Original:  R OBUSTNESS C ERTIFICATION WITH R EFINEMENT
Revised: B OOSTING R OBUSTNESS C ERTIFICATION OF N EURAL N ETWORKS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: We present a novel approach for veriﬁcation of neural networks  which combines scalable over-approximation methods with precise (mixed integer) linear programming.
Revised: We present a novel approach for the certiﬁcation of neural networks against adversarial perturbations which combines scalable overapproximation methods with precise (mixed integer) linear programming.

Original: This results in signiﬁcantly better precision than state of the art veriﬁers on feed forward neural networks with piecewise linear activation functions.
Revised: This results in signiﬁcantly better precision than state-of-the-art veriﬁers on challenging feedforward and convolutional neural networks with piecewise linear activation functions

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Neural networks are increasingly applied in critical domains such as autonomous driving Bojarski et al. (2016), medical diagnosis Amato et al. (2013), and speech recognition Hinton et al.
Revised: Neural networks are increasingly applied in critical domains such as autonomous driving (Bojarski et al., 2016), medical diagnosis (Amato et al., 2013), and speech recognition (Hinton et al., 2012).

Original: However, it has been shown by Goodfellow et al.
Revised: However, it has been shown by Goodfellow et al.

Original: (2014) that neural networks can be non-robust against adversarial attacks, i.e., small input perturbations cause neural networks to misclassify.
Revised: (2014) that neural networks can be vulnerable against adversarial attacks, i.e., imperceptible input perturbations cause neural networks to misclassify.

Original: To address this challenge and prove  a network is free of adversarial examples (usually, in a region around a given input), recent work has started investigating the use of veriﬁcation techniques.
Revised: To address this challenge and prove that a network is free of adversarial examples (usually, in a region around a given input), recent work has started investigating the use of certiﬁcation techniques.

Original: Current veriﬁers can be broadly classiﬁed in complete or incomplete ones.
Revised: Current veriﬁers can be broadly classiﬁed as either complete or incomplete.

Original: Complete veriﬁers are exact, i.e., if the veriﬁer fails to check a property then the network is nonrobust and vice-versa.
Revised: Complete veriﬁers are exact, i.e., if the veriﬁer fails to certify network robustness against an attack then the network is non-robust with respect to that attack and vice-versa.

Original: Existing complete veriﬁers are based on Mixed Integer Linear Programming (MILP) Lomuscio & Maganti (2017); Fischetti & Jo (2018); Dutta et al. ; Cheng et al. (2017); Tjeng & Tedrake (2017), SMT Katz et al. (2017); Ehlers (2017), or input reﬁnement Wang et al.
Revised: Existing complete veriﬁers are based on Mixed Integer Linear Programming (MILP) (Lomuscio & Maganti, 2017; Fischetti & Jo, 2018; Dutta et al., 2018; Cheng et al., 2017; Tjeng et al., 2019), SMT solvers (Katz et al., 2017; Ehlers, 2017), or input reﬁnement (Wang et al., 2018a;b).

Original: Although precise, theseveriﬁes have limited scalability and can only handle networks with a small number of layers and neurons.
Revised: Although precise, these have limited scalability and can only handle networks with a small number of layers and neurons.

Original: To scale, incomplete veriﬁers usually employ overapproximation methods and hence they are sound (no false negatives) but may fail to verify arobust network.
Revised: To scale, incomplete veriﬁers usually employ overapproximation methods and hence they are sound (no false negatives) but may fail to certify the robustness of a network.

Original: Incomplete veriﬁers are based on methods such as duality Raghunathan et al.

Original: ; Dvijotham et al.

Original: (2018), abstract interpretation Gehr et al. (2018), linear approximations Weng et al. ; Wong & Kolter (2018), combination of linear and non-linear approximation Xiang et al. (2017), or search space discretization Huang et al.
Revised: Incomplete veriﬁers are based on methods such as duality (Dvijotham et al., 2018), abstract interpretation (Gehr et al., 2018; Singh et al., 2018; 2019), linear approximations (Weng et al., 2018; Wong & Kolter, 2018; Zhang et al., 2018), semideﬁnite relaxations (Raghunathan et al., 2018), combination of linear and non-linear approximation (Xiang et al., 2017), or search space discretization (Huang et al., 2017).

Original: Incomplete veriﬁers are more scalable than complete ones, but can suffer from precision loss.
Revised: Incomplete veriﬁers are more scalable than complete ones, but can suffer from precision loss for deeper networks.

Original: In principle, incomplete veriﬁers can betrivially made  complete by iteratively reﬁning the input space; however, at the limit, this eliminates any scalability gains and defeats the purpose of using over-approximations in the ﬁrst place.
Revised: In principle, incomplete veriﬁers can be made asymptotically complete by iteratively reﬁning the input space; however, at the limit, this eliminates any scalability gains and defeats the purpose of using overapproximations in the ﬁrst place.

Original: This work.
Revised: This work: boosting complete and incomplete veriﬁers.

Original: A key challenge then is to design a veriﬁer which improves both, the precision of  incomplete veriﬁers and the scalability of complete ones.
Revised: A key challenge then is to design a veriﬁer which improves both the precision of the incomplete methods and the scalability of complete ones.

Original: In this work, we take a step towards addressing this challenge.
Revised: In this work, we take a step towards addressing this challenge.

Original: We present averiﬁcation method that is fast and exact for small networks, yet can analyze larger networks beyond the reach of complete veriﬁers and compute more precise bounds than incomplete ones.
Revised: We present a method that is fast and exact for small networks, yet can analyze larger networks beyond the reach of complete veriﬁers and compute more precise bounds than incomplete ones.

Original: Our method is based on two key ideas: (i) a combination of stateof-the-art over-approximation methods used in incomplete veriﬁers together with MILP solvers and LP relaxations often employed in complete veriﬁers; (ii) a novel heuristic which points to neurons whose over-approximation bounds should be reﬁned.
Revised: Our method is based on two key ideas: (i) a combination of state-of-the-art overapproximation methods used in incomplete methods, including LP relaxations, together with MILP solvers, that are often employed in complete veriﬁers; (ii) a novel heuristic, which points to neurons whose overapproximated bounds should be reﬁned.

Original: Taken together, these ideas result in a signiﬁcantprecision increase  for deeper networks (up to 9 layers and 1,800 neurons) over state-of-the-art incomplete veriﬁers (no complete veriﬁer can handle these networks, even when run for days).
Revised: Taken together, these ideas result in a signiﬁcant increase of precision for deeper networks (up tolayers and around 35 K neurons) over state [Table] [Figure] Comparison to Tjeng et al.

Original:  Main contributions.
Revised: The recent work of Tjeng et al. (2019) considers a combination of linear programming with a simple interval analysis based over-approximation. In our experimental evaluation, we show that for a complete veriﬁcation task on a 3x50 network, our method is faster, while for a larger 9x200 network their method times out (does not ﬁnish in days), while our veriﬁer ﬁnishes in 14 min. Main contributions.

Original: Our main contributions are:
Revised: Our main contributions are:

Original: • A reﬁnement-based approach for verifying neural network robustness that combines the strengths of over-approximation methods with MILP solvers and LP relaxations.
Revised: • A reﬁnement-based approach for certifying neural network robustness that combines the strengths of fast overapproximation methods with MILP solvers and LP relaxations.

Original: • A novel heuristic for selecting neurons whose bounds should be further reﬁned.
Revised: • A novel heuristic for selecting neurons whose bounds should be further reﬁned.

Revised: • A complete end-to-end implementation of our approach in a system called ReﬁneAI , publicly available at https://github.com/eth-sri/eran.

