Original: Learning Single-Index Models with Shallow Neural Networks
Revised: Learning Single-Index Models with Shallow Neural Networks

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AfﬁliationAddress
Revised: AffiliationAddress

Original: Abstract
Revised: Abstract

Original: Single-index models are a class of functions given by an unknown univariate“link” function applied to an unknown one-dimensional projection of the input.
Revised: Single-index models are a class of functions given by an unknown univariate“link” function applied to an unknown one-dimensional projection of the input.

Original: These models are particularly relevant in high dimension, when the data mightpresent low-dimensional structure that learning algorithms should adapt to.
Revised: These models are particularly relevant in high dimension, when the data mightpresent low-dimensional structure that learning algorithms should adapt to.

Original: Whileseveral statistical aspects of this model, such as the sample complexity of recovering the relevant (one-dimensional) subspace, are well-understood, they rely ontailored algorithms that exploit the speciﬁc structure of the target function.
Revised: Whileseveral statistical aspects of this model, such as the sample complexity of recovering the relevant (one-dimensional) subspace, are well-understood, they rely ontailored algorithms that exploit the specific structure of the target function.

Original: In thiswork, we introduce a natural class of shallow neural networks and study its abilityto learn single-index models via gradient descent .
Revised: In thiswork, we introduce a natural class of shallow neural networks and study its abilityto learn single-index models via gradient descent .

Original: More precisely, we considershallow networks in which the ﬁrst layer weights are tied, to mirror the singleindex model structure, and biases of the neurons are frozen at random initialization.
Revised: More precisely, we considershallow networks in which the first layer weights are tied, to mirror the singleindex model structure, and biases of the neurons are frozen at random initialization.

Original: We show that the corresponding optimization landscape is benign, which inturn leads to generalization guarantees that match the optimal sample complexityof dedicated semi-parametric methods.
Revised: We show that the corresponding optimization landscape is benign, which inturn leads to generalization guarantees that match the optimal sample complexityof dedicated semi-parametric methods.

Original:  Introduction
Revised:  Introduction

Original: High-dimensional learning with both computational and statistical guarantees remains an outstanding challenge, with even more relevance in the face of current scaling trends.
Revised: High-dimensional learning with both computational and statistical guarantees remains an outstanding challenge, with even more relevance in the face of current scaling trends.

Original: One important questionwhich has seen considerable attention recently is to understand the advantages brought by non-linearlearning models, such as neural networks, over more mature (from a theoretical standpoint) counterparts, such as kernel methods [33, 34, 53, 13], even in their simplest shallow instantiations.
Revised: One important questionwhich has seen considerable attention recently is to understand the advantages brought by non-linearlearning models, such as neural networks, over more mature (from a theoretical standpoint) counterparts, such as kernel methods [33, 34, 53, 13], even in their simplest shallow instantiations.

Original: While approximation beneﬁts of shallow neural networks over non-adaptive kernels have beenknown for decades [8, 41], an important piece of this theoretical puzzle was provided by the statistical analysis of [5], which hinted at an inherent advantage of neural networks for extracting information from high-dimensional data with a “hidden” low-dimensional structure.
Revised: While approximation benefits of shallow neural networks over non-adaptive kernels have beenknown for decades [8, 41], an important piece of this theoretical puzzle was provided by the statistical analysis of [5] which hinted at an inherent advantage of neural networks for extracting information from high-dimensional data with a “hidden” low-dimensional structure.

Original: Alas, the remainingpiece of this puzzle, namely providing computational guarantees, is still mostly unresolved.
Revised: Alas, the remainingpiece of this puzzle, namely providing computational guarantees, is still mostly unresolved.

Original: Several computational hardness results for learning functions that can be efﬁciently approximatedby shallow neural networks [18, 26, 48, 17] have been established in the literature, which rule outpositive results in the general setting.
Revised: Several computational hardness results for learning functions that can be efficiently approximatedby shallow neural networks [18, 26, 48, 17] have been established in the literature, which rule outpositive results in the general setting.

Original: On the other hand, progress has been made on the positiveside [1, 2, 45] by focusing on function classes with strong structural properties, thereby showcasingthe adaptive representation learning capabilities of neural networks.
Revised: On the other hand, progress has been made on the positiveside [1, 2, 45] by focusing on function classes with strong structural properties, thereby showcasingthe adaptive representation learning capabilities of neural networks.

Original: This work aligns with the latter effort, and focuses on the class of single-index models .
Revised: This work aligns with the latter effort, and focuses on the class of single-index models .

Original: Single-indexmodels are high-dimensional functions F : R d ! R of the form F ⇤ ( x ) = f ⇤ ( h ✓ ⇤ , x i ) , where boththe univariate “link” function f 
Revised: Single-indexmodels are high-dimensional functions F : R d → R of the form F ∗ ( x ) = f ∗ ( ⟨ θ ∗ , x ⟩ ) , where boththe univariate “link” function f ∗ : R → R and relevant (one-dimensional) subspace θ

Original: ⇤ : R ! R and relevant (one-dimensional) subspace ✓ ⇤ 2 S d  1are unknown.
Revised: ∗ ∈ S d − 1are unknown.

Original: These models have been extensively studied in the statistics literature [16, 28, 19],leading to dedicated algorithmic procedures, and can be provably approximated with shallow neuralnetworks without incurring a curse-of-dimensionality [5].
Revised: These models have been extensively studied in the statistics literature [16, 28, 19],leading to dedicated algorithmic procedures, and can be provably approximated with shallow neuralnetworks without incurring a curse-of-dimensionality [5].

Original: In contrast, the analysis of neural network learning using gradient-descent techniques has been more heavily focused on the so-called “TeacherStudent” setup [27, 24, 54, 55, 3], where the link function f is assumed to be known.
Revised: In contrast, the analysis of neural network learning using gradient-descent techniques has been more heavily focused on the so-called “TeacherStudent” setup [27, 24, 54, 55, 3], where the link function f is assumed to be known.

Original: The mix of a high-dimensional parametric component (the hidden direction) with a low-dimensionalnon-parametric one (the link function) in single-index models naturally suggests a shallow neuralnetwork architecture where the inner weights are “active”, while the biases are “lazy” [14].
Revised: The mix of a high-dimensional parametric component (the hidden direction) with a low-dimensionalnon-parametric one (the link function) in single-index models naturally suggests a shallow neuralnetwork architecture where the inner weights are “active”, while the biases are “lazy” [14].

Original: Weinstantiate such an architecture by freezing the biases at their random initialization, and analyzegradient descent dynamics on the remaining free parameters in the continuous-time limit.
Revised: Weinstantiate such an architecture by freezing the biases at their random initialization, and analyzegradient descent dynamics on the remaining free parameters in the continuous-time limit.

Original: Our main results establish that as soon as the width N of the network is bigger than a quantity, whichdepends solely on the smoothness properties of the link function f ⇤ 1 , gradient descent recovers theunknown direction ✓ ⇤ and approximates the univariate link function f ⇤ with optimal sample andtime complexity
Revised: Our main results establish that as soon as the width N of the network is bigger than a quantity, whichdepends solely on the smoothness properties of the link function f ∗ 1 , gradient descent recovers theunknown direction θ ∗ and approximates the univariate link function f ∗ with optimal sample andtime complexity

Original: O ( d s ) (Theorem 5.1), where s is the so-called information-exponent of the linkfunction [3].
Revised: O ( d s ) (Theorem 5.1), where s is the so-called information-exponent of the linkfunction [3].

Original: The information exponent roughly captures the signal strength, which here refers tothe alignment between the network direction ✓ and the hidden direction ✓ ⇤ , at typical initializations.
Revised: The information exponent roughly captures the signal strength, which here refers tothe alignment between the network direction θ and the hidden direction θ ∗ , at typical initializations.

Original: The success of gradient descent relies on the benign optimization landscape of the empirical loss,though the presence of degenerate saddles requires a careful convergence analysis leveraging uniform convergence results for the empirical landscape [37, 20].
Revised: The success of gradient descent relies on the benign optimization landscape of the empirical loss,though the presence of degenerate saddles requires a careful convergence analysis leveraging uniform convergence results for the empirical landscape [37, 20].

