review_iclr_bench/iclr_parsed/1NvflqAdoom.txt

# NEURAL NETWORKS AS KERNEL LEARNERS: THE SILENT ALIGNMENT EFFECT

**Alexander Atanasov[∗]** **, Blake Bordelon[∗]** **& Cengiz Pehlevan**
Harvard University
Cambridge, MA 02138, USA
_{atanasov,blake bordelon,cpehlevan}@g.harvard.edu_

ABSTRACT

Neural networks in the lazy training regime converge to kernel machines. Can
neural networks in the rich feature learning regime learn a kernel machine with
a data-dependent kernel? We demonstrate that this can indeed happen due to a
phenomenon we term silent alignment, which requires that the tangent kernel of
a network evolves in eigenstructure while small and before the loss appreciably
decreases, and grows only in overall scale afterwards. We empirically show that
such an effect takes place in homogenous neural networks with small initialization
and whitened data. We provide an analytical treatment of this effect in the fully
connected linear network case. In general, we find that the kernel develops a
low-rank contribution in the early phase of training, and then evolves in overall
scale, yielding a function equivalent to a kernel regression solution with the final
network’s tangent kernel. The early spectral learning of the kernel depends on
the depth. We also demonstrate that non-whitened data can weaken the silent
alignment effect.

1 INTRODUCTION

Despite the numerous empirical successes of deep learning, much of the underlying theory remains
poorly understood. One promising direction forward to an interpretable account of deep learning
is in the study of the relationship between deep neural networks and kernel machines. Several
studies in recent years have shown that gradient flow on infinitely wide neural networks with a
certain parameterization gives rise to linearized dynamics in parameter space (Lee et al., 2019; Liu
et al., 2020) and consequently a kernel regression solution with a kernel known as the neural tangent
kernel (NTK) in function space (Jacot et al., 2018; Arora et al., 2019). Kernel machines enjoy firmer
theoretical footing than deep neural networks, which allows one to accurately study their training
and generalization (Rasmussen & Williams, 2006; Sch¨olkopf & Smola, 2002). Moreover, they share
many of the phenomena that overparameterized neural networks exhibit, such as interpolating the
training data (Zhang et al., 2017; Liang & Rakhlin, 2018; Belkin et al., 2018). However, the exact
equivalence between neural networks and kernel machines breaks for finite width networks. Further,
the regime with approximately static kernel, also referred to as the lazy training regime (Chizat et al.,
2019), cannot account for the ability of deep networks to adapt their internal representations to the
structure of the data, a phenomenon widely believed to be crucial to their success.

In this present study, we pursue an alternative perspective on the NTK, and ask whether a neural
network with an NTK that changes significantly during training can ever be a kernel machine for a
_data-dependent kernel: i.e. does there exist a kernel function K for which the final neural network_
function f is f (x) _µ=1_ _[α][µ][K][(][x][,][ x][µ][)][ with coefficients][ α][µ][ that depend only on the training]_
_≈_ [P][P]
data? We answer in the affirmative: that a large class of neural networks at small initialization
trained on approximately whitened data are accurately approximated as kernel regression solutions
with their final, data-dependent NTKs up to an error dependent on initialization scale. Hence, our
results provide a further concrete link between kernel machines and deep learning which, unlike the
infinite width limit, allows for the kernel to be shaped by the data.

_∗These authors contributed equally._


-----

The phenomenon we study consists of two training phases. In the first phase, the kernel starts off
small in overall scale and quickly aligns its eigenvectors toward task-relevant directions. In the
second phase, the kernel increases in overall scale, causing the network to learn a kernel regression
solution with the final NTK. We call this phenomenon the silent alignment effect because the feature
learning happens before the loss appreciably decreases. Our contributions are the following

1. In Section 2, we demonstrate the silent alignment effect by considering a simplified model where
the kernel evolves while small and then subsequently increases only in scale. We theoretically
show that if these conditions are met, the final neural network is a kernel machine that uses the
final, data-dependent NTK. A proof is provided in Appendix B.

2. In Section 3, we provide an analysis of the NTK evolution of two layer linear MLPs with scalar
target function with small initialization. If the input training data is whitened, the kernel aligns its
eigenvectors towards the direction of the optimal linear function early on during training while
the loss does not decrease appreciably. After this, the kernel changes in scale only, showing this
setup satisfies the requirements for silent alignment discussed in Section 2.

3. In Section 4, we extend our analysis to deep MLPs by showing that the time required for alignment scales with initialization the same way as the time for the loss to decrease appreciably. Still,
these time scales can be sufficiently separated to lead to the silent alignment effect for which we
provide empirical evidence. We further present an explicit formula for the final kernel in linear
networks of any depth and width when trained from small initialization, showing that the final
NTK aligns to task-relevant directions.

4. In Section 5, we show empirically that the silent alignment phenomenon carries over to nonlinear
networks trained with ReLU and Tanh activations on isotropic data, as well as linear and nonlinear networks with multiple output classes. For anisotropic data, we show that the NTK must
necessarily change its eigenvectors when the loss is significantly decreasing, destroying the silent
alignment phenomenon. In these cases, the final neural network output deviates from a kernel
machine that uses the final NTK.

1.1 RELATED WORKS

Jacot et al. (2018) demonstrated that infinitely wide neural networks with an appropriate parameterization trained on mean square error loss evolve their predictions as a linear dynamical system
with the NTK at initalization. A limitation of this kernel regime is that the neural network internal representations and the kernel function do not evolve during training. Conditions under which
such lazy training can happen is studied further in (Chizat et al., 2019; Liu et al., 2020). Domingos
(2020) recently showed that every model, including neural networks, trained with gradient descent
leads to a kernel model with a path kernel and coefficients α[µ] that depend on the test point x. This
dependence on x makes the construction not a kernel method in the traditional sense that we pursue
here (see Remark 1 in (Domingos, 2020)).

Phenomenological studies and models of kernel evolution have been recently invoked to gain insight
into the difference between lazy and feature learning regimes of neural networks. These include
analysis of NTK dynamics which revealed that the NTK in the feature learning regime aligns its
eigenvectors to the labels throughout training, causing non-linear prediction dynamics (Fort et al.,
2020; Baratin et al., 2021; Shan & Bordelon, 2021; Woodworth et al., 2020; Chen et al., 2020; Geiger
et al., 2021; Bai et al., 2020). Experiments have shown that lazy learning can be faster but less robust
than feature learning (Flesch et al., 2021) and that the generalization advantage that feature learning
provides to the final predictor is heavily task and architecture dependent (Lee et al., 2020). Fort et al.
(2020) found that networks can undergo a rapid change of kernel early on in training after which
the network’s output function is well-approximated by a kernel method with a data-dependent NTK.
Our findings are consistent with these results.

St¨oger & Soltanolkotabi (2021) recently obtained a similar multiple-phase training dynamics involving an early alignment phase followed by spectral learning and refinement phases in the setting of
low-rank matrix recovery. Their results share qualitative similarities with our analysis of deep linear
networks. The second phase after alignment, where the kernel’s eigenspectrum grows, was studied
in linear networks in (Jacot et al., 2021), where it is referred to as the saddle-to-saddle regime.


-----

Unlike prior works (Dyer & Gur-Ari, 2020; Aitken & Gur-Ari, 2020; Andreassen & Dyer, 2020),
our results do not rely on perturbative expansions in network width. Also unlike the work of Saxe
et al. (2014), our solutions for the evolution of the kernel do not depend on choosing a specific set of
initial conditions, but rather follow only from assumptions of small initialization and whitened data.

2 THE SILENT ALIGNMENT EFFECT AND APPROXIMATE KERNEL SOLUTION

Neural networks in the overparameterized regime can find many interpolators: the precise function
that the network converges to is controlled by the time evolution of the NTK. As a concrete example,
we will consider learning a scalar target function with mean square error loss through gradient flow.
Let x ∈ R[D] represent an arbitrary input to the network f (x) and let {x[µ], y[µ]}µ[P]=1 [be a supervised]
learning training set. Under gradient flow the parameters θ of the neural network will evolve, so the
output function is time-dependent and we write this as f (x, t). The evolution for the predictions of
the network on a test point can be written in terms of the NTK K(x, x[′], t) = _[∂f]∂[(][x]θ[,t][)]_ _[∂f]_ [(]∂[x]θ[′][,t][)] as

_·_


_K(x, x[µ], t)(y[µ]_ _−_ _f_ (x[µ], t)), (1)


_dt_ _[f]_ [(][x][, t][) =][ η]


where η is the learning rate. If one had access to the dynamics of K(x, x[µ], t) throughout all t, one
could solve for the final learned function f _[∗]_ with integrating factors under conditions discussed in
Appendix A


**_Kt[′] dt[′]_** (y[ν] _f0(x[ν])) ._ (2)

_µν_ _−_




_∞_

0

Z


_t_
_dt kt(x)[µ]_ exp _η_
_−_ 0
  Z


_f_ (x) = f0(x) +

_[∗]_


_µν_


Here, kt(x)[µ] = K(x, x[µ], t), [Kt]µ,ν = K(x[µ], x[ν], t), and y[µ] _−_ _f0(x[µ]) is the initial error on_
point x[µ]. We see that the final function has contributions throughout the full training interval t ∈
(0, ∞). The seminal work by Jacot et al. (2018) considers an infinite-width limit of neural networks,
where the kernel function Kt(x, x[′]) stays constant throughout training time. In this setting where
the kernel is constant and f0(x[µ]) 0, then we obtain a true kernel regression solution f (x) =
_≈_
_µ,ν_ **_[k][(][x][)][µ][K]µν[−][1][y][ν][ for a kernel][ K][(][x][,][ x][′][)][ which does not depend on the training data.]_**
PMuch less is known about what happens in the rich, feature learning regime of neural networks,

where the kernel evolves significantly during time in a data-dependent manner. In this paper, we
consider a setting where the initial kernel is small in scale, aligns its eigenfunctions early on during
gradient descent, and then increases only in scale monotonically. As a concrete phenomenological
model, consider depth L networks with homogenous activation functions with weights initialized
with variance σ[2]. At initialization K0(x, x[′]) _O(σ[2][L][−][2]), f0(x)_ _O(σ[L]) (see Appendix B). We_
_∼_ _∼_
further assume that after time τ, the kernel only evolves in scale in a constant direction

_σ2L−2 ˜K(x, x[′], t)_ _t_ _τ_
_K(x, x[′], t) =_ _≤_ (3)
_g(t)K_ (x, x[′]) _t > τ [,]_
 _∞_

where _K[˜]_ (x, x[′], t) evolves from an initial kernel at time t = 0 to K (x, x[′]) by t = τ and g(t)
_∞_
increases monotonically from σ[2][L][−][2] to 1. In this model, one also obtains a kernel regression solution
in the limit where σ 0 with the final, rather than the initial kernel: f (x) = k (x) **_K[−][1]_**
_→_ _∞_ _·_ _∞_ **_[y][ +]_**
_O(σ[L]). We provide a proof of this in the Appendix B._

The assumption that the kernel evolves early on in gradient descent before increasing only in scale
may seem overly strict as a model of kernel evolution. However, we analytically show in Sections 3
and 4 that this can happen in deep linear networks initialized with small weights, and consequently
that the final learned function is a kernel regression with the final NTK. Moreover, we show that for
a linear network with small weight initialization, the final NTK depends on the training data in a
universal and predictable way.

