Title: UNIVERSAL APPROXIMATION WITH CERTIFIED NETWORKS

Abstract: Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, it is currently very difficult to train a neural network that is both accurate and certifiably robust. In this work we take a step towards addressing this challenge. We prove that for every continuous function f , there exists a network n such that: (i) n approximates f arbitrarily close, and (ii) simple interval bound propagation of a region B through n yields a result that is arbitrarily close to the optimal output of f on B. Our result can be seen as a Universal Approximation Theorem for interval-certified ReLU networks. To the best of our knowledge, this is the first work to prove the existence of accurate, interval-certified networks.

1. INTRODUCTION:

Much recent work has shown that neural networks can be fooled into misclassifying adversarial examples (Szegedy et al., 2014) , inputs which are imperceptibly different from those that the neural network classifies correctly. Initial work on defending against adversarial examples revolved around training networks to be empirically robust, usually by including adversarial examples found with various attacks into the training dataset (Gu and Rigazio, 2015; Papernot et al., 2016; Zheng et al., 2016; Athalye et al., 2018; Eykholt et al., 2018; Moosavi-Dezfooli et al., 2017; Xiao et al., 2018) . However, while empirical robustness can be practically useful, it does not provide safety guarantees. As a result, much recent research has focused on verifying that a network is certifiably robust, typically by employing methods based on mixed integer linear programming (Tjeng et al., 2019) , SMT solvers (Katz et al., 2017) , semidefinite programming (Raghunathan et al., 2018a) , duality (Wong and Kolter, 2018; Dvijotham et al., 2018b) , and linear relaxations (Gehr et al., 2018; Weng et al., 2018; Wang et al., 2018b; Zhang et al., 2018; Singh et al., 2018; Salman et al., 2019) .

Because the certification rates were far from satisfactory, specific training methods were recently developed which produce networks that are certifiably robust: Mirman et al. (2018) ; Raghunathan et al. (2018b) ; Wang et al. (2018a) ; Wong and Kolter (2018) ; Wong et al. (2018) ; Gowal et al. (2018) train the network with standard optimization applied to an over-approximation of the network behavior on a given input region (the region is created around the concrete input point). These techniques aim to discover specific weights which facilitate verification. There is a tradeoff between the degree of the over-approximation used and the speed of training and certification. Recently, (Cohen et al., 2019b) proposed a statistical approach to certification, which unlike the non-probabilistic methods discussed above, creates a probabilistic classifier that comes with probabilistic guarantees.

So far, some of the best non-probabilistic results achieved on the popular MNIST (Lecun et al., 1998) and CIFAR10 (Krizhevsky, 2009) datasets have been obtained with the simple Interval relaxation (Gowal et al., 2018; Mirman et al., 2019) , which scales well at both training and verification time. Despite this progress, there are still substantial gaps between known standard accuracy, experimental robustness, and certified robustness. For example, for CIFAR10, the best reported certified robustness is 32.04% with an accuracy of 49.49% when using a fairly modest l ∞ region with radius 8/255 (Gowal et al., 2018) . The state-of-the-art non-robust accuracy for this dataset is > 95% with experimental robustness > 50%. Given the size of this gap, a key question then is: can certified training ever succeed or is there a fundamental limit?

x 0 0 1 2 1 2 y 1 2 1 2 -3 2 -3 2 1 2 1 2 1 1 [0, 1] [0, 1 2 ] [0, 1 2 ] [0, 3 4 ] [0, 3 4 ] [0, 3 2 ] (a) Not certifiable network n1. 0 1 x 0.0 0.5 1.0 y f (x) (b) The function f . x 0 0 1 2 1 2 y 1 2 1 2 -1 2 -1 2 -1 2 -1 2 1 1 [0, 1] [0, 1 2 ] [0, 1 2 ] [0, 1 2 ] [0, 1 2 ] [0, 1] (c) Certifiable network n2.

Figure 2 : The ReLU networks n 1 (Figure 2a ) and n 2 (Figure 2c ) encode the same function f (Figure 2b ). Interval analysis fails certify that n 1 does not exceed [0, 1] on [0, 1] while certification succeeds for n 2 . In this paper we take a step in answering this question by proving a result parallel to the Universal Approximation Theorem (Cybenko, 1989; Hornik et al., 1989) . We prove that for any continuous function f defined on a compact domain Γ ⊆ R m and for any desired level of accuracy δ, there exists a ReLU neural network n which can certifiably approximate f up to δ using interval bound propagation. As an interval is a fairly imprecise relaxation, our result directly applies to more precise convex relaxations (e.g., Zhang et al. (2018) ; Singh et al. (2019) ). Theorem 1.1 (Universal Interval-Certified Approximation, Figure 1 ). Let Γ ⊂ R m be a compact set and let f : Γ → R be a continuous function. For all δ > 0, there exists a ReLU network n such that for all boxes

EQUATION

We recover the classical universal approximation theorem (|f (x)n(x)| ≤ δ for all x ∈ Γ) by considering boxes [a, b] describing points (x = a = b). Note that here the lower bound is not [l, u] as the network n is an approximation of f . Because interval analysis propagates boxes, the theorem naturally handles l ∞ norm bound perturbations to the input. Other l p norms can be handled by covering the l p ball with boxes. The theorem can be extended easily to functions f : Γ → R k by applying the theorem component wise.

