Mixed Nash Equilibria in the Adversarial Examples Game

Laurent Meunier∗1,2 Meyer Scetbon∗3 Rafael Pinot4Jamal Atif1 Yann Chevaleyre1

1 LAMSADE, Université Paris-Dauphine2 Facebook AI Research, Paris

3 CREST, ENSAE4 Ecole Polytechnique Fédérale de Lausanne

Abstract

This paper tackles the problem of adversarial examples from a game theoretic point of view. We studythe open question of the existence of mixed Nash equilibria in the zero-sum game formed by the attackerand the classifier. While previous works usually allow only one player to use randomized strategies, weshow the necessity of considering randomization for both the classifier and the attacker. We demonstratethat this game has no duality gap, meaning that it always admits approximate Nash equilibria. We alsoprovide the first optimization algorithms to learn a mixture of classifiers that approximately realizes thevalue of this game, i.e. procedures to build an optimally robust randomized classifier.

1 IntroductionAdversarial examples [6, 34] are one of the most dizzling problems in machine learning: state of the artclassifiers are sensitive to imperceptible perturbations of their inputs that make them fail. Last years, researchhave concentrated on proposing new defense methods [13, 24, 25] and building more and more sophisticatedattacks [11, 15, 18, 22]. So far, most defense strategies proved to be vulnerable to these new attacks or arecomputationally intractable. This asks the following question: can we build classifiers that are robust againstany adversarial attack?

A recent line of research argued that randomized classifiers could help countering adversarial attacks [17,28, 39, 40]. Along this line, [27] demonstrated, using game theory, that randomized classifiers are indeedmore robust than deterministic ones against regularized adversaries. However, the findings of these previousworks are dependent on the definition of adversary they consider. In particular, they did not investigatescenarios where the adversary also uses randomized strategies, which is essential to account for if we wantto give a principled answer to the above question. Previous works studying adversarial examples from thescope of game theory investigated the randomized framework (for both the classifier and the adversary) inrestricted settings where the adversary is either parametric or has a finite number of strategies [8, 26, 31].Our framework does not assume any constraint on the definition of the adversary, making our conclusionsindependent on the adversary the classifiers are facing. More precisely, we answer the following questions.

Q1: Is it always possible to reach a Mixed Nash equilibrium in the adversarial example game when boththe adversary and the classifier can use randomized strategies?

A1: We answer positively to this question. First we motivate in Section 2 the necessity for usingrandomized strategies both with the attacker and the classifier. Then, we extend the work of [29], byrigorously reformulating the adversarial risk as a linear optimization problem over distributions. In fact, wecast the adversarial risk minimization problem as a Distributionally Robust Optimization (DRO) [7] problemfor a well suited cost function. This formulation naturally leads us, in Section 3, to analyze adversarial riskminimization as a zero-sum game. We demonstrate that, in this game, the duality gap always equals 0,meaning that it always admits approximate mixed Nash equilibria.

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Q2: Can we design efficient algorithms to learn an optimally robust randomized classifier?A2: To answer this question, we focus on learning a finite mixture of classifiers. Taking inspiration

from robust optimization [33] and subgradient methods [9], we derive in Section 4 a first oracle algorithm tooptimize over a finite mixture. Then, following the line of work of [16], we introduce an entropic reguralizationwhich allows to effectively compute an approximation of the optimal mixture. We validate our findings withexperiments on a simulated and a real image dataset, namely CIFAR-10 [21].

Figure 1: Motivating example: blue distribution represents label −1 and the red one, label +1. The height ofcolumns represents their mass. The red and blue arrows represent the attack on the given classifier. On left:deterministic classifiers (f1 on the left, f2 in the middle) for whose, the blue point can always be attacked.On right: a randomized classifier, where the attacker has a probability 1/2 of failing, regardless of the attackit selects.

2 The Adversarial Attack Problem

2.1 A Motivating ExampleConsider the binary classification task illustrated in Figure 1. We assume that all input-output pairs (X,Y )are sampled from a distribution P defined as follows

P (Y = ±1) = 1/2 and{

P (X = 0 | Y = −1) = 1P (X = ±1 | Y = 1) = 1/2

Given access to P, the adversary aims to maximize the expected risk, but can only move each point by atmost 1 on the real line. In this context, we study two classifiers: f1(x) = −x− 1/2 and f2(x) = x− 1/21.Both f1 and f2 have a standard risk of 1/4. In the presence of an adversary, the risk (a.k.a. the adversarialrisk) increases to 1. Here, using a randomized classifier can make the system more robust. Consider f wheref = f1 w.p. 1/2 and f2 otherwise. The standard risk of f remains 1/4 but its adversarial risk is 3/4 < 1.Indeed, when attacking f , any adversary will have to choose between moving points from 0 to 1 or to −1.Either way, the attack only works half of the time; hence an overall adversarial risk of 3/4. Furthermore, if fknows the strategy the adversary uses, it can always update the probability it gives to f1 and f2 to get abetter (possibly deterministic) defense. For example, if the adversary chooses to always move 0 to 1, theclassifier can set f = f1 w.p. 1 to retrieve an adversarial risk of 1/2 instead of 3/4.

Now, what happens if the adversary can use randomized strategies, meaning that for each point it canflip a coin before deciding where to move? In this case, the adversary could decide to move points from 0to 1 w.p. 1/2 and to −1 otherwise. This strategy is still optimal with an adversarial risk of 3/4 but nowthe classifier cannot use its knowledge of the adversary’s strategy to lower the risk. We are in a state whereneither the adversary nor the classifier can benefit from unilaterally changing its strategy. In the game theoryterminology, this state is called a Mixed Nash equilibrium.

1(X,Y ) ∼ P is misclassified by fi if and only if fi(X)Y ≤ 0

2

2.2 General settingLet us consider a classification task with input space X and output space Y. Let (X , d) be a proper (i.e.closed balls are compact) Polish (i.e. completely separable) metric space representing the inputs space2.Let Y = {1, . . . ,K} be the labels set, endowed with the trivial metric d′(y, y′) = 1y 6=y′ . Then the space(X ×Y, d⊕ d′) is a proper Polish space. For any Polish space Z, we denoteM1

+(Z) the Polish space of Borelprobability measures on Z. Let us assume the data is drawn from P ∈M1

+(X × Y). Let (Θ, dΘ) be a Polishspace (not necessarily proper) representing the set of classifier parameters (for instance neural networks). Wealso define a loss function: l : Θ× (X × Y)→ [0,∞) satisfying the following set of assumptions.

Assumption 1 (Loss function). 1) The loss function l is a non negative Borel measurable function. 2) Forall θ ∈ Θ, l(θ, ·) is upper-semi continuous. 3) There exists M > 0 such that for all θ ∈ Θ, (x, y) ∈ X × Y,0 ≤ l(θ, (x, y)) ≤M .

It is usual to assume upper-semi continuity when studying optimization over distributions [7, 38]. Fur-thermore, considering bounded (and positive) loss functions is also very common in learning theory [2] and isnot restrictive.

In the adversarial examples framework, the loss of interest is the 0/1 loss, for whose surrogates aremisunderstood [1, 14]; hence it is essential that the 0/1 loss satisfies Assumption 1. In the binary classificationsetting (i.e. Y = {−1,+1}) the 0/1 loss writes l0/1(θ, (x, y)) = 1yfθ(x)≤0. Then, assuming that for all θ, fθ(·)is continuous and for all x, f·(x) is continuous, the 0/1 loss satisfies Assumption 1. In particular, it is thecase for neural networks with continuous activation functions.

2.3 Adversarial Risk MinimizationThe standard risk for a single classifier θ associated with the loss l satisfying Assumption 1 writes: R(θ) :=E(x,y)∼P [l(θ, (x, y))]. Similarly, the adversarial risk of θ at level ε associated with the loss l is defined as3

Rεadv(θ) := E(x,y)∼P

[sup

x′∈X , d(x,x′)≤εl(θ, (x′, y))

].

It is clear that R0adv(θ) = R(θ) for all θ. We can generalize these notions with distributions of classifiers.

In other terms the classifier is then randomized according to some distribution µ ∈M1+(Θ). A classifier is

randomized if for a given input, the output of the classifier is a probability distribution. The standard riskof a randomized classifier µ writes R(µ) = Eθ∼µ [R(θ)]. Similarly, the adversarial risk of the randomizedclassifier µ at level ε is4

Rεadv(µ) := E(x,y)∼P

[sup

x′∈X , d(x,x′)≤εEθ∼µ [l(θ, (x′, y))]

].