Original: We show that gradient descent overour proposed neural network architecture solves two distinct problems–univariate non-parametrickernel ridge regression and non-convex optimization in high dimension– simultaneously and efﬁciently , cementing its role as a versatile algorithm for high-dimensional learning.
Revised: We show that gradient descent overour proposed neural network architecture solves two distinct problems–univariate non-parametrickernel ridge regression and non-convex optimization in high dimension– simultaneously and efficiently , cementing its role as a versatile algorithm for high-dimensional learning.

Original: We illustrate ourtheoretical results with experiments in Appendix D.
Revised: We illustrate ourtheoretical results with experiments in Appendix D.

Original:  Related Work
Revised:  Related Work

Original: Our work builds upon methods from different areas in statistics, optimization, and machine learning.
Revised: Our work builds upon methods from different areas in statistics, optimization, and machine learning.

Original: Single–index and multi-index models.
Revised: Single–index and multi-index models.

Original: A useful modeling assumption in high-dimensional regression is that the regression function F ( x ) =
Revised: A useful modeling assumption in high-dimensional regression is that the regression function F ( x ) =

Original: E [ y | x ] only depends on one or a few directions.
Revised: E [ y | x ] only depends on one or a few directions.

Original: This leads to the single-index model F ( x ) =
Revised: This leads to the single-index model F ( x ) =

Original: f ⇤ ( h ✓ ⇤ , x i ) , and multi-index model F ( x ) =
Revised: f ∗ ( ⟨ θ ∗ , x ⟩ ) , and multi-index model F ( x ) =

Original: f ⇤ ( h ✓ ⇤ 1 , x i , . . .
Revised: f ∗ ( ⟨ θ ∗ 1 , x ⟩ , . . .

Original: , ✓ ⇤ k , x i ) , with k typically much smaller than the dimension.
Revised: , θ ∗ k , x ⟩ ) , with k typically much smaller than the dimension.

Original: Such models have along history in the statistics literature and different methods exist for various estimation problems,including projection pursuit [21], slicing [31], gradient-based estimators [32], and moment-basedestimators [16].
Revised: Such models have along history in the statistics literature and different methods exist for various estimation problems,including projection pursuit [21], slicing [31], gradient-based estimators [32], and moment-basedestimators [16].

Original: When the function f ⇤ is also to be estimated, we face a semi-parametric probleminvolving parameter recovery of ✓ ⇤ and non-parametric estimation of f ⇤ .
Revised: When the function f ∗ is also to be estimated, we face a semi-parametric probleminvolving parameter recovery of θ ∗ and non-parametric estimation of f ∗ .

Original: Our work is closely related to [19], which also characterizes the population landscape of certain objectives by leveraging
Revised: Our work is closely related to [19], which also characterizes the population landscape of certain objectives by leveraging

Original: Gaussian data.
Revised: Gaussian data.

Original: Single and multi-index models are also studied in [5] in the context of shallow neural networks, where it is shown that certain models of inﬁnite-width shallow networks can adapt tosuch low-dimensional structure, though no tractable algorithms are introduced.
Revised: Single and multi-index models are also studied in [5] in the context of shallow neural networks, where it is shown that certain models of infinite-width shallow networks can adapt tosuch low-dimensional structure, though no tractable algorithms are introduced.

Original: Finally, [1] studiesthe learnability of functions on the hypercube by shallow neural networks and stochastic gradientdescent and introduces the merged staircase property , which provides necessary and sufﬁcient conditions for learnability with linear sample complexity n =
Revised: Finally, [1] studiesthe learnability of functions on the hypercube by shallow neural networks and stochastic gradientdescent and introduces the merged staircase property , which provides necessary and sufficient conditions for learnability with linear sample complexity n =

Original: O ( d ) .
Revised: O ( d ) .

Original: While they learn a broader classof functions (including multi-index model) for a more efﬁcient sample complexity regime ( O ( d ) vs
Revised: While they learn a broader classof functions (including multi-index model) for a more efficient sample complexity regime ( O ( d ) vs

Original: O ( d s ) ), our work captures the regime of semi-parametric estimation by considering Gaussian datawithout the sparsity constraints restrictions on F implied by their merged staircase property.
Revised: O ( d s ) ), our work captures the regime of semi-parametric estimation by considering Gaussian datawithout the sparsity constraints restrictions on F implied by their merged staircase property.

Original: Teacher–student models.
Revised: Teacher–student models.

Original: In the context of neural networks, several works have considered theteacher–student setting, where the target function F takes the form of a neural network with thesame activation as network used for learning [27, 24, 54, 55, 3].
Revised: In the context of neural networks, several works have considered theteacher–student setting, where the target function F takes the form of a neural network with thesame activation as network used for learning [27, 24, 54, 55, 3].

Original: In this case the problem doesnot involve non-parametric estimation as in our setup, but this line of work often involves studyingoptimization landscapes similar to ours for estimating hidden directions.
Revised: In this case the problem doesnot involve non-parametric estimation as in our setup, but this line of work often involves studyingoptimization landscapes similar to ours for estimating hidden directions.

Original: Kernels and random features.
Revised: Kernels and random features.

Original: Our work builds on the kernel methods literature for approximation and non-parametric regression [47, 7], their links with neural networks [15, 5], and in particularon random feature approximation [42, 6, 44, 38].
Revised: Our work builds on the kernel methods literature for approximation and non-parametric regression [47, 7], their links with neural networks [15, 5], and in particularon random feature approximation [42, 6, 44, 38].

Original: Non-convex optimization landscapes.
Revised: Non-convex optimization landscapes.

Original: There is a vast literature studying tractable non-convexoptimization landscapes, arising from high-dimensional statistics and statistical physics [36, 35, 22,4, 43, 49, 10, 24].
Revised: There is a vast literature studying tractable non-convexoptimization landscapes, arising from high-dimensional statistics and statistical physics [36, 35, 22,4, 43, 49, 10, 24].

Original: A particular aspect of our setup is that the optimization landscape does not havethe strict saddle property, which is often leveraged to establish global convergence [29].
Revised: A particular aspect of our setup is that the optimization landscape does not havethe strict saddle property, which is often leveraged to establish global convergence [29].

Original:  Preliminaries
Revised:  Preliminaries

Original: We focus on regression problems under a single-index model with Gaussian input data.
Revised: We focus on regression problems under a single-index model with Gaussian input data.

Original: Speciﬁcally,we assume d -dimensional inputs x ⇠  d := N (0 , I d ) , and labels
Revised: Specifically,we assume d -dimensional inputs x ∼ γ d := N (0 , I d ) , and labels

Original: [Equation]
Revised: [Equation]

Original: where ✓ ⇤ 2 Sd  1 and ✏ is random noise such that E [ ✏ [Equation] x [Equation]
Revised: where θ ∗ ∈ S

Original: = 0 , E[Equation] ✏ [Equation]
Revised: d − 1 and ϵ is random noise such that E

Original:  x [Equation]   [Equation]
Revised: [ ϵ | x ] = 0 , E [ ϵ 2 | x ] ≤ σ 2 .

Original: Shallow networks and random features.
Revised: Shallow networks and random features.

Original: We consider learning algorithms based on shallow neural networks of the form
Revised: We consider learning algorithms based on shallow neural networks of the form

Original: [Equation]
Revised: [Equation]

Original:  with ✓ 2 S d  1 , where  ( u ) = max { 0 , u } is the ReLU activation, b i ⇠ N (0 , ⌧ 2 ) (we assume ⌧ > 1 )are random “bias” scalars that are frozen throughout training, and  ( u ) = 1 p N (  ( u  b i ))
Revised: [Equation] with θ ∈ S d − 1 , where ϕ ( u ) = max { 0 , u } is the ReLU activation, b i ∼ N (0 , τ 2 ) (we assume τ > 1 )are random “bias” scalars that are frozen throughout training, and Φ( u ) = 1 √ N ( ϕ ( u − b i ))

Original: i 2 [ N ]is the vector of random features.
Revised: i ∈ [ N ]is the vector of random features.

Original: If we also ﬁx the direction ✓ , learning c alone may be seen as arandom feature model [42] that approximates a kernel method with the following kernel:
Revised: If we also fix the direction θ , learning c alone may be seen as arandom feature model [42] that approximates a kernel method with the following kernel:

Original: [Equation]
Revised: [Equation]

Original:  where  b ( u ) :=  ( u  b ) and  ⌧ = N (0 , ⌧ 2 ) is the Gaussian measure on R with mean zero andvariance ⌧ 2 .
Revised: 1) where ϕ b ( u ) := ϕ ( u − b ) and γ τ = N (0 , τ 2 ) is the Gaussian measure on R with mean zero andvariance τ 2 .