We show empirically that our results carry over to nonlinear networks with ReLU and tanh activations under the condition that the data is whitened. For example, see Figure 1, where we show the
silent alignment effect on ReLU networks with whitened MNIST and CIFAR-10 images. We define
alignment as the overlap between the kernel and the target function _∥Ky[⊤]∥FKy |y|[2][, where][ y][ ∈]_ [R][P][ is]


-----

1.00.8 L|Kt (t)| 4 ReLU MLP on Whitened MNISTK0 ReLU MLP on Whitened CIFARK0

0.6 Alignment 2 K 5 K

0.4 0 0

0.2 2 5

Kernel and Loss

0.0 Test Prediction NTK 4 Test Prediction NTK

0 100 200 300 400 500 2 0 2 4 7.5 5.0 2.5 0.0 2.5 5.0 7.5

t Test Prediction NN Test Prediction NN


(a) Whitened Data MLP Dynamics


(b) Prediction MNIST


(c) Prediction CIFAR-10


1.00.8 L|Kt (t)| 2 Wide Res-Net on Whitened CIFAR

0.6 Alignment 1

0.4 0

0.2 1 K0

Loss and Alignment0.0 Test Prediction NTK 2 K

0 250 500 750 1000 1250 1500 2 1 0 1 2

t Test Prediction NN


(d) Wide Res-Net Dynamics


(e) Prediction Res-Net


Figure 1: A demonstration of the Silent Alignment effect. (a) We trained a 2-layer ReLU MLP
on P = 1000 MNIST images of handwritten 0’s and 1’s which were whitened. Early in training,
around t ≈ 50, the NTK aligns to the target function and stay fixed (green). The kernel’s overall
scale (orange) and the loss (blue) begin to move at around t = 300. The analytic solution for the
maximal final alignment value in linear networks is overlayed (dashed green), see Appendix E.2.
(b) We compare the predictions of the NTK and the trained network on MNIST test points. Due
to silent alignment, the final learned function is well described as a kernel regression solution with
the final NTK K . However, regression with the initial NTK is not a good model of the network’s
_∞_
predictions. (c) The same experiment on P = 1000 whitened CIFAR-10 images from the first two
classes. Here we use MSE loss on a width 100 network with initialization scale σ = 0.1. (d)
Wide-ResNet with width multiplier k = 4 and blocksize of b = 1 trained with P = 100 training
points from the first two classes of CIFAR-10. The dashed orange line marks when the kernel starts
growing significantly, by which point the alignment has already finished. (e) Predictions of the final
NTK are strongly correlated with the final NN function.

a vector of the target values, quantifying the projection of the labels onto the kernel, as discussed
in (Cortes et al., 2012). This quantity increases early in training but quickly stabilizes around its
asymptotic value before the loss decreases. Though Equation 2 was derived under assumption of
gradient flow with constant learning rate, the underlying conclusions can hold in more realistic
settings as well. In Figure 1 (d) and (e) we show learning dynamics and network predictions for
Wide-ResNet (Zagoruyko & Komodakis, 2017) on whitened CIFAR-10 trained with the Adam optimizer (Kingma & Ba, 2014) with learning rate 10[−][5], which exhibits silent alignment and strong
correlation with the final NTK predictor. In the unwhitened setting, this effect is partially degraded,
as we discuss in Section 5 and Appendix J. Our results suggest that the final NTK may be useful for
analyzing generalization and transfer as we discuss for the linear case in Appendix F.

3 KERNEL EVOLUTION IN 2 LAYER LINEAR NETWORKS

We will first study shallow linear networks trained with small initialization before providing analysis
for deeper networks in Section 4. We will focus our discussion in this section on the scalar output
case but we will provide similar analysis in the multiple output channel case in a subsequent section.
We demonstrate that our analytic solutions match empirical simulations in Appendix C.5.

We assume theΣ = _P1_ _Pµ=1_ **_[x] P[µ][x] data points[µ][⊤][. Further, we assume that the target values are generated by a linear teacher] x[µ]_** _∈_ R[D], µ = 1, . . ., P of zero mean with correlation matrix

function y[µ] = sβT **_x[µ]_** for a unit vector βT . The scalar s merely quantifies the size of the supervised learning signal: the variance ofP _·_ _|y|[2]_ = s[2]βT[⊤][Σ][β][T][ . We define the two-layer linear neu-]


-----

t = 0 t t1 t

2 2 2

1 1 1

(a) Initialization


(b) Phase 1


(c) Phase 2


Figure 2: The evolution of the kernel’s eigenfunctions happens during the early alignment phase for
_tContour plot of kernel’s norm for linear functions1 ≈_ [1]s [, but significant evolution in the network predictions happens for] f (x) = β · x. The black line represents the space[ t > t][2][ =][ 1]2 [log(][sσ][−][2][)][. (a)]

of weights which interpolate the training set, ie X _[⊤]β = y. At initialization, the kernel is isotropic,_
resulting in spherically symmetric level sets of RKHS norm. The network function is represented
as a blue dot. (b) During Phase I, the kernel’s eigenfunctions have evolved, enhancing power in the
direction of the min-norm interpolator, but the network function has not moved far from the origin.
(c) In Phase II, the network function W _[⊤]a moves from the origin to the final solution._

ral network with N hidden units as f (x) = a[⊤]W x. Concretely, we initialize the weights with
standard parameterization ai (0, σ[2]/N ), Wij (0, σ[2]/D). Understanding the role of σ
in the dynamics will be crucial to our study. We analyze gradient flow dynamics on MSE cost ∼N _∼N_
_L =_ 21P _µ_ [(][f] [(][x][µ][)][ −] _[y][µ][)][2][.]_

Under gradient flow with learning rateP _η = 1, the weight matrices in each layer evolve as_
_d_ _sβT_ **_W_** **_a_** _,_ _d_ _sβT_ **_W_** **_a_** _⊤_ **Σ.** (4)

_dt_ **_[a][ =][ −]_** _[∂L]∂a_ [=][ W][ Σ] _−_ _[⊤]_ _dt_ **_[W][ =][ −]_** _∂[∂L]W_ [=][ a] _−_ _[⊤]_

The NTK takes the following form throughout training. 
  


_K(x, x[′]; t) = x[⊤]W_ _[⊤]W x[′]_ + |a|[2]x[⊤]x[′]. (5)

Note that while the second term, a simple isotropic linear kernel, does not reflect the nature of the
learning task, the first term x[⊤]W _[⊤]W x[′]_ can evolve to yield an anisotropic kernel that has learned
a representation from the data.


3.1 PHASES OF TRAINING IN TWO LAYER LINEAR NETWORK

We next show that there are essentially two phases of training when training a two-layer linear
network from small initialization on whitened-input data.

-  Phase I: An alignment phase which occurs for t ∼ [1]s [. In this phase the weights align to their low]

rank structure and the kernel picks up a rank-one term of the form x[⊤]ββ[⊤]x[′]. In this setting, since
the network is initialized near W, a = 0, which is a saddle point of the loss function, the gradient
of the loss is small. Consequently, the magnitudes of the weights and kernel evolve slowly.

-  Phase II: A data fitting phase which begins around t ∼ [1]s [log(][sσ][−][2][)][. In this phase, the system]

escapes the initial saddle point W, a = 0 and loss decreases to zero. In this setting both the
kernel’s overall scale and the scale of the function f (x, t) increase substantially.

If Phase I and Phase II are well separated in time, which can be guaranteed by making σ small,
then the final function solves a kernel interpolation problem for the NTK which is only sensitive
to the geometry of gradients in the final basin of attraction. In fact, in the linear case, the kernel
interpolation at every point along the gradient descent trajectory would give the final solution as we
show in Appendix G. A visual summary of these phases is provided in Figure 2.

3.1.1 PHASE I: EARLY ALIGNMENT FOR SMALL INITIALIZATION

In this section we show how the kernel aligns to the correct eigenspace early in training. We focus
on the whitened setting, where the data matrix X has all of its nonzero singular values equal. We let


-----

**_β represent the normalized component of βT in the span of the training data {x[µ]}. We will discuss_**
general Σ in section 3.2. We approximate the dynamics early in training by recognizing that the
network output is small due to the small initialization. Early on, the dynamics are given by:
_d_ _d_

(6)

_dt_ **_[a][ =][ s][W β][ +][ O][(][σ][3][)][,]_** _dt_ **_[W][ =][ s][aβ][⊤]_** [+][ O][(][σ][3][)][.]

Truncating terms order σ[3] and higher, we can solve for the kernel’s dynamics early on in training

_K(x, x[′]; t) = q0 cosh(2ηst) x[⊤]_ []ββ[⊤] + I **_x[′]_** + O(σ[2]), _t ≪_ _s[−][1]_ log(s/σ[2]). (7)
where q0 is an initialization dependent quantity, see Appendix C.1. The bound on the error is ob-
tained in Appendix C.2. We see that the kernel picks up a rank one-correction ββ[⊤] which points
in the direction of the task vector β, indicating that the kernel evolves in a direction sensitive to
the target function y = sβT **_x. This term grows exponentially during the early stages of train-_**
ing, and overwhelms the original kernel · _K0 with timescale 1/s. Though the neural network has_
not yet achieved low loss in this phase, the alignment of the kernel and learned representation has
consequences for the transfer ability of the network on correlated tasks as we show in Appendix F.

3.1.2 PHASE II: SPECTRAL LEARNING

We now assume that the weights have approached their low rank structure, as predicted from the
previous analysis of Phase I dynamics, and study the subsequent NTK evolution. We will show that,
under the assumption of whitening, the kernel only evolves in overall scale.

First, following (Fukumizu, 1998; Arora et al., 2018; Du et al., 2018), we note the following conservation law _dt[d]_ **_a(t)a(t)[⊤]_** **_W (t)W (t)[⊤][]_** = 0 which holds for all time. If we assume small initial

_−_
weight variance _σ[2], aa[⊤]_ _−_ **_W W_** _[⊤]_ = O(σ[2]) ≈ 0 at initialization, and stays that way during the
training due to the conservation law. This condition is surprisingly informative, since it indicates
that W is rank-one up to O(σ) corrections. From the analysis of the alignment phase, we also have
that W _[⊤]W ∝_ **_ββ[⊤]. These two observations uniquely determine the rank one structure of W to be_**
**_aβ[⊤]_** + O(σ). Thus, from equation 5 it follows that in Phase II, the kernel evolution takes the form

_K(x, x[′]; t) = u(t)[2]x[⊤]_ []ββ[⊤] + I **_x[′]_** + O(σ), (8)

where u(t)[2] = **_a_** . This demonstrates that the kernel only changes in overall scale during Phase II.
_|_ _|[2]_

Once the weights are aligned with this scheme, we can get an expression for the evolution of u(t)[2]
analytically, u(t)[2] = se[2][st](e[2][st] _−_ 1 + s/u[2]0[)][−][1][, using the results of (Fukumizu, 1998; Saxe et al.,]
2014) as we discuss in C.4. This is a sigmoidal curve which starts at u[2]0 [and approaches][ s][. The]
transition time where active learning begins occurs when e[st] _s/u[2]0_ = _t_ _s[−][1]_ log(s/σ[2]).
_≈_ _⇒_ _≈_
This analysis demonstrates that the kernel only evolves in scale during this second phase in training
from the small initial value u[2]0

_[∼]_ _[O][(][σ][2][)][ to its asymptote.]_

Hence, kernel evolution in this scenario is equivalent to the assumptions discussed in Section 2,
with g(t) = u(t)[2], showing that the final solution is well approximated by kernel regression with
the final NTK. We stress that the timescale for the first phase t1 1/s, where eigenvectors evolve,
is independent of the scale of the initialization σ[2], whereas the second phase occurs around ∼ _t2_
effect. We illustrate these learning curves and for varyingt1 log(s/σ[2]). This separation of timescales t1 ≪ _t2 for small σ in Figure C.2. σ guarantees the silent alignment ≈_

3.2 UNWHITENED DATA


When data is unwhitened, the right singular vector of W aligns with Σβ early in training, as
we show in Appendix C.3. This happens since, early on, the dynamics for the first layer are
_d_

_dt_ **_[W][ ∼]_** **_[a][(][t][)][β][⊤][Σ][. Thus the early time kernel will have a rank-one spike in the][ Σ][β][ direction.]_**
However, this configuration is not stable as the network outputs grow. In fact, at late time W
must realign to converge to W ∝ **_aβ[⊤]_** since the network function converges to the optimum and
_f = a[⊤]W x = sβ · x, which is the minimum ℓ2 norm solution (Appendix G.1). Thus, the final_
kernel will always look like K (x, x[′]) = sx[⊤] []ββ[⊤] + I **_x[′]. However, since the realignment of_**
_∞_
**_W ’s singular vectors happens during the Phase II spectral learning, the kernel is not constant up to_**

overall scale, violating the conditions for silent alignment. We note that the learned function still is
a kernel regression solution of the final NTK, which is a peculiarity of the linear network case, but
this is not achieved through the silent alignment phenomenon as we explain in Appendix C.3.


-----

4 EXTENSION TO DEEP LINEAR NETWORKS

We next consider scalar target functions approximated by deep linear neural networks and show
that many of the insights from the two layer network carry over. The neural network function
_f : R[D]_ _→_ R takes the form f (x) = w[L][⊤]W _[L][−][1]...W_ [1]x. The gradient flow dynamics under mean
squared error (MSE) loss become


_⊤_
**_W_** _[ℓ][′]_
_ℓ[′]>ℓ_ !

Y


_⊤_
**_W_** _[ℓ][′]_
_ℓ[′]<ℓ_ !

Y


_d_

_dt_ **_[W][ ℓ]_** [=][ −][η ∂L]∂W _[ℓ]_ [=][ η]


(sβ − **_w˜)[⊤]_** **Σ**


(9)


where ˜w = W [1][⊤]W [2][⊤]...w[L] _∈_ R[D] is shorthand for the effective one-layer linear network weights.
Inspired by observations made in prior works (Fukumizu, 1998; Arora et al., 2018; Du et al., 2018),
we again note that the following set of conservation laws hold during the dynamics of gradient
descent _dtd_ **_W_** _[ℓ]W_ _[ℓ][⊤]_ **_W_** _[ℓ][+1][⊤]W_ _[ℓ][+1][]_ = 0. This condition indicates a balance in the size of

_−_
weight updates in adjacent layers and simplifies the analysis of linear networks. This balancing

condition between weights of adjacent layers is not specific to MSE loss, but will also hold for any
loss function, see Appendix D. We will use this condition to characterize the NTK’s evolution.