Practical meaning of theorem:

The practical meaning of this theorem is as follows: if we train a neural network n on a given training data set (e.g., CIFAR10) and we are satisfied with the properties of n (e.g., high accuracy), then because n is a continuous function, the theorem tells us that there exists a network n which is as accurate as n and as certifiable with interval analysis as n is with a complete verifier. This means that if we fail to find such an n, then either n did not possess the required capacity or the optimizer was unsuccessful.

Focus on the existence of a network We note that we do not provide a method for training a certified ReLU network -even though our method is constructive, we aim to answer an existential question and thus we focus on proving that a given network exists. Interesting future work items would be to study the requirements on the size of this network and the inherent hardness of finding it with standard optimization methods.

Universal approximation is insufficient We now discuss why classical universal approximation is insufficient for establishing our result. While classical universal approximation theorems state that neural networks can approximate a large class of functions f , unlike our result, they do not state that robustness of the approximation n of f is actually certified with a scalable proof method (e.g., interval bound propagation). If one uses a non scalable complete verifier instead, then the standard Universal approximation theorem is sufficient.

To demonstrate this point, consider the function f : R → R (Figure 2b ) mapping all x ≤ 0 to 1, all x ≥ 1 to 0 and all 0 < x < 1 to 1x and two ReLU networks n 1 (Figure 2a ) and n 2 (Figure 2c ) perfectly approximating f , that is n 1 (x) = f (x) = n 2 (x) for all x. For δ = 1 4 , the interval certification that n 1 maps all

x ∈ [0, 1] to [0, 1] fails because [ 1 4 , 3 4 ] ⊆ n 1 ([0, 1]) = [0, 3 2 ] ⊆ [-1 4 , 5 4

]. However, interval certification succeeds for n 2 , because n 2 ([0, 1]) = [0, 1]. To the best of our knowledge, this is the first work to prove the existence of accurate, interval-certified networks.

2. RELATED WORK:

After adversarial examples were discovered by Szegedy et al. (2014) , many attacks and defenses were introduced (for a survey, see Akhtar and Mian (2018) ). Initial work on verifying neural network robustness used exact methods (Katz et al., 2017; Tjeng et al., 2019) on small networks, while later research introduced methods based on over-approximation (Gehr et al., 2018; Raghunathan et al., 2018a; Singh et al., 2018; Salman et al., 2019) aiming to scale to larger networks. A fundamentally different approach is randomized smoothing (Li et al., 2019; Lécuyer et al., 2019; Cohen et al., 2019b) , in which probabilistic classification and certification with high confidence is performed.

As neural networks that are experimentally robust need not be certifiably robust, there has been significant recent research on training certifiably robust neural networks (Raghunathan et al., 2018b; Mirman et al., 2018; 2019; Wong and Kolter, 2018; Wong et al., 2018; Wang et al., 2018a; Gowal et al., 2018; Dvijotham et al., 2018a; Xiao et al., 2019; Cohen et al., 2019b) . As these methods appear to have reached a performance wall, several works have started investigating the fundamental barriers in the datasets and methods that preclude the learning of a robust network (let alone a certifiably robust one) (Khoury and Hadfield-Menell, 2018; Schmidt et al., 2018; Tsipras et al., 2019) . In our work, we focus on the question of whether neural networks are capable of approximating functions whose robustness can be established with the efficient interval relaxation.

Feasibility Results with Neural Networks Early versions of the Universal Approximation Theorem were stated by Cybenko (1989) and Hornik et al. (1989) . Cybenko (1989) showed that networks using sigmoidal activations could approximate continuous functions in the unit hypercube, while Hornik et al. (1989) showed that even networks with only one hidden layer are capable of approximating Borel measurable functions.

More recent work has investigated the capabilities of ReLU networks. Here, Arora et al. (2018) , based on Tarela and Martínez (1999) , proved that every continuous piecewise linear function in R m can be represented by a ReLU network. Later, He et al. (2018) reduced the number of neurons needed using ideas from finite elements methods. Relevant to our work, Arora et al. (2018) introduced a ReLU network representations of the min function. Further, we use a construction method that is similar to the construction for nodal basis functions given in He et al. (2018) .