Original: The kernel  corresponds to a Reproducing Kernel Hilbert Space (RKHS) H , whichwe precisely deﬁne as
Revised: The kernel κ corresponds to a Reproducing Kernel Hilbert Space (RKHS) H , whichwe precisely define as

Original: [Equation]
Revised: [Equation]

Original: The following lemma characterizes the corresponding RKHS norm k · k H .
Revised: The following lemma characterizes the corresponding RKHS norm ∥ · ∥ H .

Original: Lemma 3.1 (RKHS Sobolev Representation, [52, Theorem 5]) .
Revised: Lemma 3.1 (RKHS Sobolev Representation, [52, Theorem 5]) .

Original: The RKHS norm in H is given by
Revised: The RKHS norm in H is given by

Original: [Equation]
Revised: [Equation]

Original: where f 00 is deﬁned in the generalised (Sobolev) sense.
Revised: where f ′′ is defined in the generalised (Sobolev) sense.

Original: Remark 3.2.
Revised: Remark 3.2.

Original: The boundary conditions for this RKHS can be relaxed by symmetrizing the kernel, ie s ( u, v ) = 12 (  ( u, v ) +  (  u,  v )) , but we will keep the original boundary conditions and assumethem on f ⇤ for simplicity and w.l.o.g.
Revised: The boundary conditions for this RKHS can be relaxed by symmetrizing the kernel, ieκ s ( u, v ) = 12 ( κ ( u, v ) + κ ( − u, − v )) , but we will keep the original boundary conditions and assumethem on f ∗ for simplicity and w.l.o.g.

Original: Hermite decompositions.
Revised: Hermite decompositions.

Original: Given that our data is normally distributed, we consider the familyof (normalized) Hermite polynomials { h j } j 2 N , which form an orthonormal basis of L 2 (  ) , thespace of squared-integrable function under the Gaussian measure  := N (0 , 1) .
Revised: Given that our data is normally distributed, we consider the familyof (normalized) Hermite polynomials { h j } j ∈ N , which form an orthonormal basis of L 2 ( γ ) , thespace of squared-integrable function under the Gaussian measure γ := N (0 , 1) .

Original: We will denoteby f ⇤ =
Revised: We will denoteby f ∗ =

Revised: (cid:80)

Original: P j ↵ j h j the Hermite decomposition of the target link function.
Revised: j α j h j the Hermite decomposition of the target link function.

Original: We apply the following useful properties of Hermite polynomials [40]:
Revised: We apply the following useful properties of Hermite polynomials [40]:

Original: [Equation]
Revised: [Equation]

Original: where h · , · i  d is the inner product in L 2 (  d ) and  the Kronecker delta.
Revised: where ⟨· , ·⟩ γ d is the inner product in L 2 ( γ d ) and δ the Kronecker delta.

Original: We will assume throughoutthat [-k f ⇤ k 2  = P j ↵ 2 j , k f 
 ⇤ k-]
Revised: We will assume throughoutthat [-k f ⇤ k 2  = P j ↵ 2 j , k f 
 ⇤ k-]∥

Original: [Equation]
Revised: [Equation]

Original:  = P [Equation] ↵ 2 j , and k f [-
-]
Revised: j , ∥ f ′ ∗ ∥ 2 γ = (cid:80) j jα 2 j , and ∥ f [-
-]′′ ∗ ∥ 2 γ = (cid:80) j j ( j − 1)

Original: ⇤ k 2  ={+α 
+}
Revised: {+α 
+}

Original: P j j ( j  1) ↵ 2 j are all ﬁnite (see Assumption 4.1).
Revised: j are all finite (see Assumption 4.1).

Original: Deﬁnition 3.3 (Population loss) .
Revised: Definition 3.3 (Population loss) .

Original: We deﬁne the L 2 -regularized population loss by 
Revised: We define the L 2 -regularized population loss by L

Original: [Equation]
Revised: [Equation]

Revised: c, θ

Revised: [Equation]

Revised: =

Revised: E x ∼ γ d

Revised: [Equation]

Revised: ∗ ( x

Revised: [Equation]

Revised: G ( x

Revised: [Equation]

Revised: c, θ

Revised: [Equation]

Revised: +

Revised: [Equation]

Original: From features to Hermite coefﬁcients.
Revised: From features to Hermite coefficients.

Original: To precisely characterize the landscape of L ( c, ✓ ) , weintroduce notation to represent the relationship between features  b in terms of Hermite polynomialsh j .
Revised: To precisely characterize the landscape of L ( c, θ ) , weintroduce notation to represent the relationship between features ϕ b in terms of Hermite polynomialsh j .

Original: We deﬁne the linear integral operator T :
Revised: We define the linear integral operator T :

Original: L 2 (  ) ! R N by
Revised: L 2 ( γ ) → R N by

Original: [Equation]
Revised: [Equation]

Original: Note that the operator T has rank N almost surely.
Revised: Note that the operator T has rank N almost surely.

Original: We abuse notation to equivalently let T ( j, b i ) :
Revised: We abuse notation to equivalently let T ( j, b i ) :

Original: =1 p N h h j ,  b i 
Revised: =1 √ N ⟨ h j , ϕ b i ⟩ γ

Original: i  =
Revised: =

Original: ( T h j ) i and observe that ( T f ) i = P j T ( j, b i ) .
Revised: ( T h j ) i and observe that ( T f ) i = (cid:80) j T ( j, b i ) .

Original: Observe that T has an adjoint
Revised: Observe that T has an adjoint

Original: T ⇤ : R N ! L 2 (  ) deﬁned as ( T ⇤ c )( u ) = 1 p N P
Revised: T ∗ : R N → L 2 ( γ ) defined as ( T ∗ c )( u ) = 1 √ N (cid:80) Ni

Original: Ni =1 c i  ( u  b i ) =
Revised: =1 c i ϕ ( u − b i ) =

Original: c >  ( u ) .
Revised: c ⊤ Φ( u ) .

Original: Let T j :=
Revised: Let T j :=

Original: T h j =( T ( j, b i ))
Revised: T h j =( T ( j, b i ))

Original: i =1 ... N 2 R N .
Revised: i =1 ...N ∈ R N .

Original: We can then write down the Hermite expansion of the network:
Revised: We can then write down the Hermite expansion of the network:

Original: [Equation]
Revised: [Equation]

Revised: (cid:88)

Revised: [Equation]

Original: Denoting m := h ✓ , ✓ ⇤ i , this allows us to rewrite the regularized objective as
Revised: Denoting m := ⟨ θ, θ ∗ ⟩ , this allows us to rewrite the regularized objective as

Original: [Equation]
Revised: [Equation]

Original: where the term P j ↵ 2 j is a constant that can be ignored.
Revised: where the term (cid:80) j α 2 j is a constant that can be ignored.

Original: Let Q := T T ⇤ 2 R N ⇥ N be a featurecovariance matrix and Q  = Q +  I .
Revised: Let Q := T T ∗ ∈ R N × N be a featurecovariance matrix and Q λ = Q + λI .

Original: Note that Q  is positive semi-deﬁnite for  > 0 .
Revised: Note that Q λ is positive semi-definite for λ > 0 .

Original: We deﬁnethe regularized projection ˆ P  = ˆ ⌃ (ˆ ⌃ +  I )  1 onto the random feature space ˆ H for ˆ ⌃ =
Revised: We definethe regularized projection ˆ P λ = ˆΣ(ˆΣ + λI ) − 1 onto the random feature space ˆ H for ˆΣ =

Original: T ⇤ T .
Revised: T ∗ T .

Original: Observe that ( ˆ P  f )( u ) =
Revised: Observe that ( ˆ P λ f )( u ) =

Original: c ⇤>  ( u ) , where c ⇤ 2 R N is the solution to the following objective:
Revised: c ∗⊤ Φ( u ) , where c ∗ ∈ R N is the solution to the following objective:

Original: [Equation]
Revised: [Equation]

Original: Geometry on the sphere.
Revised: Geometry on the sphere.

Original: Because the direction ✓ is constrained to lie on the sphere, our optimization algorithms rely on spherical (Riemannian) gradients, which are deﬁned as follows:
Revised: Because the direction θ is constrained to lie on the sphere, our optimization algorithms rely on spherical (Riemannian) gradients, which are defined as follows:

Original: [Equation]
Revised: [Equation]

Original: where Proj ✓ ? v = v  h ✓ , v i ✓ .
Revised: where Proj θ ⊥ v = v − ⟨ θ, v ⟩ θ .

