if model predictions change. In this paper, we study stress testing using the tools of causal inference. We introduce counterfactual invariance as a formalization of the requirement that changing irrelevant parts of the input shouldn’t change model predictions. We connect counterfactual invariance to outof-domain model performance, and provide practical schemes for learning (approximately) counterfactual invariant predictors (without access to counterfactual examples). It turns out that both the means and implications of counterfactual invariance depend fundamentally on the true underlying causal structure of the data—in particular, whether the label causes the features or the features cause the label. Distinct causal structures require distinct regularization schemes to induce counterfactual invariance. Similarly, counterfactual invariance implies different domain shift guarantees depending on the underlying causal structure. This theory is supported by empirical results on text classification.

authors: D. Lopez-Paz. Invariant Risk Minimization

title: 2020

venue: arXiv:

authors:
M. Baktashmotlagh
M.T. Harandi
B.C. Lovell
M. Salzmann

title: Unsupervised domain adaptation by domain invariant projection

venue: In: Proceedings of the 2013 IEEE International Conference on Computer Vision.

authors: B. Schölkopf. Counterfactuals uncover the modular structure of deep generative models

title: 2019

authors:
J. Devlin
M.-W. Chang
K. Lee
K. Toutanova

title: Bert: pre-training of deep bidirectional transformers for language understanding

venue: In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers).

authors:
Y. Ganin
E. Ustinova
H. Ajakan
P. Germain
H. Larochelle
F. Laviolette
M. Marchand
V. Lempitsky

title: Domain-adversarial training of neural networks

venue: J. Mach. Learn. Res

authors:
S. Garg
V. Perot
N. Limtiaco
A. Taly
E.H. Chi
A. Beutel

title: Counterfactual fairness in text classification through robustness

venue: In: Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society.

authors:
A. Gretton
K.M. Borgwardt
M.J. Rasch
B. Schölkopf
A. Smola

title: A kernel two-sample test

venue: The Journal of Machine Learning Research

authors:
S. Gururangan
S. Swayamdipta
O. Levy
R. Schwartz
S. Bowman
N.A. Smith

title: Annotation artifacts in natural language inference data

venue: In: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers).

authors: S.R. Bowman. Counterfactually-Augmented SNLI Training Data Does Not Yield Better Generalization Than Unaugmented Data

title: 2020

authors: Z.C. Lipton. Learning the Difference that Makes a Difference with Counterfactually-Augmented Data

title: 2020

authors: Z.C. Lipton. Explaining The Efficacy of Counterfactually Augmented Data

title: 2021

authors:
V. Kumar
A. Choudhary
E. Cho

title: Data augmentation using pre-trained transformer models

venue: In: Proceedings of the 2nd Workshop on Life-long Learning for Spoken Language Systems.

authors:
M.J. Kusner
J. Loftus
C. Russell
R. Silva

title: Counterfactual fairness

venue: In: Advances in Neural Information Processing Systems.

authors:
F. Locatello
B. Poole
G. Raetsch
B. Schölkopf
O. Bachem
M. Tschannen

title: Weakly-supervised disentanglement without compromises

venue: In: Proceedings of the 37th International Conference on Machine Learning.

authors: E. Marchiori. Adversarial Alignment of Class Prediction Uncertainties for Domain Adaptation

title: 2019

authors:
N. Mehrabi
F. Morstatter
N. Saxena
K. Lerman
A. Galstyan. A Survey on Bias
Fairness in Machine Learning

title: 2019

title: Causality from a distributional robustness point of view

venue: In: 2018 IEEE Data Science Workshop (DSW).

authors: C. Blundell. Representation Learning via Invariant Causal Mechanisms

title: 2020

authors:
K. Muandet
D. Balduzzi
B. Schölkopf

title: Domain generalization via invariant feature representation

venue: In: Proceedings of the 30th International Conference on Machine Learning.

authors:
A. Naik
A. Ravichander
N. Sadeh
C. Rose
G. Neubig

title: Stress test evaluation for natural language inference

venue: In: Proceedings of the 27th International Conference on Computational Linguistics.

authors:
J. Ni
J. Li
J. McAuley

title: Justifying recommendations using distantly-labeled reviews and fined-grained aspects

venue: Empirical Methods in Natural Language Processing (EMNLP)

authors:
J. Peters
P. Bühlmann
N. Meinshausen

title: Causal inference by using invariant prediction: identification and confidence intervals

venue: Journal of the Royal Statistical Society. Series B (Statistical Methodology)

authors:
M.T. Ribeiro
T. Wu
C. Guestrin
S. Singh

title: Beyond accuracy: behavioral testing of nlp models with checklist

venue: In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics.

authors: H. Hassani. Model-Based Domain Generalization

title: 2021

authors:
M. Rojas-Carulla
B. Schölkopf
R. Turner
J. Peters

title: Invariant models for causal transfer learning

venue: Journal of Machine Learning Research

authors:
S. Sagawa
P.W. Koh
T.B. Hashimoto
P. Liang

title: Distributionally robust neural networks for group shifts: on the importance of regularization for worst-case generalization

venue: In: International Conference on Learning Representations.

authors:
B. Schölkopf
D. Janzing
J. Peters
E. Sgouritsa
K. Zhang
J. Mooij

title: On causal and anticausal learning

venue: In: Proceedings of the 29th International Coference on International Conference on Machine Learning.

authors:
B. Schölkopf
F. Locatello
S. Bauer
N.R. Ke
N. Kalchbrenner
A. Goyal
Y. Bengio

title: Toward causal representation learning

venue: Proceedings of the IEEE

authors: S. Saria. A Universal Hierarchy of Shift-Stable Distributions

title: 2019

authors:
A. Subbaswamy
S. Saria

title: Counterfactual normalization: proactively addressing dataset shift and improving reliability using causal mechanisms

venue: In: Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence (UAI), 2018.

authors:
D. Teney
E. Abbasnejad
A. van den Hengel

title: Learning what makes a difference from counterfactual examples and gradient supervision

venue: In: CoRR (2020). arXiv:

authors:
E. Tzeng
J. Hoffman
N. Zhang
K. Saenko
T. Darrell

title: Deep Domain Confusion: Maximizing for Domain Invariance

authors:
J. Wei
K. Zou

title: EDA: easy data augmentation techniques for boosting performance on text classification tasks

venue: In: Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP).

authors:
A. Williams
N. Nangia
S. Bowman

title: A broad-coverage challenge corpus for sentence understanding through inference

venue: In: Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers).

authors:
T. Wu
M.T. Ribeiro
J. Heer
D. Weld

title: Errudite: scalable

venue: reproducible, and testable error analysis”. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics.

authors:
T. Wu
M.T. Ribeiro
J. Heer
D.S. Weld

title: Polyjuice: generating counterfactuals for explaining

venue: evaluating, and improving models”. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics.

authors:
H. Yan
Y. Ding
P. Li
Q. Wang
Y. Xu
W. Zuo

title: Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation

authors:
K. Zhang
B. Schölkopf
K. Muandet
Z. Wang

title: Domain adaptation under target and conditional shift

heading: 1 Introduction

text:
Our focus in this paper is the sort of spurious correlations revealed by “poke it and see what happens” testing procedures for machine-learning models. For example, we might test a sentiment analysis tool by changing one proper noun for another (“tasty Mexican food” to “tasty Indian food”), with the expectation that the predicted sentiment should not change. This kind of perturbative stress testing is increasingly popular: it is straightforward to understand and offers a natural way to test the behavior of models against the expectations of practitioners [Rib+20; Wu+19; Nai+18].
Intuitively, models that pass such stress tests are preferable to those that do not. However, fundamental questions about the use and meaning of perturbative stress tests remain open. For instance, what is the connection between passing stress tests and model performance on prediction? Eliminating predictor dependence on a spurious correlation should help with domain shifts that affect the spurious correlation—but how do we make this precise? And, how should we develop models that pass stress tests when our ability to generate perturbed examples is limited? For example, automatically perturbing the sentiment of a document in a general fashion is difficult.
The ad hoc nature of stress testing makes it difficult to give general answers to such questions. In this paper, we will use the tools of causal inference to formalize what it means for models to pass stress tests, and use this formalization to answer the questions above. We will formalize passing stress tests as counterfactual invariance, a condition on how a predictor should behave when given
35th Conference on Neural Information Processing Systems (NeurIPS 2021).
certain (unobserved) counterfactual input data. We will then derive implications of counterfactual invariance that can be measured in the observed data. Regularizing predictors to satisfy these observable implications provides a means for achieving (partial) counterfactual invariance. Then, we will connect counterfactual invariance to robust prediction under certain domain shifts, with the aim of clarifying what counterfactual invariance buys and when it is desirable.
An important insight that emerges from the formalization is that the true underlying causal structure of the data has fundamental implications for both model training and guarantees. Methods for handing ‘spurious correlations’ in data with a given causal structure need not perform well when blindly translated to another causal structure.
Counterfactual Invariance Consider the problem of learning a predictor f that predicts a label Y from covariates X . In this paper, we’re interested in constructing predictors whose predictions are invariant to certain perturbations on X . Our first task is to formalize the invariance requirement.
To that end, assume that there is an additional variable Z that captures information that should not influence predictions. However, Z may causally influence the covariates X . Using the potential outcomes notation, letX(z) to denote the counterfactualX we would have seen had Z been set to z, leaving all else fixed. Informally, we can understand perturbative stress tests as a way of producing particular realizations of counterfactual pairs X(z), X(z′) that differ by an intervention on z. Then, we formalize the requirement that an arbitrary change to z does not change predictions: Definition 1.1. A predictor f is counterfactually invariant to Z if f(X(z)) = f(X(z′)) almost everywhere, for all z, z′ in the sample space of Z. When Z is clear from context, we’ll just say the predictor is counterfactually invariant.