For instance, for the 0/1 loss, the inner maximization problem, consists in maximizing the probability ofmisclassification for a given couple (x, y). Note that R(δθ) = R(θ) and Rεadv(δθ) = Rεadv(θ). In the remainderof the paper, we study the adversarial risk minimization problems with randomized and deterministic classifiersand denote

Vεrand := infµ∈M1

+(Θ)Rεadv(µ), Vεdet := inf

θ∈ΘRεadv(θ) (1)

Remark 1. We can show (see Appendix E) that the standard risk infima are equal : V0rand = V0

det. Hence,no randomization is needed for minimizing the standard risk. Denoting V this common value, we also havethe following inequalities for any ε > 0, V ≤ Vεrand ≤ Vεdet.

2For instance, for any norm ‖·‖, (Rd, ‖·‖) is a proper Polish metric space.3For the well-posedness, see Lemma 4 in Appendix.4This risk is also well posed (see Lemma 4 in the Appendix).

3

2.4 Distributional Formulation of the Adversarial RiskTo account for the possible randomness of the adversary, we rewrite the adversarial attack problem as aconvex optimization problem on distributions. Let us first introduce the set of adversarial distributions.

Definition 1 (Set of adversarial distributions). Let P be a Borel probability distribution on X ×Y and ε > 0.We define the set of adversarial distributions as

Aε(P) :={Q ∈M+

1 (X × Y) | ∃γ ∈M+1

((X × Y)2

),

d(x, x′) ≤ ε, y = y′ γ-a.s., Π1]γ = P, Π2]γ = Q}

where Πi denotes the projection on the i-th component, and g] the push-forward measure by a measurablefunction g.

For an attacker that can move the initial distribution P in Aε(P), the attack would not be a transportmap as considered in the standard adversarial risk. For every point x in the support of P, the attacker isallowed to move x randomly in the ball of radius ε, and not to a single other point x′ like the usual attackerin adversarial attacks. In this sense, we say the attacker is allowed to be randomized.

Link with DRO. Adversarial examples have been studied in the light of DRO by former works [33, 37],but an exact reformulation of the adversarial risk as a DRO problem has not been made yet. When (Z, d) isa Polish space and c : Z2 → R+ ∪ {+∞} is a lower semi-continuous function, for P,Q ∈M+

1 (Z) , the primalOptimal Transport problem is defined as

Wc(P,Q) := infγ∈ΓP,Q

∫Z2

c(z, z′)dγ(z, z′)

with ΓP,Q :={γ ∈M+

1 (Z2) | Π1]γ = P, Π2]γ = Q}. When η > 0 and for P ∈ M+

1 (Z), the associatedWasserstein uncertainty set is defined as:

Bc(P, η) :={Q ∈M+

1 (Z) |Wc(P,Q) ≤ η}

A DRO problem is a linear optimization problem over Wasserstein uncertainty sets supQ∈Bc(P,η)

∫g(z)dQ(z)

for some upper semi-continuous function g [41]. For an arbitrary ε > 0, we define the cost cε as follows

cε((x, y), (x′, y′)) :=

{0 if d(x, x′) ≤ ε and y = y′

+∞ otherwise.

This cost is lower semi-continuous and penalizes to infinity perturbations that change the label or move theinput by a distance greater than ε. As Proposition 1 shows, the Wasserstein ball associated with cε is equalto Aε(P).

Proposition 1. Let P be a Borel probability distribution on X × Y and ε > 0 and η ≥ 0, then Bcε(P, η) =Aε(P). Moreover, Aε(P) is convex and compact for the weak topology ofM+

1 (X × Y).

Thanks to this result, we can reformulate the adversarial risk as the value of a convex problem over Aε(P).

Proposition 2. Let P be a Borel probability distribution on X × Y and µ a Borel probability distribution onΘ. Let l : Θ× (X × Y)→ [0,∞) satisfying Assumption 1. Let ε > 0. Then:

Rεadv(µ) = supQ∈Aε(P)

E(x′,y′)∼Q,θ∼µ [l(θ, (x′, y′))] . (2)

The supremum is attained. Moreover Q∗ ∈ Aε(P) is an optimum of Problem (2) if and only if there exists γ∗ ∈M+

1

((X × Y)2

)such that: Π1]γ

∗ = P, Π2]γ∗ = Q∗, d(x, x′) ≤ ε, y = y′ and l(x′, y′) = supu∈X ,d(x,u)≤ε l(u, y)

γ∗-almost surely.

4

The adversarial attack problem is a DRO problem for the cost cε. Proposition 2 means that, against afixed classifier µ, the randomized attacker that can move the distribution in Aε(P) has exactly the samepower as an attacker that moves every single point x in the ball of radius ε. By Proposition 2, we also deducethat the adversarial risk can be casted as a linear optimization problem over distributions.

Remark 2. In a recent work, [29] proposed a similar adversary using Markov kernels but left as an openquestion the link with the classical adversarial risk, due to measurability issues. Proposition 2 solves theseissues. The result is similar to [7]. Although we believe its proof might be extended for infinite valued costs, [7]did not treat that case. We provide an alternative proof in this special case.

3 Nash Equilibria in the Adversarial Game

3.1 Adversarial Attacks as a Zero-Sum GameThanks to Proposition 2, the adversarial risk minimization problem can be seen as a two-player zero-sumgame that writes as follows,

infµ∈M1

+(Θ)sup

Q∈Aε(P)

E(x,y)∼Q,θ∼µ [l(θ, (x, y))] . (3)

In this game the classifier objective is to find the best distribution µ ∈ M+1 (Θ) while the adversary is

manipulating the data distribution. For the classifier, solving the infimum problem in Equation (3) simplyamounts to solving the adversarial risk minimization problem – Problem (1), whether the classifier israndomized or not. Then, given a randomized classifier µ ∈ M+

1 (Θ), the goal of the attacker is to find anew data-set distribution Q in the set of adversarial distributions Aε(P) that maximizes the risk of µ. Moreformally, the adversary looks for

Q ∈ argmaxQ∈Aε(P)

E(x,y)∼Q,θ∼µ [l(θ, (x, y))] .

In the game theoretic terminology, Q is also called the best response of the attacker to the classifier θ.

Remark 3. Note that for a given classifier µ there always exists a “deterministic” best response, i.e. everysingle point x is mapped to another single point T (x). Let T : X → X be defined such that for all x ∈ X ,l(T (x), y) = supx′, d(x,x′)≤ε l(x

′, y). Thanks to [4, Proposition 7.50], (T, id) is P-measurable. Then (T, id)]Pbelongs to BR(µ). Therefore, T is the optimal “deterministic” attack against the classifier µ.

3.2 Dual Formulation of the GameEvery zero sum game has a dual formulation that allows for a deeper understanding of the framework. Here,from Proposition 2, we can define the dual problem of adversarial risk minimization for randomized classifiers.This dual problem also characterizes a two-player zero-sum game that writes as follows,

supQ∈Aε(P)

infµ∈M1

+(Θ)E(x,y)∼Q,θ∼µ [l(θ, (x, y))] . (4)

In this dual game problem, the adversary plays first and seeks an adversarial distribution that has thehighest possible risk when faced with an arbitrary classifier. This means that it has to select an adversarialperturbation for every input x, without seeing the classifier first. In this case, as pointed out by the motivatingexample in Section 2.1, the attack can (and should) be randomized to ensure maximal harm against severalclassifiers. Then, given an adversarial distribution, the classifier objective is to find the best possible classifieron this distribution. Let us denote Dε the value of the dual problem. Since the weak duality is alwayssatisfied, we get

Dε ≤ Vεrand ≤ Vεdet. (5)

5

Inequalities in Equation (5) mean that the lowest risk the classifier can get (regardless of the game we lookat) is Dε. In particular, this means that the primal version of the game, i.e. the adversarial risk minimizationproblem, will always have a value greater or equal to Dε. As we discussed in Section 2.1, this lower boundmay not be attained by a deterministic classifier. As we will demonstrate in the next section, optimizing overrandomized classifiers allows to approach Dε arbitrary closely.

Remark 4. Note that, we can always define the dual problem when the classifier is deterministic,

supQ∈Aε(P)

infθ∈Θ

E(x,y)∼Q [l(θ, (x, y))] .

Furthermore, we can demonstrate that the dual problems for deterministic and randomized classifiers have thesame value 5; hence the inequalities in Equation (5).

3.3 Nash Equilibria for Randomized StrategiesIn the adversarial examples game, a Nash equilibrium is a couple (µ∗,Q∗) ∈M1

+(Θ)×Aε(P) where both theclassifier and the attacker have no incentive to deviate unilaterally from their strategies µ∗ and Q∗. Moreformally, (µ∗,Q∗) is a Nash equilibrium of the adversarial examples game if (µ∗,Q∗) is a saddle point of theobjective function

(µ,Q) 7→ E(x,y)∼Q,θ∼µ [l(θ, (x, y))] .