Original: We say that ( c, ✓ ) is a critical point of L if r S d  1 ✓
Revised: We say that ( c, θ ) is a critical point of L if ∇ S d − 1 θ

Original: L ( c, ✓ ) = 0 andr{+
 and∇+} c L ( c, ✓ ) = 0 .
Revised: L ( c, θ ) = {+
 and∇+} c L ( c, θ ) = 0 .

Original:  Population Landscape under Frozen Random Biases
Revised:  Population Landscape under Frozen Random Biases

Original: In order to understand the optimization and generalization properties of applying gradient descent tothe empirical loss L n ( c, ✓ ) , we ﬁrst study the optimization landscape of the population loss L ( c, ✓ ) .
Revised: In order to understand the optimization and generalization properties of applying gradient descent tothe empirical loss L n ( c, θ ) , we first study the optimization landscape of the population loss L ( c, θ ) .

Original: We precisely characterize the critical points of the population loss in order to show that a gradientbased optimizer of a neural network of sufﬁcient width N converges only when its direction vector✓ is either parallel or orthogonal to the target direction ✓ ⇤ .
Revised: We precisely characterize the critical points of the population loss in order to show that a gradientbased optimizer of a neural network of sufficient width N converges only when its direction vectorθ is either parallel or orthogonal to the target direction θ ∗ .

Original: In the subsequent section, we showthat having sufﬁciently many training samples n provides a similar optimization landscape and hasfavorable generalization properties for most initializations.
Revised: In the subsequent section, we showthat having sufficiently many training samples n provides a similar optimization landscape and hasfavorable generalization properties for most initializations.

Original: We will make the following regularity assumptions on the target link function f ⇤ , in order to ensuresmall approximation error from random features, and a benign empirical landscape.
Revised: We will make the following regularity assumptions on the target link function f ∗ , in order to ensuresmall approximation error from random features, and a benign empirical landscape.

Original: Assumption 4.1 (Regularity of f ⇤ ) .
Revised: Assumption 4.1 (Regularity of f ∗ ) .

Original: Let f ( p ) ⇤ be the p th derivative of f ⇤ .
Revised: Let f ( p ) ∗ be the p th derivative of f ∗ .

Original: We assume that f ( p ) ⇤ 2 143 L 2 (  ) (i.e. k f ( p ) ⇤ k
Revised: We assume that f ( p ) ∗ ∈

Original:  < 1 ) for all p  6 , that f ⇤ is Lipschitz and that f ⇤ is polynomially bounded.
Revised: L 2 ( γ ) (i.e. ∥ f ( p ) ∗ ∥ γ < ∞ ) for all p ≤ 6 , that f ∗ is Lipschitz and that f ∗ is polynomially bounded.

Original: The index of the ﬁrst non-zero Hermite coefﬁcient ↵ j of f ⇤ is referred to as the information exponentand is given by s :
Revised: The index of the first non-zero Hermite coefficient α j of f ∗ is referred to as the information exponentand is given by s :

Original: = min { j : ↵ j [Equation] = 0 } .
Revised: = min { j : α j ̸ = 0 } .

Original: To consider the critical points of L ( c, ✓ ) , we introduce the projected population loss at a givendirection ✓ :
Revised: To consider the critical points of L ( c, θ ) , we introduce the projected population loss at a givendirection θ :

Original: [Equation]
Revised: [Equation]

Original: Crucially, the sufﬁcient number of features N depends on the regularization parameter  2 (0 , 1) ,the variance ⌧ 2 > 1 of the random biases, and the target link function f ⇤ , but not on the inputdimension d .
Revised: Crucially, the sufficient number of features N depends on the regularization parameter λ ∈ (0 , 1) ,the variance τ 2 > 1 of the random biases, and the target link function f ∗ , but not on the inputdimension d .

Original: Theorem 4.2 (Critical Points of Population Loss) .
Revised: Theorem 4.2 (Critical Points of Population Loss) .

Original: For ⌧ > 1 , and  2 (0 , 1) , there exist constants
Revised: For τ > 1 , and δ ∈ (0 , 1) , there exist constants

Original: C ⌧ and C f ⇤ depending only on ⌧ and the target link function f ⇤ respectively such that if
Revised: C τ and C f ∗ depending only on τ and the target link function f ∗ respectively such that if

Original: [Equation]
Revised: [Equation]

Original: ⇤ and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: (5) then with probability 1   over the random biases b 1 , . . .
Revised: 5) then with probability 1 − δ over the random biases b 1 , . . .

Original: , b N , the set of ﬁrst-order critical points
Revised: , b N , the set of first-order critical points

Original: [Equation]
Revised: C L := { ( c, θ ) : ∇ S d − 1 θ

Original: [Equation] c [Equation] satisﬁes:1.
Revised: L ( c, θ ) = 0 , ∇ c L ( C, θ ) = 0 } satisfies:

Original: (orientation relative to ✓ ⇤ ) if ( c, ✓ ) 2 C L , then either ✓ 2 {± ✓ ⇤ } or h ✓ , ✓ ⇤ i = 0 ; and2.
Revised: (orientation relative to θ ∗ ) if ( c, θ ) ∈ C L , then either θ ∈ {± θ ∗ } or ⟨ θ, θ ∗ ⟩ = 0 ; and

Original: (existence and uniqueness of c ) if r S d  1 ✓ ¯ L ( ✓ ) =0 , then there exists a unique c such that
Revised: (existence and uniqueness of c ) if ∇ S d − 1 θ ¯ L θ ) = then there exists a unique c such that

Original: [Equation]
Revised: ( c, θ ) ∈ C L

Original: Theorem 4.2 thus establishes a benign optimization landscape in the population limit, rejoiningseveral known non-convex objectives with similar behavior, such as tensor decomposition [25] ormatrix completion [23].
Revised: Theorem 4.2 thus establishes a benign optimization landscape in the population limit, rejoiningseveral known non-convex objectives with similar behavior, such as tensor decomposition [25] ormatrix completion [23].

Original: It implies that all critical points are either at the equator or the poles andthus a gradient ﬂow has to either avoid or escape the “spurious” critical points in the equator, i.e., ✓ ’ssatisfying h ✓ ⇤ , ✓ i = 0 to succeed.
Revised: It implies that all critical points are either at the equator or the poles andthus a gradient flow has to either avoid or escape the “spurious” critical points in the equator, i.e., θ ’ssatisfying ⟨ θ ∗ , θ ⟩ = 0 to succeed.

Original: Importantly, this optimization landscape has the same topology asif one were using Hermite basis functions rather than random scalar features [19, Theorem 5], thetailored choice that exploits the knowledge that the data is being generated by a single-index modelon Gaussian samples.
Revised: Importantly, this optimization landscape has the same topology asif one were using Hermite basis functions rather than random scalar features [19, Theorem 5], thetailored choice that exploits the knowledge that the data is being generated by a single-index modelon Gaussian samples.

Original: We view this as an interesting robustness property of shallow neural networks– at least in the regime where biases are randomly frozen.
Revised: We view this as an interesting robustness property of shallow neural networks– at least in the regime where biases are randomly frozen.

Original: We now outline the analysis that leads to the proof of the theorem before proving it formally in
Revised: We now outline the analysis that leads to the proof of the theorem before proving it formally in

Original: Section 4.3.
Revised: Section 4.3.

Original: 4.1 Exact expression of ﬁrst-order critical points
Revised: 4.1 Exact expression of first-order critical points

Original: The ﬁrst step is to characterize the critical points of the regularized population loss.
Revised: The first step is to characterize the critical points of the regularized population loss.

Original: This is describedin the following result.
Revised: This is describedin the following result.

Original: Claim 4.3 (Critical point equation) .
Revised: Claim 4.3 (Critical point equation) .

Original: Let g m , ¯ g m 2 L 2 (  ) be deﬁned by
Revised: Let g m , ¯ g m ∈ L 2 ( γ ) be defined by

Original: [Equation]
Revised: [Equation]

Original: where m = h ✓ , ✓ ⇤ i .
Revised: where m = ⟨ θ, θ ∗ ⟩ .

