Title: APPROXIMATION AND NON-PARAMETRIC ESTIMATION OF RESNET-TYPE CONVOLUTIONAL NEURAL NET-WORKS VIA BLOCK-SPARSE FULLY-CONNECTED NEU-RAL NETWORKS

Abstract: We develop new approximation and statistical learning theories of convolutional neural networks (CNNs) via the ResNet-type structure where the channel size, filter size, and width are fixed. It is shown that a ResNet-type CNN is a universal approximator and its expression ability is no worse than fully-connected neural networks (FNNs) with a block-sparse structure even if the size of each layer in the CNN is fixed. Our result is general in the sense that we can automatically translate any approximation rate achieved by block-sparse FNNs into that by CNNs. Thanks to the general theory, it is shown that learning on CNNs satisfies optimality in approximation and estimation of several important function classes. As applications, we consider two types of function classes to be estimated: the Barron class and Hölder class. We prove the clipped empirical risk minimization (ERM) estimator can achieve the same rate as FNNs even the channel size, filter size, and width of CNNs are constant with respect to the sample size. This is minimax optimal (up to logarithmic factors) for the Hölder class. Our proof is based on sophisticated evaluations of the covering number of CNNs and the nontrivial parameter rescaling technique to control the Lipschitz constant of CNNs to be constructed.

1. INTRODUCTION:

Convolutional Neural Network (CNN) is one of the most popular architectures in deep learning research, with various applications such as computer vision (Krizhevsky et al. (2012) ), natural language processing (Wu et al. (2016) ), and sequence analysis in bioinformatics (Alipanahi et al. (2015) , Zhou & Troyanskaya (2015) ). Despite practical popularity, theoretical justification for the power of CNNs is still scarce from the viewpoint of statistical learning theory.

For fully-connected neural networks (FNNs), there is a lot of existing work, dating back to the 80's, for theoretical explanation regarding their approximation ability (Cybenko (1989) , Barron (1993) , Lu et al. (2017) , Yarotsky (2017) , and Petersen & Voigtlaender (2017) ) and generalization power ( Barron (1994) , Arora et al. (2018) , and Suzuki (2018) ). See also Pinkus (2005) and Kainen et al. (2013) for surveys of earlier works. Although less common compared to FNNs, recently, statistical learning theory for CNNs has been studied, both about approximation ability (Zhou (2018) , Yarotsky (2018) , Petersen & Voigtlaender (2018) ) and about generalization power (Zhou & Feng (2018) ). One of the standard approaches is to relate the approximation ability of CNNs with that of FNNs, either deep or shallow. For example, Zhou (2018) proved that CNNs are a universal approximator of the Barron class (Barron (1993) , Klusowski & Barron (2016) ), which is a historically important function class in the approximation theory. Their approach is to approximate the function using a 2-layered FNN (i.e., an FNN with a single hidden layer) with the ReLU activation function (Krizhevsky et al. (2012) ) and transform the FNN into a CNN. Very recently independent of ours, Petersen & Voigtlaender (2018) showed any function realizable with an FNN can extend to an equivariant function realizable by a CNN that has the same order of parameters. However, to the best of our knowledge, no CNNs that achieves the minimax optimal rate (Tsybakov (2008), Giné & Nickl (2015) ) in important function classes, including the Hölder class, can keep the number of units in each layer constant with respect to the sample size. Architectures that have extremely large depth, while moderate channel size and width have become feasible, thanks to recent methods such as identity mappings (He et al. (2016) , Huang et al. (2018) ), sophisticated initialization schemes (He et al. (2015) , Chen et al. (2018) ), and normalization techniques (Ioffe & Szegedy (2015) , Miyato et al. (2018) ). Therefore, we would argue that there are growing demands for theories which can accommodate such constant-size architectures.

In this paper, we analyze the learning ability of ResNet-type ReLU CNNs which have identity mappings and constant-width residual blocks with fixed-size filters. There are mainly two reasons that motivate us to study this type of CNNs. First, although ResNet is the de facto architecture in various practical applications, the approximation theory for ResNet has not been explored extensively, especially from the viewpoint of the relationship between FNNs and CNNs. Second, constant-width CNNs are critical building blocks not only in ResNet but also in various modern CNNs such as Inception (Szegedy et al. (2015) ), DenseNet (Huang et al. (2017) ), and U-Net (Ronneberger et al. (2015) ), to name a few. Our strategy is to replicate the learning ability of FNNs by constructing tailored ResNet-type CNNs. To do so, we pay attention to the block-sparse structure of an FNN, which roughly means that it consists of a linear combination of multiple (possibly dense) FNNs (we define it rigorously in the subsequent sections). Block-sparseness decreases the model complexity coming from the combinatorial sparsity patterns and promotes better bounds. Therefore, it is often utilized, both implicitly or explicitly, in the approximation and learning theory of FNNs (e.g., Bölcskei et al. (2017) , Yarotsky (2018) ). We first prove that if an FNN is block-sparse with M blocks (M -way block-sparse FNN), we can realize the FNN with a ResNet-type CNN with O(M ) additional parameters, which are often negligible since the original FNN already has Ω(M ) parameters. Using this approximation, we give the upper bound of the estimation error of CNNs in terms of the approximation errors of block sparse FNNs and the model complexity of CNNs. Our result is general in the sense that it is not restricted to a specific function class, as long as we can approximate it using block-sparse FNNs.

To demonstrate the wide applicability of our methods, we derive the approximation and estimation errors for two types of function classes with the same strategy: the Barron class (of parameter s = 2) and Hölder class. We prove, as corollaries, that our CNNs can achieve the approximation error of order Õ(M -D+2 2D ) for the Barron class and Õ(M -β D ) for the β-Hölder class and the estimation error of order Õp (N -D+2 2(D+1) ) for the Barron class and Õp (N -2β 2β+D ) for the β-Hölder class, where M is the number of parameters (we used M here, same as the number of blocks because it will turn out that CNNs have O(M ) blocks for these cases), N is the sample size, and D is the input dimension. These rates are same as the ones for FNNs ever known in the existing literature. An important consequence of our theory is that the ResNet-type CNN can achieve the minimax optimal estimation error (up to logarithmic factors) for β-Hölder class even if its filter size, channel size and width are constant with respect to the sample size, as opposed to existing works such as Yarotsky (2017) and Petersen & Voigtlaender (2018) , where optimal FNNs or CNNs could have a width or a channel size goes to infinity as N → ∞.

In summary, the contributions of our work are as follows:

• We develop the approximation theory for CNNs via ResNet-type architectures with constant-width residual blocks. We prove any M -way block-sparse FNN is realizable such a CNN with O(M ) additional parameters. That means if FNNs can approximate a function with O(M ) parameters, we can approximate the function with CNNs at the same rate (Theorem 1).

• We derive the upper bound of the estimation error in terms of the approximation error of FNNs and the model complexity of CNNs (Theorem 2). This result gives the sufficient conditions to derive the same estimation error as that of FNNs (Corollary 1).

• We apply our general theory to the Barron class and Hölder class and derive the approximation (Corollary 2 and 4) and estimation (Corollary 3 and 5) error rates, which are identical to those for FNNs, even if the CNNs have constant channel and filter size with respect to the sample size. In particular, this is minimax optimal for the Hölder case. 

2. RELATED WORK:

We summarize in Table 1 the differences in the CNN architectures between our work and Zhou (2018) and Petersen & Voigtlaender (2018) , which established the approximation theory of CNNs via FNNs. First and foremost, Zhou (2018) only considered a specific function class -the Barron class -as a target function class, although their method is applicable to any function class that can be realized by a 2-layered ReLU FNN. Regarding the architecture, they considered CNNs with a single channel and whose width is "linearly increasing" (Zhou (2018) ) layer by layer. For regression or classification problems, it is rare to use such an architecture. In addition, since they did not bound the norm of parameters in the approximating CNNs, we cannot derive the estimation error from this method. Petersen & Voigtlaender (2018) fully utilized the group invariance structure of underlying input spaces to construct CNNs. Such a structure makes theoretical analysis easier, especially for investigating the equivariance properties of CNNs since it enables us to incorporate mathematical tools such as group theory, Fourier analysis, and representation theory. Although their results are quite strong in that it is applicable to any function that can be approximated by FNNs, their assumption on the group structure excludes the padding convolution layer, an important and popular type of convolution operations. Another point is that if we simply apply their construction method to derive the estimation error for (equivariant) Hölder functions, combined with the approximation result of Yarotsky (2017) , the resulting CNN that achieves the minimax optimal rate has Õ(ε -D β ) channels where ε is the approximation error threshold. It is partly because their construction is not aware of the internal sparse structure of approximating FNNs. Finally, the filter size of their CNN is as large as the input dimension. As opposed to these two works, we employ padding-and ResNettype CNNs which have multiple channels, fixed-size filters, and constant widths. Like Petersen & Voigtlaender (2018) , our result is applicable to any function, as long as the FNNs to be approximated are block sparse, including the Barron and Hölder cases. If we apply our theorem to these classes, we can show that the optimal CNNs can achieve the same approximation and estimation rate as FNNs, while the number of channels is independent of the sample size. Further, this is minimax optimal up to the logarithmic factors for the Hölder class.

