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+# Deep Learning meets Nonparametric Regression: Are Weight-Decayed DNNs Locally Adaptive?
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+We study the theory of neural network (NN) from the lens of classical nonparametric regression problems with a focus on NN’s ability to adaptively estimate functions with heterogeneous smoothness — a property of functions in Besov or Bounded Variation (BV) classes. Existing work on this problem requires tuning the NN architecture based on the function spaces and sample sizes. We consider a “Parallel NN” variant of deep ReLU networks and show that the standard weight decay is equivalent to promoting the $\ell _ { p }$ -sparsity $( 0 < p < 1 )$ ) of the coefficient vector of an end-to-end learned function bases, i.e., a dictionary. Using this equivalence, we further establish that by tuning only the weight decay, such Parallel NN achieves an estimation error arbitrarily close to the minimax rates for both the Besov and BV classes. Notably, it gets exponentially closer to minimax optimal as the NN gets deeper. Our research sheds new lights on why depth matters and how NNs are more powerful than kernel methods.
+
+# 14 1 Introduction
+
+15 Why do deep neural networks (DNNs) work better? They are universal function approximators [6],   
+16 but so are splines and kernels. They learn data-driven representations, but so are the shallower and   
+17 linear counterparts such as matrix factorization. There is surprisingly little theoretical understanding   
+18 on why DNNs are superior to these classical alternatives.   
+19 In this paper, we study DNNs in nonparametric regression problems — a classical branch of statis  
+20 tical theory and methods with more than half a century of associated literature [25, 7, 46, 10, 23, 37,   
+21 33]. Nonparametric regression addresses the following fundamental problem:
+
+• Let $y _ { i } = f ( x _ { i } ) + \mathrm { N o i }$ se for $i = 1 , . . . , n$ . How can we estimate a function $f$ using data points $( x _ { 1 } , y _ { 1 } ) , . . . , ( x _ { n } , y _ { n } )$ in conjunction with the knowledge that $f$ belongs to a function class ${ \mathcal { F } }$ ?
+
+Function class $\mathcal { F }$ typically imposes only weak regularity assumptions such as smoothness, which makes nonparametric regression widely applicable to real-life applications under weak assumptions.
+
+26 Local adaptivity. A subset of nonparametric regression techniques were shown to have the property   
+27 of local adaptivity [24] in both theory and practice. These include wavelet smoothing [10], locally   
+28 adaptive regression splines [24], trend filtering [40, 47] and adaptive local polynomials [2, 3]. We   
+29 say a nonparametric regression technique is locally adaptive if it can cater to local differences in   
+30 smoothness, hence allowing more accurate estimation of functions with varying smoothness and   
+31 abrupt changes.
+
+In light of such a distinction, it is natural to consider the following question.
+
+Are NNs locally adaptive, i.e., optimal in learning functions with heterogeneous smoothness?
+
+35 This is a timely question to ask, partly because the bulk of recent theory of NN leverages its asymp  
+36 totic Reproducing Kernel Hilbert Space (RKHS) in the overparameterized regime [21, 5, 1]. RKHS  
+37 based approaches, e.g., kernel ridge regression with any fixed kernels are suboptimal in estimating   
+38 functions with heterogeneous smoothness [9]. Therefore, existing deep learning theory based on   
+39 RKHS does not satisfactorily explain the advantages of neural networks over kernel methods.   
+40 We build upon the recent work of Suzuki [39] and Parhi and Nowak [29] who provided encouraging   
+41 first answers to the question above about the local adaptivity of NNs. Specifically, Parhi and Nowak   
+42 [29, Theorem 8] showed that a two-layer truncated power function activated neural network with   
+43 a non-standard regularization is equivalent to the locally adaptive regression splines (LARS) [24].   
+44 This connection implies that such non-standard NNs achieve the minimax rate for the (higher order)   
+45 bounded variation (BV) classes. We provide a detailed discussion about this work in Section B.   
+46 Suzuki [39] showed that multilayer ReLU DNNs can achieve minimax rate for the Besov class,   
+47 but requires the width, depth and an artificially imposed sparsity-level of the DNN weights to be   
+48 carefully calibrated according to parameters of the Besov class, thus is quite different from how   
+49 DNNs are typically trained in practice.
+
+In this paper, we aim at addressing the same locally adaptivity question for a more commonly used neural network with standard weight decayed training.
+
+52 Parallel neural networks. We restrict our attention on a special network architecture called parallel   
+53 neural network [18, 15] which learns an ensemble of subnetworks — each being a multilayer ReLU   
+54 DNNs. Parallel NNs have been shown to be more well-behaved both theoretically [18, 51, 16, 15, 14]   
+55 and empirically [50, 44]. Moreover, the idea of parallel NNs was used in many successful NN   
+56 architectures such as SqueezeNet, ResNext and Inception (see [15] and the references therein).   
+57 Weight decay. Weight decay is a common method in deep learning to reduce overfitting. Em  
+58 pirically, the regularizer is not necessarily explicit. Many tricks in deep learning, including early   
+59 stopping [48], quantization [20], and dropout [45] have similar effect as weight decay. In this paper,   
+60 we make no assumption on the training method thus there is no (implicit) regularizers apart from   
+61 weight decay.
+
+62 Summary of results. Our main contributions are:
+
+1. We prove that the (standard) weight decay in training an $L$ -layer parallel ReLU-activated neural network is equivalent to a sparse $\ell _ { p }$ penalty term (where $p = 2 / L$ ) on the linear coefficients of a learned representation.   
+2. We show that neural networks can approximate B-spline basis functions of any order without the need of choosing the order parameter manually. In other words, neural networks can adapt to functions of different order of smoothness, and even functions with different smoothness in different regions in their domain.   
+3. We show that the estimation error of weight decayed parallel ReLU neural network decreases polynomially with the number of samples up to a constant error for estimating functions with heterogeneous smoothness in the both BV and Besov classes, and the exponential term in the error rate is close to the minimax rate. Notably, the method requires tuning only the weight decay parameter.   
+4. We find that deeper models achieve closer to the optimal error rate. This result helps explain why deep neural networks can achieve better performance than shallow ones empirically.
+
+77 The above results separate NNs with any linear methods such as kernel ridge regression. To the   
+78 best of our knowledge, we are the first to demonstrate that standard techniques (“weight decay”   
+79 and ReLU activation) suffice for DNNs in achieving the optimal rates for estimating BV and Besov   
+80 functions.
+
+# 2 Preliminary
+
+# 2.1 Notation and Problem Setup.
+
+83 We denote regular font letters as scalars, bold lower case letters as vectors and bold upper case letters   
+84 as matrices. $a \lesssim b$ means $a \leq C b$ for some constant $C$ that does not depend on $a$ or $b$ , and $a \stackrel { } { \sim } b$   
+85 denotes $a \lesssim b$ and $b \lesssim a$ . See Table 1 for the full list of symbols used.
+
+Table 1: Symbols used in this paper   
+
+<table><tr><td>symbol a/a/A</td><td>Meaning</td><td></td><td></td></tr><tr><td>B pq ： Bp B Mm() Mm,k,s(.) σ（.) 6 W(e j</td><td>scalars/vectors/matrices. Besov space. Besov quasi-norm. Besov norm. mth order Cardinal B-spline bases. mth order Cardinal B-spline basis function of resolution k at position s. ReLU activation function. Weight and bias in the l-th layer in the j-th subnetwork.</td><td>[a,6] [n] I:IIF 11 : p lp-norm. d M L w n # samples. R,Z,N nonnegative integers.</td><td>{x∈R:a≤x≤b} {x∈N:1≤𝑥≤n}. Frobenius norm. Dimension of input. # subnetworks in a parallel NN. #layers ina (parallel) NN. Width of a subnetwork. Set of real numbers,integers,and</td></tr></table>
+
+86 Let $f _ { 0 }$ be the target function to be estimated. The training dataset is $\mathcal { D } _ { n } : = \{ ( \boldsymbol { x } _ { i } , y _ { i } ) , y _ { i } = f _ { 0 } ( \boldsymbol { x } _ { i } ) +$ 87 $\epsilon _ { i } , i \in [ n ] \}$ , where $x _ { i }$ are fixed and $\epsilon _ { i }$ are zero-mean, independent Gaussian noises with variance $\sigma ^ { 2 }$ . In the following discussion, we assume 88 $\pmb { x } _ { i } \in [ 0 , 1 ] ^ { d }$ , $f _ { 0 } ( x _ { i } ) \in [ - 1 , 1 ] , \forall i$ .
+
+We will be comparing estimators under the mean square error (MSE), defined as
+
+$$
+\operatorname { M S E } ( { \hat { f } } ) : = \mathbb { E } _ { { \mathcal { D } } _ { n } } { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } ( { \hat { f } } ( { \pmb x } _ { i } ) - f _ { 0 } ( { \pmb x } _ { i } ) ) ^ { 2 } .
+$$
+
+89 The optimal worst-case MSE is described by $\begin{array} { r } { R ( \mathcal { F } ) : = \operatorname* { m i n } _ { \hat { f } } \operatorname* { m a x } _ { f _ { 0 } \in \mathcal { F } } \operatorname { M S E } ( \hat { f } ) } \end{array}$ , we say that   
+90 $\hat { f }$ is optimal if $\mathrm { M S E } ( \hat { f } ) \ = \ R ( \mathcal F )$ . The empirical (square error) loss is defined as $\hat { L } ( \hat { f } ) : =$   
+91 $\textstyle { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } ( { \hat { f } } ( \mathbf { x } _ { i } ) - y _ { i } ) ^ { 2 }$ . The corresponding population loss is $\begin{array} { r } { L ( \hat { f } ) : = \mathbb { E } [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { f } ( { \pmb x } _ { i } ) - y _ { i } ^ { \prime } ) ^ { 2 } | \hat { f } ] } \end{array}$   
+92 where $y _ { i } ^ { \prime }$ are new data points. It is clear that $\mathbb { E } [ L ( \hat { f } ) ] = \mathrm { M S E } [ \hat { f } ] + \sigma ^ { 2 }$ .
+
+# 2.2 Besov Spaces and Bound Variation Space
+
+Besov space, denoted as $B _ { p , q } ^ { \alpha }$ , is a flexible function class parameterized by $\alpha , p , q$ whose definition is deferred to Section C.1. Here $\alpha \geq 0$ determines the smoothness of functions, $1 \leq p \leq \infty$ determines the averaging (quasi-)norm over locations, $1 \leq q \leq \infty$ determines the averaging (quasi)norm over scale which plays a relatively minor role. Smaller $p$ is more forgiving to inhomogeneity space.9 of gen0 e other hand, it is easy to see from definition that ability, in the following discussion we will only f $p$ . Without loss $B _ { p , q } ^ { \alpha } \subset B _ { p , q ^ { \prime } } ^ { \alpha }$ $q < q ^ { \prime }$ $B _ { p , \infty } ^ { \dot { \alpha } }$
+
+$p = 1$ , the Besov space allows higher inhomogeneity, and it is more general than the Sobolev 02 or Hölder space.
+
+103 Bounded variation (BV) space is a more interpretable class of functions with spatially hetero  
+104 geneous smoothness [10]. It is defined through the total variation (TV) of a function. For   
+105 $( m + 1 ) \mathfrak { t h }$ differentiable function $f : [ 0 , 1 ] \ \to \ \mathbb { R }$ , the mth order total variation is defined as   
+106 $\begin{array} { r } { T V ^ { ( m ) } ( f ) : = T V ( f ^ { ( m + 1 ) } ) = \int _ { [ 0 , 1 ] } | f ^ { \bar { ( m + 1 ) } } ( x ) | d x } \end{array}$ , and the corresponding mth order Bounded   
+107 Variation class $B V ( m ) : = \{ f : T V ( f ^ { ( m ) } ) < \infty \}$ . The more general definition is given in Sec  
+108 tion C.2. Bounded variation class is tightly connected to Besov classes. Specifically [8]:
+
+$$
+B _ { 1 , 1 } ^ { m + 1 } \subset B V ( m ) \subset B _ { 1 , \infty } ^ { m + 1 }
+$$
+
+This allows the results derived for the Besov space to be easily applied to BV space.
+
+Minimax MSE It is well known that minimax rate for Besov and 1D BV classes are $O ( n ^ { - \frac { 2 \alpha } { 2 \alpha + d } } )$ and $O ( n ^ { - ( 2 m + 2 ) / ( 2 m + 3 ) } )$ respectively . The minimax rate for linear estimators in 1D BV classes is known to be $O ( n ^ { - ( 2 m + 1 ) / ( 2 m + 2 ) } )$ [24, 10].
+
+# 113 3 Main Results: Parallel ReLU DNNs
+
+114 Consider a parallel neural network containing $M$ multi layer perceptrons (MLP) with ReLU activa  
+115 tion functions called subnetworks. Each subnetwork has width $w$ and depth $L$ . The input is fed to   
+116 all the subnetworks, and the output of the parallel NN is the summation of the output of each sub  
+117 network. The architecture of a parallel neural network is shown in Figure 1a. This parallel neural   
+118 network is equivalent to a vanilla neural network with block diagonal weights in all but the first and   
+119 the last layers (Figure 1(c)). Let W(ℓ)j and $b _ { j } ^ { ( \ell ) }$ denote the weight and bias in the $\ell \cdot$ -th layer in the   
+120 $j$ -th subnetwork respectively. Training this model with weight decay returns:
+
+![](images/59172724a4b7a4d4938138827bc04ef9715fc77af9eb4a4cacc72d73bb7abb79.jpg)  
+Figure 1: Parallel neural network and the equivalent sparse regression model we discovered.
+
+$$
+\underset { \{ \mathbf { W } _ { j } ^ { ( \ell ) } , \pmb { b } _ { j } ^ { ( \ell ) } \} } { \operatorname { a r g m i n } } \ : \hat { L } ( f ) + \lambda \sum _ { j = 1 } ^ { M } \sum _ { \ell = 1 } ^ { L } \big \| \mathbf { W } _ { j } ^ { ( \ell ) } \big \| _ { F } ^ { 2 } ,
+$$
+
+wh121 $\begin{array} { r } { f ( x ) = \sum _ { j = 1 } ^ { M } f _ { k } ( x ) } \end{array}$ denotes the parallel neural network, $f _ { j } ( \cdot )$ denotes the $j$ -th subnetwork, $\lambda > 0$
+
+Theorem 1. For any fixed $\alpha - d / p > 1 , q \geq 1 , L \geq 3 ,$ , for any $f _ { 0 } \in B _ { p , q } ^ { \alpha } ,$ given an $L$ -layer parallel neural network satisfying
+
+• The width of each subnetwork is fixed and large enough: $w \gtrsim d$ . See Theorem 9 for the detail.
+
+• The number of subnetworks is large enough: $M \gtrsim m ^ { d } n ^ { \frac { 1 - 2 / L } { 2 \alpha / d + 1 - 2 / ( p L ) } }$ where $m = \lceil \alpha - 1 \rceil$
+
+With proper choice of the parameter of weight decay $\lambda$ , the solution $\hat { f }$ parameterized by (2) satisfies
+
+$$
+\mathrm { M S E } ( \hat { f } ) = \tilde { O } \big ( n ^ { - \frac { 2 \alpha / d ( 1 - 2 / L ) } { 2 \alpha / d + 1 - 2 / ( p L ) } } \big ) + C o n s t .
+$$
+
+where $\tilde { O }$ shows the scale up to a logarithmic factor, and the trailing constant term decreases exponentially with $L$ .
+
+We explain the proof idea in the next section,but defer the extended form of the theorem and the full proof to Section F. Before that, we comment on a few interesting aspects of the result.
+
+135 Near optimal rates and the effect of depth. The first term in the MSE bound is the estimation error   
+136 and the second term is (part of) the approximation error of this NN. Recall that the minimax rate of   
+137 a Besov class is $O ( n ^ { - \frac { 2 \alpha } { 2 \alpha + d } } )$ thus as the depth parameter $L$ increases it can get arbitrarily close to   
+138 the minimax rate. The constant term would be a negligible if we choose $L \gtrsim \log n$ .
+
+Corollary 2. Under the conditions of Theorem $I$ , for any $f _ { 0 } \in B _ { p , q } ^ { \alpha } ,$ there is a numerical constant $C$ such that when we choose $C \log n \leq L \leq 1 0 0 C \log n$ ,
+
+$$
+\mathrm { M S E } ( \hat { f } ) = \tilde { O } ( n ^ { - \frac { 2 \alpha } { 2 \alpha + d } ( 1 - o ( 1 ) ) } ) ,
+$$
+
+where 139 $\tilde { O }$ hides only logarithmic factors and the $o ( 1 )$ factor in the exponent is $O ( 1 / \log ( n ) )$
+
+40 This result says that deeper parallel neural networks achieves lower error and gets closer to the   
+141 statistical limit.   
+142 Overparameterization and sparsity. We also note that the result does not depend on $M$ as long   
+143 as $M$ is large enough. This means that the neural network can be arbitrarily overparameterized   
+144 while not overfitting. The underlying reason is sparsity. As it will become clearer in the proof   
+145 sketch, weight decayed training of a parallel L-layer ReLU NNs is equivalent to a sparse regression   
+146 problem with an $\ell _ { p }$ penalty assigned to the coefficient vector of a learned dictionary. Here $p = 2 / L$   
+147 which promotes even sparser solutions than an $\ell _ { 1 }$ penalty.   
+148 No architecture tuning. For any fixed $L$ , the required architecture of the model does not depend   
+149 on the dataset or the target function $( n , \alpha )$ expect the number of subnetworks $M$ , for which the only   
+150 requirement is being large enough. As a result, one can design a model using a large guess on $M$ ,   
+151 and achieve the claimed near-optimal error rate by only tuning the weight decay parameter.   
+52 Bounded variation classes. Thanks to the Besov space embedding of the BV class (1), our theorem   
+53 also implies the result for the BV class in $1 D$ .
+
+4 Corollary 3. If the target function is in bounded variation class $f _ { 0 } \in B V ( m )$ , For any fixed $L \geq 3$ , 5 for a neural network satisfying the requirements in Theorem $^ { l }$ with $d = 1$ and with proper choice of the parameter of weight decay 6 $\lambda$ , the NN $\hat { f }$ parameterized by (5) satisfies
+
+$$
+\mathrm { M S E } ( \hat { f } ) = \tilde { O } ( n ^ { - \frac { ( 2 m + 2 ) ( 1 - 2 / L ) } { 2 m + 3 - 2 / L } } ) + C o n s t .
+$$
+
+where $\tilde { O }$ shows the scale up to a logarithmic factor, and the trailing constant term decreases exponentially with $L$ .
+
+159 It is known that any linear estimators such as kernel smoothing and smoothing splines cannot have   
+160 an error lower than $O ( n ^ { - ( 2 m + 1 ) / ( 2 m + 2 ) } )$ for $B V ( m )$ [10]. This partly explains the advantage of   
+161 DNNs over kernels.
+
+Representation learning and adaptivity. The results also shed a light on the role of representation learning in DNN’s ability to adapt. Specifically, different from the two-layer NN in [29], which achieves the minimax rate of $B V ( m )$ by choosing appropriate activation functions using each $m$ , each subnetwork of a parallel NN can learn to approximate the spline basis of an arbitrary order, which means that if we choose $L$ to be sufficiently large, such Parallel NN with optimally tuned $\lambda$ is simultaneously near optimal for $m = 1 , 2 , 3 , \ldots .$ In fact, even if different regions of the space has different orders of smoothness, the paralle NN will still be able to learn appropriate basis functions in each local region. To the best of our knowledge, this is a property that none of the classical nonparametric regression methods possess.
+
+171 Synthesis v.s. analysis methods. Our result could also inspire new ideas in estimator design.   
+172 There are two families of methods in non-parametric estimation. One called synthesis framework   
+173 which focuses on constructing appropriate basis functions to encode the contemplated structures   
+174 and regress the data to such basis, e.g., wavelets [10]. The other is called analysis framework which   
+175 uses analysis regularization on the data directly (see, e.g., RKHS methods [37] or trend filtering   
+176 [40]). It appears to us that parallel NN is doing both simultaneously. It has a parametric family   
+177 capable to synthesizing an ${ \bar { O } } ( n )$ subset of an exponentially large family of basis, then implicitly   
+178 use sparsity-inducing analysis regularization to select the relevant basis functions. In this way the   
+179 estimator does not actually have to explicitly represent that exponentially large set of basis functions,   
+180 thus computationally more efficient.