Original: • An evaluation showing our system is more precise than existing  incomplete veriﬁers on larger networks and faster (while being complete) than complete ones on small networks.[Table] [Figure]
Revised: • An evaluation, showing that ReﬁneAI is more precise than existing state-of-the-art incomplete veriﬁers on larger networks and faster (while being complete) than complete ones on small networks.

Original:  O VERVIEW
Revised:  O VERVIEW

Original: We now demonstrate how our reﬁnement based approach improves  precision on a simple example.
Revised: We now demonstrate how our reﬁnement-based approach improves the precision of a state-of-theart certiﬁer, thus yielding better results.

Original: Our goal is to provide an intuitive understanding of the approach; full formal details are provided in the following section.
Revised: Our goal is to provide an intuitive understanding of our approach; full formal details are provided in the following section.

Original: Consider the simple fully connected  neural network with ReLU activations shown in Fig.
Revised: Consider the simple fully connected feed forward neural network with ReLU activations shown in Fig.

Original: There are two inputs to the network which are both in the range [0 , 1] .
Revised: There are two inputs to the network, which are both in the range [0 , 1] .

Original: The network consists of an input layer, two hidden layers and one output layer.
Revised: The network consists of an input layer, two hidden layers, and one output layer.

Original: Each layer consist of two neurons each.
Revised: Each layer consist of two neurons each.

Original: For our explanation, we separate each neuron into two parts: one represents the output of the afﬁne transformation while the other captures the output of the ReLU activation.
Revised: For our explanation, we separate each neuron into two parts: one represents the output of the afﬁne transformation while the other captures the output of the ReLU activation.

Original: The weights for the afﬁne transformation are represented by weights on the edges.
Revised: The weights for the afﬁne transformation are represented by weights on the edges.

Original: The bias for each node is shown above or below it.
Revised: The bias for each node is shown above or below it.

Original: Our goal is to verify that for any input in [0 , 1] × [0 , 1] , the output at neuron x 13 is greater than the output at x 14 . 
Revised: Our goal is to certify that for any input in [0 , 1] × [0 , 1] , the output at neuron x 13 is greater than the output at x 14 . [Figure]

Original: We now demonstrate  our analysis for this network.
Revised: We now demonstrate how our certiﬁer functions on this network.

Original: We assume that the analysis results after the second afﬁne transformation are reﬁned using a MILP formulation of the network whereas the results after the third afﬁne transformation are reﬁned by an LP formulation of the network.
Revised: We assume that the analysis results after the second afﬁne transformation are reﬁned using a MILP formulation of the network whereas the results after the third afﬁne transformation are reﬁned by an LP formulation of the network.

Original: In the next section, we will explain our heuristic for selectingeither MILP or LP  for each neuron.
Revised: In the next section, we will explain our heuristic for selecting MILP or LP formulations for different neurons in the network.

Original: Our analysis leverages the Zonotope domain Ghorbal et al.
Revised: Our analysis leverages the Zonotope domain (Ghorbal et al., 2009)

Original: transformers also used in AI 2 Gehr et al. (2018), a state of the art analyzer for neural networks.
Revised: transformers also used in DeepZ (Singh et al., 2018), which is a state of the art zonotope certiﬁer for neural network robustness.

Original: The Zonotope domain associates an afﬁne form  with each neuron x in the network:
Revised: The Zonotope domain associates an afﬁne form ˆ x with each neuron x in the network:

Original: [Equation]
Revised: [Equation]

Original: Here, α 0 , β 0 , α i , β i ∈ R are real coefﬁcients and η i ∈ [ a i , b i ] ⊆
Revised: Here, α 0 , β 0 , α i , β i ∈ R are real coefﬁcients and η i ∈ [ a i , b i ] ⊆

Original: [ − 1 , 1] are the noise symbols which are shared between the afﬁne forms for different neurons, which makes the domain relational.
Revised: [ − 1 , 1] are the noise symbols, which are shared between the afﬁne forms for different neurons. The sharing makes the domain relational and thus it is more precise than the non-relational Box domain.

Original: An  element in our analysis is an intersection between a Zonotope (given as conjunction of afﬁne forms) and a bounding box.
Revised: An abstract element in our analysis is an intersection between a Zonotope (given as a list of afﬁne forms) and a bounding box.

Original: Thus, for each neuron x , we keep the afﬁne form ˆ x and an interval [ l x , u x ] .
Revised: Thus, for each neuron x , we keep the afﬁne form ˆ x and an interval [ l x , u x ] .

Original: First layer.
Revised: First layer.

Original: Our analysis starts by setting:
Revised: Our analysis starts by setting

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: [Figure]

Original: [Figure]

Original: representing the inputs at x 1 and x 2 respectively.
Revised: representing the inputs at x 1 and x 2 respectively.

Original: Next, an afﬁne transformation is applied on the inputs resulting in the output:
Revised: Next, an afﬁne transformation is applied on the inputs resulting in the output

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: The Zonotope afﬁne transformer is exact for this transformation so we do not need any reﬁnement here.
Revised: The Zonotope afﬁne transformer is exact for this transformation so we do not need any reﬁnement here.

Original: Next, the Zonotope ReLU transformer is applied.
Revised: Next, the Zonotope ReLU transformer is applied.

Original: We note that as l 3 ≥ 0 , the neuron xprovably takes only positive values.
Revised: We note that as l 3 ≥ 0 , the neuron xprovably takes only non-negative values.

Original: Thus, the ReLU Zonotope transformer outputs ˆ x 5 = ˆ x 3 , l 5 = l 3 , u 5 = l 3 which is the exact result.
Revised: Thus, the ReLU Zonotope transformer outputs ˆ x 5 = ˆ xand we set l 5 = l 3 , u 5 = l 3 which is the exact result.

Original: For x 4 , l 4 < 0 and u 4 > 0 and thus neuron x 4 can take both positive and negative values.
Revised: For x 4 , l 4 < 0 and u 4 > 0 and thus neuron x 4 can take both positive and negative values.

Original: The corresponding output does not have a closed afﬁne form and hence the approximation in blue shown in Fig.
Revised: The corresponding output does not have a closed afﬁne form and hence the approximation in blue shown in Fig.

Original: 2 is used to compute the result, introducing a new noise symbol η 3 ∈ [ − 1 , 1] .
Revised: 2 is used to compute the result. This approximation minimizes the area of the result in the input-output plane and introduces a new noise symbol η 3 ∈ [ − 1 , 1] .

Original: The result is:
Revised: The result is

Original: [Equation]
Revised: [Equation]

Original: Note that due to approximation  x 6 can  take negative values. In contrast, the exact output only takes positive values.
Revised: Note that the Zonotope approximation for x 6 from Fig. 2 permits negative values whereas x 6 can only take non-negative values in the concrete.

Original: This approximation typically accumulates as the analysis progresses deeper into the network, resulting in overall imprecision and failure to prove properties that  actually hold.
Revised: This approximation typically accumulates as the analysis progresses deeper into the network, resulting in overall imprecision and failure to prove properties that may actually hold.

Original: MILP based reﬁnement at second layer.
Revised: MILP-based reﬁnement at second layer.

Original: Next, the analysis handles the second afﬁne transformation and computes:
Revised: Next, the analysis handles the second afﬁne transformation and computes [Equation] and

Original: [Equation]
Revised: [Equation]

Revised: Here, x 7 is provably positive, whereas x 8 can take both positive and negative values.

Original: Due to the approximation for x 6 , the bounds for x 7 and x 8 are imprecise.
Revised: Due to the approximation for x 6 , the bounds for x 7 and x 8 are imprecise.