Universal approximation for Lipschitz constrained networks have been considered by Anil et al. (2019) and later by Cohen et al. (2019a) . A bound on the Lipschitz constant of a network immediately yields a certified region depending on the classification margin. Anil et al. (2019) proved that the set of Lipschitz networks with the GroupSort activation is dense in the space of Lipschitz continuous functions with Lipschitz constant 1, while Cohen et al. (2019a) provide an explicit construction to obtain the network. We note that both of these works focus on Lipschitz continuous functions, a more restricted class than continuous functions, which we consider in our work.

3. BACKGROUND:

In this section we provide the concepts necessary to describe our main result.

Adversarial Examples and Robustness Verification:

Let n : R m → R k be a neural network, which classifies an input x to a label t if n(x) t > n(x) j for all j = t. For a correctly classified input x, an adversarial example is an input y such that x is imperceptible from y to a human, but is classified to a different label by n.

Frequently, two images are assumed to be "imperceptible" if there l p distance is at most . The l p ball around an image is said to be the adversarial ball, and a network is said to be -robust around x if Figure 3 : Approximating f (Figure 3a ) using a ReLU network n = ξ 0 + k n k . The ReLU networks n k (Figure 3c ) approximate the N -slicing of f (Figure 3b ), as a sum of local bumps (Figure 6 ). every point in the adversarial ball around x classifies the same. In this paper, we limit our discussion to l ∞ adversarial balls which can be used to cover to all l p balls.

The goal of robustness verification is to show that for a neural network n, input point x and label t, every possible input in an l ∞ ball of size around x (written B ∞ (x)) is also classified to t.

Verifying neural networks with Interval Analysis:

The verification technique we investigate in this work is interval analysis. We denote by B the set of boxes B = [a, b] ⊂ R m for all m, where a i ≤ b i for all i. Furthermore for Γ ⊆ R m we define B(Γ) := B ∩ Γ describing all the boxes in Γ. The standard interval-transformations for the basic operations we are considering, namely +, -, • and the ReLU function R (Gehr et al. (2018) , Gowal et al. (2018) ) are

[a, b] + [c, d] = [a + c, b + d] R ([a, b]) = [R(a), R(b)] -[a, b] = [-b, -a] λ • [a, b] = [λa, λb],

where [a, b], [c, d] ∈ B(R), and λ ∈ R ≥0 . Furthermore, we used to distinguish the function f from its interval-transformation f . To illustrate the difference between f and f , consider f (x

) := x -x evaluated on x = [0, 1]. We have f ([0, 1]) = 0, but f ([0, 1]) = [0, 1]-# [0, 1] = [0, 1]+ # [-1, 0] = [-1, 1]

illustrating the loss in precision that interval analysis suffers from.

Interval analysis provides a sound over-approximation in the sense that for all function f , the values that

f can obtain on [a, b], namely f ([a, b]) := {f (x) | x ∈ [a, b]} are a subset of f ([a, b]). If f is a composition of functions, f = f 1 • • • • • f k , then f 1 • • • • • f k is a sound interval-transformer for f . Furthermore all combinations f of +, -, • and R are monotone, that is for [a, b], [c, d] ⊆ B(R m ) such that [a, b] ⊆ [c, d] then f # ([a, b]) ⊆ f # ([c, d]) (Appendix A). For boxes [x, x] representing points f coincides with f , f ([x, x]) = f (x)

. This will later be needed.

4. PROVING UNIVERSAL INTERVAL-PROVABLE APPROXIMATION:

In this section, we provide an explanation of the proof of our main result, Theorem 4.6, and illustrate the main points of the proof.

The first step in the construction is to deconstruct the function f into slices

{f k : Γ → [0, δ 2 ]} 0≤k<N such that that f (x) = ξ 0 + N -1 k=0 f k (x)

for all x, where ξ 0 is the minimum of f (Γ). We approximate each slice f k by a ReLU network δ 2 • n k . The network n approximating f up to δ will be n(x) := ξ 0 + δ 2 k n k (x). The construction relies on 2 key insights, (i) the output of δ 2 • n k can be confined to the interval [0, δ 2 ], thus the loss of analysis precision is at most the height of the slice, and (ii) we can construct the networks n k using local bump functions, such that only 4 slices can contribute to the loss of analysis precision, two for the lower interval bound, two for the upper one. The slicing {f k } 0≤k<5 of the function f : [-2, 2] → R (Figure 3a ), mapping x to f (x) = -x 3 + 3x is depicted in Figure 3b . The networks n k are depicted in Figure 3c . In this example, evaluating the interval-transformer of n, namely n on the box 3b )). Let Γ ⊂ R m be a closed m-dimensional box and let f : Γ → R be continuous. The N -slicing of f is a set of functions {f k } 0≤k<N defined by

B = [-1, 1] results into n ([-1, 1]) = [-2, 6/5] lies is within the δ = 8 5 bound of f ([-1, 1]) = [-2, 2]. Definition 4.1 (N -slicing (Figure

f k : Γ → R, x →    0 if f (x) ≤ ξ k , f (x) -ξ k if ξ k < f (x) < ξ k+1 , ξ k+1 -ξ k if ξ k+1 ≤ f (x), ∀k ∈ {0, . . . , N -1},

where

ξ k := ξ 0 + k N (ξ N -ξ 0 ), k ∈ {1, . . . , N -1}, ξ 0 := min f (Γ) and ξ N := max f (Γ).