+
+# 181 4 Proof Overview
+
+We start by first proving that a parallel neural network trained with weight decay is equivalent to an $\ell _ { p }$ -sparse regression problem with representation learning (Section 4.1); which helps decompose its MSE into an estimation error and approxmation error. Then we bound the two terms in Section 4.2 and Section 4.3 respectively.
+
+# 4.1 Equivalence to $\ell _ { p }$ Sparse Regression with a Learned Feature Representation
+
+187 It is widely known that ReLU function is 1-homogeneous: $\sigma ( a x ) = a \sigma ( x ) , \forall a \geq 0 , x \in \mathbb { R }$ . In any   
+188 consecutive two layers in a neural network (or a subnetwork), one can multiply the weight and bias   
+189 in one layer with a positive constant, and divide the weight in another layer with the same constant.   
+190 The neural network after such transformation is equivalent to the original one:
+
+$$
+{ \mathbf W } ^ { ( 2 ) } \sigma ( { \mathbf W } ^ { ( 1 ) } \pmb x + { b } ^ { ( 1 ) } = \frac { 1 } { c } { \mathbf W } ^ { ( 2 ) } \sigma ( c { \mathbf W } ^ { ( 1 ) } \pmb x + c { \pmb b } ^ { ( 1 ) } ) , \quad \forall c > 0 , \pmb x .
+$$
+
+91 This property allows us to reformulate (2) to an $\ell _ { p }$ sparsity constraint problem:
+
+192 Proposition 4. Fix the input dataset $\mathcal { D } _ { n }$ and a constant $c _ { 1 } > 0$ . There exists an one-to-one mapping   
+193 between $\lambda > 0$ and $P ^ { \prime } > 0$ such that (2) is equivalent to the following problem:
+
+$$
+\begin{array} { r l r } { \underset { \{ \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } , \bar { \mathbf { b } } _ { j } ^ { ( \ell ) } , a _ { j } \} } { \arg \operatorname* { m i n } } } & { \hat { L } \Big ( \displaystyle \sum _ { j = 1 } ^ { M } a _ { j } \bar { f } _ { j } \Big ) = \frac { 1 } { n } \sum _ { i } ( y _ { i } - \bar { f } _ { 1 : M } ( \boldsymbol { x } _ { i } ) ^ { T } \boldsymbol { a } ) ^ { 2 } } & \\ { \quad \quad \quad \quad \quad \quad s . t . \| \bar { \mathbf { W } } _ { j } ^ { ( 1 ) } \| _ { F } \leq c _ { 1 } \sqrt { d } , \forall j \in [ M ] , } & \\ { \| \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } \| _ { F } \leq c _ { 1 } \sqrt { w } , \forall j \in [ M ] , 2 \leq \ell \leq L , } & { \| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } \leq P ^ { \prime } } & \end{array}
+$$
+
+where 194 $\bar { f } _ { j } ( \cdot )$ is a subnetwork with parameters $\bar { \mathbf { W } } _ { j } ^ { ( \ell ) } , \bar { \pmb { b } } _ { j } ^ { ( \ell ) }$
+
+195 This equivalent model is demonstated in Figure 1b. The proof can be found in Section D.1. The   
+196 constraint $\lVert \bar { \mathbf { W } } _ { j } ^ { ( 1 ) } \rVert _ { F } \lesssim \sqrt { d } , \lVert \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } \rVert _ { F } \lesssim \sqrt { \bar { w } } , \forall \ell > 1$ is typical in deep learning for better numerical   
+197 stability. The equivalent model in Proposition 4 is also a parallel neural network, but it appends one   
+198 layer with parameters $\{ a _ { k } \}$ at the end of the neural network and the constraint on the Frobenius   
+199 norm is converted to the $2 / L$ norm on the factors $\{ a _ { k } \}$ . Since $L \gg 2$ in a typical application,   
+200 $2 / L \ll 1$ and this constraint can enforce a sparser model than that in Section B.   
+201 There are two useful implications of Proposition 4. First, it gives an intuitive explanation on how   
+202 a weight decayed Parallel NN works. Specifically, it can be viewed as a sparse linear regression   
+203 with representation learning. Second, the conversion into the constrained form allows us to adapt   
+204 generic statistical learning machinery (a self-bounding argument) from Suzuki [39, Proposition 4]   
+205 for studying this constrained ERM problem.   
+206 The adaptation is nontrivial because (1) our regression problem has a fixed design (so data points are   
+207 not iid); (2) there is an unconstrained subspace with no bounded metric entropy. Specifically, our   
+208 Proposition 15 shows that the MSE of the regression problem can be bounded by
+
+$$
+\mathrm { M S E } ( \hat { f } ) = O \Bigg ( \underbrace { \underset { f \in \mathcal { F } } { \operatorname* { i n f } } \mathrm { ~ M S E } ( f ) } _ { \mathrm { a p p r o x i m a t i o n ~ e r r o r } } + \underbrace { \frac { \log \mathcal { N } ( \mathcal { F } _ { \parallel } , \delta , \| \cdot \| _ { \infty } ) + d ( \mathcal { F } _ { \perp } ) } { n } + \delta } _ { \mathrm { e s i m a t i o n ~ e r r o r } } \Bigg )
+$$
+
+209 in which $\mathcal { F }$ decomposes into $\mathcal { F } _ { \parallel } \times \mathcal { F } _ { \perp }$ , where $\mathcal { F } _ { \perp }$ is an unconstrained subspace with finite dimension,   
+210 and $\mathcal { F } _ { \| }$ is a compact set in the orthogonal complement with a $\delta$ -covering number of $\mathcal { N } ( \mathcal { F } _ { | | } , \delta , \| \cdot \| _ { \infty } )$   
+211 in $\| \cdot \| _ { \infty }$ -norm. This decomposes MSE into an approximation errorand an estimation error. The novel   
+212 analysis of these two represents the major technical contribution of this paper.
+
+# 4.2 Estimation Error Analysis
+
+The decomposition above reveals that to bound the estimation error, it suffices to compute the covering number of the constraint set in the sup-norm of the function it represents.
+
+Previous results that bound the covering number of neural networks [49, 39] depends on the width of the neural networks explicitly, which cannot be applied when analysing a potentially infinitely wide neural network. In this section, we leverage the $\ell _ { p }$ -norm bounded coefficients to avoid the dependence in $M$ in the covering number bound.
+
+Theorem 5. The covering number of the model defined in (5) apart from the bias in the last layer satisfies
+
+$$
+\log N ( \mathcal { F } , \delta ) \lesssim w ^ { 2 + 2 / ( 1 - 2 / L ) } L ^ { 2 } \sqrt { d } P ^ { \prime \frac { 1 } { 1 - 2 / L } } \delta ^ { - \frac { 2 / L } { 1 - 2 / L } } \log ( w P ^ { \prime } / \delta ) .
+$$
+
+222 The proof can be found in Section D.2. It requires the following lemma:
+
+a 6. . Th $\log \mathcal { N } ( \mathcal { G } , \delta ) \ \lesssim \ k \log ( 1 / \delta )$ $c _ { 3 }$ $g \in { \mathcal { G } } , | a | \leq 1 ,$ , wny $a g \in { \mathcal { G } }$ $\begin{array} { r } { \mathcal { F } = \Bigl \{ \sum _ { i = 1 } ^ { M } a _ { i } g _ { i } \Big | g _ { i } \in \mathcal { G } , \| a \| _ { p } ^ { p } \leq P , 0 < p < 1 \Bigr \} } \end{array}$ $P > 0$ satisfies
+
+$$
+\log \mathcal { N } ( \mathcal { F } , \epsilon ) \lesssim k P ^ { \frac { 1 } { 1 - p } } ( \delta / c _ { 3 } ) ^ { - \frac { p } { 1 - p } } \log ( c _ { 3 } P / \delta )
+$$
+
+up to a double logarithmic factor.
+
+See Section D.3 for the proof of Lemma 6. The covering number in Theorem 5 does not depend on the number of subnetworks $M$ . In other words, it provides a bound of estimation error for an arbitrarily wide parallel neural network as long as the total Frobenius norm is bounded.
+
+# 4.3 Approximation Error Analysis
+
+The approximation error analysis involves two steps. In Section 4.3.1, we analyse how a subnetwork can approximate a B-spline basis. Then in Section 4.3.2 we show that a sparse linear combination of B-spline bases approximates Besov functions. Both add up to the total error in approximating Besov functions with a parallel neural network (Theorem 9).
+
+# 4.3.1 Approximation Error of B-spline Basis Function
+
+As is shown in Section C.1, functions in Besov space can be alternatively represented in a sequence space via the coefficients of a cardinal B-spline basis. In this section we study the approximation ability of ReLU neural networks to B-spline basis function.
+
+Proposition 7. Let $M _ { m , k , s }$ be the $B$ -spline of order m with scale $2 ^ { - k }$ in each dimension and position $s \in \mathbb { R } ^ { d }$ : $M _ { m , k , s } ( { \pmb x } ) : = M _ { m } ( 2 ^ { k } ( { \pmb x } - { \pmb s } ) )$ , $M _ { m }$ is defined in (11). There exists a parallel neural network that has the structure and satisfy the constraint in Proposition $^ { 4 }$ for $d$ -dimensional input and one output, containing $M = O ( m ^ { d } )$ subnetworks, each of which has width $w = O ( d )$ and depth ${ \cal L } = { \cal O } ( \log ( c ( m , d ) / \epsilon ) )$ for some constant $w , c$ that depends only on $m$ and $d$ , denoted as $\tilde { M } _ { m } ( { \pmb x } ) , { \pmb x } \in \mathbb { R } ^ { d }$ , such that
+
+$$
+| \tilde { M } _ { m , k , s } ( { \pmb x } ) - M _ { m , k , s } ( { \pmb x } ) | \leq \epsilon , i f 0 \leq 2 ^ { k } ( x _ { i } - s _ { i } ) \leq m + 1 , \forall i \in [ d ] ,
+$$
+
+The proof can be found in Section E.1. Note that the product of the coefficients among all the layers are proportional to $2 ^ { k }$ , instead of $2 ^ { k m }$ when approximating truncated power basis functions. This is because the transformation from $M _ { m }$ to $M _ { m , k , s }$ only scales the domain of the function by $2 ^ { k }$ , while the codomain of the function is not changed. To apply the transformation to the neural network, one only need to scale weights in the first layer by $2 ^ { k }$ , which is equivalent to scaling the weights in each layer bt $2 ^ { k / L }$ and adjusting the bias according.
+
+# 4.3.2 Approximation Error in Besov Space
+
+255 With the results given in Section 4.3.1, we can estimate the approximation error of parallel ReLU   
+256 neural networks to functions in Besov space.
+
+Proposition 8. Let $\alpha - d / p > 1 , r > 0$ . For any function in Besov space $f _ { 0 } \in B _ { p , q } ^ { \alpha }$ and any $\bar { M }$ $\bar { M }$ $B$ basis of order positive integer , and the coeffi $m$ satisfyingsuch thatnts satisfy   
+$0 < \alpha < \operatorname* { m i n } ( m , m - 1 + 1 / p )$ $\begin{array} { r } { \check { f } _ { \bar { M } } = \sum _ { i = 1 } ^ { \bar { M } } a _ { k _ { i } , s _ { i } } M _ { m , k _ { i } , s _ { i } } } \end{array}$ $\bar { M }$ $\lVert \check { f } _ { \bar { M } } - f _ { 0 } \rVert _ { r } \lesssim \bar { M } ^ { - \alpha / d } \lVert f _ { 0 } \rVert _ { B _ { p , q } ^ { \alpha } }$
+
+$$
+\| \big \{ 2 ^ { k _ { i } } a _ { k _ { i } , s _ { i } } \big \} _ { k _ { i } , s _ { i } } \| _ { p } \lesssim \| f _ { 0 } \| _ { B _ { p , q } ^ { \alpha } } .
+$$
+
+257258 The proof can be found in Section E.2.
+
+259 Remark 1. The requirement in Proposition 8: $\alpha - d / p > 1$ is stronger than the condition typically   
+260 found in approximation theorem $\alpha - d / p \ge 0 \ : I \ / I$ 1], so-called “Boundary of continuity”, or the   
+261 condition in Suzuki $[ 3 9 ] \alpha > d ( 1 / p - 1 / r ) _ { + }$ . This is because although the functions in $B _ { p , q } ^ { \alpha }$ when
+
+$0 \leq \alpha - d / p < 1$ can be approximated by $B$ -spline basis, the sum of weighted coefficients may not converge. One simple example is the step function $f _ { s t e p } ( x ) = \mathbf { 1 } ( x \geq 0 . 5 )$ , $f _ { s t e p } \in B _ { 1 , \infty } ^ { 1 }$ . Although it can be decomposed using first order $B$ -spline basis as in (10), the summation of the coefficients is infinite. Actually one only needs a ReLU neural network with one hidden layer and two neurons to approximate this function to arbitrary precision, but the weight need to go to infinity.
+
+Theorem 9. Under the same condition as Proposition 8, for any positive integer $\bar { M }$ , any function in Besov space $f _ { 0 } \in B _ { p , q } ^ { \alpha }$ can be approximated by a parallel neural network with no less than $O ( m ^ { d } \bar { M } )$ number of subnetworks satisfying:
+
+1. Each subnetwork has width $w = O ( d )$ and depth $L$ .
+
+2. The weights in each layer satisfy √ $\| \bar { \mathbf { W } } _ { k } ^ { ( \ell ) } \| _ { F } \leq O ( \sqrt { w } )$ except the first layer $\| \bar { \mathbf { W } } _ { k } ^ { ( 1 ) } \| _ { F } \leq$ $O ( { \sqrt { d } } )$ ,
+
+3. The scaling factors have bounded $2 / L$ -norm: $\begin{array} { r } { \| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } \lesssim m ^ { d } e ^ { 2 m d / L } \bar { M } ^ { 1 - 2 / ( p L ) } . } \end{array}$
+
+4. The approximation error is bounded by
+
+$$
+\| \tilde { f } - f _ { 0 } \| _ { r } \leq ( c _ { 4 } \bar { M } ^ { - \alpha / d } + c _ { 5 } e ^ { - c _ { 6 } L } ) \| f \| _ { B _ { p , q } ^ { \alpha } }
+$$
+
+where $c _ { 4 } , c _ { 5 } , c _ { 6 }$ are constants that depend only on $m , d$ and $p$ .
+
+Here $\bar { M }$ is the number of “active” subnetworks, which is not to be confused with the number of subnetworks at initialization. The proof can be found in Section E.3.
+
+Using the estimation error in Theorem 5 and approximation error in Theorem 9, by choosing $\bar { M }$ to minimax the total error, we can conclude the sample complexity of parallel neural networks using weight decay, which is the main result (Theorem 1) of this paper. See Section F for the detail.
+
+# 5 Experiment
+
+We empirically compare a parallel neural network (PNN) and a vanilla ReLU neural network (NN) with smoothing spline, trend filtering (TF) [40], and wavelet denoising. Trend filtering can be viewed as a more efficient discrete spline version of locally adaptive regression spline and enjoys the same optimal rates for the BV classes. Wavelet denoising is also known to be minimax-optimal for the BV classes. The results are shown in Figure 2. We use two target functions: a Doppler function whose frequency is decreasing(Figure 2(a)-(c)), and a combination of piecewise linear function and piecewise cubic function, or “vary” function (Figure 2(d)-(f)). We repeat each experiment 10 times and take the average. The shallow area in Figure 2(b)(e) shows $9 5 \%$ confidence interval by inverting the Wald’s test. The degree of freedom (DoF) is computed based on Tibshirani [41].
+
+291 As can be shown in the figure, both TF and wavelet denoising can adapt to the different levels of   
+292 smoothness in the target function, while smoothing splines tend to be oversmoothed where the target   
+293 function is less smooth (the left side in (a)(d), enlarged in $( \mathbf { g } )$ ). The prediction of PNN is similar to   
+294 TF and wavelet denoising and shows local adaptivity. Besides, the MSE of PNN almost follows the   
+295 same trend as TF and wavelet denoising which is consistent with our theoretical understanding that   
+296 the error rate of neural network is closer to locally adaptive methods. Notably PNN, TF and wavelet   
+297 denoising achieve lower error at a much smaller degree-of-freedom than smoothing splines.   
+298 In a vanilla NN, weight decay is equivalent to $\ell _ { 1 }$ regularizer in any two successive layers, but to the   
+299 best of our knowledge it does not lead to sparse representation learning unless some specific sparse   
+300 structure is enforced. While our theory does not apply to vanilla neural networks, the results seem   
+301 to suggest the NN behaves similar to smoothing spline and is not locally adaptive.   
+302 There are some mild drops in the best MSE one can achieve with NN vs TF in both examples. We   
+303 are surprised that the drop is small because NN needs to learn the basis functions that TF essentially   
+304 hard-coded. The additional price to pay for using a more adaptive and more flexible representation   
+305 learning method seems not high at all.   
+306 In Figure 2(c)(f), we give the output all the “active” subnetwork, i.e. the subnetworks whose output is   
+307 not a constant. Notice that the number of active subnetworks is much smaller than the initialization.   
+308 This is because weight decay induces $\ell _ { p }$ sparsity and the weight in most of the subnetworks reduces   
+309 towards 0 after training. More details are shown in Section G.   
+310 In Figure 2(h)(i), we plot the MSE versus the number of training samples for “Doppler” and “Vary”   
+311 respectively. It is clear that parallel NN works the best overall. In (i), we further compare the scaling   
+312 of the MSE against the minimax rate $( n ^ { - 4 / 5 } )$ and the minimax linear rate $( n ^ { - 3 / 4 } )$ , i.e., the best rate   
+313 kernel methods could achieve. As is predicted by our theory, when $n$ is large, the MSE of parallel   
+314 neural networks and trend filtering decreases at almost the same rate as the minimax rate, while   
+315 smoothing splines, as expected, is converging at the (suboptimal) minimax linear rate. Interestingly,   
+316 vanilla NN seems to converge at the optimal rate too on this example. It remains an open question   
+317 whether vanila NN is merely “lucky” on this example, or it also achieves the minimax rate for all   
+318 functions in BV(m).
+
+![](images/eaa749bbe091ca118b8d25b970d009c10306c8e7b25a2f14ade0b3bb4e165c5f.jpg)  
+Figure 2: Numerical experiment results of the Doppler function (a-c,h), and “vary” function (d-f,g). All the “active” subnetworks are plotted in (c)(f). The horizontal axis in (b) is not linear.
+
+# 319 6 Conclusion and Discussion
+
+320 In this paper, we show that a deep parallel neural network can be locally adaptive by tuning only   
+321 the weight decay parameter. This confirms that neural networks can be nearly optimal in learning   
+322 functions with heterogeneous smoothness which separates them from kernel methods. We prove   
+323 that training an $L$ layer parallel neural network with weight decay is equivalent to an $\ell _ { 2 / L }$ -penalized   
+324 regression model with representation learning. Since in typical application $L \gg 2$ , weight decay   
+325 promotes a sparse linear combination of the learned bases. Using this method, we proved that a   
+326 parallel neural network can achieve close to the minimax rate in the Besov space and bounded   
+327 variation (BV) space. Our result reveals that one do not need to specify the smoothness parameter $\alpha$   
+328 (or $m$ ). Neural networks can adapt to different degree of smoothness, or choose different parameters   
+329 for different regions of the domain of the target function. This is a new type of adaptivity not   
+30 possessed by traditional adaptive nonparametric regression methods like locally adaptive regression   
+31 spline or trend filtering.
+
+On the other hand, as the depth of neural network $L$ increases, $2 / L$ tends to 0 and the error rate moves closer to the minimax rate of Besov and BV space. This indicates that when the sample size is large enough, deeper models have smaller error than shallower models, and helps explain why empirically deep neural networks has better performance than shallow neural networks.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work does not have potential negative societal impacts to the best of our knowledge.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] All the proofs are in the appendix.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] Although we did not publish the code, the experiments are very simple and can be easily reproduced using the information given in Section G.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section G.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We provided the confidence interval in Figure 2.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] The experiments are lightweighted and can be finished in an ordinary PC in a reasonable time.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A] We only used synthetic dataset.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
+
+# 492 A Other related works
+
+493 Besides Parhi and Nowak [29] which we discussed earlier, Parhi and Nowak [30, 31] also leveraged   
+494 the connections between NNs and splines. Parhi and Nowak [30] focused on characterizing the   
+495 variational form of multi-layer NN. Parhi and Nowak [31] showed that two-layer ReLU activated   
+496 NN achieves minimax rate for a BV class of order 1 but did not cover multilayer NNs nor BV class   
+497 with order $> 1$ , which is our focus.   