Due to its practical success, theoretical analysis for ResNet has been explored recently (e.g., Lin & Jegelka (2018) , Lu et al. (2018) , Nitanda & Suzuki (2018), and Huang et al. (2018) ). From the viewpoint of statistical learning theory, Nitanda & Suzuki (2018) and Huang et al. (2018) investigated the generalization power of ResNet from the perspective of the boosting interpretation. However, they did not discuss the function approximation ability of ResNet. To the best of our knowledge, our theory is the first work to provide the approximation ability of the CNN class that can accommodate the ResNet-type ones.

We import the approximation theories for FNNs, especially ones for the Barron class and Hölder class. The approximation theory for the Barron class has been investigated in e.g., Barron (1993) , Klusowski & Barron (2016) , and Lee et al. (2017) . Originally Barron (1993) considered the parameter s = 1 (see Definition 3) and the activation function σ satisfying σ(z) → 1 as z → ∞ and σ(z) → 0 as z → -∞. Later, Klusowski & Barron (2016) studied the approximation theory with s = 2 and proved that 2-layered ReLU FNNs with M hidden units can approximate functions of this class with the order of Õ(M -D+2 2D ). Yarotsky (2017) proved FNNs with O(S) non-zero parameters can approximate β-Hölder continuous functions with the order of Õ(S -β D ). Using this bound, Schmidt-Hieber (2017) proved that the estimation error of the ERM estimator is Õ(N -2β 2β+D ), which is minimax optimal up to logarithmic factors (see, e.g., Tsybakov (2008) ).

3. PROBLEM SETTING:



3.1. EMPIRICAL RISK MINIMIZATION:

We consider a regression task in this paper. Let X be a [-1, 1] D -valued random variable with unknown probability distribution P X and ξ be an independent random noise drawn from the Gaussian distribution with an unknown variance σ 2 : ξ ∼ N (0, σ 2 ) (σ > 0). Let f • be an unknown deterministic function f • : [-1, 1] D → R (we will characterize f • rigorously in the theorems later).

We define a random variable Y by Y := f • (X) + ξ. We denote the joint distribution of (X, Y ) by P. Suppose we are given a dataset D = ((x 1 , y 1 ), . . . , (x N , y N )) independently and identically sampled from the distribution P, we want to estimate the true function f • from the finite dataset D.

We evaluate the performance of an estimator by the squared error. For a measurable function f :

[-1, 1] D → R, we define the empirical error of f by RD (f ) := N n=1 (y n -f (x n )) 2 and the estimation error by R(f ) := E X,Y (f (X) -Y ) 2 . Given a subset F of measurable functions from [-1, 1] D → R, we consider the clipped empirical risk minimization (ERM) estimator f of F that satisfies f := clip[f min ] where f min ∈ arg min f ∈F RD (clip[f ]).

Here, clip is the clipping operator defined by clip

[f ] := (f ∨ -f • ∞ ) ∧ f • ∞ .

For a measurable function f : [-1, 1] D → R, we define the L 2 -norm (weighted by P X ) and the sup norm of f by

f L 2 (P X ) := [-1,1] D f 2 (x)dP X (x) 1 2 and f ∞ := sup x∈[-1,1] D |f (x)|, respectively. Let L 2 (P X ) be the set of measurable functions f such that f L 2 (P X ) < ∞ with the norm • L 2 (P X ) .

The task is to estimate the approximation error min f ∈F f -f • ∞ and the estimation error of the clipped ERM estimator: R( f ) -R(f • ). Note that the estimation error is a random variable with respect the choice of the training dataset D. By the definition of R and the independence of X and ξ, the estimation error equals to f -f • 2 L 2 (P X ) .

3.2. CONVOLUTIONAL NEURAL NETWORKS:

In this section, we define CNNs used in this paper. For this purpose, it is convenient to introduce 0 , the set of real-valued sequences whose finitely many elements are non-zero:

0 := {w = (w n ) n∈N>0 | ∃N ∈ N >0 s.t.

w n = 0, ∀n ≥ N }. w = (w 1 , . . . , w K ) ∈ R K can be regarded as an element of 0 by setting w n = 0 for all n > K. Likewise, for C, C ∈ N >0 , which will be the input and output channel sizes, respectively, we can think of (w k,j,i

) k∈[K],j∈[C ],i∈[C] ∈ R K×C ×C as an element of C ×C 0 . For a filter w = (w n,j,i ) n∈N>0,i∈[C],j∈[C ] ∈ C ×C 0

, we define the one-sided padding and stride-one convolution by w as an order-4 tensor

L w D = ((L w D ) β,j α,i ) ∈ R D×D×C ×C by (L w D ) β,j α,i := w (α-β+1),j,i if 0 ≤ α -β ≤ D -1 0 otherwise.

Here, i (resp. j) runs through 1 to C (resp. C ) and α and β runs through 1 to D. Since we fix the input dimension D throughout the paper, we will omit the subscript D and write as L w if it is obvious from context. Remark 1. For K ≤ K , we can embed R K into R K by inserting zeros: w = (w 1 , . . . , w K ) → w = (w 1 , . . . , w K , 0, . . . , 0). It is easy to show L w = L w . Using this equality, we can expand a size-K filter to size-K .

We can interpret L w as a linear mapping from R D×C to R D×C . Specifically, for x = (x α,i ) α,i ∈ R D×C , we define (y β,j ) β,j = L w (x) ∈ R D×C by

y β,j := i,α (L w ) β,j α,i x α,i .

Next, we define the building block of CNNs: convolutional layers and fully-connected layers. Let C, C , K ∈ N >0 be the input channel size, output channel size, and filter size, respectively. 

→ R by FC σ W,b (a) = σ(vec(W ) vec(a) -b).

Here, vec(•) is the vectorization operator that flattens a matrix into a vector.

Finally, we define the ResNet-type CNN as a sequential concatenation of one convolution block, M residual blocks, and one fully-connected layer. Figure 1 is the schematic view of the CNN we adopt in this paper. Definition 1 (Convolutional Neural Networks (CNNs)). Let M ∈ N >0 and L m ∈ N >0 , which will be the number of residual blocks and the depth of m-th block, respectively. Let

C (l) m , K (l)

m be the channel size and filter size of the l-th layer of the m-th block for m = 0, . . . , M1 and l ∈

[L m ]. We assume C (L0) 0 = C (L1) 1 = • • • = C (L M ) M . Let w (l) m ∈ R K (l) m ×C (l) m ×C (l-1) m and b (l)

m ∈ R be the weight tensors and biases of l-th layer of the m-th block in the convolution part, respectively. Finally, let W ∈ R D×C (L 0 ) 0 and b ∈ R be the weight matrix and the bias for the fully-connected layer part, respectively. For θ := ((w

EQUATION

and id :

R D×C (L 0 ) 0 → R D×C (L 0 ) 0

is the identity function.

Although CNN σ θ in this definition has a fully-connected layer, we refer to the stack of convolutional layers both with or without the final fully-connect layer as a CNN in this paper. We say a linear convolutional layer or a linear CNN when the activation function σ is the identity function and a ReLU convolution layer or a ReLU CNN when σ is ReLU defined by ReLU(x) = x ∨ 0. We borrow the term from ResNet and call Conv σ wm,bm (m > 0) and id in the above definition the m-th residual block and the m-th identity mapping, respectively. We say a 4-tuple θ is compatible with (C 

F (CNN) :=    CNN ReLU θ θ = ((w (l) m ) m,l , (b (l) m ) m,l , W, b) is compatible with (C, K), max m=0,...,M,l∈[Lm] w (l) m ∞ ∨ b (l) m ∞ ≤ B (conv) , W ∞ ∨ b ∞ ≤ B (fc)    .

Here, the domain of CNNs is restricted to [-1, 1] D . Note that we impose norm constraints to the convolution part and fully-connected part separately. We emphasize that we do not impose any sparse constraints (e.g., restricting the number of non-zero parameters in a CNN to some fixed value) to F (CNN) , as opposed to previous literature such as Yarotsky (2017) , Schmidt-Hieber (2017), and Imaizumi & Fukumizu (2018) . Since the notation is cluttered, we sometimes omit the subscripts as we do in the above. Remark 2. In this paper, we adopted one-sided padding, which is not often used practically, in order to make proofs simple. However, with slight modifications, all statements are true for equallypadded convolutions, the widely employed padding style which adds (approximately) same numbers of zeros to both ends of an input signal, with the exception that the filter size K is restricted to K ≤ D 2 instead of K ≤ D -1. We also discuss our design choice, especially the comparison with the original ResNet proposed in He et al. (2016) in Section G of the appendix.