heading: 2 Causal Structure

text: Counterfactual invariance is a condition on how the predicted label behaves under interventions on parts of the input data. However, intuitions about stress testing are based on how the true label behaves under interventions on parts of the input data. We will see that the true causal structure fundamentally affects both the implications of counterfactual invariance, and the techniques we use to achieve it. To study this phenomenon, we’ll use two causal structures that are commonly encountered in applications; see Figure 1.

heading: 2.1 Prediction in the Causal Direction

text:
We begin with the case where X is a cause of Y . Example 2.1. We want to automatically classify the quality of product reviews. Each review has a number of “helpful” votes Y (from site users). We predict Y using the text of the product review X . However, we find interventions on the sentiment Z of the text change our prediction; changing “Great shoes!” to “Bad shoes!” changes the prediction.
In the examples in this paper, the covariate X is text data. Usually, the causal relationship between the text and Y and Z will be complex— e.g., the relationships may depend on abstract, unlabeled, parts of the text such as topic, writing quality, or tone. In principle, we could enumerate all such latent variables, construct a causal graph capturing the relationships between these variables and Y, Z, and use this causal structure to study counterfactual invariance. For instance, if we think that topic causally influences the helpfulness Y , but is not influenced by sen-
timent Z, then we could build a counterfactually invariant predictor by extracting the topic and
predicting Y using topic alone. However, exhaustively articulating all possible such variables is a herculean task.
Instead, notice that the only thing that’s relevant about these latent variables is their causal relationship with Y and Z. Accordingly, we’ll decompose the observed variable X into parts defined by their causal relationships with Y and Z. We remain agnostic to the semantic interpretation of these parts. Namely, we defineX⊥Z as the part ofX that is not causally influenced by Z (but may influence Y ), X⊥Y as the part that does not causally influence Y (but may be influenced by Z), and XY ∧Z is the remaining part that is both influenced by Z and that influences Y . The causal structure is shown in Figure 1a.
We see there are two paths that lead to Y and Z being associated. The first is when Z affects XY ∧Z which, in turn, affects Y . For example, a very enthusiastic reviewer might write a longer, more detailed review, which will in turn be more helpful. The second is when a common cause or selection effect in the data generating process induces an association between Z and Y , which we denote with a dashed arrow. For example, if books tend to get more positive reviews, and also people who buy books are more likely to flag reviews as helpful, then the product type would be a common cause of sentiment and helpfulness.

heading: 2.2 Prediction in the Anti-Causal Direction

text:
We also consider the case where Y causes X .
Example 2.2. We want to predict the star rating Y of movie reviews from the text X . However, we find that predictions are influenced by the movie genre Z; e.g., changing “Adam Sandler” (a comedy actor) to “Hugh Grant” (a romance actor) changes the predictions.
Figure 1b shows the causal structure. Here, the observed X is influenced by both Y and Z. Again, we decompose X into parts defined by their causal relationship with Z and Y . Here, Z (and thus X⊥Y ) can be associated with Y through two paths. First, if XY ∧Z is non-trivial, then conditioning on it causes a dependence between Z and Y (because XY ∧Z is a collider). For example, if Adam Sandler tends to appear in good comedy movies but bad movies of other genres then seeing “Sandler” in the text induces a dependency between sentiment and genre. Second, Z and Y may be associated due to a common cause, or due to selection effects in the data collection protocol—this is represented by the dashed line between Z and Y . For example, fans of romantic comedies may tend to give higher reviews (to all films) than fans of horror movies.

heading: 2.3 Non-Causal Associations

text:
Frequently, a predictor trained to predict Y from X will rely on X⊥Y , even though there is no causal connection between Y and X⊥Y , and therefore will fail counterfactual invariance. The reason is that X⊥Y serves as a proxy for Z, and Z is predictive of Y due to the non-causal (dashed line) association.
There are two mechanisms that can induce such associations. First, Y and Z may be confounded: they are both influenced by an unobserved common causeU . For example, people who review books may be more upbeat than people who review clothing. This leads to positive sentiments and high helpfulness votes for books, creating an association between sentiment and helpfulness. Second, Y and Z may be subject to selection: there is some condition (event) S that depends on Y and Z, such that a data point from the population is included in the sample only if S = 1 holds. For example, our training data might only include movies with at least 100 reviews. If only excellent horror movies have so many reviews (but most rom-coms get that many), then this selection would induce an association between genre and score. Formally, the dashed-line causal graphs mean our sample is distributed according to P (X,Y, Z) = ∫ P(X,Y, Z, u | S = 1)dP(u) where Y,Z are caused by U and are causes of S, and (X,Y, Z) are causally related according to the graph.
In addition to the non-causal dashed-line relationship, there is also dependency induced by between Y and Z by XY ∧Z . Whether or not each of these dependencies is “spurious” is a problem-specific judgement that must be made by each analyst based on their particular use case. E.g., using genre to predict sentiment may or may not be reasonable, depending on the actual application in mind. However, there is a special case that captures a common intuition for purely spurious association.
Definition 2.3. We say that the association between Y and Z is purely spurious if Y ⊥ X |X⊥Z , Z.
That is, if the dashed-line association did not exist (removed by conditioning on Z) then the part of X that is not influenced by Z would suffice to estimate Y .

heading: 3 Observable Signatures of Counterfactually Invariant Predictors

text:
We now consider the question of how to achieve counterfactual invariance in practice. The challenge is that counterfactual invariance is defined by the behavior of the predictor on counterfactual data that is never actually observed. This makes checking counterfactual invariance impossible. Instead, we’ll derive a signature of counterfactual invariance that actually can be measured—and enforced— using ordinary datasets where Z (or a proxy) is measured. For example, the star rating of a review as a proxy for sentiment, or genre labels in the movie review case.
Intuitively, a predictor f is counterfactually invariant if it depends only on X⊥Z , the part of X that is not affected by Z. To formalize this, we need to show that such a X⊥Z is well defined: Lemma 3.1. LetX⊥Z be aX-measurable random variable such that, for all measurable functions f , we have that f is counterfactually invariant if and only if f(X) is X⊥Z -measurable. If Z is discrete 1 then such a X⊥Z exists.
Accordingly, we’d like to construct a predictor that is a function ofX⊥Z only (i.e., isX ⊥ Z measurable). The key insight is that we can use the causal graphs to read off a set of conditional independence relationships that are satisfied by Z,X⊥Z , Y . Critically, these conditional independence relationships are testable from the observed data. Thus, they provide a signature of counterfactual invariance: Theorem 3.2. If f is a counterfactually invariant predictor:
1. Under the anti-causal graph, f(X) ⊥ Z | Y . 2. Under the causal-direction graph, if Y and Z are not subject to selection (but possibly
confounded), f(X) ⊥ Z. 3. Under the causal-direction graph, if the association is purely spurious, Y ⊥ X | X⊥Z , Z,
and Y and Z are not confounded (but possibly selected), f(X) ⊥ Z | Y . Remark 3.3 (Connection to Fairness). This result has an interesting interpretation when Z is a protected attribute (e.g., sex or race) that we’d like to be fair with respect to. There are many ways of formalizing fairness, which are usually mutually incompatible. In the fairness setting, counterfactual invariance is equivalent to counterfactual fairness [Kus+17; Gar+19], the condition f(X) ⊥ Z is equivalent to demographic parity, and the condition f(X) ⊥ Z | Y is equivalent to equalized odds [Meh+19]. Theorem 3.2 then says that counterfactual fairness implies either demographic parity or equalized odds, depending on the true underlying causal structure of the problem. Hence, the relationship between disparate fairness notions is clarified by taking the underlying causal structure into account. This also suggests we can take counterfactual fairness as the fundamental notion, then use demographic parity in the causal-confounding case and equalized odds otherwise. However, we leave the (ethical) question of whether this is a sound strategy to future work.
Causal Regularization Without access to counterfactual examples, we cannot directly enforce counterfactual invariance. However, we can require a trained model to satisfy the counterfactual invariance signature of Theorem 3.2. The hope is that enforcing the signature will lead the model to be counterfactually invariant. To do this, we regularize the model to satisfy the appropriate conditional independence condition. For simplicity of exposition, we restrict to binary Y and Z. The (infinite data) regularization terms are
marginal regularization = MMD(P(f(X) | Z = 0),P(f(X) | Z = 1)) (3.1) conditional regularization = MMD(P(f(X) | Z = 0, Y = 0),P(f(X) | Z = 1, Y = 0)) (3.2)
+MMD(P(f(X) | Z = 0, Y = 1),P(f(X) | Z = 1, Y = 1)). Maximum mean discrepancy (MMD) is a metric on probability measures.2 The marginal independence condition is equivalent to (3.1) equal 0, and the conditional independence is equivalent to (3.2)
1In fact, it suffices that all potential outcomes {Y (z)}z are jointly measurable with respect to a single well-behaved sigma algebra; discrete Z is sufficient but not necessary.
2The choice of regularization by MMD is for concreteness. Any technique for enforcing the independence signatures would do in principle—e.g., adversarial methods borrowed from the fairness literature. The key point here is the observation that different causal structures imply different independence signatures.
equal 0. In practice, we can estimate the MMD with finite data samples [Gre+12]. When training with stochastic gradient descent, we compute the penalty on each minibatch.
The procedure is then: if the data has causal-direction structure and the Y ↔ Z association is due to confounding, add the marginal regularization term to the the training objective. If the data has anti-causal structure, or the association is due to selection, add the conditional regularization term instead. In this way, we regularize towards models that satisfy the counterfactual invariance signature.
A key point is that the regularizer we must use depends on the true causal structure. The conditional and marginal independence conditions are generally incompatible. Enforcing the condition that is mismatched to the true underlying causal structure may fail to encourage counterfactual invariance, or may throw away more information than is required.
Gap to Counterfactual Invariance The conditional independence signature of Theorem 3.2 is necessary but not sufficient for counterfactual invariance. This is for two reasons. First, counterfactual invariance applies to individual datapoint realizations, but the signature is distributional. In particular, the invariance P(f(X) | do(Z = z)) = P(f(X) | do(Z = z′)) for all z, z′ would also imply the conditional independence signature. But, this invariance is weaker than counterfactual invariance, since it doesn’t require access to counterfactual realizations. Second, f(X) ⊥ Z does not imply, in general, that Z is not a cause of f(X). This (unusual) behavior can happen if, e.g., there are levels of Z that we do not observe in the training data, or there are variables omitted from the causal graph that are a common cause of Z and X .
Unfortunately, the gap between the signature and counterfactual invariance is a fundamental restriction of using observational data. The conditional independence signature is in some sense the closest proxy for counterfactual invariance we can hope for. In section 5, we’ll see that enforcing the signature does a good job of enforcing counterfactual invariance in practice.

