Transition to Linearity of Wide Neural Networks is an Emerging Property of Assembling Weak Models
Authors: Chaoyue Liu
Libin Zhu
Misha Belkin
Venue: ICLR 2022 Spotlight (2022)
Keywords: Assembling
linearity
Transition to linearity
wide neural networks

Abstract: Wide neural networks with linear output layer have been shown to be near-linear, and to have near-constant neural tangent kernel (NTK), in a region containing the optimization path of gradient descent. These findings seem counter-intuitive since in general neural networks are highly complex models. Why does a linear structure emerge when the neural networks become wide? 
In this work, we provide a new perspective on this "transition to linearity" by considering a neural network as an assembly model recursively built from a set of sub-models corresponding to individual neurons. In this view, we show that the linearity of wide neural networks is, in fact, an emerging property of assembling a large number of diverse ``weak'' sub-models, none of which dominate the assembly. 

Published as a conference paper at ICLR 2022
TRANSITION TO LINEARITY OF WIDE NEURAL NET- WORKS IS AN EMERGING PROPERTY OF ASSEMBLING
WEAK MODELS
Chaoyue Liu
Libin Zhu
Mikhail Belkin
Depart. of Computer Science Ohio State University
Depart. of Computer Science UC, San Diego
HDSI
UC, San Diego mbe S abelkin@ucsddee.
1iu. 26..266666 666666600 oso edu
15zhueucad.edi
ABSTRACT
Wide neural networks with linear output layer have been shown to be near-linear, and have near-constant neural tangent kernel (NTK), in region containing the optimization path of gradient descent. These findings seem counter-intuitive since in general neural networks are highly complex models. Why does linear structure emerge when the networks become wide? In this work, we provide a new perspective on this "transition to linearity" by considering neural network as an assembly model recursively built from set of sub-models corresponding to individual neurons. In this view, we show that the linearity of wide neural networks is, in fact, an emerging property of assembling large number of diverse "weak" sub-models, none which dominate the assembly.
INTRODUCTION
Success of gradient descent methods for optimizing neural networks, which generally correspond to highly non-convex loss functions, has long been challenge to theoretical analysis. A series of recent works including Du al. (2018; 2019); Allen-Zhu al. (2019); Zou al. (2018); Oy- mak & Soltanolkotabi (2020) showed that convergence can indeed be shown for certain types of wide networks. Remarkably, Jacot et al. (2018) demonstrated that, when the network width goes to infinity, the Neural Tangent Kernel (NTK) of the network becomes constant during training with gradient flow, a continuous time limit of gradient descent. Based on that Lee et al. (2019) showed that the training dynamics of gradient flow in the space of parameters for the infinite width network equivalent to that of a linear model.
As discussed in (Liu al., 2020), the constancy of the Neural Tangent Kernel stems from the fact that wide neural networks with linear output layer "transition to linearity" as their width increases to infinity. The network becomes progressively more linear in neighneeihbooroodss around the network initialization, as the network width grows. Specifically, consider the neural network as a function f() of its parameters and write the Taylor expansion with the Lagrange remainder term:
1 ((0) f(0)) V (())) (0-00) 0o) -0 -0)0 H(()(0 -0))
where H is the Hessian, the second derivative matrix of f. and & is point between 0 and 0g. The "transition to linearity" is the phenomenon when the quadratic term { - (00)"')' to zero in an O(1) ball around du as the network width increases to infinity.
0o) tends
In particular, transition to linearity provides a method for showing that the square loss function of wide neural networks satisfy the Dajyakk-oojsiewiizz (PL) inequality which guarantees convergence of (Stochastic) Gradient Descent to global minimum (Liu al., 2022).
While the existing analyses demonstrate transition to linearity mathematically, the underlying struc- ture of that phenomenon does not appear to be fully clear. What structural properties account for it? Are they specific to neural networks, or also apply to more general models?
In this paper, we aim to explain this phenomenon from a more structural point of view and to reveal some underlying mechanisms. We provide perspective in which neural network, well as

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each of its hidden layer neurons before activation, be viewed "assembly model", built up by linearly assembling a set of sub-models, corresponding to the neurons from the previous layer. Specifically, 1)-layer pre-activation neuron
((+1))
w (1)
(1) is considered as an assembly model, whereas as a linear combination of the 1-th layer neurons of the l-layer neurons at are considered as sub-models. Furthermore, each pre-activation a!" is also an assembly model constructed from (1 1)-th layer neurons a: ((-1), In this sense, the neural network is considered as a multi-level assembly model. To show O(L)-neiihborhhod linearity, we prove that the quadratic term in the Taylor expansion (q.(1) vanishes in O(1) neighboooosss of network initialization, as a consequence of assembling sufficiently many sub-models.
To illustrate the idea of assembling, we start with a simple case: assembling independent sub-models in Section 2. The key finding is that, as long as the assembly model is not dominated by one or a few sub-models, the quadratic term in the Taylor expansion (q. becomes small in O(1)- neighborhoods of the parameter space when the number of sub-models m is tends to infinity. This means that m increases infinity, the assembly model becomes linear function of parameters. Since we put almost no requirements on the form of sub-models, it is the assembling process that results in the linearity of the assembly model. This case includes two-layer neural networks as examples.
For deep neural networks (Section 3), we consider a neural network as multi-level assembly model each neuron is considered as assembly model constructed iteratively from all the neurons from the previous layer. We then follow an inductive argument: using near-linearity of previous-layer pre-activation neurons to show the linearity of next-layer pre-activation neurons. Our key finding that, when the network width is large, the neurons within the same layer become essentially independent to each other in the sense that their gradient directions are orthogonal in the parameter space. This orthogonality allows for the existence of a new coordinate system such that these neuron gradients are parallel to the new coordinate axes. Within the new coordinate system, can apply the argument of assembling independent sub-models, and obtain the [[i--neghbooroood linearity of the pre-activation neurons, well as the output of the neural network.
We further point out that this assembling viewpoint and the O neighhoooooo linearity can be extended to general neural network architectures, e.g., DenseNet.
We end this section by commenting on a few closely related concepts and point out some of the differences.