To construct a ReLU network satisfying the desired approximation property (Equation ( 1)) if evaluated on boxes in B(Γ), we need the ReLU network nmin capturing the behavior of min as a building block (similar to He et al. (2018) ). It is given by nmin(x, y)

:= 1 2 (1 -1 -1 -1) R       1 1 -1 -1 1 -1 -1 1    x y    .

With the ReLU network nmin, we can construct recursively a ReLU network nmin N mapping N arguments to the smallest one (Definition A.8). Even though the interval-transformation loses precision, we can establish bounds on the precision loss of nmin N sufficient for our use case (Appendix A). Now, we use the clipping function R [ * ,1] := 1 -R(1x) clipping every value exceeding 1 back to 1 (Figure 5 ) to construct the local bumps φ c w.r.t. a grid G. G specifies the set of all possible local bumps we can use to construct the networks n k . Increasing the finesse of G will increases the approximation precision. Definition 4.2 (local bump, Figure 6 ).

Let M ∈ N, G := {( i1 M ), . . . , im M | i ∈ Z m } be a grid, = 2 log 2 2m +1 and let c = { i l 1 M , i u 1 M } × • • • × { i l m M , i u m

M } ⊆ G be a set of grid points describing the corner points of a hyperrectangle in G. We define a ReLU neural network φ

c : R m → [0, 1] ⊂ R w.r.t. G by φ c (x) := R   nmin 2m 1≤k≤m R [ * ,1] (M • • (x k - i l k M ) + 1), R [ * ,1] (M • • ( i u k M -x k ) + 1)   .

We will describe later how M and c get picked. A graphical illustration of a local bump for in two dimensions and c = { 6 . The local bump φ c (x) evaluates to 1 for all x that lie within the convex hull of c, namely conv(c), after which φ c (x) quickly decreases linearly to 0. φ c has 1 + 2(2d -1) + 2d ReLUs and 1 + log 2 (2d + 1) + 1 layers. Next, we give bounds on the loss of precision for the interval-transformation φ c . We can show that interval analysis can (i) never produce intervals exceeding [0, 1] and (ii) is precise if

i l 1 M , i u 1 M } × { i l 2 M , i u 2 M } = {c ll , c lu , c ul , c uu } is shown in Figure

1/M 1/M 1/M 1/M c ll c lu

B does no intersect conv(N (c)) \ conv(c). Lemma 4.3. For all B ∈ B(R m ), it holds that φ c (B) ⊆ [0, 1] ∈ B and φ c (B) = [1, 1] if B ⊆ conv(c) [0, 0] if B ⊆ Γ \ conv(N (c)).

The formal proof is given in Appendix A. The next lemma shows, how a ReLU network n k can approximate the slice f k while simultaneously confining the loss of analysis precision. Lemma 4.4. Let Γ ⊂ R m be a closed box and let f : Γ → R be continuous. For all δ > 0 there exists a set of ReLU networks {n k } 0≤k<N of size N ∈ N approximating the N -slicing of f , {f k } 0≤k<N (ξ k as in Definition 4.1) such that for all boxes B ∈ B(Γ)

n k (B) = [0, 0] if f (B) ≤ ξ k -δ 2 [1, 1] if f (B) ≥ ξ k+1 + δ 2 .

(2)

and

n k (B) ⊆ [0, 1].

It is important to note that in Equation ( 2) we mean f and not f . The proof for Lemma 4.4 is given in Appendix A. In the following, we discuss a proof sketch.

Because Γ is compact and f is continuous, f is uniformly continuous by the Heine-Cantor Theorem. So we can pick a M ∈ N such that for all x, y

∈ Γ satisfying ||y-x|| ∞ ≤ 1 M holds |f (y)-f (x)| ≤ δ 2 . We then choose the grid G = ( Z M ) m ⊆ R m . Next, we construct for every slice k a set ∆ k of hyperrectangles on the grid G: if a box B ∈ B(Γ) fulfills f (B) ≥ ξ k+1 + δ

2 , then we add a minimal enclosing hyperrectangle c ⊂ G such that B ⊆ conv(c) to ∆ k , where conv(c) denotes the convex hull of c. This implies, using uniform continuity of f and that the grid G is fine enough, that f (conv(c)) ≥ ξ k+1 . Since there is only a finite number of possible hyperrectangles in G, the set ∆ k is clearly finite. The network fulfilling Equation ( 2) is

n k (x) := R [ * ,1] c∈∆ k φ c (x) ,

where φ c is as in Definition 4.2. The n k are depicted in Figure 3c .