Original: Then, the critical points of L ( c, ✓ ) satisfy the following equations:
Revised: Then, the critical points of L ( c, θ ) satisfy the following equations:

Original: [Equation]
Revised: [Equation]

Original: Remark 4.4 (Inﬁnite-width networks) .
Revised: Remark 4.4 (Infinite-width networks) .

Original: If the kernel is universal and N ! 1 , then ˆ P  is theidentity operator when  = 0 .
Revised: If the kernel is universal and N → ∞ , then ˆ P λ is theidentity operator when λ = 0 .

Original: Hence, h h j , P  h j 0 i =  j,j 0 in this limit and the target can be exactly 176 represented when m = 1 .
Revised: Hence, ⟨ h j , P λ h j ′ ⟩ = δ j,j ′ in this limit and the target can be exactlyrepresented when m = 1 .

Original: We prove Claim 4.3 in Appendix B.1 by exactly computing the gradients and relating the criticalpoints of ¯ L ( ✓ ) and L ( c, ✓ ) .
Revised: We prove Claim 4.3 in Appendix B.1 by exactly computing the gradients and relating the criticalpoints of ¯ L ( θ ) and L ( c, θ ) .

Original: 4.2 Controlling random feature approximation
Revised: 4.2 Controlling random feature approximation

Original: We now consider (ﬁnite) random feature approximations to functions in the RKHS.
Revised: We now consider (finite) random feature approximations to functions in the RKHS.

Original: We show thatthe best possible loss of a linear combination of sufﬁciently many ﬁnite features is bounded aboveby the best approximation with inﬁnitely many features with high probability.
Revised: We show thatthe best possible loss of a linear combination of sufficiently many finite features is bounded aboveby the best approximation with infinitely many features with high probability.

Original: Let A ( f,  ) be the (regularized)
Revised: Let A ( f, λ ) be the (regularized)

Original: L 2 approximation error for functions in the space H with respect tothe target function f and measure  .
Revised: L 2 approximation error for functions in the space H with respect tothe target function f and measure γ .

Original: Formally,
Revised: Formally,

Original: [Equation]
Revised: [Equation]

Original: Lemma 4.5 (Random features approximation, adapted from [6]) .
Revised: Lemma 4.5 (Random features approximation, adapted from [6]) .

Original: Let  2 (0 , 1) , ⌧ > 1 , and f [-
-]
Revised: Let δ ∈ (0 , 1) , τ > 1 , and f [-
-]∈

Original: L 2 (  ) .
Revised: L 2 ( γ ) .

Original: There exists a constant C ⌧ > 0 such that if N  C ⌧  ln 1  , then with probability at least 1   ,
Revised: There exists a constant C τ > 0 such that if N ≥ C τ λ ln 1 λδ , then with probability at least 1 − δ ,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: The claim is immediate from Lemmas B.5 and B.6 in Appendix B.2.
Revised: The claim is immediate from Lemmas B.5 and B.6 in Appendix B.2.

Original: Next, we show that the approximation error of the RKHS corresponding to an inﬁnite number ofrandom features can be bounded in terms of the regularization  and the  -norm of the secondderivative of the target function.
Revised: Next, we show that the approximation error of the RKHS corresponding to an infinite number ofrandom features can be bounded in terms of the regularization λ and the γ -norm of the secondderivative of the target function.

Original: The bound is independent of d , as we consider ⌧ and  ﬁxed as dgrows.
Revised: The bound is independent of d , as we consider τ and λ fixed as dgrows.

Original: Lemma 4.6 (RKHS approximation error) .
Revised: Lemma 4.6 (RKHS approximation error) .

Original: Assume f 2 L 2 (  ) is in the weighted Sobolev space
Revised: Assume f ∈ L 2 ( γ ) is in the weighted Sobolev space

Original: H 2 (  ) and has polynomial growth (i.e. there exist C > 0 and M > 0 such that | f ( u ) |  C ( u M +u  M ) ).
Revised: H 2 ( γ ) and has polynomial growth (i.e. there exist C > 0 and M > 0 such that | f ( u ) | ≤ C ( u M +u − M ) ).

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: with  := ⌧ 2 ⌧ 2 +2 .
Revised: with β := τ 2 τ 2 +2 .

Original: The proof appears in Appendix B.3.
Revised: The proof appears in Appendix B.3.

Original: 4.3 Proof of Theorem  
Revised: 4.3 Proof of Theorem 4.

Original: Proof.
Revised: Proof.

Original: For ( c, ✓ ) to be a critical point of L , it must satisfy r S d  1 ✓
Revised: For ( c, θ ) to be a critical point of L , it must satisfy ∇ S d − 1 θ

Original: L ( c, ✓ ) = 0 and r c L ( ✓ , c ) = 0 .
Revised: L ( c, θ ) = 0 and ∇ c L ( θ, c ) = 0 .

Original: According to Claim 4.3, every ﬁxed ✓ has a unique complementary c such that r c L ( ✓ , c ) = 0 , andr S d  1 ✓
Revised: According to Claim 4.3, every fixed θ has a unique complementary c such that ∇ c L ( θ, c ) = 0 , and∇ S d − 1 θ

Original: L ( c, ✓ ) = 

Revised: L ( c, θ ) =

Original: if and only if ✓ 2 {  ✓ ⇤ , ✓ ⇤ } or
Revised: if and only if θ ∈ {− θ ∗ , θ ∗ } or(cid:88) [Equation]

Original: [Equation]
Revised: [Equation]

Original: The remainder of the proof involves showing that the latter condition is true only if m = 0 .
Revised: The remainder of the proof involves showing that the latter condition is true only if m = 0 .

Original: Wederive a contradiction whenever m 6 = 0 .
Revised: Wederive a contradiction whenever m ̸ = 0 .

Original: Let  :=
Revised: Let β :=

Original: ⌧ 2 ⌧ 2 [-+
-] and take N  C ⌧  ln 1  and
Revised: τ 2 τ 2 [-+
-]+2 and take N ≥ C τ λ ln 1 λδ and

Revised: [Equation]

Revised: <

Original: [Equation]
Revised: [Equation]

Original: for sufﬁciently large C 0 and C ⌧ .
Revised: for sufficiently large C ′ and C τ .

Original: We contradict the equality with probability at least 1   over therandomly sampled biases b 1 , . . .
Revised: We contradict the equality with probability at least 1 − δ over therandomly sampled biases b 1 , . . .

Original: , b N :
Revised: , b N :

Original: [Equation]
Revised: [Equation]

Original: [ Lemma 4.5 ]
Revised: [ Lemma 4.5 ]

Original: [Equation]
Revised: [Equation]

Original: [ Lemma 4.6 ]
Revised: [ Lemma 4.6 ]

Original: [Equation]
Revised: [Equation]

Original: [ Lemma 
Revised: [ Lemma B.7 ]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  Empirical Landscape and Generalization Guarantees
Revised:  Empirical Landscape and Generalization Guarantees

Original: Now that we have established that the population landscape L ( c, ✓ ) has a particularly simplestructure given sufﬁciently many random features N & O (1) , we study the optimization properties of its ﬁnite-sample counterpart.
Revised: Now that we have established that the population landscape L ( c, θ ) has a particularly simplestructure given sufficiently many random features N ≳ O (1) , we study the optimization properties of its finite-sample counterpart.

Original: For training samples { ( x i , y i ) } i 2 [ n ] generated by independently drawn x i ⇠ N (0 , I ) and y i =
Revised: For training samples { ( x i , y i ) } i ∈ [ n ] generated by independently drawn x i ∼ N (0 , I ) and y i =

Original: f ⇤ ( h x i , ✓ ⇤ i ) 2 and random features  : R ! R N [-with
-]
Revised: f ∗ ( ⟨ x i , θ ∗ ⟩ ) 2 and random features Φ : R → R N [-with
-]with

Original: For simplicity of the analysis and exposition, we assume clean labels without additive noise, but note we may incorporate additive noise to our analysis in a straightforward manner.

Original:  ( u ) j = N  1 / 2  ( u  b j ) , we deﬁne the empirical risk as
Revised: Φ( u ) j = N − 1 / 2 ϕ ( u − b j ) , we define the empirical risk as

Original: [Equation]
Revised: [Equation]

Original: We speciﬁcally consider the estimator ˆ F ( x ) :
Revised: We specifically consider the estimator ˆ F ( x ) :

Original: = ˆ f ( h x, ˆ ✓ i ) , where ( ˆ f, ˆ ✓ ) are obtained by using gradient descent on c and ✓ to minimize L n ( c, ✓ ) .
Revised: = ˆ f ( ⟨ x, ˆ θ ⟩ ) , where ( ˆ f, ˆ θ ) are obtained by using gradient descent on c and θ to minimize L n ( c, θ ) .

