Analysis of one-hidden-layer neural networks via the resolvent method

1 Introduction . Machine learning has seen many successful achievements in recent years . Applications in face identification , object and speech recognition , translation , email spam filtering , navigation , medical diagnosis , etc . have proved the enormous potential of machine learning for day-to-day live [ 16 , 11 ] . Deep neural networks have turned out to be a particularly powerful machine learning method , and understanding the theoretical underpinning of their success has received tremendous attention in mathematics , physics and computer science . A fully-connected , feed-forward neural network with L hidden layers of dimensions n1 , . . . , nL can be modelled as follows : fθ ( x ) = β ∗f ( W ( L ) f ( W ( L−1 ) f ( . . . f ( W ( 1 ) x ) . . . ) ) ) ∈ Rd , where x ∈ Rn0 denotes the input data vector and f : R → R is a non-linear activation function which is applied entry-wise . We denote the parameters of the network by θ : = ( W ( 1 ) , . . . , W ( L ) , β ) , where W ( l ) ∈ Rnl×nl−1 for 1 ≤ l ≤ L and β ∈ RnL×d are the matrices of the weights . In the classical setting of supervised learning , we are given a training set of ( say ) m samples of input feature vectors xi ∈ Rn0 with associated target vectors zi ∈ Rd . For example , xi may encode the pixels of a photograph of an animal and the target zi may label the species of the 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . animal in the image . Roughly speaking , the goal of supervised learning is to learn the mapping between the feature and the target vectors based on a given training set in order to predict the output of new unlabelled data . Let X = ( x1 . . . xm ) ∈ Rn0×m be the matrix of the data and let Z = ( z1 . . . zm ) ∈ Rd×m be the target matrix . Then , the aim of the network is to find optimal parameters θ such that fθ ( X ) = ( fθ ( x1 ) , . . . , fθ ( xm ) ) ∈ Rd×m approximates the target Z optimally . During the training phase , weights are adjusted in order to minimize the empirical risk R ( θ ) = EL ( fθ ( X ) , Z ) , where L ( · , · ) is a given loss function , usually involving some penalty for large weights θ in order to avoid over-fitting . Stochastic gradient descent ( SGD ) and its variants with back-propagation are the most commonly used algorithms for training multilayer networks by iteratively updating the parameters into the direction of the negative of the gradient of the empirical risk . For a much more complete survey , we refer the reader to [ 11 ] . In the present paper , we will focus on a single-hidden-layer neural network of the form fθ ( X ) = β∗Y with Y = f ( WX ) . This model was first studied by Louart , Liao , and Couillet [ 19 ] for the case where the data matrix X is deterministic and W is a matrix of random weights ( in particular , the weights are given by functions of standard Gaussian random variables ) , and by Pennington and Worah [ 22 ] for the case where X and W are independent random matrices with both centred Gaussian entries . In both papers , the matrix β ∈ Rn1×d is the only parameter to be learned and is chosen as the unique minimizer of the ridge-regularized least squares loss function L ( fθ ( X ) , Z ) = 1 2dm ∥Z − β∗Y ∥2F + γ∥β∥2F , where γ > 0 is the learning rate . The unique minimizing weight matrix β̂ is then equal to β̂ = Y G ( −γ ) Z∗ , where G ( −γ ) = ( 1 m Y ∗Y + γ ) −1 is the resolvent of 1mY ∗Y . As proved in [ 19 , 22 ] , the expected training loss Etrain is related to −γ ∂∂γG ( −γ ) , and thus also to the Stieltjes transform of the limiting spectral measure of 1 mY ∗Y . Here the Stieltjes transformmµ of a probability measure µ on R is defined asmµ ( z ) : = ∫ R ( x−z ) −1 dµ ( x ) for z ∈ C such that ℑz ≥ 0 , and for µn1 being the empirical probability measure of the n1 eigenvalues of 1mY ∗Y is related to the resolvent via mµn1 ( z ) = 1 n1 TrG ( z ) . The performance of one-hiddenlayer neural networks depends on the asymptotic spectral properties of the matrix 1mY ∗Y . Pennington and Worah [ 22 ] investigated the limiting spectral measure of the random matrix M = 1mY Y ∗ and derived the quartic self-consistent equation 1 + zg∞ = θ1 ( f ) g∞ ( 1− ϕ ψ ( 1 + zg∞ ) ) − θ2 ( f ) ψ g∞ ( 1 + zg∞ ) ( 1− ϕ ψ ( 1 + zg∞ ) ) + θ2 ( f ) ( θ1 ( f ) − θ2 ( f ) ) ψ g2∞ ( 1− ϕ ψ ( 1 + zg∞ ) ) 2 , ( 1 ) where g∞ ( z ) : = limn1→∞ g ( z ) and g ( z ) : = 1 n1 TrG ( z ) is the Stieltjes transform , which is approximately satisfied , g∞ ( z ) ≈ g ( z ) , by g ( z ) in case of Gaussian W , X . It is notable that the asymptotic spectrum of Y ∗Y for large dimensions such that n0/m → ϕ ∈ ( 0 , ∞ ) and n0/n1 → ψ ∈ ( 0 , ∞ ) depends on the non-linear function f only through two integral parameters θ1 ( f ) and θ2 ( f ) , where θ1 ( f ) is the Gaussian mean of f2 and θ2 ( f ) is the square of the Gaussian mean of f ′ ( c.f . ( 5 ) ) . Benigni and Péché [ 6 ] then extended this model to random matrices W and X with general i.i.d . centred entries , and obtained the same self-consistent equation ( 1 ) . We mention that ( 1 ) may be reduced , for some special cases , to the quadratic equation that is satisfied by the Stieltjes transform mµMP of the Marchenko-Pastur distribution µMP [ 20 ] . This means that for some activation functions , the non-linear random matrix model has the same limiting spectral distribution as that of sample covariance matrices XX∗ ( in other cases , the equation can simplify to the cubic equation approximately satisfied by product Wishart matrices [ 7 , 10 ] ) . This can be generalised to multilayer networks : [ 22 ] found experimentally