Published as a conference paper at ICLR 2022

ROBUST L EARNING M EETS G ENERATIVE M ODELS :C AN P ROXY D ISTRIBUTIONS I MPROVE A DVERSARIAL
ROBUSTNESS ?
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Vikash Sehwag , Saeed Mahloujifar , Tinashe Handina , Sihui Dai , Chong Xiang♣♠
Mung Chiang , Prateek Mittal

arXiv:2104.09425v3 [cs.LG] 3 Mar 2022

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Princeton University, Caltech, Purdue University

A BSTRACT
While additional training data improves the robustness of deep neural networks
against adversarial examples, it presents the challenge of curating a large numberof specific real-world samples. We circumvent this challenge by using additional
data from proxy distributions learned by advanced generative models. We firstseek to formally understand the transfer of robustness from classifiers trained onproxy distributions to the real data distribution. We prove that the difference between the robustness of a classifier on the two distributions is upper bounded by
the conditional Wasserstein distance between them. Next we use proxy distributions to significantly improve the performance of adversarial training on fivedifferent datasets. For example, we improve robust accuracy by up to 7.5% and6.7% in `∞ and `2 threat model over baselines that are not using proxy distributions on the CIFAR-10 dataset. We also improve certified robust accuracy by
7.6% on the CIFAR-10 dataset. We further demonstrate that different generativemodels bring a disparate improvement in the performance in robust training. We
propose a robust discrimination approach to characterize the impact of individual generative models and further provide a deeper understanding of why current
state-of-the-art in diffusion-based generative models are a better choice for proxy
distribution than generative adversarial networks.

1

I NTRODUCTION

Deep neural networks are powerful tools but their success depends strongly on the amount oftraining data (Sun et al., 2017; Mahajan et al., 2018). Recent works show that for improving robust training against adversarial examples (Biggio et al., 2013; Szegedy et al., 2014; Biggio & Roli,2018), additional training data is helpful (Carmon et al., 2019; Schmidt et al., 2018a; Uesato et al.,2019; Deng et al., 2021). However, curating more real-world data from the actual data distribution
is usually challenging and costly (Recht et al., 2019). To circumvent this challenge, we ask: canrobust training be enhanced using a proxy distribution, i.e., an approximation of the real data distribution? In particular, can additional samples from a proxy distribution, that is perhaps cheaper tosample from, improve robustness. If so, can generative models that are trained on limited training
images in small scale datasets (LeCun & Cortes, 2010; Krizhevsky et al., 2014), act as such a proxy1
distribution?
When training on synthetic samples from generative models, a natural question is whether robustness
on synthetic data will also transfer to real world data. Even if it does, can we determine the featuresof synthetic data that enable this synthetic-to-real robustness transfer and optimize our selection of†

Corresponding author: vvikash@princeton.edu
Our code is available at https://github.com/inspire-group/proxy-distributions.
1
Proxy distributions may not necessarily be modeled by generative models. When a proxy distribution is
the output of a generative model, we call it synthetic distribution and refer to data sampled from it as synthetic
data.

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Published as a conference paper at ICLR 2022

generative models based on these features? Finally, can we also optimize the selection of individual
synthetic samples to maximize the robustness transfer?
Q.1 When does robustness transfer from proxy distribution to real data distribution? This question
is fundamental to develop a better understanding of whether a proxy distribution will help. For aclassifier trained on only synthetic samples, we argue that in addition to empirical and generalization
error, distribution shift penalty also determines its robustness on the real data distribution. We
prove that this penalty is upper bounded by the conditional Wasserstein distance between the proxy
distribution and real data distribution. Thus robustness will transfer from proxy distributions thatare in close proximity to real data distribution with respect to conditional Wasserstein distance.
Q.2 How effective are proxy distributions in boosting adversarial robustness on real-world dataset?
Our experimental results on five datasets demonstrate that the use of samples from PrOxy distributions in Robust Training (PORT) can significantly improve robustness. In particular, PORT achieves
up to to 7.5% improvement in adversarial robustness over existing state-of-the-art (Croce et al.,2020). Its improvement is consistent across different threat models (`∞ or `2 ), network architectures, datasets, and robustness criteria (empirical or certified robustness). We also uncover thatsynthetic images from diffusion-based generative models are most helpful in improving robustness
on real datasets. We further investigate the use of proxy distributions in robust training. In particular,
we investigate why current state-of-the-art in diffusion-based models are significantly more helpful
than their counterparts, such as generative adversarial networks (GANs).
Q.3 Can we develop a metric to characterize which proxy distribution will be most helpful in robust
training? Our theory motivates the design of a measure of proximity between two distributions thatincorporates the geometry of the distributions and can empirically predict the transfer of robustness.
We propose a robust learning based approach where we use the success of a discriminator in distinguishing adversarially perturbed samples of synthetic and real data as a measure of proximity.
Discriminating between synthetic and real data is a common practice (Goodfellow et al., 2014; Guiet al., 2020), however, we find that considering adversarial perturbations on synthetic and real data
samples is the key to making the discriminator an effective measure for this task. We demonstratethat the rate of decrease in discriminator success with an increasing size of perturbations can effectively measure proximity and it accurately predicts the relative transfer of robustness from different
generative models.
We also leverage our robust discriminators to identify most helpful synthetic samples. We use theproximity of each synthetic sample to real data distribution, referred to as synthetic score, as a metricto judge their importance. This score can be computed using output probability from a discriminatorthat is robustly trained to distinguish between adversarially perturbed samples from proxy and real
data distribution. We demonstrate that selecting synthetic images based on their synthetic scores canfurther improve performance.
Contributions. We make the following key contributions.
• We provide an analytical bound on the transfer of robustness from proxy distributions to real data
distribution using the notion of conditional Wasserstein distance and further validate it experimentally.
• Overall, using additional synthetic images, we improve both clean and robust accuracy on fivedifferent datasets. In particular, we improve robust accuracy by up to 7.5% and 6.7% in `∞ and `2
threat models, respectively, and certified robust accuracy (`2 ) by 7.6% on the CIFAR-10 dataset.
• When selecting a proxy distribution, we show that existing metrics, such as FID, fail to determine
the synthetic-to-real transfer of robustness. We propose a new metric (ARC) based on the distinguishability of adversarially perturbed synthetic and real data, that accurately determines theperformance transfer.
• We also develop a metric, named synthetic score, to determine the importance of each synthetic
sample in synthetic-to-real robustness transfer. We demonstrate that choosing samples with lower
synthetic scores provide better results than randomly selected samples.

2

I NTEGRATING PROXY DISTRIBUTIONS IN ROBUST TRAINING

In this section we propose to use samples from proxy distributions to improve robustness. We
first provide analytical bounds on the transfer of adversarial robustness from proxy to real data2

Published as a conference paper at ICLR 2022

distribution. Next, using robust discriminators we provide a metric based on our analytical bound,which can accurately determine the relative ranking of different proxy distributions in terms ofrobustness transfer. Our metric can be calculated empirically (using samples from both distributions)and does not require the knowledge of the proxy or real data distribution. Finally, we present ourrobust training formulation (PORT) which uses synthetic samples generated by the generative model,
together with real samples.
Notation. We represent the input space by X and corresponding label space as Y. Data is sampledfrom a joint distribution D that is supported on X × Y. For a label y, we use D ∣ y to denote theconditional distribution of class y. We denote the proxy distribution as D̃. We denote the neuralnetwork for classification by f ∶ X → Z, parameterized by θ, which maps input images to output
probability vectors (z). We use h to refer to the classification functions that output labels. For a set
S sampled from a distribution D, we use Ŝ to denote the empirical distribution with respect to set
S. We use S ∼ D to denote the sampling of a dataset from a distribution D. We use (x, y) ← D todenote the sampling of a single point from D.
2.1

U NDERSTANDING TRANSFER OF ADVERSARIAL ROBUSTNESS BETWEEN DATA
DISTRIBUTIONS

Since our goal is to use samples from proxy distribution to improve robustness on real data distribution, we first study the transfer of adversarial robustness between two data distributions.
Definition 1 (Average Robustness). We define average robustness for a classifier h on a distribution′
D according to a distance metric d as follows: Robd (h, D) = E(x,y)←D [inf h(x′ )≠y d(x , x)]. where2classifier h ∶ X → Y predicts class label of an input sample .
This definition refers to the expected distance to the closest adversarial example for each sample.
Formalizing transfer of robustness from proxy to real data distribution. In robust learningfrom a proxy distribution, we are interested in bounding the average robustness of the classifier
obtained by a learning algorithm (L), on distribution D, when the training set is a set S of n labeled
examples sampled from a proxy distribution D̃. In particular we want to provide a lower bound on
the transferred average robustness i.e, E S∼D̃ [Robd (h, D)].
h←L(S)In order to understand this quantity better, suppose h is a classifier trained on a set S that is sampledfrom D̃, using algorithm L. We decompose Robd (h, D) to three quantities as follows:
Robd (h, D) = (Robd (h, D) − Robd (h, D̃)) + (Robd (h, D̃) − Robd (h, Ŝ)) + Robd (h, Ŝ).
Using this decomposition, by linearity of expectation and triangle inequality we can bound transferred average robustness from below by
E n [Robd (h, Ŝ)] − ∣S←D̃h←L(S)

ÍÒÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò ÑÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ï
Empirical robustness

E n [Robd (h, D̃) − Robd (h, Ŝ)]∣ − ∣S←D̃h←L(S)

ÍÒÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò ÑÒÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò ÒÏ
Generalization penalty

E n [Robd (h, D) − Robd (h, D̃)]∣ .