+498 The connection between weight-decay regularization with sparsity-inducing penalties in two-layer   
+499 NNs is folklore and used by Neyshabur et al. [27], Savarese et al. [35], Ongie et al. [28], Ergen and   
+500 Pilanci [13, 16], Parhi and Nowak [29, 31], Pilanci and Ergen [32]. The key underlying technique   
+501 — an application of the AM-GM inequality (which we used in this paper as well) — can be traced   
+502 back to Srebro et al. [38] (see a recent exposition by Tibshirani [42]). [42] also generalized the   
+503 result to multi-layered NNs, but with a simple (element-wise) connections. [14] generalized the   
+504 results to a three-layer parallel neural network, and proved its equivalence to an $\ell _ { 1 }$ sparse model, but   
+505 this requires a non-standard regularizer. Besides, [12] proved that training a two-layer convolution   
+506 neural network (CNN) with weight decay induces sparsity, and points to a potential extension to   
+507 these works including our work.   
+508 The approximation-theoretic and estimation-theoretic research for neural network has a long history   
+509 too [6, 4, 49, 36, 39]. Most existing work considered the Holder, Sobolev spaces and their exten  
+510 sions, which contain only homogeneously smooth functions and cannot demonstrate the advantage   
+511 of NNs over kernels. The only exception is Suzuki [39] which, as we discussed earlier, requires   
+512 modifications to NN architecture for each class. In contrast, we require tuning only the standard   
+513 weight decay parameter.
+
+# 514 B Two-layer Neural Network with Truncated Power Activation Functions
+
+515 We start by recapping the result of Parhi and Nowak [29] and formalizing its implication in esti  
+516 mating BV functions. Parhi and Nowak [29] considered a two layer neural network with truncated   
+517 power activation function. Let the neural network be
+
+$$
+f ( x ) = \sum _ { j = 1 } ^ { M } v _ { j } \sigma ^ { m } ( w _ { j } x + b _ { j } ) + c ( x ) ,
+$$
+
+518 where $w _ { j } , v _ { j }$ denote the weight in the first and second layer respectively, $b _ { j }$ denote the bias in the   
+519 first layer, $c ( x )$ is a polynomial of order up to $m$ , $\sigma ^ { m } ( x ) : = \bar { \operatorname* { m a x } } ( x , 0 ) ^ { m }$ . Parhi and Nowak [29,   
+520 Theorem 8] showed that when $M$ is large enough, The optimization problem
+
+$$
+\operatorname* { m i n } _ { w , v } \hat { L } ( f ) + \frac { \lambda } { 2 } \sum _ { j = 1 } ^ { M } ( | v _ { j } | ^ { 2 } + | w _ { j } | ^ { 2 m } )
+$$
+
+521 is equivalent to the locally adaptive regression spline:
+
+$$
+\operatorname* { m i n } _ { f } { \hat { L } } ( f ) + \lambda T V ( f ^ { ( m ) } ( x ) ) ,
+$$
+
+2 which optimizes over arbitrary functions that is $m$ -times weakly differentiable. The latter was   
+23 studied in Mammen and van de Geer [24], which leads to the following MSE:
+
+Theorem 10. Let 4 $M \geq n - m$ , and $\hat { f }$ be the function (7) parameterized by the minimizer of (8), 5 then
+
+$$
+\mathrm { M S E } ( \hat { f } ) = O ( n ^ { - ( 2 m + 2 ) ( 2 m + 3 ) } ) .
+$$
+
+526 We show a simpler proof in the univariate case due to Tibshirani [43]:
+
+527 Proof. As is shown in Parhi and Nowak [29, Theorem 8], the minimizer of (8) satisfy
+
+$$
+| v _ { j } | = | w _ { j } | ^ { m } , \forall k
+$$
+
+528 so the TV of the neural network $f _ { N N }$ is
+
+$$
+\begin{array} { l } { { \displaystyle T V ^ { ( m ) } ( f _ { N N } ) = T V ^ { ( m ) } c ( x ) + \sum _ { j = 1 } ^ { M } \vert v _ { j } \vert \vert w _ { j } \vert ^ { m } T V ^ { ( m ) } ( \sigma ^ { ( m ) } ( x ) ) } } \\ { ~ } \\ { { \displaystyle \qquad = \sum _ { j = 1 } ^ { M } \vert v _ { j } \vert \vert w _ { j } \vert ^ { m } } } \\ { { \displaystyle \qquad = \frac { 1 } { 2 } \sum _ { j = 1 } ^ { M } ( \vert v _ { j } \vert ^ { 2 } + \vert w _ { j } \vert ^ { 2 m } ) } } \end{array}
+$$
+
+529 which shown that (8) is equivalent to the locally adaptive regression spline (9) as long as the number   
+530 of knots in (9) is no more than $M$ . Furthermore, it is easy to check that any spline with knots no   
+531 more than $M$ can be expressed as a two layer neural network (8). It suffices to prove that the solution   
+532 in (9) has no more than $n - m$ number of knots.   
+533 Mammen and van de Geer [24, Proposition 1] showed that there is a solution to (9) ${ \hat { f } } ( x )$ such that   
+534 ${ \hat { f } } ( x )$ is a $m$ th order spline with a finite number of knots but did not give a bound. Let the number of   
+535 knots be $M$ , we can represent $\hat { f }$ using the truncated power basis
+
+$$
+\hat { f } ( x ) = \sum _ { j = 1 } ^ { M } a _ { j } ( x - t _ { j } ) _ { + } ^ { m } + c ( x ) : = \sum _ { j = 1 } ^ { M } a _ { j } \sigma _ { j } ^ { ( m ) } ( x ) + c ( x )
+$$
+
+where 536 $t _ { j }$ are the knots, $c ( x )$ is a polynomial of order up to $m$ , and define $\sigma _ { j } ^ { ( m ) } ( x ) = ( x - t _ { j } ) _ { + } ^ { m }$
+
+537 Mammen and van de Geer [24] however did not give a bound on $M$ . Parhi and Nowak [29]’s   
+538 Theorem 1 implies that $M \leq n - m$ . Its proof is quite technical and applies more generally to a   
+539 higher dimensional generalization of the BV class.   
+40 Tibshirani [43] communicated to us the following elegant argument to prove the same using elemen  
+1 tary convex analysis and linear algebra, which we present below.
+
+Define 542 $\Pi _ { m } ( f )$ as the $L ^ { 2 } ( P _ { n } )$ projection of $f$ onto polynomials of degree up to $m$ , $\Pi _ { m } ^ { \perp } ( f ) : = \begin{array} { l l l } \end{array}$ 543 $f - \Pi _ { m } ( f )$ . It is easy to see that
+
+$$
+\Pi _ { m } ^ { \perp } f ( x ) = \sum _ { j = 1 } ^ { M } a _ { j } \Pi _ { m } ^ { \perp } \sigma _ { j } ^ { ( m ) } ( x )
+$$
+
+Denote 544 $f ( x _ { 1 : n } ) : = \{ f ( x _ { 1 } ) , \ldots , f ( x _ { n } ) \} \in \mathbb { R } ^ { n }$ as a vector of all the predictions at the sample points.
+
+$$
+\Gamma _ { m } ^ { \perp } \hat { f } ( x _ { 1 : n } ) = \sum _ { j = 1 } ^ { M } a _ { j } \Pi _ { m } ^ { \perp } \sigma _ { j } ^ { ( m ) } ( x _ { 1 : n } ) \in \Pi _ { m } ^ { \perp } \mathrm { c o n v } \{ \pm \sigma _ { j } ^ { ( m ) } ( x _ { 1 : n } ) \} \cdot \sum _ { j = 1 } ^ { M } | a _ { j } | = \in \mathrm { c o n v } \{ \pm \Pi _ { m } ^ { \perp } \sigma _ { j } ^ { ( m ) } ( x _ { 1 : n } ) \}
+$$
+
+545 here conv denotes the convex hull of a set. The con hull $\begin{array} { r } { \{ \pm \sigma _ { j } ^ { ( m ) } ( x _ { 1 : n } ) \} \cdot \sum _ { j = 1 } ^ { M } | a _ { j } | } \end{array}$ is an   
+546 $n$ $m$ $m + 1$ dimensional space, so the set   
+547 defined above has dimension $n - m - 1$ . By Carathéodory’s theorem, there is a subset of points in   
+548 this space
+
+$$
+\{ \Pi _ { m } ^ { \perp } \sigma _ { j _ { k } } ^ { ( m ) } ( x _ { 1 : n } ) \} \subseteq \{ \Pi _ { m } ^ { \perp } \sigma _ { j } ^ { ( m ) } ( x _ { 1 : n } ) \} , 1 \leq k \leq n - m
+$$
+
+549 such that
+
+$$
+\Pi _ { m } ^ { \perp } f ( x ) = \sum _ { k = 1 } ^ { n - m } \tilde { a } _ { k } \Pi _ { m } ^ { \perp } \sigma _ { j _ { k } } ^ { ( m ) } ( x ) , \sum _ { k = 1 } ^ { n - m } | a _ { k } | \leq 1
+$$
+
+In other word, there exist a subset of knots 550 $\{ \tilde { t } _ { j } , j \in [ n - m ] \}$ that perfectly recovers $\Pi _ { m } ^ { \perp } { \hat { f } } ( x )$ at all the sample points, and the TV of this function is no larger than 551 $\hat { f }$ .
+
+This shows that
+
+$$
+\tilde { f } ( x ) = \sum _ { j = 1 } ^ { n - m } \tilde { a } _ { j } ( x - t _ { j } ) _ { + } ^ { m } , s . t . \tilde { f } ( x _ { i } ) = f ( x _ { i } )
+$$
+
+for all 552 $x _ { i }$ in $n$ onbservation points.
+
+53 The MSE of locally adaptivity regressive spline (9) was studied in Mammen and van de Geer [24,   
+54 Section 3], which equals the error rate given in Theorem 10. □
+
+555 This indicates that the neural network (7) is minimax optimal for $B V ( m )$ .
+
+Let us explain a few the key observations behind this equivalence. (a) The truncated power functions (together with an mth order polynomial) spans the space of an mth order spline. (b) The neural network in (7) is equivalent to a free-knot spline with $M$ knots (up to reparameterization). (c) A solution to (9) is a spline with at most $n - m$ knots [29, Theorem 8]. (d) Finally, by the AM-GM inequality
+
+$$
+| v _ { j } | ^ { 2 } + | w _ { j } | ^ { 2 m } \geq 2 | v _ { j } | | w _ { j } | ^ { m } = 2 | c _ { j } |
+$$
+
+556 where $c _ { j } = v _ { j } | w _ { j } | ^ { m }$ is the coefficient of the corresponding $j$ th truncated power basis. The mth   
+557 order total variation of a spline is equal to $\textstyle \sum _ { j } | c _ { j } |$ . It is not hard to check that the loss function   
+558 depends only on $c _ { j }$ , thus the optimal solution will always take $^ { 6 6 } = ^ { 9 9 }$ in the AM-GM inequality.
+
+# C Introduction To Common Function Classes
+
+In the following definition define $\Omega$ be the domain of the function classes, which will be omitted in the definition.
+
+# C.1 Besov Class
+
+Definition 1. Modulus of smoothness: For a function $f \in L ^ { p } ( \Omega )$ for some $1 \leq p \leq \infty$ , the $r$ -th modulus of smoothness is defined by
+
+565
+
+$$
+\begin{array} { r l r } & { } & { w _ { r , p } ( f , t ) = \displaystyle \operatorname* { s u p } _ { h \in \mathbb R ^ { d } : \| h \| _ { 2 } \leq t } \| \Delta _ { h } ^ { r } ( f ) \| _ { p } , } \\ & { } & { \Delta _ { h } ^ { r } ( f ) : = \left\{ \begin{array} { l r } { \displaystyle \sum _ { j = 0 } ^ { r } ( \stackrel { r } { j } ) ( - 1 ) ^ { r - j } f ( x + j h ) , } & { \quad \quad i f x \in \Omega , x + r h \in \Omega , } \\ { 0 , } & { \quad \quad o t h e r w i s e . } \end{array} \right. } \end{array}
+$$
+
+566 Definition 2. Besov space: For $1 \leq p , q \leq \infty , \alpha > 0 , r : = \lceil \alpha \rceil + 1 ,$ , define
+
+$$
+| f | _ { B _ { p , q } ^ { \alpha } } = \left\{ \begin{array} { l l } { \Big ( \displaystyle \int _ { t = 0 } ^ { \infty } ( t ^ { - \alpha } w _ { r , p } ( f , t ) ) ^ { q } \frac { d t } { t } \Big ) ^ { \frac { 1 } { q } } , ~ } & { \qquad q < \infty } \\ { \displaystyle \operatorname* { s u p } _ { t > 0 } t ^ { - \alpha } w _ { r , p } ( f , t ) , ~ } & { \qquad q = \infty , } \end{array} \right.
+$$
+
+567 and define the norm of Besov space as:
+
+$$
+\| f \| _ { B _ { p , q } ^ { \alpha } } = \| f \| _ { p } + | f | _ { B _ { p , q } ^ { \alpha } } .
+$$
+
+A function 568 $f$ is in the Besov space $B _ { p , q } ^ { \alpha } i f \| f \| _ { B _ { p , q } ^ { \alpha } }$ is finite.
+
+69 Note that the Besov space for $0 < p , q < 1$ is also defined, but in this case it is a quasi-Banach space   
+70 instead of a Banach space and will not be covered in this paper.   
+571 Functions in Besov space can be decomposed using $\mathbf { B }$ -spline basis functions. Any function $f$ in   
+572 Besov space $B _ { p , q } ^ { \alpha } , \alpha > d / p$ can be decomposed using B-spline of order $m , m > \alpha$ : let $\pmb { x } \in \mathbb { R } ^ { d }$ ,
+
+$$
+f ( \pmb { x } ) = \sum _ { k = 0 } ^ { \infty } \sum _ { \pmb { s } \in J ( k ) } c _ { k , \pmb { s } } ( f ) M _ { m , k , \pmb { s } } ( \pmb { x } )
+$$
+
+573 where $J ( k ) : = \{ 2 ^ { - k } s : s \in [ - m , 2 ^ { k } + m ] ^ { d } \subset \mathbb { Z } ^ { d } \}$ , $M _ { m , k , s } ( { \pmb x } ) : = M _ { m } ( 2 ^ { k } ( { \pmb x } - { \pmb s } ) )$ , and $M _ { k } ( { \pmb x } ) =$   
+574 $\textstyle \prod _ { i = 1 } ^ { d } M _ { k } ( x _ { i } )$ is the cardinal B-spline basis function which can be expressed as a polynomial:
+
+$$
+\begin{array} { l } { { \displaystyle M _ { m } ( x ) = \frac { 1 } { m ! } \sum _ { j = 1 } ^ { m + 1 } ( - 1 ) ^ { j } \binom { m + 1 } { j } ( x - j ) _ { + } ^ { m } } } \\ { { \displaystyle \quad = ( ( m + 1 ) / 2 ) ^ { m } \frac { 1 } { m ! } \sum _ { j = 1 } ^ { m + 1 } ( - 1 ) ^ { j } \binom { m + 1 } { j } \left( \frac { x - j } { ( m + 1 ) / 2 } \right) _ { + } ^ { m } } , } \end{array}
+$$
+
+575 Furthermore, the norm of Besov space is equivalent to the sequence norm:
+
+$$
+\| \{ c _ { k , s } \} \| _ { b _ { p , q } ^ { \alpha } } : = \Big ( \sum _ { k = 0 } ^ { \infty } ( 2 ^ { ( \alpha - d / p ) k } \| \{ c _ { k , s } ( f ) \} _ { s } \| _ { p } ) ^ { q } \Big ) ^ { 1 / q } \asymp \| f \| _ { B _ { p , q } ^ { \alpha } } .
+$$
+
+576 See e.g. Dung [11, Theorem 2.2] for the proof. ˜
+
+The Besov space is closely connected to other function spaces including the Hölder space $( \mathcal { C } ^ { \alpha } )$ and the Sobolev space $( W _ { p } ^ { \alpha } )$ . Specifically, if the domain of the functions is $d$ -dimensional [39, 34],
+
+579
+
+# C.2 Other Function Spaces
+
+Definition 3. Hölder space: let $m \in \mathbb { N }$ , the $m$ -th order Holder class is defined as
+
+$$
+\mathcal { C } ^ { m } = \left\{ f : \operatorname* { m a x } _ { | a | = k } \frac { | D ^ { a } f ( x ) - D ^ { a } f ( z ) | } { \| x - z \| _ { 2 } } < \infty , \forall x , z \in \Omega \right\}
+$$
+
+where $D ^ { a }$ denotes the weak derivative.
+
+Note that fraction order of Hölder space can also be defined. For simplicity, we will not cover that case in this paper.
+
+Definition 4. Sobolev space: let $m \in \mathcal { N } , 1 \leq p \leq \infty$ , the Sobolev norm is defined as
+
+$$
+\| f \| _ { W _ { p } ^ { m } } : = \left( \sum _ { | a | \leq m } \| D ^ { a } f \| _ { p } ^ { p } \right) ^ { 1 / p } ,
+$$
+
+the Sobolev space is the set of functions with finite Sobolev norm:
+
+$$
+W _ { p } ^ { m } : = \{ f : \| f \| _ { W _ { p } ^ { m } } < \infty \} .
+$$
+
+Definition 5. Total Variation (TV): The total variation (TV) of a function $f$ on an interval $[ a , b ]$ is defined as
+
+$$
+T V ( f ) = \operatorname* { s u p } _ { \mathcal { P } } \sum _ { i = 1 } ^ { n _ { \mathcal { P } } - 1 } | f ( x _ { i + 1 } ) - f ( x _ { i } ) |
+$$
+
+91 where the $\mathcal { P }$ is taken among all the partitions of the interval $[ a , b ]$ .
+
+In many applications, functions with stronger smoothness conditions are needed, which can be measured by high order total variation.
+
+Definition 6. High order total variation: the m-th order total variation is the total variation of the $( m - 1 )$ -th order derivative
+
+$$
+T V ^ { ( m ) } ( f ) = T V ( f ^ { ( m - 1 ) } )
+$$
+
+596 Definition 7. Bounded variation $( B V )$ : The $m$ -th order bounded variation class is the set of functions   
+597 whose total variation (TV) is bounded.
+
+$$
+B V ( m ) : = \{ f : T V ( f ^ { ( m ) } ) < \infty \} .
+$$
+
+# 598 D Proof of Estimation Error
+
+# 599 D.1 Equivalence Between Parallel Neural Networks and $p$ -norm Penalized Problems
+
+600 Proposition 4. Fix the input dataset $\mathcal { D } _ { n }$ and a constant $c _ { 1 } > 0$ . There exists an one-to-one mapping   
+601 between $\lambda > 0$ and $P ^ { \prime } > 0$ such that (2) is equivalent to the following problem:
+
+$$
+\begin{array} { r l r } { \underset { \{ \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } , \bar { \mathbf { b } } _ { j } ^ { ( \ell ) } , a _ { j } \} } { \arg \operatorname* { m i n } } } & { \hat { L } \Big ( \displaystyle \sum _ { j = 1 } ^ { M } a _ { j } \bar { f } _ { j } \Big ) = \frac { 1 } { n } \sum _ { i } ( y _ { i } - \bar { f } _ { 1 : M } ( \boldsymbol { x } _ { i } ) ^ { T } \boldsymbol { a } ) ^ { 2 } } & \\ { \quad \quad \quad \quad \quad \quad s . t . \| \bar { \mathbf { W } } _ { j } ^ { ( 1 ) } \| _ { F } \leq c _ { 1 } \sqrt { d } , \forall j \in [ M ] , } & \\ { \| \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } \| _ { F } \leq c _ { 1 } \sqrt { w } , \forall j \in [ M ] , 2 \leq \ell \leq L , } & { \| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } \leq P ^ { \prime } } & \end{array}
+$$
+
+where 602 $\bar { f } _ { j } ( \cdot )$ is a subnetwork with parameters $\bar { \mathbf { W } } _ { j } ^ { ( \ell ) } , \bar { \pmb { b } } _ { j } ^ { ( \ell ) }$
+
+603 Proof. Using Lagrange’s method, one can easily find (2) is equivalent to a constrained optimization   
+604 problem:
+
+$$
+\underset { \{ \mathbf { W } _ { j } ^ { ( \ell ) } , \boldsymbol { b } _ { j } ^ { ( \ell ) } \} } { \arg \operatorname* { m i n } } \hat { L } \Big ( \sum _ { j = 1 } ^ { M } f _ { j } \Big ) , s . t . \sum _ { j = 1 } ^ { M } \sum _ { \ell = 1 } ^ { L } \big \| \mathbf { W } _ { j } ^ { ( \ell ) } \big \| _ { F } ^ { 2 } \leq P
+$$
+
+605 for some constant $P$ that depends on $\lambda$ and the dataset $\mathcal { D }$ .