heading: 4 Performance Out of Domain

text:
Counterfactual invariance is an intuitively desirable property for a predictor to have. However, it’s not immediately clear how it relates to model performance as measured by, e.g., accuracy. Intuitively, eliminating predictor dependence on a spurious Z may help with domain shift, where the data distribution in the target domain differs from the distribution of the training data. We now turn to formalizing this idea.
First, we must articulate the set of domain shifts to be considered. In our setting, the natural thing is to hold the causal relationships fixed across domains, but to allow the non-causal (“spurious”) dependence between Y and Z to vary. Demanding that the causal relationships stay fixed reflects the requirement that the causal structure describes the dynamics of an underlying real-world process— e.g., the author’s sentiment is always a cause (not an effect) of the text in all domains. On the other hand, the dependency between Y and Z induced by either confounding or selection can vary without changing the underlying causal structure. For confounding, the distribution of the confounder may differ between domains—e.g., books are rare in training, but common in deployment. For selection, the selection criterion may differ between domains—e.g., we include only frequently reviewed movies in training, but make predictions for all movies in deployment.
We want to capture spurious domain shifts by considering domain shifts induced by selection or confounding. However, there is an additional nuance. Changes to the marginal distribution of Y will affect the risk of a predictor, even in the absence of any spurious association between Y and Z. Therefore, we restrict to shifts that preserve the marginal distribution of Y . Definition 4.1. We say that distributions P,Q are causally compatible if both obey the same causal graph, P (Y ) = Q(Y ), and there is a confounder U and/or selection conditions S, S̃ such that P = ∫ P(X,Y, Z | U, S=1)dP̃ (U) and Q= ∫ P(X,Y, Z | U, S̃=1)dQ̃(U) for some P̃ (U), Q̃(U).
We can now connect counterfactual invariance and robustness to domain shift. Theorem 4.2. Let F invar be the set of all counterfactually invariant predictors. Let L be either square error or cross entropy loss. And, let f∗ := argminf∈F invar EP [L(Y, f(X))] be the counterfactually invariant risk minimizer. Suppose that the target distributionQ is causally compatible with the training distribution P . Suppose that any of the following conditions hold:
1. the data obeys the anti-causal graph 2. the data obeys the causal-direction graph, there is no confounding (but possibly selection),
and the association is purely spurious, Y ⊥ X | X⊥Z , Z, or 3. the data obeys the causal-direction graph, there is no selection (but possibly confounding),
the association is purely spurious and the causal effect of X⊥Z on Y is additive, i.e., the true data generating process is
Y ← g(X⊥Z ) + g̃(U) + ξ where E[ξ | X⊥Z ] = 0, (4.1)
for some functions g, g̃.
Then, the training domain counterfactually invariant risk minimizer is also the target domain counterfactually invariant risk minimizer, f∗ = argminf∈F invar EQ[L(Y, f(X))]. Remark 4.3. The causal case with confounding requires an additional assumption (additive structure) because, e.g., an interaction between confounder and X⊥Z can yield a case where X ⊥ Z and Y have a different relationship in each domain (whence, out-of-domain learning is impossible).
This result gives a recipe for finding a good predictor in the target domain even without access to any target domain examples at training time. Namely, find the counterfactually invariant risk minimizer in the training domain. In practice, we can use the regularization scheme of section 3 to (approximately) achieve this. We’ll see in section 5 that this works well in practice.
Optimality Theorem 4.2 begs the question: if the only thing we know about the target setting is that it’s causally compatible with the training data, is the best predictor the counterfactually invariant predictor with lowest training risk? A natural way to formalize this question is to study the predictor with the best performance in the worst case target distribution. We define Q = {Q : Q causally compatible with P} and the Q-minimax predictor f∗minimax = argminf∈F maxQ∈Q EQ[L(Y, f(X)]. The question is then: what’s the relationship between the counterfactually invariant risk minimizer and the minimax predictor? Theorem 4.4. The counterfactually invariant risk minimizer is notQ-minimax in general. However, under the conditions of Theorem 4.2, if the association is purely spurious, XY ∧Z ⊥ Y |X⊥Z , Z, and P(Z, Y ) satisfies overlap, then the two predictors are the same. By overlap we mean that P(Z, Y ) is a discrete distribution such that for all (z, y), if P(z, y) > 0 then there is some y′ 6= y such that also P(z, y′) > 0.
Conceptually, Theorem 4.4 just says that the counterfactually invariant predictor excludes XY ∧Z , even when this information is useful in every domain. In the purely spurious case, XY ∧Z carries no useful information, so counterfactual invariance is optimal.

heading: 5 Experiments

text:
The main claims of the paper are:
1. Stress test violations can be reduced by suitable conditional independence regularization. 2. This reduction will improve out-of-domain prediction performance. 3. To get the full effect, the imposed penalty must match the causal structure of the data.
Setup To assess these claims, we’ll examine the behavior of predictors trained with the marginal or conditinal regularization on multiple text datasets that have either causal or anti-causal structure. We expect to see that marginal regularization improves stress test and out-of-domain performance on data with causal-confounded structure, and conditional regularization improves these on data with anti-causal structure.
For each experiment, we use BERT [Dev+19] finetuned to predict a label Y from the text as our base model. We train multiple causally-regularized models on the each dataset. The training varies by whether we use the conditional or marginal penalty, and by the strength of the regularization term. That is, we train identical architectures using CrossEntropy+λ ·Regularizer as the objective function, where we vary λ and take Regularizer as either the marginal penalty, (3.1), or conditional penalty, (3.2). We compare these models’ predictions on data with causal and anti-causal structure.
See supplement for experimental details.