S←D̃h←L(S)

ÍÒÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò ÑÒÒ Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ï
Distribution-shift penalty

As the above decomposition suggests, in order to bound the average robustness, we need to boundboth the generalization penalty and the distribution shift penalty. The generalization penalty has been
rigorously studied before in multiple works (Cullina et al., 2018; Montasser et al., 2019; Schmidt
et al., 2018b). Hence, we focus on bounding the distribution shift penalty. Our goal is to provide
a bound on the distribution-shift penalty that is independent of the classifier in hand and is only
related to the properties of the distributions. With this goal, we define a notion of distance between
two distributions.
Definition 2 (Conditional Wasserstein distance). For two labeled distributions D and D̃ supported
on X × Y, we define cwd according to a distance metric d as follows:
cwdd (D, D̃) =E(⋅,y)←D[

inf

E

J∈J (D∣y,D̃∣y) (x,x′ )←J2[d(x, x )]]′This notion of robustness is also used in Gilmer et al. (2018); Diochnos et al. (2018) and Mahloujifar et al.
(2019a).

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Published as a conference paper at ICLR 2022

where J (D, D̃) is the set of joint distributions whose marginals are identical to D and D̃.
It is simply the expectation of Wasserstein distance between conditional distributions for each class.
Now, we are ready to state our main theorem that bounds the distribution shift penalty for any
learning algorithm based only on the Wasserstein distance of the two distributions.
Theorem 1 (Bounding distribution-shift penalty). Let D and D̃ be two labeled distributions sup∗∗ported on X × Y with identical label distributions, i.e., ∀y ∈ Y, Pr(x,y)←D [y = y ] =∗Pr(x,y)←D̃ [y = y ]. Then for any classifier h ∶ X → Y∣Robd (h, D̃) − Robd (h, D)∣ ≤ cwdd (D, D̃).
Theorem 1 shows how one can bound the distribution-shift penalty by minimizing the conditional
Wasserstein distance between the two distributions. We provide its proof and empirical validationin Appendix A. Even if we successfully reduce the generalization penalty, the distribution-shift
penalty may remain the dominant factor in transferred robustness. This theorem enables us to switch
our attention from robust generalization to creating generative models for which the underlying
distribution is close to the original distribution. In Appendix A, we also provide two other theorems
about the tightness of Theorem 1 and the effect of combining clean and proxy data together.
2.2

W HICH PROXY DISTRIBUTION TO CHOOSE ? - A PPROXIMATING CONDITIONALWASSERSTEIN DISTANCE USING ROBUST DISCRIMINATORS

Our analytical results suggest that a proxy distribution that has small conditional Wasserstein distance to the real data can improve robustness. This means that optimizing for conditional Wasserstein distance in a proxy distribution (e.g., a generative model) can potentially lead to improvements
in adversarial robustness. Unfortunately, it is generally a hard task to empirically calculate the conditional Wasserstein distance using only samples from the distributions (Mahloujifar et al., 2019b;
Bhagoji et al., 2019). In this section, we introduce a metric named ARC as a surrogate for conditional
Wasserstein distance that can be calculated when we only have sample access to the data distributions. Before defining our metric, we need to define the notion of robust discrimination between two
distributions.
Robust discrimination. Our metric works based on
how well a discriminator can robustly distinguish between
samples from the real and proxy distributions. Compared
to a typical discriminator, the robust discriminator shouldbe accurate even when there is an adversary who can perReal
Real
Proxy-1
Proxy-2
Proxy-1
Proxy-2
turb the sampled instance by a perturbation of size . Intuitively, if there exists a successful and robust discrimi(a) Original samples.
(b) Perturbed samples.
nator for two distributions D and D̃, then it means that noadversary can make instances of D to look like instances Figure 1: Why robust discrimination
is effective.
(a) Without adversarof D̃ even by making perturbations of size . This means ial perturbations, a non-robust discrimthat D and D̃ are far away from each other (Figure 1b). inator can perfectly distinguish bothOn the other hand, if no robust discriminator exists, then proxy distributions from the real distriit means that the adversary can ”align“ most of the mass bution. (b) Under adversarial perturof proxy distribution with that of real distribution by mak- bations, classification between real anding perturbations of size at most . This notion is closely proxy-1 is much harder compared torelated to the optimal transport between the proxy and real and proxy-2 distribution. The sucreal distribution. In particular, the optimal adversary here cess of a robust discriminator is depencorresponds to a transport J ∈ J (D, D̃) that minimizes dent on the proximity of real and proxy′Pr(x,x′ )←J(D,D̃) [d(x, x ) ≤ ] (Compare with Definition distributions.
32). To be more specific, the goal of a robust discriminator σ is to maximize its robust accuracy :Racc (σ, D, D̃) = 21 ⋅Prx←D [∀x ∈ Ball (x); σ(x ) = 1]+ 21 ⋅Prx←D̃ [∀x ∈ Ball (x); σ(x ) = 0] .
′

′

3

′

′

All the definitions of accuracy and robust accuracy in this section are defined for true distributions. However, in the experiments we work with their empirical variant.

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We use Racc (D, D̃) to denote the accuracy of best existing robust discriminator, namely,∗
Racc (D, D̃) = minσ Racc (σ, D, D̃). Note that as  grows from 0 to ∞, the robust accuracyof any discriminator will decrease to at most 50%. If two distributions D and D̃ are close to eachother, this drop in accuracy happens much faster since the adversary can make images from two distributions to look alike by adding smaller perturbations. We use this intuition and define our metric
ARC as follows:∞
1∗
ARC(D, D̃) = ∫ (Racc (D, D̃) − )d.
20∗

In other words, the ARC metric is equal to the area under the robust discrimination accuracy v.s.
perturbation curve (See Figure 3 for a sample of this curve.) In Section 3.2, we discuss how thismetric can be calculated with only samples from the real data and proxy distributions. When selecting a proxy distribution, we choose the one with the smallest ARC value, since it is expected
to be closest to real data distribution. We also demonstrate that ARC metric is directly related toconditional Wasserstein distance. In particular, we have the following Theorem:
Theorem 2. For any two distributions D̃ and D with equal class probabilities we havecwd(D, D̃) ≥ 4 ⋅ ARC(D, D̃). Moreover, if for all labels y , (D ∣ y) and (D̃ ∣ y) are twoconcentric uniform spheres, then we have cwd(D, D̃) = 4 ⋅ ARC(D, D̃).
Why non-robust discrimination is not an effective metric. Deep neural networks are commonlyused as a discriminator between synthetic and real images (Goodfellow et al., 2014; Gui et al.,2020). In our experiments, we find that even a four-layer convolutional network can achieve near
100% classification accuracy on all selected proxy distributions (Appendix C.1). This result showsthat non-robust discrimination is not an effective measure of proximity (as highlighted in Figure 1a).
2.3

B RINGING IT ALL TOGETHER : I MPROVING ADVERSARIAL ROBUSTNESS USING PROXYDISTRIBUTIONS

Overview of robust training.
The key objective in robust training is to solve
min E Ladv (θ, x, y, Ω), where Ladv is determined by the training objective and Ω is the threatθ(x,y)∼Dmodel. In adversarial training (Madry et al., 2018; Zhang et al., 2019; Carmon et al., 2019; Panget al., 2021), we train against projected gradient descent based (PGD) adversarial examples by selecting Ladv (.) = `(θ, P GD(x, Ω), y). In randomized smoothing (Cohen et al., 2019; Salman
et al., 2019; Sehwag et al., 2020; Carmon et al., 2019), where the objective is to achieve certified
robustness (Wong & Kolter, 2018), we aim to achieve invariance to random Gaussian Noise by se2
lecting Ladv (.) = `(θ, x, y) + βDkl (fθ (x), fθ (x + δ)), where δ ∼ N(0, σ I), and Dkl (., .) is the
KL-divergence. This loss function (Zheng et al., 2016), referred to as robust stability training (RST),
performs better than using `(θ, x+δ, y) (Carmon et al., 2019). We refer the Ladv in adversarial training and randomized smoothing as Lpgd and Lsmooth , respectively. We represent the cross-entropy
loss function, which is used to train the classifier, as `(.), where `(θ, x, y) = ⟨−log(fθ (x)), y⟩ andy is the one-hot vector encoding of y.
Our objective is to integrate synthetic samples from Proxy distributions in Robust Training (PORT).
In particular, we use the following robust training formulation that combines samples from a real
training dataset with synthetic samples.
min Lagg ; Lagg = γ
θE(x,y)∼D

Ladv (θ, x, y, Ω) + (1 − γ)E(x,y)∼D̃

Ladv (θ, x, y, Ω)

where γ ∈ [0, 1] and Lagg is the aggregated adversarial loss over both D and D̃. Depending on
training objective, we select either Lpgd or Lsmooth as Ladv . We approximate the aggregated loss
using available training samples from the real data distribution (D) and a set of synthetic images
sampled from the proxy distribution (D̃). We select γ through a hyperparameter search.
Which synthetic samples to choose? In particular, which set (S) of N synthetic samples
from the proxy distribution (D̃) leads to maximum transferred robustness on real data, i.e.,
Ef ←L̂(S) [Robd (f, D)], where L̂ is a robust learning algorithm. While it can be solved in manydifferent ways, we follow a rejection sampling based approach. In particular, we measure proximityof each synthetic sample to real data distribution and use it as a metric to accept or reject a sample.
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Published as a conference paper at ICLR 2022

We use output probability score from our trained robust discriminators as a measure of proximity. In
particular we define synthetic-score(x) = minxi ∈Ball (x) σ(xi ) as the final score for each synthetic
sample, where samples with lowest score are prioritized in the selection.