4.1 NTK UNDER SMALL INITIALIZATION

We now consider the effects of small initialization. When the initial weight variance σ[2] is sufficiently
small, W _[ℓ]W_ _[ℓ][⊤]−W_ _[ℓ][+1][⊤]W_ _[ℓ][+1]_ = O(σ[2]) ≈ 0 at initialization.[1] This conservation law implies that
these matrices remain approximately equal throughout training. Performing an SVD on each matrix
and inductively using the above formula from the last layer to the first, we find that all matrices
will be approximately rank-one w[L] = u(t)rL(t), W _[ℓ]_ = u(t)rℓ+1(t)rℓ(t)[⊤], where rℓ(t) are unit
vectors. Using only this balancing condition and expanding to leading order in σ, we find that the
NTK’s dynamics look like

_K(x, x[′], t) = u(t)[2(][L][−][1)]x[⊤]_ [](L 1)r1(t)r1(t)[⊤] + I **_x[′]_** + O(σ). (10)
_−_

We derive this formula in the Appendix E. We observe that the NTK consists of a rank- 1 correction
to the isotropic linear kernel x **_x[′]_** with the rank-one spike pointing along the r1(t) direction. This
_·_
is true dynamically throughout training under the assumption of small σ. At convergence r(t) → **_β,_**
which is the unique fixed point reachable through gradient descent. We discuss evolution of u(t)
below. The alignment of the NTK with the direction β increases with depth L.

4.1.1 WHITENED DATA VS ANISOTROPIC DATA

We now argue that in the case where the input data is whitened, the trained network function is again
a kernel machine that uses the final NTK. The unit vector r1(t) quickly aligns to β since the first
layer weight matrix evolves in the rank-one direction _dtd_ **_[W][ 1][ =][ v][(][t][)][β][⊤]_** [throughout training for a]

time dependent vector function v(t). As a consequence, early in training the top eigenvector of the
NTK aligns to β. Due to gradient descent dynamics, W [1][⊤]W [1] grows only in the ββ[⊤] direction.
Since the r1 quickly aligns to β due to W [1] growing only along the β direction, then the global
scalar function c(t) = u(t)[L] satisfies the dynamics ˙c(t) = c(t)[2][−][2][/L] [s − _c(t)] in the whitened data_
case, which is consistent with the dynamics obtained when starting from the orthogonal initialization
scheme of Saxe et al. (2014). We show in the Appendix E.1 that spectral learning occurs over a
timescale on the order of t1/2 _s(LL_ 2) _[σ][−][L][+2][, where][ t][1][/][2][ is the time required to reach half the]_
_≈_ _−_

value of the initial loss. We discuss this scaling in detail in Figure 3, showing that although the
timescale of alignment shares the same scaling with σ for L > 2, empirically alignment in deep
networks occurs faster than spectral learning. Hence, the silent alignment conditions of Section 2
are satisfied. In the case where the data is unwhitened, the r1(t) vector aligns with Σβ early in
training. This happens since, early on, the dynamics for the first layer are _dtd_ **_[W][ 1][ ∼]_** **_[v][(][t][)][β][⊤][Σ][ for]_**

time dependent vector v(t). However, for the same reasons we discussed in Section 3.2 the kernel
must realign at late times, violating the conditions for silent alignment.

1Though we focus on neglecting the O(σ2) initial weight matrices in the main text, an approximate analysis for wide networks at finite σ[2] and large width is provided in Appendix H.2, which reveals additional
dependence on relative layer widths.


-----

10[0] 10[1] 10[2] 10[3]

|1.0 Alignment 0.8 0.6 0.4 and 0.2 Loss 0.0|2 = 105 2 = 104 2 = 103 2 = 102 2 = 101|
|---|---|
||2 = 10 2 = 10 2 = 10 2 = 10|
|||


2 = 10 5

2 = 10 4

2 = 10 3

2 = 10 2

2 = 10 1

t

(b) L = 3 Dynamics


10[7] L = 3

10[6] L = 4

10[5] L = 5

10[4]

t1/210[3]

10[2]

10[1]

10[0]

10 2 10 1


t1/2

align 10[3] talign L + 2
, tt1/2 10[2]

10 2 10 1


(a) ODE Time to Learn


(c) Time To Learn L = 3


Figure 3: (a) Time to half loss scales in a power law with σ for networks with L 3:
_L_ _≥_
_t1/2_ (L 2) _[σ][−][L][+2][ (black dashed) is compared with numerically integrating the dynamics]_
_∼_ _−_

_c˙(t) = c[2][−][2][/L](s −_ _c) (solid). The power law scaling of t1/2 with σ is qualitatively different than_
what happens for L = 2, where we identified logarithmic scaling t1/2 log(σ[−][2]). (b) Linear
networks with D = 30 inputs and N = 50 hidden units trained on synthetic whitened data with ∼
_|β| = 1. We show for a L = 3 linear network the cosine similarity of W_ [1][⊤]W [1] with ββ[⊤] (dashed)
and the loss (solid) for different initialization scales. (c) The time to get to 1/2 the initial loss and
the time for the cosine similarity of W [1][⊤]W [1] with ββ[⊤] to reach 1/2 both scale as σ[−][L][+2], however
one can see that alignment occurs before half loss is achieved.

4.2 MULTIPLE OUTPUT CHANNELS


We next discuss the case where the network has multiple C output channels. Each network output,
we denote as fc(x[′]) resulting in C [2] kernel sub-blocks Kc,c′ (x, x[′]) = _fc(x)_ _fc′_ (x[′]). In this
_∇_ _· ∇_
context, the balanced condition W _[ℓ]W_ _[ℓ][⊤]_ _≈_ **_W_** _[ℓ][+1][⊤]W_ _[ℓ][+1]_ implies that each of the weight matrices
is rank-C, implying a rank-C kernel. We give an explicit formula for this kernel in Appendix H.
For concreteness, consider whitened input data Σ = I and a teacher with weights β ∈ R[C][×][D]. The
singular value decomposition of the teacher weights β = _α_ _[s][α][z][α][v]α[⊤]_ [determines the evolution of]

each mode (Saxe et al., 2014). Each singular mode begins to be learned at tα = _s1α_ [log] _sαu[−]0_ [2] .

To guarantee silent alignment, we need all of the Phase I time constants to be smaller than all of[P]
 

the Phase II time constants. In the case of a two layer network, this is equivalent to the condition

timescales of spectral learning. We see that alignment precedes learning in Figure H.1 (a). Forsmin1 _[≪]_ _smax1_ [log] _smaxu[−]0_ [2] so that the kernel alignment timescales are well separated from the
 

deeper networks, as discussed in 4.1.1, alignment scales in the same way as the time for learning.

5 SILENT ALIGNMENT ON REAL DATA AND RELU NETS


In this section, we empirically demonstrate that many of the phenomena described in the previous
sections carry over to the nonlinear homogenous networks with small initialization provided that
the data is not highly anisotropic. A similar separation in timescales is expected in the nonlinear
_L-homogenous case since, early in training, the kernel evolves more quickly than the network pre-_
dictions. This argument is based on a phenomenon discussed by Chizat et al. (2019). Consider an
initial scaling of the parameters by σ. We find that the relative change in the loss compared to the
relative change in the features has the form _[|][ d]dt|∇[∇]f[f]|_ _[|]_ _|_ _dt[d]L[L|][ ≈]_ _[O][(][σ][−][L][)][ which becomes very large for]_

small initialization σ as we show in Appendix I. This indicates, that from small initialization, the
parameter gradients and NTK evolve much more quickly than the loss. This is a necessary, but not
sufficient condition for the silent alignment effect. To guarantee the silent alignment, the gradients
must be finished evolving except for overall scale by the time the loss appreciably decreases. However, we showed that for whitened data that nonlinear ReLU networks do in fact enjoy the separation
of timescales necessary for the silent alignment effect in Figure 1. In even more realistic settings,
like ResNet in Figure 1 (d), we also see signatures of the silent alignment effect since the kernel
does not grow in magnitude until the alignment has stabilized.

We now explore how anisotropic data can interfere with silent alignment. We consider the partial
whitening transformation: let the singular value decomposition of the data matrix be X = USV _[⊤]_
and construct a new partially whitened dataset Xγ = US[γ]V _[⊤], where γ ∈_ (0, 1). As γ → 0


-----

the dataset becomes closer to perfectly whitened. We compute loss and kernel aligment for depth
2 ReLU MLPs on a subset of CIFAR-10 and show results in Figure 4. As γ → 0 the agreement
between the final NTK and the learned neural network function becomes much closer, since the
kernel alignment curve is stable after a smaller number of training steps. As the data becomes more
anisotropic, the kernel’s dynamics become less trivial at later time: rather than evolving only in
scale, the alignment with the target function varies in a non-trivial way while the loss is decreasing.
As a consequence, the NN function deviates from a kernel machine with the final NTK.

|0|Partial Whitened Spectrum|
|---|---|
|100 101 102 /2 k 103 104|= 0.00 = 0.25 = 0.50 = 0.75 = 1.00|
||= 0.00 = 0.25 = 0.50 = 0.75 = 1.00|


= 0.00

= 0.25

= 0.50

= 0.75

= 1.00


Loss Classification Test Error

1.0 0.32

0.8 0.30

t0.6 t0.28
L0.4 L0.26

0.2 0.24

0.0 0.22

0 200 400 600 800 0 200 400 600 800

t t


(a) Input Spectra

800600 Kernel Norm |K 0.6 Kernel Alignment ||/|fNN10 1 NTK Predictor vs NN

)|(|tK 4002000 /|, Kyy 0.40.20.0 |ffNTKNN1010 35 KK0

0 200 400 600 800 0 200 400 600 800 0.0 0.2 0.4 0.6 0.8 1.0

t t


(d) Kernel Norm


(b) Train Loss

Kernel Alignment

0 200 400 600

t

(e) Phase I alignment


(c) Test Error

NTK Predictor vs NN

10 1

10 3

10 5

0.0 0.2 0.4 0.6 0.8

(f) Predictor Comparison


Figure 4: Anisotropy in the data introduces multiple timescales which can interfere with the silent
alignment effect in a ReLU network. Here we train an MLP to do two-class regression using Adam
at learning rate 5 × 10[−][3]. (a) We consider the partial whitening transformation on the 1000 CIFARfor unwhitened data have a multitude of timescales rather than a single sigmoidal learning curve. As10 images λk → _λ[γ]k_ [for][ γ][ ∈] [(0][,][ 1)][ for covariance eigenvalues][ Σ][v][k][ =][ λ][k][v][k][. (b) The loss dynamics]
a consequence, kernel alignment does not happen all at once before the loss decreases and the final
solution is not a kernel machine with the final NTK. (c) The network’s test error on classification. (d)
Anisotropic data gives a slower evolution in the kernel’s Frobenius norm. (e) The kernel alignment
very rapidly approaches an asymptote for whitened data but exhibits a longer timescale for the
anisotropic data. (f) The final NTK predictor gives a better predictor for the neural network when
the data is whitened, but still substantially outperforms the initial kernel even in the anisotropic case.

6 CONCLUSION

We provided an example of a case where neural networks can learn a kernel regression solution while
in the rich regime. Our silent alignment phenomenon requires a separation of timescales between
the evolution of the NTK’s eigenfunctions and relative eigenvalues and a separate phase where the
NTK grows only in scale. We demonstrate that, if these conditions are satisfied, then the final neural
network function satisfies a representer theorem for the final NTK. We show analytically that these
assumptions are realized in linear neural networks with small initialization trained on approximately
whitened data and observe that the results hold for nonlinear networks and networks with multiple
outputs. We demonstrate that silent alignment is highly sensitive to anisotropy in the input data.

Our results demonstrate that representation learning is not at odds with the learned neural network
function being a kernel regression solution; i.e. a superposition of a kernel function on the training
data. While we provide one mechanism for a richly trained neural network to learn a kernel regression solution through the silent alignment effect, perhaps other temporal dynamics of the NTK could
also give rise to the neural network learning a kernel machine for a data-dependent kernel. Further,
by asking whether neural networks behave as kernel machines for some data-dependent kernel, one
can hopefully shed light on their generalization and transfer learning capabilities (Bordelon et al.,
2020; Canatar et al., 2021; Loureiro et al., 2021; Geiger et al., 2021) and see Appendix F.


-----

ACKNOWLEDGMENTS

CP acknowledges support from the Harvard Data Science Initiative. AA acknowledges support from
an NDSEG Fellowship and a Hertz Fellowship. BB acknowledges the support of the NSF-Simons
Center for Mathematical and Statistical Analysis of Biology at Harvard (award #1764269) and the
Harvard Q-Bio Initiative. We thank Jacob Zavatone-Veth and Abdul Canatar for helpful discussions
and feedback.