heading: 5.1 Robustness to Stress Tests

text:
First, we examine whether enforcing the causal regularization actually helps to enforce counterfactual invariance. We create counterfactual (stress test) examples by perturbing the input data and compare the prediction on these. We build the experimental datasets using Amazon reviews from the product category “Clothing, Shoes, and Jewelry” [NLM19].
Synthetic To study the relationship between counterfactual invariance and the distributional signature of Theorem 3.2, we construct a synthetic confound. For each review, we draw a Bernoulli random Z, and then perturb the text X so that the common words “the” and “a” carry information about Z: for example, we replace “the” with the token “thexxxxx” when Z = 1. We take Y to be the review score, and subsample so Y is balanced. This data has anti-causal structure: the text X is written to explain the score Y . Further, we expect that the Y,Z association is purely spurious, because “the” and “a” carry little information about the label.
We train the models on data where P(Y = Z) = 0.3. We then create perturbed stress-test datasets by changing each example Xi(z) to the counterfactual Xi(1 − z) (using the synthetic model). By measuring the performance of each model on the perturbed data, we can test whether the distributional properties enforced by the regularizers result in counterfactual invariance at the instance level. Figure 2 shows that conditional regularization (matching the anti-causal structure) reduces checklist failures, as measured by the frequency that the predicted label changes due to perturbation as well as the mean absolute difference in predictive probabilities that is induced by perturbation.
Natural To study the relationship in real data, we use the review data in a different way. We now take Z to be the score, binarized as Z ∈ {1 or 2 stars, 4 or 5 stars}. We use this Z as a proxy for sentiment, and consider problems where sentiment should (plausibly) not have a causal effect on Y . For the causal prediction problem, we take Y to be the helpfulness score of the review (binarized as described below). This is causal because readers decide whether the review is helpful based on the text. For the anti-causal prediction problem, we take Y to be whether “Clothing” is included as a category tag for the product under review (e.g., boots typically do not have this tag). This is
anti-causal because the product category affects the text.
We control the strength of the spurious association between Y and Z. In the anti-causal case, this is done by selection: we randomly subset the data to enforce a target level of dependence between Y and Z. The causal-direction case with confounding is more complicated. To manipulate confounding strength, we binarize the number of helpfulness votes V in a manner determined by
the target level of association. We take Y = 1[V > TZ ] where TZ is a Z-dependent threshold, chosen to induce a target association. We choose P(Y = 0 | Z = 0) = P(Y = 1 | Z = 1) = 0.3. We balance Z by subsampling, which also balances Y .
Now, we create stress test perturbations of these datasets by randomly changing adjectives in the examples. Using predefined lists of postive sentiment adjectives and negative sentiment adjectives, we swap any adjective that shows up on a list with a randomly sampled adjective from the other list. This preserves basic sentence structure, and thus creates a limited set of counterfactual pairs that differ on sentiment.
Results for differences in predicted probabilities between original and perturbed data are shown in Figure 3. Each point is a trained model, which vary in measured MMD on the test data and on sensitivity to perturbations. Recall that the conditional independence signature of Theorem 3.2 are necessary but not sufficient for counterfactual invariance, so it’s not certain that regularizing to reduce the MMD will reduce perturbation sensitivity. Happily, we see that regularizing to reduce the MMD that matches the causal structure does indeed reduce sensitivity to perturbations.
Notice that regularizing the causally mismatched MMD can have strange effects. Regularizing marginal MMD in the anti-causal case actually makes the model more sensitive to perturbations!

heading: 5.2 Domain Shift

text:
Next, we study the effect of causal regularization on model performance under domain shift.
The plots show out-of-domain accuracy for models trained on the (natural) review data. In each row, the left figure shows out-of-domain accuracies (lines are models), with the X-axis showing the level of spurious correlation in the test data (0.3 is the training condition); the right figure shows worst out-of-domain accuracy versus in-domain test accuracy (dots are models).
Natural Product Review We again use the natural Amazon review data described above. For both the causal and anticausal data, we create multiple test sets with variable spurious correlation strength. This is done in the manner described above, varying P(Y = 0 | Z = 0) = P(Y = 1 | Z = 1) = γ. Here, γ is the strength of spurious association. The test sets are out-of-domain samples. By design, Y is balanced in each dataset, so these samples are causally compatible with the training data. For both the causal and anti-causal datasets, the training data has P(Y = 0 | Z = 0) = P(Y = 1 | Z = 1) = 0.3. We train a classifier for each regularization type and regularization strength, and measure the accuracy on each test domain. The results are shown in Figure 4.
First, the unregularized predictors do indeed learn to rely on the spurious association between sentiment and the label. The accuracy of these predictors decays dramatically as the spurious assocation moves from negative (0.3) to positive—in the causal case, the unregularized predictor is worse than chance in the 0.8 domain.
Following section 3, the regularization that matches the underlying causal structure should yield a predictor that is (approximately) counterfactually invariant. Following Theorem 4.2, we expect that good performance of a counterfactually-invariant predictor in the training domain should imply good performance in each of the other domains. Indeed, we see that this is so. Models that are regularized to have small values of the appropriate MMD do indeed have better out-of-domain performance.
Such models have somewhat worse in-domain performance, because they no longer exploit the spurious correlation.
MNLI Data For an additional test on naturally-occurring confounds, we use the multigenre natural language inference (MNLI) dataset [WNB18]. Instances are concatenations of two sentences, and the label describes the semantic relationship between them, Y ∈ {contradiction, entailment, neutral}. There is a well-known confound in this dataset: examples where the second sentence contain a negation word (e.g., “not”) are much more likely to be labeled as contradictions [Gur+18]. Following Sagawa et al. [Sag+20], we set Z to indicate whether one of a small set of negation words is present. Although Z is derived from the text X , it can be viewed as a proxy for a latent variable indicating whether the author intended to use negation in the text. This is an anti-causal prediction problem: the annotators were instructed to write text to reflect the desired label [WNB18].
Following Sagawa et al. [Sag+20], we divide the MNLI data into groups by (Y, Z) and compute the “worst group accuracy” across all such groups. Because this is an anti-causal problem, we predict that the conditional MMD is a more appropriate penalty than the marginal MMD. As shown in Figure 5, this prediction holds: conditional MMD regularization dramatically improves performance on the worst group, while only lightly impacting the overall accuracy across groups.

heading: 6 Related work

text:
Several papers draw a connection between causality and domain shifts [SS18; SCS19; Arj+20; Mei18; PBM16; RC+18; Zha+13; Sch+12]. Typically, this work considers a prediction setting where the covariates X include both causes and effects of Y , and it is unknown which is which. The goal is to learn to predict Y using only its causal parents. Zhang et al. [Zha+13] considers anti-causal domain shift induced by changing P (Y ) and proposes a data reweighting scheme. By contrast, counterfactual invariance is an example-wise notion of invariance, not an invariance across environments. In particular, learning a counterfactually invariant predictor only requires access to data from a single environment. Theorem 4.2 establishes that counterfactual invariance does imply a certain domain generalization guarantee over causally-compatible domains. Note though that the notion of causal compatibility is not generally the same as class of domain shifts previously considered. For example, we have invariant performance in the anti-causal setting, but this is ruled out by Arjovsky et al. [Arj+20].
A related body of work focuses on “causal representation learning” [Bes+19; Loc+20; Sch+21; Arj+20]. Our approach follows this tradition, but focuses on splittingX into components defined by their causal relationships with the label Y and an additional covariate Z. Rather than attempting to infer the causal relationship between X and Y , we show that domain knowledge of this relationship is essential for obtaining counterfactually-invariant predictors. The role of causal vs anti-causal data generation in semi-supervised learning and transfer learning has also been studied [Sch+21; Sch+12]. In this paper we focus on a different implication of the causal vs anti-causal distinction.
Another line of work considers the case where either the counterfactuals X(z), X(z′) are observed for at least some data points, or where it’s assumed that there’s enough structural knowledge to (learn to) generate counterfactual examples [RPH21; Wu+21; Gar+19; Mit+20; WZ19; KCC20; KHL20; TAH20]. Kusner et al. [Kus+17] and Garg et al. [Gar+19] in particular examine a notion of
counterfactual fairness that can be seen as equivalent to counterfactual invariance. In these papers, approximate counterfactuals are produced by direct manipulation of the features (e.g., change male to female names), generative models (e.g., style transfer of images), or crowdsourcing. Then, these counterfactuals can either be used as additional training data or the predictor can be regularized such that it cannot distinguish between Xi(z) and Xi(z′). This strategy can be viewed as enforcing counterfactual invariance directly; an advantage is that it avoids the necessary-but-not-sufficient nuance of Theorem 3.2. However, counterfactual examples can be difficult to obtain for language data in many realistic problem domains, and it may be difficult to learn to generalize from such examples [HLB20].
The marginal and conditional independencies of Theorem 3.2 have appeared in other contexts. As discussed in remark 3.3, if we think of Z as a protected attribute and f as a ‘fair’ classifier, then counterfactual invariance is counterfactual fairness, the marginal independence is demographic parity, and the conditional independence is equalized odds [Meh+19]. In another setting, one approach to domain adaptation is to seek representations φ such that either φ(X) [e.g., MBS13; Bak+13; Gan+16; Tze+14] or φ(X) | Y [e.g., MLM19; Yan+17] have the same distribution in each environment. The connection here is subtler; it is inappropriate to interpret Z as a domain label (we only have access to one domain at train time, and all values of Z are present in each domain). To clarify the connection, consider introducing an extra variable E that labels the domain. For concreteness, consider the anti-causal case where the spurious association between Y and Z is due to an observed confounder U . Now, suppose that E is a cause of U . Then, E labels the distribution of the unobserved confounder, and thus different values of E correspond to different causally compatible domains. Now, if we draw the causal graph we can read off that X⊥z E|Y . That is, we arrive at the same conditional independence criterion. However, it’s essentially a coincidence that this matches the criterion we’d use if we’d observed Z—the two variables have totally different causal meanings and interpretations. For example, in the review data case, we might take Z to be review score and E to be the website the review is collected from. The review quality doesn’t need to be counterfactually invariant to the website, and indeed we wouldn’t expect it to be!
Finally, this work can be viewed as part of a recent line of work using counterfactuals to examine the connection between example-wise robustness (stress testing) and distributional-level robustness (domain shift) [e.g., TAH20; RPH21; Kau+21]. Teney et al. [TAH20] use near counterfactuals with different labels in a modified training objective, and show improved performance out-of-domain empirically. Robey et al. [RPH21] use image models to generate counterfactuals, and incorporate the model into training, showing out-of-domain improvement. They assume a somewhat different causal structure—articulating the precise connection is an interesting future direction. Kaushik et al. [Kau+21] study a linear-Gaussian model and argue that out-of-domain performance can be used as a signal to distinguish ‘causal’ and ‘spurious’ features.