3

E XPERIMENTAL RESULTSFirst, we show that our proposed approach of using proxy distributions brings significant improvements in the performance of both adversarial training and randomized smoothing across both `∞ and`2 threat models (Section 3.1). We also uncover that diffusion-based generative models are more effective proxy distributions than generative adversarial networks (GANs) on multiple datasets. Next
we delve deeper into why diffusion-based generative models are highly effective proxy distributions(Section 3.2). We find that the key reason behind the success of diffusion based models is the that
there is no robust discriminator that can robustly distinguish between the samples from diffusion
based models and real data distribution. We demonstrate that our proposed metric (ARC) provides abetter characterization of the effectiveness of diffusion-based models than other widely used metrics.
Finally, we investigate the effectiveness of adaptively selecting synthetic images in our framework.
3.1

I MPROVING ROBUSTNESS USING SYNTHETIC IMAGES FROM PROXY DISTRIBUTIONS

Setup. We consider five datasets, namely CIFAR-10 (Krizhevsky et al., 2014), CIFAR100 (Krizhevsky et al., 2014), CelebA (Liu et al., 2015), AFHQ (Choi et al., 2020), and ImageNet (Deng et al., 2009). We also work with commonly used l∞ and l2 threat models for adversarial attacks and use AutoAttack (Croce & Hein, 2020) to measure robustness. We provide
extensive details on the setup in Appendix B.
Evaluation metrics for robust training. We use two key metrics to evaluate performance of trainedmodels: clean accuracy and robust accuracy. While the former refers to the accuracy on unmodified
test set images, the latter refers to the accuracy on adversarial examples generated from test set
images. We refer to the robust accuracy of any discriminator trained to distinguish between synthetic
and real data as robust discrimination accuracy. We also use randomized smoothing to measurecertified robust accuracy, i.e., robust accuracy under strongest possible adversary in the threat model.
Synthetic data. We sample synthetic images from a generative models, in particular from denoisingdiffusion-based probabilistic models (DDPM) (Ho et al., 2020; Nichol & Dhariwal, 2021). We provide extensive details on the number of images sampled and the sampling procedure in Appendix B.
We use γ = 0.4 as it achieves best results (Appendix D). We combine real and synthetic images in a
1:1 ratio in each batch, thus irrespective of the number of synthetic images, our training time is only
twice of the baseline methods.
Table 1: State-of-the-art adversarial robustness. Comparing experimental results of our framework (PORT) with baselines for adversarial training on the CIFAR-10 dataset for both `∞ and `2
threat model. We sample synthetic images from the diffusion-based generative model (DDPM).
Clean/Auto refers to clean/robust accuracy measured with AutoAttack.
(a) `∞ threat model.

(b) `2 threat model.

MethodArchitecture

Parameters (M)

Clean

Auto

MethodArchitecture

Parameters (M)

Clean

Auto

Zhang et al. (2019)
PORT

ResNet-18
ResNet-18

11.2
11.2

82.0
84.6

48.7
55.7

Rice et al. (2020)
PORT

ResNet-18
ResNet-18

11.2
11.2

88.7
89.8

67.7
74.4

Rice et al. (2020)
Gowal et al. (2020)

W RN -34-20
W RN -70-16

184.5
266.8

85.3
85.3

53.4
57.2

Madry et al. (2018)
Gowal et al. (2020)

ResNet-50
W RN -70-16

23.5
266.8

90.8
90.9

69.2
74.5

Zhang et al. (2019)
PORT

W RN -34-10
W RN -34-10

46.2
46.2

84.9
87.0

53.1
60.6

Wu et al. (2020b)
PORT

W RN -34-10
W RN -34-10

46.2
46.2

88.5
90.8

73.7
77.8

3.1.1

I MPROVING PERFORMANCE OF ADVERSARIAL TRAINING

State-of-the-art robust accuracy synthetic images (Table 1, 2). Using synthetic images fromdiffusion-based generative models (DDPM), our approach achieves state-of-the-art robust accuracy
in the category of not using any extra real world data (Croce et al., 2020). We improve robust
accuracy by up to 7.5% and 6.7% over previous works in `∞ and `2 threat model, respectively.
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Notably, the gain in performance is highest for the CIFAR-10 and Celeb-A datasets. Note that byusing only synthetic images, our work also achieves competitive performance with previous works
that use extra real-world images. For example, using additional real-data Carmon et al. (2019)achieve 59.53% robust accuracy on CIFAR-10 (`∞ ) while we achieve 60.6% robust accuracy byusing only synthetic data.
Table 2: Similar improvement on additional datasets. Our approach further improves both clean
and robust accuracy on each of datasets of Tables 1 and 2. Unless a better baseline is available inprevious works, we compare our results with baseline robust training approach from Madry et al.
(2018). Clean/Auto refers to clean/robust accuracy measured with AutoAttack. We provide resultsfor AFHQ dataset in Table 8 in Appendix.
(a) CelebA (64 × 64)

Baseline
PORT∆(c) ImageNet (64 × 64)(b) CIFAR-100 (32 × 32)`∞ ( = 8/255)`2 ( = 1.0)`∞ ( = 8/255)`2 ( = 0.5)`∞ ( = 2/255)

Clean

Auto

Clean

Auto

Clean

Auto

Clean

Auto

Clean

Auto

82.4
84.8
+2.4

60.2
63.4
+3.2

80.6
82.3
+1.7

58.9
60.0
+1.1

58.4
65.9
+7.5

27.8
31.2
+3.4

62.9
70.7
+7.8

42.1
47.8
+5.7

Baseline

54.8

28.1

PORT

55.1

28.4

∆

+0.3

+0.3

Baseline
PORT∆

Proxy distribution offsets increase in network parameters. We find that gains from using synthetic samples are Table 3: Additional baselines. Using
equivalent to ones obtained by scaling network size by an `∞ ( = 8/255) attack for both datasets.
order of magnitude (Table 1). For example, a ResNet-18
CIFAR-100
Clean
Auto
network with synthetic data achieves higher robust accuracy (`∞ ) than a W RN -34-20 trained without it, while
Wu et al. (2020a)
60.4
28.9
having 16× fewer parameters than the latter. Similar
Rade & Moosavi-Dezfooli (2021)
61.5
28.9
trend holds for W RN -34-10 networks, when comparedCui et al. (2021)
60.6
29.3
PORT
60.6
30.5
with a much larger W RN -70-16 network. This trend also
holds for both `∞ and `2 threat models.
CelebA
Clean
Auto
Simultaneous improvement in clean accuracy. Due to
TRADES
86.1
61.2
the accuracy vs robustness trade-off in adversarial trainPORT
86.1
62.7
∆
0.0
1.5
ing, improvement in robust accuracy often comes at thecost of clean accuracy. However synthetic samples provide a boost in both clean and robust accuracy, simultaneously. Using adaptively sampled synthetic
images, we observe improvement in clean accuracy by up to 7.5% and 7.8% across `∞ and `2 threat
models, respectively.
Comparison across generative models. We find that synthetic samples from diffusion-based generative models (Ho et al., 2020; Nichol & Dhariwal, 2021) provides significantly higher improvementin performance than generative adversarial networks (GANs) (Table 4). We further investigate thisphenomenon in the next section.
We further analyze sample complexity of adversarial training using synthetic data in Appendix D.1.
We also provide visual examples of real and synthetic images for each dataset at the end of the paper.
3.1.2

I MPROVING CERTIFIED ROBUSTNESS WITH RANDOMIZED SMOOTHING

We provide results on certified adversarial robustness in Table 5. We first compare the performanceof our proposed approach with the baseline technique, i.e., RST (Carmon et al., 2019). We achievesignificantly higher certified robust accuracy than the baseline approach at all `2 perturbations budgets for both ResNet-18 and W RN -28-10 network architectures. Additionally, the robustness of ourapproach decays at a smaller rate than the baseline. At `∞ perturbation of 2/255, equivalent to `2
perturbation of 111/255, our approach achieves 7.6% higher certified robust accuracy than RST. We
also significantly outperform other certified robustness techniques which aren’t based on randomized smoothing (Zhang et al., 2020; Wong et al., 2018; Balunovic & Vechev, 2019). Along with
better certified robust accuracy, our approach also achieves better clean accuracy than most previous
approaches, simultaneously.
Synthetic images vs real-world images. Using only synthetic samples (10M), we also outperform
RST where it uses an additional curated set of 500K real-world images (RST500K ). While the latterachieves 63.8% certified robust accuracy, we improve it to 66.2%.
7

Published as a conference paper at ICLR 2022

DatasetProxyDistribution

Clean

Auto

CIFAR-10

None
StyleGAN
DDPM

84.9
86.0
87.0

53.1
52.5
60.6

CelebA

None
StyleFormer
DDPM

82.4
84.1
84.8

60.2
63.1
63.4

ImageNet

None
BigGAN
DDPM

54.8
54.5
55.1

28.1
28.2
28.4

3.2

0.8

Table 5: Comparison of certified robustness. Comparing
clean accuracy (Clean) and cer0.6
tified robust accuracy (CertiOurs (WRN-28-10)
0.5
Baseline (WRN-28-10)fied) of our work with earlier
Ours (ResNet-18)
Baseline (ResNet-18)
0.4
approaches at 110/255 `2 per0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
turbation budget (equivalent to
2 radius
2/255 `∞ perturbation budget)on
the CIFAR-10 dataset.
Figure 2: Certified robustness. Certified robust accuracy
MethodClean Certifiedof baseline randomized smoothWong et al. (2018) (single)
68.3
53.9
ing technique, i.e., RST (CarWong et al. (2018) (ensemble)
64.1
63.6
(Zhang et al. (2020)) 71.5
54.0
mon et al., 2019) and our work CROWN-IBP
Balunovic & Vechev (2019)
78.4
60.5
RST (Carmon et al. (2019))
77.9
58.6
at different perturbation budgets
RST(Carmon et al. (2019))
80.7
63.8
across two different network arPORT (ResNet-18)
79.4
64.6
PORT (W RN -28-10)
80.4
66.2
chitectures.
Certified accuracy

Table 4: Comparing generative models. When choosing proxy distribution in PORT(`∞ ), DDPM model outperforms the leading generative
adversarial network (GAN) on
each dataset.