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-----

## Appendix

A DERIVATION OF EQUATION 2

A.1 TRAINING POINT PREDICTIONS WITH TIME VARYING KERNEL

Given a training set of P data points {(x[µ], y[µ])}µ[P]=1[, the dynamics of the network training er-]
rors [∆t][µ] := f (x[µ], t) − _y[µ]_ close in terms of a time-varying neural tangent kernel [Kt]µν =
_K(x[µ], x[ν], t)_

_d_

(11)
_dt_ **[∆][t][ =][ −][K][t][∆][t][.]**

We introduce the transition matrix Φt ∈ R[P][ ×][P] which has the property that ∆t = Φt∆0 and
**Φ0 = I, we obtain the matrix evolution equation** **Φ[˙]** _t = −KtΦt. This equation can be solved_
formally in terms of the Peano-Baker series (Baake & Schlaegel, 2011; Brockett, 2015)

_t_ _t_ _s1_
**Φt =I −** 0 _ds1Ks1 +_ 0 _ds1Ks1_ 0 _ds2Ks2_ (12)
Z Z Z

_t_ _s1_ _s2_
_−_ 0 _ds1Ks1_ 0 _ds2Ks2_ 0 _ds3Ks3 + ..._ (13)
Z Z Z


which can easily be verified to solve _dt[d]_ **[Φ][(][t][) =][ −][K][(][t][)][Φ][(][t][)][ with initial condition][ Φ][(0) =][ I][. Under]**

_t_
the condition that 0 **_[K][(][t][)][ commutes with][ K][(][t][)][, which is true in the settings of interest in this]_**
paper, specifically the setting discussed in Appendix B, we can simplify the Peano-Baker series into
R
a simple matrix exponential

_t_ _t_ 2 _t_ 3 _t_ _k_
**Φt = I** _dsKs_ + [1] _dsKs_ _dsKs_ _... + [(][−][1)][k]_ _dsKs_ + ...
_−_ 0 2 0 _−_ 6[1] 0 _k!_ 0
Z  Z  Z  Z 

_t_
= exp **_Ksds_** _._ (14)
_−_ 0
 Z 


_t_
Thus, under the condition that Kt commutes with 0 **_[K][s][ds][ we can exactly solve for the training]_**
error dynamics in terms of integrating factors
R

_t_
**∆t = Φt∆0 = exp** _−_ 0 **_Ksds_** **∆0.** (15)
 Z 

We expect this formula to hold approximately whenever the eigenvectors of K are approximately
_t_
equal to the eigenvectors of 0 **_[K][s][ds][.]_**
R

A.2 TEST POINT PREDICTIONS WITH TIME VARYING KERNEL

Given access to the value of the function on training points, one can evaluate the function on test
points. We have that the evolution of the function on a test point f (x) is given by

_d_

(16)
_dt_ _[f][t][(][x][) =][ −][k][t][(][x][)][∆][t][,]_

where [kt(x)][µ] := K(x, x[µ], t). This gives the final value of f to be


_∞_
_f_ (x) := f (x) = f0(x) +

_[∗]_ _∞_ 0
Z

This is exactly equation 2.


_t_
_dt kt(x)[µ]_ exp _η_
_−_ 0
  Z


**_Kt′_** _dt[′]_ (y **_f0)._** (17)
_−_



-----

B KERNEL EVOLUTION IN SCALE ONLY

We consider the model of kernel evolution introduced in Section 2 where the kernel evolves only in
scale for t > τ (ϵ) and is of small overall size for t < τ (ϵ),

_K(x, x[′], t) =_ _ϵK0(x, x′, t)_ _t ≤_ _τ_ (ϵ) (18)
_g(t)K_ (x, x[′]) _t > τ_ (ϵ) _[.]_
 _∞_

This model allows for alignment of the kernel while small in the time window t ∈ (0, τ ), followed
by scale growth only for t ∈ (τ, ∞). The time threshold τ will generally depend on the initial kernel
scale ϵ. For example, in depth L linear MLPs, ϵ _σ[2][L][−][2]_ and τ _σ[−][L][+2]_ with initialization scale
_∼_ _∼_ _t_
_σ as we show in Figure E.3. We will define a differentiable function h(t) =_ _τ_ _[g][(][t][′][)][dt][′][ so that]_
_h[′](t) = g(t), h(τ_ ) = 0, and limt→∞ _h(t) = ∞. This last condition follows fromR_ _g’s continuity_
and the assumption that limt→∞ _g(t) = 1. We will first show that the final neural network has the_
form f (x) = f0(x) + k∞(x) · K∞[−][1][(][y][ −] **_[f][0][) +][ O][(][ϵτ]_** [(][ϵ][))][. First, we need to calculate the errors]
**∆(t) = y −** **_f_** (t) ∈ R[P] made on the P training examples. These satisfy the dynamics

_d_ _ϵK0(t)∆(t)_ _t ≤_ _τ_ (19)

_dt_ **[∆][(][t][) =][ −]** _h[′](t)K_ **∆(t)** _t > τ [,]_
 _∞_

where K0(t), K R[P][ ×][P] are P _P gram matrices; e.g. [K0(t)]µν = K0(x[µ], x[ν], t). The_
_∞_ _∈_ _×_
vector k(x) has entries given by [k(x)]µ = K(x, x[µ]). For t ∈ (0, τ ), the error vector follows the
dynamics
_d_

(20)
_dt_ **[∆][(][t][) =][ −][ϵ][K][0][(][t][)][∆][(][t][)][.]**


Introducing operator norm of a matrix, |Φ|op = max|v|2=1 |Φv|2, we will now bound the operator
norm of the change in the transition matrix Φ(t) introduced in section A.1.
**Lemma 1. Let k0 = maxt** (0,τ ) **_K0(t)_** _op represent the maximum operator norm of K0 achieved_
_∈_ _|_ _|_
_on the interval (0, τ_ ). Let Φ(t) ∈ R[P][ ×][P] _be the transition matrix for the linear dynamics of equation_
_(20) so that_ _dt[d]_ **[Φ][(][t][) =][ −][ϵ][K][0][(][t][)][Φ][(][t][)][ and][ Φ][(0) =][ I][. Then,]**

_|Φ(τ_ ) − **Φ(0)|op < ϵτ** (ϵ)k0. (21)

_Proof. We begin by noting that, due to the triangle inequality,_

_τ_ _τ_

_|Φ(τ_ ) − **Φ(0)|op = ϵ** 0 **_K0(t)Φ(t)dt_** _op_ _≤_ _ϵ_ 0 _|K0(t)Φ(t)|op dt_

Z _τ_ Z _τ_ (22)

_≤_ _ϵ_ 0 _|K0(t)| |Φ(t)|op dt ≤_ _ϵk0_ 0 _|Φ(t)|op dt._
Z Z

We will now establish that |Φ(t)|op ≤ 1. Note that for any vector v ∈ R[P] that

1 _d_

2 [=][ v][⊤][Φ][⊤]Φ[˙] (t)v = _ϵv[⊤]Φ(t)[⊤]K(t)Φ(t)v_ 0, (23)

2 _dt_ _[|][Φ][(][t][)][v][|][2]_ _−_ _≤_


where the final inequality follows from the fact that K(t) is positive semidefinite for all t. Therefore
we have shown that |Φ(t)|op ≤|Φ(0)|op = |I|op = 1 τ . Using this inequality, we find that

_|Φ(τ_ ) − **Φ(0)|op ≤** _ϵk0_ 0 _|Φ(t)|op dt ≤_ _ϵk0τ_ (ϵ). (24)

Z

With the above Lemma 1, we can bound the discrepancy ∆(τ ) and ∆(0), namely


_|∆(τ_ ) − **∆(0)|2 = |(Φ(τ** ) − **Φ(0))∆(0)|2** (25)
**Φ(τ** ) **Φ(0)** _op_ **∆(0)** 2 _ϵk0τ_ (ϵ) **∆(0)** 2.
_≤|_ _−_ _|_ _|_ _|_ _≤_ _|_ _|_

This inequality must therefore hold entry-wise as well, so that

**∆(τ** ) = ∆(0) + O(ϵk0τ (ϵ)). (26)

We will now establish how the training predictions ∆(t) evolve for the second interval t ∈ (τ, ∞).


-----

**Lemma 2. Suppose that from t ∈** (τ, ∞) that ∆(t) obeys the dynamics _dt[d]_ **[∆][(][t][) =][ −][h][′][(][t][)][K][∞][∆][(][t][)]**

_where ∆(τ_ ) is as in equation 26. Then, for all t ∈ (τ, ∞),

**∆(t) = exp (−h(t)K∞) [(y −** **_f0) + O(ϵτ_** (ϵ)k0)] . (27)

_Proof. The differential equation can be solved through eigendecomposition and integrating factors._
Let ∆k(t) represent the k-th component of ∆(t) in the eigenbasis of K∞ which is static for t ∈
(τ, ∞). Let the corresponding eigenvalue of K∞ be λk. The scalar variable ∆k(t) obeys the
dynamics

_d_

(28)
_dt_ [∆][k][(][t][) =][ −][λ][k][h][′][(][t][)∆][k][(][t][)][.]

This can be solved with integrating factors, noting that _dtd_ _e[λ][k][h][(][t][)]∆k(t)_ = 0. This implies that

∆k(t) = e[−][λ][k][h][(][t][)]∆k(τ ). Written as a vector, ∆(t) = exp ( _h(t)K_ ) ∆(τ ). Since by Lemma 1
− _∞_ 
we have ∆(τ ) = ∆(0) + O(ϵk0τ (ϵ)), we obtain the desired result.

We will now combine the results of the previous two lemmas which analyze the evolution of the
network predictions on the training set to give our main silent alignment result, which specifies what
the neural network function predicts for an arbitrary test point x.

**Theorem 1. Let the kernel have dynamics of Equation 18 where g(t) is a continuous, integrable**
_function with limt→∞_ _g(t) = 1. The function learned by the neural network is_

_f_ (x) − _f0(x) = k∞(x) · K∞[−][1][y][ +][ O][ϵ][(][ϵτ]_ [(][ϵ][))][.] (29)

_Proof. Using Lemma 1 and 2, we know the full dynamics for training predictions ∆(t). Using_
**∆(t), we can solve for the final predictor f** (x) by integrating dynamics _f[˙](x, t) = k(x, t) · ∆(t)._

_τ_ _∞_
_f_ (x) − _f0(x) = ϵ_ 0 **_k0(x, t) · ∆(t)dt + k∞(x) ·_** _τ_ _h[′](t) exp (−h(t)K∞) ∆(τ_ )dt (30)
Z Z

We will now bound the first term. Taking _k[˜]0 = maxt∈(0,τ_ ),x∈RD |k0(x, t)|2, we get that
_τ_ _τ_ _τ_

**_k0(x, t)_** **∆(t)** **_k(x)_** **∆(t)** _dt_ _ϵ(1 + ϵτ_ (ϵ)k0) **∆0** **_k0(x, t)_** _dt_

Z0 _·_ _[≤]_ _[ϵ]_ Z0 _|_ _||_ _|_ _≤_ _|_ _|_ Z0 _|_ _|_

_ϵτ_ (ϵ)k[˜]0(1 + ϵτ (ϵ)k0) **∆(0)** = Oϵ(ϵτ (ϵ)).

_[ϵ]_ _≤_ _|_ _|_


We can now integrate the matrix exponential in the second term, using the fact that
_∞_ _∞_

_h[′](t) exp (_ _h(t)K_ ) dt = exp ( _hK_ ) dh = K[−][1] (31)
_τ_ _−_ _∞_ 0 _−_ _∞_ _∞_ _[.]_

Z Z

Using the fact that ∆(τ ) = ∆(0) + O(τ (ϵ)ϵ) from Lemma 1, we arrive at the desired result

_f_ (x) _f0(x) = k_ (x)K[−][1] [+][ O][ϵ][(][ϵτ] [(][ϵ][))][.] (32)
_−_ _∞_ _∞_ **[∆][0]**

We have now established that, given the kernel dynamics in Equation 18, f (x) _f0(x) converges_
_−_
to the kernel regression solution with final NTK as ϵ 0 provided limϵ 0 ϵτ (ϵ) = 0. This is
_→_ _→_
generic in the settings we consider in this paper for networks with small initialization. In this small
initialization setting, f0 is also negligible so that f (x) itself is a kernel regression solution. For
example, in a linear depth L neural network with initial weight scale σ, the initial scale of the kernel
is ϵ ∼ _σ[2][L][−][2]_ while the time to alignment scales as τ ∼ _σ[2][−][L]_ thus ϵτ ∼ _σ[L]_ can be made arbitrarily
small by taking σ 0. Lastly, the initial network outputs f0(x) _σ[L]_ can also be made arbitrarily
_→_ _∼_
small.