Alternatively, we can say that (µ∗,Q∗) is a Nash equilibrium if and only if µ∗ solves the adversarial riskminimization problem – Problem (1), Q∗ the dual problem – Problem (6), and Dε = Vεrand. In our problem, Q∗always exists but it might not be the case for µ∗. Then for any δ > 0, we say that (µδ,Q∗) is a δ-approximateNash equilibrium if Q∗ solves the dual problem and µδ satisfies Dε ≥ Rεadv(µδ)− δ.

We now state our main result: the existence of approximate Nash equilibria in the adversarial examplesgame when both the classifier and the adversary can use randomized strategies. More precisely, we demonstratethat the duality gap between the adversary and the classifier problems is zero, which gives as a corollary theexistence of Nash equilibria.

Theorem 1. Let P ∈M1+(X × Y). Let ε > 0. Let l : Θ× (X × Y)→ [0,∞) satisfying Assumption 1. Then

strong duality always holds in the randomized setting:

infµ∈M+

1 (Θ)max

Q∈Aε(P)Eθ∼µ,(x,y)∼Q [l(θ, (x, y))] (6)

= maxQ∈Aε(P)

infµ∈M+

1 (Θ)Eθ∼µ,(x,y)∼Q [l(θ, (x, y))]

The supremum is always attained. If Θ is a compact set, and for all (x, y) ∈ X × Y, l(·, (x, y)) is lowersemi-continuous, the infimum is also attained.

Corollary 1. Under Assumption 1, for any δ > 0, there exists a δ-approximate Nash-Equibilrium (µδ,Q∗).Moreover, if the infimum is attained, there exists a Nash equilibrium (µ∗,Q∗) to the adversarial examplesgame.

Theorem 1 shows that Dε = Vεrand. From a game theoretic perspective, this means that the minimaladversarial risk for a randomized classifier against any attack (primal problem) is the same as the maximalrisk an adversary can get by using an attack strategy that is oblivious to the classifier it faces (dual problem).This suggests that playing randomized strategies for the classifier could substantially improve robustness toadversarial examples. In the next section, we will design an algorithm that efficiently learn this classifier, wewill get improve adversarial robustness over classical deterministic defenses.

Remark 5. Theorem 1 remains true if one replaces Aε(P) with any other Wasserstein compact uncertaintysets (see [41] for conditions of compactness).

5See Appendix E for more details

6

4 Finding the Optimal Classifiers

4.1 An Entropic Regularization

Let {(xi, yi)}Ni=1 samples independently drawn from P and denote P̂ := 1N

∑Ni=1 δ(xi,yi) the associated empirical

distribution. One can show the adversarial empirical risk minimization can be casted as:

R̂ε,∗adv := infµ∈M+

1 (Θ)

N∑i=1

supQi∈Γi,ε

E(x,y)∼Qi,θ∼µ [l(θ, (x, y))]

where Γi,ε is defined as :

Γi,ε :={Qi |

∫dQi =

1

N,

∫cε((xi, yi), ·)dQi = 0

}.

More details on this decomposition are given in Appendix E. In the following, we regularize the aboveobjective by adding an entropic term to each inner supremum problem. Let α := (αi)

Ni=1 ∈ RN+ such that for

all i ∈ {1, . . . , N}, and let us consider the following optimization problem:

R̂ε,∗adv,α := infµ∈M+

1 (Θ)

N∑i=1

supQi∈Γi,ε

EQi,µ [l(θ, (x, y))]

− αiKL(Qi∣∣∣∣∣∣ 1

NU(xi,yi)

)where U(x,y) is an arbitrary distribution of support equal to:

S(ε)(x,y) :=

{(x′, y′) : s.t. cε((x, y), (x′, y′)) = 0

},

and for all Q,U ∈M+(X × Y),

KL(Q||U) :=

{ ∫log(dQdU )dQ + |U| − |Q| if Q� U

+∞ otherwise.

Note that when α = 0, we recover the problem of interest R̂ε,∗adv,α = R̂ε,∗adv. Moreover, we show the regularizedsupremum tends to the standard supremum when α→ 0.

Proposition 3. For µ ∈M+1 (Θ), one has

limαi→0

supQi∈Γi,ε

EQi,µ [l(θ, (x, y))]− αiKL(Q∣∣∣∣∣∣ 1

NU(xi,yi)

)= sup

Qi∈Γi,ε

E(x,y)∼Qi,θ∼µ [l(θ, (x, y))] .

By adding an entropic term to the objective, we obtain an explicit formulation of the supremum involvedin the sum: as soon as α > 0 (which means that each αi > 0), each sub-problem becomes just theFenchel-Legendre transform of KL(·|U(xi,yi)/N) which has the following closed form:

supQi∈Γi,ε

EQi,µ [l(θ, (x, y))]− αiKL(Qi||

1

NU(xi,yi)

)=αiN

log

(∫X×Y

exp

(Eθ∼µ [l(θ, (x, y))]

αi

)dU(xi,yi)

).

7

Finally, we end up with the following problem:

infµ∈M+

1 (Θ)

N∑i=1

αiN

log

(∫exp

Eµ [l(θ, (x, y))]

αidU(xi,yi)

).

In order to solve the above problem, one needs to compute the integral involved in the objective. To doso, we estimate it by randomly sampling mi ≥ 1 samples (u

(i)1 , . . . , u

(i)mi) ∈ (X × Y)mi from U(xi,yi) for all

i ∈ {1, . . . , N} which leads to the following optimization problem

infµ∈M+

1 (Θ)

N∑i=1

αiN

log

1

mi

mi∑j=1

expEµ[l(θ, u

(i)j )]

αi

(7)

denoted R̂ε,madv,α where m := (mi)Ni=1 in the following. Now we aim at controlling the error made with our

approximations. We decompose the error into two terms

|R̂ε,madv,α − R̂ε,∗adv| ≤ |R̂

ε,∗adv,α − R̂

ε,madv,α|+ |R̂

ε,∗adv,α − R̂

ε,∗adv|

where the first one corresponds to the statistical error made by our estimation of the integral, and the secondto the approximation error made by the entropic regularization of the objective. First, we show a control ofthe statistical error using Rademacher complexities [2].

Proposition 4. Let m ≥ 1 and α > 0 and denote α := (α, . . . , α) ∈ RN and m := (m, . . . ,m) ∈ RN . Thenby denoting M̃ = max(M, 1), we have with a probability of at least 1− δ

|R̂ε,∗adv,α − R̂ε,madv,α| ≤

2eM/α

N

N∑i=1

Ri + 6M̃eM/α

√log( 4

δ )

2mN

where Ri := 1mEσ

[supθ∈Θ

∑mj=1 σj l(θ, u

(i)j )]and σ := (σ1, . . . , σm) with σi i.i.d. sampled as P[σi = ±1] =

1/2.

We deduce from the above Proposition that in the particular case where Θ is finite such that |Θ| = L,with probability of at least 1− δ

|R̂ε,∗adv,α − R̂ε,madv,α| ∈ O

(MeM/α

√log(L)

m

).

This case is of particular interest when one wants to learn the optimal mixture of some given classifiers inorder to minimize the adversarial risk. In the following proposition, we control the approximation error madeby adding an entropic term to the objective.

Proposition 5. Denote for β > 0, (x, y) ∈ X × Y and µ ∈ M+1 (Θ), A(x,y)

β,µ := {u| supv∈S(ε)

(x,y)

Eµ[l(θ, v)] ≤

Eµ[l(θ, u)] + β}. If there exists Cβ such that for all (x, y) ∈ X × Y and µ ∈ M+1 (Θ), U(x,y)

(A

(x,y)β,µ

)≥ Cβ

then we have

|R̂ε,∗adv,α − R̂ε,∗adv| ≤ 2α| log(Cβ)|+ β.

The assumption made in the above Proposition states that for any given random classifier µ, and anygiven point (x, y), the set of β-optimal attacks at this point has at least a certain amount of mass dependingon the β chosen. This assumption is always met when β is sufficiently large. However in order to obtaina tight control of the error, a trade-off exists between β and the smallest amount of mass Cβ of β-optimalattacks.

Now that we have shown that solving (7) allows to obtain an approximation of the true solution R̂ε,∗adv, wenext aim at deriving an algorithm to compute it.

8

4.2 Proposed AlgorithmsFrom now on, we focus on finite class of classifiers. Let Θ = {θ1, . . . , θL}, we aim to learn the optimal mixtureof classifiers in this case. The adversarial empirical risk is therefore defined as:

R̂εadv(λ) =

N∑i=1

supQi∈Γi,ε

E(x,y)∼Qi

[L∑k=1

λkl(θk, (x, y))

]

for λ ∈ ∆L := {λ ∈ RL+ s.t.∑Li=1 λi = 1}, the probability simplex of RL. One can notice that R̂εadv(·)

is a continuous convex function, hence minλ∈∆LRεadv(λ) is attained for a certain λ∗. Then there exists a

non-approximate Nash equilibrium (λ∗,Q∗) in the adversarial game when Θ is finite. Here, we present twoalgorithms to learn the optimal mixture of the adversarial risk minimization problem.