heading: 7 Discussion

text:
We used the tools of causal inference to formalize and study perturbative stress tests. A main insight of the paper is that counterfactual desiderata can be linked to observationally-testable conditional independence criteria. This requires consideration of the true underlying causal structure of the data. Done correctly, the link yields a simple procedure for enforcing the counterfactual desiderata, and mitigating the effects of domain shift.
The main limitation of the paper is the restrictive causal structures we consider. In particular, we require that X⊥Z , the part of X not causally affected by Z, is also statistically independent of Z in the observed data. However, in practice these may be dependent due to a common cause. In this case, the procedure here will be overly conservative, throwing away more information than required. Additionally, it is not obvious how to apply the ideas described here to more complicated causal situations, which can occur in structured prediction (e.g., question answering). Extending the ideas to handle richer causal structures is an important direction for future work. The work described here can provide a template for this research program.

heading: 8 Acknowledgements

text: Thanks to Kevin Murphy, Been Kim, Molly Roberts, Justin Grimmer, and Brandon Stewart for feedback on an earlier version.

heading: 9 Funding Transparency

text: This paper was not supported by any third party funding, and the authors have no competing interests to declare.

title: Counterfactual Invariance to Spurious Correlations: Why and How to Pass Stress Tests


summaries: The submission tackles the problem of training GANs with limited data. It proposes to do so by adaptively regularizing GAN discriminators by replacing some real data by generated samples when the discriminator overfits, accoding to an overfitting heuristic.  A study of the proposed objectives show that the method keep the classical GAN optima while alleviating overfitting of the discriminator.  The method is tested on various datasets with limited data, and compared, using a StyleGAN-2 backbone, against other state-of-the-art methods for training with limited data.

summaries: This paper tries to solve discriminator overfitting problem.  The authors propose adaptive pseudo augmentation (APA). * APA employs the generator itself to augment the real data distribution with fake images.   * i.e., Fake images are presented as "real" instances to the discriminator. * Adaptiveness comes from measuring overfittingness of the discriminator. * APA has similar theoretical properties to the original GAN, but with $(1-\alpha)p_\text{data} + \alpha p_g = p_g$.  APA improves systnesis quality in the limited-data regime. * on FFHQ, AFHQ-Cat, CUB, Danbooru The authors provide a theoretical analysis of APA, similar to the original GAN.

summaries: This paper introduces a data augmentation strategy for training GANs, which proves to be especially effective in the low data regime. The essence of the idea is to replace real images by images produced by the generator, using an adaptive replacement probability in the discriminator training step. By classifying some fake images as real, discriminator overfitting is prevented, which normally leads to very poor training performance and instability when very little data is available. The proposed method (named APA) improves over or can be combined with another recent data augmentation scheme for the low data regime (such as ADA).

summaries: This paper proposes a new method for GAN training with limited data, named adaptive pseudo augmentation (APA). The key idea originates from the empirical observation that, when trained with limited data, the GAN discriminator tends to overfit easily, leading to poor generation performance. Therefore, the authors propose to adaptively add generated samples to the training data, so as to counteract the overfitting during training. Accordingly, they present the APA that is developed based on the output of the discriminator. Empirical experiments demonstrate the effectiveness of the proposed APA.

abstractText: Generative adversarial networks (GANs) typically require ample data for training in order to synthesize high-fidelity images. Recent studies have shown that training GANs with limited data remains formidable due to discriminator overfitting, the underlying cause that impedes the generator’s convergence. This paper introduces a novel strategy called Adaptive Pseudo Augmentation (APA) to encourage healthy competition between the generator and the discriminator. As an alternative method to existing approaches that rely on standard data augmentations or model regularization, APA alleviates overfitting by employing the generator itself to augment the real data distribution with generated images, which deceives the discriminator adaptively. Extensive experiments demonstrate the effectiveness of APA in improving synthesis quality in the low-data regime. We provide a theoretical analysis to examine the convergence and rationality of our new training strategy. APA is simple and effective. It can be added seamlessly to powerful contemporary GANs, such as StyleGAN2, with negligible computational cost. Code: https://github.com/EndlessSora/DeceiveD.

authors:
Martin Arjovsky
Léon Bottou

title: Towards principled methods for training generative adversarial networks

venue: In ICLR,

authors:
Martin Arjovsky
Soumith Chintala
Léon Bottou

title: Wasserstein generative adversarial networks

venue: In ICML,

title: Anonymous, and Danbooru Community. Danbooru2019 Portraits: A large-scale anime head illustration dataset. https://www.gwern.net/Crops#danbooru2019-portraits

authors:
Andrew Brock
Jeff Donahue
Karen Simonyan

title: Large scale GAN training for high fidelity natural image synthesis

title: How to train a GAN? NeurIPS workshop on adversarial training

authors:
Yunjey Choi
Minje Choi
Munyoung Kim
Jung-Woo Ha
Sunghun Kim
Jaegul Choo

title: StarGAN: Unified generative adversarial networks for multi-domain image-to-image translation

authors:
Yunjey Choi
Youngjung Uh
Jaejun Yoo
Jung-Woo Ha

title: StarGAN v2: Diverse image synthesis for multiple domains

venue: In CVPR,

authors:
Ian Goodfellow
Jean Pouget-Abadie
Mehdi Mirza
Bing Xu
David Warde-Farley
Sherjil Ozair
Aaron Courville
Yoshua Bengio

title: Generative adversarial nets

venue: NeurIPS,

authors:
Ishaan Gulrajani
Faruk Ahmed
Martin Arjovsky
Vincent Dumoulin
Aaron Courville

title: Improved training of wasserstein gans

venue: In NeurIPS,

authors:
Martin Heusel
Hubert Ramsauer
Thomas Unterthiner
Bernhard Nessler
Sepp Hochreiter

title: GANs trained by a two time-scale update rule converge to a local nash equilibrium

authors:
Xun Huang
Ming-Yu Liu
Serge Belongie
Jan Kautz

title: Multimodal unsupervised image-to-image translation

venue: In ECCV,

authors:
Phillip Isola
Jun-Yan Zhu
Tinghui Zhou
Alexei A Efros

title: Image-to-image translation with conditional adversarial networks

authors:
Simon Jenni
Paolo Favaro

title: On stabilizing generative adversarial training with noise

authors:
Liming Jiang
Bo Dai
Wayne Wu
Chen Change Loy

title: Focal frequency loss for image reconstruction and synthesis

venue: In ICCV,

authors:
Liming Jiang
Changxu Zhang
Mingyang Huang
Chunxiao Liu
Jianping Shi
Chen Change Loy

title: TSIT: A simple and versatile framework for image-to-image translation

authors:
Tero Karras
Timo Aila
Samuli Laine
Jaakko Lehtinen

title: Progressive growing of GANs for improved quality, stability, and variation

authors:
Tero Karras
Miika Aittala
Janne Hellsten
Samuli Laine
Jaakko Lehtinen
Timo Aila

title: Training generative adversarial networks with limited data

authors:
Tero Karras
Samuli Laine
Timo Aila

title: A style-based generator architecture for generative adversarial networks

authors:
Tero Karras
Samuli Laine
Miika Aittala
Janne Hellsten
Jaakko Lehtinen
Timo Aila

title: Analyzing and improving the image quality of StyleGAN

title: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. United States Governm

authors:
Hsin-Ying Lee
Hung-Yu Tseng
Jia-Bin Huang
Maneesh Singh
Ming-Hsuan Yang

title: Diverse imageto-image translation via disentangled representations

title: Divergence measures based on the Shannon entropy

authors:
Ming-Yu Liu
Thomas Breuel
Jan Kautz

title: Unsupervised image-to-image translation networks

authors:
Xihui Liu
Guojun Yin
Jing Shao
Xiaogang Wang
Hongsheng Li

title: Learning to predict layout-to-image conditional convolutions for semantic image synthesis

authors:
Xudong Mao
Qing Li
Haoran Xie
Raymond YK Lau
Zhen Wang
Stephen Paul Smolley

title: Least squares generative adversarial networks

authors:
Lars Mescheder
Andreas Geiger
Sebastian Nowozin

title: Which training methods for gans do actually converge

authors:
Mehdi Mirza
Simon Osindero

title: Conditional generative adversarial nets

venue: arXiv preprint,

authors:
Takeru Miyato
Toshiki Kataoka
Masanori Koyama
Yuichi Yoshida

title: Spectral normalization for generative adversarial networks

authors:
Sebastian Nowozin
Botond Cseke
Ryota Tomioka

title: f-GAN: Training generative neural samplers using variational divergence minimization

authors:
Xingang Pan
Xiaohang Zhan
Bo Dai
Dahua Lin
Chen Change Loy
Ping Luo

title: Exploiting deep generative prior for versatile image restoration and manipulation

authors:
Taesung Park
Ming-Yu Liu
Ting-Chun Wang
Jun-Yan Zhu

title: Semantic image synthesis with spatiallyadaptive normalization

authors:
Or Patashnik
Zongze Wu
Eli Shechtman
Daniel Cohen-Or
Dani Lischinski

title: StyleCLIP: Text-driven manipulation of stylegan imagery

title: X. on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling

authors:
Alec Radford
Luke Metz
Soumith Chintala

title: Unsupervised representation learning with deep convolutional generative adversarial networks