Now, we see that Equation (2) holds by construction: For all boxes:

B ∈ B(Γ) such that f ≥ ξ k+1 + δ 2 on B exists c ∈ ∆ k such that B ⊆ conv(c ) which implies, using Lemma 4.3, that φ c (B) = [1, 1], hence n k (B) = R [ * ,1] (φ c (B) + c∈∆ k \c φ c (B)) ∀c = c : φ c (B) ⊆ [0, 1](Lemma 4.3) = R [ * ,1] ([1, 1] + [p 1 , p 2 ]) [p 1 , p 1 ] ∈ B(R ≥0 ) = R [ * ,1] ([1 + p 1 , 1 + p 2 ]) = [1, 1]. Similarly, if f (B) ≤ ξ k -δ 2 holds, then it holds for all c ∈ ∆ k that B does not intersect N (conv(c)). Indeed, if a c ∈ ∆ k would violate this, then by construction, f (conv(c)) ≥ ξ k+1 , contradicting f (B) ≤ ξ k -δ 2 . Thus φ c (B) = [0, 0]

, and hence n (B) = [0, 0]. Theorem 4.5. Let Γ ⊂ R m be a closed box and let f : Γ → R be continuous. Then for all δ > 0, exists a ReLU network n such that for all B ∈ B(Γ)

[l + δ, u -δ] ⊆ n (B) ⊆ [l -δ, u + δ],

where l := min f (B) and u := max f (B).

Proof. Pick N such that the height of each slice is exactly δ 2 , if this is impossible choose a slightly smaller δ. Let {n k } 0≤k<N be a series of networks as in Lemma 4.4. Recall that ξ 0 = min f (Γ). We define the ReLU network

n(x) := ξ 0 + δ 2 N -1 k=0 n k (x).

(3)

EQUATION

Let p, q ∈ {0, . . . , N -1} such that

EQUATION

Figure 7 : Illustration of the proof for Theorem 4.5.

as depicted in Figure 7 . Thus by Equation ( 4) for all k ∈ {0, . . . , p -2} it holds that n k (B) = [1, 1] and similarly, by Equation ( 5) for all k ∈ {q + 2, . . . , N -1} it holds that n k (B) = [0, 0]. Plugging this into Equation (3) after splitting the sum into three parts leaves us with

n (B) = ξ 0 + δ 2 p-2 k=0 n k (B) + δ 2 q+1 k=p-1 n k (B) + δ 2 N -1 k=p+1 n k (B) = ξ 0 + (p -1)[ δ 2 , δ 2 ] + δ 2 q+1 k=p-1 n k (B) + [0, 0].

Applying the standard rules for interval analysis, leads to

n (B) = [ξ p-1 , ξ p-1 ] + δ 2 q+1 k=p-1 n k (B),

where we used in the last step, that ξ 0 + k δ 2 = ξ k . For all terms in the sum except the terms corresponding to the 3 highest and lowest k we get

EQUATION

Indeed, from Equation ( 6) we know that there is

x ∈ B such that f (x) ≤ ξ p+1 = ξ p+2 -δ 2 , thus by Lemma 4.4 n k ([x, x]) = [0, 0] for all p + 2 ≤ k ≤ q -2. Similarly, from Equation (7) we know, that there is x ∈ B such that f (x) ≥ ξ q = ξ q-1 + δ 2 , thus by Lemma 4.4 n k ([x , x ]) = [1, 1] for all p + 2 ≤ k ≤ q -2. So n k (B) is at least [0, 1],

and by Lemma 4.4 also at most [0, 1]. This leads to

n (B) = [ξ p-1 , ξ p-1 ] + δ 2 p+1 k=p-1 n k (B) + δ 2 ((q -2) -(p + 2) + 1)[0, 1] + δ 2 q+1 k=q-1 n k (B) = [ξ p-1 , ξ p-1 ] + δ 2 p+1 k=p-1 n k (B) + [0, ξ q-1 -ξ p+2 ] + δ 2 q+1 k=q-1 n k (B).