Boosting. Boosting (Schapire, 1990; Freund, 1995), which is a popular method that combines multiple "weak" models to produce a powerful ensembe model, has similar form with the assembly model, (4), see, e.g., Friedman et al. (2000). However, we note a few key differences between the two. First, in boosting, each "weak" model is trained separately on the dataset and the coefficients (i.e., n2 in (4)) of the weak models are determined by the model performance. In training an assembly model, the "weak" sub-models (i.e., hidden layer neurons), the training dataset, and the coefficients v. are considered as parameters of the assembly model and trained by gradient descent. Second, in boosting, different data samples may have different sample weights. assembling. the data samples always have a uniform weight.
never directly evaluated on
Bagging and Reservoir computing. Bagging (Breiman, 1996) and Reservoir computing (Jaeger, 2001) are two other ways of combining multiple sub-models to build single model. In bagging, each sub-model is individually trained and the ensemble model is simply the average or the max of the sub-models. In reservoir computing, each sub-model is fixed, and only the coefficients of the linear combination are trainable.
Notation.
We use bold lowercase letters, e.g., V, to denote vectors, capital letters, e.g., W. to denote matrices. We use || to denote the Euclidean norm vectors and spectral norm (i.c., matrix 2-norm) for matrices. We denote the set (1,2, n) as [n]. We use V denote the
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partial derivative of with respect to When represents all of the parameters of f, we omit the subscript, i.e., fff
ASSEMBLING INDEPENDENT MODELS
In this section, we consider assembling a sufficiently large number m independent sub-models and show that the resulting assembly model is (approximately) lincar in neighhbooooo of the model parameter space.
Let's consider set of m (sub-)modee {9, fim where each model Di. with set of model parameters wi Di C P; RP, takes an input x E Dx C IR and outputs scalar prediction 9s 9,W,;; x). Here P, the parameter space of sub-model 9, and Dx is the domain of the input. By independence, mean that any two distinct models and gj share common model parameters:
Ingredients: sub-models.
Panpy (0),
Vi E [m].
(2)
In this sense, the change of gi 's output, as result of change of it parameters w,, does not affect the outputs of the other models 9j where #
In addition, we require that there is dominating sub-models over the others. Specifically, we assume the following in this section:
Assumption (No dominating sub-models). There exists a constant e E (0, 1) independent of m such that, for any parameter setting [wr: w, E D; m and input E Dx.
median (W1: x)], 9m mi x)|]
2 c.
maxi1g (W1xx)], 19m (m: x)]
Furthermore, we assume that max maxx|gi [w]x 19m (wmi O and b >> 0.
x)] E [a, 6] C E for some constants
Remark 1. This assumption guarantees that most of the outputs of the sub-models are at the same order, typically O(1). makes sure that the resulting assembly model not dominated by one or minor portion of the sub-models. This is typically seen for the neurons of neural networks.
We further make the following technical assumption, which is common in the literature.
Assumption (Twice differentiablity and smoothness). Each sub-model 9i is twice differentiable with respect to the parameters w,, and is 3-smooth in the parameter space: there exists positive number / such that, for any E [m]]
V [V:(s;;)) ( Bl[w; w/ll-
This assumption makes sure that for each Is, the gradient is well-defined and its Hessian has bounded spectral norm.
Assembly model. Based on these sub-mon models {9.) we construct an assembly model (or super- model) f. using linear combination as follows:
[(0:x) s(m)
vagi (w;:)
(4)
1=1
where is the weight of the sub-model Ji, and 1/s(m), as function of m, is the scaling factor. Here, we denote the set of parameters of the assembly model, == (Wi...., wn)). The total number of parameters that has is > ps. Typical choices for the weights are setting is = for all i [m)) or random i..dd drawing v; from some zero-mean probability distribution.
Remark 2. For the ease of analysis and simplicity of notation, we assumed that the outputs of assembly model and sub-models 9: are scalars. It is not difficult to see that our analysis below also applies the scenarios of finite dimensional model outputs.
The scaling factor 1/s(m). The presence of the scaling factor 1/s(m) is to keep the output of assembly model at the order O(1) w.n.. m. In general, we expect that s(m) grows with m. In particular, when en are chosen i.i.d. from probability distribution with mean zero, e.g., (-1,1).
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the sum in (4) is expected to be of the order ym, under the Assumption 1. In this case, the scaling factor 1ms(m)= = O(1/vm),
Now, we will show that the assembly model has small quadratic term in its Taylor expansion, when the size of the set of sub-models ie. m, is sufficiently large.
Theorem 1. Consider the assembly model constructed in Eq.(4) with each vs either set be or randomly drawn from {-1, 1,, and suppose Assumption I and 2 hold. Given a positive number R > O and a parameter setting Bg E TR',, for any 0 E Rp such that ||0 80-66y Boll S R, the absolute value of the quadratic term in Taylor expansion (q.(1) is bounded by:
3R²
0 (0--))TT((E( - -80 - 2m( 2
(5)
Proof. In what follows, we don't explicitly write out the dependence on and x in the Hessian 0't for simplicity of notation. We further denote H :- da as the Hessian of H == 00
ow?
We decompose the assembly model Hessian H linear combination of Hg.- By the definition of the assembly model in Eq,(4), an arbitrary entry Hgs of the Hessian can be written as: Hzk
8²
s(m) Di=1
Here gg. 02 are two individual parameters of the assembly model f. Note that U/ 80, 00
8gg; is non-zero, only if both 8, and 0% are parameters of sub-model Jr. i.e., 8j. 0x E P, Hence, 00,00=
the assembly model Hessian can be decomposed as linear combination of sub-model Hessians:
H a(m) E Us Hg.- Therefore, the quadratic term becomes
1 20 -0) (E)(0- (Eu))) Bu)]
[ (w we,) (E) wi.))
Σ
2
s(m)
s(m)
2
and its absolute value is bounded by
( - ()) - Bu) S 2s(m)
lus
Hgg,
||w, - wr,oll 2s(m)
Σ
|| ww wooo
In the second inequality, we used that |ny] and that ga is 3-smooth. Because of the independence of the sub-models as seen in (2), the summation in the above equation becomes ||0 loll. I which bounded by R2, as stated in the theorem condition. Therefore, we conclude the theorem.