0.7

500K

D ELVING DEEPER INTO USE OF PROXY DISTRIBUTIONS IN ROBUST TRAINING

Earlier in Section 3.1 we showed that diffusion-based generative models are an excellent choice
for a proxy distribution for many datasets. Now we characterize the benefit of individual proxy
distributions in our robust training framework. We consider seven different generative models, which
are well-representative of the state-of-the-art on the CIFAR-10 dataset.
Using synthetic-to-real transfer of robustness as ground truth. From each generative model wesample one million synthetic images. We adversarially train a ResNet-18 (He et al., 2016) networkon these images and rank generative models in order of robust accuracy achieved on the CIFAR-10
test set. We provide further experimental setup details in Appendix C.

Robust discrimination accuracy

Calculating ARC. At each perturbation budget (), we adversarially train a ResNet-18 classifier
to distinguish between synthetic and real images. We measure robust discrimination accuracy of
trained classifier on a balanced held-out validation set. We calculate ARC by measuring the area
4
under the robust discrimination accuracy and perturbation budget curve (Figure 3). We provide a
detailed comparison of it with other metrics in Table 6.
ddpm
styleCwgan

e2gan
diffcrBigan

nda
diffBigan

1.0
0.9
0.8
0.7
0.6
0.5

0

1

2

(l )

3

4

Figure 3: Calculating ARC.
For every generative model andperturbation budget (), we first
adversarially train a binary classifier on adversarial perturbed
synthetic and CIFAR-10 images. Next we measure its robust discrimination accuracy on
the validation set at the  value
used in training. ARC is the area
under the robust discrimination
accuracy vs  curve.
4

Table 6: Comparing different generative models. We first adversarially train a ten class ResNet-18 model only on 1M synthetic images and measure its robust accuracy on the CIFAR-10
test set. We use this transferred robustness as a ground truth forbenefit of each model. Next we match ranking predicted by each
metric with the ground truth ranking. In contrast to all three
three baseline, ranking from our proposed metric (ARC) accurately matches the ground-truth.
Rank

Model

–

CIFAR-10

1
2
3
4
5
6
7

DDPM (UC)StyleGAN (C)WGAN-ALP (UC)E2GAN (UC)
DiffCrBigGAN (C)NDA (C)
DiffBigGAN (C)Robustaccuracy

FID (↓)IS (↑)

1-NN(↓)

48.7

–

–

–

–

53.1
45.0
43.5
39.6
33.7
33.4
32.4

3.17 (2)2.92 (1)12.96 (7)11.26 (5)4.30 (3)12.61 (6)4.61 (4)9.46 (2)10.24 (1)8.34 (7)8.51 (5)9.17 (3)8.47 (6)9.16 (4)9.34 (2)9.42 (3)10.10 (7)8.96 (1)9.84 (6)9.72 (5)9.73 (4)0.06 (1)0.32 (2)1.09 (3)1.19 (4)1.30 (5)1.43 (6)1.55 (7)

1.7

1.7

2.0

0.0

a
b

Mean absolute ranking differenceabUsing 50,000 training images from the CIFAR-10 dataset.
UC and C refers to unconditional and conditional generative models, respectively.

We use numpy.trapz(Racc-0.5, ) to calculate ARC.

8

ARC (↓)

Published as a conference paper at ICLR 2022

ARC is highly effective at determining synthetic to real transfer of robustness (Table 6). Eachof the baseline metrics, including FID which is widely used, fail in accurately predicting the transferof robustness from synthetic to real distributions. Even more, they also fail to identify DDPM (Hoet al., 2020) as the most successful proxy distribution. Our proposed metric (ARC), which measuresthe distinguishability of perturbed synthetic and real images, exactly predicts the ranking of different
generative models in terms of the transfer of robustness. It also provides deeper insight into why
synthetic samples from DDPM models are highly effective compared to other models. As shown in
Figure 3, discrimination between DDPM samples and real images becomes almost impossible evenat very small perturbations budgets. A more detailed comparison is provided in Appendix C.2.
Adaptive sampling of synthetic data based on ARC (Appendix C.3). We compare the benefit ofadaptive sampling with random sampling on CIFAR-10 dataset (for both `2 and `∞ threat models).
Our adaptive sampling achieves an average of 0.5% and 0.2% improvement in robust accuracy overrandom sampling in the `∞ and `∞ threat models, respectively.

4

R ELATED WORKTransfer of adversarial robustness. This line of work focuses on the transfer of adversarial robustness, i.e., correct classification even under adversarial perturbations, when testing the model on
different data distributions (Shafahi et al., 2020; Sehwag et al., 2019). Note that this is different
from just achieving correct classification on unmodified images across different distributions (Taoriet al., 2020; Hendrycks & Dietterich, 2019). Here we provide a theoretical analysis of the transfer
of adversarial robustness between data distributions.
Using extra curated real-world data in robust training. Prior works (Zhai et al., 2019; Carmon
et al., 2019; Uesato et al., 2019; Najafi et al., 2019; Deng et al., 2021) have argued for using moretraining data in adversarial training and often resort to curating additional real-world samples. In
contrast, we model the proxy distribution from the limited training images available and sample
additional synthetic images from this distribution.
Generative models for proxy distributions. State-of-the-art generative models are capable of modeling the distribution of current large-scale image datasets. In particular, generative adversarial networks (GANs) have excelled at this task (Goodfellow et al., 2014; Karras et al., 2020; Gui et al.,2020). Though GANs generate images with high fidelity, they often lack high diversity (Ravuri &Vinyals, 2019). However, samples from recently proposed diffusion process based models achieveboth high diversity and fidelity (Ho et al., 2020; Nichol & Dhariwal, 2021). Fréchet Inception Distance (FID) (Heusel et al., 2017) and Inception Score (IS) (Heusel et al., 2017) are two common
metrics to evaluate the quality of samples from generative models.
Using generative models to improve adversarial robustness. Earlier works have used generativemodels to learn training data manifold and the use it to map input samples to data manifold (Samangouei et al., 2018; Jalal et al., 2017; Xu et al., 2018). However, most of these techniques are broken
against adaptive attacks (Athalye et al., 2018; Tramèr et al., 2020). We use generative models tosample additional training samples which further improve the adversarial robustness.
Comparison with Rebuffi et al. (2021). A concurrent work by Rebuffi et al. (2021) also uses
randomly sampled synthetic images to improve adversarial robustness. In comparison, we provide atheoretical analysis of when synthetic data helps along with metrics to optimize the selection of bothgenerative models and individual samples. We also demonstrate an improvement in certified robust
accuracy using synthetic samples. Despite these differences, similar benefits of using proxy distributions in two independent and concurrent works further ascertain the importance of this research
direction.

5

D ISCUSSION AND B ROADER O UTLOOK

Using synthetic data has been a compelling solution in many applications, such as healthcare (Jordon
et al., 2018) and autonomous driving (Mayer et al., 2016) since it makes collecting a large amountof data feasible. In a similar spirit, we use synthetic data to make deep neural networks more robust
against adversarial attacks. We investigate foundational questions such as determining the transfer ofrobustness from synthetic to real data and determining selection criteria for how to choose generative9

Published as a conference paper at ICLR 2022

models or individual samples. Finally, we note that while it is crucial to improve robustness against
the threat of adversarial examples, it also has an unwanted side-effect in domains where adversarial
examples are used for good. Recent works use them to provide privacy on the web (Shan et al.,2020) or to evade website fingerprinting methods (Rahman et al., 2020). Improving defenses against
adversarial examples will negatively hurt these applications.

6

R EPRODUCIBILITY

We provide proof of each of our theorems in Appendix A. Similarly, we provide extensive details
on our experimental setup in Appendix B. Our work is also featured on RobustBench (Croce et al.,2020), an external benchmark for standardized evaluation of adversarial robustness. Our robustly
trained models are also available through the RobustBench API. For further reproducibility, we havealso submitted our code with the supplementary material.

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A

T HEORETICAL RESULTS

In this Section, we first provide the proofs of Theorem 1 and 2. Then we provide some additional
results showing the tightness of our Theorem 1 and also the effect of combining real and proxy dataon our Theorem. Finally, we provide some experimental result that validate our theory.
A.1

P ROOF OF T HEOREM 1

Theorem 1: Let D and D̃ be two labeled distributions supported on X × Y with identical label∗∗∗
distributions, i.e., ∀y ∈ Y, Pr(x,y)←D [y = y ] = Pr(x,y)←D̃ [y = y ]. Then for any classifierh∶X →Y∣Robd (h, D̃) − Robd (h, D)∣ ≤ cwdd (D, D̃).

Sketch of the proof. We first provide an informal sketch of the proof and then formalize the steps′
after that. Consider D to be the distribution that is the outcome of the following process: First′′′5
sample (x, y) from D, then find the closest x such that h(x ) ≠ y and output (x , y) . Also consider′
a similar distribution D̃ corresponding to D̃. We now prove a Lemma that shows the conditional′
Wasserestein distance between D and D is equal to Robd (h, D).
Lemma 3. We have Robd (h, D) = cwd(D, D ) and Robd (h, D̃) = cwd(D̃, D̃ ).
′′Proof. Let Jy be the optimal transport between D∣y and D ∣y. Also let Jy be the joint distribution′′′(x, x ) that is obtained by first sampling x from D ∣ y and then setting x = arg minh(x′ )≠y d(x, x ).
′′′The marginals of Jy are equal to D ∣ y and D ∣ y. Hence, Jy is a valid transport between D ∣ y′Jand D ∣ y. Also, define a potentially randomized perturbation algorithm A y that given an input x′′
samples (x, x ) ← Jy ∣ x, conditioned on x and outputs x . We have′

Robd (h, D) =
=

E

[ inf
d(x, x )]′E[ E [ inf d(x, x )]] =′′(x,y)←D h(x )≠y′E(⋅,y)←D x←D∣y h(x′ )≠y[[d(x, x )]].
′E(⋅,y)←D (x,x′ )←Jy′

On the other hand, because for all x we have h(A y (x)) ≠ y then d(x, A y (x)) ≥′inf y≠h(x′ ) d(x, x ). Therefore we haveJ

Robd (h, D) =E

[ inf d(x, x )] ≤′(x,y)←D h(x′ )≠y

E

J

J

(x,y)←D[d(x, A y (x))] =EE(⋅,y)←D (x,x′ )←Jy[d(x, x )]′Therefore we have
Robd (h, D) =E(⋅,y)←D[E′(x,x) ←Jy′[d(x, x )]] ≤
′

E

E
′

[d(x, x )].