Original: Such strategy appears to be reasonable, in light of theproperties of the population landscape studied in Section 4, namely that its only local minimisers arealso global, and correspond to ˜ f
Revised: Such strategy appears to be reasonable, in light of theproperties of the population landscape studied in Section 4, namely that its only local minimisers arealso global, and correspond to ˜ f

Original: = ˆ P  f ⇤ and ˜ ✓ = ✓ ⇤ .
Revised: = ˆ P λ f ∗ and ˜ θ = θ ∗ .

Original: Given a sufﬁciently large sample size n , oneexpects the empirical landscape L n to concentrate around its expectation L and inherit its benignoptimization properties.
Revised: Given a sufficiently large sample size n , oneexpects the empirical landscape L n to concentrate around its expectation L and inherit its benignoptimization properties.

Original: However, the presence of a degenerate saddle at ( c, m ) =
Revised: However, the presence of a degenerate saddle at ( c, m ) =

Original: (0 , 0) ﬂattensthe landscape around the “equator” { ✓ : h ✓ , ✓ ⇤ i = 0 } , and a sufﬁciently large sample complexity is 218 ultimately required to ensure that gradient descent iterates can escape from it, given its dangerouslyclose initialization | h ✓ 0 , ✓ ⇤ i | = ⇥ (1 / p d ) .
Revised: (0 , 0) flattensthe landscape around the “equator” { θ : ⟨ θ, θ ∗ ⟩ = 0 } , and a sufficiently large sample complexity isultimately required to ensure that gradient descent iterates can escape from it, given its dangerouslyclose initialization |⟨ θ 0 , θ ∗ ⟩| = Θ(1 / √ d ) .

Original: Prior works have obtained the optimal sample complexity of n =
Revised: Prior works have obtained the optimal sample complexity of n =

Original: O ( d s ) (recall that s is the information exponent of the target function f ⇤ ) for recovering the target direction ✓ ⇤ either by employinga learning algorithm that explicitly learns individual harmonic polynomials [19] or by assuming thatf ⇤ is known a priori [3].
Revised: O ( d s ) (recall that s is the information exponent of the target function f ∗ ) for recovering the target direction θ ∗ either by employinga learning algorithm that explicitly learns individual harmonic polynomials [19] or by assuming thatf ∗ is known a priori [3].

Original: Their argument goes essentially as follows:
Revised: Their argument goes essentially as follows:

Original: • The empirical optimization landscape (when regarded as a function only of the direction ✓ )near the equator | m | ⌧ 1 is of the form L ( ✓ ) = ⇥ ( m s ) .
Revised: • The empirical optimization landscape (when regarded as a function only of the direction θ )near the equator | m | ≪ 1 is of the form L ( θ ) = Θ( m s ) .

Original: • In order to certify that the optimization does not converge to a suboptimal critical point onthe equator (i.e. kr L ( ✓ ) k & ✏ ) one must require that m & ✏ 1 / ( s  1) .
Revised: • In order to certify that the optimization does not converge to a suboptimal critical point onthe equator (i e. ∥∇ L ( θ ) ∥ ≳ ϵ ) one must require that m ≳ ϵ

Revised: [Equation]

Original: • A standard uniform concentration argument of the form kr L ( ✓ )r
Revised: • A standard uniform concentration argument of the form ∥∇ L ( θ )

Original:  L n ( ✓ ) k . q dn and the
Revised: −∇ L n ( θ ) ∥ ≲ (cid:113) dn and the

Original: fact that m ⇠ d  1 / 2 at initialization together imply that n
Revised: fact that m ∼ d − 1 / 2 at initialization together imply that n

Original: = O ( d s ) samples are sufﬁcientto escape from the “inﬂuence” of the equator.
Revised: = O ( d s ) samples are sufficientto escape from the “influence” of the equator.

Original: In order to repurpose these arguments to our setting, the relative scaling of the top-layer weights crelative to the direction vector ✓ will turn out to be crucial, as has also been observed in the literatureon lazy -vs- rich regimes [14, 53] in the context of overparametrized neural networks.
Revised: In order to repurpose these arguments to our setting, the relative scaling of the top-layer weights crelative to the direction vector θ will turn out to be crucial, as has also been observed in the literatureon lazy -vs- rich regimes [14, 53] in the context of overparametrized neural networks.

Original: We consider an idealized version of a Gradient Descent algorithm over the empirical loss L n in theinﬁnitesimally small learning rate regime, resulting in the gradient ﬂow ODE:
Revised: We consider an idealized version of a Gradient Descent algorithm over the empirical loss L n in theinfinitesimally small learning rate regime, resulting in the gradient flow ODE:

Original: [Equation]
Revised: [Equation]

Original: where ⇣ is the relative time-scale between ﬁrst- and second-layer weights, and r S d  1 ✓ is the Riemannian gradient.
Revised: where ζ is the relative time-scale between first- and second-layer weights, and ∇ S d − 1 θ is the Riemannian gradient.

Original: The learning rate schedule ⇣ ( t ) is set so that only ✓ is trained ﬁrst and after aﬁxed amount of time, which we denote by T 0 , then c and ✓ is jointly optimized.
Revised: The learning rate schedule ζ ( t ) is set so that only θ is trained first and after afixed amount of time, which we denote by T 0 , then c and θ is jointly optimized.

Original: This is requiredto guarantee that the gradient ﬂow escapes the equator (see Lemma C.4), which is crucial since the(random) starting point for the ﬂow is close to the equator with high probability.
Revised: This is requiredto guarantee that the gradient flow escapes the equator (see Lemma C.4), which is crucial since the(random) starting point for the flow is close to the equator with high probability.

Original: Our main result, proved in Appendix C, establishes that this gradient ﬂow, under a simple schedulefor ⇣ , efﬁciently ﬁnds an approximate minimiser of the population loss, with an error explicitlyquantiﬁed as a function of n :
Revised: Our main result, proved in Appendix C, establishes that this gradient flow, under a simple schedulefor ζ , efficiently finds an approximate minimiser of the population loss, with an error explicitlyquantified as a function of n :

Original: Theorem 5.1 (Gradient ﬂow ﬁnds approximate minimizers) .
Revised: Theorem 5.1 (Gradient flow finds approximate minimizers) .

Original: For  > 0 and f ⇤ satisfying Assumption 4.1, suppose the following are true: (i) d
Revised: For δ > 0 and f ∗ satisfying Assumption 4.1, suppose the following are true: (i) d

Original: = ⌦ (log 1  ) , (ii) 
Revised: = Ω(log 1 δ ) , (ii) λ

Original: = O (1) (iii) n = ⌦ ( d s  4 log 1  ) , (iv)
Revised: = O (1) (iii) n = Ω( d s λ 4 log 1 δ ) , (iv)

Original: N = ⌦ ( 1  log 1  ) & N =
Revised: N = Ω( 1 λ log 1 λδ ) & N =

Original: O ( 1  (  4 nd ) 1 / (2( s +1)) ) , (v) N 0 = ⌦ (log 1  ) & N 0{+
+} =
Revised: O ( 1 λ ( λ 4 nd ) 1 / (2( s +1)) ) , (v) N 0 = Ω(log 1 δ ) & N {+
+} =

Original: O ((  4 nd ) 1 / (2( s +1)) ) ,(vi) ⇢ = ⇥ ( p NN  (2+ s ) / 2) , (vii) T 0{+
+} = ˜ ⇥ ( N 1+ s/ 2d  1+ s/ 2 ) , and (viii) T 1 =⇥
Revised: O (( λ 4 nd ) 1 / (2( s +1)) ) ,(vi) ρ = Θ( √ NN − (2+ s ) / 2) , (vii) T {+
+} = ˜Θ( N 1+ s/ 2d − 1+ s/ 2 ) , and (viii) T 1 =

Original: (  4 nd ) .
Revised: Θ( λ 4 nd ) .