3.3. BLOCK-SPARSE FULLY-CONNECTED NEURAL NETWORKS:

In this section, we mathematically define FNNs we consider in this paper, in parallel with the CNN case. Our FNN, which we coin a block-sparse FNN, consists of M possibly dense FNNs (blocks) concatenated in parallel, followed by a single fully-connected layer. We sketch the architecture of a block-sparse FNN in Figure 2 . 

Definition 2 (Fully-connected Neural Networks (FNNs)). Let M ∈ N >0 be the number of blocks in an FNN. Let D m = (D (1) m , . . . , D (Lm) m ) ∈ N Lm >0 be the sequence of intermediate dimensions of the m-th block, where L m ∈ N >0 is the depth of the m-th block for m ∈ [M ] 2 . Let W (l) m ∈ R D (l) m ×D (l-1)

EQUATION

.

We call a block-sparse FNN with M blocks a M -way block-sparse FNN. We say θ is compatible with (D 

F (FNN) :=      FNN ReLU θ θ = ((W (l) m ) m,l , (b (l) m ) m,l , (w m ) m , b) is compatible with D, max m∈[M ],l∈[Lm] ( W (l) m ∞ ∨ b (l) m ∞ ) ≤ B (bs) , max m∈[M ] w m ∞ ∨ |b| ≤ B (fin) .      .

Again, the domain is restricted to [-1, 1] D . Similar to the CNN case, we sometimes remove subscripts of the function class for simplicity.

4. MAIN THEOREMS:

With the preparation in the previous sections, we state our main results of this paper. We only describe statements of theorems and corollaries and key ideas in the main article. All complete proofs are deferred to the appendix. 

:= D-1 K-1 . Let L m ∈ N >0 , D (l) m ∈ N >0 (m ∈ [M ]), and D = (D (l) m ) m∈[M ],l∈[Lm] . Then, there exist L m ∈ N >0 (m = 0, . . . , M ), C = (C (l) m ) m=0,...,M,l∈[L m ] , and K = (K (l) m ) m=0,...,M,l∈[L m ] satisfying the following conditions: 1. L 0 ≤ 1, L m ≤ L m + L 0 (m ∈ [M ]), 2. max m=0,...,M,l∈[L m ] C (l) m ≤ 4 max m∈[M ],l∈[Lm]

D (l) m , and

3. max m=0,...,M,l∈[L m ] K (l) m ≤ K

such that, for any B (bs) , B (fin) > 0, we have

EQUATION

that is, any FNN in

F (FNN) D,B (bs) ,B (fin) can be realized by a CNN in F (CNN) C,K,B (conv) ,B (fc) . Here, B (conv) = B (bs) and B (fc) = B (fin) 1 ∨ 1 B (bs) .

An immediate consequence of this theorem is that if we can approximate a function f • with a blocksparse FNN, we can also approximate f • with a CNN.

4.2. ESTIMATION:

Our second main theorem bounds the estimation error of the clipped ERM estimator f .

Theorem 2. Let f • : R D → R be a measurable function and B (bs) , B (fin) > 0. Let M , K, L 0 , L m , D, B (conv) and B (fc) as in Theorem 1. Suppose L m , C, K satisfies the equation (1) of Theorem 1 for B (bs) and B (fin) (their existence is ensured for any B (bs) and B (fin) if they satisfy the conditions 1 -3. of Theorem 1). Suppose that the covering nubmer of

EQUATION

Here, fc) . M 1 and M 2 are defined by

F (FNN) := F (FNN) D,B (bs) ,B (fin) , C > 0 is a universal constant, F := f • ∞ σ ∨ 1 2 , and B = B (conv) ∨ B (

M 1 := (2M + 3)C (L 0 ) 0 D(1 ∨ B (fc) )(1 ∨ B (conv) ) M m=0 (1 + ρ m ) 1 + M m=0 L m ρ + m , M 2 := M m=0 L m l=1 C (l-1) m C (l) m K (l) m + C (l) m + C (L 0 ) 0 D + 1,

where

ρ m := L m l=0 C (l-1) m K (l) m B (conv) and ρ + m := L m l=0 (1 ∨ C (l-1) m K (l) m B (conv) ).

The first term of ( 2) is the approximation error achieved by F (FNN) . On the other hand, M 1 and M 2 are determined by the architectural parameters of F (CNN) -M 1 corresponds to the Lipschitz constant of a function realized by a CNN and M 2 is the number of parameters, including zeros, of a CNN. Therefore, the second term of (2) represents the model complexity of F (CNN) . There is a trade-off between the two terms. Using appropriately chosen M to balance them, we can evaluate the order of estimation error with respect to the sample size N .

Corollary 1. Under the same assumptions as Theorem 2, suppose further log M 1 (B (conv) ∨B (fc) ) = Õ(1) as a function of

M . If inf f ∈F (FNN) f -f • 2 ∞ = Õ(M -γ1

) and M 2 = Õ(M γ2 ) for some constant γ 1 , γ 2 > 0 independent of M , then, the clipped ERM estimator f of F achieves the estimation

error f • -f 2 L2(P X ) = Õp N -2γ 1 2γ 1 +γ 2

.

5. APPLICATION OF MAIN THEOREMS:



5.1. BARRON CLASS:

The Barron class is an example of the function class that can be approximated by block-sparse FNNs. We employ the definition of Barron functions used in Klusowski & Barron (2016) .

Definition 3 (Barron class). We say a measurable function f

• : [-1, 1] D → R is a Barron function with the parameter s > 0 if f • admits the Fourier representation (i.e., f • (x) = FF[f • ]) and v f • := R D w s 2 |F[f • ](w)| dw < ∞.

Here, F and F are the Fourier transformation and the inverse Fourier transformation, respectively. Klusowski & Barron (2016) studied the approximation of the Barron function f • with the parameter s = 2 by a linear combination of M ridge functions (i.e., a 2-layered ReLU FNN). Specifically, they showed that there exists a function f M of the form

f M := f • (0) + ∇f • (0)x + 1 M M m=1 b m (a m x -t m ) + (3) with |b m | ≤ 1, a m 1 = 1 and |t m | ≤ 1, such that f • -f M ∞ = Õ M -( 1 2 + 1 D )

. Using this approximator f M , we can derive the same approximation order using CNNs by applying Theorem 1 with

L 1 = • • • = L M = 1 and D (1) 1 = • • • = D (1) M = 1. Corollary 2. Let f • : [-1, 1] D →

R be a Barron function with the parameter s = 2 such that f • (0) = 0 and ∇f • (0) = 0 D . Then, for any K = 2, . . . , D, there exists a CNN f (CNN) with M residual blocks, each of which has depth O(1) and at most 4 channels, and whose filter size is at

most K, such that f • -f (CNN) ∞ = Õ M -( 1 2 + 1 D ) .

We have one design choice when we apply Corollary 1 to derive the estimation error: how to set B (bs) and B (fin) . Looking at (3), the naive choice would be B (bs) := 1 and B (fin) := 1 M . However, this cannot satisfy the assumption on M 1 of Corollary 1, due to the term M m=0 (1 + ρ m ) whose logarithm is O(M ). We want its logarithm to be Õ(1). In order to do that, we change the relative scale between parameters in the block-sparse part and the fully-connected part using the homogeneous property of the ReLU function: ReLU(ax) = aReLU(x) for a > 0. The rescaling operation enables us to choose B (bs) := 1 M and B (fin) = 1 to meet the assumption of Corollary 1. By setting such that for sufficiently large N , the clipped ERM estimator f of

γ 1 = 1 2 + 1 D and γ 2 = 1,

F := {clip[f ] | f ∈ F (CNN) C,K,S,B (conv) ,B (fc) } achieves the estimation error f • -f 2 L2(P X ) = Õp N -D+2 2(D+1)

. Here,

C (l) m = C, K (l) m = K for m = 0, . . . , M, l ∈ [L] and define C = (C (l) m ) m,l , K = (K (l) m ) m,l .

5.2. H ÖLDER CLASS:

We next consider the approximation and error rates of CNNs when the true function is a β-Hölder function.

Definition 4 (Hölder class).

Let β > 0, f • : [-1, 1] D → R is a β-Hölder function if f • β := 0≤|α|< β ∂ α f • ∞ + |α|= β sup x =y |∂ α f • (x) -∂ α f • (y)| |x -y| β-β < ∞.