that the singular value distribution is preserved through multiple layers by activation functions with θ2 ( f ) = 0 and is given by the Marchenko-Pastur distribution in each layer . This conjecture was proved in [ 6 ] for the general case of bounded activation functions . Moreover , [ 19 ] performed a spectral analysis on the Gram matrix model with general training data and proved that , in the large dimensional regime , the resolvent of Y ∗Y has a similar behaviour as that observed in sample covariance matrix models . This was extended in [ 17 ] by considering Gaussian mixture of data . We also refer to the recent paper [ 18 ] . In the context of multilayer feedforward neural networks , Fan and Wang [ 9 ] analysed the eigenvalue distribution of the Gram matrix model , where the weights are at random and the input vectors are assumed to be approximately pairwise orthogonal . In particular , they showed that the limiting spectral distribution converges to a deterministic limit and , at each intermediate layer , this limit corresponds to the Marchenko-Pastur map of a linear transformation of that of the previous layer . In recent years , there has been some progress in the asymptotic analysis of the eigenvalue distribution of another Gram matrix , the so-called Neural Tangent Kernel ( NTK ) . Consider a multilayer neural network and denote by J = ∇θfθ ( X ) the Jacobian matrix of the network outputs with respect to the weights θ . Then , the NTK is the Gram matrix of J , defined by KNTK = J∗J . It was shown in [ 14 ] that the NTK at random initialization converges , in the infinite-width limit , to a deterministic kernel and it remains constant during the whole training time of the network . Subsequently , [ 23 ] analysed the spectrum of the sample covariance matrix JJ∗ in a single-hidden-layer neural network , and provided an exact asymptotic characterization of the spectral distribution of JJ∗ with random Gaussian weights and data . Recently , [ 9 ] proved that the limiting spectral measure of the NTK converges to a deterministic measure , which may be described by recursive fixed-point equations that extend the Marchenko-Pastur distribution . The present paper is structured as follows . In the first part we consider the non-linear random matrix model studied in [ 6 ] and we compute its asymptotic spectral density . We follow the resolvent method via the cumulant expansion which , together with the moment method , is a standard approach to obtain the asymptotics for the empirical spectral density . In particular , we compute the self-consistent equation that is approximately satisfied by the Stieltjes transform of the limiting spectral distribution . This is a quartic equation and is the same as that found in [ 6 ] . In [ 6 , 22 ] the authors relied on the method of moments : they approximated general non-linear functions by polynomial ones and then computed the asymptotics of high moments ETr ( Y k ) with Y = f ( WX ) to obtain the limiting measure via its moments . Conversely , we approach matters in a more robust and less combinatorial fashion by applying the resolvent method : we consider Y as a random matrix with correlated entries and then we directly derive a self-consistent equation for its resolvent . In particular , we prove that the random matrix Y has cycle correlations , in the sense that the joint cumulant does not vanish when the random variables Yij ’ s are joined by a cycle graph . We find that the variance of Yij is given by the parameter θ1 ( f ) , whereas for k > 1 the cumulants κ ( Yi1i2 , Y ∗ i2i3 , Yi3i4 , . . . , Y ∗ i2ki1 ) are powers of θ2 ( f ) . We note that in the random matrix literature matrices with general decaying correlations have been studied previously , see e.g . [ 3 , 8 , 1 ] . However , the cycle correlations of Y considered in the present paper are much stronger compared to these previous results . The second part of this paper concerns the additive bias case which is a more realistic model for machine learning applications . More precisely , we consider the random feature matrix Y = f ( WX +B ) , where B is a rectangular rank-one Gaussian random matrix , and derive a characterization of the Stieltjes transform of the limiting spectral density . We chose B to be rank-one since for the most commonly used neural network architectures the added bias is chosen equal for each sample . [ 2 ] studied the bias case for deterministic data and i.i.d . Gaussian random weights , and computed the exact training error of a ridge-regularized noisy autoenconder in the high-dimensional regime . Interestingly we find that in the case of additive bias it is impossible to choose an activation function f such that the eigenvalue distribution is preserved throughout multiple layers , unlike in the bias-free case where θ2 ( f ) = 0 yields the Marchenko-Pastur distribution in each layer . Finally , we remark that in the bias-free case our proof via the resolvent method has no significant advantage compared to the moment method , beyond requiring less combinatorics . The main advantage of the resolvent approach is that it allows to include an additive bias without much additional effort .

Summary: This paper studies the limiting eigenvalue-distribution of the gram matrix $M$ associated with a random feature model, and extends current theory (Pennington and Worah 2017; Benigni and Péché 2019, etc.) to the case of multiple layers with additive layer-wise additive bias. Understanding the distribution of eigenvalues of the gram matrix useful for the understanding of the training dynamics and generalization in neural networks.