(1)

E
′

[d(x, x )].

(2)′(⋅,y)←D (x,x )←Jy

On the other hand, because Jy is the optimal transport, we havecwdd (D, D ) =′E[d(x, x )] ≤′E′(⋅,y)←D (x,x )←JyE′(⋅,y)←D (x,x )←Jy′

Now combining Equations 1 and 2, we conclude that
cwdd (D, D ) =′EE(⋅,y)←D (x,x′ )←Jy[d(x, x )] =′EE(⋅,y)←D (x,x′ )←Jy′[d(x, x )] = Robd (h, D).
′

Similarly, we can also prove that cwdd (D̃, D̃ ) = Robd (h, D̃).
′

5

′

′

Here, we assume that the closest point x exists. Otherwise, We can set x so that the distance is arbitrarilyclose to the infimum and the proof follows.

15

Published as a conference paper at ICLR 2022′
˜ ′ are defined we have
By the way the distributions D and D′˜ ′ ) and cwd(D̃, D̃′ ) ≤ cwd(D̃, D′ ).
cwd(D, D ) ≤ cwd(D, D(3)′
˜ ′ could be seen as an
Roughly, the reason behind this is that all examples (x , y) sampled from D′
adversarial example for all elements of D with the label y. And we know that D consists of optimal′
adversarial examples for D, therefore, the optimal transport between D and D should be smaller
˜ ′ . Also, by triangle inequality for Wasserestein distancethan the optimal transport between D and D
we have,′′cwd(D̃, D ) ≤ cwd(D̃, D) + cwd(D, D ).
(4)

Now using Lemma 3 and Equations 4 and 3 we have
Robd (h, D̃) = cwd(D̃, D̃ ) ≤ cwd(D̃, D ) ≤ cwd(D̃, D)+cwd(D, D ) = cwd(D̃, D)+Robd (h, D).
(5)With a similar argument, because of symmetry of D and D̃, we can also prove′′′

Robd (h, D) ≤ cwd(D̃, D) + Robd (h, D̃).

(6)

Combining inequalities 5 and 7 we get− cwd(D̃, D) ≤ Robd (h, D) − Robd (h, D̃) ≤ cwd(D̃, D)(7)which finishes the proof.
∗

Full proof. The following is a succinct formalization of the proof steps mentioned above, let Jy =inf J∈J (D∣y,D̃∣y) be the optimal transport between the conditional distributions D ∣ y and D̃ ∣ y. We
have
Robd (h, D) =
=

≤

=
=

E

[ E [ infd(x , x)]]
′

E

[

E

[

E

[

E

[ inf
d(x , x )] + cwdd (D, D̃)′′(.,y)←D x←D∣y h(x )≠y(.,y)←D(.,y)←D(.,y)←DE[ inf d(x , x)]]

E

[ inf
d(x , x ) + d(x , x)]]
′

E

[ inf
d(x , x )]] +′′(x,x′′ )←Jy∗ h(x′ )≠y(x,x′′ )←Jy∗(x,x′′ )←Jy∗

′′

′

′′

h(x )≠y′′h(x )≠y′′′E(.,y)←D[E(x,x′′ )←Jy∗[d(x , x)]]′′′

(x ,y)←D̃ h(x )≠y′′= Robd (h, D̃) + cwdd (D, D̃).
With a similar argument we get Robd (h, D̃) ≤ Robd (h, D)+cwdd (D, D̃) and the proof is complete.

A.2

P ROOF OF T HEOREM 2

Theorem 2.
For any two distributions D̃ and D with equal class probabilities we havecwd(D, D̃) ≥ 4ARC(D, D̃). Moreover, if for all labels y , (D ∣ y) and (D̃ ∣ y) are two con6
centric uniform lp spheres, then we have cwd(D, D̃) = 4ARC(D, D̃).
Proof. We start by proving a lemma.
6

The statement of theorem in the main body has a typographical error that is fixed here.

16

Published as a conference paper at ICLR 2022

Lemma 4. Let J ∈ J (D, D̃) be an arbitrary transport between D and D̃. Then, for any discriminator σ we have,′
1 Pr(x,x′ )←J [d(x, x ) > ]
Racc/2 (σ, D, D̃) ≤ +.
2
2
Proof. Consider an adversary AJ that on input x sampled from D, samples a pair (x, x ) ← J ∣′′′′′J[1] = x from the transport and return a center point x between x and x such that d(x, x ) =
′
′′
′
′
′
d(x , x ) = d(x, x )/2 if d(x, x ) ≤ . Otherwise, it returns x. Also, on a in input x sampled from′′′′D̃ it samples a pair (x, x ) ← J ∣ J[2] = x from the transport and return a center point x between
′
′′
′
′′
′
′
′
x and x such that d(x, x ) = d(x , x ) = d(x, x )/2 if d(x, x ) ≤ . Otherwise, it returns x .


′





Denote DA ≡ AJ #D to be the push-forward of D under AJ . And also denote D̃A ≡ AJ #D̃. Now′
we have, δ(DA , D̃A ) ≤ Pr(x,x′ )←J [d(x, x ) > ] where δ is the total variational distance. Therefore,l∗1+Pr(x,x′ )←J [d(x,x )>]′′no discriminator σ can distinguish D from D̃ with accuracy more than
∗

2

.

∗

Now, let J = ∑y=1 αy Jy be a transport between D and D̃ where Jy is the optimal transport
between D̃ ∣ y and D ∣ y and αy is the probability of class y for D and D̃. using Lemma 4 we have′
1 Pr(x,x′ )←J ∗ [d(x, x ) > 2]∗
Racc (D, D̃) − ≤.

22Therefore we have

∫

∞

1
∗
[Racc (D, D̃)− ]d ≤ ∫
2

0

′
∞ Pr
(x,x′ )←J ∗ [d(x, x ) > 2]20d = ∫′∞ Pr(x,x′ )←J ∗ [d(x, x ) > ]

0

4

d.

(8)

Now, by integration by parts we have
∫

∞
0

Pr
′

(x,x )←J[d(x, x ) > ]d = 
∗
′

∞
»»∞
′
′
»» + ∫
Pr[d(x, x ) = ]d
[d(x,
x
)
>
]
»»0
′
∗
(x,x
)←J
(x,x )←J ∗
0

Pr
′

=0+

E

(x,x′ )←J ∗[d(x, x )]′= cwd(D, D̃).
Therefore, we have∞1cwd(D, D̃)∗[Racc (D, D̃) − ]d ≤.
240

ARC(D, D̃) = ∫Now to prove the second part of theorem about the case of concentric spheres we start by following
lemma.
Lemma 5. Let Dy = D ∣ y and D̃y = D̃ ∣ y. Assume D̃y and Dy are two concentric spheres∗
according to the some lp norm. Also let J ∈ J (Dy , D̃y ) be the optimal transport between Dy andD̃y with respect to the same norm. Then we have,
1 Pr(x,x′ )←J ∗ [d(x, x ) > ]
+
.
2
2
′

Racc/2 (D, D̃) =∗Proof. let D be the uniform sphere centered at point c and with radius r and let D̃ be the uniform
sphere centered at point c and with radios r̃ > r. Observe that the optimal robust distinguisher forany  < r+r̃is the one that assigns 0 to x if d(c, x) < r̃−r
and assigns 1 otherwise and for any
2
2
we
have
 < r̃−r2Racc/2 (D, D̃) = 1.
∗

17

(9)

Published as a conference paper at ICLR 2022

On the other hand, for any  ≥ r̃−r
we have2
Racc/2 (D, D̃) = 0.5.
∗

(10)

∗

Now consider optimal transport J between D and D̃. Observe that this transport is equivalent to
the following distribution:∗J ≡ (D, (D − c) ⋅ r̃/r)Therefore, we have Pr(x,x′ )←J ∗ [d(x, x ) = r̃ − r] = 1 Which implies for any  < r̃ − r we have′Pr[d(x, x ) > ] = 1

(11)

Pr

[d(x, x ) > ] = 0.

(12)

′

(x,x′ )←J ∗and for any  ≥ r̃ − r(x,x′ )←J ∗′Putting Equations equation 9, equation 10, equation 11, and equation 12 together, we finish the proof
of Lemma.
Having Lemma 5, we can follow the same steps as before except that instead of Inequality equation 8
we have an equality of the following form.