Here, α = (α 1 , . . . , α D ) is a multi-index. That is, ∂ α f := ). Schmidt-Hieber (2017) also proved a similar statement using a different construction method. They only specified their width (Schmidt-Hieber (2017) only), depth, and non-zero parameter counts of the approximating FNN and did not write in detail how non-zero parameters are distributed explicitly in the statements (see Theorem 1 of Yarotsky (2017) and Theorem 5 of Schmidt-Hieber ( 2017)). However, if we carefully look at their proofs, we find that we can transform the FNNs they constructed into the block-sparse ones. Therefore, we can utilize these FNNs and apply Theorem 1. To meet the assumption of Corollary 1, we again rescale the parameters of the FNNs, as we did in the Barron class case, so that log M 1 = Õ(1). We can derive the approximation and estimation errors by setting γ 1 = β D and γ 2 = 1. Corollary 4. Let β > 0, and f • : [-1, 1] D → R be a β-Hölder function. Then, for any K = 2, . . . , D, there exists a CNN f (CNN) with O(M ) residual blocks, each of which has depth O(log M ) and O(1) channels, and whose filter size is at most K, such that f 

∂ |α| f ∂x α 1 1 •••∂x

EQUATION

:= {clip[f ] | f ∈ F (CNN) C,K,S,B (conv) ,B (fc) } achieves the estimation error f • -f 2 L2(P X ) = Õp N -2β 2β+D

. Here,

C (l) m = C, K (l) m = K for m = 0, . . . , M, l ∈ [L] and define C = (C (l) m ) m,l , K = (K (l) m ) m,l .

Since the estimation error rate of the β-Hölder class is O p N -2β 2β+D (see, e.g., Tsybakov (2008)), Corollary 5 implies that our CNN can achieve the minimax optimal rate up to logarithmic factors even the width D, the channel size C, and the filter size K are constant with respect to the sample size N .

6. CONCLUSION:

In this paper, we established new approximation and statistical learning theories for CNNs by utilizing the ResNet-type architecture of CNNs and the block-sparse structure of FNNs. We proved that any M -way block-sparse FNN is realizable using CNNs with O(M ) additional parameters, when the width of the FNN is fixed. Using this result, we derived the approximation and estimation errors for CNNs from those for block-sparse FNNs. Our theory is general because it does not depend on a specific function class, as long as we can approximate it with block-sparse FNNs. To demonstrate the wide applicability of our results, we derived the approximation and error rates for the Barron class and Hölder class in almost same manner and showed that the estimation error of CNNs is same as that of FNNs, even if the CNNs have a constant channel size, filter size, and width with respect to the sample size. The key techniques were careful evaluations of the Lipschitz constant of CNNs and non-trivial weight parameter rescaling of FNNs.

One of the interesting open questions is the role of the weight rescaling. We critically use the homogeneous property of the ReLU activation function to change the relative scale between the block-sparse part and the fully-connected part, if it were not for this property, the estimation error rate would be worse. The general theory for rescaling, not restricted to the Barron nor Hölder class would be beneficial for deeper understanding of the relationship between the approximation and estimation capabilities of FNNs and CNNs.

Another question is when the approximation and estimation error rates of CNNs can exceed that of FNNs. We can derive the same rates as FNNs essentially because we can realize block-sparse FNNs using CNNs that have the same order of parameters (see Theorem 1). Therefore, if we dig into the internal structure of FNNs, like repetition, more carefully, the CNNs might need fewer parameters and can achieve better estimation error rate. Note that there is no hope to enhance this rate for the Hölder case (up to logarithmic factors) because the estimation rate using FNNs is already minimax optimal. It is left for future research which function classes and constraints of FNNs, like block-sparseness, we should choose.

A NOTATION:

For tensor a, a + := a∨0 where maximum operation is performed in element-wise manner. Similarly a -:= -(-a ∨ 0). Note that a = a + -a -holds for any tensor a. For normed spaces (V, • V ), (W, • W ) and linear operator T : V → W we denote the operator norm of T by T op := sup v V =1 T v W . For a sequence w = (w (1) , . . . , w (L) ) and l ≤ l , we denote its subsequence from the l-th to l -th elements by w[l : l ] := (w (l) , . . . , w (l ) ). 1 P equals to 1 if the statement P is true, equals to 0 otherwise.

B PROOF OVERVIEW B.1 THEOREM 1:

For f (FNN) ∈ F (FNN) , we realize a CNN f (CNN) using M residual blocks by "serializing" blocks in the FNN and converting them into convolution layers.

First, we double the channel size using the m = 0 part of CNN (i.e., D (L0) 0

= 2). We will use the first channel for storing the original input signal for feeding to downstream (i.e., m ≥ 1) blocks and the second one for accumulating the output of each blocks, that is, m m=1 w m FC ReLU Wm,bm (x) where w m is the weight of the final fully-connected layer corresponding to the m-th dense block.

For m = 1, . . . , M , we create the m-th residual block from the m-th block of f (FNN) . First, we show that for any a ∈ R D and t ∈ R, there exists L 0 -layered 4-channel ReLU CNN with O(D) parameters whose first output coordinate equals to a ridge function x → (a x -t) + (Lemma 1 and Lemma 2). Since the first layer of m-th block is concatenation of D m size-1 filters made from the weight parameters of the corresponding layer of the FNN. Observing that the convolution operation with size-1 filter is equivalent to a dimension-wise affine transformation, the first coordinate of the output of l-th layer of the CNN is inductively same as that of the m-th block of the FNN. After computing the m-th block FNN using convolutions, we add its output to the accumulating channel in the identity mapping.

Finally, we pick the first coordinate of the accumulating channel and subtract the bias term using the final affine transformation.

B.2 THEOREM 2 AND COROLLARY 1:

We relate the approximation error of Theorem 2 with the estimation error using the covering number of the hypothesis class F (CNN) . Although there are several theorems of this type, we employ the one in Schmidt-Hieber (2017) due to its convenient form (Lemma 5). We can prove that the logarithm of the covering number is upper bounded by M 2 log((B (conv) ∨ B (fc) )M 1 /ε) (Lemma 4) using the similar techniques to the one in Schmidt-Hieber (2017). Theorem 2 is the immediate consequence of these two lemmas.

To prove Cororraly 1, we set M = O(N α ) for some α ≥ 0. Then, under the assumption of the corolarry, we have f

• -f 2 L 2 (Px) = Õ max N -2αγ1

, N αγ2-1 from Theorem 2. The order of the right hand side with respect to N is minimized when α = 1 2γ1+γ2 . By substituting α, we can prove Corollary 1.

C PROOF OF THEOREM 1 C.1 DECOMPOSITION OF AFFINE TRANSFORMATION:

The following lemma shows that any affine transformation is realizable with a D-1 K-1 -layered linear conventional CNN (without the final fully-connect layer).

Under review as a conference paper at 2019 Lemma 1. Let a ∈ R D , t ∈ R, K ∈ {2, . . . , D -1}, and L 0 := D-1 K-1 . Then, there exists

EQUATION

3. Conv id w,b : R D → R D satisfies Conv id w,b (x) = a x -t for any x ∈ [-1, 1] D .

Proof. First, observe that the convolutional layer constructed from u = [u 1 . . . u K ] ∈ R K×1×1 takes the inner product with the first K elements of the input signal:

L u (x) = K k=1 u k x k .

In particular, u = [0 . . . 0 1] ∈ R K×1×1 works as the "left-translation" by K -1. Therefore, we should define w so that it takes the inner product with the K left-most elements in the first channel and shift the input signal by K -1 with the second channel. Specifically, we define w = (w (1) , . . . , w (L0) ) by

(w (1) ) :,1,: =    a 1 . . . a K    , (w (1) ) :,2,: =     0 . . . 0 1     , (w (l) ) :,1,: =    0 a (l-1)K+1 . . . . . . 0 a lK    , (w (l) ) :,2,: =     0 0 . . . . . . 0 0 1 0     , (w (L0) ) :,1,: =          0 a (L0-1)K+1 . . . . . . 0 a D 0 0 . . . . . . 0 0          .

We set b := ( 0, . . . , 0 L0 -1 times , t). Then w and b satisfy the condition of the lemma.

C.2 TRANSFORMATION OF A LINEAR CNN INTO A RELU CNN:

The following lemma shows that we can convert any linear CNN to a ReLU CNN that has approximately 4 times larger parameters. This type of lemma is also found in Petersen & Voigtlaender (2017) l) . Consider the linear convolution layers constructed from w and b:

(Lemma 2.3). Lemma 2. Let C = (C (1) , . . . , C (L) ) ∈ N L >0 be channel sizes K = (K (1) , . . . , K (L) ) ∈ N L >0 be filter sizes. Let w (l) ∈ R K (l) ×C l ×C (l) and b (l) ∈ R (

f id := Conv id w,b : R D → R D×C (L)

N L >0 where w = (w (l) ) l and b = (b (l) ) l . Then, there exists a pair w = ( w(l)

) l∈[L] , b = ( b(l) ) l∈[L] where w(l) ∈ R K (l) ×2C (l) ×2C (l-1) and b(l) ∈ R 2C (l) such that 1. L l=1 w(l) 0 ≤ 4 L l=1 w (l) 0 , L l=1 b(l) 0 ≤ L l=1 b (l) 0 , 2. max l∈[L] w(l) ∞ = max l∈[L] w (l) ∞ , max l∈[L] b(l) ∞ = max l∈[L]

b (l) ∞ , and

3. f ReLU := Conv ReLU w, b : R D → R D×2C (L) , satisfies f ReLU (•) = (f id (•) + , f id (•) -).