-----

C PHASES OF LEARNING AT SMALL INITIALIZATION

C.1 PHASE I: TWO LAYER NETWORK AND KERNEL ALIGNMENT

We now present an analysis distinct from that of the previous subsection to go beyond the first step
of gradient descent. The NTK for the two layer linear network has the form K = x[⊤]Mx[′] with
**_M = W_** _[⊤]W + |a|[2]I. Our goal is to determine the eigendecomposition of M_ . Introduce the
variables q(t) = 2[1] **_[β][⊤][Mβ][ =][ 1]2_** _|a|[2]_ + β[⊤]W _[⊤]W β_ and r(t) = a[⊤]W β. These dynamics form

a closed two dimensional linear system early in training
 


0 1 _q(t)_
= 2ηs + O(σ[3]), _t, σ_ 0
1 0 _r(t)_ _→_
   

= [1] 1 _e[2][ηst]_ + [1] 1

2 [(][q][0][ +][ r][0][)] 1 2 [(][q][0][ −] _[r][0][)]_ 1
  − 


_q(t)_
_r(t)_

_q(t)_
_r(t)_


_dt_


(33)
_e[−][2][ηst]_ + O(σ[3])


The variable q(t) represents the alignment of the NTK with the optimal direction β while r(t)
defines the alignment of the network with the teacher. We see that this alignment increases exponentially with timescale t ∼ _η[−][1]s[−][1]. While the above equations hold for early time and small_
initialization for any initial condition q0, r0, we can further estimate these initial values under
random initialization provided the input dimension is large. We stress that this limit is not necessary for the silent alignment, but allows for a nice simplification. For Gaussian initialization
_ai_ (0, σ[2]/N ), Wij (0, σ[2]/D) with large D, we have
_∼N_ _∼N_

_⟨q0⟩_ = _[σ]2[2]_ 1 + _[N]D_ _, ⟨r0⟩_ = 0, _r0[2]_ = _[σ]D [4]_ _[.]_ (34)

 

_qIn the large = q0 cosh(2 Dηst limit, we have with high probability). Note that this gives the quantities q q0 ≫(t) =r0 and thus[1]2_ **_[β][⊤][M]_** [(][t] r[)][β]([, r]t) =[(][t][) =] q0[ β] sinh(2[⊤][W][ ⊤]ηst[a][ early]) and

in training. Now consider a unit vector v which is orthogonal to the solution β[⊤]v = 0. We find that
the projection of M along this direction evolves dynamically as:

_d_

_dt_ **_[v][⊤][M]_** [(][t][)][v][ = 2][v][⊤] []βa[⊤]W + β[⊤]W _[⊤]aI_ **_v_** (35)

= 2r(t). 


We can conclude that v[⊤]Mv is equal to q(t) up to an additive initialization constant. We see that
this is evolving half as quickly as β[⊤]Mβ = 2q(t). Since v[⊤]M0v _O(σ[2]) is small compared to_
_∼_
the exponentially growing M (t), the only matrix that satisfies these two conditions must necessarily
take the form

**_M_** (t) = q0 cosh(2ηst) **_ββ[⊤]_** + I + M0. (36)

The first term, which is growing exponentially in t will eventually overwhelm the randomly initial- 
ized kernel M0, which is O(σ[2]).

C.2 PHASE I: ERROR IN THE LEADING ORDER APPROXIMATION

In solving the equations of the previous section, we truncated the full gradient descent equations at
order σ[3]. It is important to confirm that the error generated by this truncation remains bounded. We
will argue by self-consistency. The full equations are

_dtd_ **_[a][ =][ W]_** _sβ −_ **_W_** _[⊤]a_ _,_ _dtd_ **_[W][ =][ a]_** _sβ −_ **_W_** _[⊤]a_ _⊤_ _._ (37)

One can use these equations to solve for the dynamics of the 
 _r, q_ variables: 



_d_

_dt_ _[q][(][t][) = 2][sr][ −]_ **_[β][⊤]_** []W _[⊤]aa[⊤]W + a[⊤]W W_ _[⊤]a_ **_β = 2sr −_** [2r[2] +



(a[⊤]W vi)[2]] (38)
**_vXi⊥β_**


_d_

(39)
_dt_ _[r][(][t][) = 2][sq][ −]_ **_[a][⊤][[][W W][ ⊤]_** [+][ aa][⊤][]][W β][ = 2][sq][ −] [2][|][a][|][2][r][ +][ a][⊤][[][aa][⊤] _[−]_ **_[W W][ ⊤][]][W][ β]_**


-----

The second equality in equation 38 comes from inserting a complete basis of states including β and
**_vi_** **_β into the last term of the left-hand side. The second equality in equation 39 comes from_**
writing ⊥ **_W W_** _[⊤]_ + aa[⊤] = 2aa[⊤] + (W W _[⊤]_ _−_ **_aa[⊤]). Note that the final term in brackets on the_**
right hand side is a conserved quantity for linear networks, and so is always of order O(σ[2]).

Assuming the solutions for q, r are valid to order σ[2], we get that _dt[d]_ **_[a][⊤][W v][i][ =][ O][(][σ][4][)][.]_**

_d_

_dt_ **_[a][⊤][W v][i][ = (][β][ −]_** **_β[ˆ])[⊤]W_** _[⊤]W vi + |a|[2](β −_ **_β[ˆ])[⊤]vi_** (40)

Further noting that because of the conservation law aa[⊤] _−_ **_W W_** _[⊤]_ = O(σ[2]) is also constant in
time. This gives us that
_|a[⊤][aa[⊤]_ _−_ **_W W_** _[⊤]]W β| ≤_ _σ[2]|a[⊤]W β| = σ[2]r._ (41)
We now note that r, |a|[2] both grow as a (σ, s-independent) constant times times σ[2]e[2][st]. The correction to the dynamics of both equations is then bounded by a constant times σ[4]e[4][st]. This will be less
than σ[2] as long as t satisfies

_s_

_t_ _._ (42)
_≪_ 4[1]s [log] _σ[2]_

For σ[2] _≪_ _s, the alignment time t = 1/s falls within this range and we are guaranteed alignment to_ 
the Ganguli-Saxe configuration.

The error of the full solution at time t can be bounded by the integral of this error bound from 0 to
_t, namely a constant times σ[4]/s. As long as s ≫_ _σ[4], we are guaranteed that the error of the kernel_
is O(σ[2]) as given in equation 7.

C.3 PHASE I: TWO LAYER ANALYSIS WITH UNWHITENED DATA

We now study the same linearization around the initial fixed point used in the main text but for the
two layer network with unwhitened data. In this case,
_d_

(43)
_dt_ **_[a][ ∼]_** _[s][W][ Σ][β][,]_

_d_

(44)
_dt_ **_[W][ ∼]_** _[s][aβ][⊤][Σ][.]_

which holds asymptotically as t/ log(σ[−][1]) → 0. We introduce the following variables which form
a closed linear dynamical system
_r1(t) = β[⊤]W_ **_a,_**

_[⊤]_

_r2(t) =_ **_a_** _,_
_|_ _|[2]_

_r3(t) = β[⊤]W_ **_W Σβ,_** (45)

_[⊤]_

_r4(t) = β[⊤]ΣW_ **_a,_**

_[⊤]_

_r5(t) = β[⊤]ΣW_ **_W Σβ._**

_[⊤]_

Introduce the constants a = β[⊤]Σβ, b = β[⊤]Σ[2]β. Using the weight dynamics, it is straightforward
to show that

0 _a_ 1 0 0
0 0 0 2 0

**_r˙(t) ∼_** _s_ b 0 0 _a_ 0 (46)

0 _b_ 0 0 1

 
0 0 0 2b 0
 
  **_[r][(][t][)][, t][ ≪]_** [log(][σ][−][2][)][.]

This matrix has eigenvalues λ ∈{0, −√b, _√b, −2√b, 2√b}. Since there are only two positive_

eigenvalues _√b, 2√b, it suffices to consider evolution along those two eigendirections, where the_

kernel and neural network function will be amplified. Evolution along these direcions give


**_r(t)_** _c1e[s]_
_∼_


_bt_





 [+][ c][2][e][2][s]


_bt_


(47)


-----

t = 0

2

1

(a) Initialization

|t t 1|Col2|Col3|
|---|---|---|
||||


1


t

2

1


(b) Transition Time


(c) Min ℓ2 norm solution


Figure C.1: Kernel evolution on anisotropic data consists of two alignment phases. (a) At initialization the level curves of β[⊤]M _[−][1]β exhibit spherical symmetry. (b) After the initial phase I_
alignment, the matrix M exhibits a spike in the Σβ direction. (c) At long times, the network function and kernel’s spiked direction need to converge to the minimum ℓ2 norm solution as we explain
in G.1. This requires realignment of the kernel at late times, eliminating the preconditions for the
silent alignment effect.

where c1, c2 are constants determined by intialization. At large time, the large eigenvalue mode
_λ = 2√b will dominate. Decomposing W = W0 + a(t) [v1(t)β + v2(t)Σβ][⊤]_ we find that the

only self consistent solution is v1(t) = 0, v2(t) = b[−][1][/][2]. This implies that the kernel evolution will
take the form
_K(x, x[′], t) ∼_ _K(x, x[′], 0) + |a(t)|[2]x[⊤]Mx[′],_

1 (48)

**_M =_** **Σββ[⊤]Σ + I** _._

" **_β[⊤]Σ[2]β_** #

We see that the kernel evolves along the directions Σβ early in training for unwhitened data. We

p

visualize the two stages of learning for unwhitened data in Figure C.1.


C.4 PHASE II: WHITENED DATA

Consider a two layer network f = a[⊤]W x where balance has been achieved W = u(t)aβˆ _[⊤]_ and
**_a(t) = u(t)aˆ. Once this balance condition is stable for fixed ˆa, we can calculate the time derivative_**
of u(t)

_d_

_u(t)aˆ = u(t)_ _s_ _u(t)[2][]_ **_aˆ._** (49)
_dt_ **_[a][(][t][) = ˙]_** _−_

Letting c(t) = u(t)[2], we find that ˙c(t) = 2u(t)[2][ ]s −u(t)[2][] = 2c(t) [s − _c(t)], which is the twose[2][st]_
layer dynamics derived in Saxe et al. (2014). This dynamics has solution c(t) = _e[2][st]−1+s/c0_ [.]


C.5 SOLUTIONS TO THE FULL TRAINING DYNAMICS OF LINEAR NETWORKS AT SMALL
INITIALIZATION

By combining the analyses of the subsection C.1 with the exact solutions discussed in C.4 we can
match both solutions to obtain formulas for r(t) and q(t) for the entire network’s training path that
are exact up to O(σ[2]) corrections. Up to O(σ[2]) we then have that

2 sinh(2st)
_r(t) = s_ _,_ (50)

(e[2][st] 1) + 2s/q0
_−_

2 cosh(2st)
_q(t) = s_ _._ (51)

(e[2][st] 1) + 2s/q0
_−_

This yields that the initialization constant q0/2 plays the effective role of c0 in the Ganguli-Saxe
solution for phase II. Equation 34 yields the expected value of this initialization constant. We have


-----

1.00.8 = 10= 10= 10= 10 5432 12001000 Experiment1 ln 2 1.00.8 L|AlignmentKt (t)|

Lt0.60.4 t1/2 800600 0.60.4

0.2 400 Kernel and Loss0.2

0.0 0 500 1000t 1500 2000 10 5 10 4 2 10 3 10 2 0.0 0 500 1000t 1500 2000


(a) Loss Dynamics


(b) Time to Lt = 0.5


(c) Alignment σ = 10[−][4].


Figure C.2: Initialization scale controls the time spent in phase I, where the network escapes the
saddle point near W, a = 0 and the kernel aligns to the task. (a) The loss curves for two-layer
linear networks with small initialization follow sigmoidal trajectories as in Saxe et al. (2014) which
transition from their maximum to minimum at a time which decreases with initialization scale.
Theory is shown in black dashed lines. (b) Verification of the Phase I time t1/2, measured as the
time for the loss to reach one half its original value. This scales logarithmically with σ[2]. (c) The
alignment of the kernel eigenfunctions happens before the loss appreciably decreases for σ = 10[−][4],
evidenced by the kernel alignment curve. The analytically obtained maximum alignment value is
overlayed in dashed green.

empirically verified that these exact equations hold to high accuracy across a variety network sizes,
initialization scales, and whitened datasets. We illustrate some of these in figure C.3.