Algorithm 1 Oracle Algorithmλ0 = 1L

L;T ; η = 2

M√LT

for t = 1, . . . , T doQ̃ s.t. ∃Q∗ ∈ Aε(P) best response to λt−1 and for all k ∈ [L], |EQ̃(l(θk, (x, y)))− EQ∗(l(θk, (x, y)))| ≤ δgt =

(EQ̃(l(θ1, (x, y)), . . . ,EQ̃(l(θL, (x, y))

)Tλt = Π∆L (λt−1 − ηgt)

end

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Prob

aliti

ty o

f the

cla

ssifi

er

0 25 50 75 100 125 150 175 200Iterations

0.75

0.76

0.77

0.78

0.79

In-s

ampl

e Ri

sk

= 0= 10 1

= 10 3

= 10 5

= 10 7

0 1 2 3 4 5 (size of the perturbation)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Adve

rsar

ial r

isk

In-sample, deterministic classifierIn-sample, randomized classifierOut-sample deterministic classifierOut-sample, randomized classifier

Figure 2: On left, 40 data samples with their set of possible attacks represented in shadow and the optimalrandomized classifier, with a color gradient representing the probability of the classifier. In the middle,convergence of the oracle (α = 0) and regularized algorithm for different values of regularization parameters.On right, in-sample and out-sample risk for randomized and deterministic minimum risk in function of theperturbation size ε. In the latter case, the randomized classifier is optimized with oracle Algorithm 1.

A First Oracle Algorithm. The first algorithm we present is inspired from [33] and the convergence ofprojected sub-gradient methods [9]. The computation of the inner supremum problem is usually NP-hard 6,but one may assume the existence of an approximate oracle to this supremum. The algorithm is presented inAlgorithm 1. We get the following guarantee for this algorithm.

Proposition 6. Let l : Θ× (X × Y)→ [0,∞) satisfying Assumption 1. Then, Algorithm 1 satisfies:

mint∈[T ]

R̂εadv(λt)− R̂ε,∗adv ≤ 2δ +

2M√L√

T

The main drawback of the above algorithm is that one needs to have access to an oracle to guaranteethe convergence of the proposed algorithm. In the following we present its regularized version in order toapproximate the solution and propose a simple algorithm to solve it.

6See Appendix E for details.

9

An Entropic Relaxation. Adding an entropic term to the objective allows to have a simple reformulationof the problem, as follows:

infλ∈∆L

N∑i=1

εiN

log

1

mi

mi∑j=1

exp

(∑Lk=1 λkl(θk, u

(i)j )

εi

)Note that in λ, the objective is convex and smooth. One can apply the accelerated PGD [3, 36] which enjoysan optimal convergence rate for first order methods of O(T−2) for T iterations.

5 Experiments

5.1 Synthetic DatasetTo illustrate our theoretical findings, we start by testing our learning algorithm on the following synthetictwo-dimensional problem. Let us consider the distribution P defined as P (Y = ±1) = 1/2, P (X | Y = −1) =N (0, I2) and P (X | Y = 1) = 1

2 [N ((−3, 0), I2) +N ((3, 0), I2)]. We sample 1000 training points from thisdistribution and randomly generate 10 linear classifiers that achieves a standard training risk lower than 0.4.To simulate an adversary with budget ε in `2 norm, we proceed as follows. For every sample (x, y) ∼ P wegenerate 1000 points uniformly at random in the ball of radius ε and select the one maximizing the risk forthe 0/1 loss. Figure 2 (left) illustrates the type of mixture we get after convergence of our algorithms. Notethat in this toy problem, we are likely to find the optimal adversary with this sampling strategy if we sampleenough attack points.

To evaluate the convergence of our algorithms, we compute the adversarial risk of our mixture for eachiteration of both the oracle and regularized algorithms. Figure 2 illustrates the convergence of the algorithmsw.r.t the regularization parameter. We observe that the risk for both algorithms converge. Moreover, theyconverge towards the oracle minimizer when the regularization parameter α goes to 0.

Finally, to demonstrate the improvement randomized techniques offer against deterministic defenses, weplot in Figure 2 (right) the minimum adversarial risk for both randomized and deterministic classifiers w.r.t.ε. The adversarial risk is strictly better for randomized classifier whenever the adversarial budget ε is biggerthan 2. This illustration validates our analysis of Theorem 1, and motivates a in depth study of a morechallenging framework, namely image classification with neural networks.

5.2 CIFAR-10 Dataset

Models Acc. APGDCE APGDDLR Rob. Acc.1 81.9% 47.6% 47.7% 45.6%2 81.9% 49.0% 49.6% 47.0%3 81.7% 49.0% 49.3% 46.9%4 82.6% 49.7% 49.8% 47.2%

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

Accu

racy

1 models2 models3 models4 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Accu

racy

1 models2 models3 models4 models

Figure 3: On left: Comparison of our algorithm with a standard adversarial training (one model). Wereported the results for the model with the best robust accuracy obtained over two independent runs becauseadversarial training might be unstable. Standard and Robust accuracy (respectively in the middle and onright) on CIFAR-10 test images in function of the number of epochs per classifier with 1 to 3 ResNet18models. The performed attack is PGD with 20 iterations and ε = 8/255.

10

Adversarial examples are known to be easily transferrable from one model to another [35]. To counterthis and support our theoretical claims, we propose an heuristic algorithm (see Algorithm 2) to train a robustmixture of L classifiers. We alternatively train these classifiers with adversarial examples against the currentmixture and update the probabilities of the mixture according to the algorithms we proposed in Section 4.2.More details on the heuristic algorithm are available in Appendix D.

Algorithm 2 Adversarial Training for MixturesL: number of models, T : number of iterations,Tθ: number of updates for the models θ,Tλ: number of updates for the mixture λ,λ0 = (λ1

0, . . . λL0 ), θ0 = (θ1

0, . . . θL0 )

for t = 1, . . . , T doLet Bt be a batch of data.if t mod (TθL+ 1) 6= 0 then

k sampled uniformly in {1, . . . , L}B̃t ← Attack of images in Bt for the model (λt,θt)θtk ← Update θt−1

k with B̃t for fixed λt with a SGD stepelse

λt ←Update λt−1 on Bt for fixed θt with oracle or regularized algorithm with Tλ iterations.end

end

Experimental Setup. To evaluate the performance of Algorithm 2, we trained from 1 to 4 ResNet18 [20]models on 200 epochs per model7. We study the robustness with regards to `∞ norm and fixed adversarialbudget ε = 8/255. The attack we used in the inner maximization of the training is an adapted (adaptative)version of PGD for mixtures of classifiers with 10 steps. Note that for one single model, Algorithm 2 exactlycorresponds to adversarial training [24]. For each of our setups, we made two independent runs and selectthe best one. The training time of our algorithm is around four times longer than a standard AdversarialTraining (with PGD 10 iter.) with two models, eight times with three models and twelve times with fourmodels. We trained our models with a batch of size 1024 on 8 Nvidia V100 GPUs. We give more details onimplementation in Appendix D.

Evaluation Protocol. At each epoch, we evaluate the current mixture on test data against PGD attackwith 20 iterations. To select our model and avoid overfitting [30], we kept the most robust against thisPGD attack. To make a final evaluation of our mixture of models, we used an adapted version of AutoPGDuntargeted attacks [15] for randomized classifiers with both Cross-Entropy (CE) and Difference of LogitsRatio (DLR) loss. For both attacks, we made 100 iterations and 5 restarts.

Results. The results are presented in Figure 3. We remark our algorithm outperforms a standardadversarial training in all the cases by more 1%, without additional loss of standard accuracy as it is attestedby the left figure. Moreover, it seems our algorithm, by adding more and more models, reduces the overfittingof adversarial training. So far, experiments are computationally very costful and it is difficult to raiseprecise conclusions. Further, hyperparameter tuning [19] such as architecture, unlabeled data [12], activationfunction, or the use of TRADES [43] may still increase the results.

6 Related Work and DiscussionsDistributionally Robust Optimization. Several recent works [23, 33, 37] studied the problem of adver-sarial examples through the scope of distributionally robust optimization. In these frameworks, the set ofadversarial distributions is defined using an `p Wasserstein ball (the adversary is allowed to have an averageperturbation of at most ε in `p norm). This however does not match the usual adversarial attack problem,where the adversary cannot move any point by more than ε. In the present work, we introduce a cost

7L× 200 epochs in total, where L is the number of models.

11

function allowing us to cast the adversarial example problem as a DRO one, without changing the adversaryconstraints.

Optimal Transport (OT). [5] and [29] investigated classifier-agnostic lower bounds on the adversarialrisk of any deterministic classifier using OT. These works only evaluate lower bounds on the primal deterministicformulation of the problem, while we study the existence of mixed Nash equilibria. Note that [29] startedto investigate a way to formalize the adversary using Markov kernels, but did not investigate the impact ofrandomized strategies on the game. We extended this work by rigorously reformulating the adversarial riskas a linear optimization problem over distributions and we study this problem from a game theoretic point ofview.