Proof. We define w and b as follows:

( w(1) ) k,:,: = (w (1) ) k,:,:

-(w (1) ) k,:,: for k = 1, . . . , K (1) , ( w(l) ) k,:,: = (w (l) ) k,:,: -(w (l) ) k,:,:

-(w (l) ) k,:,: (w (l) ) k,:,:

for k = 1, • • • K (l) , b(l) = b (l) -b (l)

By definition, a pair ( w, b) satisfies the conditions ( 1) and ( 2). For any x ∈ R D , we set

y (l) := Conv id w[1:l],b[1:l] (x) ∈ R C (l)

×D . We will prove

EQUATION

for l = 1, . . . , L by induction. Note that we obtain

f ReLU (•) = (f id+ (•), f id-(•)) by setting l = L.

For l = 1, by definition of w(1) we have, ( w(1) ) α,:,: x β,: = (w (1) ) α,:,: x β,:

-(w (1) ) α,:,: x β,:

for any α, β ∈ [D]. Summing them up and using the definition of b( 1) yield

[L w(1) (x) -1 D ⊗ b(1) ] = L w (1) (x) -1 D ⊗ b (1) -L w (1) (x) -1 D ⊗ b (1)

Suppose (4) holds up to l (l < L), by the definition of w(l+1) , ( w(l+1) ) α,:,:

(y -(w (l+1) ) α,:,: (y (l) ) β,:

for any α, β ∈ [D]. Again, by taking the summation and using the definition of b(l+1) , we get l+1) .

[L w(l+1) ([y (l) + , y (l) -]) -1 D ⊗ b(1) ] = L w (l+1) (y (l) ) -1 D ⊗ b (l+1) -L w (l+1) (y (l) ) -1 D ⊗ b (

By applying ReLU, we get ) , -y (l+1) ] .

Conv p (l+1) ,ReLU w(l+1) , b(l+1) [y (l) + , y (l) -] = ReLU [y (l+1

(5)

By using the induction hypothesis, we get

Conv ReLU w[1:(l+1)], b[1:(l+1)] (x) = Conv p (l+1) ,ReLU w(l+1) , b(l+1) [y (l) + , y (l) -] = ReLU [y (l+1) , -y (l+1) ] = [y (l+1) + , -y (l+1) - ]

Therefore, the claim holds for l + 1. By induction, the claim holds for L, which is what we want to prove.

C.3 CONCATENATION OF CNNS:

We can concatenate two CNNs with the same depths and filter sizes in parallel. Although it is almost trivial, we state it formally as a proposition. In the following proposition, C (0) and C (0) is not necessarily 1.

Proposition 1. Let C = (C (l) ) l∈[L] , C = (C (l) ) l∈[L] , and K = (K (l) ) l∈[L] ∈ N L >0 . Let w (l) ∈ R K (l) ×C (l) ×C (l-1) , b ∈ R C (l)

and denote w = (w (l) ) l and b = (b (l) ) l . We define w and b in the same way, with the exception that C (l) is replaced with C (l) . We define w = ( w(1) , . . . , w(L) ) and b = ( b(1) , . . . , b(L) ) by ) ]. Then, we have,

( w(l) ) k,:,: := w (l) 0 0 w (l) ∈ R (C (l) +C (l) )×(C (l-1) +C (l-1) ) b(l) := b (l) b (l) ∈ R (C (l) +C (l) ) for l ∈ [L] and k ∈ [K (l

EQUATION

for any x, x ∈ R D×C (0) and any σ : R → R.

Note that by the definition of • 0 and • ∞ , we have

L l=1 w(l) 0 = L l=1 w (l) 0 + w (l) 0 , L l=1 b(l) 0 = L l=1 b (l) 0 + b (l) 0 , max l∈[L] w(l) ∞ = max l∈[L] w (l) ∞ ∨ w (l)

∞ , and

max l∈[L] b(l) ∞ = max l∈[L] b (l) ∞ ∨ b (l) ∞ .

C.4 PROOF OF THEOREM 1:

By the definition of F (FNN) D,B (bs) ,B (fin) , there exists a 4-tuple θ = ((W

(l) m ) m,l , (b (l) m ) m,l , (w m ) m , b) compatible with (D (l) m ) m,l (m ∈ [M ] and l ∈ [L m ]) such that max m∈[M ],l∈[Lm] ( W (l) m ∞ ∨ b (l) m ∞ ) ≤ B (bs) , max m∈[M ] w m ∞ ∨ |b| ≤ B (fin) ,

and f (FNN) = FNN ReLU θ . We will construct the desired CNN consisting of M residual blocks, whose m-th residual block is made from the ingredients of the corresponding m-th block in f (FNN) (specifically, W m := (W

(l) m ) l∈[Lm] , b m := (b (l) m ) l∈[Lm]

, and w m ).

[The m = 0 Block]: We prepare a single convolutional layer with 2 output channels and 2 size-1 filters suth that the first filter works as the identity function and the second filter inserts zeros to the second channel. Weight parameters of this convolutional layer are all zeros except single one. We denote this block by Conv 0 .

[The m = 1, . . . , M Blocks]: For fixed m ∈ [M ], we first create a CNN realizing FC ReLU Wm,bm . We treat the first layer (i.e. l = 1) of FC ReLU Wm,bm as concatenation of D

(1)

m hinge functions R D x → f d (x) := ((W (1) m ) d x -b (1) m ) + for d ∈ [D (1) m ]. Here, (W (1) m ) d ∈ R 1×D is the d-th row of the matrix W (1) m ∈ R D (1)

m ×D . We apply Lemma 1 and Lemma 2 and obtain ReLU CNNs realizing the hinge functions. By combining them in parallel using Proposition 1, we have a learnable parameter θ . By the inductive calculation, we have

Conv m (x) 1 = FC ReLU Wm,bm (x) ⊗ 1 0 if L m = 1 FC ReLU Wm,bm (x) if L m ≥ 2 .

By definition, Conv m has the depth of L 0 + L m -1, at most 4D

(1)

m ∨ max l=2,...Lm D (l) m ≤ 4 max l∈[Lm] D (l)

m channels. The ∞-norm of its parameters does not exceed that of parameters in FC ReLU Wm,bm .

Next, we consider the filter wm ∈ R 1×2×D (out) m defined by 

( wm ) 1,:,: = B (bs) B (fin)        0 • • • 0 w m ⊗ 0 1 if L m = 1 0 • • • 0 w m if L m ≥ 2 ,

f (conv) := (Conv M • Conv M + I) • • • • • (Conv 1 • Conv 1 + I) • Conv 0 .

Then, we have

f (conv) 1 = B (bs) B (fin) M m=1 w m FC ReLU Wm,bm

(the subscript 1 represents the first coordinate).

[Final Fully-connected Layer] Finally, we set w := B (fin)

B (bs) 0 • • • 0 ∈ R D and put FC id w,b

on top of f (conv) to pick the first coordinate of f (conv) and subtract the bias term. By definition, FNN) .

f (CNN) := FC id w,b • f (conv) satisfies f (CNN) = f (

[Condition Check]: We will check f (FNN) satisfies the desired conditions. (Condition 1): By definition the 0-th residual block Conv 0 has L 0 = 1 layer. Since Conv m and Conv m has L 0 + L m -1 and 1 layers, respectively, the m(≥ 1)-th residual block of f (CNN) has

L m = L 0 + L m layers. (Condition 2): Conv m has at most 4 max l∈[Lm] D (l)

m channels and Conv m has at most 2 channels, respectively. Therefore, the channel size of f (CNN) 

is at most 4 max m∈[M ],l∈[Lm] D (l) m .

(Condition 3): Since each filter of Conv (m) and Conv m is at most K, the filter size of CNN is also at most K. (Conditions on B (conv) and B (fin) ): Parameters of f (conv) are either 0, or parameters of FC ReLU Wm,Wm , whose absolute value is bounded by B (bs) , or B (bs) B (fin) w m . Since we have w m ∞ ≤ B (fin) , the ∞-norm of parameters in f (CNN) is bounded by B (bs) . The parameters of the final fully-connected layer FC w,b is either B (fin) , 0, or b, therefore their norm is bounded by

B (fin)

B (bs) ∨ B (fin) . Remark 3. Another way to construct a CNN which is identical (as a function) to a given FNN is as follows. First, we use a "rotation" convolution with D filters, each of which has a size D, to serialize all input signals to channels of a single input dimension. Then, apply size-1 convolution layers, whose l-th layer consisting of appropriately arranged weight parameters of the l-th layer of the FNN. This is essentially what Petersen & Voigtlaender (2018) does to prove the existence of a CNN equivalent to a given FNN. To restrict the size of filters to K, we should further replace the the first convolution layer with O(D/K) convolution layers with size-K filters. We can show essentially same statement using this construction method.