Revised: Notice that the Zonotope ReLU transformer for x 8 will introduce more imprecision and although the ReLU transformer for provably positive inputs such as x 7 does not lose precision with respect to the input, it still propagates the imprecision in the computation of the abstract values for x 7 .

Original: Thus,  in our method wenow reﬁne the bounds  by formulating the network up to the second afﬁne transformation as a MILP instance based on a formulation from Tjeng & Tedrake (2017) and compute bounds for x 7 and x 8 , respectively.
Revised: Thus, to reduce precision loss, in our method we reﬁne the bounds for both x 7 and x 8 by formulating the network up to the second afﬁne transformation as a MILP instance based on a formulation from Tjeng et al. and compute bounds for x 7 and x 8 , respectively.

Original: The MILP solver improves the lower bounds for x 7 and x 8 to 1 and − 1 , respectively, which then updates the corresponding lower bounds.
Revised: The MILP solver improves the lower bounds for x 7 and x 8 to 1 and − 1 , respectively, which then updates the corresponding lower bounds in our abstraction, i.e., l 7 = 1 and l 8 = − 1 .

Original: Next, the ReLU transformer is applied.
Revised: Next, the ReLU transformer is applied.

Original: Since x 7 is provably positive, we get ˆ x 9 =
Revised: Since x 7 is provably positive, we get ˆ x 9 =

Original: ˆ x 7 , l 9 = l 7 , u 9 = u 7 .
Revised: ˆ x 7 , l 9 = l 7 , u 9 = u 7 .

Original: We note that x 8 can take both positive and negative values and is therefore approximated.
Revised: We note that x 8 can take both positive and negative values and is therefore approximated.

Original: However, the ReLU transformer now uses the reﬁned bounds instead of the original bounds and thus the approximation shown in green from Fig.is used.
Revised: However, the ReLU transformer now uses the reﬁned bounds instead of the original bounds and thus the approximation shown in green from Fig. 2 is used.

Original: This approximation has lesser area in the input-output plane compared to the blue one and thus reduces the approximation.
Revised: This approximation has smaller area in the input-output plane compared to the blue one and thus reduces the approximation error.

Original: The result is:
Revised: The result is

Original: [Equation]
Revised: [Equation]

Original: LP based reﬁnement at ﬁnal layer.
Revised: LP-based reﬁnement at ﬁnal layer.

Original: Continuing with the analysis, we now process the ﬁnal afﬁne transformation, which yields
Revised: Continuing with the analysis, we now process the ﬁnal afﬁne transformation, which yields

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: Due to the approximations from previous layers the computed values can be imprecise.
Revised: Due to the approximations from previous layers, the computed values can be imprecise.

Revised: We note that as the analysis proceeds deeper into the network, reﬁning bounds with MILP becomes expensive.

Original: Thus, we reﬁne the bounds by encoding the network up to the third afﬁne transformation using the  LP formulation from Ehlers (2017) and compute the bounds for x 11 and x 12 , respectively.
Revised: Thus, we reﬁne the bounds by encoding the network up to the third afﬁne transformation using the faster LP relaxation of the network based on Ehlers (2017) and compute the bounds for x 11 and x 12 , respectively.

Original: This leads to the following results: l 11 = 3 .
Revised: This leads to the following results: l 11 = 3 .

Original: 25 , l 12 = 2 , and u 12 = 3 .
Revised: 25 , l 12 = 2 , and u 12 = 3 .

Original: As both x 13 and x 14 are provably positive, the subsequent ReLU transformation sets ˆ x 13 =
Revised: As both x 13 and x 14 are provably positive, the subsequent ReLU transformations set ˆ x 13 =

Original: ˆ x 11 , l 13 = l 11 , u 13 = u 11 and ˆ x 14 =
Revised: ˆ x 11 , l 13 = l 11 , u 13 = uand ˆ x 14 =

Original: ˆ x 12 , l 14 = l 12 , u 14 = u 12 .
Revised: ˆ x 12 , l 14 = l 12 , u 14 = u 12 .

Original: Proving robustness.
Revised: Proving robustness.

Original: Since the lower bound l 13 for x 13 is greater than the upper bound u 14 for x 14 , our analysis can prove that the given neural network provides the same label for all inputs in [0 , 1] × [0 , 1] and is thus robust.
Revised: Since the lower bound l 13 for x 13 is greater than the upper bound u 14 for x 14 , our analysis can prove that the given neural network provides the same label for all inputs in [0 , 1] × [0 , 1] and is thus robust.

Original: In contrast, AI 2 without our reﬁnement would computeˆ x 13 = 4 . 95 + 0 . 6 ·
Revised: In contrast, DeepZ without our reﬁnement would compute

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: + 0 . 6 · η 1 + 1 . 8 · η 2 − 0 . 6 · η 3 + 0 .

Original:  4 , l 13 = 1 .
Revised: 3 · η 4 , l 13 = 1 .

Original: 65 , u 13 = 8 .and
Revised: 65 , u 13 = 8 . and

Original: ˆ x 14 = [Equation] [Equation]
Revised: ˆ x 14 = 2 . 55 + 0 . 15 · η 1 + 0 .

Original: [Equation]
Revised: 45 · η 2 − 0 .

Original: [Equation]
Revised: 15 · η 3 − 0 .

Original: 3  , l 14 = 1 . 5 , u 14 = 2 . 6 .
Revised: 3 · η 4 , l 14 = 1 . 5 , u 14 = 3 . 6 .

Original: As a result, AI 2 fails to prove that x 13 is greater than x 14 , and thus fails to prove robustness. 
Revised: As a result, DeepZ fails to prove that x 13 is greater than x 14 , and thus fails to prove robustness. Generalization to other abstractions.

Revised: We note that our reﬁnement-based approach is not restricted to the Zonotope domain.

Revised: It can be easily extended for reﬁning the results computed by other abstractions such as Polyhedra (Singh et al., 2017) or the abstraction used in DeepPoly (Singh et al., 2019).

Original:   O UR A PPROACH
Revised: For example, the ReLU transformer in Singh et al. (2019) also depends on the bounds of input neurons and thus it will beneﬁt from the precise bounds computed using our reﬁnement. Since DeepPoly often produces more precise results than DeepZ , we believe that a straightforward extension of this work to DeepPoly will further improve certiﬁcation results.  O UR A PPROACH

Original: In this section we describe our approach in more formal terms.
Revised: In this section, we describe our approach in more formal terms.

Revised: As in the previous section, we will consider afﬁne transformations and ReLU activations as separate layers.

Original: As illustrated earlier, the key idea will be to combine abstract interpretation with exact and inexact MILP formulation of the network which is then solved using a solver so to compute more precise results for neuron bounds.
Revised: As illustrated earlier, the key idea will be to combine abstract interpretation with exact and inexact MILP formulation of the network which is then solved using a solver, in order to compute more precise results for neuron bounds.

Original: We begin by describing the  abstract interpretation ingredients.
Revised: We begin by describing the ingredients of abstract interpretation.

Original: Let A n be an abstract domain  (i.e., some set whose elements can be encoded symbolically) with a bottom element ⊥ ∈
Revised: Our approach requires an abstract domain A n (i.e., some set whose elements can be encoded symbolically) such as Interval, Zonotope, the abstraction in DeepPoly , or Polyhedra. An abstract domain has a bottom element ⊥ ∈

Original: A n . We further assume the following:
Revised: A n as well as the following components:

Original: • A  concretization function γ n : A n → P ( R n ) that associates with each abstract element a ∈ A n a set of concrete vectors from R n .
Revised: • A (potentially non-computable) concretization function γ n : A n → P ( R n ) that associates with each abstract element a ∈ A n a set of concrete points from R n .