Exact vs Analytic Solutions Exact vs Analytic Solutions

10[1]

0 200 400 600 800 0 20 40 60 80 100

|Exact vs Analytic Solutions|Col2|Col3|
|---|---|---|
|q(t) empirical q(t) analytic r(t) empirical r(t) analytic|||
|||q(t) empirical q(t) analytic r(t) empirical r(t) analytic|
|0|20 40 60 80 10 t||


Exact vs Analytic Solutions

10[0]

10 1

10 2

q(t) empirical

10 3 q(t) analytic

r(t) empirical

10 4 r(t) analytic

0 200 400 600 800

t


(a) Synthetic


(b) Whitened MNIST


Figure C.3: Overlay of empirical and exact solutions for q(t), r(t) in two layer linear feedforward
networks for synthetic and two-class MNIST whitened datasets. (a) We take D = 25, N = 10. (b)
We take D = 784, N = 100.

D BALANCING OF WEIGHTS IN DEEP LINEAR NETWORKS

The balance condition discussed in the main text holds for deep linear trained with any loss function
of the form L = _µ_ _[ℓ][(][f][ µ][, y][µ][)][ (not just MSE) since, as was shown by Arora et al. (2018); Du et al.]_

(2018)

[P]

1 _d_ _∂ℓ_ _∂f_ _[µ]_ _⊤[#]_

**_W_** _[ℓ]W_ _[ℓ][⊤][
]_ =

_η_ _dt_ _∂f_ _[µ]_ _∂W_ _[ℓ]_ **_[W][ ℓ][⊤]_** [+][ W][ ℓ] _∂[∂f]W[ µ][ℓ]_

_µ_ "

 X

= _∂ℓ_ _∂f_ _[µ]_ _⊤W_ _[ℓ][+1]_ + W _[ℓ][+1][⊤]_ _∂f_ _[µ]_ (52)

_∂f_ _[µ]_ _∂W_ _[ℓ][+1]_ _∂W_ _[ℓ][+1]_

" #

X


_d_

**_W_** _[ℓ][+1][⊤]W_ _[ℓ][+1][
]_
_dt_



= [1]


-----

The second line follows from the first since the following quantities are identical

_∂f_ _[µ]_

**_x[µ][⊤]W_** [1][⊤]...W _[ℓ][−][1][⊤][
]_ **_W_** _[ℓ][⊤],_ (53)
_∂W_ _[ℓ]_ **_[W][ ℓ][⊤]_** [=][ W][ ℓ][+1][⊤][...][W][ L][−][1][⊤][w][L][ ]

_∂f_ _[µ]_
**_W_** _[ℓ][+1][⊤]_ **_W_** _[ℓ][⊤]...w[L][
]_ **_x[µ][⊤]W_** [1][⊤]...W _[ℓ][⊤]._ (54)

_∂W_ _[ℓ][+1][ =][ W][ ℓ][+1][⊤]_ []

By inspection these two quantities are equal. Thus we have, for any loss function, a deep linear
network has the following conservation laws

_d_

**_W_** _[ℓ]W_ _[ℓ][⊤]_ **_W_** _[ℓ][+1][⊤]W_ _[ℓ][+1][]_ = 0. (55)
_dt_ _−_


We show in the next section that this balancing condition is very helpful in identifying the time
evolution of the neural tangent kernel. In the case where the network has a single output, we can
inductively prove that each layer’s weight matrix is W _[ℓ]_ = u(t)rℓ+1(t)rℓ(t)[⊤]. We will assume this
formula is true for layer ℓ + 1 and prove it must hold for layer ℓ since

**_W_** _[ℓ]W_ _[ℓ][⊤]_ = W _[ℓ][+1][⊤]W_ _[ℓ][+1]_ = u(t)[2]rℓ+1(t)rℓ+1(t)[⊤]. (56)

This implies that W _[ℓ]_ is rank one with right singular vector equal to rℓ+1(t). Thus, the decomposition for each layer the form W _[ℓ]_ = u(t)rℓ+1(t)rℓ(t)[⊤] for some unit vector rℓ(t). Similar analysis
can be performed for the multi-class setting.

E NTK FORMULA FOR DEEP LINEAR NETWORKS

The neural tangent kernel for a linear network f (x) = w[L][⊤]W _[L][−][1]...W_ [1]x is defined as an innerproduct over all gradients


_∂f_ (x)

_∂W_ _[ℓ]_ _[, ∂f]∂W[(][x][ℓ][′][)]_



_K(x, x[′]) =_


_ℓ=1_


(57)


**_x · x[′]_** +


**_W_** _[ℓ][′]_
_ℓ[′]>ℓ_

Y


**_x[⊤]W_** [1][⊤]...W _[ℓ][−][1][⊤]W_ _[ℓ][−][1]...W_ [1]x[′].


**_W_** _[ℓ]_

_ℓ=2_

Y


_ℓ=2_


Under the balanced assumption that W _[ℓ]_ = u(t)rℓ+1(t)rℓ(t)[⊤] + O(σ), expanding the kernel to
leading order in σ yields the following form:

_K(x, x[′]) = u(t)[2][L][−][2]x[⊤]_ [](L 1)I + r1(t)r1(t)[⊤][] **_x[′]_** + O(σ), (58)
_−_


1.0


|1.0|Col2|Col3|
|---|---|---|
|1.0 0.8 0.6 Loss 0.4 0.2 0.0|L = 2 L = 3 L = 4 L = 5||
||L = 2 L = 3 L = 4 L = 5||

|Alignment 0.6 0.4 Kernel 0.2|L = 2 L = 3 L = 4 L = 5|
|---|---|

|1.0 0.8 Alignment 0.6 0.4 W1 0.2|L = 2 L = 3 L = 4 L = 5|
|---|---|


10[0] 10[1] 10[2] 10[3]


10[0] 10[1] 10[2] 10[3]


10[0] 10[1] 10[2] 10[3]


L = 2
L = 3
L = 4
L = 5


L = 2
L = 3
L = 4
L = 5


L = 2
L = 3
L = 4
L = 5


Figure E.1: The depth dependence of loss dynamics, kernel alignment and the alignment of first layer
weights in a linear network on synthetic whitened data in D = 30 dimensions. (a) The loss reaches
half its initial value after t ∼ _σ[−][L][+2]_ steps for L > 2. The decay rate of the loss becomes sharper
with depth. (b) Final kernel alignment increases monotonically with depth L and approaches 1 as
_L →∞. (c) The alignment of the first layer weights W1 with the optimal direction β approaches 1_
for all models.


-----

|1.0 0.8 0.6 Loss 0.4 0.2 0.0|L = 2 L = 3 L = 4 L = 5|
|---|---|
||L = 3 L = 4 L = 5|

|0.08 Alignment 0.06 0.04 0.02|Col2|Col3|
|---|---|---|
|0.00|||


10[0] 10[1] 10[2] 10[3]

L = 2
L = 3
L = 4
L = 5

t

(a) Loss


10[2] 10

t

(b) Alignment


0.25 K0

||/|fNN 0.200.15 K
fNN

0.10

|fNTK 0.05

0.00

1.5 2.0 2.5ReLU MLP Depth3.0 3.5 4.0 4.5 5.0 5.5


(c) Test Comparison


Figure E.2: ReLU networks across depths trained on two-class whitened CIFAR. The hidden widths
are all of size 100. We use 3k train points and 7k test points. (a) The loss exhibits the same scaling as
_σ[−][L][+2]_ as in the linear setting. (b) Deep networks with nonlinearities are seen to undergo the silent
alignment effect early on in training. The dashed lines indicate when the kernel has grown to 10%
of its final value. (c) The trained network outputs on test data match closely the kernel regression
with the final learned kernel, but do not match regression the initial kernel.

E.1 DEEP LINEAR NETWORK DYNAMICS UNDER BALANCE


In this section, we will consider the dynamics of the variable u once the balance condition is satisfied.
Let wL = ˆwu(t). Then the dynamics for u(t) under the balancing assumption is


_L−1_

**_W_** _[ℓ]_

_ℓ=1_

Y


_L−1_

(s − _u(t)[L])_ **_W_** _[ℓ]β = u(t)[2][L][−][2](s −_ _u(t)[L]) ˆw,_ (59)

_ℓ=1_

Y


_d_

**_w_** _[d]_
_dt_ **_[w][L][ =][ ˆ]_** _dt_ _[u][(][t][) =]_


which implies the fact ˙u(t) = u(t)[L][−][1](s − _u(t)[L]). Changing variables to c(t) = u(t)[L]_ we obtain

_c˙(t) = u(t)[L][−][1]u˙_ = u(t)[2][L][−][2](s − _u(t)[L]) = c(t)[2][−][2][/L](s −_ _c(t))._ (60)

When c[L]0 [is initialized to a very small value compared to][ s][ we can]


_d_

_dt_ _[c][(][t][)][ ∼]_ _[c][(][t][)][2][−][2][/L][s]_ =⇒ _c(t)[−][1+2][/L]_ _−_ _c[−]0_ [1+2][/L] = − _[L][ −]L_ [2] _st_

=⇒ _c(t) =_ _c−0h[L]L[−][2]_ _−_ [(][L][ −]L [2)] _st_ _−_ _LL−2i._ (61)
 


This implies a timescale to learn of t ∼ _s[−][1]_ _LL−2_ _[σ][−][L][+2][.]_

We can approximate the timescale for the first layer’s singular vector r1(t) to align to β as well. Let
**_v be a vector orthogonal to β._**

_d_ **_W_** [(1)]β

_|_ _|[2]_ _O(σ[L][−][2])._ (62)
_dt_ **_W_** [(1)]v **_W_** [(1)]v _∼_

_|_ _|[2][ = 2][β][⊤]|[W][ (1)][⊤]|[...][2]_ **_[w][L]_**


This suggests that alignment in a deep network should also occur on a timescale of t ∼ _σ[−][L][+2]._
While there is no strict separation of timescales in terms of the scaling of alignment and learning
with σ, we find that alignment tends to precede a significant drop in the loss as we show in Figure 3.

E.2 FINAL NTK FOR DEEP LINEAR NETWORKS


Independent of the structure of the data, the first vector r1 **_β and the final NTK has the following_**
form: _→_

_K(x, x[′]) = s[2][L][−][2]x[⊤]_ [](L − 1)ββ[⊤] + I **_x[′]_** + O(σ). (63)

This formula is merely a consequence of the balancing condition and convergence to the optimum
_u(t)[L]_ _→_ _s. We provide empirical support that final kernel alignment increases with depth in Figure_
E.3.


-----

|Col1|Col2|
|---|---|


L = 2 L = 4 L = 6

0.4 L = 2

L = 4 0
L = 6 K

0.3

0.2

Kernel Alignment0.1 K

0 500 1000 1500 2000 2500

t


(a) Alignment Dynamics and Depth


(b) Initial and Final NTKs


Figure E.3: The final NTK for a deep linear network aligns with the class specific directions with
strength that depends on network depth L. This experiment is a partially whitened (γ = 0.25
see Section 5) subset of 100 CIFAR-10 images. (a) The dynamics of alignment for different depth
models. (b) The final gram matrix has the form K∞ _∝_ (L−1)yy[⊤]+K0, illustrating why alignment
of the final NTK to class structure increases with depth L.

E.3 FORMULA FOR THE INTERMEDIATE NNGP KERNELS

Let β represent the unit vector pointing in the optimal direction for the scalar output case. To
calculate this, all that needs to be assumed is that the network has converged to its optimum and that
the weights satisfy the balance property so that W _[ℓ]_ = u[∗]rℓ[∗]+1[r]ℓ[∗][. By the convergence assumption]
_u[∗]_ = s[1][/L] and r1[∗] [=][ β][. We stress that this holds for arbitrary input correlation structure][ Σ][, the]
final NNGP kernel for layer ℓ can also be computed. Expanding the weights and collecting terms at
leading order in σ yields:

_Kℓ(x, x[′])_ _s[2][ℓ/L]x[⊤]ββ[⊤]x[′]_ + O(σ). (64)
_∼_

Evaluating on the training set gives Kℓ = **_yy[⊤][
][ℓ/L]_** _∈_ R[P][ ×][P] .


F GENERALIZATION ERROR IN TRANSFER LEARNING TASK

The structure of the final kernel can alter the ability of the network to flexibly transfer to new tasks
with a small amount of data. In this section, we examine how learned intermediate representations
compares with the inductive bias of the original isotropic kernel x · x[′]. In particular, we study the
offline generalization performance of kernels of the form

_K(x, x[′]; A) = x[⊤]_ []Aββ[⊤] + I **_x[′]._** (65)


In Section 4 we showed that A could be altered by changing the network depth. Concretely, our
transfer learning problem consists of training a linear probe on one of the intermediate layers of the
network (Alain & Bengio (2016); Cohen et al. (2020)). This would also produce a kernel regression
solution for kernel K(x, x[′], A) with A which depends on the chosen layer and the depth of the
network. For simplicity, we assume that the data are generated according to a simple Gaussian
distribution x ∼N (0, I) and that the target values are generated with a linear function y(x) = w·x.
We decompose the new task vector w = αβ + (1 _α[2])w_ where w **_β = 0. The expected_**

_−_ _⊥_ _⊥_ _·_
generalization error after training with P samples can be computed with methods from the physics

p

of disordered systems (Bordelon et al., 2020; Canatar et al., 2021; Loureiro et al., 2021). For any
_A > 0, the easiest transfer task is w = β (α = 1). If w = β, increasing the alignment A strictly_
decreases the generalization error. This is illustrated in Figure F.1.