D PROOF OF THEOREM 2 D.1 COVERING NUMBER OF CNNS:

The goal of this section is to prove Lemma 4, stated in Section D.1.5, that evaluates the covering number of the set of functions realized by CNNs F (CNN) .

D.1.1 BOUNDS FOR CONVOLUTIONAL LAYERS:

We assume w, w ∈ R K×J×I , b, b ∈ R, and x ∈ R D×I unless specified. We have in mind that the activation function σ is either the ReLU function or the identity function id. But the following proposition holds for any 1-Lipschitz function such that σ(0) = 0. Remember that we can treat L w as a linear operator from R D×I to R D×J . We endow R D×I and R D×J with the sup norm and denote the operator norm L w by L w op . Proposition 2. It holds that

L w op ≤ IK w ∞ . Proof. Write w = (w kji ) k∈[K],j∈[J],i∈[I] , L w = ((L w ) β,j α,i ) α,β∈[D],j∈[J],i∈[I] . For any x = (x α,i ) α∈[D],i∈[I] ∈ R D×I , the sup norm of y := (y βj ) β∈[D]j∈[J] = L w (x) is evaluated as fol- lows: y ∞ = max β,j |y β,j | ≤ max β,j α,i |(L w ) β,j α,i ||x α,i | ≤ max β,j α,i |(L w ) β,j α,i | x ∞ = max β,j α,i |w (α-β+1),j,i | x ∞ ≤ max β,j α,i 1{w (α-β+1),j,i = 0} w ∞ x ∞ ≤ IK w ∞ x ∞ Proposition 3. It holds that Conv σ w,b (x) ∞ ≤ L w op x ∞ + |b|. Proof. Conv σ w,b (x) ∞ ≤ σ(L w (x) -1 D ⊗ b) ∞ ≤ L w (x) -1 D ⊗ b ∞ ≤ L w (x) ∞ + 1 D ⊗ b ∞ ≤ L w op x ∞ + |b|.

Proposition 4. The Lipschitz constant of Conv σ w,b is bounded by L w op .

Proof. For any

x, x ∈ R D×I , Conv σ w,b (x) -Conv σ w,b (x ) ∞ = σ (L w (x) -1 D ⊗ b) -σ (L w (x ) -1 D ⊗ b) ∞ ≤ (L w (x) -1 D ⊗ b) -(L w (x ) -1 D ⊗ b) ∞ ≤ L w (x -x ) ∞ ≤ L w op x -x ∞ .

Note that the first inequality holds because the ReLU function is 1-Lipschitz.

Proposition 5. It holds that Conv σ w,b (x) -Conv σ w ,b (x) ≤ L w-w op x ∞ + |b -b |.

Proof.

Conv σ w,b (x) -Conv σ w ,b (x) = σ(L w (x) -1 D ⊗ b) -σ(L w (x) -1 D ⊗ b ) ∞ ≤ (L w (x) -1 D ⊗ b) -(L w (x) -1 D ⊗ b ) = L w (x) -L w (x) + 1 D ⊗ (b -b ) ∞ ≤ L w-w op x ∞ + |b -b | D.1.2 BOUNDS FOR FULLY-CONNECTED LAYERS

In the following propositions in this subsection, we assume W, W ∈ R D×C , b, b ∈ R, and x ∈ R D×C . Again, these propositions hold for any 1-Lipschitz function σ : R → R such that σ(0) = 0. But σ = ReLU or id is enough for us. Proposition 6. It holds that

|FC σ W,b (x)| ≤ W 0 W ∞ x ∞ + |b|.

Proof.

|FC σ W,b (x)| ≤ |vec(W ) vec(x) -b| ≤ |vec(W ) vec(x)| + |b| ≤ α,i W α,i x α,i + |b|

The number of non-zero summand in the summation is at most W 0 and each summand is bounded by W ∞ x ∞ Therefore, we have

|FC σ W,b (x)| ≤ W 0 W ∞ x ∞ + b ∞ . Proposition 7. The Lipschitz constant of FC σ W,b is bounded by W 0 W ∞ . Proof. For any x, x ∈ R D×C , |FC σ W,b (x) -FC σ W,b (x )| ≤ (vec(W ) vec(x) -b) -(vec(W ) vec(x ) -b) ≤ vec(W ) (vec(x) -vec(x )) ≤ W 0 W ∞ vec(x) -vec(x ) ∞ . Proposition 8. It holds that |FC σ W,b (x) -FC σ W ,b (x)| ≤ ( W 0 + W 0 ) W -W ∞ x ∞ + |b -b |. Proof. |FC σ W,b (x) -FC σ W ,b (x)| ≤ |(vec(W ) vec(x) -b) -(vec(W ) vec(x) -b )| = |(vec(W -W ) vec(x) -(b -b )| ≤ |(vec(W -W ) vec(x)| + |b -b | ≤ W -W 0 W -W ∞ x ∞ + |b -b | ≤ ( W 0 + W 0 ) W -W ∞ x ∞ + |b -b | D.1.3 BOUNDS FOR RESIDUAL BLOCKS

In this section, we denote the architecture of CNNs by

C = (C (l) ) l∈[L] ∈ N L >0 and K = (K (l) ) l∈[L] ∈ N L

>0 and the norm constraint on the convolution part by B (conv) (C (0) need not equal to 1 in this section). Let w l , w (l) ∈ R K (l) ×C l ×C (l-1) and b (l) , b (l) ∈ R. We denote

w := (w (l) ) l∈[L] , b := (b (l) ) l∈[L] , w := (w (l) ) l∈[L]

, and b := (b (l) ) l∈ [L] .

For 1 ≤ l ≤ l ≤ L, we denote ρ(l, l ) := l i=l (C (i-1) K (i) B (conv) ) and ρ + (l, l ) : conv) . Then, for any

= l i=l 1 ∨ (C (i-1) K (i) B (conv) ). Proposition 9. Let l ∈ [L]. We assume max l∈[L] w (l) ∞ ∨ b (l) ∞ ≤ B (

x ∈ [-1, 1] D×C (0) , we have Conv σ w[1:l],b[1:l] (x) ∞ ≤ ρ(1, l) x ∞ + B (conv) lρ + (1, l).

Proof. We write in shorthand as C 

C [1:l] (x) ∞ ≤ L w (l) op C [1:l-1] (x) ∞ + b (l) ∞ . . . ≤ x ∞ l i=1 L w (i) op + l i=2 b (i-1) ∞ l j=i L w (j) op + b (l) ∞ .

By Proposition 2 and assumptions w conv) , it is further bounded by

(i) ∞ ≤ B (conv) and b (i) ∞ ≤ B (

x ∞ l i=1 (C (i-1) K (i) B (conv) ) + B (conv) l i=2 l j=i (C (j-1) K (j) B (conv) ) + B (conv) ≤ ρ(1, l) x ∞ + B (conv) lρ + (1, l) Proposition 10. Let ε > 0, suppose max l∈[L] w (l) -w (l) ∞ ≤ ε and max l∈[L] b (l) -b (l) ∞ ≤ ε, then C [1:L] -C [1:L] (x) ∞ ≤ (Lρ(1, L) x ∞ +(1∨B (conv) )L 2 ρ + (1, l))ε for any x ∈ R D×C (0) .

Proof. For any l ∈ [L], we have

EQUATION

between the m-th and m -th block. That is,

C [m:m ] := (C m + I) • • • • • (C m + I) (if m ≥ 1) (C m + I) • • • • • C m (if m = 0)

for m, m = 0, . . . , M . We define C 

] (x) ∞ ≤ (1 ∨ B (conv) ) ( m i=0 (1 + ρ i )) 1 + m i=0 L i ρ + i .

Here, ρ m and ρ + m are constants defined in Theorem 2.