Game Theory. Adversarial examples have been studied under the notions of Stackelberg game in [10],and zero-sum game in [8, 26, 31]. These works considered restricted settings (convex loss, parametricadversaries, etc.) that do not comply with the nature of the problem. Indeed, we prove in Appendix C.3 thatwhen the loss is convex and the set Θ is convex, the duality gap is zero for deterministic classifiers. However, ithas been proven that no convex loss can be a good surrogate for the 0/1 loss in the adversarial setting [1, 14],narrowing the scope of this result. If one can show that for sufficiently separated conditional distributions,an optimal deterministic classifier always exists (see Appendix E for a clear statement), necessary andsufficient conditions for the need of randomization are still to be established. [27] studied partly this questionfor regularized deterministic adversaries, leaving the general setting of randomized adversaries and mixedequilibria unanswered, which is the very scope of this paper.

12

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Supplementary material

A NotationsLet (Z, d) be a Polish metric space (i.e. complete and separable). We say that (Z, d) is proper if for allz0 ∈ Z and R > 0, B(z0, R) := {z | d(z, z0) ≤ R} is compact. For (Z, d) a Polish space, we denoteM1

+(Z)the set of Borel probability measures on Z endowed with ‖·‖TV strong topology. We recall the notion ofweak topology: we say that a sequence (µn)n of M1

+(Z) converges weakly to µ ∈ M1+(Z) if and only if

for every continuous function f on Z,∫fdµn →n→∞

∫fdµ. Endowed with its weak topology,M1

+(Z) isa Polish space. For µ ∈ M1

+(Z), we define L1(µ) the set of integrable functions with respect to µ. Wedenote Π1 : (z, z′) ∈ Z2 7→ z and Π2 : (z, z′) ∈ Z2 7→ z′ respectively the projections on the first and secondcomponent, which are continuous applications. For a measure µ and a measurable mapping µ, we denote g]µthe pushforward measure of µ by g. Let L ≥ 1 be an integer and denote ∆L := {λ ∈ RL+ s.t.

∑Lk=1 λk = 1},

the probability simplex of RL.

B Useful LemmasLemma 1 (Fubini’s theorem). Let l : Θ×(X×Y)→ [0,∞) satisfying Assumption 1. Then for all µ ∈M1

+(Θ),∫l(θ, ·)dµ(θ) is Borel measurable; for Q ∈M1

+(X × Y),∫l(·, (x, y))dQ(x, y) is Borel measurable. Moreover:∫

l(θ, (x, y))dµ(θ)dQ(x, y) =∫l(θ, (x, y))dQ(x, y)dµ(θ)

Lemma 2. Let l : Θ × (X × Y) → [0,∞) satisfying Assumption 1. Then for all µ ∈ M1+(Θ), (x, y) 7→∫

l(θ, (x, y))dµ(θ) is upper semi-continuous and hence Borel measurable.

Proof. Let (xn, yn)n be a sequence of X × Y converging to (x, y) ∈ X × Y. For all θ ∈ Θ, M − l(θ, ·) is nonnegative and lower semi-continuous. Then by Fatou’s Lemma applied:∫

M − l(θ, (x, y))dµ(θ) ≤∫

lim infn→∞

M − l(θ, (xn, yn))dµ(θ)

≤ lim infn→∞

∫M − l(θ, (xn, yn))dµ(θ)

Then we deduce that:∫M − l(θ, ·)dµ(θ) is lower semi-continuous and then

∫l(θ, ·)dµ(θ) is upper-semi

continuous.

Lemma 3. Let l : Θ × (X × Y) → [0,∞) satisfying Assumption 1 Then for all µ ∈ M1+(Θ), Q 7→∫

l(θ, (x, y))dµ(θ)dQ(x, y) is upper semi-continuous for weak topology of measures.

Proof. −∫l(θ, ·)dµ(θ) is lower semi-continuous from Lemma 2. Then M −

∫l(θ, ·)dµ(θ) is lower semi-

continuous and non negative. Let denote v this function. Let (vn)n be a non-decreasing sequence ofcontinuous bounded functions such that vn → v. Let (Qk)k converging weakly towards Q. Then by monotoneconvergence:

∫vdQ = lim

n

∫vndQ = lim

nlimk

∫vndQk ≤ lim inf

k

∫vdQk

Then Q 7→∫vdQ is lower semi-continuous and then Q 7→

∫l(θ, (x, y))dµ(θ)dQ(x, y) is upper semi-continuous

for weak topology of measures.

Lemma 4. Let l : Θ × (X × Y) → [0,∞) satisfying Assumption 1. Then for all µ ∈ M1+(Θ), (x, y) 7→

sup(x′,y′),d(x,x′)≤ε,y=y′∫l(θ, (x′, y′))dµ(θ) is universally measurable (i.e. measurable for all Borel probability

measures). And hence the adversarial risk is well defined.

16

Proof. Let φ : (x, y) 7→ sup(x′,y′),d(x,x′)≤ε,y=y′∫l(θ, (x′, y′))dµ(θ). Then for u ∈ R̄:

{φ(x, y) > u} = Proj1

{((x, y), (x′, y′)) |

∫l(θ, (x′, y′))dµ(θ)− cε((x, y), (x′, y′)) > u

}By Lemma 3: ((x, y), (x′, y′)) 7→

∫l(θ, (x′, y′))dµ(θ)− cε((x, y), (x′, y′)) is upper-semicontinuous hence Borel

measurable. So its level sets are Borel sets, and by [4, Proposition 7.39], the projection of a Borel set isanalytic. And then {φ(x, y) > u} universally measurable thanks to [4, Corollary 7.42.1]. We deduce that φ isuniversally measurable.

C Proofs

C.1 Proof of Proposition 1Proof. Let η > 0. Let Q ∈ Aε(P). There exists γ ∈M+

1

((X × Y)2

)such that, d(x, x′) ≤ ε, y = y′ γ-almost

surely, and Π1]γ = P, and Π2]γ = Q. Then∫cεdγ = 0 ≤ η. Then, we deduce that Wcε(P,Q) ≤ η, and

Q ∈ Bcε(P, η). Reciprocally, let Q ∈ Bcε(P, η). Then, since the infimum is attained in the Wassersteindefinition, there exists γ ∈ M+

1

((X × Y)2

)such that

∫cεdγ ≤ η. Since cε((x, x′), (y, y′)) = +∞ when

d(x, x′) > ε and y 6= y′, we deduce that, d(x, x′) ≤ ε and y = y′, γ-almost surely. Then Q ∈ Aε(P). We havethen shown that: Aε(P) = Bcε(P, η).

The convexity of Aε(P) is then immediate from the relation with the Wasserstein uncertainty set.Let us show first that Aε(P) is relatively compact for weak topology. To do so we will show that Aε(P)

is tight and apply Prokhorov’s theorem. Let δ > 0, (X × Y, d⊕ d′) being a Polish space, {P} is tight thenthere exists Kδ compact such that P(Kδ) ≥ 1− δ. Let K̃δ := {(x′, y′) | ∃(x, y) ∈ Kδ, d(x′, x) ≤ ε, y = y′}.Recalling that (X , d) is proper (i.e. the closed balls are compact), so K̃δ is compact. Moreover for Q ∈ Aε(P),Q(K̃δ) ≥ P(Kδ) ≥ 1− δ. And then, Prokhorov’s theorem holds, and Aε(P) is relatively compact for weaktopology.

Let us now prove that Aε(P) is closed to conclude. Let (Qn)n be a sequence of Aε(P) converging towardssome Q for weak topology. For each n, there exists γn ∈ M1

+(X × Y) such that d(x, x′) ≤ ε and y = y′

γn-almost surely and Π1]γn = P, Π2]γn = Qn. {Qn, n ≥ 0} is relatively compact, then tight, then⋃n ΓP,Qn is

tight, then relatively compact by Prokhorov’s theorem. (γn)n ∈⋃n ΓP,Qn , then up to an extraction, γn → γ.

Then d(x, x′) ≤ ε and y = y′ γ-almost surely, and by continuity, Π1]γ = P and by continuity, Π2]γ = Q. Andhence Aε(P) is closed.

Finally Aε(P) is a convex compact set for the weak topology.