authors:
Elad Richardson
Yuval Alaluf
Or Patashnik
Yotam Nitzan
Yaniv Azar
Stav Shapiro
Daniel Cohen-Or

title: Encoding in style: a stylegan encoder for image-to-image translation

authors:
Kevin Roth
Aurelien Lucchi
Sebastian Nowozin
Thomas Hofmann

title: Stabilizing training of generative adversarial networks through regularization

authors:
Tim Salimans
Ian Goodfellow
Wojciech Zaremba
Vicki Cheung
Alec Radford
Xi Chen

title: Improved techniques for training GANs

authors:
Omry Sendik
Dani Lischinski
Daniel Cohen-Or

title: Unsupervised multi-modal styled content generation

authors:
Casper Kaae Sønderby
Jose Caballero
Lucas Theis
Wenzhe Shi
Ferenc Huszár

title: Amortised map inference for image super-resolution

authors:
Ngoc-Trung Tran
Viet-Hung Tran
Ngoc-Bao Nguyen
Trung-Kien Nguyen
Ngai-Man Cheung

title: On data augmentation for gan training

authors:
Hung-Yu Tseng
Lu Jiang
Ce Liu
Ming-Hsuan Yang
Weilong Yang

title: Regularizing generative adversarial networks under limited data

authors:
Catherine Wah
Steve Branson
Peter Welinder
Pietro Perona
Serge Belongie

title: The Caltech-UCSD Birds-200-2011 Dataset

authors:
Ting-Chun Wang
Ming-Yu Liu
Jun-Yan Zhu
Andrew Tao
Jan Kautz
Bryan Catanzaro

title: Highresolution image synthesis and semantic manipulation with conditional GANs

authors:
Ryan Webster
Julien Rabin
Loic Simon
Frédéric Jurie

title: Detecting overfitting of deep generative networks via latent recovery

authors:
Weihao Xia
Yulun Zhang
Yujiu Yang
Jing-Hao Xue
Bolei Zhou
Ming-Hsuan Yang

title: GAN inversion: A survey

authors:
Yuanbo Xiangli
Yubin Deng
Bo Dai
Chen Change Loy
Dahua Lin

title: Real or not real, that is the question

authors:
Han Zhang
Zizhao Zhang
Augustus Odena
Honglak Lee

title: Consistency regularization for generative adversarial networks

authors:
Junbo Zhao
Michael Mathieu
Yann LeCun

title: Energy-based generative adversarial networks

authors:
Shengyu Zhao
Zhijian Liu
Ji Lin
Jun-Yan Zhu
Song Han

title: Differentiable augmentation for dataefficient gan training

authors:
Zhengli Zhao
Sameer Singh
Honglak Lee
Zizhao Zhang
Augustus Odena
Han Zhang

title: Improved consistency regularization for GANs

venue: In AAAI,

authors:
Zhengli Zhao
Zizhao Zhang
Ting Chen
Sameer Singh
Han Zhang

title: Image augmentations for GAN training

authors:
Jun-Yan Zhu
Taesung Park
Phillip Isola
Alexei A Efros

title: Unpaired image-to-image translation using cycle-consistent adversarial networks

heading: 1 Introduction

text:
While state-of-the-art GANs like StyleGAN2 [20] are constantly pushing forward the fidelity and resolution of synthesized images, they usually require a large amount of training data to fully unleash their power. Training GANs with insufficient data tend to generate poor-quality images, as shown in Figure 1. Collecting sufficient data samples for these GANs is sometimes infeasible, especially in domains where data are sparse and privacy-sensitive. To ease the practical deployment of powerful GANs, it is necessary to devise new strategies for training GANs with limited data while preserving the quality of synthesis.
Recent studies have shown that the overfitting of the discriminator is the critical reason that impedes effective GAN training on limited data [2, 18, 50, 42], rendering severe instability of training dynamics. Specifically, when the discriminator starts to overfit, the distributions of its outputs for real and generated samples gradually diverge from each other [18, 42], and its feedback to the generator becomes less informative. Consequently, the generator converges to an inferior point, compromising the quality of synthesized images. Recent solutions to this problem include the use of standard data augmentations, either conventional or differentiable, to real and generated images [18, 50, 52, 41] or applying an additional model regularization term [42]. Addressing the discriminator overfitting is still an open problem. We are interested in finding an alternative way to the aforementioned approaches.
In this paper, we present a simple yet effective way to regularize a discriminator without introducing any external augmentations or regularization terms. We call our method Adaptive Pseudo Augmentation (APA). In contrast to previous standard data augmentations [18, 50, 52, 41], we exploit the
∗Corresponding author.
35th Conference on Neural Information Processing Systems (NeurIPS 2021).
generator in a GAN itself to provide the augmentation, a more natural way to regularize the overfitting of the discriminator. Compared to the model regularization, our approach is more adaptive to fit different settings and training status without manual tuning. Specifically, APA takes the fake/pseudo samples synthesized by the generator and moderately feeds them into the limited real data. Such pseudo data are adaptively presented to the discriminator as “real” instances. The goal of this pseudo augmentation for the real data is not to enlarge the real dataset but to suppress the discriminator’s confidence in distinguishing real and fake distributions. The deceit is introduced adaptively, which is moderated by a deception probability according to the degree of overfitting. To quantify overfitting, we study a series of plausible heuristics derived from the discriminator raw output logits.
The main contribution of this work is a novel adaptive pseudo augmentation method for training GANs with limited data. This approach deceives the discriminator adaptively and mitigates the problem of discriminator overfitting. The proposed APA can be readily added to existing GAN training with negligible computational cost. We conduct extensive experiments to demonstrate the effectiveness of APA for state-of-the-art GAN training with limited data. The results are comparable or even better than other types of solutions [18, 42]. APA is also complementary to existing methods based on standard data augmentations for gaining a further performance boost. Besides, we theoretically connect APA with minimizing the JS divergence [23] between the smoothed data distribution and generated distribution, proving its convergence and rationality. We hope that our approach could extend the breadth and potential of solutions to GAN training with limited data.

heading: 2 Related Work

text:
Generative adversarial networks. Generative adversarial networks (GANs) [9, 28, 35, 47] adopt an adversarial training scheme, where a generator keeps refining its capability in synthesizing images to compete with a discriminator (i.e., a binary classifier) until the discriminator fails to classify the generated samples as fakes. GANs are known to suffer from training instability [9, 38, 2, 27]. Various approaches have been proposed to stabilize the training and improve the quality of synthesis by minimizing different f -divergences of the real and fake distributions [30]. The saturated form of vanilla GAN [9] is theoretically proven to minimize the JS divergence [23] between the two distributions. LSGAN [26] and EBGAN [49] correspond to the optimizations of χ2-divergence [34] and the total variation [3], respectively. On another note, WGAN [3] is designed for minimizing the Wasserstein distance.
State-of-the-art methods, such as PGGAN [17], BigGAN [5], StyleGAN [19], and StyleGAN2 [20], employ large-scale training with contemporary techniques, achieving photorealistic results. These methods have been extended to various tasks, including face generation [17, 19, 20], image editing [1, 7, 33], semantic image synthesis [44, 32, 25], image-to-image translation [13, 53, 8, 16, 15, 36], style transfer [24, 12, 22], and GAN inversion [31, 36, 46]. Despite the remarkable success, the performance of GANs relies heavily on the amount of training data.
Training GANs with limited data. The significance and difficulty of training GANs with limited data have been attracting attention from many researchers recently. The issue of data insufficiency tends to cause overfitting in the discriminator [45, 18, 50], which in turn deteriorates the stability of training dynamics in GANs, compromising the quality of generated images.
Many recent studies [48, 51, 41, 52, 50, 18] propose to apply standard data augmentations for GAN training to enrich the diversity of the dataset to mitigate the overfitting of the discriminator. For instance, DiffAugment (DA) [50] adopts the same differentiable augmentation to both real and fake images for the generator and the discriminator without manipulating the target distribution. Adaptive discriminator augmentation (ADA) [18] shares a similar idea with DA, while it further devises an adaptive approach that controls the strength of data augmentations adaptively. In this work, we extend the study of such an adaptive approach.
Another type of solution is model regularization. Previous efforts on regularizing GANs include adding noise to the inputs of the discriminator [2, 40, 14], gradient penalty [27, 10, 37], one-sided label smoothing [38], spectral normalization [29], label noise [6], etc. These methods are designed for stabilizing training or preventing mode collapse [38]. The essence of their goals could be considered similar to our method since training GANs in the low-data regime exhibits similar behaviors as previously observed in early GANs with sufficient data. Additional discussions on these techniques are provided in the supplementary material. Under the limited data setting, a very recent study proposes an LC-regularization term [42] to regulate the discriminator predictions using two exponential moving average variables that track the discriminator outputs throughout training.
Our work explores an alternative solution from a different perspective, which is also complementary to previous approaches based on standard data augmentations.