+
+606 We make use of the property from (4) to minimize the constraint term in (12) while keeping this   
+607 neural network equivalent to the original one. Specifically, let $\mathbf { W } ^ { ( 1 ) } , \pmb { b } ^ { ( 1 ) } , , \dots \mathbf { W } ^ { ( L ) } , \pmb { b } ^ { ( L ) }$ be the   
+608 parameters of an $L$ -layer neural network.
+
+$$
+f ( \boldsymbol { x } ) = \mathbf { W } ^ { ( L ) } \boldsymbol { \sigma } ( \mathbf { W } ^ { ( L - 1 ) } \boldsymbol { \sigma } ( \ldots \boldsymbol { \sigma } ( \mathbf { W } ^ { ( 1 ) } \boldsymbol { x } + \boldsymbol { b } ^ { ( 1 ) } ) \ldots ) + \boldsymbol { b } ^ { ( L - 1 ) } ) + \boldsymbol { b } ^ { ( L ) } ,
+$$
+
+609 which is equivalent to
+
+$$
+f ( \boldsymbol { x } ) = \alpha _ { L } \tilde { \mathbf { W } } ^ { ( L ) } \sigma ( \alpha _ { L - 1 } \tilde { \mathbf { W } } ^ { ( L - 1 ) } \sigma ( \ldots \sigma ( \alpha _ { 1 } \tilde { \mathbf { W } } ^ { ( 1 ) } \boldsymbol { x } + \tilde { \boldsymbol { b } } ^ { ( 1 ) } ) \ldots ) + \tilde { \boldsymbol { b } } ^ { ( L - 1 ) } ) + \tilde { \boldsymbol { b } } ^ { ( L ) } ,
+$$
+
+as long as 610 $\begin{array} { r } { \alpha _ { \ell } > 0 , \prod _ { \ell = 1 } ^ { L } \alpha ^ { L } = \prod _ { \ell = 1 } ^ { L } \Vert \mathbf { W } ^ { ( \ell ) } \Vert _ { F } } \end{array}$ , where $\begin{array} { r } { \tilde { \mathbf { W } } ^ { ( \ell ) } : = \frac { \mathbf { W } ^ { ( \ell ) } } { \| \mathbf { W } ^ { ( \ell ) } \| _ { F } } } \end{array}$ W(ℓ)∥W(ℓ)∥F . By the AM-GM inequal611 ity, the $\ell _ { 2 }$ regularizer of the latter neural network is
+
+$$
+\sum _ { \ell = 1 } ^ { L } \| \boldsymbol { \alpha } _ { \ell } \tilde { \mathbf { W } } ^ { ( \ell ) } \| _ { F } ^ { 2 } = \sum _ { \ell = 1 } ^ { L } \boldsymbol { \alpha } _ { \ell } ^ { 2 } \geq L \left( \prod _ { \ell = 1 } ^ { L } a _ { \ell } \right) ^ { 2 / L } = L \left( \prod _ { \ell = 1 } ^ { L } \| \mathbf { W } ^ { ( \ell ) } \| _ { F } \right) ^ { 2 / L }
+$$
+
+612 and equality is reached when $\alpha _ { 1 } = \alpha _ { 2 } = \cdot \cdot \cdot = \alpha _ { L }$ . In other word, in the problem (2), it suffices to   
+613 consider the network that satisfies
+
+$$
+\| \mathbf { W } _ { j } ^ { ( 1 ) } \| _ { F } = \| \mathbf { W } _ { j } ^ { ( 2 ) } \| _ { F } = \cdots = \| \mathbf { W } _ { j } ^ { ( L ) } \| _ { F } , \forall j \in [ M ] , \ell \in [ L ] .
+$$
+
+614 Using (4) again, one can find that the neural network is also equivalent to
+
+$$
+f ( \boldsymbol { x } ) = \sum _ { j = 1 } ^ { M } a _ { j } \bar { \mathbf { W } } ^ { ( L ) } \sigma ( \bar { \mathbf { W } } _ { j } ^ { ( L - 1 ) } \sigma ( \cdot \cdot \cdot \sigma ( \bar { \mathbf { W } } _ { j } ^ { ( 1 ) } \boldsymbol { x } + \bar { b } _ { j } ^ { ( 1 ) } ) \cdot \cdot \cdot ) + \bar { b } _ { j } ^ { ( L - 1 ) } ) + \bar { b } _ { j } ^ { ( L ) } ,
+$$
+
+615 where
+
+$$
+\| \bar { \mathbf { W } } _ { j } ^ { ( \ell ) } \| _ { F } \leq \beta ^ { ( \ell ) } , a _ { j } = \frac { \prod _ { \ell = 1 } ^ { L } \| \mathbf { W } _ { j } ^ { ( \ell ) } \| _ { F } } { \prod _ { \ell = 1 } ^ { L } \beta ^ { ( \ell ) } } = \frac { \| \mathbf { W } _ { j } ^ { ( 1 ) } \| _ { F } ^ { L } } { \prod _ { \ell = 1 } ^ { L } \beta ^ { ( \ell ) } } = \frac { ( \sum _ { \ell = 1 } ^ { L } \| \mathbf { W } _ { j } ^ { ( \ell ) } \| _ { F } ^ { 2 } / L ) ^ { L / 2 } } { \prod _ { \ell = 1 } ^ { L } \beta ^ { ( \ell ) } } ,
+$$
+
+616 where the last two equality comes from the assumption (13). Choosing $\beta ^ { ( \ell ) } = c _ { 1 } \sqrt { w }$ expect $\ell = 1$   
+617 where $\beta ^ { ( 1 ) } = c _ { 1 } \sqrt { d }$ , and scaling $\bar { \boldsymbol { b } } ^ { ( \ell ) }$ accordingly and taking the constraint in (12) into (14) finishes   
+618 the proof. □
+
+# D.2 Covering Number of Parallel Neural Networks
+
+Theorem 5. The covering number of the model defined in (5) apart from the bias in the last layer satisfies
+
+$$
+\log N ( \mathcal { F } , \delta ) \lesssim w ^ { 2 + 2 / ( 1 - 2 / L ) } L ^ { 2 } \sqrt { d } P ^ { \prime \frac { 1 } { 1 - 2 / L } } \delta ^ { - \frac { 2 / L } { 1 - 2 / L } } \log ( w P ^ { \prime } / \delta ) .
+$$
+
+622
+
+The proof relies on the covering number of each subnetwork in a parallel neural network (Lemma 11), observing that $| f ( x ) | ~ \le ~ 2 ^ { L - 1 } w ^ { L - 1 } \sqrt { d }$ under the condition in Lemma 11, and then apply Lemma 6. We argue that our choice of condition on $\| \pmb { b } ^ { ( \ell ) } \| _ { 2 }$ in Lemma 11 is sufficient to analyzing the model apart from the bias in the last layer, because it guarantees that $\sqrt { w } \| \mathbf { W } ^ { ( \ell ) } A _ { \ell - 1 } ( x ) \| _ { 2 } \leq \| \pmb { b } ^ { ( \ell ) } \| _ { 2 }$ . This leads to
+
+$$
+\| \mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( { \pmb x } ) \| _ { \infty } \leq \| \mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( { \pmb x } ) \| _ { 2 } \leq \sqrt { w } \| { \pmb b } ^ { ( \ell ) } \| _ { 2 } \leq \| { \pmb b } ^ { ( \ell ) } \| _ { \infty }
+$$
+
+623 If this condition is not met, $\mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( \pmb { x } ) + b ^ { ( \ell ) }$ is either always positive or always negative   
+624 for all feasible $_ { \textbf { \em x } }$ along at least one dimension. If $( \mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( \dot { \pmb x } ) + b ^ { ( \ell ) } ) _ { i }$ is always negative,   
+625 one can replace $b ^ { ( \ell ) } ) _ { i }$ with $- \operatorname* { m a x } _ { \pmb { x } } \| \mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( \pmb { x } ) \| _ { \infty }$ without changing the output of this model   
+626 for any feasible $_ { \textbf { \em x } }$ . If $( \mathbf { W } ^ { ( \ell ) } \mathcal { A } _ { \ell - 1 } ( \pmb { x } ) + b ^ { ( \ell ) } ) _ { i }$ is always positive, one can replace $b ^ { ( \ell ) } ) _ { i }$ with   
+627 $\begin{array} { r } { \operatorname* { m a x } _ { \pmb { x } } \| \mathbf { W } ^ { ( \ell ) } \pmb { \mathcal { A } } _ { \ell - 1 } ( \pmb { x } ) \| _ { \infty } } \end{array}$ , and adjust the bias in the next layer such that the output of this model   
+628 is not changed for any feasible $_ { \textbf { \em x } }$ . In either cases, one can replace the bias $\pmb { b } ^ { ( \ell ) }$ with another one with   
+629 smaller norm while keeping the model equivalent except the bias in the last layer.   
+630 Lemma 11. Let ${ \mathcal { F } } \subseteq \{ f : R ^ { d } \to \mathbb { R } \}$ denote the set of $L$ -layer neural network (or a subnetwork in   
+631 a parallel neural network) with width $w$ in each hidden layer. It has the form
+
+$$
+\begin{array} { r l } & { f ( x ) = \mathbf { W } ^ { ( L ) } \sigma ( \mathbf { W } ^ { ( L - 1 ) } \sigma ( \cdot . . \sigma ( \mathbf { W } ^ { ( 1 ) } x + b ^ { ( 1 ) } ) \cdot . . . ) + b ^ { ( L - 1 ) } ) + b ^ { ( L ) } , } \\ & { \mathbf { W } ^ { ( 1 ) } \in \mathbb { R } ^ { w \times d } , \| \mathbf { W } ^ { ( 1 ) } \| _ { F } \leq \sqrt { d } , b ^ { ( 1 ) } \in \mathbb { R } ^ { w } , \| b ^ { ( 1 ) } \| _ { 2 } \leq \sqrt { d w } , } \\ & { \mathbf { W } ^ { ( \ell ) } \in \mathbb { R } ^ { w \times w } \| \mathbf { W } ^ { ( \ell ) } \| _ { F } \leq \sqrt { w } , b ^ { ( \ell ) } \in \mathbb { R } ^ { w } , \| b ^ { ( \ell ) } \| _ { 2 } \leq 2 ^ { \ell - 1 } w ^ { \ell - 1 } \sqrt { d w } , \quad \forall \ell = 2 , \ldots L - 1 , } \\ & { \mathbf { W } ^ { ( L ) } \in \mathbb { R } ^ { 1 \times w } , \| \mathbf { W } ^ { ( L ) } \| _ { F } \leq \sqrt { w } , b ^ { ( L ) } = 0 } \end{array}
+$$
+
+632 and $\sigma ( \cdot )$ is the ReLU activation function, the input satisfy $\| x \| _ { 2 } \leq 1$ , then the supremum norm   
+633 $\delta$ -covering number of $\mathcal { F }$ obeys
+
+$$
+\log N ( \mathcal { F } , \delta ) \leq c _ { 7 } L w ^ { 2 } \log ( 1 / \delta ) + c _ { 8 }
+$$
+
+634 where $c _ { 7 }$ is a constant depending only on $d _ { \mathrm { { ; } } }$ , and $c _ { 8 }$ is a constant that depend on $d , w$ and $L$
+
+635 Proof. First study two neural networks which differ by only one layer. Let $g _ { \ell } , g _ { \ell } ^ { \prime }$ be two neural net  
+636 works satisfying (15) with parameters $\mathbf { W } _ { 1 } , b _ { 1 } , \dots , \mathbf { W } _ { L } , b _ { L }$ and $\mathbf { W } _ { 1 } ^ { \prime } , \boldsymbol { b } _ { 1 } ^ { \prime } , \ldots , \bar { \mathbf { W } } _ { L } ^ { \prime } , \boldsymbol { b } _ { L } ^ { \prime }$ respectively.   
+637 Furthermore, the parameters in these two models are the same except the $\ell$ -th layer, which satisfy
+
+$$
+\| \mathbf { W } _ { \ell } - \mathbf { W } _ { \ell } ^ { \prime } \| _ { F } \leq \epsilon , \| \pmb { b } _ { \ell } - \pmb { b } _ { \ell } ^ { \prime } \| _ { 2 } \leq \tilde { \epsilon } .
+$$
+
+638 Denote the model as
+
+$$
+g _ { \ell } ( x ) = \mathcal { B } _ { \ell } ( \mathbf { W } _ { \ell } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ) , g _ { \ell } ^ { \prime } ( x ) = \mathcal { B } _ { \ell } ( \mathbf { W } _ { \ell } ^ { \prime } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ^ { \prime } )
+$$
+
+639 where $\mathscr { A } _ { \ell } ( \pmb { x } ) = \sigma ( \mathbf { W } _ { \ell - 1 } \sigma ( . . . \sigma ( \mathbf { W } _ { 1 } x + \pmb { b } _ { 1 } ) \dots ) + \pmb { b } _ { \ell - 1 } )$ denotes the first $\ell - 1$ layers in the neural   
+640 network, and Aℓ(x) $) = \mathbf { W } _ { L } \sigma ( \dots \sigma ( \mathbf { W } _ { \ell + 1 } \sigma ( x ) + \pmb { b } _ { \ell + 1 } ) \dots ) + \pmb { b } _ { L } )$ denotes the last $L - \ell - 1$ layers,   
+641 with definition $\mathcal { A } _ { 1 } ( \pmb { x } ) = \pmb { x } , \mathcal { B } _ { L } ( \pmb { x } ) = \pmb { x }$ .
+
+Now focus on bounding 642 √ $\| \mathcal { A } ( \pmb { x } ) \|$ . Let $\mathbf { W } \in \mathbb { R } ^ { m \times m ^ { \prime } } , \| \mathbf { W } \| _ { F } \leq \sqrt { m ^ { \prime } } , \pmb { x } \in \mathbb { R } ^ { m ^ { \prime } } , \pmb { b } \in \mathbb { R } ^ { m } , \| \pmb { b } \| _ { 2 } \leq$ 643 $\sqrt { m }$
+
+$$
+\begin{array} { r l } { \| \sigma ( \mathbf { W } x + b ) \| _ { 2 } \leq \| \mathbf { W } x + b \| _ { 2 } } & { } \\ & { \leq \| \mathbf { W } \| _ { 2 } \| x \| _ { 2 } + \| b \| _ { 2 } } \\ & { \leq \| \mathbf { W } \| _ { F } \| x \| _ { 2 } + \| b \| _ { 2 } } \\ & { \leq \sqrt { m ^ { \prime } } \| x \| _ { 2 } + \sqrt { m } } \end{array}
+$$
+
+644 where we make use of $\| \cdot \| _ { 2 } \leq \| \cdot \| _ { F }$ . Because of that,
+
+$$
+\begin{array} { r l } & { \| \boldsymbol { \mathcal { A } } _ { 2 } ( \pmb { x } ) \| _ { 2 } \leq \sqrt { d } + \sqrt { d w } \leq 2 \sqrt { d w } , } \\ & { \| \boldsymbol { \mathcal { A } } _ { 3 } ( \pmb { x } ) \| _ { 2 } \leq \sqrt { w } \| \boldsymbol { \mathcal { A } } _ { 2 } ( \pmb { x } ) \| _ { 2 } + 2 w \sqrt { d w } \leq 4 w \sqrt { d w } , } \\ & { \qquad \cdot \cdot } \\ & { \| \boldsymbol { \mathcal { A } } _ { \ell } ( \pmb { x } ) \| _ { 2 } \leq \sqrt { w } \| \boldsymbol { \mathcal { A } } _ { \ell - 1 } ( \pmb { x } ) \| _ { 2 } \leq 2 \sqrt { d w } ( 2 w ) ^ { \ell - 2 } . } \end{array}
+$$
+
+Then focus on 645 $B ( { \pmb x } )$ . Le $\mathbf { W } \in \mathbb { R } ^ { m \times m ^ { \prime } } , \| \mathbf { W } \| _ { F } \leq \sqrt { m ^ { \prime } } , \pmb { x } , \pmb { x } ^ { \prime } \in \mathbb { R } ^ { m ^ { \prime } } , \pmb { b } \in \mathbb { R } ^ { m } , \| \pmb { b } \| _ { 2 } \leq \sqrt { m } .$ . Furthermore, 646 $\| { \pmb x } - { \pmb x } ^ { \prime } \| _ { 2 } \le \epsilon$ , then
+
+$$
+\| \sigma ( \mathbf { W } x + b ) - \sigma ( \mathbf { W } x ^ { \prime } + b ) \| _ { 2 } \leq \| \mathbf { W } ( x - x ^ { \prime } ) \| _ { 2 } \leq \| \mathbf { W } \| _ { F } \| x - x ^ { \prime } \| _ { 2 }
+$$
+
+which indicates that 647 $\| \boldsymbol { B } ( \pmb { x } ) - \boldsymbol { B } ( \pmb { x } ) ^ { \prime } \| _ { 2 } \leq ( \sqrt { w } ) ^ { L - \ell } \| \pmb { x } - \pmb { x } ^ { \prime } \| _ { 2 }$
+
+Finally, for any W,648 $\mathbf { W } ^ { \prime } \in \mathbb { R } ^ { m \times m ^ { \prime } } , \pmb { x } \in \mathbb { R } ^ { m ^ { \prime } } , \pmb { b } , \pmb { b } ^ { \prime } \in \mathbb { R } ^ { m }$ , one have
+
+$$
+\begin{array} { r l } { \| ( \mathbf { W } x + b ) - ( \mathbf { W } ^ { \prime } x + b ^ { \prime } ) \| _ { 2 } = \| ( \mathbf { W } - \mathbf { W } ^ { \prime } ) x + ( b - b ^ { \prime } ) \| _ { 2 } } & { } \\ { \leq \| \mathbf { W } - \mathbf { W } ^ { \prime } \| _ { 2 } \| x \| _ { 2 } + \| b - b ^ { \prime } \| _ { 2 } . } & { } \\ { \leq \| \mathbf { W } - \mathbf { W } ^ { \prime } \| _ { F } \| x \| _ { 2 } + \sqrt { m } \| b - b ^ { \prime } \| _ { \infty } . } \end{array}
+$$
+
+649 In summary,
+
+$$
+\begin{array} { r l } & { | g _ { \ell } ( \pmb { x } ) - g _ { \ell } ^ { \prime } ( \pmb { x } ) | = | \mathcal { B } _ { \ell } ( \mathbf { W } _ { \ell } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ) - \mathcal { B } _ { \ell } ( \mathbf { W } _ { \ell } ^ { \prime } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ^ { \prime } ) | } \\ & { \qquad \leq ( \sqrt { w } ) ^ { L - \ell } \| ( \mathbf { W } _ { \ell } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ) - ( \mathbf { W } _ { \ell } ^ { \prime } \mathcal { A } _ { \ell } ( \pmb { x } ) + \pmb { b } _ { \ell } ^ { \prime } ) \| _ { 2 } } \\ & { \qquad \leq ( \sqrt { w } ) ^ { L - \ell } ( \| \mathbf { W } _ { \ell } - \mathbf { W } _ { \ell } ^ { \prime } \| _ { F } \| \mathcal { A } _ { \ell } ( \pmb { x } ) \| _ { 2 } + \| \pmb { b } _ { \ell } - \pmb { b } _ { \ell } ^ { \prime } \| _ { 2 } ) } \\ & { \qquad \leq 2 ^ { ( \ell - 1 ) } w ^ { ( L + \ell - 3 ) / 2 } d ^ { 1 / 2 } \epsilon + w ^ { ( L - \ell ) / 2 } \overline { { \epsilon } } } \end{array}
+$$
+
+Let 650 $f ( x ) , f ^ { \prime } ( x )$ be two neural networks satisfying (15) with parameters $W _ { 1 } , b _ { 1 } , \dots , W _ { L } , b _ { L }$ and 651 $W _ { 1 } ^ { \prime } , b _ { 1 } ^ { \prime } , \ldots , W _ { L } ^ { \prime } , b _ { L } ^ { \prime }$ respectively, and $\| W _ { \ell } - W _ { \ell } ^ { \prime } \| _ { F } \le \epsilon _ { \ell } , \| b _ { \ell } - b _ { \ell } ^ { \prime } \| _ { F } \le { \tilde { \epsilon } } _ { \ell } .$ . Further define $f _ { \ell }$ be the 652 neural network with parameters $W _ { 1 } , b _ { 1 } , \dots , W _ { \ell } , b _ { \ell } , W _ { \ell + 1 } ^ { \prime } , b _ { \ell + 1 } ^ { \prime } , \dots , W _ { L } ^ { \prime } , b _ { L } ^ { \prime }$ , then
+
+$$
+\begin{array} { l } { \displaystyle | f ( x ) - f ^ { \prime } ( x ) | \le | f ( x ) - f _ { 1 } ( x ) | + | f _ { 1 } ( x ) - f _ { 2 } ( x ) | + \cdots + | f _ { L - 1 } ( x ) - f ^ { \prime } ( x ) | } \\ { \displaystyle \qquad \le \sum _ { \ell = 1 } ^ { L } 2 ^ { ( \ell - 2 ) } d ^ { 1 / 2 } w ^ { ( L + \ell - 3 ) / 2 } \epsilon + w ^ { ( L - \ell ) / 2 } \bar { \epsilon } } \end{array}
+$$
+
+For any $\delta > 0$ , one can choose
+
+$$
+\epsilon _ { \ell } = \frac { \delta } { 2 ^ { \ell } w ^ { ( L + \ell - 3 ) / 2 } d ^ { 1 / 2 } } , \tilde { \epsilon } _ { \ell } = \frac { \delta } { 2 w ^ { ( L - \ell ) / 2 } }
+$$
+
+such that 653 $| f ( x ) - f ^ { \prime } ( x ) | \leq \delta$ .