Original: Then, if werun Procedure 1 for T = T 0 + T 1 time steps with the above parameters, its outcome (ˆ c, ˆ ✓ ) satisﬁeswith probability at least 12   the following:
Revised: Then, if werun Procedure 1 for T = T 0 + T 1 time steps with the above parameters, its outcome (ˆ c, ˆ θ ) satisfieswith probability at least 12 − δ the following:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Require: N 0 , ⇢ , T 0 , T 1 , N, 
Revised: Require: N 0 , ρ, T 0 , T 1 , N, λ

Original: Initialize ✓ (0) ⇠ Unif( S d  1 ) , c (0) ⇠ Unif( { c 2 R N ; k c k 2 = ⇢ ; k c k 0{+∥ 
+} = N 0 } ) .
Revised: Initialize θ (0) ∼ Unif( S d − 1 ) , c (0) ∼ Unif( { c ∈ R N ; ∥ c ∥ 2 = ρ ; ∥ c {+∥ 
+} = N 0 } ) .

Original: Run Gradient Flow (11) with ⇣ ( t ) = 1 ( t > T 0 ) up to time T
Revised: Run Gradient Flow (11) with ζ ( t ) = 1 ( t > T 0 ) up to time T

Original: = T 0 + T 1 .
Revised: = T 0 + T 1 .

Original: Set ˆ ✓ = ✓ ( T ) , ˆ c = c ( T ) .
Revised: Set ˆ θ = θ ( T ) , ˆ c = c ( T ) .

Original: Require:  n 0 , ˆ ✓ from Procedure 1.
Revised: Require: λ n ′ , ˆ θ from Procedure 1.

Original: Set ˆ c = arg min c L 0 n 0 ( c, ˆ ✓ ) as in (14).
Revised: Set ˆ c = arg min c L ′ n ′ ( c, ˆ θ ) as in (14).

Original: Corollary 5.2 (Excess risk of Procedure 1) .
Revised: Corollary 5.2 (Excess risk of Procedure 1) .

Original: Under the assumptions of Theorem 5.1, after optimizingthe regularization parameter  , one obtains an excess risk of the form
Revised: Under the assumptions of Theorem 5.1, after optimizingthe regularization parameter λ , one obtains an excess risk of the form

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Fine-tuning the second layer.
Revised: Fine-tuning the second layer.

Original: After running Procedure 1, we may include a ﬁnal ﬁne-tuningphase of training for second layer weights c alone, using a separate training sample ( x 0 i , y 0 i ) , i =1 , . . . , n 0 and a possibly different regularization parameter  n 0 .
Revised: After running Procedure 1, we may include a final fine-tuningphase of training for second layer weights c alone, using a separate training sample ( x ′ i , y ′ i ) , i =1 , . . . , n ′ and a possibly different regularization parameter λ n ′ .

Original: More precisely, we set
Revised: More precisely, we set

Original: [Equation]
Revised: [Equation]

Original: where ˆ ✓ denotes the output of the previous gradient descent phase.
Revised: where ˆ θ denotes the output of the previous gradient descent phase.

Original: Note that this is a strongly convexoptimization problem, and can thus be optimized efﬁciently using gradient methods or by solvinga linear system.
Revised: Note that this is a strongly convexoptimization problem, and can thus be optimized efficiently using gradient methods or by solvinga linear system.

Original: While this may not be needed in practice, we use a different training sample fortechnical reasons, namely to break the dependence between the data and the kernel, which dependson the initial training sample through ˆ ✓ .
Revised: While this may not be needed in practice, we use a different training sample fortechnical reasons, namely to break the dependence between the data and the kernel, which dependson the initial training sample through ˆ θ .

Original: We note that such sample splitting strategies are commonlyused in other contexts in the statistics literature (e.g., [9, 12]).
Revised: We note that such sample splitting strategies are commonlyused in other contexts in the statistics literature (e.g., [9, 12]).

Original: We obtain the following guarantee.
Revised: We obtain the following guarantee.

Original: Proposition 5.3 (Excess risk of ﬁne-tuning) .
Revised: Proposition 5.3 (Excess risk of fine-tuning) .

Original: Let  > 0 .
Revised: Let δ > 0 .

Original: Let m =
Revised: Let m =

Original: h ˆ ✓ , ✓ ⇤ i , where ˆ ✓ is obtained from the previous gradient descent phase, and let ˆ c be the ridge regression estimator obtained from a fresh dataset D 0 of n 0 samples, N random features, 3 and regularization parameter  n 0 :=
Revised: ⟨ ˆ θ, θ ∗ ⟩ , where ˆ θ is obtained from the previous gradient descent phase, and let ˆ c be the ridge regression estimator obtained from a fresh dataset D ′ of n ′ samples, N random features, 3 and regularization parameter λ n ′ :

Original:  (  2 ⌧ 2 / k f [-
-]
Revised: = ( σ 2 τ 2 / ∥ f [-
-]′′∗

Original: ⇤ k 2  n 0 ) 1 / (  +1) , and let ˆ F ( x ) =
Revised: ∥ 2 γ n ′ ) 1 / ( β +1) , and let ˆ F ( x ) =

Original: ˆ c >  ( h ˆ ✓ , x i ) .
Revised: ˆ c ⊤ Φ( ⟨ ˆ θ, x ⟩ ) .

Original: Assume
Revised: Assume

Original: [Equation]
Revised: [Equation]

Original: ⌘ With probability at least 1   over the random features, we have
Revised: With probability at least 1 − δ over the random features, we have

Original: [Equation]
Revised: [Equation]

Original: where the expectation is over the n 0 fresh samples, and is conditioned on the previously obtained ˆ ✓ .
Revised: where the expectation is over the n ′ fresh samples, and is conditioned on the previously obtained ˆ θ .

Original: Combining the two phases leads to the following generalization guarantee on the excess risk for
Revised: Combining the two phases leads to the following generalization guarantee on the excess risk for

Original: Procedure 2.
Revised: Procedure 2.

Original: Corollary 5.4.
Revised: Corollary 5.4.

Original: Excess risk of Algorithm 2 Let  > 0 .
Revised: Excess risk of Algorithm 2 Let δ > 0 .

Original: As in Theorem 5.1, assume d  C log(   1 ) ,µ s = h h s , ⌃ h s i > 0 , and assumption 4.1 on f ⇤ with a constant C f ⇤ .
Revised: As in Theorem 5.1, assume d ≥ C log( δ − 1 ) ,µ s = ⟨ h s , Σ h s ⟩ > 0 , and assumption 4.1 on f ∗ with a constant C f ∗ .

Original: Let 
Revised: Let λ

Original: = ( s 2 ↵ 2 s / 2 C f ⇤ ) 2 /  ,and assume the following on the sample sizes and number of random features for the ﬁrst phase( n, N, N 0 ) and ﬁne-tuning phase ( n 0 , N 0 ):
Revised: = ( s 2 α 2 s / 2 C f ∗ ) 2 /β ,and assume the following on the sample sizes and number of random features for the first phase( n, N, N 0 ) and fine-tuning phase ( n ′ , N ′ ):

Original: [Equation]
Revised: [Equation]

Original: and let ⇢ be as in Theorem 5.1.
Revised: and let ρ be as in Theorem 5.1.

Original: With probability at least 1 / 2  2  over the initial n samples,initialization, random features, we have
Revised: With probability at least 1 / 2 − 2 δ over the initial n samples,initialization, random features, we have

Original: [Equation]
Revised: [Equation]

Original: where C 1 and C 2 only depend on regularity of f ⇤ , ignoring the optimization error, and logarithmicfactors in n , d , and N .
Revised: where C 1 and C 2 only depend on regularity of f ∗ , ignoring the optimization error, and logarithmicfactors in n , d , and N .

Original: We can make the following remarks:
Revised: We can make the following remarks:

Original: • The empirical gradient ﬂow therefore escapes the inﬂuence of the degenerate saddle withoptimal sample complexity n = ⇥ ( d s )[3].
Revised: • The empirical gradient flow therefore escapes the influence of the degenerate saddle withoptimal sample complexity n = Θ( d s )

Original: This is an instance of gradient descent successfully optimizing a non-convex objective without the strict saddle property as soon ass > 2 , building from the simpler optimization landscapes of [19, 3].
Revised: This is an instance of gradient descent successfully optimizing a non-convex objective without the strict saddle property as soon as s > 2 , building from the simpler optimization landscapes of [19, 3].

Original: This is in contrast withe.g.
Revised: This is in contrast withe.g.