We know further, that if p + 3 ≤ q, than there is an

x ∈ B such that f (x) ≥ ξ p+3 = ξ p+2 + δ 2 , hence similar as before n p+1 ([x, x]) = [1, 1] and similarly n p ([x, x]) = [1, 1] and n ([x, x]) = [1, 1]. So we know, that δ 2 p+1 k=p-1 n k (B) includes at least [3 δ 2 , 3 δ 2 ] and at the most [0, 3 δ 2 ]. Similarly, there exists an x ∈ B such that n q-1 ([x , x ]) = [0, 0], n q ([x , x ]) = [0, 0] and n q+1 ([x , x ]) = [0, 0]. This leaves us with [3 δ 2 , 3 δ 2 ] ⊆ δ 2 p+1 k=p-1 n k (B) ⊆ [0, 3 δ 2 ] [0, 0] ⊆ δ 2 q+1 k=q-1 n k (B) ⊆ [0, 3 δ 2 ],

If p + 3 > q the lower bound we want to prove becomes vacuous and only the upper one needs to be proven. Thus we have

[l + δ, u -δ] ⊆ [ξ p+2 , ξ p-1 ] ⊆ n (B) ⊆ [ξ p-1,ξq+2 ] ⊆ [l -δ, u + δ],

where l := min f (B) and u := max f (B). Theorem 4.6 (Universal Interval-Provable Approximation). Let Γ ⊂ R m be compact and f : Γ → R d be continuous. For all δ ∈ R m ≥0 exists a ReLU network n such that for all B ∈ B(Γ)

[l + δ, u -δ] ⊆ n (B) ⊆ [l -δ, u + δ],

where l, u ∈ R m such that l k := min f (B) k and u k := max f (B) k for all k.

Proof. This is a direct consequence of using Theorem 4.5 and the Tietze extension theorem to produce a neural network for each dimension d of the codomain of f .

Note that Theorem 1.1 is a special case of Theorem 4.6 with d = 1 to simplify presentation.

5. CONCLUSION:

We proved that for all real valued continuous functions f on compact sets, there exists a ReLU network n approximating f arbitrarily well with the interval abstraction. This means that for arbitrary input sets, analysis using the interval relaxation yields an over-approximation arbitrarily close to the smallest interval containing all possible outputs. Our theorem affirmatively answers the open question, whether the Universal Approximation Theorem generalizes to Interval analysis.

Our results address the question of whether the interval abstraction is expressive enough to analyse networks approximating interesting functions f . This is of practical importance because interval analysis is the most scalable non-trivial analysis.

A PROOFS FOR THE UNIVERSAL INTERVAL-CERTIFIED APPROXIMATION:

Lemma A.1 (Monotonicity). The operations +,are monotone, that is for all

[a 1 , b 1 ], [a 2 , b 2 ], [c 1 , d 1 ], [c 2 , d 2 ] ∈ B(R) such that [a 1 , b 1 ] ⊆ [a 2 , b 2 ] and [c 1 , d 2 ] ⊆ [c 2 , d 2 ] holds [a 1 , b 1 ] + [c 1 , d 1 ] ⊆ [a 2 , d 2 ] + [c 2 , d 2 ] [a 1 , b 1 ] -[c 1 , d 1 ] ⊆ [a 2 , d 2 ] -[c 2 , d 2 ] [a 1 , b 1 ] • [c 1 , d 1 ] ⊆ [a 2 , d 2 ] • [c 2 , d 2 ].

Further the operation * and R are monotone, that is for all

[a, b], [c, d] ∈ B(R) and for all λ ∈ R ≥0 such that [a, b] ⊆ [c, d] holds λ • [a, b] ⊆ λ • [c, d] R ([a, b]) ⊆ R ([c, d]).

Proof.

[a

1 , b 1 ] + [c 1 , d 1 ] = [a 1 + c 1 , b 1 + d 1 ] ⊆ [a 2 + c 2 , b 2 + d 2 ] = [a 2 , d 2 ] + [c 2 , d 2 ] [a 1 , b 1 ] -[c 1 , d 1 ] = [a 1 -d 1 , b 1 -c 1 ] ⊆ [a 2 -d 2 , b 2 -c 2 ] = [a 2 , d 2 ] -[c 2 , d 2 ] λ • [a, b] = [λa, λb] ⊆ [λc, λd] = [λc, λd] R ([a, b]) = [R(a), R(b)] ⊆ [R(c), R(d)] = R ([c, d]).

Definition A.2 (N -slicing). Let Γ ⊂ R m be a compact m-dimensional box and let f : Γ → R be continuous. The N -slicing of f is a set of functions {f k } 0≤k≤N -1 defined by

f k : Γ → R, x →    0 if f (x) ≤ ξ k , f (x) -ξ k if ξ k < f (x) < ξ k+1 , ξ k+1 -ξ k otherwise, ∀k ∈ {0, . . . , N -1},

where ξ k := k N (ξ maxξ min ), k ∈ {0, . . . , N }, ξ min := min f (Γ) and ξ max := max f (Γ). Lemma A.3 (N -slicing). Let {f k } 0≤k≤N -1 be the N -slicing of f . Then for all x ∈ Γ we have

f (x) := ξ 0 + N -1 k=0 f k (x).