It is important to note that 3 is constant and 1/s(m)) ((100m)))))m))) Then have the following corollary in the limiting case.
Corollary 1. Consider the assembly model f under the same setting as in Theorem 1. If the number of sub-models m increases to infinity, then for all parameter Gu and input x, the quadratic term
1 22
-.) (o) (.) 00) O,
as long as 0 is within an nnighhobbbor 0o. i.e., 110 0ol12 < R for some constant R O.
Example: Two-layer neural networks. A good example of this kind of assembly model is the two-layer neural network. A two-layer neural network is mathematically defined as:
m
f(W, v:))
uoa(w? x),
m
where m the number of hidden layer neurons, a(-) the activation function, x E IRd is the network input, and W E Rmxd and u E R are the parameters for the first and second layer, respectively. Here, we can view the i-th hidden neuron and all the parameters w, and u, that connect to it as the i-th sub-model: 9; u,o u.o(wi x). We see that these sub-models do not share parameters, and each sub-model has d + parameters. In addition, the weights of the sub-models are all 1. By Corollary 1, the two-layer neural network becomes linear model in any neighbbogbsssssods, as the network width m increases to infinity. This is consistent with the previous observation that a two-layer neural network transitions to linearity Liu al. (2020).
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Gaussian distributed weights v;-. Another common way to set the weights e; of the sub-models is to independently draw each v; from.N (V, 1). In this case, U; is unbounded, but with high probability the quadratic term is still ((log(m) / /m). Please see the detailed analysis in Appendix A.
Hierarchy of assembly models. In principle, we can consider the assembly model S, together with multiple similar models independent to each other, as sub-models, and construct higher-level assembly model. Repeating this procedures, we can have a hicrarchy of assembly models. Our analysis above also applies to this case and each assembly model is also ((1) ((1))-neihboohoodd linear, when the number of its sub-models sufficiently large. However, one drawback of this hierarchy is that the total number of parameters of the highest level assembly model increase exponentially with the number levels. We will see shortly that wide neural networks, as a hierarchy of assembly models, allows overlapping between sub-models, but still keeping the -1)-nneigbborhood linearity.
DEEP NEURAL NETWORKS AS ASSEMBLING MODELS
In this section, we view deep neural networks multi-level assembly models and show how the O neighbooooo linearity arises naturally as a consequence of assembling.
Setup. We start with the definition of multi-layer neural networks. A L-layer full-connected neural network is defined as follows:
( x,
a 1 (1-1 1I 1,2,, 1,2, , mm-1
a =a(
f = (1)
where a.)) is the activation function and is applied entry-wise above. We assume (.)) is twice differentiable make sure that the network f is twice differentiable and its Hessian well-defined. We also assume that (-) is an injective function, which includes the commonly used sigmoid, tanh, softplus, etc. In the network, with my being the width of (-th hidden layer, ā!4) E IR (called pre-activations) and (9) E RT (called post-activations) represent the vectors of the 1-th hidden layer neurons before and after the activation function, respectively. Denote ;= (W(1), ..., W(2)), with ((() E m pmm, as all the parameters of the network, and ((0) = (W W(!) as the parameters before layer l. We also denote and p( as the dimension of and ((1), respectively.
Denote ((a (!) as the set of the parameters that neuron a;
(!) depends on.
This neural network is typically initialized following Gaussian random initialization: each param- eter is independently drawn from normal distribution, i.e., (W:")) )ij ~ NN(0, 1). In this paper, we focus on the ((i)-neighoorhood of the initialization 0g. As pointed by Liu al. (2020), the whole gradient descent trajectory is contained in a niighhbbbboo of the initialization 6g (more precisely, a Euclidean ball (0u, R) with finite R).
Overview of the methodology. As it is defined recursively in (8), we view the deep neural network as a multi-level assembly model. Specifically, pre-activation neuron at a certain layer I is considered as an assembly model constructed from the post-activation neurons at layer 1 - 1, at the same time its post-activation also serves as a sub-model for the neurons in layer / + 1. Hence, a pre- activation (1) is an (-level assembly model, and the post-activation a (1) is (1+ (1+1)-level sub-model. To prove the ((1))neighbooroood linearity of the neural network, we start from the linearity at the first layer, and then follow an inductive argument from layer to layer, up to the output. The main argument lies in the inductive steps, from layer to layer / + 1. Our key finding is that, in the infinite width limit, the neurons within the same layer become independent to each other, which allows us using the arguments in Section to prove the neighheighborooodd linearity. Since the exact linearity happens in the infinite width limit, in the following analysis we take my, m2..... ML-1 + do sequentially, the same setting as in Jacot et al. (2018). Note that for neural networks that finite but large network width, the linearity will be approximate; the larger the width, the closer to linear.
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3.1 BASE CASE: FIRST AND SECOND HIDDEN LAYER.
The first hidden layer pre-activations are defined as: a v W where mo - d. It is obvious that each element of a()) linear in its parameters. Each second layer pre-activation a;
two-layer neural network with parameters {W<(1) w! ). where w (2) the i-th row W As we have seen in Section 2, the sub-models two-layer neural network share no common parameters, and the assembly model, which is the network itself, is ( neighboooooo linear in the limit of m2 + co.
(1) i E [m²]] can be considered
3.2
FROM LAYER TO LAYER I ++1 1.
First, we make the induction hypothesis layer I.
Assumption (Induction hypothesis). Assume that all the 1-th layer pre-activation neurons a are 0(1) neighborhodd linear in the limit of m....... co.
Under this assumption, we will first show two key properties about the sub-models, i.c., the /-th layer post-activation neurons (!; (1) level sets of these neurons are hyper-planes with co-dimension in neighbogghboo:oods: (2) normal vectors of these hyper-planes are orthogonal with probability one the infinite width limit.