Proof. By using Proposition 9 inductively, we have

EQUATION

We will bound each term of (7). By Proposition 8 and Proposition 11,

EQUATION

On the other hand, for m = 0, . . . , M ,

EQUATION

By applying ( 8) and ( 9) to (7), we have fc) . Therefore, by Lemma 5,

|CNN ReLU θ (x) -CNN ReLU θ (x)| ≤ 3C (L0) 0 D(1 ∨ B (conv) ) M m=0 (1 + ρ m ) 1 + M m=0 L m ρ + m ε + 2M C (L0) 0 DB (fc) (1 ∨ B (conv) ) M m=0 (1 + ρ m ) 1 + M m=0 L m ρ + m ε ≤ (2M + 3)C (L0) 0 D(1 ∨ B (fc) )(1 ∨ B (conv) ) M m=0 (1 + ρ m ) 1 + M m=0 L m ρ + m ε = M 1 ε. L 2 (P X ) to conclude. Proof of Theorem 2. By definition of • ∞ , we have f -f • L 2 (P X ) ≤ f -f • ∞ for any f ∈ F. By Lemma 4, log N := log N ( 1 N , F (CNN) , • ∞ ) ≤ M 2 log(2BM 1 N ), where B = B (conv) ∨ B (

f • -f 2 L 2 (P X ) ≤ 4 inf f ∈F f -f • 2 L 2 (P X ) + (56 log N + 180) F 2 N ≤ C inf f ∈F f -f • 2 ∞ + M 2 F 2 N log(2BM 1 N ) .

Proof of Corollay 1. We only care the order with respect to N in the O-notation. Set M = N α for α ≥ 0. Using the assumptions of the corollary, the estimation error is

f • -f 2 L 2 (Px) = Õ max N -2αγ1 , N αγ2-1 .

by Theorem 2. The order of the right hand side with respect to N is minimized when α = Proof. First, we prove the lemma when the domain of f

(FNN) ∈ F (FNN) D,1,2M f • β such that f • -f (FNN) ∞ = Õ(M -β D ).

• is [0, 1] D . Let M be the largest interger satisfying (M + 1) D ≤ M . Let Γ(M ) = Z M D ∩ [0, 1] D = { m M | m ∈ {0, . . . , M } D } be the set of lattice points in [0, 1] D 3 . Note that the cardinality of Γ(M ) is (M + 1) D . Let P β a f • be the Taylor expansion of f • up to order β at a ∈ [0, 1] D : (P β a f • )(x) = 0≤|α|<β (∂ α f • )(a) α! (x -a) α .

For a ∈ [0, 1] D , we define a hat-shaped function

H a : [0, 1] D → [0, 1] by H a (x) := D j=1 (M -1 -|x j -a j | + ).

Note that we have a∈Γ(M ) H a (x) = 1, i.e., they are a partition of unity. Let P β f • be the weighted sum of the Taylor expansions at lattice points of Γ(M ):

(P β f • )(x) := M D a∈D(M ) (P β a f • )(x)H a (x).

By Lemma 7 of Schmidt-Hieber (2017), we have

P β f • -f • ∞ ≤ f • β M -β .

Let m be an interger specified later and set Then, we have

FC a - P β a f • B + 1 2 H a ∞ ≤ 2 -m + 3 D 2 -m + 3 D 2 -m ≤ 3 D+1 2 -m .

We define f (FNN) (x) := a∈Γ(M ) (BM D FC a (x))-B 2 . By construction, f (FNN) is a block-sparse FNN with (M +1) D (≤ M ) blocks each of which has depth and width at most L := 2+L * +(m+4) and D := 6(C D,β + 1)D, respectively. The norms of the block-sparse part and the finally fullyconnected layer are 1 and BM D (≤ BM ), respectively. In addition, we have

|f (FNN) (x) -(P β f • )(x)| ≤ a∈Γ(M ) BM D FC a (x) - (P β a f • )(x) B + 1 2 H a (x) + B 2 1 -M D a∈Γ(M )

H a (x)

≤ (M + 1) D × BM D 3 D+1 2 -m ≤ 3 D+1 2 -m BM 2

for any x ∈ [0, 1] D . Therefore,

|f (FNN) (x) -f • (x)| ≤ |f (FNN) -(P β f • )(x)| + |(P β f • )(x) -f • (x)| ≤ B3 D+1 M 2 2 -m + f • β M -β ≤ 2 f • β 3 D+1 M 2 2 -m + f • β M -β D 2 β .

We set m = log 2 M 2+ β D , then, we have L = O(log M ), D = O(1), and

f (FNN) -f • ≤ f • β (2 • 3 D+1 + 2 β )M -β D .

By the defnition of f (FNN) we have f (FNN) ∈ F

(FNN) D,1,2 f • β M .

When the domain of f • is [-1, 1] D , we should add the function x → 1 2 (x + 1) = 1 2 (x + 1) + -1 2 (-x -1) + as a first layer of each block to fit the range into [0, 1] D . Specifically, suppose the first layer of m-th block in f (FNN) is x → ReLU(W x -b), then the first two layers become x → ReLU( 1 2 (x + 1) -1 2 (x + 1) ) and [y 1 y 2 ] → ReLU(W y 1 -W y 2 -b), respectively. Since this transformation does not change the maximum sup norm of parameters in the block-sparse and the order of L and D , the resulting FNN is still belongs to F 

G COMPARISON OF OUR CNNS AND ORIGINAL RESNET:

There are several differences between the CNN in this paper and the original ResNet, aside from the number of layers. First and foremost, our CNN does not have pooling nor Batch Normalization (Ioffe & Szegedy (2015) ) layers. It is left for future research whether our result can extend to the ResNet-type CNNs with pooling or Batch Normalization layers. Second, our CNN does not have ReLU activation after the junction points and the final layer of the 0-th block, while they have in the original ResNet. We choose this design to make proofs simpler. We can easily extend our results to the architecture that adds the ReLU activations to those points with slight modifications using similar techniques appeared in Lemma 2 of the appendix.

:

Note that m starts from 0. It is convenient for our purpose.

Be aware that contrary to the CNN case, m starts from 1 here.

Schmidt-Hieber (2017) used D(M ) to denote this set of lattice points. We used different character to avoid notational conflict.

We prepare Qa for each a ∈ Γ(M ) as opposed to the original proof of Schmidt-Hieber (2017), in which Qa's shared the layers the except the final one and were collectively denoted by Q1.

Figures and Tables:

Figure 1: ResNet-type CNN defined in Definition 1. Variables are as in Definition 1.

For a weight tensor w ∈ R K×C ×C , a bias vector b ∈ R C , and an activation function σ : R → R, we define the convolutional layer Conv σ w,b : R D×C → R D×C by Conv σ w,b (x) := σ(L w (x) -1 D ⊗ b) where, ⊗ is the outer product of vectors and σ is applied in element-wise manner. Similarly, let W ∈ R D×C , b ∈ R, and σ : R → R, we define the fully-connected layer FC σ W,b : R D×C

(l) m ) m,l and (K (l) m ) m,l when each component of θ satisfies the aforementioned dimension conditions.

Figure 2: Schematic view of a block-sparse FNN. Variables are as in Definition 2.

be the weight matrix and the bias of the l-th layer of mth block (with the convention D (0) m = D). Let w m ∈ R D (Lm) m be the weight (sub)vector of the final fully-connected layer corresponding to the m-th block and b ∈ R be the bias for the last layer. For θ = ((W (l) m ) m,l , (b (l) m ) m,l , (w m ) m , b) and an activation function σ : R → R, we define FNN σ θ : R D → R, the block-sparse FNN constructed from θ, by

) m,l when each component of θ matches the dimension conditions determined by (D (l) m ) m,l , as we did in the CNN case. Note that when L m = 1 for all m ∈ [M ], the blocksparse FNN is a 2-layered neural network with D := M m=1 D (1) m hidden units of the form f (x) = D d=1 b d σ(a d x -t d ) -b where a d ∈ R D and b d , t d , b ∈ R. For an architecture D = (D (l) m ) m∈[M ],l∈[Lm] and norm parameters for the block part B (bs) > 0 and for the final layer B (fin) > 0, we define F (FNN) = F (FNN) D,B (bs) ,B (fin) , the set of function realizable by FNNs:

4.1 APPROXIMATION Our first main theorem claims that any M -way block-sparse FNN is realizable by a ResNet-type CNN with fixed-sized channels and filters by adding O(M ) parameters, if we treat the widths D (l) m of the FNN as constants with respect to M . Theorem 1. Let M ∈ N >0 , K ∈ {2, . . . D} and L 0

we obtain the desired estimation error. Corollary 3. There exist the number of residual blocks M = O N D 2+2D , depth of each residual block L = O (1), channel size C = O(1), filter size K ∈ {2, . . . , D}, and norm bounds for the convolution part B (conv) = O N -D 2+2D , and for the fully-connected part B (fc) = O N D 2+2D

showed that FNNs with O(S) non-zero parameters can approximate any D variate β-Hölder function (β > 0) with the order of Õ(S -β D

5. There exist the number of residual blocks M = O N D 2β+D , depth of each residual block L = Õ (1), channel size C = O(1), filter size K ∈ {2, . . . , D}, norm bounds for the convolution part B (conv) = O(1), and for the fully-connected part B (fc) > 0 (log B (fc) = O(log N )) such that for sufficiently large N , the clipped ERM estimator f of F