Proof. Pick x ∈ Γ and let l ∈ {0, . . . , N -1} such that ξ l ≤ f (x) ≤ ξ l+1 . Then Proof. We show the proof for R

ξ 0 + N -1 k=0 f k (x) = ξ 0 + l-1 k=0 f k (x) + f l (x) + N -1 k=l+1 f k (x) = ξ 0 + l-1 k=0 (ξ k+1 -ξ k ) + f l (x) = ξ l + f l (x) = f (x).

[a,b] , the proof for R [ * ,b] is similar. x < b ⇒ R [ * ,b] (x) = b -R(b -x) = b -b + x = x x ≥ b ⇒ R [ * ,b] (x) = b -R(b -x) = b -0 = b Next, R [ * ,b] ([a , b ]) = b -R (b -[a , b ]) = b -R (b + [-b , -a ]) = b -R ([b -b , b -a ]) = b -[R(b -b ), R(b -a )] = b + [-R(b -a ), -R(b -b )] = [b -R(b -a ), b -R(b -b )] = [R [ * ,b] (a ), R [ * ,b] (b )].

Definition A.6 (nmin). We define the ReLU network nmin : R 2 → R by nmin(x, y) :

= 1 2 (1 -1 -1 -1) R       1 1 -1 -1 1 -1 -1 1    x y    .

Lemma A.7 (nmin). Let x, y ∈ R, then nmin(x, y) = min(x, y).

Proof. Because nmin is symmetric in its arguments, we assume w.o.l.g. x ≥ y.

nmin(x, y) = 1 2 (1 -1 -1 -1) R       1 1 -1 -1 1 -1 -1 1    x y    = 1 2 (1 -1 -1 -1) R    x + y -x -y x -y -x + y    If x + y ≥ 0, then nmin(x, y) = 1 2 (x + y -x + y) = y. If x + y < 0, then nmin(x, y) = 1 2 (x + y -x + y) = y.

Definition A.8 (nmin N ). For all N ∈ N ≥1 , we define a ReLU network nmin N defined by

nmin 1 (x) := x nmin N (x 1 , . . . , x N ) := nmin(nmin N/2 (x 1 , . . . , x N/2 ), nmin N/2 +1 (x N/2 +1 , . . . , x N )). Lemma A.9. Let [a, b], [c, d] ∈ B(R). Then nmin ([a, b], [c, d]) = nmin ([c, d], [a, b]) and nmin ([a, b], [c, d]) =    [c + a-b 2 , d + b-a 2 ] if d ≤ a [a + c-d 2 , b + d-c 2 ] if a ≤ d and b < c [a + c -b+d 2 , b+d 2 ] if a ≤ d and b ≥ c

Proof. The symmetry on abstract elements is immediate. In the following, we omit some of to improve readability.

nmin 

EQUATION

:



Figures and Tables:

Figure 1: Illustration of Theorem 1.1.

Networks n k approximating f k .

Figure 4: Neighbors N (x) (blue dots) and N (U ) (red squares).

Figure 6: Local bump φ c , where c contains the points c ll , c lu , c ul , c uu . The points in N (conv(c)) are depicted by the red squares.

Definition A.4 (clipping). Let a, b ∈ R, a < b. We define the clipping function R [ * ,b] : R → R by R [ * ,b] (x) := b -R(bx).

Lemma A.5 (clipping). The function R [ * ,b] sends all x ≤ b to x, and all x > b to b. Further, R [ * ,b] ([a , b ]) = [R [ * ,b] (a ), R [ * ,b] (b )].