∫

∞ Pr
1
(x,x′ )←J ∗ [d(x, x ) > 2]∗[Racc (D, D̃) − ]d = ∫
d
2
2
0
0
′

∞

=∫

′
∞ Pr
(x,x′ )←J ∗ [d(x, x ) > ]
0

=

A.3

4

d

cwd(D, D̃).
4

E FFECT OF C OMBINING P ROXY AND R EAL DATA ON T HEOREM 1

A natural question is what happens when we combine the original distribution with the proxy distribution. For example, one might have access to a generative model but they want to combine thesamples from the generative model with some samples from the original distribution and train arobust classifier on the aggregated dataset. The following corollary answers this question.
Theorem 6. Let D and D̃ be two labeled distributions supported on X × Y with identical label
distributions and let D̄ = p ⋅ D + (1 − p) ⋅ D̃ be the weighted mixture of D and D̃. Then for any
classifier h ∶ X → Y∣Robd (h, D̄) − Robd (h, D)∣ ≤ (1 − p) ⋅ cwdd (D, D̃).
Note that the value of p is usually very small as the number of data from proxy distribution is usuallymuch higher than the original distribution. This shows that including (or not including) the datafrom original distribution should not have a large effect on the obtained bound on distribution-shift
penalty.
Proof of Theorem 6. We just need to show that cwdd (D, D̄) ≤ (1 − p) ⋅ cwdd (D, D̃)). Note thatsince the label distributions are equal, we haveD̄ ∣ y ≡ p ⋅ D ∣ y + (1 − p) ⋅ D̃ ∣ y.
Now let Jy be the optimal transport between D ∣ y and D̃ ∣ y. Now construct a joint distribution′′
Jy ≡ (1 − p) ⋅ J + p ⋅ (x, x)x←D∣y . Notice that Jy is a joint distribution with marginals equal to D′
and D̄. Therefor Jy is a transport between D and D̄ and we can calculate its cost. We have18

Published as a conference paper at ICLR 2022E

[d(x, x )] = (1 − p) ⋅′(x,x′ )←Jy′[d(x, x )] +E [d(x, x)] = (1 − p) ⋅′E(x,x′ )←Jy

x←D∣y

E

(x,x′ )←Jy[d(x, x )].
′Therefore, we havecwd(D, D̄) ≤

A.4

E

[

E

(⋅,y)←D (x,x′ )←Jy′[d(x, x )]] = (1−p)⋅
′

E

[

E

(⋅,y)←D (x,x′ )←Jy[d(x, x )]] = (1−p)⋅cwd(D, D̃).
′

T IGHTNESS OF T HEOREM 1

Here, we show a theorem that shows our bound on distribution-shift penalty is tight. The following
theorem shows that one cannot obtain a bound on the distribution-shift penalty for a specific classifierthat is always better than our bound.
Theorem 7 (Tightness of Theorem 1). For any distribution D supported on X × Y , any classifierh, any homogeneous distance d and any  ≤ Robd (h, D), there is a labeled distribution D̃ such that
Robd (h, D) − Robd (h, D̃) = cwd(D, D̃) = .
Proof. for α ∈ [0, 1] let D̃α be the distribution of the following process: First sample (x, y) from′′′D, then find the closest x such that h(x ) ≠ y and output (x + α(x − x), y). By definition,
the conditional Wasserestein distance between D and D̃1 is equal to Robd (h, D). We also havecwd(D, D̃α ) = α ⋅ cwd(D, D̃1 ).
Observe that for any classifier we have Robd (h, D̃α ) ≤ (1 − α)Robd (h, D) because if (x , y) is an′′adversarial example for (x, y), then x is also an adversarial example for (x + α(x − x), y) with′
distance (1 − α)d(x, x ). On the other hand we have Robd (h, D̃α ) ≥ (1 − α)Robd (h, D) because′
any adversarial example for (x + α(x − x), y) with distance r is also an adversarial example for x′′
with distance at most r+αd(x −x) and since x is the optimal adversarial example for x then r must′be at least α(x − x). Therefore, we have Robd (h, D̃α ) = (1 − α)Robd (h, D). Putting everything
together and setting α = /Robd (h, D)we have′

Robd (h, D) − Robd (h, D̃α ) = αRobd (h, D) = αcwd(D, D̃1 ) = cwd(D, D̃α ) = .

Note that Theorem 7 only shows the tightness of Theorem 1 for a specific classifier. But there might
exist a learning algorithm L that incurs a much better bound in the expectation. Namely, there might
exist L such that for any two distributions D and D̃ we have∣

E n [Robd (h, D) − Robd (h, D̃)]∣ ≪ cwd(D, D̃).

S←D̃h←L(S)We leave finding such an algorithm as an open question.
A.5

E XPERIMENTAL VALIDATION OF MAIN THEOREM

In theorem 1 we prove that transfer of adversarial robustness between two distributions is upperbounded by their conditional Wasserstein distance. Now we provide empirical validation of thisresult. We consider a 10-class classification problem where each class is modeled using a 128dimensional multivariate normal distribution. In contrast to real-world datasets, where the underlying distribution of the data is unknown, we can efficiently and exactly calculate Wasserstein distanceon this synthetic dataset. We robustly train a four-layer convolutional network on it. To constructa proxy distribution, we perturb both mean and covariance parameters across each class. On eachproxy distribution, we measure the average robustness of the pre-trained model and its conditional
19

Diff. in Avg. robustness

Published as a conference paper at ICLR 2022

101
10 1
10 3
10 1

100

101

Conditional Wasserstein distance

Figure 4: Validating the upper bound from Theorem 1. The green line is the upper bound calculatedby Wasserstein-2. Note that the Wasserstein-1 (The bound of Theorem 1) is a tighter upper-boundbut it does not have a closed form for normal distributions.
7

Wasserstein distance from the original distribution. We use Wasserstein-2 distance , which has aclosed-form expression for normal distributions (Givens et al., 1984). As shown in Figure 4, thedifference in average robustness is upped bounded by the conditional Wasserstein distance.

B

A DDITIONAL DETAILS ON EXPERIMENTAL SETUP

We describe experimental details common across most experiments in this section. We discussdesign choices pertaining to individual experiments in their respective sections.
Training setup. We use network architectures from the ResNet family, namely ResNet-18 (He et al.,2016) and variants of WideResNet (Zagoruyko & Komodakis, 2016). We train each network using
stochastic gradient descent and 0.1 learning rate with cosine learning rate decay, weight decay of−4−3
5 × 10 , batch size 128, and 200 epochs. We use 1 × 10 weight decay on CIFAR-100 dataset,since higher weight decay further reduces the generalization gap on this dataset. We work with fivedatasets, namely CIFAR-10, CIFAR-100, ImageNet (64 × 64 size images), CelebA (64 × 64), andAFHQ (224 × 224). For CelebA (Liu et al., 2015) dataset, we consider a four-class classification
problem based on the attribute smile and male, i.e., smile/male, smile/not-male, not-smile/male,not-smile/not-male (total 200K images). Animal Faces-High-Quality (AFHQ) dataset comprises
15K images for three classes: cat, dog, and wild animals. Thus we consider the task of three-classclassification on this dataset. To reduce the computational cost on ImageNet dataset, we use the freeadversarial training (Shafahi et al., 2019) procedure. We use the ResNet-18 model for CIFAR-10,
ImageNet, CelebA, and AFHQ dataset. However, for CIFAR-10 and CIFAR-100 dataset, we alsouse much larger W RN -34-10 network.
Robust training parameters. We consider both l∞ and l2 threat models. For CIFAR-10 and
CIFAR-100 dataset, we use the commonly used perturbation budget () of 8/255 and 127/255 forl∞ and l2 threat model, respectively. For other three dataset too, we choose commonly used perturbation budgets. We perform adversarial training using a 10-step projected gradient descent attack
8(PGD-10) and benchmark test set robustness with the much stronger AutoAttack (Croce & Hein,
2020). We use β = 6, σ = 0.25, 100 samples for selection, and 10, 000 samples for estimation in
randomized smoothing, as described in Cohen et al. (2019).
Generative models. We consider seven different generative models for the CIFAR-10 and two
different networks for the CelebA and ImageNet dataset. In Table 7. We provide the number ofsynthetic images generated from each model and whether the generated images are labelled or unlabeled. If the model is unconditional, i.e., only generates unlabeled images, we label the images
using an additional classifier. We use state-of-the-art LaNet (Wang et al., 2019) network to label
7

The bound in our theorem is based on W1 . Since we know W1 < W2 , using W2 provides a loose upperbound.
8
We don’t report numbers with PGD attacks as AutoAttack already captures them while also making iteasier to compare with other works (Croce et al., 2020).

20

Published as a conference paper at ICLR 2022

images for the CIFAR-10 dataset. LaNet network is trained only on the training images of the
CIFAR-10 dataset and achieves 99.0% accuracy on its test set. For CelebA dataset, we train a
ResNet50 model to label synthetic images. For adaptive sampling of synthetic images for CIFAR9
10 dataset, we first sample of set of 15M synthetic images from the DDPM model. Next we select
10M synthetic images with the lowest synthetic score from this set.
Table 7: Generative models for each dataset. In this table we list the different generative models
used for each dataset and the number of synthetic images sampled form each model. We also
indicate whether the generative models generate labeled images, i.e., class-conditioned. If not classconditioned, the model only generated unlabeled synthetic images.
Number of training
images in dataset

Generative model

Number ofsynthetic images

Class-conditioned

CIFAR-10

50K

DDPM Ho et al. (2020)
StyleGAN (Karras et al., 2020)
WGAN-ALP (Terjék, 2019)
E2GAN (Tian et al., 2020)
DiffCrBigGAN (Zhao et al., 2020)
NDA (Sinha et al., 2021)
DiffBigGAN (Zhao et al., 2020)

10M
10M
1M
1M
1M
1M
1M

×
✓
×
×
✓
✓
✓

CelebA

120K

StyleFormer (Park & Kim, 2021)
DDPM (Song et al., 2020)

1M
1M

×
×

ImageNet

1.2M

BigGAN (Brock et al., 2019)
DDPM (Nichol & Dhariwal, 2021)

1M
400K

✓
✓

CIFAR-100

50K

DDPM (Nichol & Dhariwal, 2021)

1M

✓

AFHQ

15K

StyleGAN (Karras et al., 2020)

300K

✓

DatasetEvaluation metrics for generative models. We use following existing baseline metrics to evaluate
the quality of synthetic samples. 1) Fréchet Inception Distance (FID): It measures the Fréchetdistance (Dowson & Landau, 1982) between features of synthetic and real images extracted froman Inception-V3 network. 2) Inception score (IS): Unlike FID, Inception score only uses synthetic
data with the goal to account for both fidelity and diversity of synthetic data. 3) Nearest neighbour
distance (1-NN): It computes the average distance of a synthetic image to the nearest real image inthe pixel space. 4) ARC: This is the metric we propose and it measures distance between synthetic
and real data based on the success of a robust discriminator. We compare performance of all threebaselines with our proposed metric.
Computational cost. In addition to robust training, sampling from generative models is another key
contributor to the computational cost in our approach. We sample images from the DDPM modelusing 250 steps for both CIFAR-10 and ImageNet datasets. Using an RTX 4x2080Ti GPU cluster,
it takes 23.8 hours to sample one million images on the CIFAR-10 dataset. With the same setup,
it takes 26.1 hours to sample 100K images for the 64 × 64 ImageNet dataset. On both models, we10
use the publicly available checkpoints of pretrained generative models . Note that both training
generative models and sampling from them is a one-time cost. Once the pretrained checkpoints andsynthetic images are made publicly available, they can be directly used in downstream tasks. We
will make our code and synthetic data publicly available.