+
+On the other hand, the 654 $\epsilon$ -covering number of $\{ \mathbf { W } \in \mathbb { R } ^ { m \times m ^ { \prime } } : \| \mathbf { W } \| _ { F } \leq \sqrt { m ^ { \prime } } \}$ on Frobenius norm is no larger than 655 $( 2 \sqrt { m ^ { \prime } } / \epsilon + 1 ) ^ { m \times m ^ { \prime } }$ , and the $\bar { \epsilon }$ -covering number of $\{ b \in \mathbb { R } ^ { m } : \| \pmb { b } \| _ { 2 } \leq 1 \}$ on infinity norm is no larger than 656 $( 2 / \bar { \epsilon } + 1 ) ^ { m }$ . The entropy of this neural network can be bounded by
+
+$$
+\log \mathcal { N } ( f ; \delta ) \leq w ^ { 2 } L \log ( 2 ^ { L + 1 } w ^ { L - 1 } / \delta + 1 ) + w L \log ( 2 ^ { L - 1 } w ^ { ( L - 1 ) / 2 } d ^ { 1 / 2 } / \delta + 1 )
+$$
+
+657
+
+# D.3 Covering Number of $p$ -Norm Constrained Linear Combination
+
+a 6. . Th $\log \mathcal { N } ( \mathcal { G } , \delta ) \ \lesssim \ k \log ( 1 / \delta )$ $c _ { 3 }$ $g \in { \mathcal { G } } , | a | \leq 1$ , wny $a g \in { \mathcal { G } }$ $\begin{array} { r } { \mathcal { F } = \Bigl \{ \sum _ { i = 1 } ^ { M } a _ { i } g _ { i } \Big | g _ { i } \in \mathcal { G } , \| a \| _ { p } ^ { p } \leq P , 0 < p < 1 \Bigr \} } \end{array}$ $P > 0$
+
+$\log \mathcal { N } ( \mathcal { F } , \epsilon ) \lesssim k P ^ { \frac { 1 } { 1 - p } } ( \delta / c _ { 3 } ) ^ { - \frac { p } { 1 - p } } \log ( c _ { 3 } P / \delta )$ up to a double logarithmic factor.
+
+Proof. Let $\epsilon$ be a positive constant. Without the loss of generality, we can sort the coefficients in descending order in terms of their absolute values. There exists a positive integer $\mathcal { M }$ (as a function of $\epsilon$ ), such that $| a _ { i } | \geq \epsilon$ for $i \leq \mathcal { M }$ , and $| a _ { i } | < \epsilon$ for $i > M$ .
+
+By definition, 666 $\begin{array} { r } { \mathcal { M } \epsilon ^ { p } \le \sum _ { i = 1 } ^ { M } | a _ { i } | ^ { p } \le P } \end{array}$ so $\mathcal { M } \le P / \epsilon ^ { p }$ , and $| a _ { i } | ^ { p } \leq P , | a _ { i } | \leq P ^ { 1 / p }$ for all $i$ 667 Furthermore,
+
+$$
+\sum _ { i > m } | a _ { i } | = \sum _ { i > M } | a _ { i } | ^ { p } | a _ { i } | ^ { 1 - p } < \sum _ { i > M } | a _ { i } | ^ { p } \epsilon ^ { 1 - p } \leq P \epsilon ^ { 1 - p }
+$$
+
+668 Let $\begin{array} { r } { \tilde { g } _ { i } = \arg \operatorname* { m i n } _ { g \in \tilde { \mathcal { G } } } \| g - \frac { a _ { i } } { P ^ { 1 / p } } g _ { i } \| _ { \infty } } \end{array}$ where $\tilde { \mathcal { G } }$ is the $\delta ^ { \prime }$ -convering set of $\mathcal { G }$ . By definition of the   
+669 covering set,
+
+$$
+\left\| \sum _ { i = 1 } ^ { M } a _ { i } g _ { i } ( x ) - \sum _ { i = 1 } ^ { M } P ^ { 1 / p } \tilde { g } _ { i } ( x ) \right\| _ { \infty } \leq \left\| \sum _ { i = 1 } ^ { M } ( a _ { i } g _ { i } ( x ) - P ^ { 1 / p } \tilde { g } _ { i } ( x ) ) \right\| _ { \infty } + \left\| \sum _ { i = M + 1 } ^ { M } a _ { i } g _ { i } ( x ) \right\| _ { \infty }
+$$
+
+$$
+\epsilon = ( \delta / 2 c _ { 3 } P ) ^ { \frac { 1 } { 1 - p } } , \delta ^ { \prime } \asymp P ^ { - \frac { 1 } { p ( 1 - p ) } } ( \delta / 2 c _ { 3 } ) ^ { \frac { 1 } { 1 - p } } / 2 ,
+$$
+
+we have 671 $\mathcal { M } \leq P ^ { \frac { 1 } { 1 - p } } ( \delta / 2 c _ { 3 } ) ^ { - \frac { p } { 1 - p } } , \mathcal { M } P ^ { 1 / p } \delta ^ { \prime } \leq \delta / 2 , c _ { 3 } P \epsilon ^ { 1 - p } \leq \delta / 2$ , so $( 1 7 ) \leq \delta$ . One can 672 compute the covering number of $\mathcal { F }$ by
+
+$$
+\log \mathcal { N } ( \mathcal { F } , \delta ) \leq \mathcal { M } \log \mathcal { N } ( \mathcal { G } , \delta ^ { \prime } ) \lesssim k \mathcal { M } \log ( 1 / \delta ^ { \prime } )
+$$
+
+673 Taking (18) into (19) finishes the proof.
+
+# E Proof of Approximation Error
+
+# 75 E.1 Approximation of Neural Networks to B-spline Basis Functions
+
+Proposition 7. Let $M _ { m , k , s }$ be the $B$ -spline of order $_ { m }$ with scale $2 ^ { - k }$ in each dimension and position $s \in \mathbb { R } ^ { d }$ : $M _ { m , k , s } ( { \pmb x } ) : = M _ { m } ( 2 ^ { k } ( { \pmb x } - { \pmb s } ) )$ , $M _ { m }$ is defined in (11). There exists a parallel neural network that has the structure and satisfy the constraint in Proposition 4 for $d$ -dimensional input and one output, containing $M = O ( m ^ { d } )$ subnetworks, each of which has width $w = O ( d )$ and depth ${ \cal L } = { \cal O } ( \log ( c ( m , d ) / \epsilon ) )$ for some constant $w , c$ that depends only on m and $d$ , denoted as $\tilde { M } _ { m } ( { \pmb x } ) , { \pmb x } \in \mathbb { R } ^ { d }$ , such that
+
+• $| \tilde { M } _ { m , k , s } ( { \pmb x } ) - M _ { m , k , s } ( { \pmb x } ) | \leq \epsilon , i f 0 \leq 2 ^ { k } ( x _ { i } - s _ { i } ) \leq m + 1 , \forall i \in [ d ] ,$ • $\tilde { M } _ { m , k , s } ( \pmb { x } ) = 0$ , otherwise. • The weights in the last layer satisfy $\| a \| _ { 2 / L } ^ { 2 / L } \lesssim 2 ^ { k } m ^ { d } e ^ { 2 m d / L }$ .
+
+85 We follow the method developed in Yarotsky [49], Suzuki [39], while putting our attention on bound  
+86 ing the Frobenius norm of the weights.
+
+Lemma 12 (Yarotsky [49, Proposition 3]). : There exists a neural network with two-dimensional input and one output $f _ { \times } ( x , y )$ , with constant width and depth ${ \cal O } ( \log ( 1 / \delta ) )$ , and the weight in each layer is bounded by $a$ global constant $c _ { 1 }$ , such that
+
+$$
+\begin{array} { r l } & { \bullet \ \lvert f _ { \times } ( x , y ) - x y \rvert \leq \delta , \forall 0 \leq x , y \leq 1 , } \\ & { \bullet \ f _ { \times } ( x , y ) = 0 , \forall x = 0 o r y = 0 . } \end{array}
+$$
+
+92 We first prove a special case of Proposition 7 on the unscaled, unshifted B-spline basis function by   
+93 fixing $k = 0 , s = 0$ :
+
+Proposition 13. There exists a parallel neural network that has the structure and satisfy the constraint in Proposition 4 for $d$ -dimensional input and one output, containing $M = \lceil ( m + 1 ) / 2 \rceil ^ { d } =$ $O ( m ^ { d } )$ subnetworks, each of which has width $w = O ( d )$ and depth ${ \cal L } = { \cal O } ( \log ( c ( m , d ) / \epsilon ) )$ ) for some constant $w , c$ that depends only on m and $d$ , denoted as $\tilde { M } _ { m } ( { \pmb x } ) , { \pmb x } \in \mathbb { R } ^ { d }$ , such that
+
+• $| \tilde { M } _ { m } ( { \pmb x } ) - M _ { m } ( { \pmb x } ) | \leq \epsilon ,$ , if $0 \leq x _ { i } \leq m + 1 , \forall i \in [ d ]$ , while $M _ { m } ( \cdot )$ denote $m$ -th order $B$ -spline basis function, and c only depends on m and $d$ .
+
+• The weights in the last layer satisfy $\| a \| _ { 2 / L } ^ { 2 / L } \lesssim m ^ { d } e ^ { 2 m d / L }$
+
+702 Proof. We first show that one can use a neural network with constant width $w _ { 0 }$ , depth $L \ \stackrel { } { \sim }$ 703 $\log ( m / \epsilon _ { 1 } )$ and bounded norm $\| W ^ { ( 1 ) } \| _ { F } ~ \leq ~ O ( \sqrt { d } ) , \| W ^ { ( \ell ) } \| _ { F } ~ \leq ~ O ( \sqrt { w } ) , \forall \ell ~ = ~ 2 , \ldots , L$ to 704 705 r basis function up tobe the binary digits of ccuracy , and defi $\epsilon _ { 1 }$ $[ 0 , 1 ]$ $\begin{array} { r l } { m } & { { } = } \end{array}$ $\sum _ { i = 0 } ^ { \lceil \log _ { 2 } m \rceil } m _ { i } 2 ^ { i } , m _ { i } \in \{ 0 , 1 \}$ $m$ $\begin{array} { r } { \bar { m } _ { j } = \sum _ { j = 0 } ^ { i } m _ { i } , \gamma = \lceil \log _ { 2 } m \rceil } \end{array}$
+
+706 then for any $x$
+
+$$
+\begin{array} { c } { { x _ { + } ^ { m } = x _ { + } ^ { \bar { m } _ { \gamma } } \times \left( x _ { + } ^ { 2 ^ { \gamma } } \right) ^ { m _ { \gamma } } } } \\ { { \lbrack x _ { + } ^ { \bar { m } _ { \gamma } } , x _ { + } ^ { 2 ^ { \gamma } } \rbrack = \lbrack x _ { + } ^ { \bar { m } _ { \gamma - 1 } } \times \left( x _ { + } ^ { 2 ^ { \gamma - 1 } } \right) ^ { m _ { \gamma - 1 } } , x _ { + } ^ { 2 ^ { \gamma - 1 } } \times x _ { + } ^ { 2 ^ { \gamma - 1 } } \rbrack } } \\ { { \ldots } } \end{array}
+$$
+
+$$
+[ x _ { + } ^ { \bar { m } _ { 2 } } , x _ { + } ^ { 4 } ] = [ x _ { + } ^ { \bar { m } _ { 1 } } \times \left( x _ { + } ^ { 2 } \right) ^ { m _ { 1 } } , x _ { + } ^ { 2 } \times x _ { + } ^ { 2 } ]
+$$
+
+$$
+[ x _ { + } ^ { \bar { m } _ { 1 } } , x _ { + } ^ { 2 } ] = [ x _ { + } ^ { \bar { m } _ { 0 } } \times x _ { + } ^ { m _ { 0 } } , x _ { + } \times x _ { + } ]
+$$
+
+707 Notice that each line of equation only depends on the line immediately below. Replacing the   
+708 multiply operator $\times$ with the neural network approximation shown in Lemma 12 demonstrates the   
+709 architecture of such neural network approximation. For any $x , y \in [ 0 , 1 ]$ , let $| f _ { \times } ( x , y ) - x y | \leq$   
+710 $\delta , | x - \tilde { x } | \leq \delta _ { 1 } , | y - \delta y | \leq \delta _ { 2 }$ , then $| f _ { \times } ( \bar { x } , \tilde { y } ) - x y | \leq \delta _ { 1 } + \bar { \delta } _ { 2 } + \delta$ . Taking this into (20) shows that   
+711 $\epsilon _ { 1 } \gtrsim 2 ^ { \gamma } \delta \stackrel { } { \sim } m \delta$ , where $\epsilon _ { 1 }$ is the upper bound on the approximate error to truncated power basis of   
+712 order $m$ and $\delta$ is the approximation error to a single multiply operator as in Lemma 12.   
+713 A univariate B-spline basis can be expressed using truncated power basis, and observing that it is   
+714 symmetric around $( m + 1 ) / 2$ :
+
+$$
+\begin{array} { c } { \displaystyle M _ { m } ( x ) = \frac { 1 } { m ! } \sum _ { j = 1 } ^ { m + 1 } ( - 1 ) ^ { j } \binom { m + 1 } { j } ( x - j ) _ { + } ^ { m } } \\ { \displaystyle \ = \frac { 1 } { m ! } \sum _ { j = 1 } ^ { \lceil { ( m + 1 ) / 2 } \rceil } ( - 1 ) ^ { j } \binom { m + 1 } { j } ( \operatorname* { m i n } ( x , m + 1 - x ) - j ) _ { + } ^ { m } } \\ { \displaystyle \ = \frac { ( ( m + 1 ) / 2 ) ^ { m } } { m ! } \sum _ { j = 1 } ^ { \lceil { ( m + 1 ) / 2 } \rceil } ( - 1 ) ^ { j } \binom { m + 1 } { j } \Big ( \frac { \operatorname* { m i n } ( x , m + 1 - x ) - j } { ( m + 1 ) / 2 } \Big ) _ { + } ^ { m } , } \end{array}
+$$
+
+715 A multivariate $d$ -dimensional) B-spline basis function can be expressed as the product of truncated   
+716 power basis functions and thus can be decomposed as
+
+$$
+\begin{array} { l } { { \displaystyle M _ { m } ( \pmb { x } ) = \prod _ { i = 1 } ^ { d } M _ { m } ( x _ { i } ) } } \\ { { \displaystyle \ = \frac { ( ( m + 1 ) / 2 ) ^ { m d } } { ( m ! ) ^ { d } } \prod _ { i = 1 } ^ { d } \prod _ { j = 1 } ^ { \binom { \lceil ( m + 1 ) / 2 \rceil } { j = 1 } ( - 1 ) ^ { j } \binom { m + 1 } { j } \binom { m + 1 } { j } \binom { \displaystyle \operatorname* { m i n } ( x _ { i } , m + 1 - x ) - j } { ( m + 1 ) / 2 } \sp { m } _ { + } \biggr ) } ^ { m } } } \\ { { \displaystyle \ = \frac { ( ( m + 1 ) / 2 ) ^ { m d } } { ( m ! ) ^ { d } } \sum _ { j _ { 1 } , \ldots , j _ { d } = 1 } ^ { \lceil ( m + 1 ) / 2 \rceil } \prod _ { i = 1 } ^ { d } ( - 1 ) ^ { j _ { i } } \binom { m + 1 } { j _ { i } } \Bigl ( \frac { \operatorname* { m i n } ( x , m + 1 - x ) - j _ { i } } { ( m + 1 ) / 2 } \Bigr ) _ { + } ^ { m } } } \end{array}
+$$
+
+717 Using Lemma 12, one can construct a parallel neural network containing $M = \lceil ( m + 1 ) / 2 \rceil ^ { d } =$   
+718 $O ( m ^ { d } )$ subnetworks, and each subnetwork corresponds to one polynomial term in (21). Using the   
+719 results above, the approximation of this constructed neural network can be bounded by
+
+$$
+{ \frac { ( ( m + 1 ) / 2 ) ^ { m d } } { ( m ! ) ^ { d } } } \sum _ { j _ { 1 } , \dots , j _ { d } = 1 } ^ { \lceil { ( m + 1 ) / 2 } \rceil } \prod _ { i = 1 } ^ { d } { ( - 1 ) ^ { j _ { i } } { \binom { m + 1 } { j _ { i } } } \epsilon _ { 1 } \lesssim e ^ { m d } \epsilon _ { 1 } }
+$$
+
+720 where we applied Stirling’s approximation and $\delta$ and $\epsilon _ { 1 }$ has the same definition as above. Choosing   
+721 $\begin{array} { r } { \delta = \frac { \epsilon } { d ( e ^ { 2 m } \sqrt { m } + 1 ) } } \end{array}$ , and recall $\epsilon _ { 1 } \stackrel { } { \sim } m \delta$ proves the approximation error.
+
+To bound the norm of the factors 722 $\| a \| _ { 2 / L } ^ { 2 / L }$ , first observe that
+
+$$
+\begin{array} { c } { { \displaystyle | a _ { j _ { 1 } , \dots , j _ { d } } | = \frac { ( ( m + 1 ) / 2 ) ^ { m d } } { ( m ! ) ^ { d } } \frac { 1 } { ( m + 1 ) / 2 } \prod _ { i = 1 } ^ { d } { \binom { m + 1 } { j _ { i } } } } } \\ { { \leq \frac { ( ( m + 1 ) / 2 ) ^ { m d } } { ( m ! ) ^ { d } } \frac { 2 ^ { m d } } { ( m + 1 ) / 2 } = O ( e ^ { m d } ) } } \end{array}
+$$
+
+where the first inequality is from 723 $\binom { m + 1 } { j _ { i } } \leq 2 ^ { m + 1 }$ , the last equality is from Stirling’s appropximation. 724 Finally,
+
+$$
+\Vert a \Vert _ { 2 / L } ^ { 2 / L } \leq m ^ { d } \operatorname* { m a x } _ { j } \vert a _ { j } \vert ^ { 2 / L } \lesssim m ^ { d } e ^ { 2 m d / L }
+$$
+
+which finishes the proof.
+
+The proof of the Proposition 7 for general $k , s$ follows by appending one more layer in the front, as we show below.