Original: In particular, γ n ( ⊥ ) = ∅ .
Revised: We have, γ n ( ⊥ ) = ∅ .

Original: ( γ is not required to be computable.)

Original: • An abstraction function α n : P ( R n ) → A n , where X ⊆ γ n ( α n ( X )) for all X ∈ P ( R n ) .
Revised: • An abstraction function α n : P ( R n ) → A n , where X ⊆ γ n ( α n ( X )) for all X ∈ P ( R n ) .

Original: We assume that α n ( (cid:81) i [ l i , u i ]) can be computed for all l , u ∈ R n  .
Revised: We assume that α n ( (cid:81) i [ l i , u i ]) is computable for all l , u ∈ R n , where (cid:89) i [ l i , u i ] = { x ∈ R n | l i ≤ x i ≤ u i } .

Original: • A bounding box function ι n : A n → R n × R n , where γ n ( a ) ⊆
Revised: • A bounding box function ι n : A n → R n × R n , where γ n ( a ) ⊆

Original: (cid:81) i
Revised: (cid:81) i

Original: [ l i , y i ] for ( l , u ) =
Revised: [ l i , y i ] for ( l , u ) =

Original: ι n ( a ) for all a ∈ A n .
Revised: ι n ( a ) for all a ∈ A n .

Original: • A meet operation a (cid:117) L for each a ∈ A n and linear constraints L over n real variables, where { x ∈ γ n ( a ) | L ( x ) } ⊆
Revised: • A meet operation a (cid:117) L for each a ∈ A n and linear constraints L over n real variables, where { x ∈ γ n ( a ) | L ( x ) } ⊆

Original: γ n ( a (cid:117) L ) .
Revised: γ n ( a (cid:117) L ) .

Original: • An afﬁne abstract transformer T # x
Revised: • An afﬁne abstract transformer T # x

Original: (cid:55)→
Revised: (cid:55)→

Original: Ax + b : A m → A n for eachafﬁne transformation  ( x (cid:55)→
Revised: Ax + b : A m → A n for each transformation of the form ( x (cid:55)→

Original: Ax + b ): R m → R n , where { Ax + b | x ∈ γ n ( a ) } ⊆
Revised: Ax + b ): R m → R n , where { Ax + b | x ∈ γ n ( a ) } ⊆

Original: γ n ( T # x (cid:55)→ Ax + b ) for all a ∈ A m .
Revised: γ n ( T # x (cid:55)→ Ax + b ( a )) for all a ∈ A m .

Original: • A ReLU abstract transformer T # ReLU | (cid:81) i
Revised: • A ReLU abstract transformer T # ReLU | (cid:81) i

Original: [ li,ui ] : A n → A n , where { ReLU ( x ) | x ∈ γ  ( a ) ∩ (cid:81) i
Revised: [ li,ui ] : A n → A n , where { ReLU ( x ) | x ∈ γ n ( a ) ∩ (cid:89) i

Original: [ l i , u i ] } ⊆T #
Revised: [ l i , u i ] } ⊆

Original: [Table] ) for all abstract elements a ∈ A n and for all lower and upper bounds l , u ∈ R n on input activations.
Revised: T # ReLU | (cid:81) i [ li,ui ] ( a ) for all abstract elements a ∈ A n and for all lower and upper bounds l , u ∈ R n on input activations of the ReLU operation.

Original: Abstract interpretation.
Revised: Abstract interpretation.

Original: As ﬁrst shown by Gehr et al.
Revised: As ﬁrst shown by Gehr et al.

Original: (2018), any such abstract domain induces a method for robustness certiﬁcation ofclassiﬁcation based neural networks with ReLU activations.
Revised: (2018), any such abstract domain induces a method for robustness certiﬁcation of neural networks with ReLU activations.

Original: For example, assume that we want to certify that a given neural network f : R m → R n considers class i more likely than class j for all inputs  x(cid:48) with ||  x(cid:48) − x || ∞ ≤ (cid:15) for a given x and (cid:15) .
Revised: For example, assume that we want to certify that a given neural network f : R m → R n considers class i more likely than class j for all inputs ¯ x with || ¯ x − x || ∞ ≤ (cid:15) for a given x and (cid:15) .

Original: We can ﬁrst use the function α  to compute a symbolic over-approximation of the set of possible input activations  x(cid:48) , namely
Revised: We can ﬁrst use the function α m to compute a symbolic overapproximation of the set of possible input activations ¯ x , namely

Original: [Equation]
Revised: [Equation]

Original: Given that the neural network can be written as a composition of afﬁne functions and ReLU layers we can then propagate the abstract element a in through the corresponding abstract transformers to obtain a symbolic over-approximation a out of the output activations of the neural network.
Revised: Given that the neural network can be written as a composition of afﬁne functions and ReLU layers, we can then propagate the abstract element a in through the corresponding abstract transformers to obtain a symbolic overapproximation a out of the concrete outputs of the neural network.

Original: For example, if the neural networkis a single-layer perceptron f ( x ) = A (cid:48) · ReLU ( Ax + b )+ b (cid:48)  with h hidden neurons, we ﬁrst compute a (cid:48)
Revised: For example, if the neural network f ( x ) = A (cid:48) · ReLU ( Ax + b ) + b (cid:48) has a single hidden layer with h hidden neurons, we ﬁrst compute a (cid:48)

Original: =
Revised: =

Original: T # x (cid:55)→ Ax + b ( a in ) , which is a symbolic over-approximation of the input to the ReLU activation function.
Revised: T # x (cid:55)→ Ax + b ( a in ) , which is a symbolic overapproximation of the input to the ReLU activation function.

Original: We then compute ( l , u ) =
Revised: We then compute ( l , u ) =

Original: ι h ( a (cid:48) ) to obtain opposite corners of a bounding box of all possible ReLU input activations, such that we can apply the ReLU abstract transformer:
Revised: ι h ( a (cid:48) ) to obtain opposite corners of a bounding box of all possible ReLU input activations, such that we can apply the ReLU abstract transformer:

Original: [Equation]
Revised: [Equation]

Original: Finally, we apply the afﬁne abstract transformer again to obtain a out = T 
Revised: Finally, we apply the afﬁne abstract transformer again to obtain a out = T # x (cid:55)→

Original: [Equation]
Revised: [Equation]

Original:  Using our assumptions, we can conclude that the set γ n ( a out ) contains all output activations that f can possibly produce when given any of the inputs  x(cid:48) .
Revised: ( a [Equation] Using our assumptions, we can conclude that the set γ n ( a out ) contains all output activations that f can possibly produce when given any of the inputs ¯ x .

Original: Therefore, if a out (cid:117) ( y i ≤ y j ) = ⊥ , we havesuccessfully proved our property: for all  x(cid:48) , the neural network considers class i more likely than class j .
Revised: Therefore, if a out (cid:117) ( x i ≤ x j ) = ⊥ , we have proved the property: for all ¯ x , the neural network considers class i more likely than class j .

Original: Incompleteness.
Revised: Incompleteness.

Original: While this approach is sound (i.e., whenever we prove the property, it actually holds), it is incomplete (i.e., we might not prove the property, even if it holds), because the abstract transformers produce a superset of the set of possible activations.
Revised: While this approach is sound (i.e., whenever we prove the property, it actually holds), it is incomplete (i.e., we might not prove the property, even if it holds), because the abstract transformers produce a superset of the set of concrete outputs that the corresponding concrete transformers produce.

Original: This is particularly quite imprecise for deep neural networks, because the over-approximations introduced in each layer accumulate.
Revised: This is in particular quite imprecise for deep neural networks, because the overapproximations introduced in each layer accumulate.