C

D ELVING DEEPER INTO ROBUST DISCRIMINATION AND ARC

We propose a new metric (ARC), to rank different generative models in order of robustness transferfrom their samples to real data. Now we provide experimental details on how we measure ARC andits comparison with other baselines.
9

We use the 6M randomly sampled images made available by Nakkiran et al. (2021) along with 9M more
sampled by us using improved sampling techniques (Nichol & Dhariwal, 2021) from the DDPM model.
10
https://github.com/openai/improved-diffusion

21

Published as a conference paper at ICLR 2022

Table 8: Using synthetic data also improves clean and robust accuracy on AFHQ (Choi et al.,2020) dataset. Baseline refers to adversarial training based on Madry et al. (Madry et al., 2018).
Clean/Auto refers to clean/robust accuracy measured with AutoAttack. We use a ResNet-18 network and randomly sampled synthetic images.
`∞


`2

4/255

8/255

3.0

5.0

Clean Auto

Clean Auto

Clean Auto

Clean Auto

Baseline

98.8

93.3

98.4

84.3

98.9

93.7

98.7

PORT

99.1

93.5

98.8

86.5

99.0

94.0

98.8

89.2

∆

+0.3

+0.2

+0.4

+2.2

+0.1

+0.3

+0.1

+0.4

88.8

Table 9: Success of different network architectures in classifying synthetic images from real-worldimage on CIFAR-10 dataset.
Model

DDPM

StyleGANWGAN-ALP

E2GAN

DiffCrBigGAN

NDA

DiffBigGAN

CNN-2
CNN-4
ResNet-18

59.1
97.0
98.3

94.3
99.1
99.9

99.1
99.9
99.9

97.3
99.7
99.9

92.2
94.6
99.6

95.4
99.8
99.9

97.7
99.7
99.9

Setup. A critical component in our approach is training a robust binary discriminator to distinguishbetween real and synthetic data. We use a ResNet-18 network for this task and train it for 100 epochswith a 0.1 learning rate, 128 batch size, and 1e-4 weight decay. We use a randomly sampled set ofone million images from each generative model. We keep 10, 000 synthetic images from this set
for validation and train on the rest of them. We specifically choose 10, 000 images as it’s equal to
the number of test images available in the CIFAR-10 test set, thus balancing both classes in our test
set. We train for only 391 steps per epoch, i.e., equivalent to a single pass through 50, 000 training
images in the CIFAR-10 dataset. Thus over 100 epochs, we effectively end up taking five passes
through the one million synthetic images. Given the randomness in multiple steps, we aggregate
results over three different runs of our experiments.
C.1

W HY NON - ROBUST DISCRIMINATORS ARE NOT EFFECTIVE IN MEASURING PROXIMITY ?

We argue that most synthetic samples can be easily distinguished from real data, irrespective of proximity, in absence of adversarial perturbations ( = 0). This is likely because deep neural networks
have very high expressive power. So a natural question is whether shallow networks, which havemuch lower capacity, are more suitable for this task. We test this hypothesis by using a two-layer(16 and 32 filters) and a four-layer (16, 16, 32, and 32 filters) convolutional neural network. We
refer to them as CNN-2 and CNN-4, respectively. We find that even these shallow networks achieve
more than 90% accuracy in distinguishing synthetic images from real images (Table 9). The CNN4 network itself is achieving more than 99% accuracy for four out of seven generative models. A
much larger ResNet-18 network achieves near-perfect classification accuracy for most models. Suchhigh success of even shallow networks uniformly across generative models shows that non-robust
discrimination is not an effective measure of proximity.
C.2

O N EFFECTIVENESS OF ARC IN MEASURING PROXIMITY

In this section, we delve deeper into the comparison of our proposed metric (ARC) with other baselines. We judge the success of each metric by how successfully it predicts the transfer of robustness
from synthetic to real samples on the CIFAR-10 data. (referred to as transferred robust accuracy).
We report our results in Table 10.
Is FID in CIFAR-10 feature space effective? When calculating FID, the recommended approachis to measure it in the feature space of a network trained on the ImageNet dataset (Heusel et al.,2017). One may ask, whether the reason behind the failure is using ImageNet feature space for
CIFAR-10 images. This might be true as it is well known that most image datasets have their
specific bias (Torralba & Efros, 2011). To answer this question, we measure FID in the feature of22

Published as a conference paper at ICLR 2022

a state-of-the-art LaNet (Wang et al., 2019) networks trained on the CIFAR-10 dataset. We refer tothis metric as FID (CIFAR-10). Similar to FID in ImageNet feature space, FID in CIFAR-10 feature
space also fails to predict robustness transfer. It follows a similar trend as the former, where it ranks
models like DiffBigGAN and StyleGAN much higher than other models.
Generalizing to natural training. If the objective is to maximize clean accuracy instead of robustness, one may ask whether ARC will remain effective in selecting the generative model. We
answer this question by measuring the success of ARC in predicting how much clean accuracy isachieved on the CIFAR-10 test set when we train a ResNet-18 network only on one million synthetic
images without any adversarial perturbation. Note that ARC is dependent only on samples, thus canbe directly used for this task. We find that ARC correctly ranks DDPM over StyleGAN model fortransferred clean accuracy. Similarly, it also correctly rank WGAN-ALP and E2GAN model higher
than other BigGAN models where the former achieves higher clean accuracy on the CIFAR-10 test
set. However, there are small variations in transferred clean accuracy with some models, such as
WGAN-ALP and E2GAN, which ARC doesn’t capture in its ranking.

Table 10: Testing how much each proxy distribution helps on CIFAR-10 dataset and whether FID/IS
captures it. We train on 1M synthetic images and measure transferred robustness (which determines
the rank) to cifar10 test set. UC and C refers to unconditional and conditional generative models,
respectively.
Rank

Model

Transferred clean accuracy(Natural training) (↑)

Transferred robust accuracy(Adversarial training) (↑)

FID(ImageNet) (↓)

FID(CIFAR-10) (↓)

IS (↑)

1-NN(↓)ARC (↓)

1
2
3
4
5
6
7

DDPM (UC)StyleGAN (C)WGAN-ALP (UC)E2GAN (UC)
DiffCrBigGAN (C)NDA (C)
DiffBigGAN (C)

94.8
91.2
92.3
92.7
87.4
84.9
86.7

53.1
45.0
43.5
39.6
33.7
33.4
32.4

3.17
2.92
12.96
11.26
4.30
12.61
4.61

0.90
0.55
5.84
4.52
0.96
0.78
0.64

9.46
10.24
8.34
8.51
9.17
8.47
9.16

9.34
9.42
10.10
8.96
9.84
9.72
9.73

0.06
0.32
1.09
1.19
1.30
1.43
1.55

C.3

A DAPTIVE SAMPLING OF SYNTHETIC DATA .

Robust Accuracy ( )Finding synthetic samples which leads to highest synthetic-to-real robustness transfer. We
use proximity of a synthetic sample to the original distribution, which we measure using synthetic score from our trained robust discriminators (using 0.25/255 size l∞ perturbations), asa metric to judge transfer of performance from the proxy distribution to original distribution.
We sort the synthetic score for each of 6M synthetic
53
images from the DDPM model and combine them into52
ten equal-size groups. Next, we adversarially train a
51
ResNet-18 network on images from each group, along
50
with CIFAR-10 training images, and measure robustness
49
achieved on the CIFAR-10 test set (Figure 5). Our re0 1 2 3 4 5 6 7 8 9sults validate the effectiveness of synthetic scores, where
Groups based on synthetic scoregroups with the lowest synthetic score achieve highest ro(Low High)bust test accuracy. The difference in the robust test accu- Figure 5: Effectiveness of synthetic
racy is up to 4% between the group with the lowest score score. We sort 6M DDPM images usand the one with the highest score.
ing their synthetic score and divide them
Improvement in performance with adaptive sampling. into ten equal-size groups. We find that
We select of set of 10M synthetic images (from a set of groups with lower score achieve higher
15M) with the lowest synthetic score. We compare the transferred robust accuracy (`∞ ) on the
success of these adaptively selected images with a set of CIFAR-10 test set.
10M randomly selected images. We present our results in Table 11. It shows that adaptively sampling further improves the performance of our framework (PORT) across threat models (`∞ and `2 )and network architectures.
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Published as a conference paper at ICLR 2022

Table 11: Further improvement in adversarial robustness using adaptive sampling. Experimental results with adversarial training on the CIFAR-10 dataset for both `∞ and `2 threat model. Using
additional synthetic data brings a large gain in adversarial robustness across network architectures
and threat models. We also show further improvements brought in by adaptive sampling, in comparison to random sampling, of synthetic images. Clean/Auto refers to clean/robust accuracy measured
with AutoAttack.
(a) `∞ threat model.