C.2 Proof of Proposition 2Proof. Let µ ∈ M1

+(Θ). Let f̃ : ((x, y), (x′, y′)) 7→ Eθ∼µ [l(θ, (x, y))] − cε((x, y), (x′, y′)). f̃ is upper-semicontinuous, hence upper semi-analytic. Then, by upper semi continuity of Eθ∼µ [l(θ, ·)] on the compact{(x′, y′) | d(x, x′) ≤ ε, y = y′} and [4, Proposition 7.50], there exists a universally measurable mapping Tsuch that Eθ∼µ [l(θ, T (x, y))] = sup(x′,y′), d(x,x′)≤ε,y=y′ Eθ∼µ [l(θ, (x, y))]. Let Q = T]P, then Q ∈ Aε(P). And

then E(x,y)∼P

[sup(x′,y′), d(x,x′)≤ε,y=y′ Eθ∼µ [l(θ, (x′, y′))]

]≤ supQ∈Aε(P) E(x,y)∼Q [Eθ∼µ [l(θ, (x, y))]].

Reciprocally, let Q ∈ Aε(P). There exists γ ∈M1+((X × Y)2), such that d(x, x′) ≤ ε and y = y′ γ-almost

surely, and, Π1]γ = P and Π2]γ = Q. Then: Eθ∼µ [l(θ, (x′, y′))] ≤ sup(u,v), d(x,u)≤ε,y=v Eθ∼µ [l(θ, (u, v))]

17

γ-almost surely. Then, we deduce that:

E(x′,y′)∼Q [Eθ∼µ [l(θ, (x′, y′))]] = E(x,y,x′,y′)∼γ [Eθ∼µ [l(θ, (x′, y′))]]

≤ E(x,y,x′,y′)∼γ

[sup

(u,v), d(x,u)≤ε,y=v

Eθ∼µ [l(θ, (u, v))]

]

≤ E(x,y)∼P

[sup

(u,v), d(x,u)≤ε,y=v

Eθ∼µ [l(θ, (u, v))]

]Then we deduce the expected result:

Rεadv(µ) = supQ∈Aε(P)

E(x,y)∼Q [Eθ∼µ [l(θ, (x, y))]]

Let us show that the optimum is attained. Q 7→ E(x,y)∼Q [Eθ∼µ [l(θ, (x, y))]] is upper semi continuous byLemma 3 for the weak topology of measures, and Aε(P) is compact by Proposition 1, then by [4, Proposition7.32], the supremum is attained for a certain Q∗ ∈ Aε(P).

C.3 Proof of Theorem 1Let us first recall the Fan’s Theorem.

Theorem 2. Let U be a compact convex Haussdorff space and V be convex space (not necessarily topological).Let ψ : U × V → R be a concave-convex function such that for all v ∈ V , ψ(·, v) is upper semi-continuousthen:

infv∈V

maxu∈U

ψ(u, v) = maxu∈U

infv∈V

ψ(u, v)

We are now set to prove Theorem 1.

Proof. Aε(P), endowed with the weak topology of measures, is a Hausdorff compact convex space, thanks toProposition 1. Moreover,M1

+(Θ) is clearly convex and (Q, µ) 7→∫ldµdQ is bilinear, hence concave-convex.

Moreover thanks to Lemma 3, for all µ, Q 7→∫ldµdQ is upper semi-continuous. Then Fan’s theorem applies

and strong duality holds.

In the related work (Section 6), we mentioned a particular form of Theorem 1 for convex cases. Asmentioned, this result has limited impact in the adversarial classification setting. It is still a direct corollaryof Fan’s theorem. This theorem can be stated as follows:

Theorem 3. Let P ∈M1+(X × Y), ε > 0 and Θ a convex set. Let l be a loss satisfying Assumption 1, and

also, (x, y) ∈ X × Y, l(·, (x, y)) is a convex function, then we have the following:

infθ∈Θ

supQ∈Aε(P)

EQ [l(θ, (x, y))] = supQ∈Aε(P)

infθ∈Θ

EQ [l(θ, (x, y))]

The supremum is always attained. If Θ is a compact set then, the infimum is also attained.

C.4 Proof of Proposition 3

Proof. Let us first show that for α ≥ 0, supQi∈Γi,ε EQi,µ [l(θ, (x, y))]−αKL(Qi∣∣∣∣∣∣ 1NU(xi,yi)

)admits a solution.

Let α ≥ 0, (Qnα,i)n≥0 a sequence such that

EQnα,i,µ [l(θ, (x, y))]− αKL(Qnα,i

∣∣∣∣∣∣ 1

NU(xi,yi)

)−−−−−→n→+∞

supQi∈Γi,ε

EQi,µ [l(θ, (x, y))]− αKL(Qi∣∣∣∣∣∣ 1

NU(xi,yi)

).

18

As Γi,ε is tight ((X , d) is a proper metric space therefore all the closed ball are compact) and by Prokhorov’stheorem, we can extract a subsequence which converges toward Q∗α,i. Moreover, l is upper semi-continuous

(u.s.c), thus Q → EQ,µ [l(θ, (x, y))] is also u.s.c.8 Moreover Q → −αKL(Q∣∣∣∣∣∣ 1NU(xi,yi)

)is also u.s.c. 9,

therefore, by considering the limit superior as n goes to infinity we obtain that

lim supn→+∞

EQnα,i,µ [l(θ, (x, y))]− αKL(Qnα,i

∣∣∣∣∣∣ 1

NU(xi,yi)

)= sup

Qi∈Γi,ε

EQi,µ [l(θ, (x, y))]− αKL(Qi∣∣∣∣∣∣ 1

NU(xi,yi)

)≤ EQ∗α,i,µ [l(θ, (x, y))]− αKL

(Q∗α,i

∣∣∣∣∣∣ 1

NU(xi,yi)

)from which we deduce that Q∗α,i is optimal.

Let us now show the result. We consider a positive sequence of (α(`)i )`≥0 such that α(`)

i → 0. Let us denoteQ∗α

(`)i ,i

andQ∗i the solutions of maxQi∈Γi,ε EQi,µ [l(θ, (x, y))]−α(`)i KL

(Qi∣∣∣∣∣∣ 1NU(xi,yi)

)and maxQi∈Γi,ε EQi,µ [l(θ, (x, y))]

respectively. Since Γi,ε is tight, (Q∗α

(`)i ,i

)`≥0 is also tight and we can extract by Prokhorov’s theorem a subse-quence which converges towards Q∗. Moreover we have

EQ∗i ,µ [l(θ, (x, y))]− α(`)i KL

(Q∗i∣∣∣∣∣∣ 1

NU(xi,yi)

)≤ EQ∗

α(`)i,i,µ [l(θ, (x, y))]− α(`)

i KL(Q∗α

(`)i ,i

∣∣∣∣∣∣ 1

NU(xi,yi)

)from which follows that

0 ≤ EQ∗i ,µ [l(θ, (x, y))]− EQ∗α(`)i,i,µ [l(θ, (x, y))] ≤ α(`)

i

(KL(Q∗i∣∣∣∣∣∣ 1

NU(xi,yi)

)−KL

(Q∗α

(`)i ,i

∣∣∣∣∣∣ 1

NU(xi,yi)

))Then by considering the limit superior we obtain that

lim sup`→+∞

EQ∗α(`)i,i,µ [l(θ, (x, y))] = EQ∗i ,µ [l(θ, (x, y))] .

from which follows that

EQ∗i ,µ [l(θ, (x, y))] ≤ EQ∗,µ [l(θ, (x, y))]

and by optimality of Q∗i we obtain the desired result.

C.5 Proof of Proposition 4Proof. Let us denote for all µ ∈M+

1 (Θ),

R̂ε,madv,α(µ) :=

N∑i=1

αiN

log

1

mi

mi∑j=1

expEµ[l(θ, u

(i)j )]

αi

.

Let also consider (µ(m)n )n≥0 and (µn)n≥0 two sequences such that

R̂ε,madv,α(µ(m)n ) −−−−−→

n→+∞R̂ε,madv,α, R̂εadv,α(µn) −−−−−→

n→+∞R̂ε,∗adv,α.

8Indeed by considering a decreasing sequence of continuous and bounded functions which converge towards Eµ [l(θ, (x, y))]and by definition of the weak convergence the result follows.