Setup. Note that I-th layer neurons only depend on the parameters g(), Without ambiguity, denote the parameter space spanned by g()) as P. We can write (()) as
g( ( U ( U...U W.!! mm?
where w!") is the i-th row of the weight matrix W One observation is that the set of parameters of neuron as ((a!)) 0(1-1) U (") Namely, (il-1) are shared by all I-layer neurons and each parameter in w; is privately owned by only neuron (1) Hence, we can decompose the parameter space P as: A Pi, where common space Po is spanned by ((l-1) and the private spaces P., i & 0, are spanned by w; Define a set of projection operators { any vector P. (z) P., and Σm (z) z.
m. such that, for
Property 1: Level sets of neurons are hyper-planes The first key observation is that each of the 1-th layer post-activations (4) has linear level sets in the neighboogbosssooos. Theorem 2 (Linear level sets of neurons). Assume the induction hypothesis Assumption holds. Given a fix inpul x, for each post-activation (1) i [mi], the level set
S. :0 a. (00) = (.W(o)
is linear or an empty set, for all E R, where ((o) is an neighbooogor of 0g. Moreover, these level sets are parallel each other: for any c1, E. if Sc, + and Soy 0. then So, and Se, are parallel to each other.
Remark 3.
With abuse notation, we did not explicitly write out the dependence of the level set (1) the input x, and the network initialization og-
on the neuron a
Proof. First, note that all the pre-activation neurons a( are linear in N(ep) in the limit of mM....,11-1 M + 20, as assumed in the induction hypothesis. This linearity of these function guarantees that all the level sets of the pre-activations {g()) a. (0():) c E IR and i E [mı] are either linear or empty sets, and they are parallel to each other. Since the activation function (-) is element-wisse
c} n ((()), for all
applied to the neurons, it does not change the shape and direction of the level sets. Specifically, if {0()) a (01) = c} is non-empty, then (0(0) a 1 (0(1) c) (0()) a! (0
(c)).
Therefore, (0()) : 1 (0: = c) (W(()) is linear or an empty set, and the activation function ( preserves the parallelism.

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This means that, although each of the neurons a! constructed by large number of neurons in pre- vious layers containing tremendously many param- eters and is transformed by non-linear functions, it actually has very simple geometric structure. geometric view of the post-activation neurons is il- lustrated Figure 1.
a(0)
level sets
As the neurons have scalar outputs and a-) is in- jective by assumption, each level set piece of a co-dimension hyper-plane in the p-dimensional parameter space P. Hence, at each point of the hyper-plane there exists unique (up to negative sign) unit-length normal vector n, which is perpen- dicular to the hyper-plans A direct corollary of the linearity of the level sets, Theorem 2, that the normal vector n is identical everywhere in the O(1)- neighborhood (f6)..
Figure I: A geometric view of the post- activation neurons. Level sets are parallel hyper-planes.
Corollary 2. Assume the induction hypothesis Assumption 3 holds. Given a specific neuron of and an input x, for any ( 0 PP.(()))) m(0 =n0!))
Hence, we can define the normal vector for each neuron o The next property is about the set of normal vectors {n. [=1
Property 2: Orthogonality of normal vectors nę. Now, let's look at the directions of the normal
vectors n,, Note that the neuron a does not depend the parameters w (!) for any # i, hence (n,),) for all & { 03. That means:
Proposition 1. For all i E [mi], the normal vector n, resides in the sub-space Ps + Po. and can be decomposed as n, = n; (n:) + (n:):.
As (n:) P,, and Pe n. P, {0} for + j. the components (M: orthogonal each other:
(n:)) 1 m for all + E [mu]. By Proposition 1, to show the orthogonality of the normal vectors {n:)a it suffices to show the orthogonality of (m.))
(10)
Since it is perpendicular to the corresponding level sets, the normal vector parallel to the gradient Va up to potential negative sign. Similarly, the projection (o(n) is parallel to the partial gradient Vou .)
(1)
By the definition of neurons in (8) and the constancy of n, in the neighborhood. W(0o), have
(V Vaa-naluye
1 1 (w2.) (1-11)
(11)
Vm/-1
Here, because a (1-1) is an my -yddimensionnal vector, (L-1) is an my X ((( matrix, where ( denotes the size of g(t-1), is important note that the matrix (a!!11 1) is shared by all the neurons in layer and is independent of the index i, while the vector w 2,0 which is w()) at initialization, is totally private to the (-th layer neurons.
Recall that the vectors (w!)) mi are independently drawn from N/O, mm As is well- 1 known, when the vector dimension mi-1 is large, these independent random vectors { {w!t) are nearly orthogonal. When in the infinite width limit, the orthogonality holds with probability
(-- m
VE > O,
lim
F
22) = 0, for all i i $ [m]].
(12)
(1) From (11,,,), we see that the partial gradients (V j are in fact the result of applying a universal linear transform, i.c., !!-!-), onto a set of nearly orthogonal vectors. The following lemma shows that the vectors remain orthogonal even after this linear transformation.

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Lemma 1.
Ve > 0, lim IP ((Vor-ina)))) T vo
(0 26)
] =0, for all i i [mi]. (13)
See the proof in Appendix C.
Recalling (q.(1 and the fact that the normal vectors To (n;) are parallel to the gradients Vau-n,ã we immediately have that these normal vectors are orthogonal with probability 1, in the limit of m + oo, as stated the following theorem.
Theorem (Orthogonality of normal vectors).
VE O, lim P P[nn ns 20) 20)=0, =0, for all i # [mm]-
(14)
Remark 4. As seen in Section 2, two-layer neural network Eq.(7) is an assembly model with independent sub-models. The normal vectors {n,} are exactly orthogonal to each other, even for finite hidden layer width.
See Appendix B for numerical verification of this orthogonality. This orthogonality allows the existence of new coordinate system such that all the normal vectors are along the axes. Specifically, in the old coordinate system O, each axis is along an individual neural network parameter 0g E with i [p]. After appropriate rotation, O can be transformed to new coordinate system O' such that each normal vector n; parallel one of the new axes. See Figure 2 for an illustration. Denote O' as the set new parameters that are long the axes of new coordinate system O²: 0' = R0, where R is a rotation operator.