(b) `2 threat model.

MethodArchitecture

Parameters (M)

Clean

Auto

MethodArchitecture

Parameters (M)

Clean

Auto

Zhang et al. (2019)
PORT (random)
PORT (adaptive)

ResNet-18
ResNet-18
ResNet-18

11.2
11.2
11.2

82.0
84.4
84.6

48.7
55.6
55.7

Rice et al. (2020)
PORT (random)
PORT (adaptive)

ResNet-18
ResNet-18
ResNet-18

11.2
11.2
11.2

88.7
89.9
89.8

67.7
74.0
74.4

Rice et al. (2020)
Gowal et al. (2020)

W RN -34-20
W RN -70-16

184.5
266.8

85.3
85.3

53.4
57.2

Madry et al. (2018)
Gowal et al. (2020)

ResNet-50
W RN -70-16

23.5
266.8

90.8
90.9

69.2
74.5

Zhang et al. (2019)
PORT (random)
PORT (adaptive)

W RN -34-10
W RN -34-10
W RN -34-10

46.2
46.2
46.2

84.9
86.7
87.0

53.1
60.3
60.6

Wu et al. (2020b)
PORT (random)
PORT (adaptive)

W RN -34-10
W RN -34-10
W RN -34-10

46.2
46.2
46.2

88.5
90.8
90.8

73.7
77.1
77.8

Table 12: Hyperparmeter search for γ using ResNet-18 on CIFAR-100 dataset.

D

γ

0.15

0.40

0.50

0.75

0.90

Clean Accuracy
Robust Accuracy

60.6
27.6

64.7
27.7

64.5
26.4

62.2
23.7

59.8
21.9

B RINGING IT ALL TOGETHER : U SING SYNTHETIC SAMPLES IN ROBUSTTRAINING

In this section, we first analyze the synthetic images and later provide additional analysis basedon them in robust training. We will primarily use synthetic images sampled from the DDPM andStyleGAN model.
Analyzing synthetic images. Since DDPM is an unconditional model, it only generates unlabelled
synthetic images. We label these images using a LaNet network on the CIFAR-10 dataset. We
visualize the frequency of different classes in Figure 6. It shows that the synthetic images are almost
uniformly distributed across classes. On the ImageNet dataset, we use the classifier conditioned
sampling to generate labeled images from the improved DDPM model (Nichol & Dhariwal, 2021).
We also provide samples images from different generative models in Figure 10 and 11.

0.10

Robust Clean

Fraction of images

Hyperparameter search for γ. Before using DDPM images in robust training, we perform a hyperparameter search for γ. We consider ten different values from 0 to 1 and train a ResNet-18 network
using PORT at each of them. We measure both the clean accuracy and robust accuracy of each network (Figure 7). We find that γ = 0.4 achieves the highest clean and robust accuracy. Thus we used
γ = 0.4 in our experiments.

0.05
0.00

0 1 2 3 4 5 6 7 8 9Image class

Figure 6: Histogram of class labels forsynthetic images from DDPM model.

84
82
60
58
56
54

0.2

0.4

0.6

0.8

Figure 7: Hyperparameter search for γ in
PORT.

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Published as a conference paper at ICLR 2022

D.1

S AMPLE COMPLEXITY OF ADVERSARIAL TRAINING

Given the ability to sample an unlimited amount of synthetic images from a proxy distribution,
now we investigate the performance of adversarial training with an increasing number of training
samples. We train the network only on synthetic images and measure its performance on anotherheld-out set of synthetic images. We also measure how much the clean and robust accuracy transferson the CIFAR-10 test set.
Setup. We primarily work with the CIFAR-10 dataset and its two-class subset, i.e., an easier problemof binary classification between class-1 (automobile) and class-9 (truck). We refer to the latteras CIFAR-2. We robustly train a ResNet-18 network on 2K to 10M synthetic images from theStyleGAN model, as in both 10-class and 2-class setup. We opt for StyleGAN over DDPM model as
sampling images from the former is much faster, thus we were able to generate up to 10M synthetic
from it. Note that the cost of adversarial training increases almost linearly with the number of attack
steps and training images. Thus to achieve manageable computational cost when training on millionsof images, we opt for using only a 4-step PGD attack (PGD-4) in both training and evaluation. Since
robustness achieved with this considerably weak attack may not hold against a strong attack, suchas AutoAttack, we opt for evaluating with the PGD-4 attack itself. We also perform natural training,i.e., training on unmodified images in some experiments. We test each network on a fixed set of
100K images from the StyleGAN and 10K images from the CIFAR-10 test set.
Accuracy vs robustness trade-off. We compare the clean accuracy achieved with both naturaland adversarial training in Figure 8. Indeed with a very small number of samples, clean accuracy
in adversarial training is traded to achieve robustness. This is evident from the gap between theclean accuracy of natural and adversarial training. However, with the increasing number of training
samples, this gap keeps decreasing for both CIFAR-2 and CIFAR-10 datasets. Most interestingly,
this trade-off almost vanishes when we use a sufficiently high number of training samples for the
CIFAR-2 classification.
On sample complexity of adversarial training. We report both clean and robust accuracy withadversarial training in Figure 9. We find that both clean and robust accuracy continues to improvewith the number of training samples. We also observe non-trivial generalization to test images fromthe CIFAR-10 dataset, which also improves with the number of training samples. Both of theseresults suggest that even with a small capacity network, such as ResNet-18, adversarial robustness
can continue to benefit from an increase in the number of training samples.
D.2

E XPLORING EFFECT OF IMAGE QUALITY OF SYNTHETIC SAMPLES

We explore different classifiers to label the unlabeled synthetic data generated from the DDPM
model. In particular, we use BiT (Kolesnikov et al., 2020), SplitNet (Zhao et al., 2020), and
LaNet (Wang et al., 2019) where they achieve 98.5%, 98.7%, and 99.0% clean accuracy, respectively, on the CIFAR-10 dataset. We notice that labels generated from different classifiers achieveslightly different downstream performance when used with adversarial training in the proposed approach. We find that only up to 10% of synthetic images are labeled differently by these networks,
which causes these differences. On manual inspection, we find that some of these images are ofpoor quality, i.e., images that aren’t photorealistic or wrongly labeled and remain hard to classify,
even for a human labeler. Since filtering millions of images with a human in the loop is extremely
costly, we use two deep neural networks, namely LaNet (Wang et al., 2019) and SplitNet (Zhaoet al., 2020), to solve this task. We avoid using labels from BiT as it requires transfer learning from
ImageNet (Deng et al., 2009) dataset, whereas our goal is to avoid any dependency on extra realworld data. We discard an image when the predicted class of both networks doesn’t match and it
is classified with less than 90% confidence by both networks. While the former step flags images
which are potentially hard to classify, the latter step ensures that we do not discard images whereat least one network is highly confident in its prediction. We also try the 50% and 75% confidence
threshold but find that 90% gives the best downstream results. In particular, using the filtering step
improves robust accuracy by another 0.1%.

25

100
90
80
70
60

CIFAR-2

104

105

106

Number of training images

StyleGANCIFAR-2

100

Robust accuracy

Natural training
Adversarial training

Clean accuracy

Clean accuracy

Published as a conference paper at ICLR 2022

80
60
104

105

80
60
104

106

Number of training images(a) CIFAR-2

StyleGAN

100

105

106

Number of training images

80
70

Natural training
Adversarial training

104

105

106

107

Number of training images(b) CIFAR-10

Figure 8: Reduction in accuracy vs robustness trade-off.
Accuracy vs robustness tradeoff when training on an increasing amount of synthetic images
from the StyleGAN model. It
shows that the drop in clean accuracy with adversarial training
decreases with increase in training samples.

CIFAR-10100

StyleGAN

Robust accuracy

90

Clean accuracy

Clean accuracy(a) CIFAR-2
100

80
60
40
104

105

106

CIFAR-10100

StyleGAN

80
60
40

107

Number of training images

104

105

106

107

Number of training images(b) CIFAR-10

Figure 9: Sample complexity of adversarial training. Clean
and robust accuracy on the test set of synthetic samples whentrained on an increasing number of synthetic samples from theStyleGAN model. It shows that performance of adversarial
training continues to benefit from increase in number of training samples. We also measure generalization to the CIFAR-10
dataset, which also improves with number of training samples.

26

Published as a conference paper at ICLR 2022

(a) CIFAR-10(b) DDPM(c) StyleGAN(d) WGAN-ALP

Figure 10: Visualizing images from different sets. Randomly selected images form the CIFAR-10
dataset and synthetic images from different generative models. Rows in each figure correspond tofollowing classes: Airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.

27

Published as a conference paper at ICLR 2022(e) E2GAN(f) DiffCrBigGAN(g) NDA(h) DiffBigGAN

Figure 10: Visualizing images from different sets. Randomly selected synthetic images fromdifferent generative models. Rows in each figure correspond to following classes: Airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.

28

Published as a conference paper at ICLR 2022(a) ImageNet (Deng et al., 2009)(b) Improved DDPM (Nichol & Dhariwal, 2021)(c) BigGAN-Deep (Brock et al., 2019)

Figure 11: ImageNet (64 × 64) samples along with synthetic images from two different generativemodels. Rows correspond to the following classes: hummingbird, english foxhound, arctic fox,
basketball, iPod, ping-pong ball, running shoe, and water tower.
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Published as a conference paper at ICLR 2022

(a) CelebA (Real)(b) CelebA (Synthetic)

Figure 12: Real and synthetic images for the CelebA dataset. We consider a four-class classification problem based on the attribute smile and male, i.e., not-smile/not-male, smile/not-male,not-smile/male, smile/male.

(a) AFHQ (Real)(b) AFHQ (Synthetic)

Figure 13: Real and synthetic images for the AFHQ dataset. Rows correspond to the following threeclasses: cat, dog, and wild animals.

30