9for α = 0 the result is clear, and if α > 0, note that KL(·∣∣∣∣∣∣ 1NU(xi,yi)

)is lower semi-continuous

19

We first remarks that

R̂ε,madv,α − R̂ε,∗adv,α ≤ R̂

ε,madv,α − R̂

ε,madv,α(µn) + R̂ε,madv,α(µn)− R̂εadv,α(µn) + R̂εadv,α(µn)− R̂ε,∗adv,α

≤ supµ∈M+

1 (Θ)

∣∣∣R̂ε,madv,α(µ)− R̂εadv,α(µ)∣∣∣+ R̂εadv,α(µn)− R̂ε,∗adv,α,

and by considering the limit, we obtain that

R̂ε,madv,α − R̂ε,∗adv,α ≤ sup

µ∈M+1 (Θ)

∣∣∣R̂ε,madv,α(µ)− R̂εadv,α(µ)∣∣∣

Simarly we have that

R̂ε,∗adv,α − R̂ε,madv,α ≤ R̂

ε,∗adv,α − R̂

εadv,α(µ(m)

n ) + R̂εadv,α(µ(m)n )− R̂ε,madv,α(µ(m)

n ) + R̂ε,madv,α(µ(m)n )− R̂ε,madv,α

from which follows that

R̂ε,∗adv,α − R̂ε,madv,α ≤ sup

µ∈M+1 (Θ)

∣∣∣R̂ε,madv,α(µ)− R̂εadv,α(µ)∣∣∣

Therefore we obtain that∣∣∣R̂ε,∗adv,α − R̂ε,madv,α∣∣∣ ≤ N∑i=1

α

Nsup

µ∈M+1 (Θ)

∣∣∣ log

1

mi

mi∑j=1

exp

Eθ∼µ[l(θ, u

(i)j ))

]α

− log

(∫X×Y

exp

(Eθ∼µ [l(θ, (x, y))]

α

)dU(xi,yi)

) ∣∣∣.Observe that l ≥ 0, therefore because the log function is 1-Lipschitz on [1,+∞), we obtain that

∣∣∣R̂ε,∗adv,α − R̂ε,madv,α∣∣∣ ≤ N∑i=1

α

Nsup

µ∈M+1 (Θ)

∣∣∣ 1

mi

mi∑j=1

exp

Eθ∼µ[l(θ, u

(i)j ))

]α

− ∫X×Y

exp

(Eθ∼µ [l(θ, (x, y))]

α

)dU(xi,yi)

∣∣∣.Let us now denote for all i = 1, . . . , N ,

R̂i(µ,u(i)) :=

mi∑j=1

exp

Eθ∼µ[l(θ, u

(i)j ))

]α

Ri(µ) :=

∫X×Y

exp

(Eθ∼µ [l(θ, (x, y))]

α

)dU(xi,yi).

and let us define

f(u(1), . . . ,u(N)) :=

N∑i=1

α

Nsup

µ∈M+1 (Θ)

∣∣∣R̂i(µ)−Ri(µ)∣∣∣

where u(i) := (u(i)1 , . . . , u

(m)1 ). By denoting z(i) = (u

(i)1 , . . . , u

(i)k−1, z, u

(i)k+1, . . . , u

(i)m ), we have that

|f(u(1), . . . ,u(N))− f(u(1), . . . ,u(i−1), z(i),u(i+1), . . . ,u(N))| ≤ α

N

∣∣∣ supµ∈M+

1 (Θ)

∣∣∣R̂i(µ,u(i))−Ri(µ)∣∣∣

− supµ∈M+

1 (Θ)

∣∣∣R̂i(µ, z(i))−Ri(µ)∣∣∣∣∣∣

≤ α

N

∣∣∣ 1

m

exp

Eθ∼µ[l(θ, u

(i)k ))

]α

− exp

(Eθ∼µ

[l(θ, z(i)))

]α

) ∣∣∣≤ 2 exp(M/α)

Nm

20

where the last inequality comes from the fact that the loss is upper bounded by l ≤M . Then by appling theMcDiarmid’s Inequality, we obtain that with a probability of at least 1− δ,∣∣∣R̂ε,∗adv,α − R̂ε,madv,α∣∣∣ ≤ E(f(u(1), . . . ,u(N))) +

2 exp(M/α)√mN

√log(2/δ)

2.

Thanks to [32, Lemma 26.2], we have for all i ∈ {1, . . . , N}

E(f(u(1), . . . ,u(N))) ≤ 2E(Rad(Fi ◦ u(i)))

where for any class of function F defined on Z and point z : (z1, . . . , zq) ∈ Zq

F ◦ z :={

(f(z1), . . . , f(zq)), f ∈ F}

, Rad(F ◦ z) :=1

qEσ∼{±1}

[supf∈F

q∑i=1

σif(zi)

]

Fi :={u→ exp

(Eθ∼µ [l(θ, u))]

α

), µ ∈M+

1 (Θ)}.

Moreover as x→ exp(x/α) is exp(M/α)α -Lipstchitz on (−∞,M ], by [32, Lemma 26.9], we have

Rad(Fi ◦ u(i)) ≤ exp(M/α)

αRad(Hi ◦ u(i))

where

Hi :={u→ Eθ∼µ [l(θ, u))] , µ ∈M+

1 (Θ)}.

Let us now define

g(u(1), . . . ,u(N)) :=

N∑j=1

2 exp(M/α)

NRad(Hj ◦ u(j)).

We observe that

|g(u(1), . . . ,u(N))− g(u(1), . . . ,u(i−1), z(i),u(i+1), . . . ,u(N))| ≤ 2 exp(M/α)

N|Rad(Hi ◦ u(i))− Rad(Hi ◦ z(i))|

≤ 2 exp(M/α)

N

2M

m.

By Applying the McDiarmid’s Inequality, we have that with a probability of at least 1− δ

E(g(u(1), . . . ,u(N))) ≤ g(u(1), . . . ,u(N)) +4 exp(M/α)M√

mN

√log(2/δ)

2.

Remarks also that

Rad(Hi ◦ u(i)) =1

mEσ∼{±1}

supµ∈M+

1 (Θ)

m∑j=1

σiEµ(l(θ, u(i)j ))

=

1

mEσ∼{±1}

supθ∈Θ

m∑j=1

σil(θ, u(i)j )

Finally, applying a union bound leads to the desired result.

21

C.6 Proof of Proposition 5Proof. Following the same steps than the proof of Proposition 4, let (µεn)n≥0 and (µn)n≥0 two sequences suchthat

R̂εadv,α(µεn) −−−−−→n→+∞

R̂ε,∗adv,α, R̂εadv(µn) −−−−−→n→+∞

R̂ε,∗adv.

Remarks that

R̂ε,∗adv,α − R̂ε,∗adv ≤ R̂

ε,∗adv,α − R̂

εadv,α(µn) + R̂εadv,α(µn)− R̂εadv(µn) + R̂εadv(µn)− R̂ε,∗adv

≤ supµ∈M+

1 (Θ)

∣∣∣R̂εadv,α(µ)− R̂εadv(µ)∣∣∣+ R̂εadv(µn)− R̂ε,∗adv

Then by considering the limit we obtain that

R̂ε,∗adv,α − R̂ε,∗adv ≤ sup

µ∈M+1 (Θ)

∣∣∣R̂εadv,α(µ)− R̂εadv(µ)∣∣∣.

Similarly, we obtain that

R̂ε,∗adv − R̂ε,∗adv,α ≤ sup

µ∈M+1 (Θ)

∣∣∣R̂εadv,α(µ)− R̂εadv(µ)∣∣∣,

from which follows that∣∣∣R̂ε,∗adv,α − R̂ε,∗adv∣∣∣ ≤ 1

N

N∑i=1

supµ∈M+

1 (Θ)

∣∣∣α log

(∫X×Y

exp

(Eµ[l(θ, (x, y))]

α

)dU(xi,yi)

)− supu∈Sε

(xi,yi)

Eµ[l(θ, u)]∣∣∣.

Let µ ∈M+1 (Θ) and i ∈ {1, . . . , N}, then we have∣∣∣α log

(∫X×Y

exp

(Eµ[l(θ, (x, y))]

α

)dU(xi,yi)

)− supu∈Sε

(xi,yi)

Eµ[l(θ, u)]∣∣∣

=∣∣∣α log

(∫X×Y

exp

(Eµ[l(θ, (x, y))]− supu∈Sε(xi,yi)

Eµ[l(θ, u)]

α

)dU(xi,yi)

)∣∣∣= α

∣∣∣ log

(∫A

(xi,yi)

β,µ

exp

(Eµ[l(θ, (x, y))]− supu∈Sε(xi,yi)

Eµ[l(θ, u)]

α

)dU(xi,yi)

+

∫(A

(xi,yi)

β,µ )cexp

(Eµ[l(θ, (x, y))]− supu∈Sε(xi,yi)

Eµ[l(θ, u)]

α

)dU(xi,yi)

)∣∣∣≤ α

∣∣∣ log(

exp(−β/α)U(xi,yi)

(A

(xi,yi)β,µ

)) ∣∣∣+ α

∣∣∣ log

1 +exp(β/α)

U(xi,yi)

(A

(xi,yi)β,µ

) ∫(A

(xi,yi)

β,µ )cexp

(Eµ[l(θ, (x, y))]− supu∈Sε(xi,yi)

Eµ[l(θ, u)]

α

)dU(xi,yi)

∣∣∣≤ α log(1/Cβ) + β +

α

Cβ

≤ 2α log(1/Cβ) + β

22

C.7 Proof of Proposition 6

Proof. Thanks to Danskin theorem, ifQ∗ is a best response to λ, then g∗ := (EQ∗ [l(θ1, (x, y))] , . . . ,EQ∗ [l(θL, (x, y))])T

is a subgradient of λ→ Rεadv(λ). Let η ≥ 0 be the learning rate. Then we have for all t ≥ 1:

‖λt − λ∗‖2 ≤ ‖λt−1 − ηgt − λ∗‖2

= ‖λt−1 − λ∗‖2 − 2η〈gt,λt−1 − λ∗〉+ η2‖gt‖2

≤ ‖λt−1 − λ∗‖2 − 2η〈g∗t ,λt−1 − λ∗〉+ 2η〈g∗t − gt,λt−1 − λ∗〉+ η2M2L

≤ ‖λt−1 − λ∗‖2 − 2η (Rεadv(λt)−Rεadv(λ∗)) + 4ηδ + η2M2L

We then deduce by summing:

2η

T∑t=1

Rεadv(λt)−Rεadv(λ∗) ≤ 4δηT + ‖λ0 − λ∗‖2 + η2M2LT

Then we have:

mint∈[T ]

Rεadv(λt)−Rεadv(λ∗) ≤ 2δ +4

ηT+M2Lη

The left-hand term is minimal for η = 2M√LT

, and for this value:

mint∈[T ]

Rεadv(λt)−Rεadv(λ∗) ≤ 2δ +2M√L√

T

.