0'
8
n
n2
or
0
Ihs
0'x
Figure 2: Coordinate systems: O' (new) vs. O (old). normal vectors n, and gradients Va, are essentially depends on only one new parameter along an axis of O'.
The interesting observation is that each neuron a;
0f E 0', because its gradient direction is parallel with one normal vector and is along one axis in the new coordinate system O'. Moreover, different neurons depends on different new pa- rameters, normal vectors are never parallel. Hence, these neurons are essentially independent to each other. This view actually allows us to use the analysis for assembling independent models in Section 2 to show the ((1)-neighhoria)
linearity at layer + 1, as follows.
Linearity the pre-activations layer [+1. Having the properties for neurons layer I discussed above, we are now ready to analyze the pre-activation neurons (!+1) in layer 1 as assembly models. Recall that each pre-activation (1+1)) defined
(1+1)
(15)
Without ambiguity, we omitted the index for the pre-activation ((+1) and the weights ((+1),
We want to derive the quadratic term (i.e., the Lagrange remainder term) of (1+1) in its Taylor ex- pansion and to show it is arbitrarily small, in the -II-nneigboorhhodd A((0). First, let's consider the special case where the parameters (+++)) are fixed, i.c., ((+1) = w ((+1) and (1++) only depends on ((1) This case conveys the key concepts of the 0(1) neighhoogbooroood linearity after assembling. We will relax this constraint in Appendix D.
Consider an arbitrary parameter setting 0 E ((0)), and let R = 10 - Poll = 0(1). By the definition of assembly model @!++1) in Eq.() the quadratic term in its Taylor expansion (q.()) can be decomposed as:
2
(0 -80) ga (E) - -80)
.m
1
Σ
w
8.0
(++1)
[
2
00) a (E) 8b)
(16)
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(1) (a((+1) and H (0 aa are the Hessians of ((++1) and o; respectively, and a0
8 ())
where Hg =
& E P is on the line segment between 8 and 00
We bounded the term in the square bracket using treatment analogous to Theorem 1. First, in Property I, the level sets of a, are hyper-planes with co-dimension 1, perpendicular to n, Then (1) the value of aj only depends on the component (0 - 0g)² n,, and Hessian H gos is rank 1 and can
be written H e en;n?, with some constant B. Hence, have
H (0- 00)= Ha go (E)(0 - 0o) c -(0- n.) )² S (10-00))
Here 3 is the smoothness of the sub-model, i.e., post-activation (.,), By (q.) we have:
2 100 (00-0)) 2
1 Ho0"" (E) 0o)l s 2/m
max( (
Σ (00-0)) o.)
mi
(17)
Second, by the orthogonality of normal vectors as in Theorem 3, we have
Σ (10 0o) ):)² 2 < |0 0ol R²2.
(18)
Combining the above two equations, we obtain the following theorem which upper bound the mag- nitude of the quadratic term of (1+1);
Theorem (Bounding the quadratic term). Assume that the parameters layer 1 are fixed to the initialization. For any parameter setting 0 E EN(()).
(0 - Haann (E) 0o) s 2/m
3R
max([w:
( (w
(19)
For the Gaussian random initialization, the above upper bound is of the order O(log(ma)/ /m²).. Hence, my oc, the quadratic term of pre-activation (1+1) in the (1 + 1)-th layer vanishes, and the function (1++) becomes linear.
Concluding the induction analysis. Applying the analysis in Section 3.2, Theorem 2 4, with standard induction argument, we conclude that the neural network, as well each of its hidden pre-activation neurons, is -niighbboooo linear, in infinite network width limit,
mi...., ML-1
Extension to other architectures. Appendix E, we further show that the assembly analysis and the ((1) neighborood linearity also hold for DenseNet (Huang et al., 2017).
4 CONCLUSION AND FUTURE DIRECTIONS
In this work, we viewed a wide fully-connected neural network as hierarchy of assembly models. Each assembly model corresponds pre-activation neuron and linearly assembled from a set of sub-models, the post-activation neurons from the previous layer. When the network width increases to infinity, we observed that the neurons within the same hidden layer become essentially indepen- dent. With this property, shown that the network is linear in an neighbroghbooooood around the network initialization.
We believe the assembly analysis and the principles we identified, especially the essential indepen- dence of sub- mob-models, and their iterative construction, are significantly more general than the specific structures considered in this work, and, for example, apply to broad range of neural architec- tures. In future work. we aim to apply the analysis to general feed-forward neural networks, such as architectures with arbitrary connections that form acyclic graph.

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Published as a conference paper at ICLR 2022
ACKNOWLEDGEMENTS
We are grateful for the support of the NSF and the Simons Foundation for the Collaboration on the and #814639. We also acknowledge NSF support through IIS-1815697 and the TILOS institute (NSF CCF-21
Theoretical Foundations of Deep Learning' through awards
REFERENCES
Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over- parameterization. International Conference on Machine Learning, pp. 242 142-252, 2019.
Leo Breiman. Bagging predictors. Machine learning, 24(2):123-14,, 1996.
Simon Du, Jason Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds global minima of deep neural networks. In International Conference on Machine Learning, pp. 1675- 1685, 2019.
Simon S Du, Xiyu Zhai, Barnabas Poczos, and Aarti Singh. Gradient descent provably optimizes over-parameterized neural networks. In International Conference Learning Representations, 2018.
Yoav Freund. Boosting weak learning algorithm by majority. Information and computation, 121 (2):256-285, 1995.
Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Additive logistic regression: statistical view of boosting (with discussion and rejoinder by the authors). The annals of statistics, 28(2):
337-407, 2000.
Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4700-4708, 2017.
Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and gen- eralization in neural networks. Advances neural information processing systems, pp. 8571- 8580, 2018.
Herbert Jaeger. The "echo state" approach to analysing and training recurrent neural networks-with an erratum note, Bonn, Germany: German National Research Center for Information Technology GMD Technical Report, 148(34):13, 2001.
Beatrice Laurent and Pascal Massart. Adaptive estimation of a quadratic functional by model selec- tion. Annals of Statistics, pp. 1302-1338, 2000.