Original: Reﬁnement.
Revised: Reﬁnement.

Original: To combat spurious over-approximation, we use mixed integer linear programming (MILP) to compute reﬁned lower and upper bounds l , u after applying each  abstract transformer (except  the ﬁrst).
Revised: To combat spurious overapproximation, we use mixed integer linear programming (MILP) to compute reﬁned lower and upper bounds l , u after applying each afﬁne abstract transformer (except for the ﬁrst layer).

Original: We then reﬁne the abstract element using the meet operator and the linear constraints l i ≤ x i ≤ u i for all input activations i , i.e., we replace the current abstract element a by a (cid:48)
Revised: We then reﬁne the abstract element using the meet operator and the linear constraints l i ≤ x i ≤ u i for all input activations i , i.e., we replace the current abstract element a by a (cid:48)

Original: = a (cid:117) ( (cid:86) i l i ≤ x i ≤ u i ) , and continue analysis with the reﬁned abstract element.
Revised: = a (cid:117) ( (cid:86) i l i ≤ x i ≤ u i ) , and continue analysis with the reﬁned abstract element.

Original: Furthermore, and importantly, for ReLU, we obtain a reﬁned abstract transformer using the new (and tighter) lower and upper bounds.
Revised: Furthermore, and importantly, for ReLU, we obtain a reﬁned abstract transformer using the new (and tighter) lower and upper bounds.

Original: To enable reﬁnement, we augment each abstract element with variables and constraints.
Revised: To enable reﬁnement, we augment each abstract element with variables and constraints.

Original: There is one variable for each neuron, and the constraints fully capture the behaviour of the neural network up to the last layer whose abstract transformer has been executed.
Revised: There is one variable for each neuron, and the constraints fully capture the behaviour of the neural network up to the last layer whose abstract transformer has been executed.

Original: We write x ( k ) i to denote the variable corresponding to the activation of the i -th neuron in the k -th layer, where the input layer has k = 0 .
Revised: We write x ( k ) i to denote the variable corresponding to the activation of the i -th neuron in the k -th layer, where the input layer has k = 0 .

Original: Similarly, we will write l ( k ) i and u ( k ) i to denote the best derived lower and upper bounds for this neuron.
Revised: Similarly, we will write l ( k ) i and u ( k ) i to denote the best derived lower and upper bounds for this neuron.

Original: From the input layer, we get constraints of the form a ≤ x (0) i ≤ b{+u 
 i+} , from afﬁne layers, we get constraints of the form x ( k )
Revised: From the input layer, we get constraints of the form l 0 i ≤ x (0) i ≤ {+u 
 i+} , from afﬁne layers, we get constraints of the form x ( k )

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: ) j and from ReLU layers we get constraints of the form x ( k )
Revised: and from ReLU layers we get constraints of the form x [Equation]

Original: i
Revised: k

Original: [Equation]
Revised: [Equation]

Original: MILP.
Revised: MILP.

Original: Let ϕ ( k ) denote the conjunction of all constraints up to and including those from layer k .
Revised: Let ϕ ( k ) denote the conjunction of all constraints up to and including those from layer k .

Original: To obtain the best possible lower and upper bounds for layer k with p neurons, we need to solve the following 2  p optimization problems:
Revised: To obtain the best possible lower and upper bounds for layer k with p neurons, we need to solve the following 2 · p optimization problems:

Original: [Equation]
Revised: [Equation]

Original: [Table] As was shown by Tjeng & Tedrake (2017), those optimization problems can be encoded exactly as MILPs, such that the bounds can be computed with off-the-shelf MILP solvers.
Revised: As was shown by Tjeng et al. (2019), those optimization problems can be encoded exactly as MILPs, such that the bounds can be computed with off-the-shelf MILP solvers.

Original: Note that the encoding of subsequent layers depends on the bounds computed in previous layers, where tighter bounds produce simpler MILP instances.
Revised: Note that the encoding of subsequent layers depends on the bounds computed in previous layers, where tighter bounds produce simpler MILP instances.

Original: LP relaxation.
Revised: LP relaxation.

Original: While not introducing any approximation, unfortunately, current MILP solvers do not scale to larger deep neural networks.
Revised: While not introducing any approximation, unfortunately, current MILP solvers do not scale to larger deep neural networks.

Revised: It becomes increasingly more expensive to reﬁne bounds with the MILP-based formulation as the analysis proceeds deeper into the network.

Original: However, for soundness it is not crucial that the produced bounds are the best possible: for example, plain abstract interpretation uses sound bounds produced by the bounding box function ι instead.
Revised: However, for soundness it is not crucial that the produced bounds are the best possible: for example, plain abstract interpretation uses sound bounds produced by the bounding box function ι instead.

Original: We explore the trade-off between precision and scalability by also considering an intermediate method, which is faster than exact MILP, but also more precise than abstract interpretation.
Revised: Therefore, for deeper layers in the network, we explore the trade-off between precision and scalability by also considering an intermediate method, which is faster than exact MILP, but also more precise than abstract interpretation.

Original: We relaxa subset of the constraints in ϕ ( k ) in the same way as Ehlers (2017) to obtain a set of weaker linear constraints ϕ ( k ) LP .
Revised: We relax the constraints in ϕ ( k ) in the same way as Ehlers (2017) to obtain a set of weaker linear constraints ϕ ( k ) LP .

Original: We then use the solver to solve the relaxed optimization problems that are constrained by ϕ ( k ) LP instead of ϕ ( k ) , producing possibly looser bounds l ( k ) and u ( k ) .
Revised: We then use the solver to solve the relaxed optimization problems that are constrained by ϕ ( k ) LP instead of ϕ ( k ) , producing possibly looser bounds l ( k ) and u ( k ) .

Original: Note that the encoding of subsequent layers depends on the bounds computed in previous layers, where tighter bounds reduce the amount of newly introduced approximation.
Revised: Note that the encoding of subsequent layers depends on the bounds computed in previous layers, where tighter bounds reduce the amount of newly introduced approximation.

Original: Anytime MILP relaxation.
Revised: Anytime MILP relaxation.

Original: The MILP solvers usually provide the option to provide an explicit timeout after which the solver must terminate.
Revised: MILP solvers usually provide the option to provide an explicit timeout after which the solver must terminate.

Original: In return, the solver may not be able to solve the instance exactly, but it will instead provide lower and upper bounds on the objective function in a best-effort fashion.
Revised: In return, the solver may not be able to solve the instance exactly, but it will instead provide lower and upper bounds on the objective function in a best-effort fashion.

Original: This provides another way to compute sound but inexact bounds l ( k ) and u ( k ) .
Revised: This provides another way to compute sound but inexact bounds l ( k ) and u ( k ) .

Original: In practice, we choose a fraction α ∈ (0 , 1] of neurons in a given layer k and compute bounds for them using MILP with a timeout T in a ﬁrst step.
Revised: In practice, we choose a fraction α ∈ (0 , 1] of neurons in a given layer k and compute bounds for them using MILP with a timeout T in a ﬁrst step.

Original: In the second step, for a fraction δ ∈ [0 , 1 − α ] of neurons in the layer, we set the timeout to β · T , where T is the average time taken by the MILP solver to solve one of the instances from the ﬁrst step and β ∈ [0 , 1] is a parameter.
Revised: In the second step, for a fraction δ ∈ [0 , 1 − α ] of neurons in the layer, we set the timeout to β · T , where T is the average time taken by the MILP solver to solve one of the instances from the ﬁrst step and β ∈ [0 , 1] is a parameter.

Original: Neuron selection heuristic.
Revised: Neuron selection heuristic.