a + c) -R(-ac), R(b + d) -R(-bd)] + [-R(bc) -R(da), -R(ad) -R(cb)])Claim: R(a+c)-R(-a-c) = a+c. If a+c > 0 then -a-c < 0 thus the claim in this case. Indeed:If a + c ≤ 0 then -ac ≥ 0 thus R(a + c) -R(-ac) = -R(-ac) = -(-ac) = a + c. Similarly R(b + d) -R(-bd) = b + d. So the expression simplifies to nmin ([a, b], [c, d]) = 1 2 ([a + c, b + d] + [-R(bc) -R(da), -R(ad) -R(cb)])We proceed by case distinction:Case 1: bc ≤ 0: Then a ≤ b ≤ c ≤ d: nmin ([a, b], [c, d]) = 1 2 ([a + c, b + d] + [ad, bc]) c + ad, b + d + bc]) = [a + c-d 2 , b + d-c 2 ] Case 2: ad ≥ 0: Then c ≤ d ≤ a ≤ b. By symmetry of nmin equivalent to Case 1. Hence nmin ([a, b], [c, d]) = [c + a-b 2 , d + b-a 2 ]. Case 3: ad < 0 and bc > 0: nmin ([a, b], [c, d]) = 1 2 ([a + c, b + d] + [cbd + a, 0]) = 1 2 ([a + c + cbd + a, b + d]) = [a + c -b+d 2 , b+d 2 ]Lemma A.18. Let Γ ⊂ R m be a closed box and let f : Γ → R be continuous. For all δ > 0 exists a set of ReLU networks {n k } 0≤k≤N -1 of size N ∈ N approximating the N -slicing of f , {f k } 0≤k≤N -1 (ξ k as in Definition A.2) such that for all boxes B ∈ B(Γ)n k (B) = [0, 0] if f (B) ≤ ξ k -δ 2 [1, 1] if f (B) ≥ ξ k+1 + δ 2 .andn k (B) ⊆ [0, 1]. Proof. Let N ∈ N such that N ≥ 2 ξmax-ξmin δwhere ξ min := min f (Γ) and ξ max := max f (Γ). For simplicity we assume Γ = [0, 1] m . Using the Heine-Cantor theorem, we get that f is uniformly continuous, thus there exists a δ > 0 such that ∀x,y ∈ Γ.||y -x|| ∞ < δ ⇒ ||f (y)f (x)|| < δ 2 . Further, let M ∈ N such that M ≥ 1δ and let G be the grid defined byG := ( Z M ) m ⊆ R m . Let C(B) be the set of corner points of the closest hyperrectangle in G confining B ∈ B(Γ). We construct the set ∆ k := {C(B) | B ∈ B(Γ) : f (B) ≥ ξ k+1 + δ 2 }.We claim that {n k } 0≤k≤N -1 defined byn k (x) := R [ * ,1] c∈∆ k φ c (x)satisfies the condition.Case1: Let B ∈ B(Γ) such that f (B) ≥ ξ k+1 + δ 2 . Then for all g ∈ N (B) holds f k (g) = δ 2 . By construction exists a c ∈ ∆ k such that B ⊆ conv(c ). Using Lemma 4.3 we get n k (B) = R [ * ,1] * ,1] ([1, 1] + [p 1 , p 2 ]) = [1, 1],where [p 1 , p 2 ] ∈ B(R ≥0 ). Indeed, by case distinction:Case 2: Let B ∈ B(Γ) such that f (B) ≤ ξ k -δ 2 . Then for all g ∈ N (B) holds f k (g) = 0.Further, B ∩ conv(N (c)) = ∅ for all c ∈ ∆ k because G is fine enough. Using Lemma 4.3 we obtain n k (B) = R [ * ,1] c∈∆ k φ c (B) = R [ * ,1] ([0, 0]) = [0, 0]. By construction we have n k (B) ⊆ [0, 1].

Peer review (iclr2020):
Summary: This paper proves a theoretical result, that for every continuous function f, there exists a ReLU network approximating f well, and for which interval propagation will result in tight bounds. The proof is by construction: the function to approximate is decomposed into a sum of functions (slices) with bounded range, which will be approximable to the correct precision. Each slices is approximated as a sum of local bumps, who essentially approximates an indicator function over a grid. Comments: * Equation (1) of Theorem 1.1, I assume that the left inclusion could be replaced by [l, u], given that interval propagation generate valid bounds? Or is the case that due to the approximating network being only an approximation, interval propagation on it might not give actual bounds on the real function? Opinion: The paper is very technical and takes some effort to parse to understand the construction, but the authors do a good job of providing examples and illustrations to make it at least intuitive. The content is clearly novel and the result is far from trivial. The only thing that i*m slightly wary is the significance of what is proven. The paper shows that approximating a continuous function with a network such that bound propagation works well on it is feasible. On the other hand, the motivation given is *given a dataset and a specification, is there a network that is both certified and accurate with respect to these*, which is a slightly different problem. If this was proven to be yes, then we would know that when our robust training fails, this is due to optimization not being good enough or our not network not having enough capacity. The result proven here does not suffice, because it pre-suppose the existence of a correct function, that would match the dataset and specification, which is not guaranteed (see for example the proofs that robustness and accuracy are at odds, or you can just imagine verification of robustness for too-large radius which would lead to infeasible specification). I still like the paper and I think it might just be a matter of framing the abstract and introduction. Do the authors have an implementation, even for only toy-size examples, of the generation of a Neural Network approximating a given function or is this purely theoretical? Especially when the proof is constructive like this, this can strengthen the confidence that no subtle mistakes have been made. If so, this should be mentioned. Potential typos: Page 4: *Further* -> *Furthermore*? Page 5: *Increasing the fines* -> *Increasing the finesse?* Page 6, just before Lemma 4.3, missing a mathcal before the N? Disclaimer: I carefully checked the derivations in the paper but only assessed the sensibility of the proofs in the appendix.