Jaehoon Lee, Lechao Xiao, Samuel Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl- Dickstein, and Jeffrey Pennington. Wide neural networks of any depth evolve linear models under gradient descent. In Advances in neural information processing systems, pp. 8570-8581, 2019.
Chaoyue Liu, Libin Zhu, and Mikhail Belkin. the linearity of large non-linear models: when and why the tangent kernel is constant. Advances in Neural Information Processing Systems, 33, 2020.
Chaoyue Liu, Libin Zhu, and Mikhail Belkin. Loss landscapes and optimization in over- parameterized non-linear systems and neural networks. Applied and Computational Harmonic Analysis, 2022.
Samet Oymak and Mahdi Soltanolkotabi. Toward moderate overparameterization: Global con- vergence guarantees for training shallow neural networks. IEEE Journal on Selected Areas in Information Theory, 1(1):84-105, 2020.
Robert E Schapire. The strength of weak learnability. Machine learning, 5(2):197-227, 1990.
hhttps://deefoouddaat
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Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. arXiv preprint arXiv:1011.3027, 2010.
Roman Vershynin. Hligh-iimensional probability: An introduction with applications in data science, volume 47, Cambridge university press, 2018.
Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep relu networks. arXiv preprint arXiv:1811.08890 2018.

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Published as a conference paper at ICLR 2022
A RANDOM SUB-MODEL WEIGHTS WITH GAUSSIAN DISTRIBUTION
Let's consider the case where the weights 8; of the sub-models are randomly drawing from Gaussian distributions, for example the normal distribution NO,, 1). That
f(0;x)=
Σ
4 (w;x),
(20)
s(m)
where 0z 's are randomly drawn from 1).
In this case, there no strict upper bounded on the absolute values of weights |oy]. But with high probability, we can still bound them using the following lemma.
Lemma 2. Let Ui,..., Um be ii.d 2c 0.5 m lng
Then with probability at least Gaussian variables.
max kmmm]
[
S log m.
The above lemma can be obtained by letting t logm m in (2.10)))) Vershynin (2018) and using union bound.
Using the same analysis as the proof of Theorem I, we have, with probability at least 1 - -0.5..0o
0.5 log" m++log m.
(0 0o) ()) 00)
1 Jel -1
|[w, w.,ol? Blog m
Σ ww wooll?
2
2s(m)
2s (m)
Under Assumption 2, 3 is constant. Note that as the number of m goes to infinity, the probability 1 2e 0.5 Ing" m converges to 1. Since 1/s(m) is ((1/ hence, get
1 (I) 0b)
log m vm
(21)
When m + co, the quadratic term of the Taylor expansion converges to O, with probability 1.
B EXPERIMENTAL VERIFICATION OF THE ORTHOGONALITY IN THEOREM 3
In this section, we run experiments to verify the theoretical finding in Theorem 3, that the normal vectors n; within the same hidden layer tends to become more and more orthogonal, the layer width increases.
two experiments. In the first one, we use full-batch gradient descent to train 4-layer fully-connected neural networks on a simple dataset D: 20 images randomly selected from CIFAR-10 of the classes "airplane" or "bird". In the network, all the hidden layers have the same width m. Given training sample and any two neurons à; and â in the same layer, we compute the cosine cos fy between the gradients Vä, and Vā as
Specifically,
(Vā Vāy) cos O.j V Vā.l
(22)
As a metric measure the orthogonality of the gradients, the average absolute cosine |cos B| is calculated by averaging cos over all pairs of (i,j) with i = and over all data samples. Left panel of Figure shows the numerical results of the average absolute cosine cos 0) as a function of network width m. We see that, as increases, cos 0) monotonically decreases towards zero, verifying that the neuron gradients becomes more and more orthogonal.
In the second experiment, consider bottleneck network, which has three hidden layers, with the 1st and 3rd layers with large width m 1000 and 2nd layer with width m, as the bottleneck layer. We train this bottleneck network using full-batch gradient descent on the same dataset D as in the first experiment. Given training sample and any two neurons ă; and å in the bottleneck layer (i.e., 2nd hidden layer), compute the cosine of the angle between their gradients using Eq. (22),
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03
0.25
0.-
Average [cod]
Average
0.13
0.10
0.85
0.00
10
10F
10
10
10P
10'
Network Width
Bottleneck Width
Figure 3: Absolute cosine of the angle between two neurons, averaged over all neuron pairs and data samples. Left: 4-layer fully-connected neural network, all hidden layers of which have equal width m. Right:
Bottleneck network with three hidden layers, my Mg 1000 and m2 = mp, In both cases, the average absolute cosine cos 6| monotonically decreases towards zero, consistent with the theoretical prediction of orthogonality between neuron gradient in Theorem 3.
and then take the average of its absolute value over all pairs of different neurons and over all data samples. We use this averaged absolute cosine cos 0 as the metric to measure the orthogonality of the gradients. Right panel of Figure 3 shows the relation between cos and the bottleneck width my- is clear that cos 0| monotonically decreases towards zero as my increase, verifying the orthogonality of gradients of bottleneck neurons.
C PROOF OF LEMMA
As in Eq. (11), the partial gradient V can be written as: (Vou-)ja0)r w (1) 1,0
Va Hence, we have
oc n ( " BC 1) a
(w (0) Va (Va (w)))
mi-1
Note that w/o and w
3.0 are independent to each other when i * j. and (a is a fixed matrix
(1)
independent of (!!)) and w 1.0 1.0
Denote a, = Vall-!((aall-))) w j.0 E Rm--1, Then,
(o nā TV 0a n mi-1
( a,.
(23)
the inner product of two independent vectors w 1 and ay with a scaling factor 1/m--1
When conditioned on the vector ay, the quantity (2.) is a Gaussian variable:
a az
(T aj N 0,
my-1
mi_1
(1) is initialized following NNO,, I). In the limit of mi-1 - co, if the variance because w
af a
m,
converges to 0, then this Gaussian variable should also converges to zero with probability 1, from which we conclude the lemma. In the following, will show that the variance ay converges to with probability in this limit.