D Additional Experimental Results

D.1 Experimental setting.Optimizer. For each of our models, The optimizer we used in all our implementations is SGD with learningrate set to 0.4 at epoch 0 and is divided by 10 at half training then by 10 at the three quarters of training.The momentum is set to 0.9 and the weight decay to 5× 10−4. The batch size is set to 1024.

Adaptation of Attacks. Since our classifier is randomized, we need to adapt the attack accordingly. Todo so we used the expected loss:

l̃ ((λ,θ), (x, y)) =

L∑k=1

λkl(θk, (x, y))

to compute the gradient in the attacks, regardless the loss (DLR or cross-entropy). For the inner maximizationat training time, we used a PGD attack on the cross-entropy loss with ε = 0.03. For the final evaluation, weused the untargeted DLR attack with default parameters.

Regularization in Practice. The entropic regularization in higher dimensional setting need to be adaptedto be more likely to find adversaries. To do so, we computed PGD attacks with only 3 iterations with 5different restarts instead of sampling uniformly 5 points in the `∞-ball. In our experiments in the main paper,we use a regularization parameter α = 0.001. The learning rate for the minimization on λ is always fixed to0.001.

23

Alternate Minimization Parameters. Algorithm 2 implies an alternate minimization algorithm. Weset the number of updates of θ to Tθ = 50 and, the update of λ to Tλ = 25.

D.2 Effect of the RegularizationIn this subsection, we experimentally investigate the effect of the regularization. In Figure 4, we notice, thatthe regularization has the effect of stabilizing, reducing the variance and improving the level of the robustaccuracy for adversarial training for mixtures (Algorithm 2). The standard accuracy curves are very similarin both cases.

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Accu

racy

1 models2 models3 models4 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Ac

cura

cy

1 models2 models3 models4 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

Accu

racy

1 models2 models3 models4 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

Accu

racy

1 models2 models3 models4 models

Figure 4: On left and middle-left: Standard accuracies over epochs with respectively no regularization andregularization set to α = 0.001. On middle right and right: Robust accuracies for the same parametersagainst PGD attack with 20 iterations and ε = 0.03.

D.3 Additional Experiments on WideResNet28x10We now evaluate our algorithm on WideResNet28x10 [42] architecture. Due to computation costs, we limitourselves to 1 and 2 models, with regularization parameter set to 0.001 as in the paper experiments section.Results are reported in Figure 5. We remark this architecture can lead to more robust models, corroboratingthe results from [19].

Models Acc. APGDCE APGDDLR Rob. Acc.1 85.2% 49.9% 50.2% 48.5%2 86.0% 51.5% 52.1% 49.6%

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

Accu

racy

1 models2 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.2

0.4

0.6

0.8

Accu

racy

1 models2 models

Figure 5: On left: Comparison of our algorithm with a standard adversarial training (one model) onWideResNet28x10. We reported the results for the model with the best robust accuracy obtained over twoindependent runs because adversarial training might be unstable. Standard and Robust accuracy (respectivelyin the middle and on right) on CIFAR-10 test images in function of the number of epochs per classifier with 1and 2 WideResNet28x10 models. The performed attack is PGD with 20 iterations and ε = 8/255.

D.4 Overfitting in Adversarial RobustnessWe further investigate the overfitting of our heuristic algorithm. We plotted in Figure 6 the robust accuracyon ResNet18 with 1 to 5 models. The most robust mixture of 5 models against PGD with 20 iterations arrivesat epoch 198, i.e. at the end of the training, contrary to 1 to 4 models, where the most robust mixture occursaround epoch 101. However, the accuracy against AGPD with 100 iterations in lower than the one at epoch

24

101 with global robust accuracy of 47.6% at epoch 101 and 45.3% at epoch 198. This strange phenomenonwould suggest that the more powerful the attacks are, the more the models are subject to overfitting. Weleave this question to further works.

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Accu

racy

1 models2 models3 models4 models5 models

0 25 50 75 100 125 150 175 200Epochs per model

0.0

0.1

0.2

0.3

0.4

0.5

Accu

racy

1 models2 models3 models4 models5 models

Figure 6: Standard and Robust accuracy (respectively on left and on right) on CIFAR-10 test images infunction of the number of epochs per classifier with 1 to 5 ResNet18 models. The performed attack is PGDwith 20 iterations and ε = 8/255. The best mixture for 5 models occurs at the end of training (epoch 198).

E Additional Results

E.1 Equality of Standard Randomized and Deterministic Minimal RisksProposition 7. Let P be a Borel probability distribution on X × Y, and l a loss satisfying Assumption 1,then:

infµ∈M1

+(Θ)R(µ) = inf

θ∈ΘR(θ)

Proof. It is clear that: infµ∈M1+(Θ)R(µ) ≤ infθ∈ΘR(θ). Now, let µ ∈M1

+(Θ), then:

R(µ) = Eθ∼µ(R(θ)) ≥ essinfµ

Eθ∼µ (R(θ))

≥ infθ∈ΘR(θ).

where essinf denotes the essential infimum.

We can deduce an immediate corollary.

Corollary 2. Under Assumption 1, the dual for randomized and deterministic classifiers are equal.

E.2 Decomposition of the Empirical Risk for Entropic Regularization

Proposition 8. Let P̂ := 1N

∑Ni=1 δ(xi,yi). Let l be a loss satisfying Assumption 1. Then we have:

1

N

N∑i=1

supx, d(x,xi)≤ε

Eθ∼µ [l(θ, (x, y))] =

N∑i=1

supQi∈Γi,ε

E(x,y)∼Qi,θ∼µ [l(θ, (x, y))]

25

where Γi,ε is defined as :

Γi,ε :={Qi |

∫dQi =

1

N,

∫cε((xi, yi), ·)dQi = 0

}.

Proof. This proposition is a direct application of Proposition 2 for diracs δ(xi,yi).

E.3 On the NP-Hardness of Attacking a Mixture of ClassifiersIn general, the problem of finding a best response to a mixture of classifiers is in general NP-hard. Let usjustify it on a mixture of linear classifiers in binary classification: fθk(x) = 〈θ, x〉 for k ∈ [L] and λ = 1L/L.Let us consider the `2 norm and x = 0 and y = 1. Then the problem of attacking x is the following:

supτ, ‖τ‖≤ε

1

L

L∑k=1

1〈θk,τ〉≤0

This problem is equivalent to a linear binary classification problem on τ , which is known to be NP-hard.

E.4 Case of Separated Conditional DistribtionsProposition 9. Let Y = {−1,+1}. Let P ∈ M1

+(X × Y). Let ε > 0. For i ∈ Y, let us denote Pi thedistribution of P conditionally to y = i. Let us assume that dX (supp(P1+1), supp(P−1)) > 2ε. Let usconsider the nearest neighbor deterministic classifier : f(x) = d(x, supp(P+1)) − d(x, supp(P−1)) and the0/1 loss l(f, (x, y)) = 1yf(x)≤0. Then f satisfies both optimal standard and adversarial risks: R(f) = 0 andRεadv(f) = 0.

Proof. Let Let denote pi = P(y = i). Then we have

Rεadv(f) = p+1EP+1

[sup

x′, d(x,x′)≤ε1f(x′)≤0

]+ p−1EP−1

[sup

x′, d(x,x′)≤ε1f(x′)≥0

]

For x ∈ supp(P+1), we have, for all x′ such that d(x, x′) 6= 0, f(x′) > 0, then: EP+1

[supx′, d(x,x′)≤ε 1f(x′)≤0

]=

0. Similarly, we have EP−1

[supx′, d(x,x′)≤ε 1f(x′)≥0

]= 0. We then deduce the result.

26