heading: 3 Methodology

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In GAN’s adversarial training, the goal of the generator G is to deceive the discriminator D and maximize the probability that D makes a wrong judgment. Therefore, G keeps refining its generated samples to better deceive D over time. When the training only accesses a limited amount of data, one would observe that D turns out to be overly confident and hardly makes any mistake, causing its feedback to G to become meaningless.
In Figure 2, we show the training “snapshots” of two StyleGAN2 [20] models on the FFHQ dataset [19]. The settings of the two models differ only by the amount of data available to them for
training. As can be observed, both training processes start smoothly, and distributions of discriminator outputs for the real and generated images overlap at the early stage. As the training progresses, the discriminator, which only has access to limited data (7k images, 10% of full data), experiences diverged predictions much more rapidly, and the average sign boundary turns out to be more apparent. This divergence in prediction shows that D becomes increasingly confident in classifying real and fake samples. At the late stage of training, D can even judge all the input samples correctly with high confidence. Meanwhile, the evaluation FID [11] scores (lower is better) deteriorate, consistent with the divergence of D’s predictions. The phenomena above demonstrate how a discriminator gets overfitted quickly with limited data. As can be seen from the FID curves, the overfitting directly influences training dynamics and the convergence of G, leading to poor generation performance.

heading: 3.1 Adaptive Pseudo Augmentation

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The generator itself naturally possesses the capability to counteract discriminator overfitting. To harness this capability, our method employs a GAN to augment itself using the generated samples (see Figure 3). Specifically, we feed the samples generated by G into the limited real data to form a pseudo-real set. The fake images in this set will be adaptively presented to D, pretending themselves as real data. The goal here is to deceive D with the pseudo-real set and consequently suppressing its confidence in distinguishing real and fake images.
Blindly presenting the fake images as reals to D may weaken the fundamental ability of D in adversarial training. In our approach, the deceit is introduced adaptively to avoid any potential adverse effects. To moderate the deception, we perform pseudo augmentation based on a probability, p ∈ [0, 1), that quantifies the deception strength. Specifically, the pseudo augmentation will be applied with the probability p or be skipped with the probability 1− p. We note that the overfitting state of D is dynamic throughout the training (see Figure 2). It is intuitive to let the deception probability p be adjusted adaptively based on the degree of overfitting. Ideally, p should be adjusted without manual tuning regardless of data scales and properties. To achieve this goal, inspired by ADA [18], we apply an overfitting heuristic λ that quantifies the degree of D’s overfitting. We extend the control scheme of ADA and provide three plausible variants:
λr = E (sign (Dreal)) , λf = −E (sign (Dfake)) , λrf = E (sign (Dreal))− E (sign (Dfake))
2 , (1)
where Dreal and Dfake are defined as Dreal = logit (D (x)) , Dfake = logit (D (G (z))) , (2)
where logit denotes the logit function. As shown in Figure 2, the λr in Eq. (1) estimates the portion of real images that obtain positive logit predictions by D, and that of generated images is captured by λf . Besides, λrf indicates half of the distance between the signs of the real and fake logits. For all these heuristics, λ = 0 represents no overfitting, and λ = 1 means complete overfitting. We use λr in our main experiments and study other variants in the ablation study.
The strategy of using λ to adjust p is as follows. We set a threshold t (in most cases of our experiments, t = 0.6) and initialize p to be zero. If λ signifies too much/little overfitting regarding t (i.e., larger/smaller than t), the probability p will be increased/decreased by one fixed step. Using this step size, p can increase from zero to one in 500k images shown to D. We adjust p once every four iterations and clamp p from below to zero after each adjustment. In this way, the strength of pseudo augmentation can be adaptively controlled based on the degree of overfitting (see Figure 3).

heading: 3.2 Theoretical Analysis

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Let pz (z) be the prior on the input noise variable. The mapping from the latent space to the image space is denoted as G (z). For sample x, D (x) represents the estimated probability of x coming from the real data. To examine the rationality of APA, we analyze it in a non-parametric setting [9], where a model is represented with infinite capacity by exploring its convergence in the space of probability density functions. Ideally, the estimated probability distribution pg defined by G should perfectly model the real data distribution pdata without bias if given enough capability and training time.
Since the deception strength p is adaptively adjusted, to facilitate this theoretical analysis, we make a mild assumption that α is the expected strength that approximates the effect of dynamic adjustment during the entire training procedure. Since p ∈ [0, 1), we have 0 ≤ α < pmax < 1, where pmax is the maximum of deception strength throughout training. Hence, the value function V (G,D) for the minimax two-player game of APA can be reformulated as:
min G max D V (G,D) = (1− α)Ex∼pdata(x) [logD (x)] + α Ez∼pz(z) [logD (G (z))]
+ Ez∼pz(z) [log (1−D (G (z)))] , (3)
First, let us consider the optimal discriminator [9] for any given generator.
Proposition 1. If the generator G is fixed, the optimal discriminator D for APA is:
D∗G (x) = (1− α) pdata (x) + α pg (x)
(1− α) pdata (x) + (1 + α) pg (x) (4)
Proof. Applying APA, given any generator G, the training objective of the discriminator D is to maximize the value function V (G,D) in Eq. (3):
V (G,D) = (1− α) ∫ x pdata (x) logD (x)dx+ α ∫ z pz (z) logD (g (z))dz + ∫ z pz (z) log (1−D (g (z)))dz
= ∫ x [(1− α) pdata (x) logD (x) + α pg (x) logD (x) + pg (x) log (1−D (x))] dx
= ∫ x
[((1− α) pdata (x) + α pg (x)) logD (x) + pg (x) log (1−D (x))] dx (5)
For any (m,n) ∈ R2 \ {0, 0}, the function f (y) = m log (y) + n log (1− y) achieves its maximum in the range [0, 1] at mm+n . Besides, the discriminator D is defined only inside of supp (pdata) ∪ supp (pg), where supp is the set-theoretic support. Therefore, we conclude the proof for Proposition 1.
We have got the optimal discriminator D∗G (x) in Eq. (4) that maximizes the value function V (G,D) given any fixed generator G. The goal of generator G in the adversarial training is to minimize the value function V (G,D) in Eq. (3) when D achieves the optimum. Since the training objective of D can be interpreted as maximizing the log-likelihood for the conditional probability P (Y = y|x), where Y estimates that x comes from pdata (i.e., y = 1) or from pg (i.e., y = 0), we reformulate virtual training criterion [9] as:
C (G) = (1− α)Ex∼pdata [logD ∗ G (x)] + α Ez∼pz [logD ∗ G (G (z))] + Ez∼pz [log (1−D ∗ G (G (z)))]
= (1− α)Ex∼pdata [logD ∗ G (x)] + α Ex∼pg [logD ∗ G (x)] + Ex∼pg [log (1−D ∗ G (x))]
(6)
Then, let us consider the global minimum of C (G) trained with the proposed APA.
Proposition 2. Applying APA, the global minimum of the virtual training criterion C (G) is still achieved if and only if pg = pdata, where C (G) = − log 4.
Proof. 1) If pg = pdata, we have D∗G (x) = 1 2 according to Eq. (4). By inspecting Eq. (6) at D∗G (x) = 1 2 , we get C ∗ (G) = (1− α) log 12 + α log 1 2 + log 1 2 = log 1 2 + log 1 2 = − log 4. 2) To verify C∗ (G) is the global minimum of C (G), and it can only be achieved when pg = pdata, as in the derivation of Eq. (5), we obtain:
C (G) = ∫ x ((1− α) pdata (x) + α pg (x)) logD∗G (x)dx+ ∫ x pg (x) log (1−D∗G (x))dx (7)
Observe that − log 4 = (1− α)Ex∼pdata [− log 2] + α Ex∼pg [− log 2] + Ex∼pg [− log 2]
= − ∫ x ((1− α) pdata (x) + α pg (x)) log 2 dx− ∫ x pg (x) log 2 dx (8)
Subtracting Eq. (8) from Eq. (7), C (G) = − log 4 + ∫ x ((1− α) pdata (x) + α pg (x)) log 2 ·D∗G (x)dx+ ∫ x
pg (x) log 2 · (1−D∗G (x))dx (9)
By substituting Eq. (4) into Eq. (9), we achieve: C (G) = − log 4 + KLD ( ((1− α) pdata + α pg) ∥∥∥∥ (1− α) pdata + (1 + α) pg2 ) + KLD ( pg ∥∥∥∥ (1− α) pdata + (1 + α) pg2 ) ,
(10)
where KLD is the Kullback-Leibler (KL) divergence. Moreover, Eq. (10) further implies that the generation process of G by APA can be regarded as minimizing the Jensen-Shannon (JS) divergence between the smoothed data distribution and the generated distribution:
C (G) = − log 4 + 2 · JSD (((1− α) pdata + α pg) ‖pg) . (11)
For the two distributions P and Q, their JS divergence JSD (P‖Q) ≥ 0 and JSD (P‖Q) = 0 if and only if P = Q. Therefore, for 0 ≤ α < pmax < 1, we obtain that C∗ (G) = − log 4 is the global minimum of C (G), and the only solution is (1− α) pdata + α pg = pg , i.e., pg = pdata. Q.E.D. Given the proof in [9], if G and D have enough capacity to reach their optimum, Proposition 2 indicates that the generated distribution pg can ideally converge to the real data distribution pdata. So far, we have proved the convergence of G trained with our proposed APA, which can perfectly model the real data distribution given sufficient capability and training time. Besides, the JS divergence term between the smoothed data distribution and the generated distribution in Eq. (11) implies that the judgment of D may be moderated to alleviate overfitting. These conclusions explain the rationality of the proposed APA for training GANs with limited data.