First, we need the following lemma to upper bound the spectral norm of () The proof is deferred to Section C.I.
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Lemma 3. Consider the neural networks defined in Eq.(8) with random Gaussian initialized pa-
rameters. There exist a constant C >> O, such that, with probability at least 1 - 4(1 + 1)e m/2 we have
IVa (Va))))
with m = min[mi, 12, m-
Since aj - (Valtt)Va J. using Lemma 3, we can bound ||a] ayll by
a ay - Va (Va (1) S C 1330 12
(1)
Recall that w 3.0 R follows the Gaussian distribution (0, Then || s.ol1 ~ xm By Lemma in Laurent Massart (2000), we have with probability at least 1 E mi-1,
|| 00 < 5m
By union bound, have with probability
- 2(L + 1)e - m--m-1, the variance
a a²'a aj
5C
<
mf_ m4-1
In the infinite network width limit mı,..... m1-1 oo, that the variance converges to 0 and the probability converges to 1.
C.I PROOF OF LEMMA 3
Proof. First, note that Va (1- ((-1)((( (Vaal-11) can be decomposed as
Va
Way a way a1-1)r
Then, its spectral norm can be bounded by
(113 (V waya aa 11
I Way a a1 1) aaagaull 111)
Dall's
dall")
dall)
dall")
T
aw(")
11
dall"-1)
1=
1-=141
)all"))
II
(24)
@W(")
P=#++
In the following, we need to bound the terms (')) //0(())))))))))) and ||aa ((") /aa!! (1)[[
To simplify the presentation of the proof, assume in the following that each network hidden layer has the same width m. We leave the general analysis for the readers.
We use the following lemma bound the spectral norm of the weight matrices at initialization.
Lemma 4. If each component of ! I E [L], is iid. drawn from N 1), then with probability at least 26 m/2
|| < 3
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L3 Published as a conference paper at ICLR 2022
See the proof in Section C.2.
There is also lemma that bounds the Euclidean norm of each hidden layer neuron vector (!))
Lemma (Modified Lemma F.3 of (Liu al., 2020)), There exists constants Ca, Ba such that, with probability at least 1 2(L + 1)e m/2 for all E [L].
|a |a S C. m B..
(25)
Using the above two lemmas, we can bounded those two terms. At initialization, we have, with probability least 1 2e me/2
Oa
D 03 ) o
0all11)
sup
[|v|
1=1
Wo v11
sup
||v|=1 m
< |2
< 9L
and with probability least 1 2(L 1)e m/2 for all / [4].
(
a 1) T V18 2
awws
sup
(a )ay
[V]
3.3
Va
(1-))2 |
sup
|ill== m
S ( m
11)
L.
(26)
m
Here yy(1) is a diagonal matrix, with the diagonal entry x1( o'(a")) and La is the degree of Lipschitz continuity the activation function a(-).
Now, using the above results, we can upper bound the spectral norm of the matrix
Va (ta With probability 1 4(T + 1)e mm/2 we have
1-1
(a)
da dal
11 ~ Σ
II
ow
Dall -1)
r==++1
<
(Ca
1
BB.
(27)
m Since 1 < L. we can see that, for sufficient large network width m, the spectral norm [|Va (1-1) (Val()))) is bounded by constant.
C.2
PROOF OF LEMMA 4
Proof. Consider an arbitrary random matrix A Rmm Xma with each entry Ar, N/0 1). By Corollary 5.35 of Vershynin (2010), for any t > 0, we have with probability at least 1 - 2exp( - _ )
||||| & vma + Vms + t.
(28)
For the initial neural network weight matrices, we have
[ ||2 S vd+ Vm+1.
(1) 0 2| < 2/m +t, lE (2,3,
|| 112 S Vm +ltt
Letting / = /m and noting that m > d, we finish the proof.
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D INCLUDING ((++1) AS PARAMETERS
Now, we consider the more general case where parameters in layer / + are not fixed. Then, the quadratic term of the pre-activation (1++1) is written as:
1 (0 0a)² H Haain) (E) 0n)
2
1 w((+1) w
(++1)
I
(owiL+Iry (5)
8 (1+1)
80 30 IT (5)
8 (1+3)
]
w(++))
w (1+1)
2
(() 8
Ow!tF-100u (E)
(0013T (E)
g(0) 80
1
(w((+1)
w 0
u+1) Pa((1)
(ōwli+1)22 (5)
w )
(=: A)
2
+ + (/((00)
) pau+1)
10(00-09 (=:
(20W))
wo 11 Pall+1)
+(w(++) w
Ow(+))000)) (E(() - =00 (=:C)
where & is some point between 0g and 0.
Note that ((++1) is linear in w(++1) hence, 0 aa++)
(wyl is always matrix. Hence, term A 0. Moreover, by Theorem 4, the term B becomes zero, as the width my increases infinity. For the term C, we note that
Pa Ow!++1/000
Vmm Va
Here, a = (a] ami) is the vector of the neurons in layer and (a is the matrix of the first-order derivative of the vector (t)), Using Lemma 3, we have upper bound on the spectral norm: || < vC, with C' >> 0 being constant. Therefore, we have
000 I RVVC
CCI
(4+1)
wo (1+1) V| va
(29)
m
VmL
In the limit of my - oc, term C converges to O. Therefore, we have proved that each (1 + 1)-th layer pre-activation (1+1) has a vanishing quadratic term in its Taylor expansion Eq.(I), and hence becomes linear, the limit of my + 0c.
E DENSENET
In DenseNet (Huang al., 2017), 1-th layer neuron takes all previous layer neurons inputs:
a 1 = oa)
(()) Concat(a (!-1) Vm
a (1) x
(30)
where the "Concat" function concatenates all the vector arguments into a single long vector with being the size, and is the matrix multiplication. The pre-activation can be viewed as the sum of linear functions, ( = E ((') where each a = //m
LT(") has a similar form as
the pre-activation in full-connected neural network (8), but with parameters U(')) being a subset of W Hence, using the analysis in Section 3, in the infinite network width limit and in O(1)- neighborhoods of the initialization, each of the pre-activations of the DenseNet is summation of I < T, linear functions, and hence linear. Therefore, the DenseNet is also linear in 0(1)- neighborhoods of the initialization.