+
+28 Proof of Proposition 7. Using the neural network proposed in Proposition 13, one can construct a   
+729 neural network for appropximating $M _ { m , k , s }$ by adding one layer before the first layer:
+
+$$
+\sigma ( 2 ^ { k } \mathbf { I } _ { d } \pmb { x } - 2 ^ { k } s )
+$$
+
+The unused neurons in the first hidden layer is zero padded. The Frobenius norm of the weight is $2 ^ { k } \| \mathbf { I } _ { d } \| _ { F } = 2 ^ { k } { \sqrt { d } }$ . Following the proof of Proposition 4, rescaling the weight in this layer by $2 ^ { - k }$ , and the weight matrix in the last layer by $2 ^ { k }$ , and scaling the bias properly, one can verify that this neural network satisfy the statement. □
+
+# 734 E.2 Sparse approximation of Besov functions using B-spline wavelets
+
+Proposition 8. Let $\alpha - d / p > 1 , r > 0$ . For any function in Besov space $f _ { 0 } \in B _ { p , q } ^ { \alpha }$ and any $\bar { M }$ $\bar { M }$ $B$ -spline basis of order r any positive integer $m$ satisfyingsuch that $0 < \alpha < \operatorname* { m i n } ( m , m - 1 + 1 / p )$ $\begin{array} { r } { \check { f } _ { \bar { M } } = \sum _ { i = 1 } ^ { \bar { M } } a _ { k _ { i } , s _ { i } } M _ { m , k _ { i } , s _ { i } } } \end{array}$ $\bar { M }$ the approximation error is bounded as $\lVert \check { f } _ { \bar { M } } - f _ { 0 } \rVert _ { r } \lesssim \bar { M } ^ { - \alpha / d } \lVert f _ { 0 } \rVert _ { B _ { p , q } ^ { \alpha } }$ , and the coefficients satisfy
+
+$$
+\| \big \{ 2 ^ { k _ { i } } a _ { k _ { i } , s _ { i } } \big \} _ { k _ { i } , s _ { i } } \| _ { p } \lesssim \| f _ { 0 } \| _ { B _ { p , q } ^ { \alpha } } .
+$$
+
+735
+
+36 The proof is divided into three steps:
+
+1. Bound the 0-norm and the 1-norm of the coefficients of B-spline basis in order to approximate an arbitrary function in Besov space up to any $\epsilon > 0$ .   
+2. Bound $p$ -norm of the coefficients of B-spline basis functions where $0 < p < 1$ using the results above .   
+3. Add the approximation to neural network to B-spline basis computed in Section 4.3.1 into Step 2.
+
+743 Proof. Dung [11, Theorem 3.1] Suzuki [39, Lemma 2] proposed an adaptive sampling recovery ˜   
+744 method that approximates a function in Besov space. The method is divided into two cases: when   
+745 $p \geq r$ , and when $p < r$ .   
+746 When $p \geq r$ , there exists a sequence of scalars $\lambda _ { j } , j \in P ^ { d } ( \mu ) , P _ { d } ( \mu ) : = \{ j \in \mathbb { Z } ^ { d } : | j _ { i } | \leq \mu , \forall i \in$   
+747 $d \}$ for some positive $\mu$ , for arbitrary positive integer $\bar { k }$ , the linear operator
+
+$$
+Q _ { \bar { k } } ( f , \pmb { x } ) = \sum _ { s \in J ( \bar { k } , m , d ) } a _ { \bar { k } , s } ( f ) M _ { \bar { k } , s } ( \pmb { x } ) , \quad a _ { \bar { k } , s } ( f ) = \sum _ { j \in \mathbb { Z } ^ { d } , P ^ { d } ( \mu ) } \lambda _ { j } \bar { f } ( s + 2 ^ { - \bar { k } } j )
+$$
+
+748 has bounded approximation error
+
+$$
+\| f - Q _ { \bar { k } } ( f , x ) \| _ { r } \leq C 2 ^ { - \alpha \bar { k } } \| f \| _ { B _ { p , q } ^ { \alpha } } ,
+$$
+
+where 749 $\bar { f }$ is the extrapolation of $f$ , $J ( \bar { k } , m , d ) : = \{ s : 2 ^ { \bar { k } } s \in \mathbb { Z } ^ { d } , - m / 2 \leq 2 ^ { \bar { k } } s _ { i } \leq 2 ^ { \bar { k } } + m / 2 , \forall i \in$ 750 $[ d ] \}$ . See Dung [11, 2.6-2.7] for the detail of the extrapolation as well as references for options of ˜ 751 sequence $\lambda _ { j }$ .
+
+Furthermore, 752 $Q _ { \bar { k } } ( f ) \in B _ { p , q } ^ { \alpha }$ so it can be decomposed in the form (10) with $\begin{array} { r } { M = \sum _ { k = 0 } ^ { \bar { k } } ( 2 ^ { k } + m - } \end{array}$ 753 $1 ) ^ { d } \lesssim 2 ^ { \bar { k } d }$ components and $\| \{ \tilde { c } _ { k , s } \} _ { k , s } \| \lesssim \| Q _ { \bar { k } } ( f ) \| _ { B _ { p , q } ^ { \alpha } } \lesssim \| f \| _ { B _ { p , q } ^ { \alpha } }$ where $\tilde { c } _ { k , s }$ is the coefficients of 754 the decomposition of $Q _ { \bar { k } } ( f )$ . Choosing $\bar { k } \approx \log _ { 2 } M / d$ leads to the desired approximation error.
+
+755 On the other hand, when $p < r$ , there exists a greedy algorithm that constructs
+
+$$
+G ( f ) = Q _ { \bar { k } } ( f ) + \sum _ { k = \bar { k } + 1 } ^ { k ^ { * } } \sum _ { j = 1 } ^ { n _ { k } } c _ { k , s _ { j } } ( f ) M _ { k , s _ { j } }
+$$
+
+where 756 $\bar { k } \ : \asymp \log _ { 2 } ( M ) , k ^ { * } \ : = \ : [ \epsilon ^ { - 1 } \log ( \lambda M ) ] + \bar { k } + 1 , n _ { k } \ : = \ : [ \lambda M 2 ^ { - \epsilon ( k - \bar { k } ) } ]$ for some $0 ~ < ~ \epsilon ~ <$ 757 $\alpha / \delta - 1 , \delta = d ( 1 / p - 1 / r ) , \lambda > 0$ , such that
+
+$$
+\| f - G ( f ) \| _ { r } \leq \bar { M } ^ { - \alpha / d } \| f \| _ { B _ { p , q } ^ { \alpha } }
+$$
+
+758 and
+
+$$
+\sum _ { k = 0 } ^ { \bar { k } } ( 2 ^ { k } + m - 1 ) ^ { d } + \sum _ { k = \bar { k } + 1 } ^ { { k ^ { * } } } n _ { k } \leq \bar { M } .
+$$
+
+759 See Dung [11, Theorem 3.1] for the detail. ˜
+
+760 Finally, since $\alpha - d / p > 1$ ,
+
+$$
+\begin{array} { l } { \displaystyle \| \{ 2 ^ { k _ { i } } c _ { k _ { i } , s _ { i } } \} _ { k _ { i } , s _ { i } } \| _ { p } \leq \displaystyle \sum _ { k = 0 } ^ { \bar { k } } 2 ^ { k } \| \{ c _ { k _ { i } , s _ { i } } \} _ { s _ { i } } \| _ { p } } \\ { = \displaystyle \sum _ { k = 0 } ^ { \bar { k } } 2 ^ { ( 1 - ( \alpha - d / p ) ) k } ( 2 ^ { ( \alpha - d / p ) k } \| \{ c _ { k _ { i } , s _ { i } } \} _ { s _ { i } } \| _ { p } ) } \\ { \lesssim \displaystyle \sum _ { k = 0 } ^ { \bar { k } } 2 ^ { ( 1 - ( \alpha - d / p ) ) k } \| f \| _ { B _ { p , q } ^ { \alpha } } } \\ { \quad \sim \| f \| _ { B _ { p , q } ^ { \alpha } } } \end{array}
+$$
+
+where the first line is because for arbitrary vectors 761 $\begin{array} { r } { \pmb { a } _ { i } , i \in [ n ] , \| \sum _ { i = 1 } ^ { n } \pmb { a } _ { i } \| _ { p } \leq \sum _ { i = 1 } ^ { n } \| \pmb { a } _ { i } \| _ { p } } \end{array}$ , the 762 third line is because the sequence norm of B-spline decomposition is equivalent to the norm in Besov space (see Section C.1) . □
+
+Note that when $\alpha - d / p = 1$ , the sequence norm (22) is bounded (up to a factor of constant) by $k ^ { * } \| f \| _ { B _ { p , q } ^ { \alpha } }$ , which can be proven by following (22) except the last line. This adds a logarithmic term with respect to $\bar { M }$ compared with the result in Proposition 8. This will add a logarithmic factor to the MSE. We will not focus on this case in this paper of simplicity.
+
+# E.3 Sparse approximation of Besov functions using Parallel Neural Networks
+
+Theorem 9. Under the same condition as Proposition 8, for any positive integer $\bar { M }$ , any function in Besov space $f _ { 0 } \in B _ { p , q } ^ { \alpha }$ can be approximated by a parallel neural network with no less than $O ( m ^ { d } \bar { M } )$ number of subnetworks satisfying:
+
+1. Each subnetwork has width $w = O ( d )$ and depth $L$ .
+
+2. The weights in each layer satisfy √ $\| \bar { \mathbf { W } } _ { k } ^ { ( \ell ) } \| _ { F } \leq O ( \sqrt { w } )$ except the first layer $\| \bar { \mathbf { W } } _ { k } ^ { ( 1 ) } \| _ { F } \leq$ $O ( { \sqrt { d } } )$ ,
+
+3. The scaling factors have bounded $2 / L$ -norm: $\begin{array} { r } { \| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } \lesssim m ^ { d } e ^ { 2 m d / L } \bar { M } ^ { 1 - 2 / ( p L ) } . } \end{array}$
+
+4. The approximation error is bounded by
+
+$$
+\| \tilde { f } - f _ { 0 } \| _ { r } \leq ( c _ { 4 } \bar { M } ^ { - \alpha / d } + c _ { 5 } e ^ { - c _ { 6 } L } ) \| f \| _ { B _ { p , q } ^ { \alpha } }
+$$
+
+777
+
+where $c _ { 4 } , c _ { 5 } , c _ { 6 }$ are constants that depend only on $m , d$ and $p$ .
+
+778 We first prove the following lemma.
+
+Lemma 14. For any $a \in \mathbb { R } ^ { \bar { M } }$ , $0 < p ^ { \prime } < p $ , it holds that:
+
+$$
+\begin{array} { r } { \| a \| _ { p ^ { \prime } } ^ { p ^ { \prime } } \leq \bar { M } ^ { 1 - p ^ { \prime } / p } \| a \| _ { p } ^ { p ^ { \prime } } . } \end{array}
+$$
+
+Proof.
+
+$$
+\sum _ { i } | a _ { i } | ^ { p ^ { \prime } } = \langle \mathbf { 1 } , | a | ^ { p ^ { \prime } } \rangle \leq \left( \sum _ { i } 1 \right) ^ { 1 - { \frac { p ^ { \prime } } { p } } } \left( \sum _ { i } ( | a _ { i } | ^ { p ^ { \prime } } ) ^ { \frac { p } { p ^ { \prime } } } \right) ^ { \frac { p ^ { \prime } } { p } } = { \bar { M } } ^ { 1 - { \frac { p ^ { \prime } } { p } } } \| a \| _ { p } ^ { p ^ { \prime } }
+$$
+
+The first inequality uses a Holder’s inequality with conjugate pair 779 $\textstyle { \frac { p } { p ^ { \prime } } }$ and $1 / ( 1 - { \frac { p ^ { \prime } } { p } } )$ .
+
+780 Proof of Theorem 9. Using Proposition 8, one can construct $\bar { M }$ number of PNN each $O ( m ^ { d } )$ sub  
+781 networks according to Proposition 7, and in each PNN, such that each PNN represents one B-spline   
+782 basis function.The weights in the last layer of each PNN is scaled to match the coefficients in Propo  
+783 sition 8. Taking $p ^ { \prime }$ in Lemma 14 as $2 / \dot { L }$ and combining with Proposition 7 finishes the proof. □
+
+# F Proof of the Main Theorem
+
+Theorem 1 extended form. For any fixed $\alpha - d / p > 1 , q \geq 1 , L \geq 3 ,$ , for any $f _ { 0 } \in B _ { p , q } ^ { \alpha } ,$ , given an $L$ -layer parallel neural network satisfying
+
+• The width of each subnetwork is fixed and large enough: $w \gtrsim d .$ . See Theorem 9 for the detail. • The number of subnetworks is large enough: $M \gtrsim m ^ { d } n ^ { \frac { 1 - 2 / L } { 2 \alpha / d + 1 - 2 / ( p L ) } }$ where $m = \lceil \alpha - 1 \rceil$ .
+
+With proper choice of the parameter of weight decay $\lambda$ , the solution $\hat { f }$ parameterized by (2) satisfies
+
+$$
+\mathrm { M S E } ( \hat { f } ) = \tilde { O } \left( \Big ( \frac { w ^ { 4 - 4 / L } L ^ { 2 - 4 / L } } { n ^ { 1 - 2 / L } } \Big ) ^ { \frac { 2 \alpha / d } { 2 \alpha / d + 1 - 2 / ( p L ) } } + e ^ { - c _ { 6 } L } \right)
+$$
+
+where $\tilde { O }$ shows the scale up to a logarithmic factor, and $c _ { 6 }$ is the constant defined in Theorem 9.
+
+Proof. First recall the relationship between covering number (entropy) and estimation error:
+
+Proposition 15. Let ${ \mathcal { F } } \subseteq \{ \mathbb { R } ^ { d } \to [ - F , F ] \}$ be a set of functions. Assume that $\mathcal { F }$ can be decomposed into two orthogonal spaces $\mathcal { F } = \mathcal { F } _ { \parallel } \times \mathcal { F } _ { \perp }$ where $\mathcal { F } _ { \perp }$ is an affine space with dimension of $N .$ Let $f _ { 0 } \in \{ \mathbb { R } ^ { d } \to [ - F , F ] \}$ be the target function and $\hat { f }$ be the least squares estimator in $\mathcal { F }$ :
+
+$$
+\hat { f } = \underset { f \in \mathcal { F } } { \arg \operatorname* { m i n } } \sum _ { i = 1 } ^ { n } ( y _ { i } - f ( x _ { i } ) ) ^ { 2 } , y _ { i } = f _ { 0 } ( x _ { i } ) + \epsilon _ { i } , \epsilon _ { i } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } ) i . i . d . ,
+$$
+
+796 then it holds that
+
+$$
+\mathrm { M S E } ( \hat { f } ) \leq \tilde { O } \Big ( \underset { f \in \mathcal { F } } { \arg \operatorname* { m i n } } \mathrm { M S E } ( f ) + \frac { N + \log \mathcal { N } ( \mathcal { F } _ { \parallel } , \delta ) + 2 } { n } + ( F + \sigma ) \delta \Big ) .
+$$
+
+797 The proof of Proposition 15 is defered to the section below. We choose $\mathcal { F }$ as the set of functions   
+798 that can be represented by a parallel neural network as stated, the (null) space ${ \mathcal { F } } _ { \perp } = \{ f : f ( { \pmb x } ) =$   
+799 constant $\}$ be the set of functions with constant output, which has dimension 1. This space captures   
+800 the bias in the last layer, while the other parameters contributes to the projection in $\mathcal { F } _ { \| }$ . See Sec  
+801 tion D.2 for how we handle the bias in the other layers. One can find that $\mathcal { F } _ { \| }$ is the set of functions   
+802 that can be represented by a parallel neural network as stated, and further satisfy $\textstyle \sum _ { i = 1 } ^ { n } f ( { \pmb x } _ { i } ) = 0$ .   
+803 Because $\mathcal { F } _ { \parallel } \tilde { \subseteq \mathcal { F } } , \mathcal { N } ( \mathcal { F } _ { \parallel } , \delta ) \overset { \cdot } { \leq } \mathcal { N } ( \mathcal { F } , \delta )$ for all $\delta > 0$ , and the latter is studied in Theorem 5.   
+804 In Theorem 1, the width of each subnetwork is no less than what is required in Theorem 9, while the   
+805 depth and norm constraint are the same, so the approximation error is no more that that in Theorem 9.   
+806 Choosing $r = 2 , p = 2 / L$ , and taking Theorem 5 and Theorem 9 into this Proposition 15, one gets
+
+$$
+\mathrm { M S E } ( \hat { f } ) \lesssim \bar { M } ^ { - 2 \alpha / d } + \frac { w ^ { 2 + 2 / ( 1 - 2 / L ) } L ^ { 2 } } { n } \bar { M } ^ { \frac { 1 - 2 / ( p L ) } { 1 - 2 / L } } \delta ^ { - \frac { 2 / L } { 1 - 2 / L } } ( \log ( \bar { M } / \delta ) + 3 ) + \delta ,
+$$
+
+where $\| f \| _ { B _ { p , q } ^ { \alpha } } , m$ and $d$ taken as constants. The stated MSE is obtained by choosing
+
+$$
+\delta \approx \frac { w ^ { 4 - 4 / L } L ^ { 2 - 4 / L } \bar { M } ^ { 1 - 2 / ( p L ) } } { n ^ { 1 - 2 / L } } , \bar { M } \asymp \Big ( \frac { n ^ { 1 - 2 / L } } { w ^ { 4 - 4 / L } L ^ { 2 - 4 / L } } \Big ) ^ { \frac { 1 } { 2 \alpha / d + 1 - 2 / ( p L ) } }
+$$
+
+807 Note that there exists a weight decay parameter $\lambda ^ { \prime }$ such that the $( 2 / L )$ -norm of the coefficients   
+808 of the parallel neural network satisfy that $\| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } \ : = \ : m ^ { d } e ^ { 2 m d / L } \| \{ \tilde { a } _ { j , \bar { M } } \} \| _ { 2 / L } ^ { 2 / L }$ where $\{ \tilde { a } _ { j , \tilde { M } } \}$   
+809 is the coefficient of the particular $\bar { M }$ -sparse approximation, although $\{ a _ { j } \}$ is not necessarily $\bar { M }$   
+810 sparse. Empirically, one only need to guarantee that during initialization, the number of subnetworks   
+811 $\dot { M } \geq \bar { M }$ such that the $\bar { M }$ -sparse approximation is feasible, thus the approximation error bound   
+812 from Theorem 9 can be applied. Theorem 9 also says that $\| \{ a _ { j } \} \| _ { 2 / L } ^ { 2 / L } = m ^ { d } e ^ { 2 m d / L } \| \{ \tilde { a } _ { j , \bar { M } } \} \| _ { 2 / L } ^ { 2 / L } \lesssim$   
+813 $\bar { M } ^ { 1 - 2 / p L }$ , thus we can apply the covering number bound from Theorem 5 with $P ^ { \prime } = \bar { M } ^ { 1 - 2 / p L }$ .   
+814 Finally, if $\lambda$ is optimally chosen, then it achieves a smaller MSE than this particular $\lambda ^ { \prime }$ , which has   
+815 been proven to be no more than $O ( \bar { M } ^ { - \alpha / d } )$ and completes the proof.