heading: 4 Experiments

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Datasets. We use four datasets in our main experiments: Flickr-Faces-HQ (FFHQ) [19] with 70, 000 human face images, AFHQ-Cat [8] with 5, 153 cat faces, Caltech-UCSD Birds-200-2011 (CUB) [43] with 11, 788 images of birds, and Danbooru2019 Portraits (Anime) [4] with 302, 652 anime portraits. We exploit some of their artificially limited subsets under different settings. All the images are resized to a moderate resolution of 256× 256 using a high-quality Lanczos filter [21] to reduce the energy consumption for large-scale GAN training while preserving image quality. Additional dataset details, including the data source and license information, can be found in our supplementary material.
Evaluation metrics. We follow the standard evaluation protocol [5, 42] for the quantitative evaluation. Specifically, we use the Fréchet Inception Distance (FID, lower is better) [11], which quantifies the distance between distributions for the real and generated images. FID evaluates the realness of synthesized images. Following [11, 18], we calculate FID for the models trained with limited
datasets using 50k generated images and all the real images in the original datasets. We also apply the Inception Score (IS, higher is better) [38]. IS considers the clarity and diversity of generated images.
Implementation details. We choose the state-of-the-art StyleGAN2 [20] as the backbone to verify the effectiveness of APA on limited data. We use the default setups of APA provided in Section 3.1 unless specified otherwise. For a fair and controllable comparison, we reimplement all baselines and run the experiments from scratch using official code. All the models are trained on 8 NVIDIA Tesla V100 GPUs. Please refer to the supplementary material for more implementation details and additional benchmark results (e.g., performance on BigGAN [5]).

heading: 4.1 The Effectiveness of APA

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Effectiveness on various datasets. The comparative results of StyleGAN2 on various datasets with limited data amounts are shown in Figure 4. The quality of images synthesized by StyleGAN2 deteriorates under limited data. Ripple artifacts appear on the cat faces and human faces, and the facial features of the anime faces are misplaced. On the bird dataset with heavy background clutter, the generated images are completely distorted albeit trained with more data. The proposed APA significantly ameliorates image quality on all these datasets, producing much more photorealistic results. The quantitative evaluation results are reported in Table 1. Applying APA contributes to a performance boost of FID and IS in all cases, suggesting that the synthesized images are with higher quality and diversity on different datasets.
Effectiveness given different data amounts. The comparative results on subsets of FFHQ [19], with varying amounts of data, are shown in Figure 5. The corresponding quantitative test results are
presented in Table 2. APA improves the image quality and metric performance in all cases. Notably, the quality of synthesized images by APA on 5k/7k data is visually close to StyleGAN2 results on the full dataset while with an order of magnitude fewer training samples. As for the quantitative results, the IS score of APA on 1k data is better than that of StyleGAN2 on 5k data, and both metrics of APA on 5k data outperform StyleGAN2 results on 7k data. APA can even improve StyleGAN2 performance on the full dataset, further indicating its potential.
Overfitting and convergence analysis. As shown in Figure 6, the divergence of StyleGAN2 discriminator predictions can be effectively restricted on FFHQ-7k (10% of full data) by applying the proposed APA. The curves of APA on FFHQ-7k become closer to that of StyleGAN2 on FFHQ-70k, suggesting the effectiveness of APA in curbing the overfitting of the discriminator. Besides, APA improves the training convergence of StyleGAN2 on limited data, shown by the FID curves. More overfitting and convergence analysis can be found in the supplementary material.

heading: 4.2 Comparison with Other Solutions for GAN Training with Limited Data

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Performance and compatibility. We compare the proposed APA with representative approaches designed for the low-data regime, including ADA [18] and LC-regularization (LC-Reg) [42], which perform standard data augmentations and model regularization, respectively. The results are reported in Table 3. As a single method, APA outperforms previous solutions in most cases, effectively
improving the StyleGAN2 baseline on both the limited and full datasets. Although ADA [18] achieves slightly better results than our method on FFHQ-5k, it yields a worse FID score with StyleGAN2 on the full dataset. Applying LC-Reg [42] needs careful manual tuning, and its own effect on limited data is not apparent compared to other methods.
It is noteworthy that APA is also complementary to existing methods based on standard data augmentations, e.g., ADA [18]. As can be observed in Table 3, APA can further boost the performance of StyleGAN2 with ADA [18] given limited training data, suggesting the compatibility of our approach with standard data augmentations. Combining ADA [18] and APA on FFHQ full data outperforms StyleGAN2 but is slightly inferior to applying APA solely. The degraded performance is mainly affected by ADA [18], which we empirically found might slightly harm the performance when the training data is sufficient. Overall, these methods are not dedicated to improving performance under sufficient data. Nevertheless, as a beneficial side effect, the proposed APA may have this potential. More comparative results are included in our supplementary material.
Training cost. We compare the computational cost of APA against ADA [18], using the same basic official codebase. There is no parameter or memory increment for both methods. As for the time consumption, we test the training cost on 8 NVIDIA Tesla V100 GPUs. On FFHQ-5k (256× 256), the average training time of the StyleGAN2 [20] baseline is (4.740± 0.100) sec/kimg (i.e., seconds per thousand of images shown to the discriminator). The cost of our method is negligible, slightly increasing this value to (4.789 ± 0.078) sec/kimg. As a reference, the value for ADA [18] is (5.327± 0.116) sec/kimg, spending additional time for applying external augmentations.

heading: 4.3 Comparison with Previous Techniques for Regularizing GANs

text: As mentioned in Section 2, APA is closely related to previous techniques for regularizing GANs. The comparative results on APA and some representative conventional techniques, i.e., instance noise [40] and one-sided label smoothing (LS) [38], are shown in Table 4. Applying instance noise [40] may not boost the performance of StyleGAN2 much under limited data. One-sided label smoothing (LS) [38] (with the real label of 0.9) outperforms StyleGAN2 but still has a huge performance gap with our method. This further suggests the effectiveness and usefulness of APA.

heading: 4.4 Ablation Studies

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Ablation studies on variants of APA. We study three key elements of APA, i.e., the overfitting heuristic λ, the deception strength p, and the deception strategy. The version used in our main experiments is denoted as the “main” version, where λ = λr, and p is adjusted adaptively. Besides, the “main” version applies the deception strategy that is analogous to one-sided label flipping. As reported in Table 5, when using other variants of λwe suggested in Eq. (1) (i.e., λ = λf and λ = λrf ), the models achieve comparable performance as the main version. The FID score becomes even better for λ = λrf , indicating the flexibility of our provided heuristics to extend and modify APA. More interestingly, even a fixed moderate deception probability (e.g., p = 0.5) can still work much better
than original StyleGAN2 on limited data, albeit slightly inferior to the adaptively adjusted p. This implies the importance of the pseudo augmentation, and the adaptive control scheme can further boost performance without manual tuning. As for the deception strategy, we empirically observe that two-sided label flipping can still outperform StyleGAN2 but is inferior to the main version.
Ablation studies on the threshold t. We further provide the ablation studies on the threshold value t in Table 6. The version used in our main experiments is denoted as the “main” version, where t = 0.6. It can be seen that the models with different values of t achieve comparable results, outperforming the StyleGAN2 baseline. On FFHQ-5k (256× 256), t = 0.6 could be a more plausible choice. For a further explanation, we use t = 0.6 as the default value since it works well in most cases. In practice, the value of t could be further adjusted to achieve even better results. Empirically, a smaller t can be chosen when one has fewer data. This means the deception strength p can be adjusted to increase more rapidly since the discriminator is more prone to overfitting when the data amount is fewer.

heading: 5 Discussion

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We have shown the effectiveness of the proposed adaptive pseudo augmentation (APA) for stateof-the-art GAN training with limited data empirically. With negligible computational cost, APA achieves comparable or even better performance than other types of solutions on various datasets. APA is also complementary to existing methods based on standard data augmentations.
Limitations. Despite promising results, the quality of synthesized images by APA on the datasets with extremely limited data amount (e.g., hundreds of images) can still be improved. Besides, on certain datasets such as FFHQ-5k, applying APA solely may be slightly inferior to approaches based on standard data augmentations. Since we do not apply any external augmentations, these two limitations are both due to the insufficiency of the dataset’s intrinsic diversity. These limitations may be approached in the future in two ways: 1) Incorporating better standard data augmentations to APA. 2) Exploring the issue of data insufficiency from the generator aspect, e.g., using a multi-modal generator [39] to enhance diversity. In addition, we only theoretically verified the convergence and rationality of APA. In future work, the theoretical analysis on the effectiveness of APA could be further explored.
Broader impact. On the one hand, the effectiveness of APA with negligible computational cost will benefit the practical deployment of GANs, especially in the low-data regime. APA may also extend the breadth and potential of solutions to training GANs with limited data and benefit downstream tasks, such as conditional synthesis. On the other hand, APA may also bring potential concerns on its capability to ease the higher-quality fake media synthesis using only limited data, as technology is usually a double-edged sword. However, we believe that these concerns can be resolved by developing better media forensics methods and datasets as countermeasures.
Acknowledgments and funding disclosure. This study is supported under the RIE2020 Industry Alignment Fund – Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s).

title: Deceive D: Adaptive Pseudo Augmentation for GAN Training with Limited Data