F.1 DERIVATION OF LEARNING CURVES

We will discuss the average case generalization error in the transfer learning setting. Prior work has
shown that the generalization performance of kernel regression can be calculated through a kernel


-----

Kernel Alignment Strength Transfer Task Alignment Eg with P = 10

1.0 1.0 1.0 0.8

0.9 0.9

0.75

0.8 0.6

0.8

Eg 0.70.6 A = 0 Eg 0.7 = 0.0 0.5 0.4

0.5 AA = 1 = 5 0.6 = 0.2= 0.5 0.25

0.4 A = 10 0.5 = 0.7 0.2

0.0

0 5 10 15 20 25 0 5 10 15 20 25 0.0 2.5 5.0 7.5 10.0

P P A


(a) α = 0.75


(b) A = 1


(c) Fixed P


Figure F.1: The offline generalization error in a transfer task with the learned linear kernel K =
**_x[⊤]_** []Aβ[⊤]β[⊤] + I **_x[′]_** and ynew = _αβ +_ (1 _α[2])w_ **_x. (a) For a new transfer task which_**

_−_ _⊥_ _·_
is correlated with the learned function h _β, neural networks with large feature learningp_ i _A give lower_
generalization error at small sample sizes. (b) For fixed A = 1, the tasks w which are strongly
correlated with β are easier to learn during transfer. (c) The lowest generalization error in a transfer
learning setup occurs when feature learning strength A and correlation between tasks α are both
large.

eigenvalue problem (Bordelon et al., 2020; Canatar et al., 2021; Loureiro et al., 2021)


_⟨K(x, x[′])φk(x)⟩x = λkφk(x)._ (66)

Once this integral eigenvalue problem is solved for eigenvalues λk and orthonormal eigenfunctions
_φk, the average case generalization error at P training examples is_


_y(x)φk(x)_
_⟨_ _⟩[2]_ _, κ = λ + κ_

(λkP + κ)[2]


_λ[2]k_ (67)

(λkP + κ)[2][ .]


_λk_

_λkP + κ [, γ][ =][ P]_


_Eg =_


1 − _γ_


In our case, we are interested in the generalization performance of the linear kernel

_K(x, x[′]) = x[⊤]_ []ββ[⊤] + I **_x[′]._** (68)

Since K is a linear kernel, its eigenfunctions should be linear functions _φk(x) = φk_ **_x. Assuming_**
that the data distribution has identity covariance, we find _·_
**_x[⊤]_** []Aββ[⊤] + I **_x[′]φ[⊤]k_** **_[x]_** = φ[⊤]k _Aββ[⊤]_ + I **_x[′]_** = λkφ[⊤]k **_[x][′][.]_** (69)

This implies that the φk vectors are eigenvectors of  M . The first eigenvector is **_φ1 = β with_**
eigenvalue λ1 = A + 1. The other D − 1 eigenvectors can be chosen as any frame in the D − 1
dimensional subspace orthogonal to β. Each of these D − 1 eigenvectors has eigenvalue λk = 1.
Using these results, and the fact that w = αφ1 + _√1_ _α[2]w_, we can calculate the expected

_−_ _⊥_
generalization error.


_α[2]_ 1 − _α[2]_

((1 + A)P + κ)[2][ +] (P + κ)[2]


_Eg =_


_g_

1 − _γ_ ((1 + A)P + κ)[2][ +] (P

1 + A
_κ = λ + κ_

(1 + A)P + κ [+][ D]P +[ −] κ[1]




(1 + A)[2] _D_ 1
_, γ = P_ _−_

((1 + A)P + κ)[2][ +] (P + κ)[2]




By the result proven in Canatar et al. (2021), the lowest possible error for fixed A occurs by maximizing the fraction of variance along the large eigenvalue direction, corresponding to α = ±1.

G LINEAR NTKS DURING GD LEARN THE SAME FUNCTION

In the overparameterized setting where D > P, all linear networks discussed in the Section 4
converge to the minimum norm interpolator when the data is whitened. Specifically, letting the
learned neural network function be written as f (x) = **_β[ˆ][⊤]x, and the data matrix X ∈_** R[D][×][P] and


-----

target labels y ∈ R[P] represent the training data. The solution vector β solves the constrained
optimization problem

min **_β[ˆ]_** _, s.t. X_ _[⊤]β[ˆ] = y,_ (70)
**_βˆ_** _||_ _||[2]_

which is the kernel regression solution for the initial kernel K0(x, x[′]) = x **_x[′]._** This is un_·_
surprising due to a symmetry argument: when β0 0, the only privileged point on the affine
space X _[⊤]β = y is the point closest to the origin, which is precisely the solution above. Surpris- ≈_
ingly, the final pseudo-inverse solution sβ also minimizes the RKHS norms for any of the kernels
throughout gradient descent. Up to an overall scale, the kernels throughout evolution take the form
_K(x, x[′]; t) = x[⊤]_ []A(t)ββ[⊤] + I **_x[′]_** (see Section 4.1) which would induce the following kernel
interpolation problems


min **_βˆ[⊤]_** []A(t)ββ[⊤] + I _−1 ˆβ, s.t. X_ _[⊤]β[ˆ] = y._ (71)
**_βˆ_**


The solution to this optimization problem is indeed the kernel regression solution with kernel K(t)
since the learned function takes the form f (x) = _µ_ _[α][µ][K][(][x][,][ x][µ][, t][)][ with][ α][ =][ K][(][t][)][−][1][y][. Using]_

the Sherman-Morrison rule, we show that the solution to each of these problems t ≥ 0 gives the
same result, namely the pseudo-inverse solution. This can be seen from the following

[P]

**_βˆ[⊤]_** []A(t)ββ[⊤] + I _−1 ˆβ =_ **_β[ˆ]_** _A(t)_ **_β_** **_β[ˆ]_** 2 _._ (72)
_|_ _|[2]_ _−_ 1 + A(t) _·_
  

Now, we let **_β[ˆ] = sβ + β_**, where β **_β_** = 0. This is the general decomposition for the set of
_⊥_ _·_ _⊥_
interpolators which have the property X _[⊤]_ [β + β ] = y.
_⊥_

min (73)
**_β⊥_** _[|][β][⊥][|][2][.]_

The solution is merely to set β = 0. Thus the optimal solution is therefore the same for any finite value of A. However, the final RKHS norm of the learned function⊥ **_β[⊤]_** []A(t)ββ[⊤] + I _−1 β_
decreases with time, indicating that the kernel becomes more aligned with the pseudo-inverse direction as A increases. 

G.1 DEEP LINEAR NETWORKS FROM SMALL INITIALIZATION LEARN PSEUDO-INVERSE

In this subsection of the appendix, we will use our theoretical technology for balanced linear networks to demonstrate the universal learned function for any data, not just whitened input, providing an alternative derivation to the result proven in Theorem 7 of Yun et al. (2020). This
analysis is performed in the σ 0 limit, where Wℓ = u(t)rℓ+1(t)rℓ(t)[⊤] as we showed in
_→_
Section D. Under this condition the learned function f (x) = **_β[ˆ] · x is defined through weights_**
**_βˆ = W1[⊤][W][ ⊤]2_** _[...][w][L]_ [=][ u][(][t][)][L][r][1][(][t][)][. We see that the direction of the learned function is controlled]
entirely by r1(t). It suffices to prove that r1(t) span **_x1, ..., xP_** for all t to show that the network
learns the pseudo-inverse solution β[∗] = X(X ∈[⊤]X)[−]{[1]y, where X } _∈_ R[D][×][P] and y ∈ R[P] are the
training data and targets respectively. Note that by gradient descent, we have


_d_

2 _[...][w][L]_
_dt_ **_[W][1][(][t][) =][ W][ ⊤]_**


(yµ − _fµ(t))x[⊤]µ_ [=][ u][(][t][)][L][−][1][r][2][(][t][)]


(yµ − _fµ(t))x[⊤]µ_ _[.]_ (74)


From the balance condition W1(t) = u(t)r2(t)r1(t)[⊤], we also have

_d_

_u(t)r2(t) + u(t) ˙r2(t)] r1(t)[⊤]_ + u(t)r2(t) ˙r1(t)[⊤]. (75)
_dt_ **_[W][1][(][t][) = [ ˙]_**

Equating the two above expressions for _dtd_ **_[W][1][(][t][)][ and taking an inner product with][ r][2][(][t][)][ from the]_**

left gives the following

_u(t) ˙r1(t) = u(t)[L][−][1][ X](yµ_ _fµ(t))x[µ]_ [ ˙u(t) + u(t)r2(t) ˙r2(t)] r1(t). (76)

_µ_ _−_ _−_ _·_


-----

Thus, if r1(t) span **_x1, ..., xP_** then ˙r1(t) span **_x1, ..., xP_** so that the full dynamics
of r1(t) lie in the subspace spanned by the training data. At initialization, we have ∈ _{_ _}_ _∈_ _{_ _}_ **_W˙_** 1(t)
_∝_
**_z(t)_** _µ_ **_[y][µ][x]µ[⊤]_** [+][ O][(][σ][3][)][ so the initial][ r][1] [vector will indeed align with the span of the training data]

in the σ → 0 limit.

By the fact that[P] **_β[ˆ] = u(t)[L]r1(t), the learned linear coefficients_** **_β[ˆ] must also be in span_** **_x1, ..., xP_**
_{_ _}_
so **_β[ˆ] =_** _µ_ _[α][µ][x][µ][. These must also interpolate the data provided][ D][ ≥]_ _[P]_ [, giving the following]

condition
**_βˆ · x[P][ν]_** = _µ_ **_xν · xµαµ = yν =⇒_** **_α = (X_** _[⊤]X)[−][1]y =⇒_ **_β[ˆ] = X(X_** _[⊤]X)[−][1]y._ (77)
X

This is exactly the minimum ℓ2 norm interpolating solution which solves

min **_β[ˆ]_** 2 _[,][ s.t.][,][ X]_ _[⊤]β[ˆ] = y._ (78)
**_βˆ_** _|_ _|[2]_

While anisotropy of the data makes no impact on what function is ultimately learned in the linear
network case, the anisotropy can have a signficant influence on whether the preconditions for silent
alignment are satisfied in a nonlinear network, which can prevent the final function from being a
NTK regressor with final NTK.

H FINAL KERNEL IN MULTI-CLASS NETWORKS

For a network with C output channel, balancing and alignment guarantee that the configuration of
the network is orthogonal and balanced as in the setting of Saxe et al. (2014). One can then integrate
each mode separately to obtain the final kernel as


_uα(t)rℓ[α]+1[(][t][)][r]ℓ[α][(][t][)][⊤]_ _[, K][c,c][′]_ [(][x][,][ x][′][) =][ x][⊤][M][c,c][′] **_[x][′][,]_**
_α=1_

X


**_W_** _[ℓ]_ =


(79)


_L−1_

_ℓ=1_

X


_u[2(]β_ _[ℓ][−][1)](t)r1[β][r]1[β][⊤][,]_


_u[2(]α_ _[L][−][1)](t)r1[α][r]1[α]_ [+]


_u[2(]α_ _[L][−][ℓ][)](t)e[⊤]c_ **_[r]L[α][r]L[α][⊤][e][c][′]_**


_Mc,c[′] = δc,c[′]_


where the Cartesian unit vectorscontributions to the kernel depend on how well the class output channels align with the unit vectors ec ∈ R[C] are one-hot on class output c. This shows that the
**_rL[α][. Further, the singular values][ u][α][ can evolve at different timescales depending on the structure of]_**
the data.

Specifically, for a depth L network, both the the alignment time t[(]α[L][)] and the time to learn a given
singular value sα scale as s[−]α [1][σ][2][−][L][, as shown in appendix E.1. The differences in alignment times]
∆tαβ := t[(]α[L][)] _−_ _t[(]β[L][)]_ for modes sα, sβ therefores scales as ∆t[(]αβ[L][)] [=][ σ][2][−][L][∆][t]αβ[(2)][.]

H.1 FINAL NNGP IN MULTI-OUTPUT CASE

We can also gain intuition about the learned representations in each layer by looking at the NNGP
kernels, which merely take inner-products between layer activations for different inputs. Let β ∈
R[C][×][D] represent the optimal weight matrix which has the property (in the over-parameterized case)
**_βx[µ]_** = y[µ]. At the optimum, the neural network must learn β = W _[L]...W_ [1]. Computing the SVD
of β = _α_ _[β][α][z][α][v]α[⊤]_ [reveals that][ u]α[∗] [=][ β]α[1][/L] and rα[1] [=][ v][α] [and][ r]α[L] [=][ z][α][. Using these facts, it is]
easy to derive the final NNGP kernel for layer ℓ.