+
+816
+
+817 Proof of Proposition 15. For any function $f ~ \in ~ \mathcal { F }$ , define $\begin{array} { r } { f _ { \perp } \ = \ \arg \operatorname* { m i n } _ { h \in \mathcal { F } _ { \perp } } \sum _ { i = 1 } ^ { n } ( f ( \pmb { x } _ { i } ) \ - } \end{array}$   
+818 $h ( \pmb { x } _ { i } ) ) ^ { 2 }$ be the projection of $f$ to $\mathcal { F } _ { \perp }$ , and define $f _ { \parallel } = f - f _ { \perp }$ be the projection to the orthogo  
+819 nal complement. Note that $f _ { \parallel }$ is not necessarily in $\ddot { \mathcal { F } _ { \parallel } }$ . However, if $f \in { \mathcal { F } }$ , then $f _ { \parallel } \in \mathcal { F } _ { \parallel }$ . $y _ { i \perp }$ and   
+820 $y _ { i \parallel }$ are defined by creating a function $f _ { y }$ such that $f _ { y } ( { \pmb x } _ { i } ) = y _ { i } , \forall i$ , e.g. via interpolation. Because   
+821 $\mathcal { F } _ { \parallel }$ and $\mathcal { F } _ { \perp }$ are orthononal, the empirical loss and population loss can be decomposed in the same   
+822 way:
+
+$$
+\begin{array} { r l r } { \displaystyle L _ { \parallel } ( f ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \parallel } ( \boldsymbol { x } ) - f _ { 0 \parallel } ( \boldsymbol { x } ) ) ^ { 2 } + \frac { n - N } { n } \sigma ^ { 2 } , } & { \ } & { L _ { \perp } ( f ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \perp } ( \boldsymbol { x } ) - f _ { 0 \perp } ( \boldsymbol { x } ) ) ^ { 2 } + \frac { N } { n } } \\ { \displaystyle \hat { L } _ { \parallel } ( f ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \parallel } ( \boldsymbol { x } ) - y _ { i \parallel } ) ^ { 2 } , } & { \ } & { \hat { L } _ { \perp } ( f ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \perp } ( \boldsymbol { x } ) - y _ { i \perp } ( \boldsymbol { x } ) ) ^ { 2 } , } \\ { \displaystyle f S E _ { \parallel } ( f ) = \mathbb { E } _ { \mathcal { D } } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \parallel } ( \boldsymbol { x } ) - f _ { 0 \parallel } ( \boldsymbol { x } ) ) ^ { 2 } \Big ] , } & { \ } & { M S E _ { \perp } ( f ) = \mathbb { E } _ { \mathcal { D } } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { \perp } ( \boldsymbol { x } ) - f _ { 0 \perp } ( \boldsymbol { x } ) ) ^ { 2 } \Big ] , } \end{array}
+$$
+
+823 such that $L ( f ) = L _ { \parallel } ( f ) + L _ { \perp } ( f ) , \hat { L } ( f ) = \hat { L } _ { \parallel } ( f ) + \hat { L } _ { \perp } ( f )$ . This can be verified by de  
+824 825 $\begin{array} { r } { \sum _ { i = 1 } ^ { n ^ { - } } \int _ { 1 \perp } ( \pmb { x } _ { i } ) f _ { 2 \parallel } ( \pmb { x } _ { i } ) = 0 , \forall f _ { 1 } , f _ { 2 } } \end{array}$ $\hat { f } , f _ { 0 }$ $y$ orthogonal components as shown above, and observing that.
+
+# First prove the following claim
+
+Claim 16. Assume that $\begin{array} { r } { \hat { f } ~ = ~ \arg \operatorname* { m i n } _ { f \in \mathcal { F } } \hat { L } ( f ) } \end{array}$ is the empirical risk minimizer. Then $\hat { f } _ { \perp } ~ =$ $\begin{array} { r } { \arg \operatorname* { m i n } _ { f \in \mathcal { F } _ { \bot } } \hat { L } _ { \bot } ( f ) , \hat { f } _ { \parallel } \ = \ \arg \operatorname* { m i n } _ { f \in \mathcal { F } _ { \parallel } } \hat { L } _ { \parallel } ( f ) } \end{array}$ , where $\hat { f } _ { \perp }$ is the projections of $\hat { f }$ in $\mathcal { F } _ { \perp }$ , and $\hat { f } _ { \parallel } = \hat { f } - \hat { f } _ { \perp }$ respectively.
+
+Proof. Since 830 $\hat { f } \in \mathcal { F }$ , by definition $\hat { f } _ { \parallel } \in \mathcal { F } _ { \parallel }$ . Assume that there exist $\hat { f } _ { \perp } ^ { \prime } , \hat { f } _ { \parallel } ^ { \prime }$ , and either $\hat { L } _ { \perp } \big ( \hat { f } _ { \perp } ^ { \prime } \big ) <$ 831 $\hat { L } _ { \perp } ( \hat { f } _ { \perp } )$ , or $\hat { L } _ { \parallel } ( \hat { f } _ { \parallel } ^ { \prime } ) < \hat { L } _ { \parallel } ( \hat { f } _ { \parallel } )$ . Then
+
+$$
+\begin{array} { r l } & { \hat { L } ( \hat { f } ^ { \prime } ) = \hat { L } ( \hat { f } _ { \perp } ^ { \prime } + \hat { f } _ { \parallel } ^ { \prime } ) = \hat { L } _ { \parallel } ( \hat { f } _ { \perp } ^ { \prime } + \hat { f } _ { \parallel } ^ { \prime } ) + \hat { L } _ { \perp } ( \hat { f } _ { \perp } ^ { \prime } + \hat { f } _ { \parallel } ^ { \prime } ) = \hat { L } _ { \parallel } ( \hat { f } _ { \parallel } ^ { \prime } ) + \hat { L } _ { \perp } ( \hat { f } _ { \perp } ^ { \prime } ) } \\ & { \qquad < \hat { L } _ { \parallel } ( \hat { f } _ { \parallel } ) + \hat { L } _ { \perp } ( \hat { f } _ { \perp } ) = \hat { L } _ { \parallel } ( \hat { f } _ { \perp } + \hat { f } _ { \parallel } ) + \hat { L } _ { \perp } ( \hat { f } _ { \perp } + \hat { f } _ { \parallel } ) = \hat { L } ( \hat { f } ) } \end{array}
+$$
+
+which shows that 832 $\hat { f }$ is not the minimizer of $\hat { L } ( f )$ and violates the assumption.
+
+834 Then we bound $M S E _ { \bot } ( f )$ . We convert this part into a finite dimension least square problem:
+
+$$
+\begin{array} { r l } {  { \hat { f } _ { \perp } = \underset { f \in \mathcal { F } _ { \perp } } { \mathrm { a r g m i n } } \hat { L } _ { \perp } ( f ) } } \\ & { = \underset { f \in \mathcal { F } _ { 1 } } { \mathrm { a r g m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f ( \boldsymbol { x } _ { i } ) - f _ { 0 1 } ( \boldsymbol { x } _ { i } ) - \epsilon _ { i 1 } ) ^ { 2 } } \\ & { = \underset { f \in \mathcal { F } _ { \perp } } { \mathrm { a r g m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f ( \boldsymbol { x } _ { i } ) - f _ { 0 1 } ( \boldsymbol { x } _ { i } ) - \epsilon _ { i 1 } ) ^ { 2 } + \epsilon _ { i 1 } ^ { 2 } } \\ & { = \underset { f \in \mathcal { F } _ { \perp } } { \mathrm { a r g m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f ( \boldsymbol { x } _ { i } ) - f _ { 0 1 } ( \boldsymbol { x } _ { i } ) - \epsilon _ { i 1 } ) ^ { 2 } + \epsilon _ { i 1 } ^ { 2 } } \\ & { = \underset { f \in \mathcal { F } _ { \perp } } { \mathrm { a r g m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f ( \boldsymbol { x } _ { i } ) - f _ { 0 1 } ( \boldsymbol { x } _ { i } ) - \epsilon _ { i \perp } - \epsilon _ { i 1 } ) ^ { 2 } } \\ & { = \underset { f \in \mathcal { F } _ { \perp } } { \mathrm { a r g m i n } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f ( \boldsymbol { x } _ { i } ) - f _ { 0 \perp } ( \boldsymbol { x } _ { i } ) - \epsilon _ { i } ) ^ { 2 } } \end{array}
+$$
+
+835 The forth line comes from our assumption that $\mathcal { F } _ { \perp }$ is orthogonal to $\mathcal { F } _ { \parallel }$ , so $\forall f \in \mathcal { F } _ { \perp } , f + f _ { 0 \perp } + \epsilon _ { \perp }$   
+836 is orthogonal to $\epsilon _ { \parallel }$ .
+
+837 Let the basis function of $\mathcal { F } _ { \perp }$ be $h _ { 1 } , h _ { 2 } , \ldots , h _ { N }$ , the above problem can be reparameterized as
+
+$$
+\operatorname* { a r g m i n } _ { \pmb { \theta } \in \mathbb { R } ^ { N } } \frac { 1 } { n } \| \mathbf { X } \pmb { \theta } - \pmb { y } \| ^ { 2 }
+$$
+
+838 where $\mathbf { X } \in \mathbb { R } ^ { n \times N } : X _ { i } = h _ { j } ( x _ { i } ) , y = y _ { 0 \perp } + \epsilon , y _ { 0 \perp } = \left[ f _ { 0 \perp } ( x _ { 1 } ) , \ldots , f _ { 0 \perp } ( x _ { n } ) \right] , \epsilon = \left[ \epsilon _ { 1 } , \ldots , \epsilon _ { n } \right] .$   
+839 This problem has a closed-form solution
+
+$$
+\pmb { \theta } = ( \mathbf { X } ^ { T } \mathbf { X } ) ^ { - 1 } \mathbf { X } ^ { T } \pmb { y }
+$$
+
+Observe that 840 $f _ { 0 \perp } \in \mathcal { F } _ { \perp }$ , let $\pmb { y } _ { 0 \perp } = \mathbf { X } \pmb { \theta } ^ { * }$ ,The MSE of this problem can be computed by
+
+$$
+\begin{array} { l } { { \displaystyle { \cal L } \big ( \hat { f } _ { \perp } \big ) = \frac { 1 } { n } \| { \bf X } \theta - y _ { 0 \perp } \| ^ { 2 } = \frac { 1 } { n } \| { \bf X } ( { \bf X } ^ { T } { \bf X } ) ^ { - 1 } { \bf X } ^ { T } ( { \bf X } \theta ^ { * } + \epsilon ) - { \bf X } \theta ^ { * } \| ^ { 2 } } \ ~ } \\ { { \displaystyle ~ = \frac { 1 } { n } \| { \bf X } ( { \bf X } ^ { T } { \bf X } ) ^ { - 1 } { \bf X } ^ { T } \epsilon \| ^ { 2 } } \ ~ } \end{array}
+$$
+
+841 Observing that $\boldsymbol { \Pi } : = \mathbf { X } ( \mathbf { X } ^ { T } \mathbf { X } ) ^ { - 1 } \mathbf { X } ^ { T }$ is an idempotent and independent projection whose rank is   
+842 $N$ , and that $\mathbb { E } [ \epsilon \epsilon ^ { T } ] = \sigma ^ { 2 } \mathbf { I }$ , we get
+
+$$
+\mathrm { M S E } _ { \perp } ( \hat { f } _ { \perp } ) = \mathbb { E } [ L ( \hat { f } _ { \perp } ) ] = \frac { 1 } { n } \| \Pi \boldsymbol { \epsilon } \| ^ { 2 } = \frac { 1 } { n } \mathrm { t r } ( \Pi \boldsymbol { \epsilon } \boldsymbol { \epsilon } ^ { T } ) = \frac { \sigma ^ { 2 } } { n } \mathrm { t r } ( \Pi )
+$$
+
+843 which concludes that
+
+$$
+\mathrm { M S E } _ { \perp } ( \hat { f } ) = O \Big ( \frac { N } { n } \sigma ^ { 2 } \Big ) .
+$$
+
+844 See also [19, Proposition 1].
+
+Next we study 845 $\mathrm { M S E } _ { \parallel } ( \hat { f } )$ . Denote $\begin{array} { r } { \tilde { \sigma } _ { \parallel } ^ { 2 } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i \parallel } ^ { 2 } , E = \operatorname* { m a x } _ { i } \left| \epsilon _ { i } \right| } \end{array}$ . Using Jensen’s inequality and 846 union bound, we have
+
+$$
+\exp ( t \mathbb { E } [ E ] ) \leq \mathbb { E } [ \exp ( t E ) ] = \mathbb { E } [ \operatorname* { m a x } \exp ( t | \epsilon _ { i } | ) ] \leq \sum _ { i = 1 } ^ { n } \mathbb { E } [ \exp ( t | \epsilon _ { i } | ) ] \leq 2 n \exp ( t ^ { 2 } \sigma ^ { 2 } / 2 )
+$$
+
+847 Taking expectation over both sides, we get
+
+$$
+\mathbb { E } [ E ] \leq \frac { \log { 2 n } } { t } + \frac { t \sigma ^ { 2 } } { 2 }
+$$
+
+848 maximizing the right hand side over $t$ yields
+
+$$
+\mathbb { E } [ E ] \leq \sigma { \sqrt { 2 \log { 2 n } } } .
+$$
+
+Let 849 $\tilde { \mathcal { F } } _ { \parallel }$ be the covering set of $\mathcal { F } _ { \parallel } = \{ f _ { \parallel } : f \in \mathcal { F } \}$ . For any $\tilde { f _ { \parallel } } \in \tilde { \mathcal { F } } _ { \parallel }$ ,
+
+$$
+\begin{array} { l } { { \displaystyle { \cal L } _ { \parallel } ( f _ { j } ) - \hat { \cal L } _ { \parallel } ( f _ { j } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( f _ { j \parallel } ( { \bf x } _ { i } ) - f _ { 0 \parallel } ( { \bf x } _ { i } ) ) ^ { 2 } - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \tilde { f } _ { \parallel } ( { \bf x } _ { i } ) - y _ { i \parallel } ) ^ { 2 } + \frac { n - N } { n } \sigma ^ { 2 } } } \\ { ~ = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i \parallel } ( 2 \tilde { f } _ { \parallel } ( { \bf x } _ { i } ) - f _ { 0 \parallel } ( { \bf x } _ { i } ) - y _ { i \parallel } ) + \frac { n - N } { n } \sigma ^ { 2 } } \\ { ~ = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 \tilde { f } _ { \parallel } ( { \bf x } _ { i } ) - f _ { 0 \parallel } ( { \bf x } _ { i } ) - y _ { i \parallel } ) + \frac { n - N } { n } \sigma ^ { 2 } } \\ { ~ = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 \tilde { f } _ { \parallel } ( { \bf x } _ { i } ) - 2 f _ { 0 \parallel } ( { \bf x } _ { i } ) ) + \frac { n - N } { n } \sigma ^ { 2 } - \tilde { \sigma } _ { \parallel } ^ { 2 } } \end{array}
+$$
+
+850 The first term can be bounded using Bernstein’s inequality: let $h _ { i } = \epsilon _ { i } ( f _ { j \parallel } ( \pmb { x } _ { i } ) - f _ { 0 \parallel } ( \pmb { x } _ { i } ) )$ , by definition 851 $| h _ { i } | \leq 2 E F$ ,
+
+$$
+\begin{array} { r } { \mathrm { V a r } [ h _ { i } ] = \mathbb { E } [ \epsilon _ { i } ^ { 2 } ( \tilde { f } _ { \parallel } ( \pmb { x } _ { i } ) - f _ { 0 \parallel } ( \pmb { x } _ { i } ) ) ^ { 2 } ] } \\ { = ( \tilde { f } _ { \parallel } ( \pmb { x } _ { i } ) - f _ { 0 \parallel } ( \pmb { x } _ { i } ) ) ^ { 2 } \mathbb { E } [ \epsilon _ { i } ^ { 2 } ] } \\ { = ( \tilde { f } _ { \parallel } ( \pmb { x } _ { i } ) - f _ { 0 \parallel } ( \pmb { x } _ { i } ) ) ^ { 2 } \sigma ^ { 2 } } \end{array}
+$$
+
+using Bernstein’s inequality, for any 852 $\tilde { f } _ { \parallel } \in \tilde { \mathcal { F } } _ { \parallel }$ , with probably at least $1 - \delta _ { p }$ ,
+
+$$
+\begin{array} { l l } { \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 \tilde { f } _ { \| } ( { \pmb x } _ { i } ) - 2 f _ { 0 \| } ( { \pmb x } _ { i } ) ) = \frac { 2 } { n } \sum _ { i = 1 } ^ { n } h _ { i } } \\ { \displaystyle \qquad \le \frac { 2 } { n } \sqrt { 2 \sum _ { i = 1 } ^ { n } \left( \tilde { f } _ { \| } ( { \pmb x } _ { i } ) - f _ { 0 \| } ( { \pmb x } _ { i } ) \right) ^ { 2 } \sigma ^ { 2 } \log ( 1 / \delta _ { p } ) } + \frac { 8 E F \log ( 1 / \delta _ { p } ) } { 3 n } } \\ { \displaystyle \qquad = 2 \sqrt { \left( L _ { \| } ( \tilde { f } _ { \| } ) - \frac { n - N } { n } \sigma ^ { 2 } \right) \frac { 2 \sigma ^ { 2 } \log ( 1 / \delta _ { p } ) } { n } } + \frac { 8 E F \log ( 1 / \delta _ { p } ) } { 3 n } } \\ { \displaystyle \qquad \le \epsilon \Big ( L _ { \| } ( \tilde { f } _ { \| } ) - \frac { n - N } { n } \sigma ^ { 2 } \Big ) + \frac { 8 \sigma ^ { 2 } \log ( 1 / \delta _ { p } ) } { n \epsilon } + \frac { 8 E F \log ( 1 / \delta _ { p } ) } { 3 n } } \end{array}
+$$
+
+853 the last inequality holds true for all $\epsilon > 0$ . The union bound shows that with probably at least $1 - \delta$ , for all 854 $\tilde { f } _ { \parallel } \in \tilde { \mathcal { F } } _ { \parallel }$ ,
+
+$$
+\begin{array} { r l } & { \displaystyle \mathbb { E } _ { \| } ( \tilde { f } _ { \| } ) - \hat { L } _ { \| } ( \tilde { f } _ { \| } ) \leq \epsilon \Big ( L _ { \| } ( \tilde { f } _ { \| } ) - \frac { n - N } { n } \sigma ^ { 2 } \Big ) + \frac { 8 \sigma ^ { 2 } \log ( \mathcal { N } ( \mathcal { F } _ { \| } , { \delta } ) / { \delta _ { p } } ) } { n \epsilon } + \frac { 8 E F \log ( \mathcal { N } ( \mathcal { F } _ { \| } , { \delta } ) / { \delta _ { p } } ) } { 3 n } } \\ & { \quad \quad \quad \quad + \frac { n - N } { n } \sigma ^ { 2 } - \tilde { \sigma } _ { \| } ^ { 2 } . } \end{array}
+$$
+
+By rearanging the terms and using the definition of 855 $L ( \tilde { f } _ { \parallel } )$ , we get
+
+$$
+1 - \epsilon ) \Big ( L _ { | | } ( \tilde { f } _ { | | } ) - \frac { n - N } { n } \sigma ^ { 2 } \Big ) \leq \hat { L } _ { | | } ( \tilde { f } _ { | | } ) + \frac { 8 \sigma ^ { 2 } \log ( N ( \mathcal F _ { | | } , \delta ) / \delta _ { p } ) } { n \epsilon } + \frac { 8 E F \log ( N ( \mathcal F _ { | | } , \delta ) / \delta _ { p } ) } { 3 n } - \tilde { \sigma } _ { \| } ^ { 2 } .
+$$
+
+856 Taking the expectation (over $\mathcal { D }$ ) on both sides, and notice that $\begin{array} { r } { \mathbb { E } [ \tilde { \sigma } _ { \parallel } ^ { 2 } ] = \frac { n - N } { n } \sigma ^ { 2 } } \end{array}$ . Furthermore, for any random variable 857 $\begin{array} { r } { X , \mathbb { E } [ X ] = \int _ { - \infty } ^ { \infty } x d P ( X \leq x ) } \end{array}$ , we get
+
+$$
+\begin{array} { r l } & { \displaystyle \operatorname* { m a x } _ { \bar { f } _ { \| } \in \tilde { \mathcal { F } } _ { \| } } \Big ( ( 1 - \epsilon ) M S E _ { \| } ( \tilde { f } _ { \| } ) - \mathbb { E } [ \hat { L } _ { \| } ( \tilde { f } _ { \| } ) ] \Big ) } \\ & { \le \Big ( \displaystyle \frac { 8 \sigma ^ { 2 } } { n \epsilon } + \frac { 8 F \sigma \sqrt { 2 \log { 2 n } } } { 3 n } \Big ) \Big ( \log \mathcal { N } ( \mathcal { F } _ { \| } , \delta ) - \displaystyle \int _ { \delta = 0 } ^ { 1 } \log ( \delta _ { p } ) d \delta _ { p } \Big ) - \frac { n - N } { n } \sigma ^ { 2 } } \\ & { = \Big ( \displaystyle \frac { 8 \sigma ^ { 2 } } { n \epsilon } + \frac { 8 F \sigma \sqrt { 2 \log { 2 n } } } { 3 n } \Big ) ( \log \mathcal { N } ( \mathcal { F } _ { \| } , \delta ) + 1 ) - \displaystyle \frac { n - N } { n } \sigma ^ { 2 } . } \end{array}
+$$
+
+858 where the integration can be computed by replacing $\delta$ with $e ^ { x }$ . Though it is not integrable under   
+859 Riemann integral, it is integrable under Lebesgue integration.