16


Reviews:
[{"review_id": "pR56a3f4NLx", "reviewer": "Reviewer_r126", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper is very well written, with proper style and easy to follow. The findings look interesting and expand the work of other manuscripts. My main concerns are as follow:\n\n1. The introduction mainly focuses on the proposed new perspective (assembly model) without much discussion of the benefits of it. This could also be solved at the end of the manuscript with a discussion and limitations section that explains better which future work does this work provide.\n\n2. The related work at the end of the introduction is a bit short. I think further discussion of the similarities and limitations of the proposed perspective with works like Liu et al. 2020a and Lee et al. 2019 is needed. I also miss some references to works that analyse the mean field gradient descent dynamics of training neural networks in the large-width limit and the relation to the submitted manuscript.\n\n3. The key finding of this manuscript is \"[...], when the network width is large, the neurons within the same layer become essentially independent to each other in the sense that their gradient directions are orthogonal in the parameter space.\". This is linked to assembling independent sub-models to obtain the O(1)-neighbourhood linearity. It seems like the proposed perspective relies directly onto Assumption 1. However, my impression is that if Assumption 1 is not there, other properties could emerge that explain the transition to linearity. The authors start by fixing the property of the neural network being an assembly model and work from there, but does that imply that there are no other structural properties that could explain the transition to linearity on wide neural networks? Is it guaranteed that the constructed assembly model is not dominated by one or a minor group of sub-models?\n\nMinor typos:\npage 3: Further more --> Furthermore\npages 4, 5, 7 and 8: the the --> the\n", "summary_of_the_paper": "This paper aims to explore what structural properties account for transition to linearity when a neural network becomes wider. The authors propose a new perspective where the neural network is considered as a multi-level assembly model. Each hidden layer can be viewed as an assembly model built up from a set of independent sub-models. Furthermore, they assume that the proposed assembly model is not dominated by a single or minor group of sub-models. This leads to neurons within the same hidden layer being independent when the network width grows to infinity, and to show that the network is linear in an O(1)-neighbourhood around the initialization.", "summary_of_the_review": "The paper looks solid and well written, although the contribution and usefulness of the findings is not made totally clear. I am not an expert on this field, thus some more relevant related work might be missing. Most of the mathematical formulation seems correct, although I might have missed some details. My main concern is the way in which the findings are presented seem to point as to the proposed perspective (assembly model) being one of the explanations under Assumption 1, but not necessarily the only one, or valid when said assumption is not true.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "PYeVkA9H5yO", "reviewer": "Reviewer_nTFG", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strength:\n\nThe authors managed to providing a new perspective on why the wider neural networks tend to be linear, which looks interesting and also consistent with previous results. They also extend the basic results to more complex multi-layer cases. \n\nAs this paper is out of my expertise and purely theoretical, I hope other reviewers with more background can comment more on its novelty and significance.\n\n\n\nWeakness:\n\nIt's not clear to me how the main theoretical results compare against previous methods on linearity of neural networks. Taking Theorem 1 for example, is the convergence term s(m) faster or slower when compared with previous results? Another general comment is that a specific section on \"related work\" may make the connection with previous work more clear. ", "summary_of_the_paper": "This paper aims to explain the phenomenon that neural networks with infinite width tend to be linear in the neighborhood of initial optimization points, from an assemble model perspective. The key point is to prove that the quadratic term from Taylor expansion disappears when the width approaches infinity. The authors first proved in the simple two-layer case and then extend the results to deep neural networks with L layers.", "summary_of_the_review": "This paper provides a new perspective towards an interesting observation in deep neural networks, while the connection with previous work could be stated more explicitly. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "I1Ia1SKAjoN", "reviewer": "Reviewer_rv6p", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Authors present a very interesting and novel theoretical contribution to the understanding of contemporary deep learning architectures with increasing network width. Contributions of this study neatly follows the line of investigations presented by [Lee et al. 2019], and [Liu et al., 2020a]. The paper is very clearly written, and I find this work to be of interest for presentation at ICLR.\n\nThe paper provides a rigorous theoretical understanding of the phenomenon, while it naturally lacks empirical contributions and demonstrations overall. Some comments that the authors can discuss/include:\n- What does the results tell us about the impact of common practices in network regularization (e.g., with dropout)?\n- Any discussions on how can the performed assembly analysis extend to residual networks (e.g., WideResNets)?\n- Is it possible to perform any synthetic data based empirical/numerical validation (e.g., for wide neural networks with linear output and a bottleneck) of the finding on the orthogonality of the gradient directions for neurons within the same layer of a network with large width?", "summary_of_the_paper": "Authors present a theoretical understanding for the \"transition to linearity\" phenomenon of wide neural networks with linear output layer under gradient descent. Presented line of thought develops from the perspective where one considers a neural network as a recursively-built assembly of sub-models which correspond to individual neurons from the preceding layers. Authors prove that the transition to linearity phenomenon when the number of sub-models converge to infinity (i.e., the network becomes wider) is a result of assembling weak sub-models that do not dominate the assembly from network initialization.\n", "summary_of_the_review": "The paper is well written and presents a novel theoretical contribution to the understanding of the recently highlighted phenomenon on how wide neural network architectures with linear output layer inherently evolve as linear models. I find this work to be of good quality in terms of theoretical contribution, whereas on the other side empirically weak. My rating would be to accept the paper for presentation at ICLR, as long as the authors can present some numerical verifications or discussions on the how to make an understanding of these results for a broader audience.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors provide in this manuscript a theoretical analysis to explain why deep neural networks become linear in the neighbourhood of the initial optimisation point as their width tends to infinity. They approach this question by viewing the network as a multi-level assembly model.

All reviewers agree that this is an interesting, novel, and relevant study. The paper is very well-written.

Initially, a weak point raised by a reviewer was that an empirical evaluation of the theory was missing. The authors addressed this issue in a satisfactory manner in their response.

In conclusion, this is a strong contribution worth publication.
Decision: Accept (Spotlight)