Original: spiked tensor recovery problems [4] where the signal strength is substantially weaker,leading to complexity in the optimization landscape.
Revised: spiked tensor recovery problems [4] where the signal strength is substantially weaker, leading to complexity in the optimization landscape.

Original: • While the recovery of the high-dimensional direction is optimal in statistical terms,
Revised: • While the recovery of the high-dimensional direction is optimal in statistical terms,

Original: the
Revised: the

Original: joint recovery of the link function is sub-optimal, by comparing Corollaries 5.2 and 5.4.In this sense, the ﬁne-tuning stage recovers the optimal sample complexity, and where thenon-parametric rate no longer depends on the ambient dimension d .•
Revised: joint recovery of the link function is sub-optimal, by comparing Corollaries 5.2 and 5.4. In this sense, the fine-tuning stage recovers the optimal sample complexity, and where the non-parametric rate no longer depends on the ambient dimension d .

Original:  The time-scale separation scheduling for ⇣ is sufﬁcient but possibly not necessary.
Revised: • The time-scale separation scheduling for ζ is sufficient but possibly not necessary.

Original: The
Revised: The

Original: analysis of vanilla dynamics ( ⇣ ( t ) ⌘ 1 ) is challenging, since during the initial phase oftraining there may be adverse interaction effects between c and ✓ , which under naive analysis lead to sub-optimal sample complexity of n  O ( d 2 s ) .•
Revised: analysis of vanilla dynamics ( ζ ( t ) ≡ 1 ) is challenging, since during the initial phase oftraining there may be adverse interaction effects between c and θ , which under naive anal- ysis lead to sub-optimal sample complexity of n ≥ O ( d 2 s ) .

Original:  The time discretisation to turn Procedure 1 into a proper algorithm should follow fromstandard time discretisation arguments, although the case where  = ReLU requires specialcare due to the non-smoothness of the loss.
Revised: • The time discretisation to turn Procedure 1 into a proper algorithm should follow fromstandard time discretisation arguments, although the case where ϕ = ReLU requires special care due to the non-smoothness of the loss.

Original: In such setting, such discretisation arguments
Revised: In such setting, such discretisation arguments

Original: do not hold for vanilla gradient descent in the worst-case [30], although may be recoveredby
Revised: do not hold for vanilla gradient descent in the worst-case [30], although may be recovered

Original:  appropriately smoothing the objective prior to computing the gradient.
Revised: by appropriately smoothing the objective prior to computing the gradient.

Original:  Conclusions
Revised:  Conclusions

Original: This work studies the ability of shallow neural networks to learn single-index models with gradientdescent.
Revised: This work studies the ability of shallow neural networks to learn single-index models with gradientdescent.

Original: Our main results are positive, and demonstrate their ability to solve a semi-parametric problem with nearly optimal guarantees.
Revised: Our main results are positive, and demonstrate their ability to solve a semi-parametric problem with nearly optimal guarantees.

Original: Interestingly, this success story combines elements from thefeature-learning regime —the ability to efﬁciently identify the hidden direction in high-dimensionsunder a non-convex objective, with ingredients from the lazy-regime, which offer better computational tradeoffs to solve low-dimensional non-parametric problems.
Revised: Interestingly, this success story combines elements from thefeature-learning regime —the ability to efficiently identify the hidden direction in high-dimensionsunder a non-convex objective, with ingredients from the lazy-regime, which offer better computational tradeoffs to solve low-dimensional non-parametric problems.

Original: Our technical analysis leverages tools from high-dimensional probability (such as uniform gradient concentration) and RKHSapproximation theory, and complements the growing body of theoretical work certifying the efﬁciency of gradient methods on non-convex objectives.
Revised: Our technical analysis leverages tools from high-dimensional probability (such as uniform gradient concentration) and RKHSapproximation theory, and complements the growing body of theoretical work certifying the efficiency of gradient methods on non-convex objectives.

Original: We have followed the standard approach ofﬁrst establishing benign topological properties of the population loss, and then extend them to theempirical loss.
Revised: We have followed the standard approach offirst establishing benign topological properties of the population loss, and then extend them to theempirical loss.

Original: That said, there are several unanswered questions that our work has not addressed, such as:
Revised: That said, there are several unanswered questions that our work has not addressed, such as:

Original: • Extension to Multi-Index Models.
Revised: • Extension to Multi-Index Models.

Original: Multi-index models are natural extensions of single-
Revised: Multi-index models are natural extensions of single-

Original: index models where the hidden direction is replaced by a hidden low-dimensional subspace.
Revised: index models where the hidden direction is replaced by a hidden low-dimensional subspace.

Original: They enjoy similar statistical guarantees as single-index models [19, 5]
Revised: They enjoy similar statistical guarantees as single-index models [19, 5]

Original: and thus a natural
Revised: and thus a natural

Original: question is whether the same algorithmic tools developed here extend to the multi-index
Revised: question is whether the same algorithmic tools developed here extend to the multi-index

Original: setting.
Revised: setting.

Original: • Gradient dynamics without warmstart.
Revised: • Gradient dynamics without warmstart.

Original: An unsatisfactory aspect of our results is the re-
Revised: An unsatisfactory aspect of our results is the re-

Original: quirement that the algorithm starts by only optimizing ✓ for t < T 0 .
Revised: quirement that the algorithm starts by only optimizing θ for t < T 0 .

Original: It would be interestingto understand whether the vanilla dynamics could also succeed.
Revised: It would be interesting to understand whether the vanilla dynamics could also succeed.

Original: • Trainable biases and untied directions.
Revised: • Trainable biases and untied directions.

Original: Our proposed neural network architecture isnon-standard, in the sense that its biases are frozen at initialization and all neurons sharethe same inner weight.
Revised: Our proposed neural network architecture is non-standard, in the sense that its biases are frozen at initialization and all neurons sharethe same inner weight.

Original: For the purposes of learning single-index models, removing theserestrictions would not bring any statistical beneﬁts.
Revised: For the purposes of learning single-index models, removing theserestrictions would not bring any statistical benefits.

Original: However, it would be interesting toextend our analysis to the general setting.
Revised: However, it would be interesting toextend our analysis to the general setting.

Original: Checklist
Revised: Checklist

Original: The checklist follows the references.
Revised: The checklist follows the references.

Original: Please read the checklist guidelines carefully for informationon how to answer these questions.
Revised: Please read the checklist guidelines carefully for informationon how to answer these questions.

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Revised: For each question, change the default [TODO] to [Yes] , [No] , or[N/A] .

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Revised: You are strongly encouraged to include a justification to your answer , either by referencingthe appropriate section of your paper or providing a brief inline description.

Original: For example:
Revised: For example:

Original: • Did you include the license to the code and datasets?
Revised: • Did you include the license to the code and datasets?

Original: [Yes] See Section ?? .
Revised: [Yes] See Section ?? .•

Original: • Did you include the license to the code and datasets?
Revised: Did you include the license to the code and datasets?

Original: [No] The code and the data areproprietary.•
Revised: [No] The code and the data areproprietary.•

Original: Did you include the license to the code and datasets?
Revised: Did you include the license to the code and datasets?

Original: [N/A]
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Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See Conclusions.(c)
Revised: [Yes] See Conclusions.(c)

Original: Did you discuss any potential negative societal impacts of your work?
Revised: Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: This is apurely theoretical work with no practical applications.(d)
Revised: This is apurely theoretical work with no practical applications.(d)

Original: Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]2.

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] See Sections 4and 5.(b)
Revised: [Yes] See Sections 4and 5.(b)

Original: Did you include complete proofs of all theoretical results?
Revised: Did you include complete proofs of all theoretical results?

Original: [Yes] See Section 4.3.
Revised: [Yes] See Section 4.3.3.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] Supplementalzip ﬁle.
Revised: [Yes] Supplementalzip file.(b)

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
Revised: Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See Section D.
Revised: [Yes] See Section D.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)?

Original: [Yes] See Section D.
Revised: [Yes] See Section D.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Section D.
Revised: [Yes] See Section D.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A]
Revised: [N/A]

Original: (d) Did you discuss whether and how consent was obtained from people whose data
Revised: (d) Did you discuss whether and how consent was obtained from people whose data

Original: you’re using/curating?
Revised: you’re using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁ-
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifi-

Original: able information or offensive content?
Revised: able information or offensive content?

Original: [N/A]
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Original: If you used crowdsourcing or conducted research with human subjects...
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Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
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Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
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Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]