+
+Similarly, let 860 $\begin{array} { r } { \check { f } _ { \parallel } = \arg \operatorname* { m i n } _ { f \in \mathcal { F } _ { \parallel } } L _ { \parallel } ( f ) } \end{array}$ ,
+
+$$
+L _ { \parallel } ( { \check { f } } _ { \parallel } ) - \hat { L } _ { \parallel } ( { \check { f } } _ { \parallel } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 { \check { f } } _ { \parallel } ( \pmb { x } _ { i } ) - 2 f _ { 0 \parallel } ( \pmb { x } _ { i } ) ) + \frac { n - N } { n } \sigma ^ { 2 } - \tilde { \sigma } _ { \parallel } ^ { 2 }
+$$
+
+861 with probably at least $1 - \delta _ { q }$ , for any $\epsilon > 0$ ,
+
+$$
+\begin{array} { r l } & { - \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 \check { f } _ { \| } ( { \boldsymbol x } _ { i } ) - 2 f _ { 0 \| } ( { \boldsymbol x } _ { i } ) ) \leq \epsilon \Big ( L _ { \| } ( \check { f } _ { \| } ) - \frac { n - N } { n } \sigma ^ { 2 } \Big ) + \frac { 8 \sigma ^ { 2 } \log ( 1 / \delta _ { p } ) } { n \epsilon } + \frac { 8 E F \log ( 1 / \delta _ { p } ) } { 3 n } , } \\ & { \hat { \boldsymbol { c } } _ { \| } ( \check { f } _ { \| } ) \leq ( 1 + \epsilon ) \Big ( L _ { \| } ( \check { f } _ { \| } ) - \frac { n - N } { n } \sigma ^ { 2 } \Big ) + \frac { 8 \sigma ^ { 2 } \log ( 1 / \delta _ { p } ) } { n \epsilon } + \frac { 8 E F \log ( 1 / \delta _ { q } ) } { 3 n } + \tilde { \sigma } _ { \| } ^ { 2 } . } \end{array}
+$$
+
+862 Taking the expectation on both sides,
+
+$$
+\mathbb { E } [ \hat { L } _ { \| } ( \check { f } _ { \| } ) ] \leq ( 1 + \epsilon ) \mathrm { M S E } _ { \| } ( \check { f } _ { \| } ) + \frac { 8 \sigma ^ { 2 } } { n \epsilon } + \frac { 8 F \sigma \sqrt { 2 \log { 2 n } } } { 3 n } + \frac { n - N } { n } \sigma ^ { 2 } .
+$$
+
+863 Finally, let $\begin{array} { r } { \hat { f } _ { * } : = \arg \operatorname* { m i n } _ { f \in \tilde { \mathcal { F } } _ { \parallel } } \sum _ { i = 1 } ^ { n } ( \hat { f } _ { \parallel } ( \pmb { x } _ { i } ) - f ( \pmb { x } _ { i } ) ) ^ { 2 } } \end{array}$ be the projection of $\hat { f } _ { \parallel }$ in its $\delta$ -covering   
+864 space,
+
+$$
+\begin{array} { l l } { \displaystyle \mathcal { A } \mathrm { S } \mathrm { E } _ { \parallel } ( \hat { f } _ { \parallel } ) = \mathbb { E } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { f } _ { \parallel } ( x _ { i } ) - f _ { 0 \parallel } ( x _ { i } ) ) ^ { 2 } \Big ] } \\ { \displaystyle = \mathbb { E } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { f } _ { * } ( x _ { i } ) - f _ { 0 \parallel } ( x _ { i } ) ) ^ { 2 } + \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { f } _ { \parallel } ( x _ { i } ) - \hat { f } _ { * } ( x _ { i } ) ) ( \hat { f } _ { \parallel } ( x _ { i } ) + \hat { f } _ { * } ( x _ { i } ) - 2 f _ { 0 \parallel } ( x _ { i } ) ) } \\ { \displaystyle \leq \mathbb { E } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( \hat { f } _ { * } ( x _ { i } ) - f _ { 0 \parallel } ( x _ { i } ) ) ^ { 2 } \Big ] + 4 F \delta } \\ { \displaystyle = \mathrm { M S E } _ { \parallel } ( \hat { f } _ { * } ( x _ { i } ) ) + 4 F \delta , } \end{array}
+$$
+
+865 and similarly
+
+$$
+\hat { L } _ { \parallel } ( \hat { f } _ { * } ) \leq \hat { L } _ { \parallel } ( \hat { f } _ { \parallel } ) + ( 4 F + 2 E ) \delta .
+$$
+
+866 We can conclude that
+
+$$
+\begin{array} { r l } & { \mathrm { M S F } _ { 1 } ( \hat { j } _ { 1 } ) \leq \frac { 1 } { 1 - e } \bigg \{ \mathrm { R } [ \hat { \varepsilon } _ { 1 } ( \hat { j } _ { 1 } ) ] + \big ( \frac { 8 e ^ { 2 } } { n \alpha } + \frac { 8 \beta e \sqrt { 2 } \log 2 \pi } { 3 \alpha } \big ) ( \log \mathcal { N } ( \mathcal { F } ( \boldsymbol { F } _ { 1 } , \boldsymbol { \delta } ) + 1 ) - \frac { \pi - N } { \alpha } \sigma ^ { 2 } } \\ & { \qquad + 4 F \delta } \\ & { \qquad \leq \frac { 1 } { 1 - e } \bigg ( \mathrm { E } [ \hat { \varepsilon } _ { 1 } ( \hat { j } _ { 1 } ) ] + ( 4 F - \sigma \sqrt { 8 \log 2 \pi } ) \delta } \\ & { \qquad + \big ( \frac { 8 e ^ { 2 } } { \mathrm { N e } } + \frac { 8 F \sigma \sqrt { 2 } \log 2 \pi } { 3 \alpha } \big ) ( \log \mathcal { N } ( \mathcal { F } _ { 1 } , \boldsymbol { \delta } ) + 1 ) - \frac { \pi - N } { \alpha } \sigma ^ { 2 } \bigg ) + 4 F \delta } \\ & { \qquad \leq \frac { 1 } { 1 - e } \bigg ( \mathrm { E } [ \hat { \varepsilon } _ { 1 } ( \hat { j } _ { 1 } ) ] + ( 4 F + \sigma \sqrt { 8 \log 2 \pi } ) \delta } \\ & { \qquad + \big ( \frac { 8 e ^ { 2 } } { \mathrm { N e } } + \frac { 8 F \sigma \sqrt { 2 } \log 2 \pi } { 3 \alpha } \big ) ( \log \mathcal { N } ( \mathcal { F } _ { 1 } , \boldsymbol { \delta } ) + 1 ) - \frac { n - N } { \pi } \sigma ^ { 2 } \bigg ) + 4 F \delta } \\ & { \qquad \leq \frac { 1 + e } { 1 - e } \operatorname { W S F } ( \hat { j } _ { 1 } ) + \frac { 1 } { n } \bigg ( \frac { 8 e ^ { 2 } } { \mathrm { N e } } + \frac { 8 F \sigma \sqrt { 2 } \log 2 \pi } { 3 \alpha } \bigg ) \big ( \frac { \log \mathcal { N } ( \mathcal { F } _ { 1 } , \boldsymbol { \delta } ) + 1 } { \pi } - \frac { n - N } { \pi } \sigma ^ { 2 } \big ) } \\ &  \qquad \leq \frac { 1 + e } { 1 - e } \operatorname { W S F } ( \hat { j } _ { 1 } ) + \frac { 1 } { n } \big ( \frac { 8 e ^ { 2 } } { \mathrm { N e } } + \frac  8 F \sigma ^ { 2 }  \end{array}
+$$
+
+867 where the first line comes from (24), and second comes from (26), the thid line is because   
+868 $\begin{array} { r } { \hat { f } _ { \parallel } = \arg \operatorname* { m i n } _ { f \in \mathcal { F } _ { \parallel } } \hat { L } _ { \parallel } ( f ) } \end{array}$ , and the last line comes from (25). We also use that fact that $\hat { L } _ { \parallel } ( \hat { f } ) \leq$   
+869 $\hat { L } _ { \parallel } ( f ) , \forall f$ . Noticing that $\mathrm { M S E } ( \hat { f } ) = \mathrm { M S E } _ { \parallel } ( \hat { f } ) + \mathrm { M S E } _ { \perp } ( \hat { f } )$ , combining this with (23) finishes the   
+870 proof. □
+
+# 871 G Detailed experimental setup
+
+# G.1 Target Functions
+
+873 The doppler function used in Figure 2(d)-(f) is
+
+$$
+f ( x ) = \sin ( 4 / ( x + 0 . 0 1 ) ) + 1 . 5 .
+$$
+
+874 The “vary” function used in Figure $2 ( \mathbf { g } )$ -(i) is
+
+$$
+\begin{array} { r l } & { f ( x ) = M _ { 1 } ( x / 0 . 0 1 ) + M _ { 1 } ( ( x - 0 . 0 2 ) / 0 . 0 2 ) + M _ { 1 } ( ( x - 0 . 0 6 ) / 0 . 0 3 ) } \\ & { \qquad + M _ { 1 } ( ( x - 0 . 1 2 ) / 0 . 0 4 ) + M _ { 3 } ( ( x - 0 . 2 ) / 0 . 0 2 ) + M _ { 3 } ( ( x - 0 . 2 8 ) / 0 . 0 4 ) } \\ & { \qquad + M _ { 3 } ( ( x - 0 . 4 4 ) / 0 . 0 6 ) + M _ { 3 } ( ( x - 0 . 6 8 ) / 0 . 0 8 ) , } \end{array}
+$$
+
+875 where $M _ { 1 } , M _ { 3 }$ are first and third order Cardinal B-spline bases functions respectively. We uni  
+876 formly take 256 samples from 0 to 1 in the piecewise cubic function experiment, and uniformly   
+877 1000 samples from 0 to 1 in the doppler function and “vary” function experiment. We add zero   
+878 mean independent (white) Gaussian noise to the observations. The standard derivation of noise is   
+879 0.4 in the doppler function experiment and 0.1 in the “vary” function experiment.
+
+# G.2 Training/Fitting Method
+
+In the piecewise polynomial function (“vary”) experiment, the depth of the PNN $L = 1 0$ , the width of each subnetwork $w = 1 0$ , and the model contains $M = 5 0 0$ subnetworks. The depth of NN is also 10, and the width is 200 such that the NN and PNN have almost the same number of parameters. In the doppler function experiment, the depth of the PNN $L = 1 2$ , the width of each subnetwork $w =$ 10, and the model contains $M = 2 0 0 0$ subnetworks, because this problem requires a more complex model to fit. The depth of NN is 12, and the width is 400. We used Adam optimizer with learning rate of $1 0 ^ { - 3 }$ . We first train the neural network layer by layer without weight decay. Specifically, we start with a two-layer neural network with the same number of subnetworks and the same width in each subnetwork, then train a three layer neural network by initializing the first layer using the trained two layer one, until the desired depth is reached. After that, we turn the weight decay parameter and train it until convergence. In both trend filtering and smoothing spline experiment, the order is 3, and in wavelet denoising experiment, we use sym4 wavelet with soft thresholding. We implement the trend filtering problem according to Tibshirani [40] using CVXPY, and use MOSEK to solve the convex optimization problem. We directly call R function smooth.spline to solve smoothing spline.
+
+# 96 G.3 Post Processing
+
+7 The degree of freedom of smoothing spline is returned by the solver in R, which is rounded to the   
+98 nearest integer when plotting. To estimate the degree of freedom of trend filtering, for each choice   
+99 of $\lambda$ , we repeated the experiment for 10 times and compute the average number of nonzero knots as   
+00 estimated degree of freedom. For neural networks, we use the definition [41]:
+
+$$
+2 \sigma ^ { 2 } \mathrm { d } \mathbf { f } = \mathbb { E } \| \pmb { y } ^ { \prime } - \pmb { \hat { y } } \| _ { 2 } ^ { 2 } - \mathbb { E } \| \pmb { y } - \pmb { \hat { y } } \| _ { 2 } ^ { 2 }
+$$
+
+901 where df denotes the degree of freedom, $\sigma ^ { 2 }$ is the variance of the noise, $\textbf {  { y } }$ are the labels, $\hat { \pmb { y } }$ are   
+902 the predictions and $\mathbf { \Delta } _ { \mathbf { \boldsymbol { y } } ^ { \prime } }$ are independent copy of $y$ . We find that estimating (27) directly by sampling   
+903 leads to large error when the degree of freedom is small. Instead, we compute
+
+$$
+2 \sigma ^ { 2 } \hat { \mathrm { d } } \mathbf { f } = \hat { \mathbb { E } } \Vert \pmb { y } _ { 0 } - \hat { \pmb { y } } \Vert _ { 2 } ^ { 2 } - \hat { \mathbb { E } } \Vert \pmb { y } - \hat { \pmb { y } } \Vert _ { 2 } ^ { 2 } + \hat { \mathbb { E } } \Vert \pmb { y } - \bar { y } _ { 0 } \Vert _ { 2 } ^ { 2 } - \Vert \pmb { y } _ { 0 } - \bar { y } _ { 0 } \Vert _ { 2 } ^ { 2 }
+$$
+
+where df is the estimated degree of freedom, ˆ $\mathbb { E }$ denotes the empirical average (sample mean), $\scriptstyle { \pmb { y } } _ { 0 }$ is the target function and $\bar { y } _ { 0 }$ is the mean of the target function in its domain.
+
+906 Proposition 17. The expectation of (28) over the dataset $\mathcal { D }$ equals (27).
+
+Proof.
+
+$$
+\begin{array} { r l } { 2 \sigma ^ { 2 } \hat { \mathrm { d } } \mathbf { f } = \mathbb { E } _ { \mathcal { D } } [ \hat { \mathbb { E } } \| y _ { 0 } - \hat { y } \| _ { 2 } ^ { 2 } - \hat { \mathbb { E } } \| y - \hat { y } \| _ { 2 } ^ { 2 } + \hat { \mathbb { E } } \| y - \bar { y } _ { 0 } \| _ { 2 } ^ { 2 } - \| y _ { 0 } - \bar { y } _ { 0 } \| _ { 2 } ^ { 2 } ] } & { } \\ { = \mathbb { E } \| y _ { 0 } - \hat { y } \| _ { 2 } ^ { 2 } - \mathbb { E } \| y - \hat { y } \| _ { 2 } ^ { 2 } + \mathbb { E } _ { \mathcal { D } } [ \hat { \mathbb { E } } [ ( y - y _ { 0 } ) ( y + y _ { 0 } - 2 \bar { y } _ { 0 } ) ] ] } & { } \\ { = \mathbb { E } \| y _ { 0 } - \hat { y } \| _ { 2 } ^ { 2 } - \mathbb { E } \| y - \hat { y } \| _ { 2 } ^ { 2 } + \mathbb { E } \Big [ \displaystyle \sum _ { i = 1 } ^ { n } \epsilon _ { i } ( 2 y _ { i } + \epsilon _ { i } - 2 \bar { y } _ { 0 } ) \Big ] } & { } \\ { = \mathbb { E } \| y _ { 0 } - \hat { y } \| _ { 2 } ^ { 2 } - \mathbb { E } \| y - \hat { y } \| _ { 2 } ^ { 2 } + n \sigma ^ { 2 } } & { } \\ { = \mathbb { E } \| y ^ { \prime } - \hat { y } \| _ { 2 } ^ { 2 } - \mathbb { E } \| y - \hat { y } \| _ { 2 } ^ { 2 } } & { } \end{array}
+$$
+
+where 907 $\mathcal { D }$ denotes the dataset. In the third line, we make use of the fact that $\mathbb { E } [ \epsilon _ { i } ] = 0 , \mathbb { E } [ \epsilon _ { i } ^ { 2 } ] = \sigma ^ { 2 }$ and in the last line, we make use of 908 $\mathbb { E } [ \epsilon _ { i } ^ { \prime } ] = 0 , \mathbb { E } [ { \epsilon _ { i } ^ { \prime } } ^ { 2 } ] = \sigma ^ { 2 }$ , and $\epsilon _ { i } ^ { \prime }$ are independent of $y _ { i }$ and $y _ { 0 , i }$
+
+909 One can easily check that a “zero predictor” (a predictor that always predict $\bar { y } _ { 0 }$ , and it always predicts   
+910 0 if the target function has zero mean) always has an estimated degree of freedom of 0.   
+911 In Figure 2(h)(i), we take the minimum MSE over different choices of $\lambda$ , and plot the average over   
+912 10 runs. Due to optimization issue, sometimes the neural networks are stuck at bad local minima   
+913 and the empirical loss is larger than the global minimum by orders of magnitude. To deal with this   
+914 problem, in Figure 2(h)(i), we manually detect these results by removing the experiments where   
+915 the MSE is larger than 1.5 times the average MSE under the same setting, and remove them before   
+916 computing the average.
+
+# G.4 More experimental results
+
+# G.4.1 Regularization weight vs degree-of-freedom
+
+19 As we explained in the previous section, the degree of freedom is the exact information-theoretic   
+920 measure of the generalization gap. A Larger degree-of-freedom implies more overfitting.   
+921 In figure Figure 3, we show the relationship between the estimated degree of freedom and the scaling   
+922 factor of the regularizer $\lambda$ in a parallel neural network and in trend filtering. As is shown in the   
+923 figure, generally speaking as $\lambda$ decreases towards 0, the degree of freedom should increase too.   
+924 However, for parallel neural networks, if $\lambda$ is very close to 0, the estimated degree of freedom will   
+925 not increase although the degree of freedom is much smaller than the number of parameters —   
+926 actually even smaller than the number of subnetworks. Instead, it actually decreases a little. This   
+927 effect has not been observed in other nonparametic regression methods, e.g. trend filtering, which   
+928 overfits every noisy datapoint perfectly when $\lambda  0$ . But for the neural networks, even if we do   
+929 not regularize at all, the among of overfitting is still relatively mild 30/256 vs $8 0 / 1 0 0 0$ . In our   
+930 experiments using neural networks, when $\lambda$ is small, we denoise the estimated degree of freedom   
+931 using isotonic regression.
+
+![](images/ed50621e49b377249e98a1b965b05a247f84f9850c364b87763c7d1f04af761b.jpg)  
+Figure 3: The relationship between degree of freedom and the scaling factor of the regularizer $\lambda$ . The solid line shows the result after denoising. (a)(b)in a NN. (c)(d) In trend filtering. (a)(c): the piecewise cubic function. (b)(d) the doppler function.
+
+![](images/7b7a208b457a680c3ffd08befa37293b617a1e19ece4c7701ad1dd82d7cb3fde.jpg)  
+Figure 4: More experiments results of Doppler function.
+
+![](images/414b78b9f40a51c9b6ac424501092f7c4b97fe51fc6f21a91e8cb9e8ad4764ae.jpg)  
+Figure 5: More experiments results of the “vary” function.
+
+We do not know the exact reason of this curious observation. Our hypothesis is that it might be related to issues with optimization, i.e., the optimizer ends up at a local minimum that generalizes better than a global minimum; or it could be connected to the “double descent” behavior of DNN [26] under over-parameterization.
+
+# G.4.2 Detailed numerical results
+
+37 In order to allow the readers to view our result in detail, we plot the numerical experiment results of   
+38 each method separately in Figure 4 and Figure 5.
+
+In this section we investigate the equivalence of two-layer NN and the locally adaptive regression splines from Section B. In the special case when $m = 1$ the special regularization reduces to weight decay and the non-standard truncated power activation becomes ReLU. We compare L1 trend filtering [22] (shown to be equivalent to locally adaptive regression splines by Tibshirani [40]) and an overparameterized version of the neural network for all regularization parameter $\lambda > 0$ , i.e., a regularization path. The results are shown in Figure 6. It is clear that as the weight decay increases, it induces sparsity in the number of knots it selects similarly to L1-Trend Filtering, and the regularization path matches up nearly perfectly even though NNs are also learning knots locations.
+
+![](images/0919b7bb471d06baec77dba3c11b02372b2be0431bcccc40755d122531decebc.jpg)  
+x  lambda Figure 6: Comparison of the weight decayed ReLU neural networks (Top row) and L1 Trend Filtering (Bottom row) with different regularization parameters. The left column shows the fitted functions and the right column shows the regularization path (in the flavor of [17]) of the coefficients of the truncated power basis at individual data points (the free-knots learned by NN are snapped to the nearest input $x$ to be comparable).
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