Original: To select the α -fraction of neurons for the ﬁrst step of the anytime
Revised: To select the α -fraction of neurons for the ﬁrst step of the anytime

Original: MILP relaxation for the k -th layer, we rank the neurons.
Revised: MILP relaxation for the k -th layer, we rank the neurons.

Original: If the next layer is a ReLU layer, we ﬁrst ignore all neurons whose activations can be proven to be non-positive using abstract interpretation (i.e., using the bounds produced by ι ), because in this case it is already known that ReLU will map the activation to 0 .
Revised: If the next layer is a ReLU layer, we ﬁrst ignore all neurons whose activations can be proven to be non-positive using abstract interpretation (i.e., using the bounds produced by ι ), because in this case it is already known that ReLU will map the activation to 0 .

Original: The remaining neurons are ordered in up to two different ways, once by width (i.e. neuron i has key u ( k ) i − l ( k ) i ), and possibly once by the sum of absolute output weights.
Revised: The remaining neurons are ordered in up to two different ways, once by width (i.e. neuron i has key u ( k ) i − l ( k ) i ), and possibly once by the sum of absolute output weights.

Original: i.e., if the next layer is a fully connected layer x
Revised: i.e., if the next layer is a fully connected layer x

Original: (cid:55)→ Ax + b , the key of neuron i is (cid:80)
Revised: (cid:55)→ Ax + b , the key of neuron i is (cid:80)

Original: j
Revised: j

Original: | A
Revised: | A

Original: i,j | .
Revised: i,j | .

Original: If the next layer is a ReLU layer, we skip the ReLU layer and use the weights from the fully connected layer that follows it (if any).
Revised: If the next layer is a ReLU layer, we skip the ReLU layer and use the weights from the fully connected layer that follows it (if any).

Original: The two ranks of a neuron in both orders are added, and the α -fraction with smallest rank sum is selected and their bounds  reﬁned with a timeout of T whereas the next δ -fraction of neurons are reﬁned with a timeout of β · T .
Revised: The two ranks of a neuron in both orders are added, and the α -fraction with smallest rank sum is selected and their bounds are reﬁned with a timeout of T whereas the next δ -fraction of neurons are reﬁned with a timeout of β · T .

Original: End-to-end approach.
Revised: End-to-end approach.

Original: To certify robustness of deep neural networks, we combine MILP, LP relaxation, and abstract interpretation.
Revised: To certify robustness of deep neural networks, we combine MILP, LP relaxation, and abstract interpretation.

Original: We ﬁrst pick numbers of layers k MILP , k LP , k AI that sum to the total number of layers of the neural network.
Revised: We ﬁrst pick numbers of layers k MILP , k LP , k AI that sum to the total number of layers of the neural network.

Original: For the analysis of theﬁrst k MILP layers, we reﬁne bounds using anytime MILP relaxation with the neuron selection heuristic.
Revised: For the analysis of the k MILP layers, we reﬁne bounds using anytime MILP relaxation with the neuron selection heuristic.

Original: As an optimization, we do not perform reﬁnement after theﬁrst abstract transformer  in case it is an afﬁne transformation and the abstract domain computes the tightest possible bounding box for an afﬁne transformation of a box (this is always the case in our experiments).
Revised: As an optimization, we do not perform reﬁnement after the abstract transformer for the ﬁrst layer in case it is an afﬁne transformation as the abstract domain computes the tightest possible bounding box for an afﬁne transformation of a box (this is always the case in our experiments).

Original: For the k LP layers after that, we reﬁne bounds usingboth MILP and LP relaxation, i.e., the ﬁrst k MILP layers are encoded as MILP and the next k LP use the LP relaxation.
Revised: For the k LP layers, we reﬁne bounds using LP relaxation combined with the neuron selection heuristic, i.e., the network up to the layer to be reﬁned is encoded using linear constraints.

Original: For the ﬁnal k AI layers, we use abstract interpretation without additional reﬁnement (however, this also beneﬁts from reﬁnement that was performed in previous layers), and compute the bounds using ι .
Revised: For the remaining k AI layers, we use abstract interpretation without additional reﬁnement (however, this also beneﬁts from reﬁnement that was performed in previous layers), and compute the bounds using ι .

Original: Final property veriﬁcation.
Revised: Final property certiﬁcation.

Original: In case the reﬁned abstract interpretation is already able to verify the ﬁnal property on its own, we terminate. Otherwise, we can encode the ﬁnal veriﬁcation problem using ϕ ( k ) LP .
Revised: Let k be the index of the last layer and p be the number of output classes. We can encode the ﬁnal certiﬁcation problem using the output abstract element a out obtained after applying the abstract transformer for the last layer in the network.

Original: If we want to prove that class i is assigned a higher probability than class j , it sufﬁces to show thatthe constraints ϕ ( k ) LP ( x (0)1 , . . .
Revised: If we want to prove that class i is assigned a higher probability than class j , it sufﬁces to show that

Original: , x ( k ) p )
Revised: a out (cid:117) ( x ( k ) i

Original: ∧ x ( k ) i ≤ x ( k ) j are unsatisﬁable, where k is the index of the last layer and p is the number of output classes.
Revised: [Equation]

Original: If this fails as well, one can resort to complete veriﬁcation using MILP: the property is satisﬁed if and only if  ϕ ( k ) ( x (0)1 , . . .
Revised: If this fails, one can resort to complete veriﬁcation using MILP: the property is satisﬁed if and only if the set of constraints ϕ ( k ) ( x (0)1 , . . .

Original: , x ( k ) p ) ∧  x ( k ) i ≤ x ( k ) j  is unsatisﬁable.
Revised: , x ( k ) p ) ∧ ( x ( k ) i ≤ x ( k ) j ) is unsatisﬁable.

Original:  E VALUATION
Revised:  E VALUATION

Original: We now evaluate the effectiveness of our approach for the problem of robustness veriﬁcation of
Revised: We now evaluate the effectiveness of our approach for the problem of robustness veriﬁcation of

Original: ReLU-based feed forward neural networks.
Revised: ReLU-based feedforward and convolutional neural networks.

Revised: Our results show that our approach enables faster complete veriﬁcation than the state-of-the-art complete certiﬁer from Tjeng et al.

Revised: (2019) and produces more precise results than state-of-the-art incomplete certiﬁers: DeepZ (Singh et al., 2018) and DeepPoly (Singh et al., 2019), when complete certiﬁcation becomes infeasible.

Original: We implemented our approach in the form of a Python analyzer which we call ReﬁneAI.
Revised: We implemented our approach in the form of a Python analyzer which we call ReﬁneAI .

Revised: ReﬁneAI uses the Zonotope transformers from DeepZ for performing abstract interpretation over the networks and Gurobi (Gurobi Optimization, LLC, 2018) for solving MILP and LP instances.

Original: ReﬁneAI uses the Zonotope and Interval implementation from the ELINA eli; Singh et al. (2017) library.
Revised: ReﬁneAI builds on the Zonotope and Interval abstractions implemented in the ELINA library (eli, 2018; Singh et al., 2017).

Original: Gurobi gur is used for solving MILP instances.
Revised: All of our code, neural networks, and images used in our experiments are publicly available at https://github.com/eth-sri/eran.

Original: Evaluation datasets.
Revised: Evaluation datasets.

Original: We used the popular MNIST Lecun et al. (1998) dataset of hand-written digits for our experiments.
Revised: We used the popular MNIST (Lecun et al., 1998) and CIFAR10 (Krizhevsky, 2009) datasets in our experiments.

Original: The dataset includes a training set of 60 000 and a testing set of 10gray scale images.
Revised: Both datasets include a training set of 60 000 and a testing set of 10 000 images.