ReLU CNN with L 0 -layers.For the l-th layer of the m-th block (m ∈ [M ], l = 2, . . . , L

l+1) ) α,:,: -(w (l+1) ) α,:,: -(w (l+1) ) α,:,: (w (l+1) ) α,:,: l+1) ) α,:,: (y (l) ) β,:

Conv m := Conv id wm,0 adds the output of m-th residual block, weighted by w m , to the second channel in the identity connections, while keeping the first channel intact. Note that the final layer of each residual block does not have the ReLU activation. By definition, Conv m has D (Lm) m parameters. Given Conv m and Conv m for each m ∈ [M ], we construct a CNN realizing FNN ReLU θ . Let f (conv) : R D → R D be the sequential interleaving concatenation of Conv m and Conv m , that is,

[s:t] := Conv σ w[s:t],b[s:t] . Using Proposition 3 recursively, we get

m and C [m:m ] similarly for θ . Proposition 11. For m = 0, . . . M and x ∈ [-1, 1] D , we have C [0:m

-1 B(bs) ,k L B(fin) . Also, by the homogeneous property of the ReLU function (i.e., ReLU(ax) = aReLU(x) for a > 0), we have FNN ReLU θ prove the existence of a block-sparse FNN with constant-width blocks that optimally approximates a given β-Hölder function. It is almost same as the proof of Theorem 5 of Schmidt-Hieber (2017). However, we need to construct the FNN so that it has a block-sparse structure. Lemma 7 (cf. Schmidt-Hieber (2017) Theorem 5 ). Let β > 0, M ∈ N >0 and f • : [-1, 1] D → R be a β-Hölder function. Then, there exists D := O(1) ∈ N >0 , L := O(log M ) > 0 (C 1 and C 2 are constants independent of M ) and a block-sparse FNN f

Here, we set L m := L and D (l) m := D for all m ∈ [M ] and l ∈ [L m ] and define D := (D (l) m ) m,l .

L * := (m + 5) log 2 D . By the proof of Lemma 8 of Schmidt-Hieber (2017), for any a ∈ Γ(M ), there exists an FNN Hat a : [0, 1] D → [0, 1] whose depth and width are at most 2 + L * and 6D, respectively and whose parameters have sup-norm 1, such that Hat a -H a ∞ ≤ 3 D 2 -m . Next, let B := 2 f • β and C D,β be the number of distinct D-variate monomials of degree up to β . By the equation (7.11) of Schmidt-Hieber (2017), for any a ∈ Γ(M ), there exists an FNN Q a : [0, 1] D → [0, 1] 4 whose depth and width are 1 + L * and 6DC D,β respectively and whose parameters have sup-norm 1, such that 2 -m . Thirdly, by Lemma 4 of Schmidt-Hieber (2017), there exists an FNN Mult : [0, 1] 2 → [0, 1], whose depth and width are m + 4 and 6, respectively and whose parameters have sup-norm 1 such that |Mult(x, y) -xy| ≤ 2 -m for any x, y ∈ [0, 1]. For each a ∈ Γ(M ), we combine Hat a and Q a using Mult and constitute a block of the block-sparse FNN corresponding to a ∈ Γ(M ) by FC a := Mult(Q a (•), Hat a (•)).

2 f • M . Proof of Corollary 4 and Corollary 5. In this proof, we only care the dependence on M in the O-notation. Let M := 2 f • β M . By Lemma 7, there exists f (FNN) D , and D as in Lemma 7). Let k := 16D K∧ 1) -1 ≥ 1 where C is a constant such that L = C log M . Using, Lemma 6, there exists f (FNN) ∈ F (FNN) D,k -1 ,k L M such that f (FNN) = f (FNN) . We apply Theorem 1 toF (FNN) D,k -1 ,k L M and find f (CNN) ∈ F (CNN) C,K,B (conv) ,B (fc) such that L ≤ M (L + L 0 ), C := (C (l) m ) m=0,...,M,l∈[Lm] with C (l) m ≤ 4D , K := (K (l) m ) m=0,...,M,l∈[Lm] with K (l) m ≤ K, B (conv) = k -1 , B (fc) = k L (k ∨ 1) M = k L +1 M ,and f (CNN) = f (FNN) . By definition, we have B (conv) = k -1 = O(1) and log B (fc) = (L + 1)k + log( M ) = O(log M ). This proves Corollary 4. By the definition of k and the bound on C large M , we have ρ + m = 1 for sufficiently large M . In addition, we have log(B (conv) ∨ B (fc) ) = Õ(1). Combining them, we have log M 1 = Õ(1) and hence log M 1 (B (conv) ∨ B (fc) ) = Õ(1). For M 2 , we can bound it by M 2 = O(M log M ) using bounds for C and L . Therefore, we can apply Corollary 2 with γ 1 = β D , γ 2 = 1 and obtain the desired estimation error. Since we have M = O N 1 2γ 1 +γ 2 by the proof of Corollary 1, we can derive the bounds for L m , B(conv) , and B (fc) with respect to N .

Comparison of CNN architectures. "Channel size (Dense FNN case)": The number of channels needed to realize a function represented by a fixed-width dense FNN. "Channel size (β-Hölder case)": The number of channles needed to approximate a β-Hölder function with accuracy

Table content: <table><tr><td/><td>Zhou (2018)</td><td>Petersen &amp; Voigtlaender</td><td>Ours</td></tr><tr><td/><td/><td>(2018)</td><td/></tr><tr><td>CNN type</td><td>Conventional</td><td>Conventional</td><td>ResNet</td></tr><tr><td>Function type</td><td>Barron (s = 2)</td><td>Any (FNNs)</td><td>Any (block-sparse FNNs)</td></tr><tr><td>Channel size (Dense FNN case)</td><td>1</td><td>≥ 1</td><td>≥ 1</td></tr><tr><td>Channel size (β-Hölder case)</td><td>N.A.</td><td>Õ(ε -D β )</td><td>O(1)</td></tr><tr><td>Width</td><td>Increasing</td><td>Fixed</td><td>Fixed</td></tr><tr><td>Filter size</td><td>Fixed</td><td>Full</td><td>Fixed</td></tr><tr><td>Norm bound</td><td>No</td><td>Yes</td><td>Yes</td></tr><tr><td>Padding</td><td>Yes</td><td>No</td><td>Yes</td></tr></table>

Peer review (iclr2019):
 The paper studies approximation and estimation properties of CNNs with residual blocks in the context of non-parametric regression, by constructing equivalent fully-connected architectures (with a block-sparse structure), and leveraging previous approximation results for such functions. Explicit risk bounds are obtained for regression functions in Barron and Holder classes. The main contribution of the paper is Theorem 1, which shows that a class of ResNet-type CNNs contains a class of "block-sparse" fully-connected networks, with appropriate constraints on various size quantities. This result allows the authors to obtain a general risk bound for the ResNet CNN that minimizes empirical risk (Theorem 2, which mostly follows Schmidt-Hieber (2017)), as well as adaptations of the bound for the Barron and Holder classes, by relying on existing approximation results. The construction of Theorem 1 is interesting, and shows that ResNet CNNs can be quite powerful function approximators, even with a filter size that is arbitrarily fixed. However, the obtained CNN approximating architectures look quite unrealistic compared to most practical use-cases of CNNs, since they specifically try to reproduce a fully-connected architecture, leading to residual blocks of depth ~= D/K, which is very deep compared to usual CNNs/ResNets (considering, e.g. K=3 and D in the hundreds for images). In particular, CNNs are typically used when there is some relevant inductive bias such as equivariance to translations (and invariance with pooling operations) to take advantage of, so removing this inductive bias by approximating fully-connected architectures seems a bit twisted. The approach of reducing the function class to be approximated would seem more relevant here, as in the cited papers Petersen & Voigtlaender (2018) and Yarotsky (2018), and perhaps the results of the present paper can be useful in such a scenario as well. Separately, the presentation of the paper could be significantly improved, for instance by introducing relevant notions more clearly in the introduction and related work sections, and by providing more insight and discussion of the obtained results in the main paper. More specific comments: - Section 1, p.2: define M? define D? M seems to be used for different things in different paragraphs - Section 2: Explain what is "s" in the Barron class, or at least point to the relevant definition in the paper - Section 3.1: * *estimation error* is usually called *(expected) risk* in the statistical literature (also in the introduction). estimation error would have to do with relating R and R^hat * why is the estimator "regularized"? - Definition 2: shouldn*t it be D_m^(0) = D instead of 1? - Theorem 1: What is L? Also, it would be helpful to sketch the construction in the main paper given that this is the main result. - Section 4.2: M_1 is the Lipschitz constant of what function? - Section 5.1: "M = 1" this is confusing, maybe use a different letter for the ridge expansion? The discussion on *relative scale* could be made clearer. - Section 5.2, *if we carefully look at their proofs*: more details on this should be provided.