[P]

_Kℓ(x, x[′]) = x[⊤]W_ [1][⊤]...W _[ℓ][⊤]W_ _[ℓ]...W_ [1]x[′]

_ℓ/L_ (80)

= x[⊤] (u[∗]α[)][2][ℓ] **_[r]α[1]_** **_[r]α[1][⊤]_** **_x[′]_** = x[⊤] []β[⊤]β **_x′._**

_α_ #

"X 

_ℓ/L_
Evaluating on the training set X ∈ R[D][×][P] gives Kℓ = X _[⊤]_ []β[⊤]β **_X, which interpolates_**
between X _[⊤]X at layer ℓ_ = 0 and Y _[⊤]Y at layer L._ 


-----

Linear Depth 2 Linear Depth 4 Tanh Depth 3

1.0 1.0 1.0

0.8 0.8 0.8

0.6 L 0.6 0.6

|K|

0.4 Kernel Alignment 0.4 0.4

Kernel and Loss0.2 Kernel and Loss0.2 Kernel and Loss0.2

0.0 0.0 0.0

0 50 100 150 200 250 300 0 1000 2000 3000 4000 5000 6000 0 200 400 600 800 1000 1200 1400

t t t


(a) Linear depth 2


(b) Linear depth 4


(c) Tanh depth 3


Figure H.1: Demonstration of a separation between alignment and spectral learning phases across
networks trained on multi-class data. Here we train on whitened MNIST. Each network has σ so that
_σ[L]_ = 10[−][4] for L the depth of the network. (a) Depth two multi-class dynamics are very similar to
single class. The analytically predicted final alignment is in dashed green. (b) For deeper multi-class
networks, each singular values is learned far apart in time, and alignment does not as clearly precede
loss decay. (c) Similar dynamics are obtained for tanh networks. Note for deeper networks there is
a stronger separation of the times to learn each singular value, resulting in the ten separated drops in
the loss.

H.2 MORE REFINED BALANCING ANALYSIS

We can derive corrections to our decomposition of the kernel by including the initial conditions in
our derived conservation laws. In particular, we will consider balancing for large width networks.
Note that the network does not need to be in the lazy regime. The balancing condition is

**_W_** _[ℓ](t)W_ _[ℓ][⊤](t)_ **_W_** _[ℓ][+1][⊤](t)W_ _[ℓ][+1](t) = W0[ℓ][W]0[ ℓ][⊤]_ **_W0[ℓ][+1][⊤]W0[ℓ][+1]._** (81)
_−_ _−_

Note that W0[ℓ]
rameterization. The products of initial matrices are therefore Wishart distributed. For sufficiently[∈] [R][N][ℓ][+1][×][N][ℓ] [has entries with zero mean and variance][ σ][2][/N][ℓ] [in the standard pa-]
large widths, we can approximate the initial weight matrix products with their expectation over the
random initialization


**_W0[ℓ][W]0[ ℓ][⊤]_** _−_ **_W0[ℓ][+1][⊤]W0[ℓ][+1]_** _≈_ _σ[2]_ 1 − _[N]N[ℓ]ℓ[+2]+1_



**_I._** (82)


This concentration becomes more accurate as the widths Nℓ, Nℓ+1 . For the last layer if the
number of classes C = NL is sufficiently large, we also obtain similar concentration for the last →∞
layer W _[L][⊤]W_ _[L]_ _≈_ _σ[2]_ _NNLL−1_ **_[I][. Repeating the backward induction on the conservation law, we find]_**

the following recursively defined singular value decompositions


**_W_** _[ℓ]_ =


_u[ℓ]α[(][t][)][r]ℓ[α]+1[(][t][)][r]ℓ[α][(][t][)][⊤][,]_


_u[ℓ]α[(][t][)][2][ =][ u][ℓ]α[+1](t)[2]_ + σ[2] 1 − _N[N][ℓ]ℓ[+2]+1_



= u[L]α[(][t][)][2][ +][ σ][2][g][ℓ][,]


(83)


_L−1_

_k=ℓ_

X


_Nk+1_

_Nk_


_gℓ_ = L _ℓ_
_−_ _−_


We note that the corrections gℓ vanish if all layers k > ℓ have the same width. Let uα = u[(]α[L][)][(][t][)][.]
The condition for convergence is

2
_u[(]α[ℓ][)]_ = (uα)[2] + σ[2]gℓ = s[2]α[.] (84)

_ℓ_ _ℓ_

Y h i Y  

In the σ[2] 0 limit, we can solve that uα _s[1][/L]_ as before. However, we can now obtain leading
order corrections (in→ _σ[2]) which take the form of the form ∼_


_gσ[2]_

[uα][2] _s[2]α_ _, σ[2]_ 0, g =
_∼_ _[−]_ 2Ls[1]α[/L] _→_


_L_

_gℓ_ = _[L][(][L][ −]_ [1)]

2

_ℓ=1_

X


_Nk+1_

_,_ (85)
_Nk_


_ℓ=1_


_k=ℓ_


-----

which reveals that the size of the correction depends not only on σ but also on the depth and network
widths. Suppose all network widths were equal Nℓ = Nk, then the term g = 0 and there is no
contribution from the first moment of the random weights.

I LAZINESS IN HOMOGENOUS NETWORKS

In this section we recapitulate the argument found in Chizat et al. (2019). The goal is to estimate how
rapidly the gradient features on a test point ∇f (x) change compared to the loss early in training.
Let f ∈ R[P] represent the function outputs on the training set. We will compute the time derivatives
of the loss and the network gradients.

_d_

_dt_ _[∇][θ][f]_ [(][x][)] [=] _dt_ [=] (y[µ] _−_ _f_ _[µ])∇f_ _[µ]_ _[,]_ (86)

_µ_

X

2

_d_ [=] _[∇]dθ[2][f]_ [(][x]=[)][ ·][ d]f[θ] (y _[∇]f[2][f])[(][x].[)][ ·]_ (87)

_dt_ _[L]_ _dt_ _|∇_ _·_ _−_ _|[2]_


Here, | · |op denotes the operator norm of a matrix. We are interested in the ratio of the loss’ time
derivative to the gradient’s time derivative. With an initialization scale of f ∼ _O(σ[L]) we find_

_dt[d]_ _[∇][f]_ [(][x][)] _L_ _|∇[2]f_ (x) · _µ[(][y][µ][ −]_ _[f][ µ][)][∇][f][ µ][||][y][ −]_ **_[f]_** _[|][2]_ _|∇[2]f_ (x) · _µ_ _[y][µ][∇][f][ µ][||][y][|][2]_ _,_

**_f_** _[d]_ [=] **_f_** **_f_** (y **_f_** ) _≈_ **_f_** **_f_** **_y_**
_|∇_ _|_ _dt_ _[L]_ _|∇_ _||∇_ _·_ _−_ _|[2]_ _|∇_ _||∇_ _·_ _|[2]_

[P] [P] (88)

where in the last step we approximated y[µ] _−_ _f_ _[µ]_ _≈_ _y[µ]_ for small initialization scale since y[µ] _∼_ _O(1)_
and f _[µ]_ _∼_ _O(σ[L]). Now we will estimate the scale of each of the terms above. For a homogenous_
model ∇f ∼ _O(σ[L][−][1]) and ∇[2]f ∼_ _O(σ[L][−][2]). Counting powers of σ in numerator and denominator,_
we find that this quantity of interest scales as

_[d]_

_dt_ _[∇][f]_ [(][x][)]

_L_ (89)
**_f_** _[d]_ [=][ O][(][σ][−][L][)][.]
_|∇_ _|_ _dt_ _[L]_

This result indicates that, from small initialization, the gradient NTK features and thus the kernel
itself will evolve much more rapidly than the loss. This effect can be amplified by increasing depth
and decreasing initialization scale.

J RESNET EXPERIMENTAL DETAILS

Below, we provide the alignment and loss dynamics for wide resnet for CIFAR-10 with 100 training
points. Because the loss decreases significantly before the kernel reaches its final alignment value,
the final NTK is not perfectly correlated with the final neural network function. The wide ResNet
model is taken from Novak et al. (2020) and is based on the original architecture of Zagoruyko &
Komodakis (2017) with a widening factor of k = 4 and a single block per ResNet group b = 1,
giving a final network with 8 trainable layers. For both Figure 1 (d) and (e) as well as Figure
J.1 use Adam with a learning rate of η = 10[−][5] and initial weight scale of σ = 0.3 in standard
parameterization for all intermediate blocks. For the first conv layer, we used σ = 6.0. We find
that small initial weight variance in the first layer gives rise to less stable learning and worse kernel
alignment.

Below, in Figure J.2, we provide comprehensive results for different depths which we control by
increasing the number of blocks per group b, corresponding to WRNs with 6b + 1 trainable conv
layers.

J.1 ADAPTIVE OPTIMIZERS AND THE RELEVANT KERNEL

Many adaptive gradient methods compute updates to parameters θj according to


_θ˙j(t) =_ _ηj(t)_ _[∂][L]_ (90)
_−_ _∂θj,t_


-----

1.0 Lt Wide Res-Net on Unwhitened CIFAR

|K(t)| 2 K0

0.8 Alignment K

1

0.6

0

0.4

1

Loss and Alignment0.2 Test Prediction NTK

2

0.0

0 500 1000 1500 2000 2500 2 1 0 1 2

t Test Prediction NN


(a) Dynamics


(b) Predictions


Figure J.1: The dynamics and predictions of a Wide-Resnet with k = 4 and b = 1 on P = 100
unwhited CIFAR-10 images from the first two classes.


1.0 5

b = 1

0.8 b = 2 4

b = 3

Loss0.6 b = 4 3

0.4 2

Kernel Norm

0.2 1

0.0 0

0 1000 2000 3000 0 1000 2000 3000

t t


(a) Training Loss

0.5 |0.8 K0

fNN K

0.4

|/|0.6

0.3 fNN0.4

0.2

Alignment

0.1 |fNTK0.2

0.0

0.0

0 1000 2000 3000 1 2 3 4

t Wide Res-Net Blocks


(c) Alignment Dynamics


(b) Kernel Norm

1 2 3

Wide Res-Net Blocks

(d) Predictor Comparison


Figure J.2: The silent alignment effect is preserved across a large range of depths in WideResNet
trained on whitened CIFAR-10 images. The number of blocks per group b alters the total number
of conv layers (6b + 1 total conv layers). (a) The deeper models train faster with Adam. (b) The
final NTK norm increases with depth. (c) The alignment achieves close to its final value by the time
the kernel norm reaches 10% of its final value (dashed line), indicating successful silent alignment.
(d) The neural network predictions are very close to the predictions of the final NTK but is not
accurately predicted by the initial NTK.

where ηj(t) are time-varying functions which are computed in terms of the history of gradient moments for parameter θj or in terms of its instantaneous gradient. The relevant kernel at time t which
governs instantaneous evolution of network predictions is


_∂f_ (x[′], t)

(91)
_∂θj_


_ηj(t)_ _[∂f]_ [(][x][, t][)]

_∂θj_


_K(x, x[′], t) =_


since _f[˙](x) =_ _µ_ _[K][(][x][,][ x][µ][, t][)(][y][µ][ −]_ _[f]_ [(][x][µ][, t][))][. Though we do not calculate this kernel which is]

relevant to the adaptive learning rate scheme since it is not supported in Neural Tangents API, this
could be a worthy future investigation.

[P]


-----

1.0 k = 1 7

k = 2
k = 3 6

0.8 k = 4

5

Loss0.60.4 43

2

Kernel Norm

0.2

1

0.0 0

0 1000 2000 3000 4000 5000 6000 7000 0 500 1000 1500 2000 2500 3000 3500

t t


(a) Train Loss Dynamics

0.5 1.0 K0

K

0.40.3 ||/|fNN 0.80.6

fNN

0.2 0.4

Alignment

0.1 |fNTK 0.2

0.0

0.0

0 500 1000 1500 2000 2500 3000 3500 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

t Wide Res-Net Width


(c) Alignment


(b) Kernel Norm

1.0 1.5 2.0 2.5 3.0 3.5

Wide Res-Net Width

(d) Predictor Comparison


Figure J.3: Varying the ResNet widening parameter k also alters the kernel and loss dynamics. (a)
The loss curve for b = 2 WRNs with widening factor k. Wider networks train more quickly. (b)
The kernel norm increases more rapidly for wider networks but changes by a smaller amount. (c)
Alignment reaches close to its asymptote by the time the kernel norm grows to 10% its final value
(dashed). (d) The final kernel is a much better predictor of the NN function than the initial kernel.


-----