Original: All images  are  of size 28 × 28 pixels.
Revised: All images in the MNIST dataset are grayscale images of size 28 × 28 pixels whereas those in CIFAR10 are RGB images of size 32 × 32 . [Table]

Original: Neural networks.
Revised: Neural networks.

Original: We use 5 different feed forward networks (FNNs)  withthe ReLU activation function trained for the MNIST dataset.
Revised: Table 1 shows 11 different MNIST and CIFAR10 feedforward (FNNs) and convolutional networks (CNNs) with ReLU activations used in our experiments.

Original: The sizes of the FNNs are: (a) 3 layers with 50 units each, (b) 5 layers with 100 units each, (c) 6 layers with 100 units each, (d) 6 layers with 200 units each, and (e) 9 layers with 200 units each.

Original: Among these, the 5 × 100 network is trained to be robust using the adversarial training from Mirman et al. (2018) whereas the other networks were taken from Gehr et al. and are not trained to be robust.
Revised: Out of thesewere trained to be robust against adversarial attacks using DiffAI (Mirman et al., 2018) whereas the remaining 7 did not have any adversarial training. The largest network in our experiments contains > 88 K neurons whereas the deepest network contains 9 layers.

Original: Robustness properties.
Revised: Robustness properties.

Original: We consider the adversarial region deﬁned by the L ∞ -norm Carlini &
Revised: We consider the adversarial region deﬁned by the L ∞ -norm (Carlini &

Original: Wagner (2017) perturbation.
Revised: Wagner, 2017) perturbation.

Original: The region is parameterized by (cid:15) ∈ R and contains all images in which each pixel x(cid:48) i hasa distance of at most (cid:15) from the corresponding pixel x i in the original input image x .
Revised: The region is parameterized by (cid:15) ∈ R and contains all images in which each pixel x i has distance ≤ (cid:15) from the corresponding pixel x i in the original input image x .

Original: Our goal is to verify that the neural network produces the same label  on all images in the adversarial region as on the original image.
Revised: Our goal is to certify that the neural network produces the same label as on the original image on all images in the adversarial region.

Original: The adversarial attack considered here is untargeted and therefore stronger than the targeted attack in Weng et al. ; Tjeng & Tedrake (2017) where the goal is to prove that the score for the correct label is more than some other label on all images in the adversarial region Benchmarks.
Revised: Thus, the adversarial attack considered here is untargeted.

Original: We selected the ﬁrst 100 images from the MNIST test set.

Original: Any image on which the neural network misclassiﬁed without perturbation was not considered for evaluation.

Original: Experimental setup.
Revised: Experimental setup.

Original: All experiments for the 3 × 50 FNN  were carried out on a 2.6 GHz Intel
Revised: All experiments for the 3 × 50 FNN and all CNNs were carried out on a 2.

Original:  Xeon CPU E5-2690 with 512 GB of main memory; the remaining FNNs were evaluated on a 3.
Revised: GHz Intel Xeon CPU E5-2690 with 512 GB of main memory; the remaining FNNs were evaluated on a 3.

Original:  GHz 10 Core Intel i9-7900X Skylake CPU with a main memory of 64 GB.
Revised: 3 GHz 10 Core Intel i9-7900X Skylake CPU with a main memory of 64 GB Benchmarks.

Original: For each network, we chose an (cid:15) for which AI 2 veriﬁed <
Revised: We selected the ﬁrst 100 images from the test set of both MNIST and CIFAR10. Any image on which the neural network misclassiﬁed without perturbation was not considered for robustness certiﬁcation in our experiments. We consider complete certiﬁcation with ReﬁneAI on the 3 × 50 network and chose an (cid:15) for which the incomplete certiﬁer

Original:  40% of all candidate images.
Revised: DeepZ certiﬁed < 40% of all candidate images.

Original: We used a time limit of 40 minutes per run in our experiments. We experimented with different values of the analysis parameters k MILP , k LP , k AI , α, δ, β, T and chose those values that offered the best tradeoff between performance and precision for the robustness veriﬁcation of larger neural networks.
Revised: We consider incomplete certiﬁcation for the remaining networks and chose an (cid:15) for which complete certiﬁcation with ReﬁneAI becomes infeasible.

Original: Complete veriﬁcation of small network.
Revised: 4. C OMPLETE CERTIFICATION

Original: We ﬁrst consider the exact robustness veriﬁcation of  3 × 50 networks against L ∞ -norm attack with (cid:15) = 0 .
Revised: We ﬁrst consider the exact robustness certiﬁcation of the MNIST 3 × 50 networks against L ∞ -norm attack with (cid:15) = 0 .

Original: AI 2 veriﬁes 37% of images in the test set for this value of (cid:15) .
Revised: DeepZ veriﬁes 37% of images in the test set for this value of (cid:15) .

Original: ReﬁneAI  runs the analysiswith AI 2 on the whole network collecting the bounds for all neurons in the network.
Revised: ReﬁneAI ﬁrst runs DeepZ analysis on the whole network collecting the bounds for all neurons in the network.

Original: If AI 2 does not verify the property then the  boundsfrom AI 2 are used to encode the robustness veriﬁcation as a MILP problem as discussed in section 3.
Revised: If DeepZ fails to certify, then the collected bounds are used to encode the robustness certiﬁcation as a MILP instance (discussed in section 3).

Original: We compare ReﬁneAI against the approach from Tjeng & Tedrake (2017) (state-of-the-art for complete veriﬁers) which also uses a MILP formulation and uses a presolve analysis  to formulate the veriﬁcation as a
Revised: We compare ReﬁneAI against the approach from Tjeng et al. (2019) which is a state-of-the-art complete certiﬁer. This approach is also MILP-based like ours, but it uses Interval analysis and LP to determine neuron bounds.

Original: MILP problem.

Original: We implemented the Interval analysis and LP-based analysis to determine the initial bounds.
Revised: We implemented the Interval analysis and LP-based analysis to determine the initial bounds.

Original: We note that LP analysis is the default presolve in Tjeng & Tedrake (2017).

Original:  If LP analysis (or Interval analysis) can already verify an image then the MILP solver is not called.
Revised: We call the MILP solver only If LP analysis (or Interval analysis) fails to certify.

Original: All complete veriﬁers verify 85% of  images.
Revised: All complete certiﬁers certify the neural network to be robust against L ∞ -norm perturbations for 85% of the images.

Original: The average runtimes of ReﬁneAI, MILP with presolve bounds from
Revised: The average runtimes of ReﬁneAI ,

Original:  Interval analysis, and MILP with bounds from LP analysis are 28 , 123 , and 35 seconds respectively.
Revised: MILP with presolve bounds from Interval analysis, and MILP with bounds from LP analysis are 28 , 123 , and 35 seconds respectively.

Revised: Based on our result, we believe that the Zonotope analysis offers a good middle ground between the speed of the Interval analysis and the precision of LP for bound computation as it produces precise bounds faster than LP.

Original: Incomplete veriﬁcation of larger networks.
Revised: [Table] 4. I NCOMPLETE CERTIFICATION We next consider incomplete certiﬁcation for the remaining networks and compare against DeepZ and DeepPoly . We measure the average runtime per image in seconds and the precision measured by the % of images for which the certiﬁer certiﬁed the network to be robust.

Original: Complete veriﬁcation does not scale for  the remaining networks.
Revised: We note that DeepZ has same precision as Fast-Lin (Weng et al., 2018) and DeepPoly has the same precision as CROWN (Zhang et al., 2018). The (cid:15) values used for deﬁning the adversarial regions are shown in Table [Table] The (cid:15) values for networks trained to be robust are larger than for networks that are not.

Original: We note that running LP analysis to determine initial boundsfor the network does not scale for  larger networks whereas the bounds from the Interval analysis are quite imprecise.
Revised: We note that running LP analysis to obtain initial bounds does not scale for these larger networks with (cid:15) values considered in our experiments. Similarly, the bounds from the Interval analysis are quite imprecise.

Original: The values of(cid:15) considered for the L ∞ -norm attack on 5 × 100 , 6 × 100 , 6 × 200 ,  and 9 × 200 are 0 . 02 , 0 . 015 , and 0 .
Revised: Parameter values. We experimented with different values of the analysis parameters k MILP , k LP , k AI , α, δ, β, T and chose those values that offered the best tradeoff between performance and precision for the robustness certiﬁcation for each neural network.

Original: 015 respectively.
Revised: We reﬁne the output of all afﬁne transformations in the network that are followed by a ReLU except the ﬁrst one. In a given layer, we consider all neurons µ that can take non-negative values after applying the afﬁne transformation as candidates for reﬁnement. For the MNIST 5 × 100 , 6 × 100 , 9 × 100 , 6 × 200 , and 9 × 200 FNNs, we reﬁne the candidate neurons in the output of the afﬁne transformations in layer 2 - 4 with MILP and those in the remaining layers using LP.

Original: Forall networks, we set k MILP  = 3 , k LP = n − kMILP − 1 , k AI =
Revised: For MILP based reﬁnement, we use α = µ 5 k − 2 · p in layer k with p neurons.

Original: 1 , T = 1 s, β = 0 . 
Revised: For LP based reﬁnement, we use α = µ 2 k − 5 · p . We use timeout T = 1 second, β = 0 . 5 , and δ = µp − α for both MILP and LP based reﬁnements. For the CIFAR10 6 × 100 FNN, we use the same values except that we use α = µ 2 k − 2 · p for MILP reﬁnement and set T = 6 seconds for both MILP and LP based reﬁnement as it is more time consuming to reﬁne for CIFAR10 networks.

Original: 5 ( n is the number of layers). The value of α for the ﬁrst two layers is equal to number of  neurons in  the layer that can take positive values.
Revised: For the CNNs, we do not reﬁne convolutional layers as the number of candidates in these layers are very large and instead reﬁne all candidate neurons in fully connected layers with larger timeout to compensate for more difﬁcult problem instances for the solver.

Original: For the remaining layers, we choose a fraction of neuronstaking positive values for MILP reﬁnement with  timeout T .
Revised: For the MNIST ConvSmall, ConvBig and CIFAR10 ConvSmall networks, we reﬁne all the candidate neurons with MILP using a timeout T = 10 seconds.

Original: The remaining δ neurons with positive values are reﬁned with MILP using a timeout ofβ · T .
Revised: For the MNIST ConvSuper network, we reﬁne similarly but use LP instead of MILP with a timeout of T = 15 seconds.

Original: In Fig. 3, we compare the precision and the average runtime(in log scale) of ReﬁneAI against AI 2 per image for different neural networks.
Revised: Results for incomplete certiﬁcation. Table 2 shows the precision and the average runtime of all three certiﬁers.

Original: It can be seen that ReﬁneAI is signiﬁcantly more precise than AI 2 .
Revised: It can be seen that ReﬁneAI is signiﬁcantly more precise than both DeepZ and DeepPoly .

Original: ReﬁneAI veriﬁes > 2 x more images than AI 2 on 3 out of 4 neural networks.
Revised: It certiﬁes signiﬁcantly more images than DeepZ on all networks except the MNIST ConvSuper network.

Original: Speciﬁcally, for the largest 9 ×neural network, AI 2 manages to verify the robustness of only 12% of all images. In contrast, ReﬁneAI veriﬁes 35% of images to be robust.
Revised: This is because that for the (cid:15) considered, DeepZ is already very precise. ReﬁneAI certiﬁes network to be more robust on more images than DeepPoly on 6 out of 10 networks.

Original: The average runtimeper image of ReﬁneAI is <minutes for all neural networks.
Revised: It can be seen that the number of neurons in the network is not the determining factor for the average runtime of ReﬁneAI .

Original: [Figure]
Revised: We observe that ReﬁneAI runs faster on the networks trained to be robust and the top three networks with the largest runtime for ReﬁneAI are all networks not trained to be robust. This is because robust networks are relatively easier to certify and produce only a small number of candidate neurons for reﬁnement, which are easier to reﬁne by the solver. For example, even though the same parameter values are used for reﬁning the results on the MNIST ConvSmall and ConvBig networks, the average runtime of ReﬁneAI on the robustly trained ConvBig network with around 35 K neurons, 6 layers and a perturbation region deﬁned using (cid:15) = 0 . 2 is almost 4 times less than on the non-robust ConvSmall network with only 3 , 604 neurons, 3 layers and a smaller (cid:15) = 0 .

Original: Effect of neuron selection heuristic.
Revised: 4. E FFECT OF NEURON SELECTION HEURISTIC

Original: Reﬁning all neurons in a layer with MILP can signiﬁcantly slow down the analysis. We use the neuron selection heuristic from section 3 to determine neurons which need to be reﬁned more than others.
Revised: We use the neuron selection heuristic from section 3 to determine neurons which need to be reﬁned more than others for FNNs as reﬁning all neurons in a layer with MILP can signiﬁcantly slow down the analysis.

Original: To check whether our heuristic is able to identify important neurons, we ran the analysis on the 6 × 100 FFN by keeping all analysis parameters the same, except instead of selecting the neurons with the smallest rank sum ﬁrst we selected the neurons with the largest rank sum ﬁrst (thus reﬁningunimportant neurons as per our heuristic more).
Revised: To check whether our heuristic is able to identify important neurons, we ran the analysis on the MNIST 9 × 200 FNN by keeping all analysis parameters the same, except instead of selecting the neurons with the smallest rank sum ﬁrst we selected the neurons with the largest rank sum ﬁrst (thus reﬁning neurons more if our heuristic deeps them unimportant).

Original: With this change the modiﬁed analysis loses precision and veriﬁes 62% of images instead of 65% that the analysis reﬁning with our neuron selection heuristics veriﬁes.
Revised: With this change, the average runtime does not change signiﬁcantly.

Original: Although the number of images veriﬁed change by only 3% , the analysis  with our  heuristic produces tighter output bounds on all images than without it.
Revised: However, the modiﬁed analysis loses precision and fails to certify two images that the analysis reﬁning with our neuron selection heuristic certiﬁes.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper we presented a reﬁnement based approach for increasing the precision of incomplete veriﬁers for feed forward neural networks with ReLU activations.
Revised: In this paper, we presented a reﬁnement-based approach for increasing the precision of incomplete veriﬁers for feedforward and convolutional neural networks with ReLU activations.

Original: This enabled us to verify robustness properties that existing state-of-the-art complete veriﬁers cannot handle (due to scalability issues) whereas state-of-the-art incomplete veriﬁers could not prove due to using over-approximations which are too coarse.
Revised: This enabled us to certify robustness properties that existing state-of-the-art complete veriﬁers cannot handle (due to scalability issues) whereas state-of-the-art incomplete veriﬁers could not prove as they use overapproximations that are too coarse.

Original: We believethe direction of combining the strengths of over-approximation methods with that of mixed integer linear programming as done in this work is a promising direction for  advancing the state-of-the-art in neural network veriﬁcation.
Revised: We believe combining the strengths of overapproximation methods with that of mixed integer linear programming as done in this work is a promising direction for further advancing the state-of-the-art in neural network certiﬁcation